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Adaptive rate control for time-varying communication channels with a feedback link Cavers, James Kennedy 1970

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ADAPTIVE RATE CONTROL FOR TIME-VARYING COMMUNICATION CHANNELS WITH A FEEDBACK LINK , by JAMES K. CAVERS B.A. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard Research Supervisor Members of the Committee A c t i n g Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In p resen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s tudy. I f u r t h e r agree tha t permiss ion f o r ex tens i ve copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n pe rm iss i on . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ^ c ? /2 /9?Q ABSTRACT Many communication channels have a power-to-noise r a t i o (PNR) which i s not constant i n time, producing a time-varying e r r o r p r o b a b i l i t y . I f a feedback channel i s a v a i l a b l e , the r e c e i v e r can request changes i n c e r t a i n t r a n s m i t t e r parameters i n response to the changing channel c o n d i t i o n s . I n t h i s t h e s i s a scheme f o r a d a p t i v e l y changing the data r a t e by v a r y i n g the d u r a t i o n of the t r a n s m i t t e d pulses i n order to compensate f o r t h i s f l u c t u a t i n g PNR i s described and analyzed. I m p l i c i t equations f o r the optimum r a t e request as a f u n c t i o n of past and c u r r e n t instantaneous PNR have been derived f o r an a r b i t r a r y p r o b a b i l i t y d e n s i t y f u n c t i o n of the PNR. The e f f e c t s of a bandwidth c o n s t r a i n t , of time delay i n the feedback l i n k , and of time and amplitude d i s c r e t e r a t e requests have been i n c l u d e d i n the a n a l y s i s . A p p l i c a t i o n of adaptive r a t e c o n t r o l to the Rayleigh f a d i n g channel can produce an enormous r e d u c t i o n i n r e q u i r e d t r a n s m i t t e r power over a f i x e d r a t e n o n - d i v e r s i t y system, up to 50 db f o r t y p i c a l values of e r r o r p r o b a b i l i t y . For the same values of bandwidth, data r a t e , and e r r o r p r o b a b i l i t y , and f o r t y p i c a l values of feedback delay, the v a r i a b l e r a t e system can s t i l l e f f e c t a power r e d u c t i o n i n the range 15-18 db, or a f a c t o r of 30-60, over the best a l t e r n a t i v e scheme, known as maximal-ratio p r e d e t e c t i o n combined frequency d i v e r s i t y . A method i s given which allows t r a d e o f f s between power, bandwidth and data r a t e f o r two-way communication over Rayleigh f a d i n g channels to be examined g r a p h i c a l l y . Adaptive r a t e c o n t r o l on m u l t i - u s e r channels produces a s m a l l e r improvement. For the range of parameters considered l i k e l y , there i s a maximum of about 1.7 db improvement over a f i x e d r a t e system. Although the magnitude of the improvement introduced by adaptive i-LX r a t e c o n t r o l i s s t r o n g l y dependent on the p r o b a b i l i t y d e n s i t y f u n c t i o n of the PNR, i t has been shown that f o r at l e a s t one commonly o c c u r r i n g c l a s s of time-v a r y i n g channels the savings are w e l l worth the cost of implementation. TABLE OF CONTENTS Page 1 . INTRODUCTION . . . . 1 1 . 1 M o t i v a t i o n 1 1 . 2 Background 1 1 . 3 O u t l i n e of Thesis 3 2 . MODELLING AND ANALYSIS OF THE VARIABLE RATE SYSTEM 4 2 . 1 System Model 4 2 . 1 . 1 The Forward Channel 4 2 . 1 . 2 The Feedback Channel 9 2 . 2 Mathematical Formulation of the V a r i a b l e Rate Problem 1 0 2 . 3 S o l u t i o n f o r the Continuous Rate Function 1 2 2 . 4 S o l u t i o n f o r the D i s c r e t e Rate Function 1 4 3 . ADAPTIVE RATE CONTROL FOR THE RAYLEIGH FADING CHANNEL 1 9 3 . 