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A hybrid-level digital correlation-spectrometer for radio-astronomy Hovey, Gary J. 1998

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A Hybrid-Level Digital Correlation-Spectrometer for Radio-Astronomy by Gary J. Hovey B. Eng., Elec. Eng., Lakehead University, 1983 Diploma, Elec. Tech., Okanagan College, 1973 A THESIS SUBMITTED TN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in T H E FACULTY OF GRADUATE STUDIES Department of Electrical Engineering .We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April, 1998 © Gary J. Hovey, 1998. All Rights Reserved. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract This thesis describes the design, development, testing, and construction of a hybrid-level digital correlation-spectrometer built for the seven-element synthesis and 26m paraboloid radio-telescopes at the Dominion Radio-Astrophysical Observatory near Penticton, British Columbia. Included is an analysis of the theoretical performance of the correlator and formulae for linearizing its output response. This new instrument replaces an existing, circa 1975, spectrometer system at DRAO and while it is functionally similar to its predecessor, it is more capable and flexible, and costs less to construct. These advantages were achieved largely by improvements in correlator architecture and are not solely due to, say, scaling-up an existing design using advanced technology. Following a review of correlator architectures, it is shown that the architecture developed has a number of advantages over many correlation-spectrometers. Unlike many correlation-spectrometers, this one can auto-correlate or cross-correlate both Nyquist and double-Nyquist sampled data using all lags so that spectral resolution is always maximized. As well, it is a hybrid-level correlator—as it uses 14-level correlators, for measuring total power, and three-level (multi-lag) correlators for measuring spectral power. A review of the theoretical performance of a three-level correlator is given. These results are extended for a 14-level correlator. It is shown that the sensitivity of a 14-level correlator is close to a perfect analog correlator; however, its input-output response is non-linear (over a 3dB range). These non-linear effects are analyzed and inversion formulae for correcting them are developed. The resulting corrected, or linearized, hybrid-level correlator is nearly ideal for measuring total power, while its system cost is low as three-level correlators are used for making spectral measurements. Ill The main component in the system is a four-lag correlator gate-array IC, that was designed by the author. Its architectural advantages are: a) the multiplier load present at a correlator delay tap is minimal and b) it has a readout scheme that allows sampling the most significant accumulator bits in parallel. These features yield a high-performance flexible architecture that uses IC gates efficiently and maximizes sample rate and data-transfer performance. The basic system fits in a 19-inch rack and consists of a microprocessor based controller, eight analog to digital converters (ADCs), several support-logic cards, eight (single-lag) 14-level auto-correlators, and twenty (512-lag) three-level correlators. The system is normally controlled by a host computer that sends ASCII commands to the system and receives and stores integrated correlation data. For testing purposes, the system can also be operated directly via a VT-100 type terminal. A two-rack cross-correlator system was constructed for the synthesis telescope, while a smaller half-rack auto-correlation system was built for the 26m telescope. These systems are now in routine service and make astronomical measurements almost 24 hours per day. iv Table of Contents ABSTRACT ii LIST OF FIGURES vi LIST OF TABLES. . . . . viii GLOSSARY ix ACKNOWLEDGMENTS x 1. INTRODUCTION 1 L l A BRIEF INTRODUCTION TO RADIO ASTRONOMY ; 2 1.2 AN OVERVIEW OF THE DRAO RADIO-TELESCOPES ...4 1.2.1 The DRAO 26 m Radio-Telescope : .4 1.2.2 The DRAO Synthesis-Telescope 5 1.2.2.1 An Overview of Aperture Synthesis 5 1.2.2.2 The DRAO Synthesis Telescope (ST) 12 1.3 SPECTROMETER DESIGN CONSIDERATIONS 15 1.3.1 Astronomical Requirements...! 15 1.3.2 Computing the Power Spectrum: F X versus X F Correlators 1,7 1.3.3 Quantization Issues 18 1.3.4 Previous Designs: A Historical Review of Spectrometers at DRAO 20 2. A DESCRIPTION OF T H E HYBRID-LEVEL CORRELATION-SPECTROMETER 25 2.1 THE ANALOG TO DIGITAL CONVERTERS.... 28 2.2 THE CORRELATOR 29 2.2.1 The Control Interface Card (CIC) ........ 31 2.2.2 The Auto-correlator Card (AC) 32 2.2.3 The Spectrometer Correlator Card (SCC) :....34 2.2.4 Spectrometer-Correlator Card Support-Circuitry 40 2.3 THE MICROPROCESSOR CONTROLLER 42 2.4 THE SYSTEM CLOCK 45 V 3. A REVIEW OF CORRELATION-SPECTROMETER I C ARCHITECTURES. 46 3.1 A OVERVIEW OF THE DRAO, CANARIS, ANDBOSICS : 47 3.1.1 Delay Architecture 51 3.1.2 Multiplier Architecture 54 3.1.3 Accumulator Architecture 54 3.1.4 Accumulator Readout Architecture 54 3.2 THE DRAO-IC ARCHITECTURE VERSUS THE CANARIS AND BOS ICS 56 4. T H E THEORETICAL PERFORMANCE OF T H E HYBRID-LEVEL CORRELATOR 62 4. l THE IDEAL CORRELATOR 64 4.2 THE DIGITAL CORRELATOR 67 4.3 OPTIMIZING THE SNR PERFORMANCE OF A DIGITAL CORRELATOR 72 4.4 THE OUTPUT RESPONSE OF THE THREE- AND FOURTEEN-LEVEL CORRELATOR 76 4.5 LINEARIZING THE OUTPUT RESPONSE OF A DIGITAL CORRELATOR 83 4.5.1 Linearizing the Auto-Correlator Response:;.....; 84 4.5.2 Linearizing the Cross-Correlator Response.. 87 5. TESTS AND RESULTS 96 5.1 DRAO CORRELATOR IC TESTS 96 5.2 SPECTROMETER CORRELATOR CARD TESTS ...97 5.3 LINEARITY TESTS 100 5.3.1 The A.21-cm Continuum System Linearity Tests and Results 100 5.3.2 26M Telescope Linearity Test and Results '. 106 5.4 SYSTEM TESTS AND RESULTS 109 6. SUMMARY AND CONCLUSIONS 114 6.1 AREAS REQUIRING FURTHER WORK 116 CITED REFERENCES 117 UNCITED REFERENCES 121 vi List of Figures Figure 1.1: The point source response of a paraboloid 7 Figure 1.2: A block diagram of an adding interferometer 9 Figure 1.3: The spatial response of an interferometer 10 Figure 1.4: The complex interferometer 12 Figure 1.5: The Synthesis Telescope antenna positions 14 Figure 1.6: The Research Frontier of Radio-Astronomy Correlators 24 Figure 2.1: A block diagram of the spectrometer system 28 Figure 2.2: A block diagram of the correlator sub-system 30 Figure 2.3: A block diagram of the correlator ASIC 36 Figure 2.4: A data-flow diagram of the controller software 44 Figure 3.1: A block diagram of the DRAO-IC architecture 48 Figure 3.2: A block diagram of the (scaled) Canaris IC architecture 49 Figure 3.3: A block diagram of the (scaled) Bos IC architecture; : 50 Figure 3.4: A block diagram showing how two DRAO-IC can correlate Nyquist or double-Nyquist sampled data 53 Figure 3.5: A block diagram of the double-Nyquist asymmetric pipeline 58 Figure 4.1: The input-output response of an ideal n-level quantizer 69 Figure 4.2: The relative sensitivity of several digital correlators compared to an ideal correlator ...75 Figure 4.3: The normalized output response of an auto-correlator 77 Figure 4.4: The normalized output response of a three-level and a 14-level cross-correlator 79 Figure 4.5: The output response of a three-level cross-correlator versus the normalized source temperature 81 Figure 4.6: The output response of a 14-level cross-correlator versus the normalized source temeperature 82 V l l Figure 4.7: The relative error in determining the threshold level, 1, using various inversion functions. 88 Figure 4.8: The relative error in determining the ideal correlation, R x x , using various inversion functions. 89 Figure 4.9: The response of the linearized three-level cross-correlator using a two-point Taylor series inversion 92 Figure 4.10: The response of the linearized 14-level cross-correlator using a 3-dimensional Taylor series inversion. :95 Figure 5.1: The output of the spectrometer correlator card with uniform pseudo random-noise test vectors 99 Figure 5.5: The measured relative linearization error of a 14-level cross-correlator 105 Figure 5.6: The measured nominal and linearized input-output response of a 14-level auto-correlator v........\......: 107 Figure 5.7: The measured relative linearization error of a 14-level cross-correlator 108 Figure 5.8: A cross-power absorption spectrum of the radio-source 3C147 I l l Figure 5.9: An auto-power spectrum of the emission from radio-source S7 , 112 Figure 5.10: Auto-power spectrum of the comet Hale-Bopp 113 List of Tables Table 1.1: The DRAO Synthesis Telescope Specifications 13 Table 1.2: The Signal to Noise Ratio Degradation of a Digital Correlator 19 Table 2.1: Correlator bandwidth, HI velocity, and resolution specifications of the hybrid-level correlator..... 26 Table 3.1: A comparison of delay architecture specifications for an N lag correlator 56 Table 3.2: A comparison of readout architectures '•: 61 Table 4.1: The optimum correlator sensitivity for several quantization schemes 74 ix Glossary 26M - The DRAO 26 m paraboloid radio-telescope. A/D - Analog to digital (converter). AC - (14-level) auto-correlator circuit card. ADC - Analog to digital converter. ASIC -application specific integrated circuit CIC - correlator interface circuit card. CMOS - complementary metal-oxide semiconductor. crate - a sub-system that contains a CIC, an AC, and up to ten SCCs. ECL - Emitter coupled logic. GPIB - general purpose interface bus defined by the IEEE-488 bus specification. Host - a high level commercial computer and operating system (e.g., an IBM 550/AIX computer). IC - integrated circuit. PAL - programmable array logic. SCC - (three-level) spectrometer correlator circuit card. ST - The DRAO seven-element synthesis radio-telescope. STD bus - standard (computer) bus defined by the IEEE-961P bus specification. Stream - a data path that carries sampled data. Task - a independent program unit that is scheduled by a real-time kernel. TTL - transistor-transistor logic. VRTX - a commercial real-time software kernel made by Microtec (formerly Ready Systems). X Acknowledgments The spectrometer systems installed took several person-years to develop and construct, and I was assisted by the diligent work of many of the DRAO staff. I would like to show my appreciation by thanking all the members of the development team who I supervised—as it was their work that helped transform ideas into a working and reliable system. Thank you to Ev Sheehan who constructed most of system and laid-out the correlator circuit card. His prompt and top-notch workmanship was key to installing the instrument on time and making it a reliable piece of hardware. I also thank Ralph Webber for writing much of the microprocessor control software, and Jennifer McKinley for doing most of the detailed digital design and for testing the production correlator ICs. As well, I would like to thank Ron Casorso for his quick and effective help in debugging the production hardware. I also thank Dr. Tony Willis, Dave Karpa, and Mike Davies for integrating the spectrometer systems with the DRAO ST and 26m telescope observing software. This project and thesis could not have been initiated and completed without the generous support and confidence expressed by the DRAO and NRC management staff. I thank Dr. Lloyd Higgs, Dr. Brian Andrew and Dr. Don Morton for arranging and supporting my education leave. As well, I thank the DRAO director, Dr. Tom Landecker, for his help, support, guidance, and confident patience with all aspects of this project and my thesis work. And I especially thank Dr. Peter Dewdney for contributing ideas and advise for my thesis, and for supporting my idea to design a new spectrometer for the observatory and my aspirations to pursue a graduate degree. I sincerely thank my thesis advisor, Professor Mabo R. Ito, at the University of British Columbia, for his guidance, interest, and support of my thesis work. 1 1. Introduction I n 1 9 3 2 t h e s c i e n c e o f r a d i o - a s t r o n o m y w a s b o r n w h e n i t w a s d i s c o v e r e d t h a t n a t u r a l l y o c c u r r i n g r a d i o - e m i s s i o n w a s c o m i n g f r o m s p a c e . M e a s u r i n g t h e p o w e r s p e c t r u m o f r a d i o - s i g n a l s , a r o u n d 1 4 2 0 M H z , c a n y i e l d i n f o r m a t i o n c o n c e r n i n g t h e n a t u r e o f a s o u r c e a n d t h e n a t u r e o f h y d r o g e n g a s i n t h e i n t e r v e n i n g m e d i u m . S o m e c h a r a c t e r i s t i c s o f t h e s e s i g n a l s a r e t h a t t h e y a r e g e n e r a l l y w e a k a n d w i d e b a n d , a s t h e y c a n b e s e v e r a l h u n d r e d k H z w i d e a n d m o r e t h a n 3 0 d B b e l o w t h e r e c e i v e r n o i s e . A n e f f i c i e n t w a y t o c o m p u t e t h e s p e c t r u m o f r a d i o - s i g n a l s i s t o a u t o - o r c r o s s - c o r r e l a t e t h e m , w i t h a d i g i t a l c o r r e l a t o r , a n d t h e n F o u r i e r t r a n s f o r m t h e i n t e g r a t e d c o r r e l a t i o n f u n c t i o n w i t h a c o m p u t e r . B e c a u s e t h e s i g n a l t o n o i s e r a t i o ( S N R ) o f t h e s e s i g n a l s i s l o w , t h e y c a n b e c o a r s e l y q u a n t i z e d u s i n g a n a l o g t o d i g i t a l c o n v e r t e r s ( A D C s ) t h a t h a v e , o n l y a f e w l e v e l s . T h i s c o r r e l a t i o n - s p e c t r o m e t e r a p p r o a c h i s q u i t e e f f i c i e n t a s i t i s r e l a t i v e l y e a s y t o i m p l e m e n t a h i g h - s p e e d c o r r e l a t o r t h a t m u l t i p l i e s a n d i n t e g r a t e s d e l a y e d v e r s i o n s o f s i g n a l s t h a t a r e r e p r e s e n t e d d i g i t a l l y b y a f e w b i t s . T h i s t h e s i s d e s c r i b e s a c o r r e l a t i o n - s p e c t r o m e t e r t h a t u s e s 1 4 - l e v e l a u t o - c o r r e l a t o r s t o m e a s u r e t o t a l p o w e r a n d t h r e e - l e v e l c o r r e l a t o r s t o m e a s u r e p o w e r s p e c t r a . T h i s / i y £ > m / - l e v e l a p p r o a c h h a s s o m e o f t h e h i g h p e r f o r m a n c e c h a r a c t e r i s t i c s o f a 1 4 - l e v e l c o r r e l a t o r w i t h t h e l o w e r c o s t a n d s i m p l i c i t y o f a t h r e e - l e v e l c o r r e l a t o r . T h e c h a r a c t e r i s t i c s o f t h i s h y b r i d s y s t e m a r e a n a l y z e d a n d m e t h o d s f o r l i n e a r i z i n g t h e i n p u t - o u t p u t r e s p o n s e o f t h e s e c o r r e l a t o r s a r e g i v e n . T h e m o t i v a t i o n a n d g e n e r a l s p e c i f i c a t i o n s f o r t h i s s y s t e m a r e p r o v i d e d i n t h e r e m a i n d e r o f t h i s c h a p t e r . H e r e a n i n t r o d u c t i o n t o r a d i o a s t r o n o m y i s g i v e n a n d t h e r a d i o t e l e s c o p e s a t t h e D R A O a r e d i s c u s s e d . I n c h a p t e r 2 a n o v e r v i e w o f t h e n e w s y s t e m i s g i v e n , w h i l e i n c h a p t e r 3 t h e m e r i t s o f t w o p o p u l a r c o r r e l a t i o n - s p e c t r o m e t e r I C a r c h i t e c t u r e s a r e c o m p a r e d w i t h t h e o n e d e v e l o p e d f o r t h i s s y s t e m . I n c h a p t e r 4 t h e t h e o r e t i c a l s e n s i t i v i t y a n d i n p u t - o u t p u t r e s p o n s e o f t h e c o r r e l a t o r s a r e a n a l y z e d a n d m e t h o d s f o r l i n e a r i z i n g t h e o u t p u t r e s p o n s e o f t h r e e - l e v e l a n d 1 4 - l e v e l c o r r e l a t o r s a r e 2 g i v e n . T e s t s a n d r e s u l t s a r e p r e s e n t e d a n d d i s c u s s e d i n c h a p t e r 5 , w h i l e s u m m a r y a n d c o n c l u d i n g r e m a r k s a r e g i v e n i n t h e f i n a l c h a p t e r . 1.1 A Brief Introduction to Radio Astronomy I n J a n u a r y 1 9 3 2 K a r l J a n s k y , a r a d i o e n g i n e e r w i t h B e l l T e l e p h o n e L a b o r a t o r i e s , w a s c o n d u c t i n g t e s t s a t 2 0 . 5 M H z t o i d e n t i f y t h e s o u r c e s o f s t a t i c n o i s e i n r e c e i v e r s w h e n h e n o t i c e d a s i g n i f i c a n t n o i s e s i g n a l c o m i n g f r o m s p a c e i n t h e d i r e c t i o n o f t h e c e n t e r o f o u r g a l a x y . T h e r e c e p t i o n o f t h i s c o s m i c s i g n a l i s c o n s i d e r e d , b y m a n y , t o b e t h e b i r t h o f r a d i o a s t r o n o m y . S i n c e J a n s k y ' s d i s c o v e r y a s t r o n o m e r s h a v e l e a r n e d m u c h a b o u t t h e n a t u r e o f c o s m i c r a d i o s i g n a l s a n d t h e s o u r c e s t h a t e m i t t h e m . T h e s e s i g n a l s a r e g e n e r a t e d b y n a t u r a l p r o c e s s e s a n d b e h a v e a s G a u s s i a n r a n d o m - n o i s e s i g n a l s . T h e y a r e a l s o e r g o d i c a n d c a n o f t e n b e t r e a t e d a s s t a t i o n a r y o v e r t i m e s c a l e s o f s e v e r a l w e e k s b e c a u s e m o s t c o s m i c s o u r c e s v a r y e x t r e m e l y s l o w l y w i t h t i m e . P e r h a p s t h e m o s t s t r i k i n g t h i n g a b o u t t h e s e s i g n a l s i s t h a t t h e y a r e v e r y w e a k c o m p a r e d t o t y p i c a l c o m m u n i c a t i o n s s i g n a l s . F o r e x a m p l e , m o s t c o s m i c r a d i o - s i g n a l s h a v e a f l u x d e n s i t y o f l e s s t h a n o n e J a n s k y ( J y ) , w h i c h h a s u n i t s o f 1 0 ' 2 6 W m ' 2 H z " 1 . T h i s i s m i n u s c u l e c o m p a r e d t o a 1 0 U.V m " ' c o m m u n i c a t i o n s i g n a l — w h i c h i s a b o u t 2 . 7 0 b i l l i o n t i m e s m o r e p o w e r f u l . C o s m i c r a d i o - s i g n a l s c a n b e c a t e g o r i z e d i n t o t w o g r o u p s b a s e d o n t h e s h a p e o f t h e i r p o w e r s p e c t r a . T h e f i r s t a r e c o n t i n u u m s i g n a l s , w h i c h h a v e r e l a t i v e l y b r o a d a n d f e a t u r e l e s s s p e c t r a , w h i l e t h e s e c o n d a r e l i n e s i g n a l s , w h i c h h a v e n a r r o w s p e c t r a a n d a r e f e a t u r e - f u l l . C o n t i n u u m s i g n a l s a r e c r e a t e d i n a n u m b e r o f w a y s . O n e w a y i s b y a t h e r m a l p r o c e s s , w h e r e f r e e -f r e e c o l l i s i o n s b e t w e e n u n b o u n d e l e c t r o n s a n d i o n s i n a p l a s m a e m i t r a d i a t i o n . T h e p o w e r s p e c t r a o f t h e s e s i g n a l s h a v e a s h a p e s i m i l a r t o t h o s e e m i t t e d f r o m a b l a c k b o d y , a s t h e y f o l l o w t h e R a y l e i g h -J e a n l a w w h e r e p o w e r i s p r o p o r t i o n a l t o f r e q u e n c y s q u a r e d . - C o n t i n u u m s i g n a l s a r e a l s o c r e a t e d b y n o n - t h e r m a l p r o c e s s e s w h i c h g e n e r a l l y h a v e p o w e r s p e c t r a t h a t d e c r e a s e w i t h f r e q u e n c y . A 3 c o m m o n n o n - t h e r m a l p r o c e s s i s t h e s y n c h r o t r o n m e c h a n i s m , w h e r e h i g h e n e r g y e l e c t r o n s i n a m a g n e t i c f i e l d r a d i a t e a s a r e s u l t o f t h e i r o r b i t a l m o t i o n . B o t h o f t h e s e p r o c e s s e s a r e r a n d o m a n d . o c c u r o v e r a c o n t i n u o u s r a n g e o f e n e r g i e s , a n d s o p r o d u c e a f e a t u r e l e s s s p e c t r u m o v e r a b r o a d f r e q u e n c y r a n g e . B y m e a s u r i n g h o w a p o w e r s p e c t r u m c h a n g e s w i t h f r e q u e n c y ( i . e . , i t s s p e c t r a l i n d e x ) a s t r o n o m e r s c a n d e t e r m i n e i f a c o n t i n u u m s o u r c e i s t h e r m a l o r n o n - t h e r m a l . S p e c t r a l l i n e s ( o r l i n e s ) , o n t h e o t h e r h a n d , h a v e e x t r e m e l y n a r r o w s p e c t r a c o m p a r e d t o c o n t i n u u m s i g n a l s . T h e r e a s o n f o r t h i s i s t h a t a p a r t i c l e c a n o n l y c h a n g e e n e r g y l e v e l s i n d i s c r e t e s t e p s ; s o o n l y p h o t o n s o f a s p e c i f i c e n e r g y , o r f r e q u e n c y , c a n b e e m i t t e d o r a b s o r b e d b y a p a r t i c l e . T h i s n a r r o w c h a r a c t e r o f l i n e s m a k e s t h e m q u i t e u s e f u l t o a s t r o n o m e r s , a s t h e y c a n b e u s e d t o u n d e r s t a n d b o t h t h e c h e m i s t r y a n d p h y s i c s o f a c e l e s t i a l r e g i o n — i n s o m e s e n s e l i n e s , a r e l i k e p r o b e s . F o r e x a m p l e , t h e s p e c i e s o f a p a r t i c l e c a n b e i d e n t i f i e d f r o m t h e l i n e ( s ) i n a p o w e r s p e c t r u m , a s t h e e n e r g y l e v e l d i s t r i b u t i o n o f a p a r t i c l e c a u s e s i t t o e m i t o r a b s o r b p h o t o n s a t c h a r a c t e r i s t i c f r e q u e n c i e s . A l s o , t h e o b s e r v e d f r e q u e n c y c a n b e u s e d t o d e t e r m i n e t h e r a d i a l v e l o c i t y o f p a r t i c l e s u s i n g t h e D o p p l e r r e l a t i o n . U s i n g t h i s i n f o r m a t i o n o n e c a n e s t i m a t e t h e d i s t a n c e o f a r e g i o n f r o m e a r t h a n d i n s o m e c a s e s e s t i m a t e i t s s t r u c t u r e . F o r e x a m p l e , a s t r o n o m e r s h a v e d e t e r m i n e d t h a t o u r g a l a x y h a s a s p i r a l s t r u c t u r e a n d h a v e d e t e r m i n e d o u r l o c a t i o n i n t h e g a l a x y u s i n g t h e r a d i a l v e l o c i t y i n f o r m a t i o n d e r i v e d f r o m s p e c t r a l l i n e o b s e r v a t i o n s . O n e o f t h e m o s t c o m m o n l i n e s i s t h a t p r o d u c e d b y n e u t r a l a t o m i c - h y d r o g e n ( H I ) , w h i c h e m i t s o r a b s o r b s p h o t o n s a r e s t f r e q u e n c y o f 1 4 2 0 M H z 1 . B e c a u s e H I i n o u r g a l a x y o c c u p i e s a s u b s t a n t i a l f r a c t i o n ( 2 0 % t o 9 0 % ) o f i n t e r - s t e l l a r s p a c e [ K u l k a r n i 8 8 ] , a n d r e p r e s e n t s 4 . 4 % o f t h e m a s s o f t h e v i s i b l e m a t t e r [ B a h c a l l 8 3 ] , i t i s o n e o f t h e m o s t u s e f u l s p e c t r a l l i n e p r o b e s , a n d i s t h e p r i m a r y l i n e f r e q u e n c y o b s e r v e d b y t h e r a d i o t e l e s c o p e s a t t h e D R A O . 4 1.2 An Overview of the DRAO Radio-Telescopes T h e D R A O h a s t w o r a d i o - t e l e s c o p e s f o r o b s e r v i n g H I s p e c t r a l l i n e s , t h e a p e r t u r e - s y n t h e s i s t e l e s c o p e ( S T ) a n d t h e 2 6 m r a d i o - t e l e s c o p e ( 2 6 M ) . T h e 2 6 M c o n s i s t s o f a s i n g l e 2 6 m p a r a b o l o i d , w h i l e t h e S T u s e s a n a r r a y o f s e v e n a n t e n n a s t o s y n t h e s i z e a 6 0 0 - m a p e r t u r e . I n t h i s s e c t i o n t h e s e t w o t e l e s c o p e s a r e d e s c r i b e d i n o u t l i n e a n d t h e b a s i c p r i n c i p l e s o f a p e r t u r e s y n t h e s i s , c o r r e l a t o r s , a n d i n t e r f e r o m e t r y a r e d i s c u s s e d . 1.2.1 The D R A O 26 m Radio-Telescope T h e 2 6 M c o n s i s t s o f a 2 5 . 6 m p a r a b o l i c r e f l e c t o r , s t e e r a b l e i n h o u r a n g l e a n d d e c l i n a t i o n . A , r e c e i v e r i s m o u n t e d a t t h e p r i m e f o c u s o f t h e p a r a b o l o i d , a n d a m p l i f i e s a n d d o w n c o n v e r t s t h e r a d i o s i g n a l t o a n i n t e r m e d i a t e f r e q u e n c y . T h e s i g n a l i s f u r t h e r d o w n c o n v e r t e d t o n e a r b a s e b a n d s o t h a t a n a u t o -c o r r e l a t i o n s p e c t r o m e t e r c a n p r o d u c e t h e c o r r e l a t i o n c o e f f i c i e n t s f o r e a c h s i g n a l p o l a r i z a t i o n . A U N I X c o m p u t e r c o n t r o l s t h e t e l e s c o p e a n d s t o r e s d a t a f r o m t h e s y s t e m o n a m a g n e t i c d i s k d r i v e . S e v e r a l r e c e i v e r s , w h i c h o p e r a t e a t s p e c i f i c c e n t e r f r e q u e n c i e s , " a r e a v a i l a b l e . W i t h t h e s e r e c e i v e r s o n e c a n m e a s u r e e i t h e r : t h e 1 . 4 2 0 G H z a t o m i c - h y d r o g e n ( H I ) l i n e , t h e 1 . 4 2 5 G H z i o n i z e d h y d r o g e n ( H I I ) l i n e , t h e 1 . 6 2 7 G H z H y d r o x y l ( O H ) l i n e , o r t h e 6 . 6 8 6 G H z M e t h a n o l l i n e , a m o n g o t h e r s . A l l t h e r e c e i v e r s a r e s e n s i t i v e t o b o t h l e f t - a n d right-hand c i r c u l a r p o l a r i z a t i o n s , e x c e p t t h e H I r e c e i v e r w h i c h m e a s u r e s t w o o r t h o g o n a l l i n e a r p o l a r i z a t i o n s . W i t h t h e 1 . 4 2 0 G H z r e c e i v e r t h e h a l f - p o w e r a n t e n n a b e a m s i z e i s 3 6 a r c m i n a n d t h e s y s t e m t e m p e r a t u r e i s a p p r o x i m a t e l y 5 0 ° K [ G a l t 9 4 ] . W h i l e t h e 2 6 M i s a p o w e r f u l i n s t r u m e n t , i t s b e a m i s t o o l a r g e f o r r e s o l v i n g t h e d e t a i l i n h i g h l y s t r u c t u r e d o b j e c t s , s u c h a s n e a r b y c l o u d s o f H I g a s i n o u r g a l a x y . S u c h o b j e c t s c a n h a v e 1 - a r c m i n f e a t u r e s [ K u l k a r n i 8 8 ] w h i c h c a n n o t b e r e s o l v e d w i t h a 3 6 - a r c m i n b e a m . A more precise frequency of the HI line, made by measuring hydrogen masers, is: 1,420,405,751.786 +.01 Hz. 5 I n c r e a s i n g t h e r e s o l u t i o n o f a p a r a b o l o i d t o r e s o l v e 1 - a r c m i n f e a t u r e s a t 1 . 4 2 0 G H z i s i m p r a c t i c a l — a s i t w o u l d r e q u i r e a p a r a b o l o i d w i t h a 7 5 0 m d i a m e t e r . T h e r e a s o n i s t h a t a t a g i v e n f r e q u e n c y t h e s p a t i a l r e s o l u t i o n o f a p a r a b o l o i d i s i n v e r s e l y p r o p o r t i o n a l t o i t s d i a m e t e r ; t h u s , h i g h r e s o l u t i o n i m p l i e s a l a r g e a p e r t u r e a t a g i v e n f r e q u e n c y . F u r t h e r m o r e , a s i n g l e r e c e i v e r c a n o n l y m e a s u r e o n e p o i n t i n t h e f o c a l p l a n e a t a t i m e , s o t o p r o d u c e a n i m a g e t h e t e l e s c o p e m u s t s c a n t h e r e g i o n o f i n t e r e s t i n a s e r i a l f a s h i o n ; T h i s i s n o t o n l y t i m e c o n s u m i n g , b u t i s a l s o h a m p e r e d b y i n s t r u m e n t a l e r r o r s , t h a t c a n v a r y s l o w l y a s t h e f i e l d i s b e i n g s c a n n e d t h e r e b y c o r r u p t i n g t h e i m a g e . 1.2.2 The DRAO Synthesis-Telescope A n a l t e r n a t i v e t o i m p l e m e n t i n g a l a r g e f i l l e d - a p e r t u r e i s t o s y n t h e s i z e o n e u s i n g a n a r r a y o f s m a l l e r a n t e n n a s . T h e c o n c e p t i n v o l v e s i m p l e m e n t i n g p o r t i o n s o f t h e a p e r t u r e , u s i n g s m a l l e r a n t e n n a s , a n d f o r m i n g a s y n t h e t i c a p e r t u r e , e l e c t r o n i c a l l y , u s i n g c o m p u t e r s a n d c u s t o m c i r c u i t s . A p e r t u r e - s y n t h e s i s r a d i o - t e l e s c o p e s , w i t h b a s e l i n e l e n g t h s o f s e v e r a l - h u n d r e d k m a n d s u b - a r c s e c r e s o l u t i o n , h a v e b e e n c o n s t r u c t e d t h a t e x c e e d t h e r e s o l u t i o n o f e v e n t h e l a r g e s t p a r a b o l i c a n t e n n a b y s e v e r a l o r d e r s o f m a g n i t u d e . I n f a c t t h e r e c e n t V S O P m i s s i o n , w h i c h p l a c e d a r a d i o t e l e s c o p e i n e a r t h o r b i t , p r o m i s e s t o d e l i v e r 1 0 0 u . - a r c s e c r e s o l u t i o n w h i c h w i l l e x c e e d t h e r e s o l u t i o n o f c u r r e n t o p t i c a l t e l e s c o p e s b y t w o o r d e r s o f m a g n i t u d e [ H i r a b a y a s h i 9 2 ] . I n t h i s s e c t i o n a n o v e r v i e w o f a p e r t u r e s y n t h e s i s i s f i r s t g i v e n f o l l o w e d b y a d e s c r i p t i o n o f t h e D R A O S T . 1.2.2.1 An Overview of Aperture Synthesis F o r b r e v i t y t h e f o l l o w i n g i s a n a b r i d g e d v e r s i o n o f h o w a p e r t u r e s y n t h e s i s i s a c c o m p l i s h e d . F o r a d e t a i l e d t r e a t m e n t r e f e r t o T h o m p s o n e t a l . ( 1 9 8 6 ) , a n d C h r i s t i a n s e n a n d H o g b o m ( 1 9 8 5 ) . B e f o r e i n v e s t i g a t i n g h o w a n a p e r t u r e i s s y n t h e s i z e d , i t i s u s e f u l r e v i e w h o w t h e f i e l d i n t h e f o c a l -p l a n e r e g i o n o f a n i d e a l p a r a b o l i c - a n t e n n a i s f o r m e d i n r e s p o n s e t o a d i s t a n t m o n o - c h r o m a t i c p o i n t -source in the direction of the symmetry axis of an antenna. As shown in Figure 1.1, the source radiation distribution, or structure, s(x,y), arrives as a plane wave, S(l,m), and strikes the uniform circular aperture, G(l,m). The aperture truncates the extent of the wave, and the antenna surface reflects and focuses it on a plane to form the field, f[x,y). To gain insight to what is happening, consider the above operations in signal analysis terms. First the point source, s(x,y), is a two-dimensional impulse and its Fourier transform produces the wave front S(l,m). Given that x and y have units of one wavelength, and / and m have units of cycles per unit wavelength, these operations can be written as S(l,m) = ]\s{x,y)-ei2K{U^y)dxdy = K$j d{x,y)-ej2«(U+my)dxdy = K. (1.1) As expected in this case, S(l,m) has an infinite and uniform spatial-frequency spectrum, K. Next, the aperture, G(l,m), spatially low-pass filters S(l,m) to produce F(l,m) = G(l,m)-S(l,m) = K-G(l,m), (1.2) where F(l,m) is the spatial spectrum of the field in the focal-plane. The focal-plane field, j\x, y), is just the Fourier transform of F(7,m). And for the case where the aperture is uniform, circular, and illuminated by a point source, " f(x,y)= \\F(l,m)-eilK(lx+my)dxdy = K - D ^ ^ , (1.3) where D is the diameter of the aperture and r2 = x1 + y2. In this case, f(x, y), is also the point spread function of the aperture and has the same form as the far field beam pattern, g(x, y) [Padman95]. 