1 The System Model f o r Rayleigh Fading Channels 2 0 3 . 1 . 1 Modulation System 2 0 3 . 1 . 2 The S c a t t e r Channel : 2 3 3 . 1 . 3 P r o b a b i l i t y of E r r o r 2 6 3 . 1 . 4 Prediction 2 7 3 . 2 C a l c u l a t i o n s f o r P e r f e c t Feedback 2 9 3 . 3 I n d i v i d u a l E f f e c t of the Feedback Parameters 3 5 3 . 3 . 1 Bounds on the Instantaneous Rate 3 5 3 . 3 . 2 E f f e c t of Feedback Time-Delay . 3 9 3 . 3 . 3 E f f e c t of Rate Change Spacing 4 4 3 . 3 . 4 E f f e c t of Rate Q u a n t i z a t i o n 4 5 3 . 4 Adaptive Rate C o n t r o l i n a Two-Way S i t u a t i o n 5 0 Page 3.5 A Numerical Example 55 3.6 D i s c u s s i o n 56 4. ADAPTIVE RATE CONTROL FOR THE MULTI-USER CHANNEL 58 4.1 System Model f o r M u l t i - U s e r Channels 58 4.1.1 Forward Channel Model 58 4.1.2 Channel Usage S t a t i s t i c s 61 4.2 P r o b a b i l i t y of E r r o r s 64 4.2.1 The L i n e a r Channel . . . 64 4.2.2 The Hard L i m i t i n g Channel 69 4.3 C a l c u l a t i o n s 70 4.3.1 O p t i m i z a t i o n Equations . . . . 70 4.2.2 Re s u l t s . . . . 72 4.4 Two Numerical Examples 78 4.4.1 M u l t i - U s e r S a t e l l i t e Channel . 78 4.4.2 M u l t i - U s e r Cable 80 4.5 D i s c u s s i o n 81 5. CONCLUSIONS 81 APPENDIX I . Glossary of Symbols 84 APPENDIX I I Proofs of the Lemmas of Chapter 2 . . . . . . . .-. . . 86 APPENDIX I I I Optimum D i v e r s i t y f o r Maximal R a t i o Pre D e t e c t i o n Combining . . 89 APPENDIX IV C a l c u l a t i o n of the D i s c r e t e Rate Function 91 REFERENCES 93 LIST OF ILLUSTRATIONS F i g . 2.1.1 General V a r i a b l e Rate System Model F i g . 2.1.2 E s t i m a t i o n of Channel PNR F i g . 2.4.1 A (non-optimum) 4 - I n t e r v a l Rate Function F i g . 2.4.2 R e l a t i o n Between Continuous and D i s c r e t e S o l u t i o n s F i g . 3.1.1 Modulation System f o r the Rayleigh Fading Channel F i g . 3.1.2 A u t o c o r r e l a t i o n Function of u ( t ) F i g . 3.2.1 Comparison of Modulation Techniques f o r Coherent Reception of Rayleigh Faded Binary Othogonal Pulses w i t h P e r f e c t . Feedback F i g . 3.2.2 Comparison of Modulation Techniques f o r Incoherent Reception of Rayleigh Faded Binary Orthogonal Pulses w i t h P e r f e c t Feedback F i g . 3.3.1 Rate Function w i t h Upper and Lower Bounds F i g . 3.3.2 Rate Function w i t h r = 0 Allowed F i g . 3.3.3 E f f e c t of a Rate Range C o n s t r a i n t on the Performance of a V a r i a b l e Rate System Using Incoherent D e t e c t i o n of Ra y l e i g h Faded Pulses F i g . 3.3.4 E f f e c t of a Rate Range C o n s t r a i n t on the Performance of a s V a r i a b l e Rate System Using Coherent D e t e c t i o n of Rayleigh Faded Pulses F i g . 3.3.5 E f f e c t of Bandwidth Expansion C o n s t r a i n t r Q on the Performance of a V a r i a b l e Rate System Using Incoherent D e t e c t i o n of Rayleigh Faded Pulses F i g . 3.3.6 E f f e c t of Bandwidth Expansion C o n s t r a i n t r on the Performance o of a V a r i a b l e Rate System Using Coherent D e t e c t i o n of Rayleigh Faded Pulses v i i F i g . 3.3.7 E f f e c t of Feedback Time Delay vx Fading Time Constants on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses F i g . 3.3.8 E f f e c t of Rate Change Spacing of vA Fading Time Constants on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses F i g . 3.3.9 E f f e c t of Rate Quantization to N non-zero Rates on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses F i g . 3.3.10 E f f e c t of Rate Quantization to N non-zero Rates on the Performance of a Variable Rate System Using Coherent Detection of Rayleigh Faded Pulses F i g . 3.4.1 Two-Way Communication System F i g . 3.4.2 Time Frame A l l o c a t i o n F i g . 