7 Figure 1.1: A perspective view of how a parabolic antenna responds to a distant point source, s(x,y). At a great distant s(x,y) becomes an infinite plane wave, S(l,m), which is the Fourier transform ofs(x,y). S(l,m) is then multiplied by the aperture G(l,m) which reduces the spatial frequency content ofS(l,m) to G(l,m). The product is then Fourier transformed by the antenna's reflective surface on to the focal plane, f(x,y) where it can be detected. 8 T h e F o u r i e r r e l a t i o n s h i p s g i v e n a b o v e a r e n o t e w o r t h y f o r a c o u p l e o f r e a s o n s . F i r s t t h e y d e m o n s t r a t e w h y a l a r g e a p e r t u r e h a s m o r e r e s o l u t i o n t h a n a s m a l l o n e . M o r e i m p o r t a n t l y t h o u g h , t h e s e r e l a t i o n s i m p l y t h a t s p a t i a l f r e q u e n c i e s c a n b e m e a s u r e d s e p a r a t e l y a n d t h e n l a t e r c o m b i n e d — a n d t h i s i s t h e k e y p r i n c i p l e o f s y n t h e s i z i n g a n a p e r t u r e . T h e r a d i o i n t e r f e r o m e t e r , s h o w n i n F i g . 1 . 2 , e n a b l e s t w o p o i n t s , o r o n e s p a t i a l f r e q u e n c y , o f a f u l l , a p e r t u r e t o b e i m p l e m e n t e d . T h e i n t e r f e r o m e t e r , a s t h e n a m e i m p l i e s , m e a s u r e s t h e i n t e r f e r e n c e p a t t e r n p r o d u c e d b y r a d i a t i o n f a l l i n g o n i t s t w o - p o i n t a p e r t u r e . A s s h o w n i n F i g . 1 . 2 , w h e n a p l a n e w a v e s t r i k e s t h e i n t e r f e r o m e t e r i t w i l l e x p e r i e n c e a r e l a t i v e d e l a y , is, t h a t d e p e n d s o n t h e i n c i d e n t a n g l e , 0 . W h e n t h e s i g n a l s v j a n d v 2 a r e a d d e d , t h e y w i l l c o m b i n e , c o n s t r u c t i v e l y o r d e s t r u c t i v e l y , t o p r o d u c e . . . B sm(ax) + sm(w(t + Te)) = sm(ax) + sm(co(t +—sin(#))). ( 1 . 4 ) s c N o r m a l l y , t h e f a s t e r , c o t, t e r m s c a n b e f i l t e r e d o u t a n d t h e s l o w e r , c o tg, t e r m c a n b e d e t e c t e d . S q u a r i n g E q . ( 1 . 4 ) y i e l d s \ + cos(o)Tg)-cos(2ox + (OTg)-^cos(2m) + cos(2m + 2Q)rs)^ ( 1 . 5 ) F i l t e r i n g t h e c o n s t a n t a n d h i g h f r e q u e n c y t e r m s ( i n v o l v i n g (Of), a n d s u b s t i t u t i n g 2%B/k s i n ( 0 ) f o r vng, r e s u l t s i s t h e q u a s i - s i n u s o i d a l i n t e r f e r o m e t e r r e s p o n s e s h o w n i n F i g . 1 . 3 a n d g i v e n b y cos(a)Tg) = cos(2n — s i n ( # ) ) . ( 1 . 6 ) T h e f r e q u e n c y o f t h i s r e s p o n s e d e p e n d s o n t h e r a d i a t i o n w a v e l e n g t h , X ; t h e d i s t a n c e , o r b a s e l i n e , b e t w e e n t h e a n t e n n a s ; a n d t h e s p a t i a l d i m e n s i o n , 6 . B e c a u s e s p a t i a l r e s o l u t i o n i s a f u n c t i o n o f t h e 9 V Vi + Vi Figure 1.2: A block diagram of an adding interferometer. baseline length, and not the antenna diameter, aperture-synthesis radio-telescopes have a huge resolution advantage over filled-aperture antennas. Several assumptions concerning the source radiation and receiving electronics simplify the use of a synthesis telescope. As mentioned earlier, a source's structure is normally invariant over time scales of several weeks and the signals are ergodic. Therefore, interferometers can operate in parallel, to increase speed, or in serial, to reduce equipment cost. Minimally, a single interferometer, with an adjustable baseline length, can be used to measure a source's spatial frequency structure in serial fashion. To do this the source must be scanned, as the interferometer response varies in only one dimension, 8, while the source structure varies in two dimensions. A simple way to scan the source is to use the earth's rotation, or so called "rotation synthesis". .. Figure 1.3: The spatial response of an interferometer whose baseline lies parallel to the x axis and is pointed in the direction z( B = 4 X). U s i n g r o t a t i o n s y n t h e s i s t h e s o u r c e c a n b e s c a n n e d i n 1 2 h o u r s ( 1 8 0 ° ) , i n s t e a d o f 2 4 h o u r s . T h e r e a s o n i s t h a t t h e s o u r c e s t r u c t u r e i s r e a l , a n d i n g e n e r a l a s y m m e t r i c a l , s o i t s s p a t i a l f r e q u e n c y c o m p o n e n t s a r e c o m p l e x a n d s y m m e t r i c a l ( i . e . , H e r m i t i a n ) . T h e r e f o r e , t h e i n t e r f e r o m e t e r r e s p o n s e f o r t h e f i r s t 1 2 h o u r s w i l l b e t h e s a m e a s t h e s e c o n d 1 2 h o u r s , e x c e p t f o r a p h a s e r e v e r s a l . T h e a d d i n g i n t e r f e r o m e t e r , s h o w n i n F i g . 1 . 2 , i s o n e w a y t o m a k e t h e m e a s u r e m e n t s d e s c r i b e d i n t h e p r e v i o u s p a r a g r a p h s ; h o w e v e r , i t h a s s o m e d i s a d v a n t a g e s . O n e i s t h a t i t o n l y m e a s u r e s t h e r e a l s p a t i a l - f r e q u e n c y c o m p o n e n t o f a s o u r c e . I n o r d e r t o m e a s u r e t h e i m a g i n a r y c o m p o n e n t s e p a r a t e c i r c u i t s a r e r e q u i r e d t o p h a s e s h i f t o n e s i g n a l b y 9 0 ° a n d a d d t h e m . A n o t h e r p r o b l e m i t t h a t a n a d d e r i s u s e d t o c o m b i n e t h e s i g n a l s , a n d t h i s a p p r o a c h i s p r o n e t o a d d i n g i n s t r u m e n t a l o f f s e t e r r o r s t h a t c a n b e d i f f i c u l t t o r e m o v e . A n a l t e r n a t i v e i s t o u s e a m u l t i p l i e r , i n s t e a d o f a n a d d e r , a n d t o m o d u l a t e t h e p h a s e o f o n e s i g n a l b y 1 8 0 ° . W h e n t h e m u l t i p l i e r o u t p u t p h a s e i s d e - m o d u l a t e d a n d a v e r a g e d o v e r a n i n t e g r a t i o n p e r i o d , T , 11 the phase modulation, or phase switching, will remove offsets errors (including cross-talk) and the averaging will improve the signal to noise ratio of the measurement. Adding these improvements yields the complex phase-switched correlating-interferometer shown in Fig. 1.4. The output of the correlating interferometer is described by Eq. (1.7), and is equivalent to the cross-correlation, c(0), of vi and v2 at a relative delay, x, of zero. This quantity is also commonly called a visibility vector, V(u,v), as it represents the complex power of a spatial frequency at a particular baseline orientation described by (w,v). c(0) = Xv,(f,-)-v2a+0) + yX v i(O : v 2 (f / +0)=V0i,v). (1.7) The general procedure used to observe, or map, a source is to point the antennas at the source and record the visibility vectors, V(u,v), in the spatial frequency plane (w,v). Then maximally after twelve hours—depending on, the source structure, the number of interferometers and their orientations—the interferometer baselines may be changed so that a new set of visibilities can be collected. This process is repeated until the spatial frequency structure of the source is adequately sampled, or the (u,v) plane is filled. The (u, v) plane describes the spatial frequency content of the source multiplied by the synthesized response of a telescope. To re-construct an image of the source the (w,v) plane is Fourier transformed and then de-convolved to remove the synthesized response. The result is an estimate of the brightness distribution, or image, of the source observed. These principles are used by the DRAO synthesis telescope which is described in the next section. 1 2 o Baseline Length =B 6 90 Phase Switch M = l . . M = -l 180 Figure 1.4: The complex interferometer. MVi 1.2.2.2 The DRAO Synthesis Telescope (ST) T h e D R A O - S T u s e s s e v e n 9 - m p a r a b o l i c - a n t e n n a s d i s t r i b u t e d a l o n g a n e a s t - w e s t l i n e t o s y n t h e s i z e a 6 0 0 - m a p e r t u r e . F i l l i n g i t s (u,v) p l a n e t a k e s t w e l v e , t w e l v e - h o u r o b s e r v a t i o n s . T h e t e l e s c o p e s i m u l t a n e o u s l y p r o d u c e s c o n t i n u u m v i s i b i l i t i e s a t 1 4 2 0 M H z a n d 4 0 8 M H z , a n d 2 5 6 H l - l i n e v i s i b i l i t i e s a t 1 4 2 0 M H z [ L a n d e c k e r 9 7 ] . T h e 1 4 2 0 M H z c o n t i n u u m - s y s t e m p r o d u c e s v i s i b i l i t i e s f o r a l l f o u r S t o k e s - p a r a m e t e r s a n d c a n m e a s u r e t h e p o l a r i z a t i o n s t r u c t u r e o f a r a d i o s o u r c e [ K a r p a 8 9 ] [ S m e g a l 9 5 ] . T h e 1 4 2 0 M H z l i n e - s y s t e m m e a s u r e s l e f t - a n d r i g h t - h a n d c i r c u l a r p o l a r i z a t i o n s , w h i c h 1 3 a r e c o m b i n e d t o i m p r o v e s e n s i t i v i t y . T h e 4 0 8 M H z c o n t i n u u m - s y s t e m m e a s u r e s e i t h e r r i g h t - o r l e f t -h a n d c i r c u l a r p o l a r i z a t i o n s . A n i n t e r e s t i n g a s p e c t o f t h e 4 0 8 - s y s t e m i s t h a t i t u s e s r e a l - t i m e c o m p u t e r s t o e s t i m a t e c o m p l e x - p o w e r . T h i s i s d o n e b y i n t e r p o l a t i n g a n d H i l b e r t t r a n s f o r m i n g t h e 1 6 - p o i n t c r o s s - c o r r e l a t i o n f u n c t i o n f o r e a c h b a s e l i n e [ L o 8 2 ] [ V e i d t 8 4 ] . T h e S T s p e c i f i c a t i o n s a r e s u m m a r i z e d i n T a b l e 1 . 1 . T h e a n t e n n a s a r e a r r a n g e d s o t h a t f o u r a r e f i x e d i n p o s i t i o n a n d t h r e e a r e m o v a b l e a l o n g a 3 0 0 - m s e c t i o n o f r a i l w a y t r a c k ( s e e F i g . 1 . 5 ) . T h e r a i l w a y t r a c k a l l o w s t h e d i s t a n c e b e t w e e n a n t e n n a s , o r b a s e l i n e s , t o b e c h a n g e d s o t h a t d i f f e r e n t s p a t i a l f r e q u e n c i e s c a n b e m e a s u r e d . I n g e n e r a l , a s e v e n - e l e m e n t a r r a y h a s 2 1 b a s e l i n e - c o m b i n a t i o n s . H o w e v e r , o n l y t w e l v e o f t h e p o s s i b l e 2 1 D R A O - S T b a s e l i n e s a r e u n i q u e , a s t h e d i s t a n c e s a r e n o t c h a n g e d b e t w e e n m o v a b l e a n t e n n a s a n d t h e d i s t a n c e s a r e , o f c o u r s e , f i x e d b e t w e e n t h e o t h e r a n t e n n a s . Table 1.1: The DRAO Synthesis Telescope Specifications. , Parameter Continuum HI Line 408 MHz 1420 MHz Polarization left circular All Stokes left + right (Stokes I) Bandwidth 4 MHz 30 MHz 4 MHz - 128 kHz (in powers of two) Frequency Channels 1 4 sub-bands 256 Central Velocity Range 0 -840 to + 3000 km/s Field Size (to 20% response) 8.1° 2.6° Synthesized beam (EW x NS) 3.5 x3.5 cosec(S) arcmin2 1.0 x 1.0 cosec(8) arcmin2 System temperature 150 K 65 K 65 K Brightness temperature sensitivity 670 sin(8) mK 49 sin(5) mK 3.8sin(8) VA / /MHz Flux density sensitivity 3.3 mJy/beam .23 mJy/beam 18 mJy vW /MHzbeam 14 West East 1 2 3 4 . 5 6 7 ) v & % ( - 3 c H 7 0 d = 3 0 0 m 1 — 1 5 0 m 1 1 5 0 m 1 Figure 1.5: The Synthesis Telescope antenna positions. A d u a l - f r e q u e n c y f o c u s b o x i s m o u n t e d t o t h e a p e x o f a t r i p o d a t t h e p r i m e f o c u s o f e a c h a n t e n n a . T h e f o c u s b o x c o n t a i n s 4 0 8 M H z a n d 1 4 2 0 M H z r e c e i v e r s . E a c h r e c e i v e r a m p l i f i e s a n d d o w n -c o n v e r t s i t s r e c e i v e d s i g n a l ( s ) t o a n i n t e r m e d i a t e f r e q u e n c y ( I F ) o f 3 0 M H z . T h e s e s i g n a l s a r e s e n t t o a c e n t r a l l o c a t i o n w h e r e t h e y a r e f i l t e r e d a n d d o w n - c o n v e r t e d t o n e a r b a s e b a n d s o t h a t t h e y c a n b e p r o c e s s e d b y d i g i t a l c o r r e l a t o r s . T h e v i s i b i l i t i e s f o r e a c h f r e q u e n c y a r e c o m p u t e d b y s e p a r a t e c o r r e l a t o r s y s t e m s . W h i l e t h e a g e a n d d e s i g n o f e a c h s y s t e m a r e q u i t e d i f f e r e n t , t h e y a l l p r o c e s s s i g n a l s i n a s i m i l a r w a y . I n g e n e r a l , a n a l o g v o l t a g e - s i g n a l s f r o m e a c h a n t e n n a a n d p o l a r i z a t i o n a r e c o n v e r t e d t o d i g i t a l f o r m b y a n a l o g t o d i g i t a l c o n v e r t e r s ( A D C s ) l o c a t e d i n e a c h c o r r e l a t o r s y s t e m . T h e d i g i t a l s i g n a l s a r e t h e n r o u t e d t o a n a r r a y o f d i g i t a l c r o s s - c o r r e l a t o r s t h a t c o m p u t e t h e v i s i b i l i t i e s f o r e a c h b a s e l i n e . T h e i n t e g r a t i o n p e r i o d f o r e a c h c o r r e l a t o r s y s t e m i s s y n c h r o n i z e d b y a c e n t r a l c l o c k . ( S o t h a t a p h a s e s w i t c h c a n o c c u r b e t w e e n e a c h a n t e n n a p a i r , t h e i n t e g r a t i o n p e r i o d i s m u c h f a s t e r t h a n t h e N y q u i s t p e r i o d t h a t i s r e q u i r e d t o s a m p l e t h e v i s i b i l i t i e s . ) A t t h e e n d o f a n i n t e g r a t i o n e a c h v i s i b i l i t y i s d e - m o d u l a t e d a n d a v e r a g e d , a n d t h e n s e n t t o a h o s t c o m p u t e r w h e r e i t i s s t o r e d o n m a g n e t i c d i s k . T h i s h o s t c o m p u t e r a l s o m a n a g e s o t h e r t e l e s c o p e a c t i v i t i e s s u c h a s p o i n t i n g t h e a n t e n n a s , m a i n t a i n i n g i n t e g r a t i o n t i m i n g , a n d c o n t r o l l i n g t h e c e n t r a l o b s e r v i n g f r e q u e n c y , p h a s e a n d d e l a y . 15 1.3 Spectrometer Design Considerations E n g i n e e r i n g a n d b u i l d i n g t h e h y b r i d - l e v e l s p e c t r o m e t e r w i t h l i m i t e d r e s o u r c e s r e q u i r e d m a t c h i n g t h e a s t r o n o m i c a l r e q u i r e m e n t s a n d c h a r a c t e r i s t i c s o f t h e s i g n a l s w i t h a v a i l a b l e t e c h n o l o g y . A s w e l l , i t w a s i m p o r t a n t t o b u i l d u p o n t h e e x p e r i e n c e o f p a s t s p e c t r o m e t e r d e s i g n e r s b y i m p r o v i n g t h e v a r i o u s a s p e c t s o f t h e i r d e s i g n s . I n t h i s s e c t i o n t h e s e t h i n g s a r e c o n s i d e r e d t o j u s t i f y t h e d e s i g n o f t h e h y b r i d - l e v e l s p e c t r o m e t e r d e t a i l e d i n c h a p t e r s 2 a n d 3 . I n t h e i n i t i a l s e c t i o n t h e r e s o l u t i o n a n d b a n d w i d t h s p e c i f i c a t i o n s a r e j u s t i f i e d b y c o n s i d e r i n g t h e a s t r o n o m i c a l r e q u i r e m e n t s o f s p e c t r o m e t e r s a t D R A O . A l t h o u g h o t h e r s a t D R A O h a v e c o n s i d e r e d t h e s e r e q u i r e m e n t s b e f o r e ( a s t h i s s p e c t r o m e t e r r e p l a c e s a n e x i s t i n g o n e ) , i t i s f e l t t h a t r e v i e w i n g t h e s e r e q u i r e m e n t s p r o v i d e s i n s i g h t i n t o h o w t h e s e s p e c t r o m e t e r s p e c i f i c a t i o n s w e r e l i k e l y s e t . I n t h e s e c o n d s e c t i o n t w o m e t h o d s f o r i m p l e m e n t i n g a d i g i t a l s p e c t r o m e t e r a r e r e v i e w e d t o j u s t i f y t h e m e t h o d c h o s e n . S i n c e t h e s y s t e m i s d i g i t a l , a n a l o g s i g n a l s m u s t b e q u a n t i z e d i n t o d i g i t a l f o r m a n d t h e s c h e m e u s e d i s r a t i o n a l i z e d i n t h e t h i r d s e c t i o n . I n t h e f i n a l s e c t i o n p r e v i o u s D R A O s p e c t r o m e t e r s a r e r e v i e w e d a n d t h e m o t i v a t i o n f o r t h i s p r o j e c t i s d i s c u s s e d . 1.3.1 Astronomical Requirements A s m e n t i o n e d p r e v i o u s l y , a r a d i o - a s t r o n o m y s p e c t r o m e t e r i s u s e d t o m e a s u r e t h e s p e c t r a l s t r u c t u r e o f r a d i a t i o n f r o m a c o s m i c s o u r c e , a s f r o m s u c h m e a s u r e m e n t s o n e c a n i d e n t i f y t h e c h e m i c a l c o m p o s i t i o n a n d d e t e r m i n e t h e p h y s i c a l p r o p e r t i e s o f a s o u r c e . T h e o b j e c t i v e o f t h i s s e c t i o n i s t o r e v i e w t h e n a t u r e o f t y p i c a l s o u r c e s o b s e r v e d w i t h t h e D R A O t e l e s c o p e s , s o t h a t t h e s p e c i f i c a t i o n s o f t h e h y b r i d - l e v e l s p e c t r o m e t e r , d e t a i l e d i n C h a p t e r 2 , a r e j u s t i f i e d i n t e r m s o f s o m e i m p o r t a n t a s t r o n o m i c a l p a r a m e t e r s . 1 6 A t t h e D R A O s p e c t r o m e t e r s a r e t y p i c a l l y u s e d t o m e a s u r e t h e v e l o c i t y s t r u c t u r e o f g a l a c t i c H y d r o g e n - g a s c l o u d s b y m e a s u r i n g t h e D o p p l e r s h i f t f r o m t h e g a s ' s r e s t f r e q u e n c y . W h i l e o t h e r l i n e s ( s u c h a s O H a n d M e t h a n o l u s i n g t h e 2 6 M ) a r e m e a s u r e d , t h e a s t r o n o m i c a l r e q u i r e m e n t s c a n b e u s e f u l l y s e t b y c o n s i d e r i n g t h e c h a r a c t e r o f g a l a c t i c H l - l i n e s . T h e t w o m o s t i m p o r t a n t s p e c i f i c a t i o n s a r e t h e m a x i m u m b a n d w i d t h a n d t h e r e s o l u t i o n o f t h e s p e c t r o m e t e r . T h e v e l o c i t y r a n g e o f g a s c l o u d s d e t e r m i n e s t h e b a n d w i d t h r e q u i r e d , a s i t i s s o m e t i m e s d e s i r e d t o o b s e r v e t h e e n t i r e r a n g e i n a s i n g l e o b s e r v a t i o n . K u l k a r n i a n d H e i l e s ( 1 9 8 8 ) s t a t e t h a t t h e m a j o r i t y o f g a l a c t i c H I l i e s w i t h i n ± 4 0 k m / s e c a n d t h e r e i s l i t t l e b e y o n d - 9 0 k m / s e c , e x c e p t a t h i g h g a l a c t i c l a t i t u d e s w h e r e t h e v e l o c i t y c a n b e - 5 0 0 k m / s e c . U s i n g t h e D o p p l e r r e l a t i o n , g i v e n i n E q . ( 1 . 8 ) , m o s t H I c a n b e s e e n w i t h i n a b a n d w i d t h o f 4 0 0 k H z w h i l e a b a n d w i d t h o f 3 M H z c o v e r s t h e e n t i r e v e l o c i t y r a n g e . T o a i d c a l i b r a t i o n i t i s u s e f u l t o h a v e s o m e a d d i t i o n a l b a n d w i d t h , s o t h a t t h e H I s p e c t r a l - p r o f i l e c a n b e d i s t i n g u i s h e d f r o m c o n t i n u u m r a d i a t i o n . T h e b a s e b a n d c o n v e r s i o n s y s t e m s f o r e x i s t i n g s p e c t r o m e t e r s a t D R A O p r o v i d e s b a n d w i d t h s f r o m 4 t o 1 / 8 M H z i n p o w e r s o f t w o , a n d s o s p a n s t h e r e q u i r e d g a l a c t i c H I v e l o c i t y r a n g e . F o r c o m p a t i b i l i t y t h e h y b r i d - l e v e l s p e c t r o m e t e r u s e s t h e s e s a m e r a n g e s a n d a d d i t i o n a l l y c a n o p e r a t e a t b a n d w i d t h s o f 8 M H z a n d 1 / 1 6 M H z . frest - f -Af v = c r e s t . = c — ; v « c , (1.8) f f r e s t rest • T h e v e l o c i t y d i s p e r s i o n , o r f e a t u r e s s i z e , o f c l o u d s d e t e r m i n e s t h e f r e q u e n c y r e s o l u t i o n r e q u i r e d . F o r c o l d H I i n a b s o r p t i o n t h e d i s p e r s i o n d i s t r i b u t i o n p e a k s a t . 7 5 k m / s e c a n d h a s a m e a n o f - 1 . 7 k m / s e c . E m i s s i o n f e a t u r e s a r e g e n e r a l l y b r o a d e r a n d t h e d i s p e r s i o n p e a k s a t - 2 . 2 k m / s e c . I n g e n e r a l , n o c l o u d h a s b e e n o b s e r v e d w i t h a d i s p e r s i o n l e s s t h a n . 4 k m s " 1 , a s a n u p p e r l i m i t w a r m H I c a n h a v e d i s p e r s i o n s a s h i g h a s 1 7 k m s " 1 [ K u l k a r n i 8 8 ] . T h u s c o v e r i n g t h e H I d i s p e r s i o n , w i t h s a y f o u r p o i n t s , r e q u i r e s a f r e q u e n c y r e s o l u t i o n r a n g e f r o m - . 5 k H z t o 2 0 k H z . One problem is that it is expensive build a spectrometer that covers the widest bandwidth at the highest.resolution, as it requires 4 MHz / 500 Hz-channeF1, or 8000 spectral channels. A compromise is to provide the highest resolution at the lowest bandwidth; and the lowest resolution at the widest bandwidth. Astronomers cope with this restriction by judiciously choosing an appropriate velocity-range and resolution before observing a source. The hybrid-level spectrometer can produce 256 cross-power spectral-channels, and so it has a resolution of-15 kHz at 4 MHz and -.5 kHz at 1/8 MHz bandwidth. Thus it meets the astronomical requirement and has twice the resolution and bandwidth of the previous generation of spectrometers at DRAO. 1.3.2 Computing the Power Spectrum: FX versus XF Correlators A spectrometer computes the power spectrum of two input signals V](t) and v2(t). The power spectrum, V^fyV^f), is desired as it yields a set of visibilities that are a function of frequency, or equivalently a function of relative velocity via the Doppler relation. The power spectrum can be computed directly in the frequency domain by Fourier transforming the input signals, using the FFT,.and then multiplying the results [Chicada84]. This approach, referred to as an FX correlator, has that advantage that the Fourier transform is performed on only N antennas inputs and some types of corrections are more easily performed in the frequency domain [Romney86]. Alternatively, an XF correlator computes the power spectrum by correlating two inputs and then Fourier transforming the integrated result. The discrete correlation function, . •• • . JL • C(T) = fjVl(ti)-V2(t.l+Tl, (1.9) 1 8 i s c o m p u t e d b y m u l t i p l y i n g v i ( t ) a n d v 2 ( t ) a t d i f f e r e n t r e l a t i v e d e l a y s , x , a n d s u m m i n g t h e s e r e s u l t s o v e r a n i n t e g r a t i o n p e r i o d , T . T h e h y b r i d - l e v e l c o r r e l a t o r u s e s a n X F a r c h i t e c t u r e . T h i s a r c h i t e c t u r e i s a t t r a c t i v e b e c a u s e t h e m u l t i p l i c a t i o n a n d a d d i t i o n o p e r a t i o n s a r e p e r f o r m e d o n d a t a w i t h f e w b i t s , t y p i c a l l y l e s s t h a n t h r e e . C o m p a r e t h i s t o t h e F X a p p r o a c h w h e r e t h e o p e r a t i o n s i n t h e F F T k e r n e l c a u s e s t h e d a t a t o c o n t a i n m o r e b i t s , m a k i n g a d d i t i o n a n d m u l t i p l i c a t i o n o p e r a t i o n s m o r e d i f f i c u l t t o i m p l e m e n t , e s p e c i a l l y a t h i g h s a m p l e - r a t e s . T h e X F c o r r e l a t o r , o n t h e o t h e r h a n d , i n v o l v e s c o m p u t i n g s i m p l e o p e r a t i o n s a t h i g h s p e e d a n d d e f e r s t h e F F T u n t i l t h e d a t a h a s b e e n i n t e g r a t e d f o r l o n g p e r i o d s , t y p i c a l l y g r e a t e r t h a n 1 0 5 s a m p l e s . T h e c h i e f d i s a d v a n t a g e o f t h e X F c o r r e l a t o r i s t h a t i n g e n e r a l f o r N a n t e n n a i n p u t s N ( N - l ) / 2 c o r r e l a t o r o u t p u t s m u s t b e F o u r i e r t r a n s f o r m e d , w h e r e a s t h e F X a p p r o a c h r e q u i r e s F o u r i e r t r a n s f o r m i n g o n l y N a n t e n n a i n p u t s . F o r l a r g e N t h i s a d v a n t a g e c a n b e c o m e s i g n i f i c a n t . T h e X F a p p r o a c h w a s u s e d t o c o n s t r u c t t h e f i r s t g e n e r a t i o n o f r a d i o - a s t r o n o m y d i g i t a l c o r r e l a t i o n -s p e c t r o m e t e r s u s i n g d i s c r e t e l o g i c I C s [ W e i n r e b 6 3 ] [ B o w e r s 7 3 ] [ D e w d n e y 7 8 ] [ B o s 8 1 ] . S i n c e t h e e a r l y 1 9 8 0 ' s c u s t o m a n d a p p l i c a t i o n s p e c i f i c c o r r e l a t o r I C s h a v e b e e n u s e d t o i m p l e m e n t t h e X F c o r r e l a t o r b e c a u s e o f t h e i r l o w c o s t a n d h i g h p e r f o r m a n c e [ N a p i e r 8 3 ] [ H o v e y 8 8 ] [ B o s 9 0 ] [ V o n H e r z e n 9 1 ] [ P a d i n 9 3 ] [ T i m o c 9 3 ] [ S E R C 9 3 ] [ B u r g e s s 9 6 ] . B e c a u s e o f t h e i r c o m p l e x i t y , o n l y a f e w r a d i o - a s t r o n o m y s p e c t r o m e t e r s h a v e b e e n d e v e l o p e d u s i n g t h e F X a p p r o a c h . 1.3.3 Quantization Issues T h e s i g n a l t o n o i s e r a t i o ( S N R ) o f r a d i o - a s t r o n o m y s i g n a l s i s e x t r e m e l y l o w , t y p i c a l l y < - 2 0 d B . C o n s e q u e n t l y , t h e n u m b e r o f d i g i t a l l e v e l s r e q u i r e d t o r e p r e s e n t t h e s i g n a l c a n b e v e r y l o w , i f f a c t a s l o w a s t w o , a s t h e n o i s e a d d e d b y c o a r s e q u a n t i z a t i o n ( i n a n a n a l o g t o d i g i t a l t o c o n v e r t e r ) i s s m a l l c o m p a r e d t o n o i s e a d d e d t o t h e s i g n a l f r o m t h e f r o n t - e n d r e c e i v e r . 1 9 T h e s e i s s u e s a r e i m p o r t a n t , a s f e w e r q u a n t i z a t i o n l e v e l s t r a n s l a t e i n t o l o w e r c o s t a n d s i m p l i c i t y . W i t h f e w e r l e v e l s d a t a p a t h s a r e n a r r o w e r , a n d t h e m u l t i p l i e r s a n d a d d e r s s m a l l e r . W h i l e t h e t w o -l e v e l c o r r e l a t o r i s t h e s i m p l e s t t o i m p l e m e n t , D ' A d d a r i o e t a l ( 1 9 8 4 ) h a v e s h o w n t h a t a t h r e e - l e v e l -q u a n t i z a t i o n s c h e m e i s o p t i m u m i n t h e s e n s e t h a t i t m i n i m i z e s t h e a m o u n t o f i n f o r m a t i o n t h a t m u s t b e p r o c e s s e d i n o r d e r t o d e t e r m i n e t h e c o r r e l a t i o n w i t h i n a g i v e n r m s e r r o r . W h i l e q u a n t i z a t i o n n o i s e i s h i g h e r w i t h f e w e r q u a n t i z a t i o n l e v e l s i t c a n b e l o w e r e d b y i n c r e a s i n g t h e s a m p l e r a t e a b o v e t h e N y q u i s t r a t e . T h e r e a s o n t h i s w o r k s i s t h a t e v e n t h o u g h t h e n o i s e i n t h e i n p u t d a t a c a n n o t b e r e d u c e d b y a v e r a g i n g , s i n c e t h e s a m p l e s a r e n o t i n d e p e n d e n t , t h e q u a n t i z a t i o n n o i s e c a n b e r e d u c e d — a s q u a n t i z a t i o n n o i s e i s n o t b a n d l i m i t e d a n d s o e v e r y s a m p l e h a s s o m e i n d e p e n d e n t n o i s e [ V a n V l e c k 6 6 ] . T h u s a v e r a g i n g t h e c o r r e l a t i o n s o f o v e r - s a m p l e d d i g i t a l d a t a i m p r o v e s t h e s e n s i t i v i t y o f a d i g i t a l c o r r e l a t o r [ B o w e r s 7 4 ] . T a b l e 1 . 2 , l i s t s t h e S N R d e g r a d a t i o n o f a d i g i t a l c o r r e l a t o r , c o m p a r e d t o a n i d e a l o n e , a s a f u n c t i o n o f t h e q u a n t i z a t i o n l e v e l s a n d o v e r - s a m p l i n g f a c t o r , K , [ K l i n g l e r 7 2 ] . D e g r a d a t i o n i s d e f i n e d b y E q . Table 1.2: The Signal to Noise Ratio Degradation of a Digital Correlator, with N quantization levels and over-sampling factor K, compared to a perfect analog correlator. Q u a n t i z a t i o n L e v e l s ( N ) D e g r a d a t i o n f o r o v e r - s a m p l i n g f a c t o r K K = l K = 2 1 . 5 6 1 . 3 5 . 1 . 2 3 1 . 1 2 1 . 1 6 1 . 0 6 2 0 ( 1 . 1 0 ) , a n d i s t h e r a t i o b e t w e e n t h e S N R o f a n i d e a l a n a l o g c o r r e l a t o r c o m p a r e d t o a d i g i t a l c o r r e l a t o r w i t h a g i v e n q u a n t i z a t i o n s c h e m e . Degradation = D= 5 M ? i d e a l c ° r r e l a l o r .. ( 1 . 1 0 ) " ^ ^ d i g i t a l c o r r e l a t o r W h i l e c o a r s e q u a n t i z a t i o n d e g r a d e s t h e s e n s i t i v i t y o f a c o r r e l a t o r i t a l s o c a u s e s t h e c o r r e l a t o r o u t p u t r e s p o n s e t o b e n o n - l i n e a r . T h e s e e f f e c t s c a n b e a m e l i o r a t e d b y l i n e a r i z i n g t h e r e s p o n s e o f t h e d i g i t a l c o r r e l a t o r ; h o w e v e r , t o d o t h i s o n e n e e d s a n e s t i m a t e o f t h e q u a n t i z e r t h r e s h o l d s o r a n e s t i m a t e o f t h e v a r i a n c e o f t h e a n a l o g t o d i g i t a l c o n v e r t e r ( A D C ) o u t p u t . T h e s e q u a n t i z a t i o n a n d s a m p l i n g i s s u e s a n d t h e i r a f f e c t s o n t h e s e n s i t i v i t y a n d l i n e a r i t y p e r f o r m a n c e o f t h e h y b r i d - l e v e l c o r r e l a t o r a r e d i s c u s s e d m o r e f u l l y i n C h a p t e r 4 . 1.3.4 Previous Designs: A Historical Review of Spectrometers at D R A O T h i s s e c t i o n p r o v i d e s a h i s t o r i c a l p e r s p e c t i v e b y r e v i e w i n g p a s t D R A O s p e c t r o m e t e r d e s i g n s . A s w e l l , a t t r i b u t e s g l e a n e d f r o m t h e s e s p e c t r o m e t e r s a r e p o i n t e d o u t a n d t h e 1 m o t i v a t i o n f o r b u i l d i n g a n e w s p e c t r o m e t e r i s d i s c u s s e d . T h e f i r s t D R A O s p e c t r o m e t e r w a s a n a n a l o g d e s i g n [ A r g y l e 6 3 ] t h a t o p e r a t e d o n t h e 2 6 M , a n d c o n s i s t e d o f a n a r r a y o f 1 0 0 n a r r o w - b a n d f i l t e r s . T h e f i l t e r o u t p u t s w e r e d e t e c t e d , i n t e g r a t e d , a n d t h e n r e a d i n a m u l t i p l e x e d f a s h i o n b y a n A D C c o n n e c t e d t o a c o m p u t e r . W h i l e t h i s s p e c t r o m e t e r w a s a n e x t r e m e l y p r o d u c t i v e i n s t r u m e n t , i t h a d s e v e r a l p r o b l e m s t h a t w e r e i n h e r e n t w i t h t h i s a p p r o a c h . F i r s t i t w a s d i f f i c u l t t o m a t c h t h e r e s p o n s e s o f t h e i n d i v i d u a l f i l t e r c h a n n e l s . A s w e l l , c h a n g i n g r e s o l u t i o n w a s i m p r a c t i c a l a s s e p a r a t e f i l t e r a r r a y s w o u l d b e r e q u i r e d . A n o t h e r p r o b l e m w a s t h a t t h e o r i g i n a l d e s i g n u s e d a n a r r a y o f m e c h a n i c a l s w i t c h e s , t o m u l t i p l e x e a c h c h a n n e l o u t p u t t o t h e A D C , a n d t h e s e s w i t c h e s w e r e u n r e l i a b l e . E v e n a f t e r t h e s e s w i t c h e s w e r e r e p l a c e d w i t h r e l i a b l e e l e c t r o n i c o n e s , t h e s y s t e m s u f f e r e d f r o m d r i f t p r o b l e m s a n d h a d t o b e r e -c a l i b r a t e d f r e q u e n t l y . 2 1 R e l i a b i l i t y , s t a b i l i t y a n d f l e x i b i l i t y a r e i m p o r t a n t d e s i g n a t t r i b u t e s a n d t h e s e c o n d g e n e r a t i o n o f D R A O s p e c t r o m e t e r s o v e r c a m e s u c h p r o b l e m s b y c o r r e l a t i n g t h e i n p u t s i g n a l s u s i n g d i g i t a l c i r c u i t s . B e c a u s e o f t h e c h a r a c t e r i s t i c s o f d i g i t a l c i r c u i t s , s u c h c o r r e l a t i o n - s p e c t r o m e t e r s w e r e i n h e r e n t l y m o r e s t a b l e a n d r e l i a b l e . T h e y w e r e a l s o m o r e f l e x i b l e , a n d c h a n g i n g t h e r e s o l u t i o n a n d b a n d w i d t h , f o r e x a m p l e , c o u l d b e a c c o m p l i s h e d b y s i m p l y c h a n g i n g t h e c o r r e l a t o r s a m p l i n g f r e q u e n c y . T h e i n i t i a l D R A O d e s i g n w a s a 2 5 6 - l a g c o r r e l a t o r [ W h y t e 7 2 ] , w h i c h u s e d a 3 b y 5 l e v e l q u a n t i z a t i o n s c h e m e , i n v e s t i g a t e d b y K l i n g l e r ( 1 9 7 2 ) . T h i s d e s i g n r e p o r t e d l y h a d s o m e i n h e r e n t t i m i n g p r o b l e m s t h a t m a d e i t d i f f i c u l t t o b u i l d a n d m a i n t a i n . I t i s l i k e l y t h a t t h e l i m i t e d a v a i l a b i l i t y o f c o m m o d i t y h i g h - s p e e d a n d l a r g e - s c a l e i n t e g r a t e d ( L S I ) l o g i c a l s o h a m p e r e d i t s i m p l e m e n t a t i o n . T h e s e p r o b l e m s w e r e r e c o g n i z e d a n d a n e w t h r e e - l e v e l d i g i t a l c o r r e l a t o r w a s d e s i g n e d b y D e w d n e y ( 1 9 7 8 ) . A l t h o u g h D e w d n e y ' s c o r r e l a t o r h a d l e s s s e n s i t i v i t y t h a n W h y t e ' s , i t w a s s i m p l e r , h a d a s y m m e t r i c a r c h i t e c t u r e ( t h a t i n h e r e n t l y c o r r e l a t e d d o u b l e - N y q u i s t s a m p l e d d a t a ) , a n d i n c o r p o r a t e d f a s t e r a n d d e n s e r l o g i c , s u c h a s S c h o t t k y T T L a n d L S I C M O S . B e c a u s e o f i t s r e l i a b i l i t y i t b e c a m e t h e p r i m a r y s p e c t r o m e t e r u s e d o n t h e D R A O S T a n d w a s u s e d i n 1 9 8 4 ( b y t h e a u t h o r ) t o r e p l a c e t h e a n a l o g s p e c t r o m e t e r o n t h e 2 6 M . T h i s c o r r e l a t o r w a s a l s o u s e d t o i m p l e m e n t t h e 4 0 8 M H z c o n t i n u u m -c o r r e l a t o r ( m e n t i o n e d i n s e c t i o n 1 . 2 . 2 ) w h i c h d e r i v e d c o m p l e x - p o w e r b y i n t e r p o l a t i n g a n d H i l b e r t t r a n s f o r m i n g a 1 6 - l a g c r o s s - c o r r e l a t i o n f u n c t i o n u s i n g r e a l - t i m e s o f t w a r e [ L o 8 2 ] . S o m e d e s i r a b l e c h a r a c t e r i s t i c s g l e a n e d f r o m D e w d n e y ' s c o r r e l a t o r w e r e i t s d o u b l e - N y q u i s t s y m m e t r i c p i p e l i n e a n d i t s s e r i a l - r e a d o u t s c h e m e . O n l y r e c e n t l y , h o w e v e r , w e r e t h e f u l l b e n e f i t o f t h e s e c h a r a c t e r i s t i c s r e c o g n i z e d . O r i g i n a l l y , t h e s e r i a l r e a d o u t s c h e m e w a s s e e n a s u n d e s i r a b l e , a s i t r e q u i r e d e x t r a h a r d w a r e t o b i t - t r a n s p o s e s e r i a l d a t a f r o m t h e C M O S c o u n t e r s u s e d i n t h e d e s i g n . I n C h a p t e r 3 , h o w e v e r , a s e r i a l r e a d o u t s c h e m e i s p r e s e n t e d t h a t s h o w s h o w t h e a v a i l a b l e d a t a t r a n s f e r -b a n d w i d t h o f a c o r r e l a t o r c a n b e b e t t e r u t i l i z e d . I n t h i s c h a p t e r i t i s a l s o s h o w n t h a t a d o u b l e -N y q u i s t p i p e l i n e c a n b e u s e d t o a d v a n t a g e i n C M O S i n t e g r a t e d - c i r c u i t d e s i g n s . 2 2 D e w d n e y ' s c o r r e l a t o r w a s a v e r y p r o d u c t i v e i n s t r u m e n t a n d p r o v i d e d m u c h i n s p i r a t i o n ; h o w e v e r , i t i s u s e f u l t o r e v i e w s o m e o f t h e r e a s o n s w h y i t w a s r e p l a c e d , a s t h e s e r e a s o n s p r o v i d e t h e m a i n m o t i v a t i o n f o r t h i s p r o j e c t . O n e r e a s o n w a s t h a t ( b y c u r r e n t s t a n d a r d s ) t h i s c o r r e l a t o r w a s r e l a t i v e l y c o s t l y a n d l a b o u r i n t e n s i v e t o m a n u f a c t u r e ( ~ $ 1 0 / l a g ) . I t s h i g h c o s t w a s d u e i n p a r t t o i t s l o w c i r c u i t d e n s i t y , w h i c h w a s n o t a t y p i c a l o f d i s c r e t e l o g i c d e s i g n s o f t h i s e r a . F o r i n s t a n c e , i t r e q u i r e d s i x t e e n c i r c u i t c a r d s t o i m p l e m e n t a 2 5 6 l a g c o r r e l a t o r . T h e s y s t e m w a s a l s o l a b o u r i n t e n s i v e t o m a n u f a c t u r e , a s m a n y c o m p o n e n t s r e q u i r e d h a n d . a s s e m b l y . F o r e x a m p l e , a s i n g l e 2 5 6 - l a g c o r r e l a t o r u s e d 8 0 c a b l e s , i n s t e a d o f e d g e c o n n e c t o r s , t o i n t e r c o n n e c t 1 7 b o a r d s v i a a p s e u d o f o u r - l a y e r b a c k p l a n e . T h i s b a c k p l a n e h a d t o b e h a n d - f a s h i o n e d f r o m t w o d o u b l e - s i d e d b o a r d s . B e s i d e s b e i n g t e d i o u s t o m a k e , t h e c a b l e s a n d b a c k p l a n e w e r e n o t i m p e d a n c e c o n t r o l l e d w h i c h m a d e i t d i f f i c u l t t o p r o p e r l y t e r m i n a t e t h e c l o c k a n d d a t a s i g n a l s t h a t w e r e f a n n e d - o u t a c r o s s m u l t i p l e c o r r e l a t o r - b o a r d s . A s a r e s u l t s i g n a l q u a l i t y b e t w e e n b o a r d s w a s m a r g i n a l a n d g l o b a l s y s t e m t i m i n g a d j u s t m e n t s w e r e o f t e n d i f f i c u l t t o m a k e . H o w e v e r , o n c e a d j u s t e d t h e s y s t e m w a s v e r y s t a b l e a n d r e l i a b l e . T h e s y s t e m w a s a l s o s u b j e c t t o c a l i b r a t i o n p r o b l e m s a n d s o m e w h a t i n f l e x i b l e . F o r i n s t a n c e , i t c o u l d n o t b e e a s i l y e x t e n d e d t o c o r r e l a t e m o r e t h a n 2 5 6 l a g s , c o n f i g u r e d a s a n a u t o - c o r r e l a t o r , n o r c o n f i g u r e d t o c o r r e l a t e N y q u i s t - s a m p l e d d a t a . H o w t h e s e i s s u e s w e r e a d d r e s s e d i n t h e h y b r i d - l e v e l c o r r e l a t o r a r e d e s c r i b e d i n C h a p t e r s 2 a n d 3 . A l s o , c a l i b r a t i o n o f t h e s y s t e m w a s a p r o b l e m b e c a u s e t h e s y s t e m w o u l d n o t r e s p o n d c o r r e c t l y t o c h a n g e s i n t h e t o t a l i n p u t - p o w e r a n d t h e c i r c u i t s n e e d e d t o m e a s u r e t h e c o r r e c t i o n s w e r e n o t p r o v i d e d . ; F o r i n s t a n c e , t h e t o t a l i n p u t - p o w e r c o u l d n o t b e m e a s u r e d b y t h e c o r r e l a t o r , a s a n a u t o m a t i c l e v e l c o n t r o l ( A L C ) k e p t t h e c o r r e l a t o r i n p u t - p o w e r c o n s t a n t ( a t t h e o p t i m u m s e n s i t i v i t y l e v e l f o r a t h r e e - l e v e l c o r r e l a t o r ) . A s w e l l , t h e A L C i n d u c e d a c a l i b r a t i o n g a i n - e r r o r , a s i t w o u l d c h a n g e t h e s y s t e m g a i n i n r e s p o n s e t o a c h a n g e i n t h e s y s t e m t e m p e r a t u r e ( i . e . , u n c o r r e l a t e d n o i s e ) . S i n c e t h e t o t a l p o w e r w a s n o t m e a s u r e d b y s e p a r a t e c i r c u i t s i n f r o n t o f t h e A L C , t h i s i n f o r m a t i o n w a s l o s t a n d A L C i n d u c e d c a l i b r a t i o n g a i n - e r r o r s c o u l d n o t b e 23 removed. These issues are discussed more fully in chapter 4 and are avoided in the hybrid-level correlator. Before concluding this chapter it is useful to summarize the main contributions of this thesis relative to the work carried out by other authors. Figure 1.6 shows the main research areas associated with the contributions made by various authors. The areas in gray indicate the contributions presented in this thesis. The main contributions of this thesis concern the architecture of the correlator, and the performance and linearization analysis of 14-level correlators. 00 K OS 3 CD N' o 3 W D C Q C Z 5 ? 5 N" O 03 2. 3 cn i— © CD < CD CO 7\ O CQ —* CD CO CO D > C L C L CO o cr — - CD ~t C O 4^ I OJ ^ < » 9 5= »-C D < ( Q ^ " V l ^ | _ L 3 w ro co """" 03 —* T J 03 CO-CO CD o o o —1 CO CD < CD CO if) O 7 3 2. » 3 DO a » o CD ^ oca CD ~7\ CO 55 TJ CD O 0) o CD co" CO c CD CO TJ o o ZT 3 o' CU CD C L *< 0> —^V ^ s. CO CD C O C O •t^ " — ' J X o o —* JD o CO 00 P .3 ^ g D CD ? CL CD CD CD CO O " ro TJ -a" CD 5' CD ro d gJ £ 3 8 CQ O «—. CD O - L CO ^-v CD CO — L CO — (O O -i. to — CD 03 CO - — CD m X3. 5 O -rj S oS. C L ± ZS O ^ t D 5- -Q c co' CD CD CT CO CD CO O 0 5 -n CO TJ T J ' CD •5" CD -o z JB 03 o.-g. 5" CD 3 03 < O n: CD —t N CD 3 CO CO S-M-CD O 03 03 —s. co' T I 03 CO i—*• TJ Tj' CD ^ — ^ CO' CD c co' O -O c CT Z z: c CO o CD CD i o CQ < o CQ o" > Q. U O CD CL O CD O CD CO to 2 5 2. A Description of the Hybrid-Level Correlation-Spectrometer T h e h y b r i d - l e v e l c o r r e l a t o r u s e s 1 4 - l e v e l a u t o - c o r r e l a t o r s , f o r m e a s u r i n g t h e m e a n a n d v a r i a n c e o f e a c h s i g n a l i n p u t , a n d t h r e e - l e v e l c o r r e l a t o r s f o r c o m p u t i n g m u l t i - l a g c o r r e l a t i o n f u n c t i o n s . T h e s y s t e m c a n c o r r e l a t e 2 0 - p a i r s o f i n p u t s f r o m o n e t o e i g h t a n t e n n a s . E a c h o f t h e t w e n t y t h r e e - l e v e l c o r r e l a t o r s p r o d u c e s a 5 1 2 - l a g c o r r e l a t i o n f u n c t i o n t h a t m a y b e F o u r i e r t r a n s f o r m e d t o p r o d u c e a p o w e r s p e c t r u m . B e i n g a h y b r i d - l e v e l s y s t e m i t h a s s o m e o f t h e p e r f o r m a n c e b e n e f i t s o f a 1 4 - l e v e l c o r r e l a t o r w i t h t h e s i m p l i c i t y a n d l o w c o s t o f a t h r e e - l e v e l c o r r e l a t o r - s p e c t r o m e t e r . B e c a u s e t h e s y s t e m o p e r a t e s o n d i f f e r e n t r a d i o t e l e s c o p e s , s o m e c o n f i g u r a t i o n f l e x i b i l i t y w a s d e s i g n e d i n t o i t . F o r a p e r t u r e s y n t h e s i s w o r k t h e s y s t e m c a n b e c o n f i g u r e d t o c r o s s - c o r r e l a t e u p t o 2 0 b a s e l i n e s f r o m e i g h t a n t e n n a i n p u t s , w h i l e f o r s i n g l e a n t e n n a s t h e 2 0 c o r r e l a t o r s c a n b e c o m b i n e d t o f o r m t w o 5 1 2 0 - l a g a u t o - c o r r e l a t o r s . I n t h e a u t o - c o r r e l a t i o n c o n f i g u r a t i o n o n l y n e g a t i v e l a g s a r e c o r r e l a t e d , a s t h e a u t o - c o r r e l a t i o n f u n c t i o n h a s e v e n s y m m e t r y . O n e f e a t u r e o f t h e s y s t e m i s t h a t t h e c o r r e l a t o r c a n b e c o n f i g u r e d t o s a m p l e t h e i n p u t d a t a a t t h e e i t h e r t h e N y q u i s t r a t e o r t w i c e t h e N y q u i s t r a t e — w i t h n o l o s s i n f r e q u e n c y r e s o l u t i o n 2 . T h i s m e a n s t h a t t h e c o r r e l a t i o n f u n c t i o n i s a l w a y s N y q u i s t s a m p l e d , e v e n t h o u g h t h e s a m p l e r a t e o f t h e i n p u t d a t a c a n b e a f a c t o r o f t w o h i g h e r . T h e b e n e f i t o f t h e d o u b l e N y q u i s t s a m p l i n g m o d e i s t h a t t h e S N R i s i m p r o v e d b y 7 % , a s q u a n t i z a t i o n n o i s e i s r e d u c e d . T h e o n l y d i s a d v a n t a g e o f t h e d o u b l e N y q u i s t s a m p l i n g c o n f i g u r a t i o n i s t h a t t h e m a x i m u m b a n d w i d t h i s r e d u c e d b y a f a c t o r o f t w o . I n t h e N y q u i s t s a m p l e d m o d e , t h e c o r r e l a t o r h a s a m a x i m u m b a n d w i d t h o f 8 M H z a n d a r e l a t i v e s e n s i t i v i t y o f 8 1 % ( c o m p a r e d t o a n i d e a l c o r r e l a t o r ) . In many correlator designs resolution is reduced when the data is over-sampled as the correlation function must be over-sampled as well. Since over-sampling implies some of the correlator channels (i.e., lags) do not contain independent signal information, resolution is reduced. 2 6 W h i l e i n t h e d o u b l e N y q u i s t m o d e i t h a s a m a x i m u m b a n d w i d t h o f 4 M H z a n d a r e l a t i v e s e n s i t i v i t y o f 8 8 % . T h e f o l l o w i n g t a b l e l i s t s t h e b a s i c b a n d w i d t h a n d r e s o l u t i o n o p e r a t i n g m o d e s o f a s i n g l e 5 1 2 - l a g c r o s s - c o r r e l a t o r . Table 2.1: Correlator bandwidth, HI velocity, and resolution specifications of the hybrid-level correlator. Note that these specifications can be applied to the auto-correlator as it has twice the resolution of the cross-correlator. Frequency Bandwidth Resolution HI Velocity Range Resolution Nyquist Sampling Mode Single Double (MHz) (kHz) (km/sec) (km/sec) 8 . 3 1 . 2 5 1 6 8 9 6 . 6 0 Y e s N o 4 1 5 . 6 3 8 4 4 . 5 3 . 3 0 Y e s Y e s 2 7 . 8 2 4 2 2 . 2 5 1 . 6 5 Y e s Y e s 1 3 . 9 1 2 1 1 . 1 3 0 . 8 3 Y e s Y e s . 5 1 . 9 6 1 0 5 . 5 7 0 . 4 2 Y e s Y e s . 2 5 0 . 9 8 5 2 . 7 9 0 . 2 1 Y e s Y e s . 1 2 5 0 . 4 9 2 6 . 4 0 . 1 1 Y e s Y e s . 0 6 2 5 0 . 2 4 1 3 . 2 0 . 0 5 N o Y e s I n s t e a d o f d e v e l o p i n g a c u s t o m s y s t e m a r c h i t e c t u r e f o r t h e n e w s y s t e m , t h e s y s t e m a r c h i t e c t u r e o f t h e D R A O X 2 1 - c m c o n t i n u u m - c o r r e l a t o r s y s t e m , d e s i g n e d b y K a r p a ( 1 9 8 9 ) , w a s u s e d . T h e a d o p t i o n o f t h i s a r c h i t e c t u r e r e d u c e d d e v e l o p m e n t t i m e a s m a n y c o m p o n e n t s d i d n o t h a v e t o b e d e s i g n e d , i n s t e a d t h e y w e r e e i t h e r u s e d o r m o d i f i e d f r o m t h o s e c o m p o n e n t s u s e d i n t h e c o n t i n u u m s y s t e m . T h i s a p p r o a c h s h o u l d a l s o r e d u c e m a i n t e n a n c e b u r d e n , a s m a i n t e n a n c e s t a f f a r e a l r e a d y f a m i l i a r w i t h t h i s h a r d w a r e a n d a s p a r e s c a n b e s h a r e d b e t w e e n t h e t w o s y s t e m s . 2 7 T h e p r i m a r y d i f f e r e n c e b e t w e e n t h e t w o s y s t e m s i s t h e t y p e a n d t h e n u m b e r o f c o r r e l a t o r s t h e y u s e . I n t h e c o n t i n u u m s y s t e m t h e r e a r e f e w e r c o r r e l a t o r s , a s o n l y o n e c o m p l e x l a g i s r e q u i r e d p e r a n t e n n a p a i r . T h e c o n t i n u u m s y s t e m i s l i k e a v e c t o r p o w e r m e t e r a n d p r o v i d e s n o s p e c t r a l i n f o r m a t i o n . T h e s p e c t r o m e t e r s y s t e m , h o w e v e r , i s l i k e a v e c t o r s p e c t r u m a n a l y z e r . A n d c a n ( v i a a F o u r i e r t r a n s f o r m ) m e a s u r e t h e c o m p l e x c o r r e l a t e d p o w e r a s a f u n c t i o n o f f r e q u e n c y b e t w e e n 2 0 p a i r s o f i n p u t s . T o d o t h i s , a t t h e r e s o l u t i o n s n e e d e d , t h e s p e c t r o m e t e r r e q u i r e s 2 5 6 m o r e c o r r e l a t o r l a g s p e r i n p u t p a i r t h a n t h e c o n t i n u u m s y s t e m . A m a j o r d e s i g n c h a l l e n g e w a s a d a p t i n g t h e c o n t i n u u m a r c h i t e c t u r e s o t h a t i t c o u l d h a n d l e m a n y m o r e c o r r e l a t o r c h a n n e l s a n d t h e i r a c c o m p a n y i n g c i r c u i t r y . A b l o c k d i a g r a m o f t h e c o r r e l a t i o n - s p e c t r o m e t e r s y s t e m i s s h o w n i n F i g . 2 . 1 . T h e m a j o r s p e c t r o m e t e r s u b - s y s t e m s a r e : • T h e A / D c o n v e r t e r s , w h i c h c o n d i t i o n a n d c o n v e r t u p t o e i g h t a n t e n n a s i g n a l s f r o m a n a l o g t o d i g i t a l f o r m . • T h e c o r r e l a t o r s u b - s y s t e m , w h i c h c a n c o m p u t e u p t o t w e n t y 5 1 2 - p o i n t t h r e e - l e v e l c o r r e l a t i o n f u n c t i o n s , a n d c o m p u t e t h e m e a n a n d v a r i a n c e o f e i g h t i n p u t s u s i n g 1 4 - l e v e l a u t o - c o r r e l a t o r s . • T h e m i c r o p r o c e s s o r b a s e d c o n t r o l l e r , w h i c h p r o c e s s e s d a t a a n d c o n t r o l s t h e s y s t e m u s i n g r e a l - t i m e m u l t i - t a s k i n g s o f t w a r e . • T h e s y s t e m c l o c k , w h i c h p r o v i d e s c l o c k s t o s y n c h r o n i z e t h e o p e r a t i o n o f t h e s y s t e m . 2 8 Baseband Antenna Signals (-20,dBm) A/D Data Commands & Control 8 8 N / Accumulator Clock Data& Status Telescope f N Control Micro-processorLG P I B B u s ^ Computer Controller RS-PI? KHS-Z3Z ^ VT200 Terminal Sample Rate Control Figure 2.1: A block diagram of the hybrid-level correlation-spectrometer system. 2.1 The Analog to Digital Converters A n t e n n a s i g n a l s e n t e r t h e s y s t e m v i a e i g h t 4 - b i t a n a l o g t o d i g i t a l c o n v e r t e r s ( A D C s ) w h e r e t h e y a r e a m p l i f i e d , f i l t e r e d , q u a n t i z e d a n d s a m p l e d . A n t i - a l i a s i n g f i l t e r s a r e n o t p r o v i d e d , s o t h e i n p u t s i g n a l s m u s t b e b a n d l i m i t e d b e f o r e e n t e r i n g a n A D C . T h e s i g n a l l e v e l r e q u i r e d b y t h e A D C s i s - 2 0 d B m w h i c h r e d u c e s t h e p o w e r r e q u i r e m e n t s o f t h e ( 5 0 o h m ) s i g n a l d i s t r i b u t i o n s y s t e m f e e d i n g t h e s y s t e m . A c o r r e l a t o r i s v e r y s e n s i t i v e t o c o m m o n m o d e s i g n a l s a n d c a n d e t e c t v e r y w e a k a s t r o n o m i c a l s o u r c e s , w h i c h a r e t y p i c a l l y - 3 0 d B b e l o w t h e n o i s e f l o o r o f t h e r e c e i v i n g s y s t e m . T h e r e f o r e , a n i m p o r t a n t A D C d e s i g n c r i t e r i a w a s t o m i n i m i z e t h e p i c k - u p o f e x t r a n e o u s c o m m o n - m o d e s i g n a l s , a s t h e y c a n s w a m p t h e a s t r o n o m i c a l s i g n a l u n l e s s s p e c i a l c a r e i s t a k e n . R a d i a t e d a n d c o n d u c t e d n o i s e c o u p l i n g w a s m i n i m i z e d i n t h e A D C b y s h i e l d i n g t h e a n a l o g s e c t i o n f r o m t h e d i g i t a l s e c t i o n . A l s o t h e g r o u n d p l a n e o f t h e a n a l o g s e c t i o n w a s i s o l a t e d f r o m t h e d i g i t a l s e c t i o n , e x c e p t a t o n e p o i n t , i n o r d e r t o m i n i m i z e t h e p a t h l e n g t h o f c u r r e n t l o o p s i n t h e g r o u n d s y s t e m . U p o n e n t e r i n g a n A D C t h e s i g n a l i s f i r s t p a d d e d b y - 3 d B , a n d t h e n b a n d - l i m i t e d b y a 5 - p o l e l o w -p a s s f i l t e r t h a t h a s a - 3 d B c u t o f f o f 1 0 M H z . T h e s i g n a l i s t h e n a m p l i f i e d b y a v i d e o a m p l i f i e r t h a t h a s a v o l t a g e g a i n o f 2 7 . T h i s s t a g e r a i s e s t h e s i g n a l l e v e l i n o r d e r t o m i n i m i z e t h e e f f e c t o f n o i s e p i c k e d u p i n t h e f o l l o w i n g s t a g e s . N e x t t h e s i g n a l i s h i g h - p a s s f i l t e r e d t o r e m o v e a n y l o w f r e q u e n c y 2 9 n o i s e b e l o w 8 0 k H z , s u c h a s p o w e r - s u p p l y r i p p l e , a n d t h e n d i g i t i z e d b y a r e l a t i v e l y f a s t ( 2 0 0 M s a m p l e s / s e c ) c o m m e r c i a l f o u r - b i t E C L f l a s h - c o n v e r t e r . T h e d a t a o u t p u t f r o m t h e c o n v e r t e r a r e s e n d t o t h e c o r r e l a t o r s u s i n g d i f f e r e n t i a l E C L b u f f e r s . 2.2 The Correlator T h e c o r r e l a t o r s u b - s y s t e m r e c e i v e s d i g i t a l d a t a f r o m u p t o e i g h t A D C - s t r e a m s a n d c o r r e l a t e s i t u n d e r c o n t r o l o f t h e m i c r o p r o c e s s o r c o n t r o l l e r . T h e c o r r e l a t o r s u b - s y s t e m i s o r g a n i z e d a s a s y s t o l i c a r r a y , a s A D C d a t a a n d s o m e c o n t r o l i n f o r m a t i o n i s t r a n s m i t t e d f r o m c a r d t o c a r d i n a p i p e l i n e d f a s h i o n . T h i s a r c h i t e c t u r e m a k e s t h e s y s t e m e x t e n d a b l e a n d a v o i d s t h e f a n o u t p r o b l e m s t h a t w o u l d o c c u r i f d a t a w e r e d i s t r i b u t e d b y b r o a d c a s t i n g i t . T h e c o r r e l a t o r s u b - s y s t e m c o n s i s t s o f t w o c r a t e s ( s e e F i g . 2 . 2 ) . E a c h c r a t e p r o c e s s e s f o u r s t r e a m s o f A D C d a t a a n d c o n t a i n s a c o n t r o l i n t e r f a c e c a r d ( C I C ) , a n a u t o - c o r r e l a t i o n c a r d ( A C ) a n d u p t o t e n 5 1 2 - l a g s p e c t r o m e t e r - c o r r e l a t o r c a r d s ( S C C s ) . A n 8 - b i t p a r a l l e l i n t e r f a c e - b u s c o n n e c t s t h e C I C s t o t h e e x t e r n a l w o r l d , s o t h a t a m i c r o p r o c e s s o r c a n s e n d c o m m a n d s a n d r e c e i v e d a t a f r o m t h e s u b -s y s t e m c r a t e s . I n e a c h c r a t e a b a c k p l a n e w a s d e s i g n e d t o i n t e r c o n n e c t t h e c a r d s v i a a 9 6 - p i n D I N c o n n e c t o r . T h e b a c k p l a n e i s u s e d t o t r a n s m i t d a t a s t r e a m s b e t w e e n c a r d s ; d i s t r i b u t e p o w e r , c o n t r o l , a n d c l o c k s i g n a l s ; a n d p r o v i d e a n e i g h t b i t t r i - s t a t e d a t a b u s f o r t r a n s m i t t i n g c o r r e l a t i o n d a t a f r o m a n y c a r d t o t h e c o n t r o l i n t e r f a c e c a r d . I n o r d e r t o e n s u r e h i g h s i g n a l f i d e l i t y a l l s i g n a l c o n d u c t o r s a r e t e r m i n a t e d i n t o t h e i r c h a r a c t e r i s t i c i m p e d a n c e a n d s e p a r a t e c l o c k s i g n a l s a r e t o d e l i v e r e d t o e a c h c a r d . S t r e a m s a r e c o n v e y e d i n t h e c o r r e l a t o r s u b - s y s t e m i n t w o w a y s , e i t h e r a l o n g t h e b a c k p l a n e o r t h r o u g h c r a t e c a b l e s . T h e b a c k p l a n e h a s 1 6 d a t a - l i n e s t h a t c a n c a r r y e i t h e r f o u r 4 - b i t s t r e a m s o r e i g h t 2 - b i t s t r e a m s . W h i l e o n l y f o u r s t r e a m s a r e n e e d e d f o r t h e A D C d a t a , t h e b a c k p l a n e h a s t h e 30 Crate 1 Correlator Data and Control Bus A/D Data A/DData Figure 2.2: A block diagram of the correlator sub-system. capacity to carry four extra streams. The crate cables are normally used to carry streams between crates. But they can also be used to form multi-board correlators within a crate. Each cable has eight data-lines andean carry either two (CIC or AC) 4-bit streams or four (SCC) 2-bit streams. Large correlators, with multiples of 512 lags, are formed by using one of the extra streams on the backplane, and by arranging the crate cables within a crate. In this scheme one of the extra backplane-streams carries a suitably delayed version of the correlator x output. By feeding this stream into and out of adjacent SCCs, properly delayed versions of an x stream will propagate through the correlators. To form a multi-board correlator the y stream is connected in a similar manner, but its direction of delay is opposite from the x stream. This is accomplished by using the crate cables to connect the outputs of downstream SCCs to the cable inputs of upstream SCCs. Doing this reverses the direction of the y stream. And since the direction of the y stream is opposite of the x stream a single large correlation function is formed. 3 1 2.2.1 The Control Interface Card (CIC) T h e c o n t r o l i n t e r f a c e c a r d ( C I C ) c o n t r o l s t h e o p e r a t i o n o f t h e c o r r e l a t o r s a n d c o n n e c t s t h e m i c r o p r o c e s s o r a n d t h e A / D c o n v e r t e r s t o t h e c o r r e l a t o r s i n a c r a t e . B y s e n d i n g c o m m a n d s t o a C I C t h e m i c r o p r o c e s s o r c a n c o n f i g u r e t h e s u b - s y s t e m , s t a r t a n d s t o p t h e c o r r e l a t o r s ( i . e . , c o n t r o l i n t e g r a t i o n ) , a n d r e a d c o r r e l a t o r d a t a . T h e C I C a l s o r e c e i v e s u p t o f o u r s t r e a m s o f A D C d a t a , a n d r e f o r m a t s i t f r o m 1 6 l e v e l s t o 1 4 l e v e l s [ K a r p a 8 9 ] . R e f o r m a t t i n g i s d o n e b e c a u s e t w o o f t h e l e v e l s , 0 a n d 1 5 , a r e u s e d t o e m b e d c o n t r o l i n f o r m a t i o n i n t h e d a t a . T h e C I C a l s o h a s r e g i s t e r s t h a t c o n f i g u r e t h e a u t o - c o r r e l a t o r s , a n d a s w e l l , h a s a R A M t h a t c a n s e n d t e s t v e c t o r s t o t h e c o r r e l a t o r s , o r s a v e d a t a s a m p l e s f r o m t h e A / D c o n v e r t e r s . U s i n g t h i s R A M t h e m i c r o p r o c e s s o r c a n b e u s e d t o t e s t t h e c o r r e l a t o r s o r b e u s e d t o a n a l y z e t h e p e r f o r m a n c e o f t h e A / D c o n v e r t e r s . T h e c o n t r o l i n t e r f a c e c a r d a l s o c o n t r o l s t h e i n t e g r a t i o n p e r i o d o f t h e c o r r e l a t o r s . T o s t o p c o r r e l a t i o n t h e C I C s u b s t i t u t e s a s t o p c o d e ( 1 5 ) f o r t h e n o r m a l A D C d a t a . W h e n a c o r r e l a t o r s e e s a s t o p c o d e i n t h e d a t a s t r e a m , i t s t o p s c o r r e l a t i n g a n d t h e c o n t e n t o f i t s a c c u m u l a t o r s a r e s a v e d i n b u f f e r s . T o s t a r t a n i n t e g r a t i o n p e r i o d t h e C I C s t o p s i n s e r t i n g s t o p c o d e s a n d c o r r e l a t i o n r e s u m e s . T h e i n t e g r a t i o n p e r i o d m a y b e c o n t r o l l e d e i t h e r l o c a l l y b y s o f t w a r e c o m m a n d s , o r e x t e r n a l l y b y a h a r d w a r e s i g n a l . I n l o c a l m o d e t h e m i c r o p r o c e s s o r c a n s e n d c o m m a n d s t o s t a r t a n d s t o p a n i n t e g r a t i o n . I n l o c a l m o d e r a n d o m s o f t w a r e l a t e n c i e s c a n c a u s e t h e i n t e g r a t i o n p e r i o d - t o j i t t e r , t o a v o i d t h i s t h e i n t e g r a t i o n p e r i o d m a y c o n t r o l l e d e x t e r n a l l y b y a h a r d w a r e t i m e r . T h e c o n t r o l i n t e r f a c e c a r d i s a l s o u s e d t o r e a d d a t a f r o m t h e c o r r e l a t o r s u n d e r c o n t r o l o f t h e m i c r o p r o c e s s o r . D a t a i s t r a n s f e r r e d f r o m t h e c o r r e l a t o r s t o t h e c o n t r o l i n t e r f a c e c a r d u s i n g a n e i g h t -b i t p a r a l l e l b u s t h a t u s e s t h r e e s i g n a l s t o c o n t r o l d a t a t r a n s f e r . T h i s r e a d - o u t p r o t o c o l i s b a s e d o n a t o k e n p a s s i n g s c h e m e a n d u s e s a n ENABLE IN s i g n a l , a n ENABLE OUT s i g n a l , a n d a READ STROBE. ENABLE IN i n d i c a t e s t h e a v a i l a b i l i t y o f t h e t o k e n . W h e n i t i s t r u e , i t m e a n s a c a r d h a s t h e t o k e n a n d c a n s e n d d a t a o n t h e b u s w h e n READ STROBE i s t r u e . A c a r d u s e s 3 2 ENABLE OUT to pass the token on, as when it is true it indicates that the card is finished reading and has no more data to send. By connecting the cards so that the ENABLE OUT of one card is connected to the ENABLE IN of another, reading is carried out in a sequential fashion from card to card. The CIC is the first card in a crate and controls the token using its ENABLE OUT signal. It can also determine when all cards (both SCCs and AC) have been read because the ENABLE OUT signal on the last card is feed back to the CIC. A status register on the CIC contains the state of this signal so that the microprocessor can determine when all cards have been read. To illustrate how this works, assume all cards have been read. This means the CIC has the token as the last card in the crate has ENABLE OUT asserted. The microprocessor, software monitors this signal by polling a status register on the CIC. A card indicates that it has new data by de-asserting ENABLE OUT. This essentially takes the token away from the CIC as ENABLE OUT false propagates to down-stream boards. Soon after the token is taken, the CIC will pre-fetch the data for the microprocessor by activating READ STROBE. When the microprocessor recognizes that the CIC no longer has the token it will read the pre-fetched data stored in the CIC. After the data has been read, the CIC will re-examine ENABLE OUT from the last card and if it is still false the read sequence will repeat. Reading continues until all the data has been read and ENABLE OUT from the last card is asserted. 