3.4.4 Two-Way System Parameters b, N, A which Produce an Erro r P r o b a b i l i t y of 10 ^ on a Rayleigh Fading Channel with Feedback Delay x; = 0.01 Fading Time Constants, and Maximum Bandwidth Expansion r = 2 o F i g . 3.4.5. Two-Way System Parameters b, N, A which Produce an Erro r P r o b a b i l i t y of 10 ^ on a Rayleigh Fading Channel with Feedback Delay T = 0.02 Fading Time Constants, and Maximum Bandwidth Expansion r = 2 o F i g . 4.1.1 Pseudonoise Modulation System F i g . 4.1.2 Ty p i c a l Usage Function F i g . 4.2.1 The Linear Channel F i g . 4.2.2 Microscopic View of I n t e r f e r i n g Waveforms F i g . 4.2.3 The Hard L i m i t i n g Channel F i g . 4.3.1 Combinations of Thermal Noise C o e f f i c i e n t C^ and I n t e r f e r e n c e Noise C o e f f i c i e n t C^ which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users M^ , = 100, Duty Cycle of Users ct^ = 0.5 F i g . 4.3.2 Combinations of Thermal Noise C o e f f i c i e n t C^ and I n t e r f e r e n c e Noise C o e f f i c i e n t C^ which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users M^ , = 100, Duty Cycle of Users = 0.25 F i g . 4.3.3. Combinations of Thermal Noise C o e f f i c i e n t and I n t e r f e r e n c e Noise C o e f f i c i e n t C^ which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users M^ , = 100, Duty Cycle of Users = 0.1 F i g . 4.3.4 E f f e c t of Feedback Delay on Combinations of Thermal Noise C o e f f i c i e n t and I n t e r f e r e n c e Noise C o e f f i c i e n t C^ which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users M-j, = 20, Duty Cycle of Users a 1 = 0.5 ACKNOWLEDGEMENT I would l i k e to thank my research s u p e r v i s o r , Dr. R.W. Donaldson, f o r h i s encouragement and many suggestions during the course of t h i s t h e s i s , and f o r h i s guidance through the a d m i n i s t r a t i v e mazes. Thanks are due a l s o to Miss Heather DuBois who q u i c k l y and n e a t l y typed the manuscript. G r a t e f u l acknowledgement i s given to the N a t i o n a l Research C o u n c i l f o r f i n a n c i a l support from 1966 to 1970. F i n a l l y , I would l i k e to thank my w i f e , Penny, f o r her patience and understanding during these y e a r s , and f o r her uncomplaining t y p i n g of the d r a f t copy. 1. INTRODUCTION 1.1 M o t i v a t i o n Many communication channels have f l u c t u a t i n g c h a r a c t e r i s t i c s and q u a l i t y , and are not the s t a t i c e n t i t i e s found i n textbooks. This time-v a r y i n g f e a t u r e can a r i s e from such sources as changing noise l e v e l s , jamming, or time-varying f i l t e r i n g induced by the channel to change the r e c e i v e pulse shape or introduce echoes. The t r a d i t i o n a l approach to t h i s problem has been e i t h e r worst case design or d i v e r s i t y t r a n s m i s s i o n . The f i r s t i s ob-v i o u s l y i n e f f i c i e n t and the second may not be p o s s i b l e , as i n t r a n s m i s s i o n from deep space. In such c o n d i t i o n s , i t may be d e s i r a b l e to a l l o w the modula-t i o n system to adapt to these changes through appropriate use of a feedback channel. This t h e s i s contains a proposal f o r one such method, i n which the t r a n s m i t t e d p u l s e d u r a t i o n w i l l be adjusted to compensate f o r changes i n the forward channel s i g n a l to noise r a t i o . 