2.2.2 The Auto-correlator Card (AC) The auto-correlator card (AC) contains eight single-lag cross-correlators that can be configured to provide the mean and variance for up to four ADC streams [Karpa89]. This statistical information is necessary for normalizing the output of a correlator and is useful for monitoring the behavior of the input signals. Equations ( 2 . 1 ) through ( 2 . 4 ) below, illustrate how the mean and variance are computed by the auto-correlator. Note that the constant, 3 3 3 , and the scalar, 4096, in the equations are artifacts of the AC design. Eq. ( 2 . 1 ) shows that a when the CIC applies a constant, K, to one 3 3 i n p u t o f a n a u t o - c o r r e l a t o r t h e A C c o m p u t e s t h e s u m o f t h e A D C d a t a o y e r t h e i n t e g r a t i o n p e r i o d . S i m i l a r l y , E q . ( 2 . 2 ) s h o w s t h a t a n a u t o - c o r r e l a t o r c o m p u t e s t h e s u m o f t h e A D C d a t a s q u a r e d w h e n t h e s a m e d a t a i s s e n t t o b o t h i n p u t s . £ ( £ * , . + 3 3 3 ) 4 0 9 6 5>,2 + 3 3 3 ) r» (0) = — (2-2) 4 0 9 6 T h e A D C s a m p l e m e a n a n d v a r i a n c e m a y b e c o m p u t e d f r o m r f a a n d u s i n g - _ 4 0 9 6 - f i , ( 0 ) - 3 3 a . t ( 2 3 ) nK a n d , 4 0 9 6 r t I ( 0 ) - ( 3 3 3 + 3 t 2 ) n (n - 1 ) I n o r d e r t o c o m p u t e E q s . ( 2 . 1 ) a n d ( 2 . 2 ) , t h e c r o s s - c o r r e l a t o r s o n a n A C a r e o r g a n i z e d i n a 4 b y 2 m a t r i x ( w h e r e t h e x i n p u t o f a r o w o f t w o c o r r e l a t o r s i s c o n n e c t e d t o o n e d a t a - s t r e a m a n d t h e y i n p u t o f a c o l u m n o f f o u r c o r r e l a t o r s i s c o n n e c t e d t o a 4 - b i t r e g i s t e r o n a C I C ) . T h e c o r r e l a t o r s i n a m a t r i x r o w p r o c e s s o n e d a t a s t r e a m , w h i l e t h e c o r r e l a t o r s i n a m a t r i x c o l u m n a r e c o m m o n t o o n e ( o f t w o ) C I C 4 - b i t r e g i s t e r s . T h e v a l u e i n a r e g i s t e r c o n t r o l s t h e o p e r a t i o n o f t h e c o r r e l a t o r s i n a c o l u m n : w h e n t h e v a l u e i s z e r o t h e n t h e c o r r e l a t o r s c o m p u t e E q ( 2 . 2 ) , a n d w h e n i t i s n o n - z e r o t h e y c o m p u t e E q . ( 2 . 1 ) . T h e r e g i s t e r v a l u e c a n b e s e t b y t h e m i c r o p r o c e s s o r c o n t r o l l e r , s o t h a t e a c h c o r r e l a t o r r o w , o r p a i r , c a n b e u s e d t o c o m p u t e t h e s a m p l e m e a n a n d v a r i a n c e o f a n A D C d a t a s t r e a m . 34 2.2.3 The Spectrometer Correlator Card (SCC) The spectrometer correlator card (SCC) is the primary card in the system and was the major hardware component designed. This card computes a 512-lag correlation function for a pair of three-level inputs. The operations required to compute a 512-lag correlation function can be derived by inspecting the following equation: n r^T) = X*.-x^ ; T =-255,-254...256., (2.5) where, i is the sample number from the start of an integration period, xi and y; are ADC data samples, X is the delay, or sample time difference, between x and y, n is the total number of samples during an integration period, rxy (T) is the correlation function between x and y as a function of delay. This equation shows that to compute one delay, or lag, of rxy (T) requires a multiplier, an accumulator, and samples of x and y that have a relative delay of x. To compute the entire function requires 512 multiplier-accumulator pairs, and delayed version of x and y from -255x to 256x. Also needed is a way to start and stop an integration, zero the accumulator, and read the accumulator. The operations above are implemented in a custom correlator ASIC, designed for this project. While a single ASIC computes a four point correlation function, its interface is designed so that arrays of ASICs can be used to compute a large correlation function. In the case of the spectrometer system, 128 ASICs implement a 512-lag correlator. 3 5 T h e b l o c k d i a g r a m i n F i g . 2 . 3 s h o w s h o w t h e A S I C i s o r g a n i z e d , w h i l e a m o r e d e t a i l e d v e r s i o n i s g i v e n F i g u r e 3 . 1 . T h e x a n d y A D C d a t a i n p u t s a r e t h r e e - l e v e l q u a n t i t i e s ( 1 , 0 , - 1 ) a n d a r e p i p e l i n e d t h r o u g h t h e A S I C u s i n g a s h i f t r e g i s t e r c r e a t e d f r o m f o u r 4 - b i t r e g i s t e r s . T h e r e g i s t e r s a r e a r r a n g e d s o t h a t t h e x a n d y s t r e a m s f l o w i n o p p o s i t e d i r e c t i o n s , t h e r e b y c r e a t i n g a r e l a t i v e d e l a y b e t w e e n t h e t w o s t r e a m s . A s a m p l e c l o c k d r i v e s t h e s h i f t r e g i s t e r a n d c o n t r o l s t h e s a m p l e r a t e o f t h e c o r r e l a t o r . T h e c o r r e l a t o r l a g s a r e f o r m e d b y c o n n e c t i n g a m u l t i p l i e r - a c c u m u l a t o r a c r o s s e a c h s h i f t - r e g i s t e r o u t p u t . T h e m u l t i p l i e r o u t p u t i s b i a s e d s o t h a t o n l y p o s i t i v e p r o d u c t s ( i . e . , 0 , 1 , 2 ) a r e p r o d u c e d . T h i s a l l o w s t h e a c c u m u l a t o r t o b e i m p l e m e n t e d a s a s i m p l e u p - c o u n t e r . E a c h a c c u m u l a t o r ( o r c o u n t e r i n F i g . 2 . 3 ) i s i m p l e m e n t e d u s i n g a 2 - b i t a d d e r w h o s e c a r r y o u t p u t s y n c h r o n o u s l y c l o c k s a 2 3 - b i t r i p p l e - c o u n t e r . T h e a d d e r i s c o n n e c t e d t o t h e t w o - b i t m u l t i p l i e r o u t p u t a n d a d d s t h e s e b i t s i n p a r a l l e l t o t h e p r e v i o u s L S B s s t o r e d i n a r e g i s t e r . A l l t h e c a r r y - o u t p u t s a r e c l o c k e d b y a f i x e d - r a t e a c c u m u l a t o r c l o c k , s o t h a t t h e m e a n v a l u e o f t h e a c c u m u l a t o r i s c o n s t a n t r e g a r d l e s s o f t h e s a m p l e r a t e . C a r r i e s g e n e r a t e d b y a n a d d e r a r e a c c u m u l a t e d u s i n g a 2 3 - b i t r i p p l e c o u n t e r . E a c h a c c u m u l a t o r i s b u f f e r e d u s i n g a 1 6 - b i t l a t c h w h i c h i s c o n n e c t e d t o t h e u p p e r 1 6 b i t s o f a ' r i p p l e c o u n t e r . T h i s a l l o w s a c c u m u l a t o r r e s u l t s t o b e s a v e d q u i c k l y a n d r e a d s l o w l y w h i l e t h e a c c u m u l a t o r i s i n t e g r a t i n g . A l a t c h e n a b l e s i g n a l i s u s e d t o s t o p t h e a c c u m u l a t o r a n d l a t c h t h e r e s u l t s i n t h e b u f f e r . T h e A S I C c o r r e l a t o r h a s a n u m b e r o f i n t e r e s t i n g a s p e c t s . O n e i s i t s a c c u m u l a t o r s c h e m e d i f f e r s f r o m t h e a p p r o a c h u s e d i n t h e p r e v i o u s D R A O s p e c t r o m e t e r d e s i g n . I n t h e p r e v i o u s d e s i g n a s y n c h r o n o u s p a r a l l e l - a d d e r w a s n o t u s e d . I n s t e a d t h e m u l t i p l i e r o u t p u t w a s s e r i a l i z e d ( e f f e c t i v e l y d o u b l i n g i t s o u t p u t f r e q u e n c y ) , s o t h a t i t c o u l d d i r e c t l y c l o c k t h e i n p u t o f a r i p p l e c o u n t e r . T h i s w a s d o n e b y g a t i n g t w o p u l s e s t o c l o c k t h e r i p p l e c o u n t e r . T h e n u m b e r o f p u l s e s g a t e d d e p e n d e d o n t h e m u l t i p l i e r v a l u e . S o f o r e x a m p l e , a m u l t i p l i e r v a l u e o f t w o r e s u l t e d i n t w o c l o c k s . T h e c l o c k p u l s e s a . S " § Si s-o o co 7> T3 o a n> CO O 3 V D a •< X O O < X D D D O x o U3 37. w e r e d e r i v e d f r o m a n a c c u m u l a t o r c l o c k u s i n g a n a s y n c h r o n o u s c i r c u i t . B e c a u s e o f t h e a s y n c h r o n o u s n a t u r e o f t h e c i r c u i t , i t w a s s e n s i t i v e t o v a r i a t i o n s i n p r o p a g a t i o n d e l a y . T o e n s u r e c o r r e c t o p e r a t i o n , t h e t i m e b e t w e e n p u l s e s a n d t h e i r r e l a t i o n t o t h e s h i f t r e g i s t e r c l o c k h a d t o b e " a d j u s t e d . S u c h a n a d j u s t m e n t i s d i f f i c u l t t o i m p l e m e n t , a s i t i s n e e d e d f o r e a c h m u l t i p l i e r , s o i n s t e a d a g l o b a l a d j u s t m e n t w a s p r o v i d e d . H o w e v e r , t h i s s o l u t i o n w a s n o t i d e a l a s f i n d i n g a g l o b a l o p t i m u m f o r a l a r g e n u m b e r o f m u l t i p l i e r s w a s s o m e t i m e s q u i t e d i f f i c u l t . T h e a c c u m u l a t o r s c h e m e u s e d i n t h e A S I C a v o i d s t h e a b o v e d i f f i c u l t i e s b y u s i n g a s y n c h r o n o u s p a r a l l e l - a d d e r b e t w e e n t h e m u l t i p l i e r a n d t h e ripple c o u n t e r . I t s s y n c h r o n o u s n a t u r e i s m o r e t o l e r a n t o f p r o p a g a t i o n d e l a y v a r i a t i o n s . F u r t h e r m o r e , t h e m u l t i p l i e r o u t p u t r a t e r e q u i r e d i s a f a c t o r o f t w o l o w e r b e c a u s e o f i t s p a r a l l e l o p e r a t i o n . S i n c e t h e m a x i m u m c o r r e l a t i o n r a t e i s p r i m a r i l y d e t e r m i n e d b y t h e m u l t i p l i e r - a c c u m u l a t i o n s t a g e s , t h e a p p r o a c h u s e d i n t h e A S I C i s a n i n h e r e n t l y f a s t e r a n d m o r e s t a b l e a r c h i t e c t u r e ; A n o t h e r a s p e c t o f t h e A S I C i s t h a t d a t a i s r e a d o u t o f t h e a c c u m u l a t o r s u s i n g a b i t s e r i a l a p p r o a c h . T h e s c h e m e i s i m p l e m e n t e d u s i n g a 2 o f 6 4 b i t m u l t i p l e x e r , a r e a d c o u n t e r , a n d t w o t r i - s t a t e b u f f e r s . D e p e n d i n g o n t h e r e a d c o u n t e r v a l u e , t h e m u l t i p l e x e r r o u t e s o n e b i t f r o m t w o a c c u m u l a t o r b u f f e r s t o t h e t r i - s t a t e b u f f e r s . A s s e r t i n g READ STROBE c a u s e s t h e t r i - s t a t e b u f f e r s t o p r e s e n t t h e s e l e c t e d b i t o n a b u s . W h e n READ STROBE i s s e t f a l s e t h e c o u n t e r i s a d v a n c e d s o t h a t a n e w p a i r o f b i t s m a y b e o u t p u t o n s u b s e q u e n t t r a n s f e r s . T h e r e a d o r d e r i s m o s t s i g n i f i c a n t b i t f i r s t , l e a s t s i g n i f i c a n t b i t l a s t . T h i s r e a d o u t s c h e m e h a s s e v e r a l a d v a n t a g e s . F i r s t , i t u s e s o n l y t h r e e s i g n a l l i n e s t o t r a n s m i t 6 4 b i t s o f d a t a . T h i s r e d u c e s t h e n u m b e r o f o u t p u t p i n s o n t h e A S I C w h i c h l o w e r s t h e c o s t a n d t h e s i z e o f t h e p a c k a g e r e q u i r e d . F u r t h e r m o r e , t h e d a t a t r a n s f e r r a t e o f a n a r r a y o f A S I C s i s n o t r e d u c e d b e c a u s e t h e o u t p u t s o f s e v e r a l A S I C s c a n b e r e a d i n p a r a l l e l ( f o r e x a m p l e , e i g h t o u t p u t s c o u l d b e r e a d i n p a r a l l e l b y c o n n e c t i n g f o u r A S I C s t o a p a r a l l e l b u s ) . T h e s c h e m e i s a l s o f l e x i b l e , s i n c e w i d e , o r n a r r o w b u s e s c a n b e f o r m e d d e p e n d i n g o n t h e n u m b e r o f A S I C s r e a d i n p a r a l l e l . 38 A f u r t h e r a d v a n t a g e i s t h a t i t a l l o w s o n e t o e f f i c i e n t l y e x t e n d t h e l e n g t h o f t h e a c c u m u l a t o r s . T h i s i s d o n e b y m o n i t o r i n g t h e M S B o f e a c h a c c u m u l a t o r a n d d e t e c t i n g a n d c o u n t i n g a c c u m u l a t o r o v e r f l o w s . T h e c h i e f a d v a n t a g e i s t h a t t h e e x t e r n a l c i r c u i t r y r e q u i r e d t o p e r f o r m t h i s f u n c t i o n c o s t s l e s s t h a n c i r c u i t r y o n t h e A S I G . A n o t h e r a d v a n t a g e i s t h a t i t i s m o r e f l e x i b l e , s i n c e t h e u l t i m a t e a c c u m u l a t o r l e n g t h r e q u i r e d d o e s n o t h a v e t o b e d e t e r m i n e d i n t h e A S I C . T h e m e t h o d i s a l s o e f f i c i e n t b e c a u s e a g r o u p o f A S I C s c a n o u t p u t t h e i r M S B d a t a i n p a r a l l e l a n d s o t h e M S B d a t a i s n o t . m i x e d w i t h b i t s o f d i f f e r e n t w e i g h t s , a s i t i s w i t h o t h e r s c h e m e s . A l s o t h e e x t e r n a l c i r c u i t r y c a n b e a r r a n g e d s o t h a t s e q u e n t i a l r e a d s o n l y t r a n s f e r M S B d a t a . T h i s i s d o n e o n t h e S C C b y g e n e r a t i n g 1 5 r e a d - s t r o b e s , o r d u m m y r e a d s , a f t e r e a c h s e t o f M S B s a r e r e a d . B e c a u s e o n l y M S B d a t a i s t r a n s f e r r e d t h i s c a n b e d o n e v e r y q u i c k l y . A n o t h e r a s p e c t o f t h e A S I C i s t h a t e v e n d e l a y p r o d u c t s a r e n o t f o r m e d , i n s t e a d i t p r o d u c e s t h e f o l l o w i n g c o r r e l a t i o n f u n c t i o n : N ^ ( ^ ) = Z J C . - X ^ ^ = - 3 , - 1 , 1 , 3 . (2.6) i = l A t f i r s t t h i s g l a n c e t h i s m a y s e e m p r o b l e m a t i c , s i n c e t h e c o r r e l a t o r f u n c t i o n i s u n d e r - s a m p l e d . O n e w a y t o o v e r c o m e t h i s i s t o a d d a s e c o n d c o r r e l a t o r A S I C a n d d e l a y i t s y ; i n p u t w i t h D f l i p - f l o p s . T h i s c a u s e s t h e r e l a t i v e d e l a y s b e t w e e n x a n d y t o b e s h i f t e d o n e l a g a n d y i e l d s t h e e v e n l a g c o r r e l a t i o n f u n c t i o n . JV -rXy^eVen) = rxy(Todd-l) = ^xixyi_hTadj ; Teven = -4,-2,0,2 . (2.7) / = i E q . (2.8) s h o w s t h a t i f t h e o u t p u t o f b o t h c o r r e l a t o r s a r e c o m b i n e d t h e n w e o b t a i n a N y q u i s t s a m p l e d c o r r e l a t i o n f u n c t i o n f r o m N y q u i s t s a m p l e d d a t a . rxy(*") e rv(Teven)urxy(rodd) (2.8) 3 9 A s e c o n d w a y t o o b t a i n a N y q u i s t s a m p l e d c o r r e l a t i o n f u n c t i o n i s t o i n c r e a s e t h e s a m p l e r a t e b y a f a c t o r o f t w o ( i . e . , b y d o u b l e - N y q u i s t s a m p l i n g t h e d a t a ) . N o w t h e p e r i o d b e t w e e n c o r r e l a t o r l a g s i s t h e N y q u i s t p e r i o d . T h e c o r r e l a t i o n f u n c t i o n p r o d u c e d i s g i v e n i n E q . ( 2 . 9 ) a n d s h o w s t h a t e v e n t h o u g h t h e p e r i o d b e t w e e n l a g s i s N y q u i s t t h e l a g s a r e s h i f t e d b y h a l f a p e r i o d . N (^?") = 5 > , X y ( . + r ; T = - 1 . 5 , - - 5 , . 5 , 1 . 5 ( 2 . 9 ) T h i s , h a l f p e r i o d s h i f t , d o e s n o t c a u s e a p r o b l e m f r o m a n i n f o r m a t i o n s t a n d p o i n t b e c a u s e t h e c o r r e l a t i o n f u n c t i o n i s N y q u i s t s a m p l e d . A s b e f o r e o n e m a y c r e a t e a c o r r e l a t i o n f u n c t i o n w i t h a z e r o l a g , s h o w n i n E q . ( 2 . 1 0 ) , b y s i m p l y d e l a y i n g y ; b y h a l f a l a g w i t h D f l i p - f l o p s . N ^ ( 7 ) = 5 > , x y , + r ; T = - 2 , - 1 , 0 , 1 ( 2 . 1 0 ) A l t e r n a t i v e l y , o n e c a n m a k e t h e c o r r e c t i o n p r i o r t o c o m p u t i n g t h e F F T b y i n s e r t i n g z e r o s b e t w e e n e a c h l a g [ D e w d n e y 7 8 ] . H o w e v e r , t h i s m e t h o d c o s t s a f a c t o r o f 2 i n s p a c e a n d a f a c t o r o f 2 . 2 5 i n t h e t i m e t o c o m p u t e t h e F F T . A s w e l l , t h e t o t a l p o w e r i n f o r m a t i o n f r o m t h e z e r o l a g c h a n n e l i s l o s t s i n c e i t i s s e t t o z e r o . T h e f a c t t h a t t h e A S I C c a n b e u s e d t o p r o d u c e a N y q u i s t s a m p l e d c o r r e l a t i o n f u n c t i o n f r o m e i t h e r N y q u i s t o r d o u b l e - N y q u i s t s a m p l e d d a t a h a s s o m e i n t e r e s t i n g a s p e c t s . N y q u i s t s a m p l i n g h a s t h e a d v a n t a g e t h a t t h e c o r r e l a t o r c a n o p e r a t e a t i t s m a x i m u m s a m p l e r a t e , a n d s o i t s m a x i m u m b a n d w i d t h i s a f a c t o r o f t w o h i g h e r t h a n d o u b l e - N y q u i s t s a m p l i n g . T h e c h i e f a d v a n t a g e o f d o u b l e -N y q u i s t s a m p l i n g i s t h a t t h e S N R o f t h e c o r r e l a t o r i s i m p r o v e d b y a b o u t 7 % a n d i s e q u i v a l e n t t o t h e s e n s i t i v i t y o f a f o u r - l e v e l c o r r e l a t o r ( s e e t a b l e 4 . 1 i n s e c t i o n 4 . 3 ) . 4 0 2.2.4 Spectrometer-Correlator Card Support-Circuitry W h i l e 1 2 8 c o r r e l a t o r - A S I C s i m p l e m e n t t h e f u n c t i o n s r e q u i r e d f o r c o r r e l a t i o n , a n u m b e r o f I C s a r e n e e d e d t o i n t e r f a c e t h e A S I C s t o a C I C a n d b a c k p l a n e . T h i s i n t e r f a c e c i r c u i t r y m a y b e d i v i d e d i n t o t h r e e p a r t s , t h e d a t a p a t h l o g i c , t h e i n t e g r a t i o n c o n t r o l l o g i c a n d t h e r e a d - o u t l o g i c . T h e d a t a p a t h l o g i c a d a p t s t h e b a c k p l a n e d a t a s t r e a m s t o t h a t r e q u i r e d b y t h e A S I C s . A s m e n t i o n e d i n p r e v i o u s s e c t i o n s a n u m b e r o f d a t a s t r e a m s a r e p i p e l i n e d i n t o a n d o u t o f a n S C C . W h e n t h e s t r e a m s a r e r e c e i v e d b y t h e S C C t h e y a r e f i r s t l a t c h e d b y t w o r e g i s t e r s . S i n c e a d a t a s t r e a m c a n b e i n e i t h e r a 1 4 - l e v e l o r t h r e e - l e v e l f o r m a t a n d t h e A S I C s r e q u i r e a t h r e e - l e v e l f o r m a t , t w o c o n v e r s i o n P A L s ( P r o g r a m m a b l e A r r a y L o g i c ) c a n b e e n a b l e d t o r e - q u a n t i z e 1 4 - l e v e l d a t a t o t h r e e - l e v e l s . T h e c o r r e l a t o r A S I C s a l s o r e q u i r e a n x a n d y s t r e a m i n p u t a n d t w o m u l t i p l e x e r P A L s a r e u s e d t o s e l e c t t w o s t r e a m s f r o m t h e c o n v e r s i o n P A L o u t p u t s . T h e c o n v e r s i o n P A L o u t p u t s a r e a l s o s e n t o u t t o t h e b a c k p l a n e a n d c r a t e c a b l e s v i a t w o r e g i s t e r s . T h e s e r e g i s t e r s b u f f e r a n d r e - t i m e t h e d a t a s o t h a t i t c a n b e p r o p e r l y r e c e i v e d b y a d j a c e n t S C C s i n a b a c k p l a n e . T h e x a n d y o u t p u t s a r e a l s o c o n n e c t e d t o o n e o f t h e r e g i s t e r s , s o t h a t a n a d j a c e n t c o r r e l a t o r c a n u s e t h e m t o f o r m a m u l t i - b o a r d c o r r e l a t i o n f u n c t i o n . T h e i n t e g r a t i o n c o n t r o l l o g i c ( i m p l e m e n t e d u s i n g a P A L ) a l l o w s t h e c o n t r o l i n t e r f a c e c a r d ( C I C ) t o c o n t r o l t h e o p e r a t i o n o f t h e c o r r e l a t o r A S I C s . T h e w a y t h e C I C s t o p s i n t e g r a t i o n i s b y i n s e r t i n g STOP CODEs i n t h e d a t a s t r e a m s . W h e n t h i s P A L d e t e c t s t h e p r e s e n c e o f a STOP CODE i t a s s e r t s t h e A S I C LATCH ENABLE l i n e s w h i c h s t o p c o r r e l a t i o n a n d c a u s e s t h e a c c u m u l a t o r v a l u e s t o b e l a t c h e d . W h e n STOP CODES a r e r e m o v e d , t h e P A L w i l l d e - a s s e r t t h e e n a b l e l i n e s . I f t h e a c c u m u l a t o r s s h o u l d b e r e s e t , t h e C I C w i l l i n d i c a t e t h i s w h i l e t h e STOP CODEs a r e p r e s e n t b y a s s e r t i n g a r e s e t s i g n a l o n t h e b a c k p l a n e . T h e P A L d o e s n o t r e s e t t h e A S I C s r i g h t a w a y , a s LATCH ENABLE i s a s s e r t e d a t t h i s t i m e a n d z e r o s w o u l d b e l a t c h e d i n t h e b u f f e r s . I n s t e a d , t h e P A L w a i t s u n t i l STOP CODEs a r e n o l o n g e r p r e s e n t a n d t h e n r e s e t s t h e A S I C s . 41 A second function of the integration control logic is that it is used to implement a method for extending the length of the ASIC accumulators (discussed in the previous section). This is done with three programmable-counters that determine the MSB sample-period. The counters are preset with a time-out value whenever the accumulators are reset. Counting is started at the beginning of an integration. When the time-out value is reached the counters assert the SAMPLE-MSB signal, which causes the integration control PAL to toggle the ASIC LATCH ENABLE lines. This causes the correlator ASICs to latch the content of their accumulators and resume integration, so that the most-significant accumulator bits (MSBs) can be read. The readout logic on the SCC allows the control interface card (CIC) to read the data stored in the ASIC accumulator buffers, or latches. But, before discussing this logic we need to examine how the ASICs are organized on the SCC. The correlator ASICs are arranged in a readout matrix with four columns and eight rows. A group of four ASICs make up a matrix element whose outputs are tied to an eight-bit data bus (four ASICs x two output bits each). The READ STROBE lines of four ASICs (in an element) are tied together, so that when the strobe is asserted the four ASICs output eight data-bits on the bus. Data is readout by asserting READ STROBEs to each matrix element (of four ASICs) in a seqeuntial fashion. The read logic is implemented using a read control PAL, a read counter PAL, and four decoders. The read counter is used to address the decoders which are used to route a read strobe to a matrix element of four ASICs. The counter PAL keeps track of the number of READ STROBEs send to each matrix element, and uses a decoder to address each element in the matrix. The read control PAL determines when to respond to READ STROBEs sent by a control interface card, and determines when new data is available. New data is ready whenever the integration control PAL asserts its LATCH ENABLE line (as mentioned before this signal causes the ASIC accumulator contents to be latched). When this occurs the read control PAL sets the SCC backplane ENABLE 4 2 OUT l i n e f a l s e , w h i c h s i g n a l s t h e C I C t h a t n e w d a t a i s r e a d y . A t s o m e p o i n t t h e S C C ENABLE IN l i n e i s a s s e r t e d a n d t h e C I C w i l l b e g i n a s s e r t i n g READ STROBES. T h e r e a d c o n t r o l P A L r e s p o n d s t o t h e s e e v e n t s b y e n a b l i n g t h e r e a d c o u n t e r P A L a n d d e c o d e r s w h i c h r o u t e t h e s t r o b e s t o a m a t r i x e l e m e n t o f f o u r A S I C s . T h e r e a d o u t s e q u e n c e i s c o m p l e t e a f t e r 1 0 2 4 READ-STROBEs h a v e b e e n r e c e i v e d . A t t h i s p o i n t t h e r e a d c o n t r o l P A L a s s e r t s t h e S C C ENABLE OUT l i n e i n d i c a t i n g t h a t t h e S C C h a s n o m o r e d a t a t o s e n d . A m e t h o d f o r t r a n s f e r r i n g M S B d a t a t o t h e C I C i s a l s o i m p l e m e n t e d i n t h e r e a d c o n t r o l P A L . B y m o n i t o r i n g t h e M S B s a m p l e l i n e , t h e r e a d c o n t r o l P A L c a n d e t e r m i n e i f M S B d a t a i s t o b e e n s e n t t o t h e C I C . I f i t i s , t h e n a f t e r e a c h M S B b y t e i s r e a d b y t h e C I C , t h e r e a d c o n t r o l P A L w i l l g e n e r a t e 1 5 ( d u m m y ) READ STROBES. T h i s a c t i o n r e d u c e s t h e a m o u n t o f d a t a s e n d t o t h e C I C b y a f a c t o r o f 1 6 , w h i c h s a v e s d a t a t r a n s f e r t i m e a n d c o m p u t e r s t o r a g e s p a c e . 2.3 The Microprocessor Controller T h e m i c r o p r o c e s s o r c o n t r o l l e r h a r d w a r e i s i m p l e m e n t e d u s i n g a M o t o r o l a 6 8 0 0 0 S T D - b u s b a s e d c o m p u t e r w i t h s e v e r a l i n t e r f a c e c a r d s . T h e s e c a r d s i n t e r f a c e t h e c o m p u t e r t o t h e c o r r e l a t o r s u b -s y s t e m , a h o s t c o m p u t e r , a n d a v i d e o t e r m i n a l . T h e s o f t w a r e f o r t h i s s y s t e m e x e c u t e s u n d e r c o n t r o l o f V R T X ( a c o m m e r c i a l r e a l - t i m e m u l t i - t a s k i n g k e r n e l ) a n d i s w r i t t e n i n t h e C p r o g r a m m i n g l a n g u a g e . T h e c o n t r o l l e r p r o v i d e s h i g h - l e v e l s e r v i c e s f o r t h e c o r r e l a t i o n - s p e c t r o m e t e r s y s t e m . U s i n g t h e c o n t r o l l e r ' s G P I B b u s i n t e r f a c e , a h o s t c o m p u t e r c a n s e t s y s t e m p a r a m e t e r s , r e c e i v e c o r r e l a t o r d a t a , a n d c o n t r o l t h e o p e r a t i o n o f t h e s y s t e m . A HOST INTERFACE t a s k d o e s m o s t o f t h i s . I t o p e r a t e s , b y r e a d i n g a n d e x e c u t i n g c o m m a n d s f r o m t h e G P I B bus. T h e s y s t e m m a y a l s o b e c o n t r o l l e d b y a v i d e o t e r m i n a l . F r o m t h i s t e r m i n a l a u s e r m a y s e t a n d s h o w s y s t e m p a r a m e t e r s , a n d c o n t r o l t h e o p e r a t i o n o f t h e s y s t e m . S o f t w a r e i s a l s o p r o v i d e d s o t h a t a u s e r 43 can test the system. The USER INTERFACE task does this, by reading and executing commands coming from a video terminal. Three tasks are used to control the operation of the system while it is correlating, they are: the OBSERVE task, the PROCESS DATA task, and the HOST WRITE task (see Fig. 2.4). The OBSERVE task performs all the time critical operations in the system. This task is used to synchronize the integration period with the host, and read out data from the correlator sub-system. When signaled to start an observation it reads the current operating parameters and initializes the correlator sub-system. It then synchronizes to an external integration control signal and waits for an integration to begin. When the integration starts this task monitors the data ready status bit from the first control interface card in system. Whenever data is ready it is read into an internal buffer and passed on to the PROCESS DATA task. The OBSERVE task continues in this manner until the observation is completed. Two types of read operations are required: one for reading MSB data and one for reading all the data. MSB reads are shorter and are performed during an integration period every 500 ms or so. Integrations last for about 5.6 sec and when they end a full readout is performed. The PROCESS DATA task processes the data in buffers sent by the observe task. When MSB data is received the data is scanned for carry transitions by comparing the last (MSB) buffer received to those in the current buffer. When a carry is detected it is added to a carry buffer which accumulates the carries for each correlator lag. When a full-readout buffer is received it is first scanned for carry transitions and then it is transposed from bit serial to bit parallel form. 44 Figure 2.4: A data-flow diagram of the controller software. Note that dashed and solids lines indicate control and data flow, respectively. T h e f u l l - r e a d o u t d a t a i s t h e n a d d e d t o t h e c a r r i e s a c c u m u l a t e d d u r i n g a n M S B r e a d o u t . N e x t t h e d a t a i s s c a l e d b y t h e n u m b e r o f s a m p l e s r e c o r d e d i n t h e i n t e g r a t i o n p e r i o d . T h i s v a l u e i s p r o v i d e d b y t h e f i r s t C I C i n t h e c o r r e l a t o r s u b - s y s t e m a n d i s r e q u i r e d b e c a u s e t h e i n t e g r a t i o n p e r i o d c a n v a r y . T h e c o r r e l a t i o n d a t a f o r e a c h S C C i s t h e n p h a s e d e m o d u l a t e d b y m u l t i p l y i n g i t b y p l u s o r m i n u s o n e 3 . T h e r e s u l t i s t h e n a d d e d t o a n a c c u m u l a t i o n b u f f e r w h i c h i s ( t y p i c a l l y ) h e l d f o r 1 6 i n t e g r a t i o n s o r 9 0 s e c o n d s . A t t h e e n d o f e v e r y 1 6 i n t e g r a t i o n s t h e a c c u m u l a t i o n b u f f e r i s p l a c e d o n t o t h e HOST WRITE t a s k q u e u e . T h e HOST WRITE t a s k i s u s e d t o s e n d c o r r e l a t o r d a t a t o a h o s t c o m p u t e r v i a t h e G P I B b u s . I t w a i t s a t i t s q u e u e f o r d a t a a n d w h e n i t a r r i v e s i t f o r m a t s t h e d a t a a n d s e n d s i t t o t h e h o s t . This is required because the antenna input signal phases are switched by 180 degrees and so are de-modulated here. Phase switching is done to remove instrumental biases between the antenna signals. 