1.2 Background The use of a feedback channel to decrease the e r r o r p r o b a b i l i t y i n t r a n s m i s s i o n over a t i m e - i n v a r i a n t channel has been e x t e n s i v e l y i n v e s t i g a t e d [1-5]. Most of these schemes i n v o l v e r e p e t i t i v e transmissions of the d i g i t , w i t h instantaneous power and t o t a l number of retransmissions under c o n t r o l of the r e c e i v e r , which forms s u c c e s s i v e estimates of the d i g i t ' s i d e n t i t y . I t should be emphasized that these schemes assume a forward channel of f i x e d c h a r a c t e r i s t i c s , and use the feedback channel to augment or r e p l a c e coding i n order to transmit i n f o r m a t i o n at a ra t e approaching the channel c a p a c i t y . Much l e s s work has been done on the use of a feedback channel to enable the system to adapt parameters such as power, p u l s e shape or data r a t e 2 to changes i n a time-variant forward channel. In one experiment, [6,7] a sequential decoder was connected' through a t o l l grade long distance telephone l i n e to a convolutional coder which had several p a r i t y check nets with d i f f e r e n t degrees of redundancy. The r e c e i v e r monitored the channel q u a l i t y , as deter-mined by the queue length i n the decoder, and could send requests over a feedback l i n k f o r a decrease i n redundancy or a repeat at increased redundancy. No a n a l y t i c a l optimization of the set of rates and the way i n which they were used was performed, since bounds on the performance of sequential decoders are too loose. An a n a l y t i c a l i n v e s t i g a t i o n of the adaptive use of feedback was conducted by Hayes [8], who considered the transmission of v a r i a b l e amplitude, f i x e d duration pulses over a slowly varying Rayleigh fading channel. Under the assumptions of e r r o r - f r e e estimation of the instantaneous channel gain, and n o i s e l e s s , delayless feedback, he determined the pulse amplitude as a function of the instantaneous gain which minimized the average error p r o b a b i l i t y f o r a given average transmitter power, and demonstrated a s u b s t a n t i a l improve-ment i n performance. Another i n v e s t i g a t i o n was begun by Clowes [9], who also considered a Rayleigh fading channel and p e r f e c t feedback, but treated v a r i a b l e duration, rather than v a r i a b l e amplitude pulses. Although no general r e s u l t s were reported, experiments were performed both i n the laboratory and through a s a t e l l i t e [10]. A v a r i a b l e data rate scheme has been described and analyzed by Metzner and Morgan [35]. They consider a system using long block codes which induce an approximate threshold on the data rate, below which the code word can almost c e r t a i n l y be decoded, and above which i t i s almost c e r t a i n to be rejected. This threshold i s a function of the signal-to-noise r a t i o (SNR) i n the channel. In the case of a time-varying SNR, the r e c e i v e r accurately measures the current SNR and sends t h i s i n f o r m a t i o n to the t r a n s m i t t e r over a n o i s e - f r e e d e l a y l e s s feedback channel, a l l o w i n g the system to use a data r a t e which i s always f r a c t i o n a l l y l e s s than the current t h r e s h o l d . For the R a y l e i g h f a d i n g channel they found that t h i s method could a t t a i n an average data r a t e t r i p l e that of an automatic repeat system. 1.3 O u t l i n e of Thesis This t h e s i s a t t acks the problem of how to use a feedback channel i n a scheme to o p t i m a l l y vary t r a n s m i t t e d pulse d u r a t i o n i n order to combat a f l u c t u a t i n g power-to-noise r a t i o (PNR) i n the forward channel. In Chapter 2, a model f o r a general feedback communication system w i t h a time-varying PNR i s presented, and i m p l i c i t equations are derived f o r the optimum t r a n s m i s s i o n r a t e as a f u n c t i o n of past and current PNR f o r an a r b i t r a r y penalty w e i g h t i n g of the e r r o r p r o b a b i l i t y , and an a r b i t r a r y pro-b a b i l i t y d e n s i t y f u n c t i o n of