4 5 2.4 The System Clock T h e s y s t e m c l o c k i s u s e d t o g e n e r a t e t h e s a m p l e c l o c k a n d a c c u m u l a t o r c l o c k . T h e s a m p l e c l o c k d e t e r m i n e s t h e s a m p l e r a t e o f t h e s y s t e m , a n d i s u s e d b y t h e A / D c o n v e r t e r s a n d t h e s p e c t r o m e t e r -c o r r e l a t o r c a r d s . T h e r a t e i s a d j u s t a b l e i n p o w e r s o f t w o f r o m 1 6 M H z t o 2 5 6 k H z a n d i s c o n t r o l l e d by t h e m i c r o p r o c e s s o r c o n t r o l l e r . T h e a c c u m u l a t o r c l o c k o p e r a t e s a t a f i x e d r a t e o f 1 6 M H z a n d i s u s e d b y t h e a c c u m u l a t o r s i n t h e s p e c t r o m e t e r c o r r e l a t o r c a r d s . T h i s c l o c k i s a l s o u s e d b y t h e c o n t r o l i n t e r f a c e c a r d a n d t h e a u t o -c o r r e l a t o r c a r d s . 46 3. A Review of Correlation-Spectrometer IC Architectures I n c o m p u t i n g t h e d i s c r e t e c o r r e l a t i o n f u n c t i o n a r a d i o - a s t r o n o m y c o r r e l a t o r p e r f o r m s t h r e e o p e r a t i o n s : d e l a y , m u l t i p l i c a t i o n , a n d a c c u m u l a t i o n . A f o u r t h o p e r a t i o n i s r e a d o u t w h i c h i n v o l v e s t h e t r a n s f e r o f d a t a f r o m t h e a c c u m u l a t o r s t o a h o s t c o m p u t e r . T h e s e o p e r a t i o n s a r e v e r y e a s y t o i m p l e m e n t u s i n g d i g i t a l l o g i c , e s p e c i a l l y f o r r a d i o - a s t r o n o m y a p p l i c a t i o n s w h e r e t h e p r e c i s i o n o f t h e d a t a i s l i m i t e d t o a f e w b i t s . A l t h o u g h t h e s e o p e r a t i o n s a r e i n d i v i d u a l l y s i m p l e , t h e y m a y b e c o m b i n e d i n s e v e r a l w a y s t o i m p l e m e n t t h e a r c h i t e c t u r e o f a d i g i t a l c o r r e l a t o r . F r o m t h e n u m b e r o f d i f f e r e n t c o r r e l a t o r a r c h i t e c t u r e s t h a t h a v e b e e n i m p l e m e n t e d i t m i g h t a p p e a r t h a t t h e r e i s o n l y a p h i l o s o p h i c a l d i f f e r e n c e b e t w e e n t h e m . H o w e v e r , t h i s i s n o t t h e c a s e , a s t h e r e a r e s o m e i m p o r t a n t t e c h n i c a l a d v a n t a g e s o f s o m e a r c h i t e c t u r e s o v e r o t h e r s . I n f a c t t h e r e i s e v i d e n c e t h a t o n e V L S I d e s i g n m a y h a v e r u n i n t o t r o u b l e b e c a u s e t h e s e i s s u e s w e r e i g n o r e d . I n t h i s c h a p t e r t h e a r c h i t e c t u r e o f t h e c o r r e l a t o r I C d e v e l o p e d f o r t h i s p r o j e c t i s c o m p a r e d t o t w o o t h e r C M O S I C s , t h e B o s I C [ B o s 9 0 ] a n d t h e C a n a r i s I C [ S E R C 9 3 ] . T h e m e t r i c s o f i n t e r e s t a r e t h e g a t e c o u n t , t h e s p e e d , a n d t h e f l e x i b i l i t y o f t h e a r c h i t e c t u r e s u s e d i n t h e s e I C s . T h e c h a p t e r i s d i v i d e d i n t o t w o s e c t i o n s . I n t h e f i r s t s e c t i o n a n o v e r v i e w o f t h e s e I C s i s g i v e n , w h i l e i n t h e s e c o n d s e c t i o n t h e m e r i t s o f t h e i r a r c h i t e c t u r e s a r e d i s c u s s e d . T h e m a i n p r e m i s e i s t h a t t h e a r c h i t e c t u r e o f D R A O c o r r e l a t o r I C h a s s e v e r a l t e c h n i c a l a d v a n t a g e s . T h e s e a d v a n t a g e s a r e : • t h a t i t s d e l a y a r c h i t e c t u r e i s f l e x i b l e a n d f a s t . I t i s f l e x i b l e a s , l i k e t h e C a n a r i s I C a n d u n l i k e t h e B o s I C , i t c a n b e u s e d t o c o r r e l a t e N y q u i s t o r d o u b l e - N y q u i s t s a m p l e d d a t a w i t h o u t c o r r e l a t i n g r e d u n d a n t d a t a . I t i s f a s t , a s f o r a n N l a g c o r r e l a t o r t h e r e i s o n l y o n e m u l t i p l i e r l o a d p e r d e l a y t a p , w h e r e a s , t h e C a n a r i s I C h a s N l o a d s p e r d e l a y t a p a n d t h e B o s I C h a s t w o ( l o a d s p e r d e l a y t a p ) . 47 • t h a t i t s d a t a t r a n s f e r a r c h i t e c t u r e h a s h i g h e r t h r o u g h p u t w h e n i n t e g r a t i o n i s p e r f o r m e d " o f f - c h i p " . U n l i k e o t h e r c o r r e l a t o r s , t h e M S B s f r o m m a c c u m u l a t o r s c a n b e s a m p l e d i n p a r a l l e l t o d e t e c t a n d a c c u m u l a t e c a r r i e s o f f - c h i p . O n l y a t t h e e n d o f a n i n t e g r a t i o n a r e t h e L S B s r e a d a n d a d d e d t o t h e c a r r i e s . T h i s s c h e m e r e q u i r e s f e w e r t r a n s f e r s {21m) t h a n s i m p l y r e a d i n g a c c u m u l a t o r b i t s o f m i x e d s i g n i f i c a n c e , a n d s o c a n b e d o n e m/2 t i m e s f a s t e r . • t h a t i t m a k e s e f f i c i e n t u s e o f a v a i l a b l e I C g a t e s , a s i t s d e l a y a r c h i t e c t u r e r e q u i r e s f e w e r g a t e s p e r l a g , f o r t h e s a m e f u n c t i o n a l i t y , t h a n e i t h e r t h e B o s o r C a n a r i s I C . I t i s i m p o r t a n t t o n o t e t h a t t h e a r c h i t e c t u r a l c h a r a c t e r i s t i c s d i s c u s s e d a r e i n h e r e n t a n d m o s t l y t e c h n o l o g y i n d e p e n d e n t . H o w e v e r , t o m a k e q u a n t i t a t i v e c o m p a r i s o n s i t i s u s e f u l t o c o n s i d e r a v c o m m o n s e t o f c o r r e l a t o r s p e c i f i c a t i o n s i m p l e m e n t e d i n t h e s a m e I C p r o c e s s t e c h n o l o g y . C o n s e q u e n t l y , i n t h i s c h a p t e r c o m p a r i s o n s a r e m a d e w i t h r e s p e c t t o a t h r e e - l e v e l c o r r e l a t o r w i t h f o u r c h a n n e l s , o r l a g s , i m p l e m e n t e d u s i n g L S I L o g i c ' s 3 0 0 0 - s e r i e s C M O S g a t e - a r r a y t e c h n o l o g y [ L S I 9 3 ] . 3.1 An Overview of the DRAO, Canaris, and Bos ICs I n F i g . 3.1 a b l o c k d i a g r a m o f t h e D R A O - I C i s g i v e n . F o r c o m p a r i s o n p u r p o s e s F i g . 3 . 2 s h o w s a b l o c k d i a g r a m o f a s c a l e d v e r s i o n o f t h e C a n a r i s I C . T h e C a n a r i s I C w a s c h o s e n f o r c o m p a r i s o n a s i t i s f u n c t i o n a l l y s i m i l a r t o t h e D R A O - I C . A s w e l l , i t s a r c h i t e c t u r e i s c o m m o n p l a c e y e t d i f f e r e n t f r o m t h e D R A O - I C . A n o t h e r c o r r e l a t o r , w h o s e d e l a y a r c h i t e c t u r e i s s i m i l a r t o t h e D R A O - I C , i s t h e B o s I C ( s e e F i g . 3 . 3 ) . F o r c o m p a r i s o n p u r p o s e s t h e C a n a r i s a n d B o s I C s s h o w n i n t h e f i g u r e s h a v e b e e n s c a l e d t o t h e s a m e n u m b e r o f l a g s a n d a c c u m u l a t o r b i t s a s t h e D R A O - I C . T h e m a j o r d i f f e r e n c e b e t w e e n t h e s e I C a r c h i t e c t u r e s i s t h e w a y r e l a t i v e d e l a y s a r e f o r m e d a n d t h e w a y t h e a c c u m u l a t o r i s r e a d o u t . T h e s e d i f f e r e n c e s a r e e x a m i n e d f u r t h e r i n t h e f o l l o w i n g s e c t i o n s . 48 Xin > Sample Clk > Yout < Accumulator > Clock Reset >-Latch > Read > > Xout < Yin A Dual B 32:1 C Mux D So-4 En Data Out Figure 3.1: A block diagram of the DRAO-IC architecture. For simplicity, buffering and some control logic are not shown. 49 Sampling Mode Sample Clk Sample Clk ; D Q s Mux B D Q PC D Q D Q t>C s Mux D Q H>c. D Q \A+B_Cout/ Accumulator Clock Reset Latch >_ Read > Q D I—1> C R s Mux A C B D Q D Q H>c \A-tS Cout/ Q D ^ C R 23 Bit Ripple Counter Ch Q 2 3 s Mux A C r B D Q ri>c D Q H>c Xout A v B A+B Cout Q D R 23 Bit Ripple Counter Ch Q 2 3 7 YCXJI A v B \A+B Cout Q D r - t > C "L9_ 16 Bit Latch D1.16 Q l - 1 6 23 Bit Ripple Counter Q 7 (>C R Q 2 3 7 4 Bit Counter R 16 Bit Latch I 1 D..I6 Q M >c 23 Bit Ripple Counter l>C R 16 Bit Latch t>C 16 Bit Latch A Hex B 4:1' C Mux D Q T , S0-3 En Data Out Figure 3.2: A block diagram of the (scaled) Canaris IC architecture. The IC shown has been scaled to the same specifications as the DRAO-IC. The actual Canaris IC has 32 bit accumulators, a 32 bit data bus, and 1024 lags. For simplicity, buffering and some control logic are not shown. 5 0 Xin Sample Clk ' . >~ D Q f—f> C Yout <- Q D D Q r — > C A B \A+S Coat/ Accumulator > Clock Reset Latch >-Read Q D R Q D A B \A+B Coul/ I Q D > c R 23 Bit Ripple Counter Or -> Xout r A B vA+B Coul/ Q D H R 23 Bit Ripple Counter ^ C Or Q23 Yin A B \A+B Coul/ Q D M R 16 Bit Latch 23 Bit Ripple Counter > C Q 7 » ... R Cte 4 Bit Counter R Qo Q 3 16 Bit Latch DMB Q v i a t>C 23 Bit Ripple Counter R Q « 16 Bit Latch D1-16 Q i - i 16 Bit Latch Q M >0 A Hex B 4:1 C Mux D Q T StK3 En Data Out Figure 3.3: A block diagram of the (scaled) Bos IC architecture. The IC shown has been scaled to the same specifications as the DRAO-IC. The actual Bos IC has 32 bit accumulators, a 16 bit data bus, and 16 lags. For simplicity, buffering and some control logic are not shown. 51 3.1.1 Delay Architecture The Canaris IC uses an asymmetric delay pipeline where only one of the two signals is delayed. Correlator lags are formed by multiplying the undelayed signal with delayed versions in the other signal path. This architecture inherently produces positive lags, so it may be directly used to implement an auto-correlator. Another correlator IC that uses this scheme is described by Von Herzen (1991). A cross-correlator can easily can be formed by using two asymmetric pipelines: one for positive lags and one for negative lags [Padin93]. By contrast the DRAO-IC, like the Bos IC, uses a symmetrical delay pipeline that is implemented with two opposing shift-registers. This delay architecture inherently produces a cross-correlator, as both positive and negative lags are formed. Another correlator IC, which uses a symmetric pipeline is given by Timoc (1993). With this pipeline an auto-correlator is formed by feeding back a (suitably) delayed version of the x stream output into the y stream, as this produces a correlator with all negative lags. Unlike many correlators, both the Canaris- and DRAO-ICs can use all their lags to produce a Nyquist sampled correlation function from Nyquist or double-Nyquist sampled data. Correlating double-Nyquist sampled data improves the SNR (see section 4.3), while correlating Nyquist sampled data provides maximum bandwidth. Other ICs can in principle correlate double-Nyquist sampled data; however, only half of the lags are useful as the other half contains redundant information. In the Canaris IC over-sampling is accomplished by switching in additional delays, between the correlator lags, using multiplexers. In this mode, when data are sampled at twice the Nyquist rate, the time between lags is the Nyquist period (i.e., i = 1 / 2v). This produces a Nyquist sampled correlation function from double-Nyquist sampled data. Maximum bandwidth is achieved by 5 2 r e m o v i n g t h e a d d i t i o n a l d e l a y s u s i n g t h e m u l t i p l e x e r s . I n t h i s m o d e b o t h t h e t i m e b e t w e e n l a g s a n d t h e t i m e b e t w e e n s a m p l e s i s t h e N y q u i s t p e r i o d . O n e m a y n o t e f r o m F i g . 3 . 1 t h a t t h e D R A O - I C i n h e r e n t l y c o r r e l a t e s d o u b l e - N y q u i s t s a m p l e d d a t a . E s s e n t i a l l y t h e D R A O - I C d e l a y a r c h i t e c t u r e i s l i k e t h e B o s I C a n d T i m o c ' s ( 1 9 9 3 ) , e x c e p t t h a t a n a d d i t i o n a l d e l a y s t a g e i s p e r m a n e n t l y i n s e r t e d b e t w e e n e a c h ( B o s I C ) o d d - l a g m u l t i p l i e r . F o r N y q u i s t s a m p l e d d a t a t h e D R A O - I C c o m p u t e s t h e o d d l a g s a t : { - 3 x , - x , x , 3 x } ; a n d f o r d o u b l e -N y q u i s t s a m p l e d d a t a i t c o m p u t e s t h e l a g s a t : { - 1 . 5 x , - . 5 x , . 5 x , 1 . 5 x ) . U n l i k e t h e C a n a r i s I C , t h e s e d e l a y s c a n n o t b e b y p a s s e d , a s o n - c h i p m u l t i p l e x e r s a r e n o t u s e d t o s w i t c h t h e s e d e l a y s i n a n d Out. I n s t e a d a m o r e e f f i c i e n t s c h e m e u s i n g a s i n g l e e x t e r n a l d e l a y a n d m u l t i p l e x e r i s u s e d . T o f l e x i b l y c o r r e l a t e e i t h e r N y q u i s t o r d o u b l e - N y q u i s t s a m p l e d d a t a a t l e a s t t w o D R A O - I C a r e u s e d w i t h a n e x t e r n a l d e l a y a n d m u l t i p l e x e r ( s e e F i g . 3 . 4 ) . F o r N y q u i s t s a m p l e d d a t a i n s e r t i n g a n e x t r a d e l a y i n t h e y d a t a s t r e a m c a u s e s a D R A O - I C t o c o m p u t e t h e e v e n l a g s {-4T, -2T, OT, 2X } . B y u s i n g t w o I C s , o n e w i t h a n e x t r a d e l a y i n t h e y d a t a s t r e a m a n d o n e w i t h o u t , a n d f e e d i n g t h e p a i r t h e s a m e d a t a , a N y q u i s t s a m p l e d c o r r e l a t i o n f u n c t i o n i s p r o d u c e d , a s t o g e t h e r t h e t w o I C s c o m p u t e t h e l a g s {-3T, - x , x , 3 x } u { - 4 x , - 2 x , O x , 2 x } . F o r d o u b l e - N y q u i s t s a m p l i n g t w o I C s a r e c o n n e c t e d i n s e r i e s s o t h a t t h e o u t p u t s o f o n e I C a r e f e d t o t h e o t h e r . T h e m u l t i p l e x e r s h o w n i n F i g . 3 . 4 m a k e s s w i t c h i n g b e t w e e n c o n f i g u r a t i o n s i s e a s y . I n t h i s s a m p l i n g m o d e t h e e x t r a d e l a y c a u s e s t h e d a t a t o b e s h i f t e d b y . 5 x . T h i s c a u s e s o n e I C t o c o m p u t e t h e l a g s { - 4 x , - 3 x , - 2 x , - x } , w h i l e t h e o t h e r c o m p u t e s t h e l a g s { O x , l x , 2 x , 3 x } . T o g e t h e r , t h e I C s c o m p u t e a N y q u i s t s a m p l e c o r r e l a t i o n f u n c t i o n , w i t h l a g s { - 4 x , -3 x , - 2 x , - x } u { O x , l x , 2 x , 3 x } , f r o m d o u b l e - N y q u i s t s a m p l e d d a t a . 53 Double Nyquist Symmetric Delay Pipeline Sampling Mode X in> -Y out Sample Clk -Double Nyquist X l Symmetric X o ^ Y i Delay Yo Pipeline \> Sample Clk Mul out S Quad Mux At Ci C2 Double Nyquist X 1 Symmetric X o Yi Delay Yo Pipeline b> Sample Clk Mul out [—, Q D C < -> Xout -< Yin-Sample Clk Double Nyquist Mode : i = {-4, -3, -2, -1) Nyquist Mode : r= [ -3, -1, 1, 3) x={0, 1,2,3} x={-4, -2, 0,2) Figure 3.4: A block diagram showing how two DRAO-IC can correlate Nyquist or double-Nyquist sampled data. 54 3.1.2 Multiplier Architecture The multiplier and accumulator architectures of the DRAO, BOS, and Canaris ICs are all similar. All use hard-wired logic to implement the multiplier. The multiplier output is also biased to produce non-negative values. This simplifies the accumulator circuitry as only addition, and not subtraction, is required. A look-up table, using a programmable device such as a RAM, can also be used to implement a multiplier. While this approach requires more circuitry, and is typically slower, it is more flexible, as the multiplier can be programmed to handle data with different quantization schemes. 3.1.3 Accumulator Architecture The accumulator architectures in the DRAO, BOS, and Canaris ICs are also similar. The accumulator is implemented in two stages. In the first stage the multiplier output is synchronously added, in parallel, to the LSBs of the accumulator that are stored in a register. In the second stage the carry output of the adder asynchronously toggles a long ripple counter. An alternative to using a parallel first stage is to use a serial one. This scheme was used by the previous correlation-spectrometer at DRAO [Dewdney87]. In this scheme the multiplier output is serialized and used to clock a ripple counter. While this architecture eliminates the need for a parallel adder, the ripple counter must operate a factor of N higher than the sample rate, where N is the maximum (biased) multiplier weight. Thus this architecture can limit the maximum sample rate of the system, as for a three-level correlator the ripple counter must operate a factor of two higher than any other circuitry in the correlator. 3.1.4 Accumulator Readout Architecture With all of these ICs the accumulator contents can be latched and then transferred sequentially to a host computer. These latches provide buffering so that accumulator data can be transferred while . 5 5 t h e c o r r e l a t o r i s c o m p u t i n g t h e n e x t i n t e g r a t i o n . A h o s t c o m p u t e r i s t y p i c a l l y u s e d t o p e r f o r m l o n g e r t e r m i n t e g r a t i o n , F o u r i e r t r a n s f o r m a t i o n , a n d d a t a s t o r a g e . A d a t a t r a n s f e r , o r r e a d o u t , s e q u e n c e c o n s i s t s o f t o g g l i n g a l a t c h s t r o b e a n d t h e n a s s e r t i n g NxF r e a d s t r o b e s t o s e q u e n t i a l l y r e a d N l a g s , w h e r e F e q u a l s t h e t o t a l n u m b e r o f a c c u m u l a t o r b i t s , n, d i v i d e d b y t h e n u m b e r o f b i t s t r a n s f e r r e d i n p a r a l l e l , m , d u r i n g a r e a d , ( i . e . , F - nlm). A m a j o r d i f f e r e n c e b e t w e e n t h e D R A O - I C a n d o t h e r c o r r e l a t o r s , i s t h e w a y t h e a c c u m u l a t o r d a t a a r e t r a n s f e r r e d . B o t h t h e C a n a r i s a n d B o s I C s t r a n s f e r t h e b i t s f r o m a s i n g l e a c c u m u l a t o r - l a t c h d u r i n g a r e a d . I n t h e s c a l e d v e r s i o n s o f t h e s e I C s , s h o w n i n f i g u r e s F i g s . 3 . 2 a n d 3 . 3 , t h e s e a r c h i t e c t u r e s t r a n s f e r a l l 1 6 a c c u m u l a t o r b i t s i n o n e r e a d ( i . e . , F = 1 ) . S o a n N l a g c o r r e l a t o r r e q u i r e s N r e a d s t r o b e s . -T h e k e y p o i n t i s t h a t w i t h t h i s r e a d o u t a r c h i t e c t u r e b i t s o f d i f f e r i n g s i g n i f i c a n c e , f r o m a c o m m o n a c c u m u l a t o r , a r e t r a n s f e r r e d d u r i n g e a c h r e a d . , • C o n v e r s e l y , t h e D R A O - I C t r a n s f e r s b i t s o f e q u a l s i g n i f i c a n c e f r o m t w o a c c u m u l a t o r s d u r i n g a r e a d . I n t h i s s c h e m e a c c u m u l a t o r d a t a a r e t r a n s f e r r e d , s e r i a l l y , m o s t s i g n i f i c a n t b i t f i r s t a n d l e a s t s i g n i f i c a n t b i t l a s t . O n e m i g h t t h i n k t h a t t h i s s e r i a l r e a d s c h e m e w o u l d h a v e a l o s s o f t h r o u g h p u t c o m p a r e d t o a p a r a l l e l o n e ; h o w e v e r , t h e r e i s n o l o s s o f t h r o u g h p u t w i t h t h i s s c h e m e ( s e e s e c t i o n 3 . 2 a n d T a b l e 3 . 2 ) , a s s e t s o f I C s c a n b e g r o u p e d t o g e t h e r s o t h a t e i g h t , s i x t e e n , o r t h i r t y - t w o a c c u m u l a t o r - b i t s c a n b e t r a n s f e r r e d , i n p a r a l l e l , d u r i n g a r e a d . W i t h t h i s s c h e m e b i t s o f equal significance f r o m m u l t i p l e a c c u m u l a t o r s a r e t r a n s f e r r e d i n p a r a l l e l . O n t h e D R A O s p e c t r o m e t e r c i r c u i t c a r d ( s e e s e c t i o n 2 . 2 . 3 ) f o u r I C o u t p u t s a r e r e a d i n p a r a l l e l s o t h a t e i g h t a c c u m u l a t o r b i t s a r e t r a n s f e r r e d d u r i n g e a c h r e a d s t r o b e ( i . e . , F = 2 ) . 5 6 3.2 The DRAO-IC Architecture versus the Canaris and Bos ICs I n t h i s s e c t i o n t h e a r c h i t e c t u r a l a d v a n t a g e s o f t h e D R A O - I C a r e d i s c u s s e d . A s w e l l , i t i s s h o w n t h a t t h e a s y m m e t r i c p i p e l i n e , u s e d i n t h e C a n a r i s I C , c a n b e i m p l e m e n t e d w i t h f e w e r g a t e s . O n e a d v a n t a g e o f t h e D R A O - I C i s t h a t i t s d e l a y p i p e l i n e a r c h i t e c t u r e i s f a s t e r t h a n t h e o t h e r I C s , y e t i t o f f e r s m o r e f l e x i b i l i t y a n d i s i m p l e m e n t e d w i t h f e w e r g a t e s . A n o t h e r a d v a n t a g e i s t h a t i t s r e a d o u t a r c h i t e c t u r e u s e s t h e d a t a - t r a n s f e r b a n d w i d t h m o r e e f f i c i e n t l y w h e n l o n g - t e r m i n t e g r a t i o n i s p e r f o r m e d " o f f - c h i p " . L i k e t h e B o s I C , t h e D R A O - I C u s e s a s y m m e t r i c d e l a y p i p e l i n e w h i c h h a s s o m e a d v a n t a g e s o v e r t h e a s y m m e t r i c o n e . A s s h o w n i n T a b l e 3 . 1 , t h e D R A O - I C d e l a y a r c h i t e c t u r e h a s t h e f a s t e s t c l o c k t o o u t p u t d e l a y a n d u s e s f e w e s t g a t e s i n t h e d o u b l e - N y q u i s t m o d e . T h e B o s I C i s a b o u t 6 % s l o w e r b e c a u s e i t h a s t w o m u l t i p l i e r l o a d s p e r d e l a y t a p . A l t h o u g h t h i s I C u s e s t h e f e w e s t g a t e s w h e n c o r r e l a t i n g N y q u i s t s a m p l e d d a t a , i t r e q u i r e s t w i c e a s m a n y g a t e s i n t h e d o u b l e - N y q u i s t s a m p l e d m o d e , a s h a l f t h e l a g s c o r r e l a t e r e d u n d a n t i n f o r m a t i o n . Table 3.1: A comparison of delay architecture specifications for an N lag correlator. The values shown are based on data-book values [LSI93]. For comparison purposes the Bos and Canaris ICs have been scaled to the same number of lags, N. Parameter DRAO Bos (scaled) Canaris (scaled) Number of equivalent loads per delay pipeline tap (including wire lengths). 1.78 3.55 . 1.78N Worst case (nominal) clock to pipeline output delay (in ns). 1.13 1.20 .07N+T.06 Number of pipeline and multiplier gates for: Nyquist sampled data Double-Nyquist sampled data 34N+22 34N+22 20N+22 40N+22 38N+14 38N+14 57 The Canaris IC has the slowest performance and requires more gates to implement than the DRAO-IC. The reason is that the y delay, of an AMag asymmetric delay pipeline, sees N multiplier and wire length loads which increase its clock to output propagation-delay. Although the propagation delay can be reduced by using a high current buffer, it cannot eliminate this inherent problem. A secondary problem is that for large N the integrity of the y delay output-signal can be degraded. Since these problems are more sensitive to layout and process variation than a simple point load, they make it more difficult to estimate the on-chip performance of an IC during design. There is some evidence that initial versions of the Canaris IC did not perform to specification because of these effects [Hayward94]. The DRAO-IC has the highest delay pipeline performance because each delay output sees only one multiplier and wire length load (i.e., a point load). Since the loading is less, and the wire length to each multiplier can be kept relatively short, signal integrity is preserved, performance is improved, and estimation of on-chip performance is easier. For a four-lag IC this architecture is 16% faster than the Canaris architecture. The DRAO-IC delay pipeline also uses fewer gates than the Canaris IC. For an AMag correlator the Canaris architecture requires N multiplexers, one for each multiplier, while the DRAO architecture requires only two components: a multiplexer and an additional delay. The Canaris IC asymmetric pipeline could have been designed with fewer gates using an approach used in the DRAO-IC. This can be done by eliminating the multiplexers between each delay stage to create the double-Nyquist asymmetric pipeline shown in Fig. 3.5. This pipeline is similar to the DRAO-IC architecture as multiplexers are not used to by-pass the delays. As shown in the lower half of Fig. 3.5, one can use two of these pipelines to create two asymmetric pipelines: one with an extra delay and one without. Furthermore, by adding two multiplexers one can switch between Nyquist or double-Nyquist sampling modes, as one can: join the two pipelines and bypass an extra delay (for the double-Nyquist mode), or separate the pipelines and include an extra delay (for the 58 Double Nyquist Asymmetric Delay Pipeline i Sample ! Clk >-D Q Sample Clk ; D Q D Q D Q r > C T = 0 D Q > C D Q D Q D Q H>c T = 1 T = 2 D Q H> Xc-1 Xo r > C x = 3 Yo Sampling Mode > " Sample Clk >" X in ; Yin>-Sample Clk >~ r — D Q > C s Mux Double Nyquist X 0-1 Asymmetric Xo Delay Yo Pipeline t> Sample Clk Mul out — i LJ Xi Y i Double Nyquist x 0-1 Asymmetric Delay * ° Pipeline Yo f> Sample Clk Mul out S Mux -> X out -> Yout Double Nyquist Mode x= {0,1,2,3} Nyquist Mode x={0, 2,4, 6} T = { 4, 5, 6, 7) x= { 1,3,5,7} Figure 3.5: A block diagram of the double-Nyquist asymmetric pipeline. The basic architecture is shown in the upper part of the figure, while the lower half shows how the architecture can be used to correlate Nyquist or double-Nyquist sampled data. 59 N y q u i s t m o d e ) . L i k e t h e D R A O - I C i n t h e N y q u i s t s a m p l e d m o d e o n e d e l a y - l i n e c o m p u t e s t h e o d d l a g s { IT, 3T, 5T, 7T}, w h i l e t h e o t h e r c o m p u t e s t h e e v e n l a g s { O x , 2T, 4T, 6T}. I n t h e d o u b l e - N y q u i s t s a m p l e d m o d e o n e d e l a y l i n e c o m p u t e s t h e l a g s {-4T, -3T, -2T, -T}, w h i l e t h e o t h e r c o m p u t e s t h e l a g s {OT, IT, 2T, 3T}. F o r a n N l a g a s y m m e t r i c p i p e l i n e t h i s s c h e m e u s e s N - 1 f e w e r m u l t i p l e x e r s t h a n t h e C a n a r i s I C , a s s w i t c h i n g m o d e s i s a c c o m p l i s h e d u s i n g t w o m u l t i p l e x e r i n s t e a d o f N m u l t i p l e x e r s . T h u s , f u n d a m e n t a l l y , b o t h t h e a s y m m e t r i c a n d t h e s y m m e t r i c d e l a y p i p e l i n e s c a n b e i m p l e m e n t e d w i t h r o u g h l y t h e s a m e n u m b e r o f g a t e s . A n o t h e r a d v a n t a g e o f t h e D R A O - I C i s i t s r e a d o u t a r c h i t e c t u r e c a n u s e t h e a v a i l a b l e d a t a - t r a n s f e r b a n d w i d t h m o r e e f f i c i e n t l y . I n s i t u a t i o n s w h e r e t h e i n t e g r a t i o n p e r i o d i s s h o r t , f e w e r b i t s n e e d t o b e t r a n s f e r r e d ( s i n c e t h e h i g h - o r d e r a c c u m u l a t o r b i t s a r e z e r o ) . I n s u c h c a s e s t h e C a n a r i s a n d B o s I C a r c h i t e c t u r e s w a s t e t r a n s f e r b a n d w i d t h , a s t h e y c a n n o t a v o i d r e a d i n g t h e h i g h - o r d e r b i t s . T h e r e a s o n f o r t h i s i s t h e y i n h e r e n t l y t r a n s f e r a l l t h e b i t s ( o f m i x e d s i g n i f i c a n c e ) f r o m a n a c c u m u l a t o r d u r i n g a r e a d . T h e D R A O - I C , h o w e v e r , t r a n s f e r s b i t s o f e q u a l s i g n i f i c a n c e f r o m m u l t i p l e a c c u m u l a t o r s , s o t h e h i g h - o r d e r b i t s c a n e a s i l y b e s k i p p e d o v e r ( u s i n g a p p r o p r i a t e e x t e r n a l l o g i c ) a n d o n l y t h e b i t s w i t h d a t a c a n b e t r a n s f e r r e d . B e c a u s e f e w e r b i t s a r e t r a n s f e r r e d , t h e D R A O - I C u s e s t h e d a t a - t r a n s f e r b a n d w i d t h m o r e e f f i c i e n t l y . A s w e l l , t h e D R A O - I C c a n e f f i c i e n t l y i n t e g r a t e c o r r e l a t o r d a t a " o f f - c h i p " 4 . A c o n v e n t i o n a l w a y o f d o i n g t h i s i s t o u s e a h o s t c o m p u t e r t o r e a d e a c h c o r r e l a t o r a c c u m u l a t o r a n d s u m t h e o u t p u t i n a R A M . H o w e v e r , a n o t h e r w a y i s t o e m u l a t e a r i p p l e c o u n t e r , u s i n g s o f t w a r e o r s p e c i a l p u r p o s e h a r d w a r e . D o i n g t h i s r e q u i r e s s a m p l i n g t h e M S B o f e a c h a c c u m u l a t o r a n d r e c o r d i n g h i g h t o l o w 4 Integrating off-chip can be more cost effective. Since a host computer is required for other-purpose (e.g., storage and control) the incremental cost in compute power and RAM, for accumulating the slow changing higher accumulator bits in software , is lower than using relatively expensive high-performance IC gates. i 6 0 . t r a n s i t i o n s , o r o v e r f l o w s . A t t h e e n d o f a n i n t e g r a t i o n t h e b i t s f r o m t h e a c c u m u l a t o r s a r e r e a d a n d a d d e d t o t h e o v e r f l o w s . A n a d v a n t a g e o f t h i s s c h e m e i s d a t a t r a n s f e r b a n d w i d t h i s u s e d m o r e e f f i c i e n t l y , a s o n l y t h e m o s t - s i g n i f i c a n t b i t n e e d s t o b e t r a n s f e r r e d i n s t e a d o f a l l m a c c u m u l a t o r b i t s . A l t h o u g h t h e M S B m u s t b e t r a n s f e r r e d t w i c e a s o f t e n ( f o r s a m p l i n g r e a s o n s ) , t h i s s c h e m e r e q u i r e s 21m f e w e r t r a n s f e r s . T o i l l u s t r a t e t h e a d v a n t a g e s o f t h e D R A O - I C r e a d o u t a r c h i t e c t u r e c o n s i d e r t h e f o l l o w i n g e x a m p l e s . A s i n d i c a t e d i n t a b l e 3 . 2 , t h e D R A O - I C c a n t r a n s f e r t h e M S B s f r o m a f o u r - l a g c o r r e l a t o r i n t w o r e a d s , w h i l e t h e o t h e r t w o I C s b o t h r e q u i r e f o u r . H o w e v e r , c o n s i d e r a 3 2 - l a g c o r r e l a t o r c o n s i s t i n g o f e i g h t I C s . A s m e n t i o n e d i n t h e p r e v i o u s s e c t i o n , g r o u p s o f D R A O - I C s c a n b e r e a d i n p a r a l l e l t o i n c r e a s e t h r o u g h p u t . S i n c e e a c h D R A O - I C h a s t w o o u t p u t s , a l l t h e M S B s c a n b e t r a n s f e r r e d i n j u s t t w o r e a d s . T h e o t h e r I C s h a v e 1 6 o u t p u t s b u t t r a n s f e r o n l y o n e M S B p e r r e a d . S o t h e y r e q u i r e 3 2 r e a d s t o t r a n s f e r t h e M S B s . I n t h i s c a s e t h e D R A O - I C a r c h i t e c t u r e i s 1 6 t i m e s f a s t e r . E x t e n d i n g t h i s e x a m p l e , c o n s i d e r t h e n u m b e r o f t r a n s f e r s r e q u i r e d t o i n t e g r a t e o f f - c h i p . F o r t h e c o n v e n t i o n a l a p p r o a c h ( d i s c u s s e d i n t h e p r e v i o u s p a r a g r a p h ) a l l t h e I C s r e q u i r e 3 2 r e a d s ( s e e T a b l e 3 . 2 ) . H o w e v e r , o n l y f o u r r e a d s a r e r e q u i r e d i f t h e M S B s a r e s a m p l e d u s i n g t h e D R A O - I C . I n t h i s c a s e t h e D R A O - I C a r c h i t e c t u r e i s e i g h t t i m e s m o r e e f f i c i e n t . T h e b e n e f i t s o f t h i s i n c r e a s e d t h r o u g h p u t a r e t h a t o n e c a n r e d u c e t h e s i z e o f t h e a c c u m u l a t o r s b y i n t e g r a t i n g " o f f - c h i p " . S i n c e t h e M S B s a m p l i n g s c h e m e i n t h e a b o v e e x a m p l e m a k e s i n t e g r a t i n g o f f - c h i p e i g h t t i m e s f a s t e r , o n e c a n ( p o t e n t i a l l y ) r e d u c e t h e a c c u m u l a t o r l e n g t h b y t h r e e b i t s a n d s a v e a b o u t 2 1 g a t e s p e r l a g . A n o t h e r b e n e f i t i s t h a t f o r a n I C w i t h o n l y f o u r l a g s t h e D R A O - I C m a k e s m o r e e f f i c i e n t u s e o f o u t p u t p i n s a n d d r i v e r s , a s 3 2 b i t s a r e t r a n s f e r r e d p e r o u t p u t p i n c o m p a r e d t o f o u r b i t s p e r p i n f o r t h e o t h e r I C s . T h i s e f f i c i e n t u s e o f o u t p u t p i n s l o w e r s p a c k a g e c o s t , g r o u n d b o u n c e a n d p a c k a g e s i z e . 61 Table 3.2: A comparison of readout architectures. Parameter DRAO Bos Canaris (scaled) (scaled) O u t p u t p i n u t i l i z a t i o n ( i n b i t s / p i n ) F o r a 4 l a g I C 3 2 4 4 F o r a 3 2 l a g c o r r e l a t o r 3 2 3 2 3 2 N u m b e r o f r e a d s t r o b e s r e q u i r e d t o t r a n s f e r a l l a c c u m u l a t o r s a n d a l l b i t s . F o r a 4 l a g I C 3 2 4 4 F o r a 3 2 l a g c o r r e l a t o r 3 2 3 2 3 2 N u m b e r o f r e a d s s t r o b e s r e q u i r e d t o t r a n s f e r t h e m o s t s i g n i f i c a n t a c c u m u l a t o r b i t s f r o m a l l a c c u m u l a t o r s . • F o r a 4 l a g I C 2 4 4 F o r a 3 2 l a g c o r r e l a t o r 2 3 2 • 3 2 6 2 4. The Theoretical Performance of The Hybrid-Level Correlator T h e h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r h a s 1 4 - l e v e l a u t o - c o r r e l a t o r s a n d t h r e e - l e v e l 5 1 2 - l a g c o r r e l a t o r s ; T h e c o r r e l a t o r s a r e v e r y r e l i a b l e a s t h e y a r e i m p l e m e n t e d u s i n g d i g i t a l c i r c u i t r y . F o r i n s t a n c e , g i v e n a s e t o f d i g i t a l i n p u t s t h e n e w c o r r e l a t o r c a n c o m p u t e > 1 0 1 7 c o r r e l a t i o n s w i t h z e r o e r r o r ( s e e s e c t i o n 5 . 2 ) . D i g i t a l c o r r e l a t o r s , h o w e v e r , d o n o t b e h a v e a s i d e a l c o r r e l a t o r s — a s t h e i r o u t p u t r e s p o n s e i s n o n -l i n e a r a n d t h e y d e g r a d e t h e s i g n a l t o n o i s e r a t i o o f t h e i n p u t . T h e s e n o n - i d e a l b e h a v i o r s r e s u l t f r o m t h e a n a l o g t o d i g i t a l c o n v e r s i o n ( A D C ) p r o c e s s , a n d t h e y o c c u r e v e n i f t h e A D C p r o c e s s i s p e r f e c t . M a n y a u t h o r s h a v e a n a l y z e d t h e b e h a v i o r o f r a d i o - a s t r o n o m y d i g i t a l - c o r r e l a t o r s w i t h l e s s t h a n s i x q u a n t i z a t i o n l e v e l s ; h o w e v e r , l i t t l e h a s b e e n r e p o r t e d c o n c e r n i n g t h e b e h a v i o r o f t h e 1 4 - l e v e l c o r r e l a t o r . A s w e l l , m o s t a u t h o r s h a v e i n v e s t i g a t e d t h e o u t p u t r e s p o n s e o f a c r o s s - c o r r e l a t o r u n d e r t h e c o n d i t i o n t h a t t h e q u a n t i z e r v a r i a n c e i s n e a r l y c o n s t a n t , a n d t h i s c o n d i t i o n i s n o t t r u e f o r t h e h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r a t D R A O . I n t h i s , c h a p t e r t h e o u t p u t r e s p o n s e a n d t h e s i g n a l t o n o i s e r a t i o ( S N R ) a r e i n v e s t i g a t e d f o r t h e t h r e e -a n d 1 4 - l e v e l d i g i t a l c o r r e l a t o r . T h e f i r s t t w o s e c t i o n s a r e i n t r o d u c t o r y . E q u a t i o n s d e s c r i b i n g t h e i d e a l c o r r e l a t o r a r e g i v e n i n s e c t i o n 4 . 1 , w h i l e t h e e q u i v a l e n t d i g i t a l o n e s a r e g i v e n i n s e c t i o n 4 . 2 . T h e s e e q u a t i o n s a r e t h e n u s e d i n s e c t i o n s 4 . 3 a n d 4 . 4 w h e r e t h e d i g i t a l c o r r e l a t o r S N R p e r f o r m a n c e a n d o u t p u t r e s p o n s e , r e s p e c t i v e l y , a r e d i s c u s s e d . I n t h e f i n a l s e c t i o n ( 4 . 5 ) m e t h o d s a r e d e v e l o p e d f o r l i n e a r i z i n g t h e d i g i t a l c o r r e l a t o r o u t p u t r e s p o n s e , f o r t h e g e n e r a l c a s e w h e r e t h e q u a n t i z e r v a r i a n c e i s n o t c o n s t a n t . W h i l e m o s t o f t h e c o n t e n t o f t h i s c h a p t e r i s a s u m m a r y o f t h e e x i s t i n g l i t e r a t u r e , t h a t c o n c e r n i n g t h e 1 4 - l e v e l c o r r e l a t o r i s m o s t l y n e w . F o r c o m p l e t e n e s s t h e 1 4 - l e v e l c r o s s - c o r r e l a t o r i s a l s o a n a l y z e d e v e n t h o u g h i t w a s n o t u s e d i n t h i s p r o j e c t . A l s o a n i m p r o v e d m e t h o d f o r d e t e r m i n i n g t h e v a r i a n c e • 63 of an over-sampled correlator is given in section 4.2. It is also shown that the output response of the hybrid auto-correlation spectrometer, discussed in section 4.4, is nearly linear. A common and reasonable assumption, used throughout, is that the correlator inputs are zero-mean Gaussian signals that are band-limited by a rectangular filter. Under this assumption the major differences between a digital correlator and an ideal one are: • the sensitivity of the Nyquist sampled three- and 14-level cross-correlator are about 19% and 1.5% less, respectively, than an ideal correlator. When the sample rate is twice the Nyquist rate the sensitivity is 11.8% and .8 % less, respectively. • the digital auto-correlation output response is non-linear. For a 30% increase in the optimum input variance the three-level correlator output increases by only 9%, while the 14-level auto-correlator output increases by about 26%. However, when a 14-level ADC is set to the three-level optimum (as in the hybrid-level correlator), the output of the 14-level auto-correlator increases by 29% and is nearly linear. • the digital cross-correlation output response is nearly linear for low levels of correlation when the quantizer variance is held constant at the optimum value. For example, if the correlated input variance is 30% of the uncorrelated input variance, then the normalized three-level and 14-level cross-correlator Outputs are 30.18% and 30.005%, respectively, of the uncorrelated input variance. • the digital cross-correlation output response is non-linear when the quantizer variance is not constant. For example, if the correlated input variance is 30% of the uncorrelated input variance, such that the total (quantizer) variance is 30% above the optimum, then the normalized three-level and 14-level cross-correlator outputs are 25.3% and 28.9%, respectively, of the uncorrelated input variance. 6 4 • t h e d i g i t a l c r o s s - c o r r e l a t i o n o u t p u t r e s p o n s e i s s e n s i t i v e t o c h a n g e s i n t h e u n c o r r e l a t e d i n p u t l e v e l . U s i n g t h e p r e v i o u s e x a m p l e , w h e n t h e u n c o r r e l a t e d i n p u t v a r i a n c e i s i n c r e a s e d b y 3 0 % f r o m i t s i n i t i a l v a l u e , t h e n t h e n o r m a l i z e d t h r e e - l e v e l a n d 1 4 - l e v e l c r o s s - c o r r e l a t o r o u t p u t s a r e 2 1 . 7 % a n d 2 7 . 7 % , r e s p e c t i v e l y , o f t h e i n i t i a l u n c o r r e l a t e d i n p u t v a r i a n c e . 4.1 The Ideal Correlator I n t h i s s e c t i o n e q u a t i o n s d e s c r i b i n g t h e i d e a l c o r r e l a t o r a r e d e v e l o p e d . T h e i d e a l c o r r e l a t o r i s i m p o r t a n t , a s i t i s t h e s t a n d a r d t h a t a d i g i t a l c o r r e l a t o r i s c o m p a r e d t o . T h e c h a r a c t e r i s t i c s o f i n t e r e s t a r e t h e s i g n a l t o n o i s e r a t i o ( S N R ) a n d t h e o u t p u t r e s p o n s e . T h e i d e a l c r o s s - c o r r e l a t o r i n t e g r a t e s t h e N s a m p l e p r o d u c t o f t h e i n p u t s i g n a l s , x a n d y , a t u n i f o r m i n t e r v a l s , At. T h e e x p e c t e d v a l u e , d e n o t e d b y E, o f t h e d i s c r e t e c o r r e l a t i o n f u n c t i o n , rxy, i s Rxy(r) = E[rxy(r)] = E 1 " " — ^x(iAt)y(iAt + r) ( 4 . 1 ) w h e r e Rxy(i) i s t h e a v e r a g e c r o s s - c o r r e l a t i o n f u n c t i o n f o r a r e l a t i v e d e l a y , x, a n d T i s t h e i n t e g r a t i o n p e r i o d ( T = N At). S i n c e r a d i o - a s t r o n o m y s i g n a l s a r e G a u s s i a n , t h e y c a n b e d e s c r i b e d b y t h e r a n d o m v a r i a b l e s X a n d Y w i t h j o i n t G a u s s i a n B i - v a r i a t e p r o b a b i l i t y d e n s i t y [ P e e b l e s 8 0 ] pXY(x,y,p) = 7 = T e 2 0 V ) 2KO~ xoY-^\- p w h e r e t h e c r o s s - c o r r e l a t i o n c o e f f i c i e n t , p , i s d e f i n e d a s ("-fx)2 , (y-Mr)1 2p(x-MxXy~Mr> (4.2) In the above equation, Tsrc, is the temperature of the signal received from the source, and Tsys is the independent system noise added to Tsrc. The source temperature is, of course, common to both inputs, while Tsys is independent in each input. If we assume x and y are ergodic, then time correlation is equivalent to ensemble correlation, so we can write Rxy(p) = RXY(p) = J jxypXY(x,y,p)dxdy = p<7xcrY+jUxjUY . (4.4) It can be shown that the cross-correlator output, Riy, is linearly related to the temperature of the source, Tsrc, which is the quantity astronomers wish to measure. This can be done by first noting that the total power input to the correlator is proportional to Tsys + Tsrc, that is (7xcrY=K(Tsys+Tsrc), (4.5) where K is some constant. Now by substituting Eq. (4.4) and Eq. (4.5) into Eq. (4.3) and re-arranging yields Thus the ideal cross-correlator output is linearly related to, Tsrc.. Another quantity of interest is the auto-correlation output response, which can be used to estimate the variance of an input signal. Auto-correlation is a special case of Eq. (4.4) where p = 1, and is given by the second moment relation R»(0)=^H^=CT-+^ (4'7) Note that when the input mean is zero = .a2,. The other quantity of interest is the standard deviation, orxy, of xxy for a finite integration time, T— as arxy affects the sensitivity and SNR, of the cross-correlator. If the integration time is infinite then 66 csrxy is zero, since the system noise, Tsys, is independent in each input. However, if the integration period is finite crxy depends on the number of independent samples, N. For a band-limited signal the number of independent samples is limited by the sampling theorem to 772 Av. The variance of rxy is given by (4.8) Since the SNR, and hence crxy, is more important at the threshold of detection (i.e., when ~ arxy), it is reasonable to simplify the previous equation by letting R ,^ = 0 [Klingler72]. In this case the variance of rxy can be written as: " N N-l Rxx (0) RYY (G) + 2 £ (1 - i) Rxx (i At) RYY (i At) (4.9) Since for zero mean Gaussian signals, band-limited by a rectangular filter, the normalized auto-correlation function is Rxxir) RYY(r) sin(2^Avr) xx YY 2K A V T sinc(2/rAvr), (4.10) we can substitute Eq. (4.10) into Eq. (4.9) and note that RJQ) = aj and Ryy(0) = ayy2. Doing this yields 2 2 ° XX°YY N N-l 1 + 2 ^ (1 - ^> sine2 (/• 2A: A u Ar) 1=1 (4.11) Now using Eqs. (4.6) and Eq. (4.11), and letting N = Ks Av T, the signal to noise ratio is given by SNR = -R Ps/KsAvT I KsAvT-l (1 + 2 ^ a - T ^ > s m c \ ^ ) (4.12) • 6 7 W h e n Ks = 2 , s a m p l i n g i s p e r f o r m e d a t t h e N y q u i s t r a t e , w h i l e v a l u e s o f K„ = { 2 / 2 , 2 / 3 , 2 / 4 , . . . } c o r r e s p o n d t o u n d e r - s a m p l i n g . I n e i t h e r c a s e t h e s u m m a t i o n t e r m i n E q . ( 4 . 1 2 ) i s z e r o a n d t h e S N R i s g i v e n b y t h e f o l l o w i n g , c o m m o n e q u a t i o n : SNRK. { 2 m = PvXA\)T = pV/V . ( 4 . 1 3 ) O v e r - s a m p l i n g o c c u r s w h e n A T , i s g r e a t e r t h a n t w o , a n d i n t h i s c a s e t h e s u m m a t i o n t e r m i n E q . ( 4 . 1 2 ) h a s a l i m i t i n g v a l u e o f ( K - 2 ) / 4 [ T h o m p s o n 8 6 ] . T h u s t h e d e n o m i n a t o r o f E q . ( 4 . 1 2 ) s i m p l i f i e s t o V K s / 2 a n d t h e S N R i s g i v e n b y t h e f o l l o w i n g , f a m i l i a r , f o r m u l a : R SNRK>2 = -^- = pJ2AvT . ( 4 . 1 4 ) T h i s r e s u l t i s i n a g r e e m e n t w i t h t h e s a m p l i n g t h e o r e m , a s s a m p l i n g f a s t e r t h a n t h e N y q u i s t r a t e d o e s n o t y i e l d n e w i n f o r m a t i o n . 4.2 The Digital Correlator F o l l o w i n g t h e s a m e t h e m e a s t h e p r e v i o u s s e c t i o n , t h i s s e c t i o n e x a m i n e s t h e d i g i t a l c o r r e l a t o r s o t h a t i n t h e f o l l o w i n g s e c t i o n s i t c a n b e c o m p a r e d t o a n i d e a l o n e . T h e c h a r a c t e r i s t i c s o f i n t e r e s t a r e , a g a i n , t h e s i g n a l t o n o i s e r a t i o a n d t h e o u t p u t r e s p o n s e o f t h e a u t o - a n d c r o s s - c o r r e l a t o r s . I n t h e f o l l o w i n g , e q u a t i o n s f o r a g e n e r i c d i g i t a l c o r r e l a t o r a r e g i v e n , a s w e l l a s s p e c i f i c e q u a t i o n s f o r t h e 3 x 3 l e v e l a n d t h e 1 4 x 1 4 l e v e l c o r r e l a t o r s u s e d i n t h i s p r o j e c t . S i m i l a r t o t h e i d e a l c o r r e l a t o r , t h e d i g i t a l o n e i n t e g r a t e s t h e N s a m p l e p r o d u c t o f t h e q u a n t i z e d i n p u t s i g n a l s , qx a n d qy, a t u n i f o r m i n t e r v a l s , At. T h e e x p e c t e d v a l u e , d e n o t e d b y E, o f t h e d i g i t a l c o r r e l a t i o n f u n c t i o n , rDxy, i s RD(r) = E[rD (T)] = E 1 1 1=1 ( 4 . 1 5 ) 6 8 w h e r e RDxy(T) i s t h e a v e r a g e c r o s s - c o r r e l a t i o n f u n c t i o n f o r a r e l a t i v e d e l a y , x, a n d T i s t h e i n t e g r a t i o n p e r i o d ( T = N At). T h e e q u a t i o n a b o v e i s s i m i l a r t o E q . ( 4 . 1 ) e x c e p t t h a t t h e a m p l i t u d e o f t h e i n p u t s i g n a l h a s b e e n b i n n e d b y a n n - l e v e l q u a n t i z e r . B e f o r e c o n t i n u i n g w e n e e d t o i n v e s t i g a t e t h e b e h a v i o r o f a q u a n t i z e r . T h e o u t p u t r e s p o n s e o f a n i d e a l n - l e v e l q u a n t i z e r i s s h o w n i n F i g . 4 . 1 , a n d c a n b e e x p r e s s e d a s n-l j q{x) = £ (w.+I - w.) &(x - /,.) i=i ( l -n )w, +^w ( . i=2 ( 4 . 1 6 ) w h e r e O i s t h e H e a v i s i d e s t e p f u n c t i o n , n i s t h e n u m b e r o f l e v e l s , a n d w , i s t h e w e i g h t c o r r e s p o n d i n g t o t h r e s h o l d l e v e l /,. E x c l u d i n g t h e t w o t h r e s h o l d s a t ± i n f i n i t y , t h e r e a r e n w e i g h t s a n d ( n - l ) t h r e s h o l d s . T h e t h r e e - l e v e l q u a n t i z e r h a s t h r e s h o l d s {-/, / } , a n d i n t e g e r w e i g h t s { - 1 , 0 , 1 } . T h e o u t p u t r e s p o n s e i s q3(x) = Q(x + l) + ®(x-l)-l. (4.17) I n p r i n c i p l e a f o u r t e e n - l e v e l q u a n t i z e r c a n h a v e f l o a t i n g p o i n t w e i g h t s a n d t h r e s h o l d s t h a t a r e n o t l i n e a r l y r e l a t e d . M o s t c o m m e r c i a l A D C s , h o w e v e r , a r e m u c h s i m p l e r , a n d h a v e t h r e s h o l d s t h a t d i f f e r b y a c o n s t a n t Al a n d w e i g h t s t h a t a r e r e s t r i c t e d t o i n t e g e r s . I n t e g e r w e i g h t s s i m p l i f y t h e c o r r e l a t o r c i r c u i t r y . T h e f o u r t e e n - l e v e l q u a n t i z e r u s e d i n t h i s s y s t e m h a s t h r e s h o l d s { - 6 / , - 5 / , . . . - / , 0 , /, ... 6 / } a n d o d d i n t e g e r w e i g h t s { - 1 3 , - 9 , . . . , - 1 , 1 , . . . , 1 3 } [ K a r p a 8 9 ] . T h e t r a n s f e r f u n c t i o n i s n- l qu(x) = -\3 + 2\Yj<I>(x + (i-l)-l). ( 4 . 1 8 ) 69 wn -w3-w2 1 1 1 I li h K\ h x Figure 4.1: The input-output response of an ideal n-level quantizer with thresholds, I, and weights, w. Now returning our attention to RDxy, again since x and y are ergodic Gaussian processes, then time correlation equals ensemble correlation and the expected value of the generic digital correlator, RDxy, can be written as: n-l n-i 'l+i 'tt+i RD„ (P) = R D x r (P) = £ X w>;+, J J Pxv (*> y. P) dxdy. (4.19) i=0 ;=0 Where the w's and /'s are the quantizer weights and threshold levels (see Fig. 4.1), and p(x,y) is the Bi-variate probability density function given by Eq. (4.2). Note that the thresholds at / 0 and /„ in Eq. (4.19) are equal to minus and plus infinity, respectively. 70 Alternatively, we can simplify the integral in Eq. (4.19) by integrating with respect to p [Hagen73], and by assuming a = ox = Gy. Using this form we can write J n-l n-\ P 2 o - 2 ( l - p 2 ) R °» { P ) = W 5 5 K " } ' ( W y i ~ } ' > V l - r 2 ^ ' ( 4 ' 2 0 ) Assuming the thresholds are equal for both x and y quantizers (i.e., lx = ly), the output of the 3x3 and 14x14 level cross-correlators, respectively, aregiven by R™sy (P) = Y I 7 = ^ d r > and by, (4.21) 2 6 6 P 2 < r 2 ( l - r 2 ) (P) = — r 2 S J Y- 2 ^ • (4-22) " ° ,=-6>-6 V l - r While the output of the generic, 3x3, and 14x14 digital auto-correlators (at T = 0) are given by ^(0) = ^ | < P - . (4-23) R3x, (0) = = f e 2 a l dx = erfc (4.24) and by, i W ( 0 ) = 169-85>erf V2tT (4.25) 71 The equation for the variance of a digital correlator, aDxy 2, is similar to the one for an ideal correlator (see Eq. (4.9)), except the ideal correlation functions are replaced by the equivalent digital ones. Using the same assumptions given in section 4.1 for Eq. (4.9) (that RDxy = 0), and assuming the mean quantizer-output is zero, the variance of a digital correlator is , 1 <7 n = — D" N RDxx (0) RDyy (0) + -22 (1 - i) RDxx (pB (0) RDyy (pB (0) i = i (4.26) where RDxx (p(0) and RUyy(p(i)) are given by Eq. (4.20) and pB(/) = sinc(i 2n Av At). Note that pB is the auto-correlation coefficient and describes the effect of bandlimiting each input. One should not confuse pB with the cross-correlation coefficient, p, of x and y used elsewhere in this chapter. By letting N = Ks Av T, and using the fact that RDxx = RDyy, then the SNRs for the 3x3 and 14x14 level correlators are •,2 S N R ^ = ^ = " 2pV5A5Te ( 4 . 2 7 ) + 2 J^a-Trh?)RL (sinc(^)) 3 x 3 " T t o M e r f c V2a. and 6 6 ,2« 2+J 2> 5A//?, 4 x M = ' 4 2 x l 4 " p < < 1 = . , = - 6 j = - 6 (4.28) 14X14 ^ 2 K,AuT-l ' V y respectively. As in the previous section, when the sampling factor, Ks, is greater than two, over-sampling occurs. When Ks is two, samples are taken at the Nyquist rate. Values of Ks = {2/2, 2/3, 2/4, ...}, correspond to under-sampling and in both this and the prior case the summation term involving R2Dxx in Eq. (4.26) through Eq. (4.28) is zero. 72 One problem with Eq. (4.26) is that the summation term makes computing a2DRXy expensive, when N is large and Ks is greater than two. Thompson et al (1986) have shown a way around this. They note that for i/K large the /?X),(sinc( i/Ks)) terms are essentially zero, and that the infinite sum for an ideal correlator equals (Ks - 2)1 A. ' Thus for N large G2Dxy can be approximated by N R2D(T = 0) + 2 p=o> £ s i n c 2 ( ^ f ) i=i (4.29) One may recognize that the derivative in Eq. (4.29) represents a first order approximation of RDxy in terms of p. A problem with this is that when p (i.e., i 2n I Ks) is large the approximation has considerable error. For example if Ks = 4, Eq. (4.29) is limited to single digit accuracy for the three-level correlator. These problems can be overcome by replacing the initial, say Ks, approximated-terms with the exact formula. Doing this yields 1 D" N R2D(0) + 2 dRD(p) dp ^ r ^ - i s i n c2 ^ )1 + X ^ ( s i n c ( ^ ) ) p=0) 1=1 •J . (4.30) This equation is a new result and superior to that given by Thompson et al, as it is more accurate with only a modest increase in computational effort. 4.3 Optimizing the SNR Performance of a Digital Correlator The equations in the preceding section all depend on the quantizer thresholds and weights, and these parameters can be optimized to maximize the SNR. A number of authors have investigated optimum values for correlators with less than five levels. For the 14-level correlator Karpa (1989) found that .374 a is the optimum threshold for a 14-level correlator with odd integer weights and uniform 7 3 i n t e g e r l e v e l s . I n t h i s s e c t i o n t h e w o r k o f t h e s e a u t h o r s i s e x t e n d e d i n o r d e r t o i n v e s t i g a t e t h e p e r f o r m a n c e o f t h e 1 4 - l e v e l c o r r e l a t o r . A u s e f u l m e t r i c i s t h e r e l a t i v e s e n s i t i v i t y , r | , o f a d i g i t a l c o r r e l a t o r . T h e r e l a t i v e s e n s i t i v i t y i s t h e i n v e r s e o f t h e d e g r a d a t i o n f a c t o r i n t r o d u c e d b y K l i n g l e r ( 1 9 7 2 ) a n d i s d e f i n e d a s • ( 4 . 3 I ) SNRU„, D A l t h o u g h K l i n g l e r i n t r o d u c e d t h e r e l a t i o n s h i p , t h e r e l a t i v e s e n s i t i v i t y i s m o r e p o p u l a r , p r e s u m a b l y , b e c a u s e i t d i r e c t l y r e p r e s e n t s t h e l o s s i n S N R d u e t o q u a n t i z a t i o n . K l i n g l e r ( 1 9 7 2 ) f o u n d t h a t { - 1 , 0 , 1 } a r e o p t i m u m i n t e g e r w e i g h t s f o r a t h r e e - l e v e l q u a n t i z e r , a n d t h a t t h e o p t i m u m t h r e s h o l d i s . 6 1 2 c. T h e o p t i m u m t h r e s h o l d c a n b e f o u n d , n u m e r i c a l l y , b y s o l v i n g t h e f o l l o w i n g n o n - l i n e a r e q u a t i o n f o r / „ . : • = 0 . ( 4 . 3 2 ) T a b l e 4 . 1 c o m p a r e s t h e o p t i m u m c o r r e l a t o r s e n s i t i v i t y , T| 0, a n d t h e c o r r e s p o n d i n g o p t i m u m t h r e s h o l d s , f o r s e v e r a l q u a n t i z a t i o n s c h e m e s a s a f u n c t i o n o f t h e o v e r - s a m p l i n g f a c t o r Ks. K l i n g l e r ( 1 9 7 2 ) n o t e s t h a t t h e o p t i m u m t h r e s h o l d c h a n g e s s l i g h t l y f o r Ks > 2 ; h o w e v e r , u s i n g t h e o p t i m u m a t Ks = 2 d o e s n o t c h a n g e T| s i g n i f i c a n t l y . F o r a 1 4 - l e v e l c o r r e l a t o r i t w a s f o u n d t h a t t h e s e n s i t i v i t y i s h i g h e s t f o r o d d i n t e g e r w e i g h t s . F o r e i t h e r i n t e g e r o r e v e n i n t e g e r w e i g h t s ( i . e . , { - 6 , - 5 ... - 1 , 1 , ... 6 } , o r { - 1 4 , - 1 2 ... - 2 , 2 , ... 1 4 } ) t h e r e l a t i v e s e n s i t i v i t y i s . 9 7 5 c o m p a r e d t o . 9 8 5 f o r o d d i n t e g e r w e i g h t s . I n a l l t h r e e c a s e s t h e o p t i m u m t h r e s h o l d i s . 3 7 4 c. 7 4 Table 4.1: The optimum digital correlator sensitivity, T | 0 ) and corresponding threshold, 1„, (at Ks = 2), for several quantization schemes, as a function of the over-sampling factor, K s. Number of Optimum Quantizer Threshold, /„ Sensitivity Relative to an Ideal Correlator, T|„ Quantization Levels (at K s = 2) Ks = 2 Ks = 4 2 Oa .637 .744 3 .612 a .810 .882 4 .995 a .881 .930 14 .374 a .985 .992 A c h o i c e m u s t b e m a d e i n s e t t i n g t h e o p t i m u m A D C t h r e s h o l d s i n t h e h y b r i d - l e v e l c o r r e l a t o r , a s a n A D C o u t p u t i s u s e d b y b o t h a 1 4 l e v e l a u t o - c o r r e l a t o r a n d t h r e e - l e v e l c r o s s - c o r r e l a t o r . S i n c e i t i s m o r e i m p o r t a n t t o m a x i m i z e t h e S N R o f t h r e e - l e v e l c r o s s - c o r r e l a t o r , t h e t h r e s h o l d s a r e s e t t o . 6 1 2 o ( t h e t h r e e - l e v e l o p t i m u m ) . U s i n g t h e t h r e e - l e v e l o p t i m u m h a s t h e a d d e d b e n e f i t o f m a k i n g t h e o u t p u t r e s p o n s e o f t h e 1 4 - l e v e l a u t o - c o r r e l a t o r n e a r l y l i n e a r ( s e e s e c t i o n 4 . 4 ) . I n F i g . 4 . 2 t h e r e l a t i v e s e n s i t i v i t y o f t h e t h r e e - a n d f o u r t e e n - l e v e l c r o s s - c o r r e l a t o r s i s p l o t t e d a s a f u n c t i o n o f t h e q u a n t i z e r t h r e s h o l d s . N o t e t h a t t h e S N R i s f a i r l y i n s e n s i t i v e t o c h a n g e s i n t h e o p t i m u m t h r e s h o l d l e v e l . F o r a 1 0 % v a r i a t i o n a b o u t t h e o p t i m u m t h r e s h o l d t h e r e l a t i v e s e n s i t i v i t y c h a n g e s b y o n l y . 2 5 % f o r a t h r e e -l e v e l c o r r e l a t o r a n d b y . 1 0 % f o r a 1 4 - l e v e l c o r r e l a t o r . F o r a ± V2 c h a n g e a b o u t t h e o p t i m u m t h r e s h o l d t h e S N R c h a n g e s b y a m a x i m u m o f 4 . 0 % f o r a t h r e e - l e v e l c o r r e l a t o r a n d 1 . 3 2 % f o r a f o u r t e e n - l e v e l c o r r e l a t o r . 75 0.2 -0.1 --0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized Quantizer Threshold, / / / „ Figure 4.2: The relative sensitivity of several digital correlators compared to an ideal correlator. The abscissa is the quantizer threshold, 1, in units of <3, the standard deviation of the input. The two sets of curves (a) and (b) correspond to different sample rates, K-2 and K=4, respectively. The curves are for: (1) the three-level correlator, (2) the four-level correlator, and (3) the 14-level correlator. 