the PNR. Both d i s c r e t e and continuously v a r i a b l e r a t e s are considered, and the e f f e c t s of delay i n the feedback loop and t i m e - d i s c r e t e r a t e requests are taken i n t o account. I t i s shown that under p e r f e c t feedback c o n d i t i o n s , adaptive r a t e c o n t r o l completely e l i m i n a t the e f f e c t of the time-varying PNR, r e s u l t i n g i n an e q u i v a l e n t constant channel w i t h energy-to-noise r a t i o given by the r a t i o of average PNR to average r a t e . In Chapters 3 and 4, more p r e c i s e models are developed f o r two important time-varying channels, the Rayleigh f a d i n g channel and the m u l t i -user channel. The o p t i m i z a t i o n equations are presented, along w i t h the c a l c u l a t e d e f f e c t s of forward and feedback channel parameters on the e r r o r p r o b a b i l i t y . I t i s shown t h a t , f o r the Rayleigh f a d i n g channel, adaptive r a t e c o n t r o l can e f f e c t an improvement i n performance g r e a t e r than any 4 e x i s t i n g scheme when feedback c o n d i t i o n s are p e r f e c t . The saving amounts to 10 to 50 db and can be r e a l i z e d i n the form of reduced t r a n s m i t t e r power, in c r e a s e d data r a t e , o r decreased e r r o r p r o b a b i l i t y . A graph i s presented which can be used to determine the optimum f r a c t i o n of the b i t stream to be used f o r the s e r v i c e i n f o r m a t i o n i n a true two-way communication s i t u a t i o n , i n which two s t a t i o n s are simultaneously t r a n s m i t t i n g to each other and the forward channel of one acts as the feedback channel f o r the other. For the m u l t i - u s e r channel i t i s shown that the savings are not as g r e a t , and i n f a c t are s m a l l enough to make the a d d i t i o n a l complexity of the feedback scheme of questionable v a l u e . F i n a l l y , Chapter 5 contains a summary of the work, and an o u t l i n e of p o s s i b l e d i r e c t i o n s f o r f u t u r e research i n t h i s area. 2. MODELLING AND ANALYSIS OF THE VARIABLE RATE SYSTEM 2.1 System Model 2.1.1 The Forward Channel The communication system i s modelled as i n F i g . -2.1.1. The source b u f f e r , which i s assumed to be i n f i n i t e to avoid d i f f i c u l t i e s w i t h overflow and underflow, s u p p l i e s the modulator w i t h message b i t s at a time v a r y i n g r a t e R(t) b i t s / s e c , which i s under the c o n t r o l of the r e c e i v e r . I t w i l l be understood that t h i s r a t e w i l l not be changed during the t r a n s m i s s i o n of a b i t ; indeed R(t) can be changed only p e r i o d i c a l l y , and must be h e l d constant over i n t e r v a l s of d u r a t i o n A which i s much longer than the d u r a t i o n of i n d i -v i d u a l b i t s . The modulator i n t u r n emits the bandpass s i g n a l ^ ( t ) cosu^t , where k i s 1 or 0, f o r each b i t . The lowpass modulating waveforms ^ ( t ) c o n s i s t of equal power pulses whose d u r a t i o n at time t i s T ( t ) = 1/R(t). Source Buffer + 1 Modulator T M^Ct) s i n to i r R(t) Forward Channel, time-varying PNR, S(t) Feedback Channel N , f , A Demodulator H q(t) Rate Control F i g . 2.1.1 General Variable Rate System Model sample^ at T S e l e c t Larger 1 T k = 1 or 0 C a l c u l a t e Averages <e(t)>-< e 2 ( t ) > - < e ( t ) / Sink B u f f e r S ( t ) F i g . 2.1.2 E s t i m a t i o n of Channel PNR, S ( t ) Passage of the s i g n a l through the channel produces a time-varying instantaneous PNR, S ( t ) . I t i s assumed that the bandwidth v of S(t) i s much l e s s than the bandwidth of ^ ( t ) so that there i s n e g l i g i b l e change i n S(t) over the duration of a s i n g l e pulse. In Chapters 3 and 4 the channel model w i l l be developed further but, f o r g e n e r a l i t y , the analysis of t h i s s e c t i o n w i l l assume only that some unknown mechanism causes the f l u c t u a t i o n s i n the PNR. At the rece i v e r , the demodulator and the dec i s i o n device ( F i g . 