4.4 The Output Response of The Three- and Fourteen-Level Correlator The output response of a digital correlator is non-linear. In this section the response of the three-and 14-level auto-correlator and cross-correlator is investigated. The auto-correlator is essentially a power detector, as it measures the mean-square quantizer output. The output response of the three- and 14-level auto-correlators, as well as their ideal responses are plotted in Fig. 4.3. For comparison purposes each curve has been normalized to unity at the optimum threshold level, / 0. At the optimum quantizer threshold, the auto-correlator output, given by Eq. 4.23, is .541 and 28.2 for a three- and a 14-level auto-correlator, respectively. In the new correlator the ADCs are set to .612 o, the optimum for a three-level cross-correlator. In this case the output of the 14-level auto-correlator is 11. Note that the 14-level correlator response is very linear in the region below the optimum input. At input levels above the optimum the response begins to saturate, as the probability of exceeding the maximum threshold increases. About the optimum input level, the three-level correlator is not nearly as linear and has a lower small-signal gain than the 14-level correlator. From this it is clear that it is better to use Ruxuxx(0) instead of R^xx(0) as the central, or zero lag, channel in the hybrid-level auto-correlator configuration. Assuming the input mean, |J., is zero this can be done by simple substitution. However, a nearly linear correlator response can be achieved if Ri^uxxiO) is linearized using the methods discussed in section 4.5.1. The resulting hybrid-level auto-correlator is a very linear device. The reason is that the input power is dominated by broad-band receiver (Gaussian) noise and so is concentrated in the central channel where it is measured by the (almost linear) 14-level auto-correlator. Additionally, the remaining (three-level) non-zero lag channels are very linear for low levels of correlation (discussed below). And since the 77 Figure 4.3: The normalized output response f Rxxf0,)/Rxx (0) 0 =;) of an auto-correlator as a function of the input variance, a 2. The curves are for: (1) the ideal correlator, (2) the three-level correlator (l0 = .612, R, = n = .541), (3) the 14 level correlator (la = .374, R x x f a =/;) = 28.2), and (4) the 14 level correlator at the three-level optimum ( l0 = .612, R X X(„ =i>) = H). 78 c o r r e l a t i o n i s t y p i c a l l y l o w i n t h e n o n - z e r o l a g c h a n n e l s o f a N y q u i s t s a m p l e d c o r r e l a t i o n f u n c t i o n , t h e o u t p u t r e s p o n s e o f t h i s h y b r i d d i g i t a l a u t o - c o r r e l a t o r i s a l m o s t l i n e a r . A s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n , t h e r e l a t i v e s e n s i t i v i t y , r | , o f t h e d i g i t a l c r o s s - c o r r e l a t o r i s o p t i m u m w h e n t h e q u a n t i z e r t h r e s h o l d s a r e a t l0. I n i n v e s t i g a t i n g t h e r e s p o n s e o f t h e d i g i t a l c r o s s -c o r r e l a t o r t h e r e a r e t w o c a s e s t o c o n s i d e r , o n e w h e r e t h e q u a n t i z e r v a r i a n c e , <7D, i s c o n s t a n t a n d o n e w h e r e oD i s a l l o w e d t o v a r y a b o u t t h e o p t i m u m . M a i n t a i n i n g o ~ D c o n s t a n t m e a n s t h a t t h e r e l a t i o n s h i p b e t w e e n t h e i n p u t l e v e l a n d t h e q u a n t i z e r t h r e s h o l d i s h e l d c o n s t a n t ( i d e a l l y a t l„). I n t h i s c a s e w e c a n w r i t e pDxy = Roxy I K a n d u s e t h e n o r m a l i z e d c r o s s - c o r r e l a t o r r e s p o n s e , pr>xy, t o i n v e s t i g a t e t h e b e h a v i o r o f t h e d i g i t a l c r o s s - c o r r e l a t o r , RDxy. F i g u r e 4 . 4 s h o w s t h a t t h e n o r m a l i z e d r e s p o n s e o f t h e t h r e e - l e v e l a n d 1 4 - l e v e l c r o s s - c o r r e l a t o r i s w i t h i n 1 % o f b e i n g l i n e a r f o r p < .38 a n d p < .99, r e s p e c t i v e l y . E x a m i n i n g E q . (4 .3) s h o w s t h a t p c a n b e u s e d t o e s t i m a t e r m . ( t h e q u a n t i t y a s t r o n o m e r s w i s h t o m e a s u r e ) , f o r p < < 1 . U s i n g p t o e s t i m a t e Tsrc h a s t h e a d v a n t a g e t h a t i t i s i n s e n s i t i v e t o s y s t e m g a i n c h a n g e s ( i . e . , A G ( T s r c ) / A G ( T s y s + T s r c ) ) . H o w e v e r , t h e e f f e c t o f s y s t e m g a i n v a r i a t i o n s c a n b e m i n i m i z e d b y c a r e f u l d e s i g n a n d b y c a l i b r a t i n g f r e q u e n t l y — s o u s i n g p t o e s t i m a t e Tsrc, i n t h i s c a s e , h a s n o s i g n i f i c a n t a d v a n t a g e . T h e r e a r e a l s o a c o u p l e o f p r o b l e m s w i t h u s i n g p t o e s t i m a t e . Tsrc. O n e i s t h a t p i s i n v e r s e l y p r o p o r t i o n a l t o v a r i a t i o n s i n s y s t e m t e m p e r a t u r e , T y ) V , w h e n Tsrc « Tsyx. V a r i a t i o n s o f s e v e r a l p e r c e n t a r e c o m m o n a n d a r e d i f f i c u l t t o a v o i d . A n o t h e r p r o b l e m i s t h a t p i s n o t l i n e a r l y r e l a t e d t o Tsrc w h e n Tsrc i s o f t h e s a m e o r d e r a s Tsys. F o r t u n a t e l y , f o r a n i d e a l c o r r e l a t o r , R^ i s i n s e n s i t i v e t o c h a n g e s i n Tsys a n d s o i s a l i n e a r m e a s u r e o f Tsrc. T h u s t h e p r o b l e m s m e n t i o n e d a b o v e c a n b e a v o i d e d . O f c o u r s e , w e w a n t t o k n o w i f t h e o u t p u t o f t h e t h r e e - l e v e l a n d 1 4 - l e v e l d i g i t a l c o r r e l a t o r s a r e a l i n e a r m e a s u r e o f Tsrc. T o i n v e s t i g a t e t h i s a s s u m e t h a t ox - oy - Tsys + Tsrc, a n d t h a t t h e 79 0.00% 0.20% 2.00% 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Normalized Ideal Correlator Output, p 0.9 Figure 4.4: The normalized output response of a three-level and a 14-level cross-correlator. Where the quantizer thresholds have been held at their optimum values, l0, and the slope at p = 0 has been normalized to 1. The curves are for: (1) a three-level cross-correlator, and (2) a fourteen-level cross-correlator. Where: (a) denotes the output response, and (b) denotes the relative error compared to an ideal correlator. 80 optimum quantizer thresholds are fixed at some quiescent value of the system temperature, say TSysmin- Furthermore, assume that the source temperature, Tsrc, is varied from zero to Tsys'„„•„ for two cases, one where Tsys = Tsysmin, and one where Tsys = 130% Tsysmin. Figures 4.5 and 4.6 show the output response of the three-level and 14-level cross-correlators, respectively, under the conditions described above. In the figures the noise temperature was normalized by the minimum system temperature, Tsys m,„, so that a2 = (Tsys + TSK)/ Tsys min. As well, the quantizer thresholds were set so that the optimum is at cTsysmin- In each figure two curves are shown, one where Tsys = Tsysmin and one where Tsys = 130% Tsysmin. Figs. 4.5 and 4.6 show that the digital cross-correlators do respond to changes in Tsys. The response is also non-linear, and causes the correlator small signal gain (d RDxy I d Tsrc) to change with both Tsrc and T,^. Another problem is that the SNR of the correlator will be reduced, albeit slightly, as the input level varies about the optimum. These effects can be understood by considering what happens to a quantizer, with fixed thresholds, as the input power is increased. As the input power approaches infinity all the thresholds approach zero, the threshold of a two level correlator. Thus the performance of any digital correlator ultimately degrades to that of a two-level correlator as the input levels increase with respect to the quantizer thresholds.. Before leaving this section it should be noted that Fig. 4.4 shows that the normalized responserpD, of both digital correlators is nearly linear. Thus one way to obtain a linear measure of Tsrc for small p, mentioned by D'Addario (1984), is to hold the thresholds constant with respect to the input level and use Eqs. (4.3) and (4.5) to compute dp _ n ° x a y dp ; p « l , (4.33) src dpD H D K dp D 81 1 rr 5 0 % . 4 5 % + 4 0 % + 3 5 % 4- 3 0 % 2 5 % + 2 0 % -5 1 5 % 4-10% 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 Normalized Source Temperature, Tm /Ts„mln Figure 4.5: The output response of a three-level cross-correlator, R 3x3xy, versus the normalized source temperature, Tsrc / Tsys min. The curves are: (la) the response where Tsys = Tsysmin, and (2a) the response where Tsys = Tgysmin 130%. In both curves the quantizer threshold, I, is fixed at I =.612 G T s y s m i n . Curves (lb) and (2b) show the relative error, compared to an ideal correlator, for curves (la) and (2q) respectively. 82 Normalized Source Temperature, r,« / T, Figure 4.6: The output response of a 14-level cross-correlator, R/4 x/4xy, versus the normalized source temperature, T s r c / T s y s m i n . The curves are: (la) the response where T s y s = T s y s m i n , and (2a) the response where T s y s = T s y s m i n 130%. In both curves the quantizer threshold, 1, is fixed at \-.374 aTsys min- Curves (lb) and (2b) show the relative error, compared to an ideal correlator, for curves (la) and (2a) respectively. 8 3 w h e r e K i s a p r o p o r t i o n a l i t y c o n s t a n t , p D i s t h e n o r m a l i z e d d i g i t a l c o r r e l a t i o n c o e f f i c i e n t , a n d t h e p a r t i a l d e r i v a t i v e p r o v i d e s u n i t y g a i n . I m p l e m e n t i n g t h i s s c h e m e , h o w e v e r , r e q u i r e s f i x i n g t h e r e l a t i o n s h i p b e t w e e n t h e q u a n t i z e r t h r e s h o l d s a n d t h e i n p u t l e v e l , p r e f e r a b l y a t / „ , a n d m e a s u r i n g ax a n d cjy. w i t h a n i d e a l d e v i c e . T h i s o f c o u r s e r e q u i r e s a d d i t i o n a l c i r c u i t r y w h i c h w e w i s h t o a v o i d . A n a l t e r n a t i v e i s t o l i n e a r i z e t h e d i g i t a l c o r r e l a t o r r e s p o n s e , a n d t h i s a p p r o a c h i s d i s c u s s e d i n t h e n e x t s e c t i o n . 4.5 Linearizing the Output Response of a Digital Correlator A s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n s t h e o u t p u t r e s p o n s e o f a d i g i t a l c o r r e l a t o r i s n o n - l i n e a r , a n d t h i s r e s u l t s i n a n u m b e r o f u n d e s i r a b l e s i d e - e f f e c t s . O n e i s t h a t c h a n g e s i n t h e i n p u t v a r i a n c e , a 2 , a r e n o t d i r e c t l y r e l a t e d t o c h a n g e s i n RDxx- T h i s m a k e s i t d i f f i c u l t t o d e t e r m i n e c 2 f r o m RDxx a s t h e d i g i t a l a u t o - c o r r e l a t o r g a i n (ARDxx(0)/ A a 2 ) c h a n g e s w i t h t h e i n p u t l e v e l . S e c o n d , t h e c r o s s - c o r r e l a t o r o u t p u t , RDXy, i s o n l y a p p r o x i m a t e l y l i n e a r f o r p « 1 a n d o n l y i f t h e q u a n t i z e r v a r i a n c e i s c o n s t a n t . I n t h e h y b r i d - l e v e l c o r r e l a t o r t h e q u a n t i z e r v a r i a n c e i s n o t c o n s t a n t , a n d t h i s c a u s e s Rpxy t o b e s e n s i t i v e t o t h e i n p u t l e v e l . T h i s m a k e s i t d i f f i c u l t t o m e a s u r e t h e s o u r c e t e m p e r a t u r e , T s r c , f r o m RDxy, a s t h e c r o s s - c o r r e l a t o r g a i n ( A / ? 0 x y / ATsrc) c h a n g e s w h e n Tsrc i s o f t h e s a m e o r d e r o f t h e Tsys. A f u r t h e r p r o b l e m i s t h a t RDxy, a n d s o t h e c r o s s - c o r r e l a t o r g a i n (ARDxy I ATsrc) i s a l s o a f f e c t e d b y c h a n g e s i n Tsys. C l e a r l y , i t i s d e s i r a b l e t o l i n e a r i z e t h e c o r r e l a t o r r e s p o n s e o v e r s o m e t y p i c a l o p e r a t i n g r a n g e . F o r t h e D R A O - S T t h e s y s t e m t e m p e r a t u r e a n d s o u r c e t e m p e r a t u r e t y p i c a l l y i n c r e a s e b y l e s s t h a n 2 0 K w h i l e t h e s y s t e m t e m p e r a t u r e i s a b o u t 6 0 K [ L a n d e c k e r 9 6 ] . I f w e a s s u m e a b e s t - c a s e m i n i m u m s y s t e m -t e m p e r a t u r e o f 4 0 K , t h e n t h e m a x i m u m c h a n g e i n i n p u t l e v e l w o u l d b e + 3 d B a n d t h e m a x i m u m c o r r e l a t i o n c o e f f i c i e n t , p , w o u l d b e . 3 3 . S o a r e a s o n a b l e g o a l i s t o m a k e t h e r e s p o n s e o f a d i g i t a l 84 correlator linear within .1% relative error over a 3dB input change. In this section some methods to do this are investigated. What is required is a function that transforms a digital correlator output, RD, to an ideal (linear) correlator output, R. One way this can be done is to use a series approximation to invert the equations describing the output response of a digital correlator. Doing this allows one to compute the quantizer threshold, /, given RDxx, and to compute the ideal cross-correlation coefficient, p, given RDxy. This information can then be used to compute and Rxy, the ideal correlator quantities. Kulkarni [1980] and D'Addario [1984] have investigated series inversion schemes for three-level correlators and these results are summarized. Using these ideas an inversion scheme was developed for the 14-level auto- and cross-correlator. These results are then applied to approximate the ideal correlator quantities, i?^ and Rxy. 4.5.1 Linearizing the Auto-Correlator Response The response of a three-level auto-correlator, under the assumptions in section 4.2, is given by Eq. (4.24). Kulkarni (1980) has shown that the threshold of a three-level auto-correlator, R^^, can be determined by simply using the polynomial inversion approximation to the error function given by Hastings (1966). However the approach used by D'Addario (1984) uses a rational inversion approximation, given by Blair (1976), and this approach is preferred as it can be computed more efficiently if it is expressed in continued fraction form. Using this later approach the quantizer threshold, /, can be determined from, + e(R3x3J, (4.34) R l3x3 - 1 -XX R, L3x3 - 1 -R, L3x3 - 1 -XX R l3x3 -1 • ., 85 where the coefficients5 are a , = .37253461 a2= 1.322842377 •• a 3 = 2.784504703 aA= .104200328 , a5= 1.249431331 I e| < 8.3 x IO"7 for RDxx > .25 . The three-level auto-correlator response can be linearized by simply fixing the optimum threshold, l0, at .612 cj0 (where o"02 is the optimum input power). Now changes i n i 3 x 3 can be interpreted as a change in the input level (J. Since, for zero mean signals, the input variance equals the auto-correlation at zero delay, the required ideal auto-correlator output can be determined from M2a0 Y (4.35) Unfortunately, the 14-level correlator response cannot be inverted as easily because its response depends on the sum of error functions at different limits (see Eq..(4.25)). So the 14-level quantizer threshold, /, cannot be determined using the inversion approximation to the error function above. Instead Eq. (4.25) must be inverted. . , The fifth-order Taylor series inversion of Eq. (4.25) about the optimum threshold, l0, is k^AR )=an+a,R +a7R2 +a,R3 +a,R* +a,R5 , (4.36) V 1 4 X l 4 \ 14x14 „ / 0 1 14xl4xt , 4 x l 4 a J , 4 X 1 4 „ 4 3 where the coefficients for a 14-level auto-correlator with /0 = .612 o 0 are a0= 1.745894800 a, =-2.783954682 xlO 4 a 2= 3.100062250xlO"2 a 3 = -2.033974134xl0"3 a4 = 7.232209770 x 10 5 a 5 = -1.079948802 x l O 6 . 5 The precision of these coefficients is high in order to avoid numerical errors in the approximations. 86 And for a 14-level auto-correlator with / 0 = .374 o0 the coefficients are ao= 1.043145145 ax =- 6.272200448 xlO"2 a2 = 2.704212927 x l O 3 a 3 =-6.980713121x10s a 4 = 9.742015325xl0"7 a5 = -5.698004717x10"9. However, a rational function often has a smaller error than a Taylor series approximation of the same order. As well, it is more efficient to compute if it is expressed in continued fraction form. A Pade approximation, discussed in many texts on numerical analysis, is one way of finding a rational approximating function [Burden85]. The Pade approximation, with numerator of degree 3 and denominator of degree 2, of Eq. (4.25) in continued fraction form is / = a 0 + V ? 1 4 x l 4 i i + — ^ , (4.37) a3 + RUxU„ + a5 + ^ 1 4 x l 4 „ where the coefficients for a 14-level auto-correlator with l0 = .612 o~0 are 0 0 = 2.794649679x 10"' a, = 2.320599066 x 10"3 a2= 5.010266877 « 3 = 3.334156924 a4 = 4.186134369 a 5 = 8.007219850 x 10" And for a 14-level auto-correlator with /„ = .374 o 0 the coefficient are a0 = 9.827728450 x 10"2 a, = -1.084977131 x 10"3 a2 = 20.42555175 a 3 = 54.24037458 a4 = -669.1910204 a5 = 14.33351404. 87 As in the three-level case, the response of the 14-level auto-correlator can be linearized, using the interpreted as a change in the input level c. Figures 4.7 and 4.8 compare the relative error in determining /, and R& , respectively, for the approximation functions discussed in this section. Examining the figures the rational (Pade) approximations have less relative error than the Taylor series ones and are more efficient to compute. The performance of these approximations for the three-level auto-correlator are very good. The relative errors are less than 10"58 in R^, and less than 10"61 in /, for a 3dB change in input power about the optimum level. The performance of the rational approximations for the 14-level auto-correlator are also quite good. For a 3dB change about the optimum input power, G 0 2 , the relative errors are less than 10' 4 2 in R^, and 10"45 in /, for an optimum threshold of .374 c 0; and less than 10"3 8 in R^, and 10"41 in /, for an optimum threshold of .612 o 0. • 4.5.2 Linearizing the Cross-Correlator Response The digital cross-correlator output can be linearized by inverting Eq. (4.20) so that p can be approximated from RDxy. The ideal correlator output, RIy, can then be determined by multiplying p by ^RXx ^ Ryy (using Eqs. 4.7 and 4.6). For the three-level correlator Kulkarni (1980) gives a Taylor series inversion, while D' Addario et al (1984) have shown that a two-point Taylor series inversion is more accurate over a wider range of p. The two-point Taylor-series inversion of p, given R D ^ = R3 x3x>» about p = 0 and p = pu is relation in Eq. (4.35), by simply fixing the optimum threshold, /„. So now changes in / 1 4 x l 4 can be (4.38) 88 2 T -8 -I 1 -i 1 H 1 1 1 1 0.5 1 1.5 2 2.5 Normalized Quantizer Threshold, / / io Figure 4.7: The relative error in approximating the quantizer threshold level, I, using various inversion functions. In the figure the quantizer threshold has been normalized by, l/l0. The curves are for: (I) the rational inversion function for three-level auto-correlator, l0 - .612, (2 a) a Taylor series and (2b) a Pade, inversion function for a 14-level auto-correlator, l0 = .374, and (3 a) a Taylor series and (3b) a Pade, inversion function for a 14-level auto-correlator, l0 = .612. 89 4 T 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 \ 4 Normalized Ideal Auto-Correlation, Rxx Figure 4.8: The relative error in determining the ideal auto-correlation function, RX X ) using various inversions functions. The curves are for: (1) the rational inversion function for three-level auto-correlator, l0 = .612, (2a) a Taylor series and (2b) a Pade, inversion function for a 14-level auto-correlator, 10 = .612, and (3 a) a Taylor series and (3b) a Pade, inversion function for a 14-level auto-correlator, l0 = .612. 90 where AkR = (I RDxy I - R o i )* and the coefficients are given by a, = a 4a, —7~ = L K 2 R* a3 = • y/R2D+60ax 6Ri (4.39) «=? e xp[ ic, 2 +0] r = - c 2 - DC,2 - D* 3 • (4.40) and by, a-ApJR^ u _B+(20Pl/RDi-Sa)/RD> (4.41) y + (-120+(60a-120/7, /RDi)/RD>)/RD> 6 < (4.42) Where a, (3, and y are given by 1 a B 3B2-ay (4.43) dp p=p\ d2Rr dp2 d3Rr y = . P=PI dp3 p=p\ (4.44) And the required derivatives are dR D„ UW dp K(\-p) 2x1/2 ' (4.45) d 2 R D v _ ~[p3 + (l]+1) - \)p]uw- lxly(p2 +1) vw dp2 7C(\-f) 2x5/2 (4.46) 91 and 2p6+[lX+5(l2x+l2y)-3]p^ ^u:+i;+4(i2xi2y-i2-i2y)]P2 d 3 RD x y _ l+(/,2-i)(/;-D 2p5+(l2+l2+2)p* +(l2x+l2y-A)p • (4.47) K(l-p2) The terms u, v, and w are u = 2 cosh PUy v - -2sihh w = exp ~(l2x+l2y) 2(\-p2) (4.48) Note that RDJ = /?3)<3jry(pi) and that it can be computed using Eq. (4.21). Figure 4.9 shows that the linearized response of the three-level cross-correlator to the normalized source temperature, Tsrc, under the same conditions used to create Fig. 4.5 in section 4.4. Note that the approximation has a relative error of less than .03% over the full range of Tsrc and Tsys. The curves shown Figure 4.9 can be generated using the following procedure. First, simulate the digital correlator responses by computing R^xx and R^xy as a function of Tsrc and Tsys. Now approximate the quantizer thresholds and the ideal auto-correlation functions using Eq. (4.34) and Eq. (4.35), respectively, and then use Eq. (4.21) to compute RDX (p\ = .75). At this point the two-point Taylor series coefficients can be computed and used to approximate the ideal cross-correlation coefficient, p(7?3x3^,) using Eq. (4.38). The last step is to approximate the ideal cross-correlation, Rxy, using the relation p(/?3x3x>) (Rxx(R\ixx) Ryy(R3x3yy))m given by Eqs. (4.7) and (4.6). One problem is that computing the coefficients for the two-point Taylor series inversion is more expensive than computing the approximation itself. This scheme is therefore most useful for a multi-lag correlator, such as this one, where the coefficients can be computed once and then used repeatedly, to correct a large number of lags. A further problem is that computing the two-point 92 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Source Temperature, Figure 4.9: The response of the linearized three-level cross-correlator, R3xy, versus the normalized source temperature, T s r c / T s y s m j n . , using a two-point Taylor series inversion. The nearly identical curves are: (la) the response where T s y s = T s y s m j n , and (2a) the response where T s y s = T s y s m i n 130%. For both curves pi = .75 and the quantizer threshold, 1, is fixed at \=.612 Ojsys m i n . Curves (lb) and (2b) show the relative error, compared to an ideal correlator, for curves (la) and (2a) respectively. 9 3 T a y l o r s e r i e s c o e f f i c i e n t s i s e v e n m o r e e x p e n s i v e f o r a l a r g e n u m b e r o f l e v e l s , s u c h a s t h e f o u r t e e n -l e v e l c o r r e l a t o r , a s c o m p u t i n g t h e h i g h o r d e r d e r i v a t i v e s o f RDxy i s r e l a t i v e l y e x p e n s i v e a n d u n w i e l d y . I t w a s f o u n d t h a t t h e s e e f f i c i e n c y p r o b l e m s c a n b e a v o i d e d b y a p p r o x i m a t i n g t h e s e r i e s c o e f f i c i e n t s . F o r t h e 1 4 - l e v e l c r o s s - c o r r e l a t o r t h i s w a s d o n e b y u s i n g a t h i r d o r d e r M a c l a u r i n s e r i e s t o i n v e r t Z ? ^ ^ a n d t h e n u s i n g a t w o - d i m e n s i o n a l T a y l o r s e r i e s t o a p p r o x i m a t e t h e c o e f f i c i e n t s . T h e t h i r d o r d e r M a c l a u r i n s e r i e s o f p a b o u t R^u^y = p 0 = 0 i s P(RHx^) = alRu,uxy+^RLuxy- ( 4 - 4 9 ) S i n c e t h e s e c o n d d e r i v a t i v e i s z e r o , t h e c o e f f i c i e n t s a r e a = d p Q = 3 - ) — — , ( 4 . 5 0 ) dp a n d 1 <?3/>0 1 ^ 1 4 X 1 4 (A),/,,/,) 3 ! ^ i 4 x u 6(a,)4 dp" N o w u s i n g a t h i r d - o r d e r t w o - d i m e n s i o n a l T a y l o r s e r i e s e x p a n d e d a b o u t t h e o p t i m u m q u a n t i z e r t h r e s h o l d s , lx, = ly = .31'4 a0, t o a p p r o x i m a t e t h e c o e f f i c i e n t s , y i e l d s a ~b0 +*,,(/,+/)+&, lj +b3(l2x + l2)+b4(l2xl+1xll)+b5(ll + Z 3 ) , ( 4 . 5 2 ) w h e r e t h e c o e f f i c i e n t s f o r a p p r o x i m a t i n g ax a n d <z 3 a r e al a 3 b0 . 0 1 1 9 9 5 6 5 5 0 9 9 . 5 8 9 0 5 7 6 6 7 2 6 x l O " 6 bi - . 0 4 9 4 0 6 0 4 3 0 4 3 1 - . 4 1 0 7 2 9 6 2 3 9 5 x l O " 5 b2 . 0 1 1 5 0 1 3 6 5 0 1 6 . 2 1 2 2 8 2 1 4 5 7 9 x l O ' 5 b3 . 1 9 9 0 7 4 8 6 8 1 4 . 1 2 3 7 4 3 9 9 3 3 3 x l O " 4 b4 . 1 3 1 4 4 6 2 8 2 0 9 - . 3 6 2 4 4 9 7 7 1 5 9 x l O ' 5 bi - . 1 7 2 5 1 9 5 9 4 0 5 - . 9 3 2 0 9 8 9 1 9 7 3 x l O 5 U s i n g t h e a b o v e i n v e r s i o n s c h e m e t h e l i n e a r i z e d r e s p o n s e o f t h e 1 4 - l e v e l c r o s s - c o r r e l a t o r h a s a r e l a t i v e e r r o r o f l e s s t h a n . 1 % w h e n T s r c i s l e s s t h a n . 3 T s y s m j n , a n d a r e l a t i v e e r r o r o f l e s s t h a n 1 % o v e r t h e f u l l r a n g e o f T s r c a n d T s y s . F i g u r e 4 . 1 0 s h o w s t h e l i n e a r i z e d r e s p o n s e . T h e c u r v e s w e r e g e n e r a t e d u s i n g t h e s a m e p r o c e d u r e u s e d t o c r e a t e F i g u r e 4 . 9 a n d u n d e r t h e s a m e c o n d i t i o n s u s e d t o c r e a t e F i g . 4 . 6 i n s e c t i o n 4 . 4 . 9 5 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 . 1 Normalized Source Temperature, Tsrc I Tsys min Figure 4.10: The response of the linearized 14-level cross-correlator, R i 4 x y , versus the normalized source temperature, T s r c / T s y s m i n . , using a 3-dimensional Taylor series inversion. The nearly identical curves are: (la) the response where T s y s = T s y s m i n , and (2a) the response where T s y s = T s y s m i n 130%. For both curves the approximation was expanded about p = 0, and /x = ly =.374 a T s y s m i n . Curves (lb) and (2b) show the relative error, compared to an ideal correlator, for curves (la) and (2a) respectively. 9 6 5. Tests and Results T h e g o a l o f t h i s c h a p t e r i s t o s h o w t h a t t h e h y b r i d - l e v e l c o r r e l a t o r b e h a v e s a s d e s i g n e d a n d t o d e m o n s t r a t e t h a t i t i s a r e l i a b l e a s t r o n o m i c a l i n s t r u m e n t . A s e c o n d a r y o b j e c t i v e i s t o o u t l i n e t h e t e s t s t h a t w e r e u s e d t o c h e c k t h e k e y s y s t e m c o m p o n e n t s d e s i g n e d . T h e s e t e s t s w e r e u s e f u l d u r i n g d e v e l o p m e n t a n d w e r e u s e d d u r i n g p r o d u c t i o n t o c h e c k t h e c o n d i t i o n o f p a r t s i n t h e s y s t e m . T h e c h a p t e r i s o r g a n i z e d b o t t o m u p , t h a t i s i n t h e i n i t i a l s e c t i o n s c o m p o n e n t a n d s u b - s y s t e m t e s t s a r e d e s c r i b e d f o l l o w e d b y h i g h e r l e v e l s y s t e m t e s t s . 5.1 DRAO Correlator IC Tests I t i s q u i t e i m p o r t a n t t o r e c e i v e a h i g h I C y i e l d f r o m t h e v e n d o r , a s i t i s c o s t l y t o f i x a n I C d e s i g n . e r r o r o n c e d e v i c e s h a v e b e e n m a d e , o r t o r e p a i r c i r c u i t b o a r d s t h a t h a v e b a d I C s i n s t a l l e d . T o e n s u r e t h e I C y i e l d w a s h i g h s e v e r a l s t a n d a r d t e s t s w e r e d e v e l o p e d t o : a ) v e r i f y t h e d e s i g n , u s i n g s i m u l a t i o n t o o l s , a n d b ) t o d i s c r i m i n a t e g o o d I C s f r o m t h e b a d o n e s d u r i n g m a n u f a c t u r e . T h e t e s t s w e r e p e r f o r m e d o v e r t h e c o m m e r c i a l r a n g e 6 d u r i n g d e s i g n v e r i f i c a t i o n a n d a t w o r s t a n d b e s t c a s e v o l t a g e s a t 2 5 ° C d u r i n g m a n u f a c t u r e . T h e t e s t s c o n s i s t e d o f : • a n i n p u t s c a n t e s t , t h a t s h o w e d e a c h I C i n p u t w o u l d r e s p o n d c o r r e c t l y t o a l o w o r a h i g h i n p u t . • a b a s i c f u n c t i o n a l i t y t e s t , t h a t e x e r c i s e d t h e m u l t i p l i e r s o v e r e v e r y i n p u t c o m b i n a t i o n , a n d e x e r c i s e d i t s a c c u m u l a t o r s a n d r e a d - o u t c i r c u i t r y . 6 T h e c o m m e r c i a l r a n g e s p e c i f i e s t e m p e r a t u r e , v o l t a g e , a n d p r o c e s s f a c t o r ( P F ) a t t h r e e p o i n t s : 1) b e s t c a s e = { O ' C , 5 . 2 5 V , .5 P F } , 2 ) t y p i c a l c a s e = { 2 5 ' C , 5 . 0 0 V , 1 . 0 P F } , a n d 3 ) w o r s t c a s e = V , 1.5 P F } . { 5 0 ' C , 4 . 7 5 9 7 • a r a n d o m n o i s e t e s t , t h a t s h o w e d t h e I C w o r k e d w h e n c o r r e l a t i n g p s e u d o - r a n d o m G a u s s i a n - n o i s e i n p u t s ( w h e r e a = 1 , u , = 0 ) . A d d i t i o n a l l y , d u r i n g m a n u f a c t u r e t h e g a t e - a r r a y v e n d o r c h e c k e d t h a t t h e c u r r e n t i n t o e a c h i n p u t w a s w i t h i n s p e c i f i c a t i o n . A s w e l l , a s p e e d t e s t w a s u s e d t o c h e c k t h a t t h e p r o c e s s f a c t o r ( P F ) f o r e a c h I C w a s w i t h i n s p e c i f i c a t i o n ( i . e . , . 5 < P F < 1 . 5 ) . T h e p r o c e s s f a c t o r w a s m e a s u r e d i n d i r e c t l y b y m e a s u r i n g t h e p r o p a g a t i o n d e l a y ( a t 2 5 ° C a n d 5 V ) t h r o u g h a l a r g e n u m b e r o f g a t e s , o r l o n g p a t h , i n t h e I C . A l o n g p a t h i s n e e d e d t o m i n i m i z e t h e i n s t r u m e n t a l d e l a y u n c e r t a i n t y o f a b o u t ± 1 n s , a n d t o a v e r a g e t h e p r o c e s s f a c t o r ( w h i c h c a n v a r y ) o v e r t h e d i e . O n e o f t h e f o u r 2 3 - s t a g e r i p p l e - c o u n t e r s p r o v i d e d a c o n v e n i e n t c i r c u i t f o r t h i s t e s t , a s w h e n t h e r i p p l e - c o u n t e r o v e r f l o w s a o n e t o z e r o c h a n g e r i p p l e s t h r o u g h 2 3 t o g g l e f l i p - f l o p s . O n e c o u n t e r p r o v i d e s a l o n g d e l a y ( ~ 4 5 n s ) , a n d r e p r e s e n t s a l a r g e f r a c t i o n o f t h e g a t e s u s e d o n t h e d i e ( a b o u t 1 0 % ) . T h e e f f i c a c y o f t h e s e t e s t s w a s q u i t e g o o d . T h e c o r r e l a t o r I C w o r k e d o n t h e f i r s t d e s i g n c y c l e a n d o f t h e 8 , 0 0 0 I C s d e l i v e r e d , i n s e v e r a l b a t c h e s o v e r f o u r y e a r s a n d i n s e r v i c e s i n c e 1 9 9 5 , o n l y s i x h a v e f a i l e d . T h u s t h e t e s t s h e l p e d p r o d u c e a h i g h A S I C y i e l d o f 9 9 . 