2.1.2) process the received s i g n a l to form the decoded message, which has a b i t error p r o b a b i l i t y dependent on both R(t) and S ( t ) . F i n a l l y the decoded b i t stream i s fed to the sink b u f f e r , which i s also assumed to be i n f i n i t e . In order f o r the system to use a v a r i a b l e - r a t e strategy the receiver must monitor the instantaneous PNR of the channel, upon which depends the choice of data rate. I t i s assumed here, and j u s t i f i e d l a t e r i n t h i s s e c t i o n , that the estimation device can produce a near perfect r e p l i c a , that i s , one with n e g l i g i b l e estimation e r r o r , of the channel PNR, S ( t ) . Now i f the feedback channel contains a delay T which i s s i g n i f i c a n t compared to the "time constant" v ^ of S(t) or i f the spacing A between rate changes i s s i g n i f i c a n t compared to v \ further processing i s necessary. This processing i s performed by the p r e d i c t i o n device which extracts from S(t) those time varying parameters which describe the p r o b a b i l i t y density function of S(t+y), the PNR y seconds l a t e r , where x '<_ y <_ T + A. The output of the p r e d i c t i o n device, i n the form of z ( t ) , i s passed to the important rate c o n t r o l device, which p e r i o d i c a l l y (at A second i n t e r -vals) forms an instantaneous function p ( z ) , which w i l l be c a l l e d the "rate function". The r e s u l t i n g waveform i s the rate request R^(t) = p [ z ( t ) ] which w i l l be the system data rate i n the i n t e r v a l [t+x, t+x+A]. The request f o r a p o s s i b l e rate change i s sent back to the transmitter over a feedback channel, which w i l l be discussed i n the next s e c t i o n , and the t r a n s m i t t e r then a l t e r s i t s data r a t e a c c o r d i n g l y . The o p t i m i z a t i o n i s concerned w i t h s e l e c t i n g the f u n c t i o n p ( z ) , and i n the case of the two-way communication, the r a t e change p e r i o d A, according to a c r i t e r i o n d e fined i n s e c t i o n 2.2. Let us consider the e s t i m a t i o n device i n more d e t a i l . The demodulator and d e c i s i o n devices are shown i n Figure 2.1.2 as a p a i r of c o r r e l a t o r s f o l l o w e d by a device which s e l e c t s the index 1 or 0 which has the l a r g e r c o r r e l a t i o n . The estimator has a v a i l a b l e the sampled c o r r e l a t o r outputs and the d e c i s i o n device output, and must know which c o r r e l a t o r output a c t u a l l y contains a s i g n a l component, r a t h e r than j u s t channel n o i s e . Since a communication system i s designed to operate at a low e r r o r p r o b a b i l i t y , -3 t y p i c a l l y 10 or l e s s , almost a l l of the decoded d i g i t s are c o r r e c t , and a switch operated by the d e c i s i o n output w i l l s e l e c t the r i g h t c o r r e l a t o r output to give e ( t ) . The low e r r o r p r o b a b i l i t y a l s o i m p l i e s a f a i r l y l a r g e energy-to-noise r a t i o per b i t . Assuming that s u c c e s s i v e n o i s e samples are independent, and r e c a l l i n g that the time constant v 1 of S ( t ) i s much gr e a t e r than the pulse d u r a t i o n , i t f o l l o w s that averaging of e ( t ) over many adjacent pulses should produce an e x c e l l e n t estimate of the s i g n a l s t r e n g t h . The sample va r i a n c e should, f o r the same reasons, be a good estimate of the channel noise 2 v a r i a n c e . Thus — » r- ; where the brackets denote .time averaging <e (t)> - <e(t)> over a p e r i o d comparable to the time constant v 1 of S ( t ) , i s a good estimate of the energy-to-noise r a t i o per b i t which, upon d i v i s i o n by the b i t d u r a t i o n T, becomes the PNR S ( t ) . I f e i t h e r s i g n a l s t r e n g t h or noise power i s constant, as i n the examples of Chapters 3 and 4 , e s t i m a t i o n of S ( t ) i s even more accurate because there i s only one random parameter to estimate, r a t h e r than the r a t i o of two random parameters. 9 2.1.2 The Feedback Channel In order f o r the system to operate p r o p e r l y , both "transmitter and r e c e i v e r must know the current data r a t e e x a c t l y . This requirement i m p l i e s e i t h e r that the feedback channel i s noise f r e e o r , more r e a l i s t i c a l l y , that the number of a l l o w a b l e r a t e s i s f i n i t e and the r a t e request Rg(t) i s d i g i t i z e d and p r o t e c t e d by coding from feedback channel n o i s e . Thus one parameter d e s c r i b i n g the e f f e c t of the feedback channel i s N, the number of non-zero r a t e s . As we s h a l l see, the r a t e R=0, where no data i s t r a n s m i t t e d , can sometimes be used to good e f f e c t . The e f f e c t s of the parameters x, the round t r i p delay, and A, the time i n t e r v a l between the p e r i o d i c r a t e requests, can a l s o be a s c r i b e d to the feedback channel. As x or A become very l a r g e compared to the PNR time constant v \ i t i s c l e a r that the system performance w i l l d e t e r i o r a t e to that of a f i x e d or constant r a t e system. In a two-way communication s i t u a t i o n , the two s t a t i o n s transmit to each other s i m u l t a n e o u s l y , w i t h the forward channel f o r one a c t i n g as the feedback channel f o r the other. Both channel 1 and channel 2 thus c a r r y s e r v i c e ( r a t e c o n t r o l ) i n f o r m a t i o n as w e l l as messages. I f an e r r o r i n the s e r v i c e i n f o r m a t i o n ( f o r channel 1, say) occurs, the r e s u l t s can be q u i t e unpleasant; the t r a n s m i t t e r and r e c e i v e r on channel 1 do not operate at the same r a t e and a l l the b i t s have e r r o r p r o b a b i l i t y ^> i n c l u d i n g the s e r v i c e i n f o r m a t i o n f o r channel 2. Thus the t r a n s m i t t e r and r e c e i v e r on channel 2 w i l l a l s o operate at d i f f e r e n t r a t e s , and the system has broken down. This s i t u a t i o n can be detected e i t h e r by the abnormally low estimated PNR, or by the nonsense emerging from the decoder. The remedy i s a p o l i c y whereby each s t a t i o n , upon d e t e c t i n g a breakdown, transmits at the lowest r a t e u n t i l normal f u n c t i o n i n g i s r e - e s t a b l i s h e d . In t h i s t h e s i s , i t w i l l be assumed that the 10 r a t e request i s p r o t e c t e d to an extent that any e r r o r s i n i t s t r a n s m i s s i o n occur w i t h s u f f i c i e n t l y s m a l l p r o b a b i l i t y t h a t the e f f e c t s can be ignored. We are faced w i t h a p o t e n t i a l t r a d e o f f : as the s e r v i c e b i t r a t e i n c r e a s e s , w i t h i n c r e a s i n g N , or decreasing A , the performance on the forward channel improves, but at the expense of message r a t e i n the reverse d i r e c t i o n . The optimum r a t e f u n c t i o n , given N and A , w i l l be derived i n s e c t i o n 2.4, and the optimum choice of N and A w i l l be examined i n d e t a i l i n s e c t i o n 3.4. 