9 2 % . 5.2 Spectrometer Correlator Card Tests A c o m p a r i s o n t e s t a n d a d i f f e r e n t i a l t e s t w e r e u s e d t o v e r i f y t h e o p e r a t i o n a n d r e l i a b i l i t y , r e s p e c t i v e l y , o f t h e s p e c t r o m e t e r c o r r e l a t o r c a r d ( S C C ) . T h e c o m p a r i s o n t e s t i s c o n v e n i e n t f o r d e b u g g i n g a n S C C i n t h e l a b , a s i t r e q u i r e s a m i n i m u m n u m b e r o f s y s t e m c o m p o n e n t s t h a t c a n b e s e t - u p o n a w o r k b e n c h . T h e d i f f e r e n t i a l t e s t i s u s e f u l f o r t e s t i n g a l l t h e S C C s i n a s y s t e m i n p a r a l l e l . I t a l s o i n v o l v e s e a c h c o m p o n e n t i n t h e s y s t e m a n d s o d o u b l e s a s a s y s t e m l e v e l t e s t . T h e c o m p a r i s o n t e s t i n v o l v e d i n j e c t i n g a k n o w n s e t o f i n p u t s ( t e s t v e c t o r s ) a n d c h e c k i n g t h a t t h e S C C o u t p u t i s t h e s a m e a s t h e e x p e c t e d o u t p u t . B e c a u s e t h e e x p e c t e d o u t p u t i s k n o w n i t c a n t e s t t h e f u n c t i o n a l i t y o f a n S C C . A s a n a d d e d c h e c k t h e S C C w a s n o m i n a l l y c o n f i g u r e d a s a n a u t o -9 8 c o r r e l a t o r w i t h r e d u n d a n t l a g s ( i . e . , i n t h e S C C c r o s s - c o r r e l a t o r c o n f i g u r a t i o n w i t h i d e n t i c a l x a n d y i n p u t s ) , s o t h a t t h e p o s i t i v e a n d n e g a t i v e l a g s c o u l d b e c h e c k e d f o r e q u a l i t y . T o p e r f o r m t h i s t e s t t h e R A M o n t h e c o r r e l a t o r i n t e r f a c e c a r d ( C I C ) w a s l o a d e d w i t h t e s t v e c t o r s t h a t w e r e t h e n c o r r e l a t e d b y t h e S C C . T h e s p e c t r o m e t e r c o n t r o l m i c r o p r o c e s s o r w a s u s e d t o g e n e r a t e t h e t e s t v e c t o r s , c o n t r o l t h e t e s t s e q u e n c e , a n d s a v e t h e S C C o u t p u t . S i n c e a n S C C c a n p l u g d i r e c t l y i n t o t h e C I C , w i t h o u t a b a c k p l a n e , o n e c a n p e r f o r m t h i s t e s t o n a b e n c h w i t h a C I C , S C C , c l o c k g e n e r a t o r , m i c r o p r o c e s s o r a n d a p o w e r s u p p l y . T h e c o m p a r i s o n t e s t w a s p e r f o r m e d f o r m a n y h o u r s o n s e v e r a l S C C s a t t h e b e s t , t y p i c a l , a n d w o r s t c a s e c o m m e r c i a l v o l t a g e s . A s w e l l , a n u m b e r o f t e s t v e c t o r p a t t e r n s w e r e t r i e d ( e . g . , c o n s t a n t c o m b i n a t i o n s o f - 1 , 0 , 1 a n d u n i f o r m p s e u d o r a n d o m - n o i s e ) , a n d i n a l l c a s e s t h e S C C o u t p u t a g r e e d w i t h t h e e x p e c t e d o u t p u t w i t h n o e r r o r . F i g u r e 5 . 1 s h o w s t h e S C C o u t p u t r e s p o n s e t o a s e t o f u n i f o r m p s e u d o r a n d o m - n o i s e i n p u t s . A s e x p e c t e d t h e a u t o - c o r r e l a t i o n f u n c t i o n o u t p u t h a s e v e n s y m m e t r y a s t h e r e i s n o d i f f e r e n c e b e t w e e n t h e p o s i t i v e a n d n e g a t i v e l a g s . A s e c o n d t e s t p e r f o r m e d o n t h e S C C w a s t h e d i f f e r e n t i a l t e s t . I n t h i s t e s t a l l t w e n t y S C C s i n a r a c k ( i . e . , s y s t e m ) w e r e c o n f i g u r e d t h e s a m e a n d t h e S C C o u t p u t s w e r e c h e c k e d f o r d i f f e r e n c e s . T h i s t e s t c a n a l s o t e s t t h e f u n c t i o n a l i t y o f S C C s i f a k n o w n g o o d ( c o m p a r i s o n ) b o a r d i s i n c l u d e d i n t h e s e t o f S C C s u n d e r t e s t . T h e p r i m a r y a d v a n t a g e o f t h i s t e s t i s t h a t a l l S C C s i n a r a c k c a n b e t e s t e d i n s i t u a n d i n p a r a l l e l , m a k i n g i t e a s y t o t e s t a l a r g e n u m b e r o f S C C s . A n o t h e r a d v a n t a g e i s t h a t i t i s a r e l a t i v e t e s t , s o t h a t t e s t v e c t o r s a n d t h e e x p e c t e d o u t p u t r e s p o n s e d o n o t h a v e t o b e c o m p u t e d . T h i s m a k e s i t e a s y t o t e s t a s e t o f S C C s w i t h d i f f e r e n t i n p u t s i g n a l s a n d i n d i f f e r e n t c o n f i g u r a t i o n s . T h e d i f f e r e n t i a l t e s t w a s u s e d t o c o n f i r m t h e r e l i a b i l i t y o f t e n p r o t o t y p e - S C C s w h e n s u b j e c t e d t o d i f f e r e n t i n p u t s t i m u l i a t s e v e r a l s a m p l e r a t e s . T h e t e s t s w e r e c o n d u c t e d o v e r s e v e r a l h o u r s u s i n g t h e f o l l o w i n g i n p u t s t i m u l i : 1 . i n d e p e n d e n t G a u s s i a n n o i s e . 2 . c o r r e l a t e d G a u s s i a n n o i s e . 3 . i n d e p e n d e n t G a u s s i a n n o i s e p l u s a c o m m o n m o d e s i n e w a v e . re O Q ' 3 s re 43 3 S ( V «»*. a re S- S &0 r. re 3 a-re •s re re 3 3 re re Q "1 re a. 3 C 3 I C-5 re c &• 2 I. 3 re re re I ' II o t o I re re r? a. S. re re* Spectrometer Correlator Card Output — 1 —I - I ro ro ro ro ro u CO u O l ~1 CO — i co cn ~j CD L u cn o o o o o o o o O o o o o o o o o o o o o o o o o o o> CD VO VO 1 0 0 T h i s t e s t w a s a l s o u s e d t o c h e c k t h e r e l i a b i l i t y o f t h e p r o d u c t i o n s p e c t r o m e t e r s y s t e m , o n t h e D R A O S T , w h i c h c o n s i s t e d o f f o r t y S C C s i n t w o r a c k s . D u r i n g t h i s t e s t t h e A D C s w e r e s t i m u l a t e d w i t h i n d e p e n d e n t r a n d o m n o i s e , f r o m t h e D R A O - S T R F - s y s t e m , a n d t h e S C C o u t p u t s w e r e c h e c k e d f o r d i f f e r e n c e s a f t e r e v e r y 5 . 6 2 5 - s e c o n d i n t e g r a t i o n . T h i s t e s t r a n f o r 5 d a y s , a t a s a m p l e r a t e o f 1 6 M H z , a n d n o d i f f e r e n c e s w e r e d e t e c t e d i n a n y o f 2 0 , 4 8 0 l a g s i n t h e 4 0 S C C s t e s t e d . T h e s i g n i f i c a n c e o f t h i s r e s u l t i s t h a t i t d e m o n s t r a t e s t h a t t h e h y b r i d - l e v e l c o r r e l a t o r i s a r e l i a b l e i n s t r u m e n t a s i t c a n p e r f o r m m o r e t h a n 1 0 1 7 m u l t i p l y - a c c u m u l a t e ( c o r r e l a t i o n ) o p e r a t i o n s w i t h o u t e r r o r . 5.3 Linearity Tests L i n e a r i t y t e s t s w e r e c o n d u c t e d t o c h e c k t h e e f f e c t i v e n e s s o f t h e l i n e a r i t y c o r r e c t i o n f o r m u l a e d e v e l o p e d , f o r t h e 1 4 - l e v e l c r o s s - a n d a u t o - c o r r e l a t o r s , i n C h a p t e r 4 . T w o t e s t s w e r e c o n d u c t e d : o n e t o c h e c k t h e c o r r e c t e d l i n e a r i t y o f t h e 1 4 - l e v e l c r o s s a n d a u t o - c o r r e l a t o r u s e d i n D R A O - S T 1 4 2 0 M H z c o n t i n u u m s y s t e m , a n d t h e o t h e r t o c h e c k t h e c o r r e c t e d l i n e a r i t y o f t h e 1 4 - l e v e l a u t o -c o r r e l a t o r u s e d i n t h e h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r . R e c a l l t h a t t h e d i f f e r e n c e b e t w e e n t h e 1 4 - l e v e l c o r r e l a t o r s i n t h e s e t w o s y s t e m s i s t h a t i n t h e f o r m e r s y s t e m t h e A D C t h r e s h o l d s a r e s e t t o t h e 1 4 - l e v e l o p t i m u m ( i . e . , a t / o = . 3 7 4 a o ) , w h i l e i n t h e l a t t e r t h e y a r e s e t t o t h e t h r e e - l e v e l o p t i m u m ( i . e . , a t / o = . 6 1 2 a o ) . 5.3.1 The ^ 21-cm Continuum System Linearity Tests and Results I n t h i s t e s t t h e l i n e a r i z e d r e s p o n s e o f t h e 1 4 - l e v e l a u t o - c o r r e l a t o r a n d c r o s s - c o r r e l a t o r , o n t h e D R A O - S T 1 4 2 0 M H z C o n t i n u u m S y s t e m , w a s c h e c k e d . T h e t e s t w a s p e r f o r m e d b y a d d i n g c o r r e l a t e d n o i s e , Pc, t o t h e u n c o r r e c t e d n o i s e , Pv, i n t o t w o A D C i n p u t s , x a n d y . T h e a u t o - a n d c r o s s - c o r r e l a t o r r e s p o n s e w a s t h e n l i n e a r i z e d a n d p l o t t e d a s a f u n c t i o n o f t h e t o t a l a n d c o r r e l a t e d i n p u t - p o w e r , r e s p e c t i v e l y . 1 0 1 T o c h e c k t h e l i n e a r i t y e r r o r t h e c o r r e l a t e d n o i s e w a s v a r i e d i n 1 0 % i n c r e m e n t s , f r o m z e r o t o a l e v e l e q u a l t o t h e i n i t i a l ( u n c o r r e l a t e d ) i n p u t p o w e r . T h i s c a u s e d t h e t o t a l p o w e r t o c h a n g e b y 3 d B , a n d t h e c o r r e l a t i o n c o e f f i c i e n t , p , t o c h a n g e f r o m 0 t o . 5 . A s w e l l , s i n c e t h e 1 4 - l e v e l c r o s s - c o r r e l a t o r r e s p o n s e i s a l s o a f u n c t i o n o f t h e u n c o r r e l a t e d i n p u t p o w e r , t w o t r i a l s w e r e c o n d u c t e d : o n e w h e r e t h e i n i t i a l i n p u t p o w e r w a s o p t i m u m ( i . e . , Tsys a t 0"02), a n d o n e w h e r e i t w a s 3 0 % h i g h e r . T o g e t h e r t h e t w o t r i a l s s p a n n e d t h e i n p u t p o w e r r a n g e o f t h e l i n e a r i z a t i o n f u n c t i o n s , E q s . 4 . 3 7 a n d 4 . 4 9 , . d e v e l o p e d i n c h a p t e r 4 . A t t h e e n d o f t h e t e s t t h e a u t o - a n d c r o s s - c o r r e l a t o r d a t a w e r e t h e n l i n e a r i z e d , u s i n g E q s . 4 . 3 7 a n d 4 . 4 9 , a n d t h e r e s u l t s w e r e p l o t t e d i n f i g u r e s 5 . 2 t h r o u g h 5 . 5 . F i g u r e s 5 . 2 a n d 5 . 3 s h o w t h a t E q . 4 . 3 7 i s e f f e c t i v e i n l i n e a r i z i n g t h e r e s p o n s e o f a 1 4 - l e v e l a u t o -c o r r e l a t o r . T h e m e a s u r e d r e l a t i v e l i n e a r i t y e r r o r w a s l e s s t h a n . 6 % a n d t y p i c a l l y l e s s t h a n . 2 % f o r b o t h a u t o - c o r r e l a t o r s t e s t e d . A s w e l l , f i g u r e s 5 . 4 a n d 5 . 5 s h o w t h a t E q . 4 . 4 9 i s e f f e c t i v e i n l i n e a r i z i n g t h e r e s p o n s e o f a 1 4 - l e v e l c r o s s - c o r r e l a t o r . T h e m e a s u r e d r e l a t i v e l i n e a r i t y e r r o r w a s l e s s t h a n . 8 5 % a n d t y p i c a l l y l e s s t h a n . 2 % . N o t i c e t h a t t h e c r o s s - c o r r e l a t o r r e s p o n s e i s a l s o n o t o n l y a f u n c t i o n o f t h e c o r r e l a t e d i n p u t p o w e r , b u t a l s o a f u n c t i o n o f t h e u n c o r r e l a t e d i n p u t p o w e r ; a n d t h a t t h e r e l a t i v e l i n e a r i t y e r r o r i n c r e a s e s a s a f u n c t i o n o f t h e i n p u t p o w e r . T h e s e r e s u l t s a r e c o n s i s t e n t w i t h t h e t h e o r e t i c a l l i n e a r i t y e r r o r o f E q . 4 . 4 9 d i s c u s s e d i n s e c t i o n 4 . 5 . 1 a n d s h o w n i n F i g 4 . 7 . H o w e v e r , t h e m e a s u r e d l i n e a r i t y e r r o r s a r e s o m e w h a t h i g h e r t h a n e x p e c t e d . T h e r e a s o n f o r t h i s d i s c r e p a n c y i s l i k e l y d u e t o a s y s t e m a t i c e r r o r , a s t h e r e s i d u a l e r r o r s a r e n o t d o m i n a t e d b y r a n d o m ( m e a s u r e m e n t ) n o i s e . I 1 0 2 2.3 2.1 + B 1-9 + c o a. 1.7 + O o o O ~ 1.5 < 0 > 1.3 1.1 + 0.9 • (1a) o (1b) X (2a) X (2b) 0.9 1.1 1.3 1.5 1.7 1.9 Normalized Total Input Power (P u + Pc) 2.1 2.3 Figure 5.2: The measured nominal and linearized input-output response of a 14-level auto-correlator, R D x x (la = .374aj. The lower curve is a second order fit to the measurements of R DXX> while the upper curve is a linear fit to the linearized values ofRDxx. The points (la) and (lb) are the measured and linearized values of RDxx at 100% Tsys, while (2a) and (2b) are the values at 130% Tsys. 1 0 3 Figure 5.3: The measured relative linearization error of a 14-level auto-correlator (lo = .374 aj. The points shown are the relative deviations from a linear fit. The curves (la) and (lb) are the linearization errors ofRDxx and RDyy at 100% Tsys, while (2a) and (2b) are the errors at 130% Tsys. 1 0 4 0 0.1 0.2 0.3 0.4 0.5 0.6 . 0.7 0.8 0.9 1 Normalized Correlated Input Power, P„ Figure 5.4: The measured nominal and linearized input-output response of a 14-level cross-correlator, R Dxy (l0 = .374 aj. Curves (la) and (lb) are the nominal and linearized response at 100% Tsyx, while curves (2a) and (2b) are the nominal and linearized response at 130% Tsys. Figure 5.5: The measured relative linearization error of a 14-level cross-correlator, RDxy ( lo = .374 a0). Curve (1) is the error at 100% Tsys, while curve (2) is the error at 130% Tsy,. 1 0 6 O n e c a n s e e e v i d e n c e o f a s y s t e m a t i c e r r o r b y e x a m i n i n g f i g u r e 5 . 3 , a s t h e a u t o - c o r r e l a t i o n l i n e a r i t y e r r o r s a r e c o r r e l a t e d b e t w e e n t h e t w o a u t o - c o r r e l a t o r s , Rxx a n d / ? y r S u c h a n e r r o r w o u l d o c c u r i f , s a y , a s m a l l v a r i a t i o n i n t h e c o r r e l a t e d i n p u t p o w e r w a s n o t m e a s u r e d d u r i n g t h e t e s t . I n t h i s c a s e b o t h a u t o - c o r r e l a t o r s w o u l d c h a n g e , b u t t h e i n p u t p o w e r w o u l d n o t , t h u s c a u s i n g a s m a l l s y s t e m a t i c e r r o r . T h i s e f f e c t i s p r o b a b l y d u e t o t h e n a t u r e o f t h e i n p u t p o w e r s o u r c e s a n d t h e w a y t h e y w e r e m e a s u r e d . D u e t o a l a c k o f e q u i p m e n t , i t w a s n o t p o s s i b l e t o c o n t i n u o u s l y m e a s u r e t h e i n p u t p o w e r d u r i n g t h e t e s t . I n s t e a d t h e i n p u t p o w e r w a s m e a s u r e d b e f o r e a n d a f t e r t h e t e s t w a s p e r f o r m e d , a n d i t w a s a s s u m e d t h a t t h e i n p u t n o i s e p o w e r w o u l d e i t h e r b e s t a t i o n a r y o r t h a t i t w o u l d c h a n g e l i n e a r l y d u r i n g t h e t e s t . A l s o t h e i n p u t p o w e r w a s d e r i v e d f r o m t h r e e ( i n d e p e n d e n t ) o u t p u t s f r o m t h e D R A O - S T R F - s y s t e m , a n d i t i s q u i t e p o s s i b l e f o r t h e s e o u t p u t s t o d r i f t b y . 4 % o v e r a n h o u r ( d u e t o s y s t e m t e m p e r a t u r e a n d g a i n v a r i a t i o n s ) [ W y s l o u z i l 9 7 ] . I f t h e i n p u t n o i s e v a r i e d u n e x p e c t e d l y d u r i n g t h e t e s t i t w o u l d c a u s e a n e r r o r o f t h i s m a g n i t u d e . 5.3.2 26M Telescope Linearity Test and Results I n t h i s t e s t t h e l i n e a r i z e d r e s p o n s e o f t h e 1 4 - l e v e l a u t o - c o r r e l a t o r u s e d i n h y b r i d - l e v e l c o r r e l a t o r w a s c h e c k e d . T h e t e s t w a s p e r f o r m e d i n t h e s a m e w a y a s t h e t e s t i n t h e p r e v i o u s s e c t i o n , e x c e p t t h a t o n e c o r r e l a t o r w a s t e s t e d a n d t h e i n p u t p o w e r w a s m e a s u r e d c o n t i n u o u s l y d u r i n g t h e t e s t . A s w e l l , t o c h e c k t h e l i n e a r i t y o v e r a b r o a d r a n g e t h e i n p u t p o w e r w a s v a r i e d f r o m - 3 d B t o + 3 d B a b o u t t h e o p t i m u m i n p u t l e v e l , ao2, i n ~ 1 d B s t e p s . T h e 1 4 - l e v e l a u t o - c o r r e l a t o r r e s p o n s e w a s t h e n l i n e a r i z e d , u s i n g E q 4 . 3 7 , a n d p l o t t e d a s a f u n c t i o n o f t h e n o r m a l i z e d i n p u t p o w e r . F i g u r e s 5 . 6 a n d 5 . 7 s h o w t h a t E q . 4 . 3 7 l i n e a r i z e d t h e r e s p o n s e o f t h e 1 4 - l e v e l a u t o - c o r r e l a t o r q u i t e w e l l . T h e a b s o l u t e v a l u e o f t h e m e a s u r e d r e l a t i v e - l i n e a r i t y e r r o r w a s t y p i c a l l y l e s s t h a n 2 x 1 0 " 4 a n d w i t h i n 1 0 o f t h e e x p e c t e d m e a s u r e m e n t e r r o r . 107 Figure 5.6: The measured nominal and linearized input-output response of a 14-level auto-correlator (l„ = .612 a j . The curves are: (1) the nominal response, RDxx, and (2) the linearized response, R^. 1 0 8 Figure 5.7: The measured relative linearization error of a 14-level cross-correlator ( l„ curves are: (1) the theoretical error, and (2) the measure error = .612 a„). The 1 0 9 5.4 System Tests and Results S e v e r a l s y s t e m - l e v e l t e s t s w e r e c o n d u c t e d t o c h e c k t h e o p e r a t i o n a n d r e l i a b i l i t y o f t h e h y b r i d - l e v e l c o r r e l a t o r - s y s t e m h a r d w a r e a n d s o f t w a r e . T h e s e t e s t s w e r e r u n o n v a r i o u s s y s t e m c o n f i g u r a t i o n s a n d i n c l u d e d : 1 . A b a s i c s y s t e m t e s t , t o c h e c k t h a t a r e a s o n a b l e b a n d s h a p e w a s p r o d u c e d a t a l l s a m p l e r a t e s w i t h G a u s s i a n n o i s e i n p u t s . T h e c r o s s - c o r r e l a t o r c o n f i g u r a t i o n w a s c h e c k e d b y i n j e c t 1 0 % c r o s s - t a l k ( i . e . , c o r r e l a t e d n o i s e ) i n t o e a c h i n p u t , w h i l e t h e a u t o - c o r r e l a t o r c o n f i g u r a t i o n w a s c h e c k e d b y a u t o - c o r r e l a t i n g b r o a d - b a n d n o i s e . 2 . A s i n e - w a v e t e s t t o c h e c k t h a t a c o s i n e s h a p e w a s o b s e r v e d i n t h e l a g d o m a i n a n d t h a t t h e a p p r o p r i a t e s i n e w a v e p o w e r w a s d e t e c t e d i n t h e c o r r e c t f r e q u e n c y c h a n n e l a f t e r t r a n s f o r m a t i o n t o t h e f r e q u e n c y d o m a i n . 3 . A n i n d e p e n d e n t r a n d o m - n o i s e t e s t t o c h e c k f o r s p u r i o u s c r o s s - c o r r e l a t i o n o v e r 1 2 h o u r s . I n t h i s t e s t i n d e p e n d e n t r a n d o m n o i s e w a s i n j e c t e d i n t o t w o i n p u t s a n d a c h e c k w a s m a d e t h a t t h e a v e r a g e n o i s e i n e a c h c h a n n e l d e c r e a s e d b y V/V i n t e g r a t i o n s a n d t h a t n o s i g n i f i c a n t c o r r e l a t i o n w a s d e t e c t e d . 4 . A c r o s s - t a l k r e j e c t i o n t e s t t o c h e c k t h a t t h e c r o s s - c o r r e l a t o r w o u l d p r o d u c e z e r o c o r r e l a t i o n w h e n t h e p h a s e o f a c o r r e l a t e d G a u s s i a n i n p u t s i g n a l w a s a l t e r n a t e d b y 1 8 0 ° w i t h a 5 0 % d u t y c y c l e . 5 . A n a s t r o n o m i c a l s o u r c e t e s t t o c h e c k t h a t t h e s y s t e m c o u l d p r o d u c e t h e s p e c t r u m o f a w e l l - k n o w n a s t r o n o m i c a l s o u r c e . T h e n e w c o r r e l a t i o n - s p e c t r o m e t e r p a s s e d t h e a b o v e t e s t s , a s w e l l a s t h o s e p e r f o r m e d i n d e p e n d e n t l y b y D e w d n e y ( 1 9 9 4 ) . T h e f o l l o w i n g f i g u r e s , 5 . 8 t h r o u g h 5 . 9 , a r e s p e c t r a m e a s u r e d b y t h e h y b r i d - l e v e l c o r r e l a t i o n -s p e c t r o m e t e r . T h e s e f i g u r e s d e m o n s t r a t e t h a t t h e h y b r i d - l e v e l s p e c t r o m e t e r i s a w o r k i n g s y s t e m . 1 1 0 T h e f i r s t t w o a r e t h e w e l l - k n o w n r a d i o - s o u r c e s 3 C 1 4 7 a n d S 7 a n d w e r e o b s e r v e d u s i n g t h e D R A O S T a n d 2 6 M , r e s p e c t i v e l y . T h e l a s t f i g u r e i s a n O H s p e c t r u m o f t h e r e c e n t c o m e t H a l e - B o p p t h a t w a s o b s e r v e d u s i n g t h e D R A O 2 6 m r a d i o - t e l e s c o p e [ G A L T 9 7 ] . 1 1 4 6. Summary and Conclusions T h i s t h e s i s d e s c r i b e s t h e d e s i g n o f a h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r f o r t h e D R A O s y n t h e s i s t e l e s c o p e ( S T ) a n d t h e 2 6 m e t e r s i n g l e - a n t e n n a t e l e s c o p e ( 2 6 M ) , w h i c h r e p l a c e s t h e o l d e r c o r r e l a t i o n - s p e c t r o m e t e r s , b u i l t c i r c a 1 9 7 5 . T h e n e w s y s t e m , l i k e i t s p r e d e c e s s o r , i s u s e d b y a s t r o n o m e r s t o m e a s u r e t h e p o w e r s p e c t r u m o f c e l e s t i a l r a d i o - s o u r c e s : M e a s u r i n g t h e p o w e r -s p e c t r u m o f a s o u r c e i s i m p o r t a n t a s , v i a t h e D o p p l e r s h i f t f o r m u l a , i t p r o v i d e s i n f o r m a t i o n a b o u t i t s v e l o c i t y s t r u c t u r e . O n t h e S T t h i s c o r r e l a t o r i s u s e d t o s y n t h e s i s a 6 0 0 m a p e r t u r e b y c r o s s -c o r r e l a t i n g p a i r s o f c o h e r e n t a n t e n n a s i g n a l s . E a c h 5 1 2 - l a g c r o s s - c o r r e l a t o r m e a s u r e s t h e c r o s s -p o w e r s p e c t r u m , v i a t h e F o u r i e r t r a n s f o r m , o f a c o m p l e x s p a t i a l f r e q u e n c y c o m p o n e n t i n t h e b r i g h t n e s s d i s t r i b u t i o n o f a s o u r c e . O n t h e 2 6 M t h e s y s t e m a u t o - c o r r e l a t e s t w o p o l a r i z e d o u t p u t s o f a 2 6 m p a r a b o l o i d . T h e t h e s i s w o r k c o v e r e d m a n y a s p e c t s o f e n g i n e e r i n g a n d c o n s i s t e d o f : r e v i e w i n g t h e l i t e r a t u r e ; d e s i g n i n g t h e s y s t e m ; a n d , s u p e r v i s i n g a n d p a r t i c i p a t i n g i n t h e d e t a i l e d d e s i g n , c o n s t r u c t i o n a n d t e s t i n g o f t h e s y s t e m . W h i l e t h e n e w s y s t e m i s f u n c t i o n a l l y s i m i l a r t o i t s p r e d e c e s s o r , t h e n e w s y s t e m h a s t w i c e a s m a n y l a g s p e r c o r r e l a t o r , f i v e t i m e s m o r e c o r r e l a t o r s p e r r a c k , a n d c o s t s ~ $ 3 p e r l a g v e r s u s ~ $ 1 0 p e r l a g t o c o n s t r u c t . T h e n e w s y s t e m i s a l s o m o r e f l e x i b l e , a s t h e c o r r e l a t o r c a n b e c o n f i g u r e d a s a n a u t o - o r c r o s s - c o r r e l a t o r , a n d u p t o t e n c o r r e l a t o r s c a n b e d a i s y - c h a i n e d t o f o r m a s i n g l e 5 1 2 0 l a g c o r r e l a t o r . A n o t h e r a d v a n t a g e i s t h a t t h e s y s t e m c a n e f f i c i e n t l y c o r r e l a t e s a m p l e s t a k e n a t t h e N y q u i s t r a t e , o r a t t w i c e t h e N y q u i s t r a t e . T h u s m a x i m u m s e n s i t i v i t y c a n b e t r a d e d f o r m a x i m u m p e r f o r m a n c e w i t h n o l o s s i n s p e c t r a l r e s o l u t i o n . T h e m a i n c o m p o n e n t t h a t g i v e s t h e c o r r e l a t o r s t h e a d v a n t a g e s d i s c u s s e d a b o v e i s a f o u r - l a g c o r r e l a t o r C M O S g a t e - a r r a y I C t h a t w a s d e s i g n e d f o r t h i s p r o j e c t . T h i s a p p l i c a t i o n s p e c i f i c I C , o r A S I C , h a s s e v e r a l i n n o v a t i v e a r c h i t e c t u r a l f e a t u r e s w h i c h a r e n o t p r e s e n t ( t o t h e a u t h o r ' s 1 1 5 k n o w l e d g e ) i n o t h e r c o r r e l a t o r s . F i r s t , u n l i k e o t h e r a r c h i t e c t u r e s , t h i s a r c h i t e c t u r e h a s i n h e r e n t l y s u p e r i o r s a m p l e - r a t e p e r f o r m a n c e , a s e a c h d e l a y p i p e l i n e o u t p u t h a s o n l y o n e m u l t i p l i e r -l o a d — r e g a r d l e s s o f t h e c o r r e l a t o r s a m p l e - r a t e c o n f i g u r a t i o n ( i . e . , i n e i t h e r N y q u i s t o r d o u b l e -N y q u i s t s a m p l i n g ) . S e c o n d , t h e a r c h i t e c t u r e i s f l e x i b l e a n d e f f i c i e n t , a s i t c a n c o r r e l a t e N y q u i s t o r d o u b l e - N y q u i s t s a m p l e d d a t a w i t h f e w e r g a t e s t h a n o t h e r a r c h i t e c t u r e s . T h i r d , t h e s e r i a l r e a d - o u t a n d m o s t s i g n i f i c a n t b i t s a m p l i n g s c h e m e d e v e l o p e d r e q u i r e s f e w e r o u t p u t p i n s p e r I C a n d u t i l i z e s t h e d a t a t r a n s f e r b a n d w i d t h m o r e e f f i c i e n t l y t h a n o t h e r a r c h i t e c t u r e s . L a s t , t h e a r c h i t e c t u r e i s i n h e r e n t l y t w i c e a s f a s t , a n d r e q u i r e s n o c l o c k a d j u s t m e n t s , c o m p a r e d t o i t s p r e d e c e s s o r — - a s , u n l i k e i t s p r e d e c e s s o r , i t u s e s a p a r a l l e l m u l t i p l y a n d a d d c i r c u i t i n t h e f i r s t a c c u m u l a t o r s t a g e a n d s o i t d o e s n o t r e q u i r e a n a s y n c h r o n o u s c l o c k d o u b l i n g c i r c u i t w h i c h c a n b e d i f f i c u l t t o a d j u s t . T o t h e a u t h o r ' s k n o w l e d g e , t h e h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r i s n e w i n t h a t i t u s e s a 1 4 - l e v e l a u t o - c o r r e l a t o r t o m e a s u r e t h e t o t a l p o w e r o f e a c h i n p u t a n d r e - q u a n t i z e s t h e 1 4 - l e v e l t o t h r e e - l e v e l s . A 1 4 - l e v e l c o r r e l a t o r h a s t h e a d v a n t a g e t h a t i t i s m o r e l i n e a r a n d m o r e s e n s i t i v e t h a n a t h r e e - l e v e l c o r r e l a t o r . T h e s e a s p e c t s w e r e i n v e s t i g a t e d f o r b o t h t h e 1 4 - l e v e l a u t o - a n d c r o s s - c o r r e l a t o r a n d i n v e r s i o n f o r m u l a e f o r l i n e a r i z i n g t h e 1 4 - l e v e l c o r r e l a t o r w e r e d e v e l o p e d . F u r t h e r , a m o r e a c c u r a t e a n d e f f i c i e n t m e t h o d f o r c o m p u t i n g t h e s e n s i t i v i t y o f a c o r r e l a t o r w a s f o u n d . A n u m b e r o f t e s t s w e r e p e r f o r m e d t o v e r i f y t h e o p e r a t i o n o f t h e n e w s y s t e m a n d c h e c k t h e p e r f o r m a n c e o f t h e l i n e a r i z a t i o n f o r m u l a e d e v e l o p e d . T h e s e t e s t s d e m o n s t r a t e t h a t t h e n e w s y s t e m f u n c t i o n s c o r r e c t l y a n d t h a t t h e l i n e a r i z a t i o n i n v e r s i o n f u n c t i o n s w o r k q u i t e w e l l . T h e h y b r i d - l e v e l c o r r e l a t i o n - s p e c t r o m e t e r h a s b e e n i n f u l l - t i m e u s e o n t h e D R A O S T s i n c e M a y 1 9 9 6 a n d s i n c e J a n u a r y 1 9 9 7 o n t h e 2 6 M t e l e s c o p e . A l l i n d i c a t i o n s a r e t h e s e s y s t e m s a r e f u n c t i o n i n g r e l i a b l y a n d m e e t i n g a s t r o n o m i c a l e x p e c t a t i o n s . S o m e i n t e r e s t i n t h e r e s u l t s o f t h i s t h e s i s h a s b e e n s h o w n b y t h e a s t r o n o m y a n d e n g i n e e r i n g c o m m u n i t y . A d e s c r i p t i o n o f t h e s y s t e m w a s g i v e n a t t h e C A S C A c o n f e r e n c e i n J u l y 1 9 9 5 1 1 6 [ H o v e y 9 5 ] a n d a s u m m a r y o f t h e a n a l y t i c a l r e s u l t s i n c h a p t e r 4 w e r e p r e s e n t e d a t t h e U R S I c o n f e r e n c e i n J u l y 1 9 9 7 [ H o v e y 9 7 ] . A s w e l l , a s u m m a r y o f t h e r e s u l t s o f c h a p t e r 3 ( c o n c e r n i n g c o r r e l a t o r I C a r c h i t e c t u r e ) w a s p r e s e n t e d a t t h e U R S I c o n f e r e n c e i n J a n u a r y 1 9 9 8 [ H o v e y 9 8 ] . 6.1 Areas Requiring Further Work A n u m b e r o f a r e a s c o n c e r n i n g t h e 1 4 - l e v e l c o r r e l a t o r c o u l d b e i n v e s t i g a t e d f u r t h e r . O n e p o t e n t i a l p r o b l e m w i t h t h e c a l c u l a t i o n s m a d e , c o n c e r n i n g t h e 1 4 - l e v e l c o r r e l a t o r , i s t h a t t h e y a r e b a s e d o n a n i d e a l 1 4 - l e v e l A D C w h o s e t h r e s h o l d s a r e l i n e a r l y s e p a r a t e d . R e a l A D C s , h o w e v e r , h a v e s m a l l s a m p l i n g a n d t h r e s h o l d e r r o r s w h i c h c a n a f f e c t t h e s e c a l c u l a t i o n s . F u r t h e r w o r k i s r e q u i r e d t o i n v e s t i g a t e t h e s e n s i t i v i t y o f t h e s e c a l c u l a t i o n s t o s m a l l n o n - i d e a l A D C e r r o r s . W h i l e t h e l i n e a r i z a t i o n f o r m u l a e d e v e l o p e d w o r k e d w e l l , e r r o r b o u n d s o n t h e s e f o r m u l a e w e r e n o t c o m p u t e d . T h i s c a n b e d o n e b y c o m p u t i n g h i g h o r d e r d e r i v a t i v e s o f t h e i n v e r s i o n a p p r o x i m a t i o n s g i v e n i n c h a p t e r 5 . L a s t l y , t h e i n v e r s i o n f o r m u l a d e v e l o p e d f o r t h e 1 4 - l e v e l c r o s s - c o r r e l a t o r u s e d a t w o - d i m e n s i o n a l T a y l o r s e r i e s a p p r o x i m a t i o n w h o s e c o e f f i c i e n t s w e r e a l s o a p p r o x i m a t e d . W h i l e t h e p e r f o r m a n c e o f t h i s f u n c t i o n i s s a t i s f a c t o r y , t h e a p p r o x i m a t i o n e r r o r i n c r e a s e s q u i t e r a p i d l y f r o m t h e p o i n t w h e r e t h e a p p r o x i m a t i o n i s e x a c t . T h i s i s t y p i c a l o f T a y l o r s e r i e s a p p r o x i m a t i o n a n d a b e t t e r a p p r o a c h c o u l d b e i n v e s t i g a t e d . 1 1 7 Cited References [ A r g y l e 6 3 ] E . 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