2.2 Mathematical Formulation of the V a r i a b l e Rate Problem I f S ( t ) and z ( t ) are e r g o d i c , then i n a long time i n t e r v a l I , z w i l l be between X, and ? + dt, f o r I P (?) dt, seconds, where P (?) i s the p r o b a b i l i t y z z d e n s i t y f u n c t i o n (p.d.f.) of z. During t h i s time p (?) I p z ( £ ) d? b i t s w i l l be t r a n s m i t t e d , and thus the average r a t e or t o t a l number of b i t s t r a n s m i t t e d during I d i v i d e d by I i s given by R = / p(?) P (?) d? 2.2.1 av z For a s p e c i f i e d R we wish to minimize the average v a l u e of some av ° p e n a l t y weighting f u n c t i o n f [ P ] which r e f l e c t s the r e l a t i v e c ost of the v a r i o u s values of the e r r o r p r o b a b i l i t y P . This penalty f u n c t i o n accounts f o r the q u a l i t y o f , say, PCM v o i c e t r a n s m i s s i o n at d i f f e r e n t b i t - e r r o r p r o b a b i l i t i e s . One p o s s i b l e choice f o r f [ P ] i s a d i f f e r e n t i a b l e approximation to u .[P -p ], where u ,(a) i s 1 f o r a > 0 and 0 f o r a < 0. The r e s u l t i s -1 e o - i — m i n i m i z a t i o n of the f r a c t i o n of r e c e i v e d b i t s w i t h e r r o r p r o b a b i l i t y g r e a t e r than p , which turns out to be the "grade of s e r v i c e " c r i t e r i o n proposed by 2 Clowes [ 9 ] . Another choice could be f [ P ] = P , i n which case we minimize the second moment of the e r r o r p r o b a b i l i t y . The examples of Chapters 3 and 4 use f [ P ] = P , the average e r r o r p r o b a b i l i t y c r i t e r i o n , because of i t s 11 simple i n t e r p r e t a t i o n and general use as a performance measure. The average penalty i s c a l c u l a t e d by summing the cost a s s o c i a t e d w i t h the e r r o r p r o b a b i l i t y of each b i t i n a very long s t r i n g and d i v i d i n g the r e s u l t by the t o t a l number of b i t s i n the s t r i n g . We w i l l f i r s t c o nsider the s i t u a t i o n where the r a t e change spacing A i s n e g l i g i b l e compared to the PNR time constant v \ Then the p e n a l t y c o n t r i b u t i o n when z i s between £ and £ + d£ i s given as f o l l o w s : 00 I P (e) p(Odc If[P (ct,p(e))]P I (a |s)da o ' where P e ( a , p ) i s the e r r o r p r o b a b i l i t y as a f u n c t i o n of the data r a t e p and the PNR a at the time the r a t e changes takes e f f e c t , T seconds l a t e r . The f u n c t i o n p S | Z ( a U ) i s t n e P-d.f. of S(t+t) c o n d i t i o n e d on z ( t ) . The average pen a l t y f i s then f = ~- / PJO p(c) f[?,p(c)]dc 2.2.2 AY Z av z where the " p r e d i c t e d p e n a l t y " f i s defined as f o l l o w s : CO f f C , p ] = / f [ P e ( a , p ) ] P s i ( a | c ) d a 2.2.3 I f A i s comparable to v \ then the system must m a i n t a i n a constant r a t e over i n t e r v a l s of d u r a t i o n A , even though S ( t ) may change a p p r e c i a b l y during that time, and the p r e d i c t e d p e n a l t y must be modified to account f o r t h i s change. I f we average the p e n a l t i e s over the p A b i t s t r a n s m i t t e d i n the time i n t e r v a l [t + T, t + T+ A ] we o b t a i n , p A °° f [ ? , p ] = -T I f f [ P ( a ,p)] P I (a \c.) da 2.2.4 i = l o s ± \ z 1 i where a. = S(t + T + i / p ) and P i (a.|?) i s the p.d.f. of S ( t + T + i/p) i ' c o n d i t i o n e d on z ( t ) . A convenient n o r m a l i z a t i o n can be performed i f P e ( S , p ) has the form P e ( S / p ) . This i s not a very r e s t r i c t i v e assumption, s i n c e i t s t a t e s that the e r r o r p r o b a b i l i t y i s a f u n c t i o n only of the energy-to-noise r a t i o per b i t . I f we d e f i n e r = p/R and X = S/R then we can w r i t e the e r r o r p r o b a b i l i t y av av r P e ( X , r ) as P e ( X / r ) . The v a r i a b l e r a t e problem now c o n s i s t s of f i n d i n g the r a t e f u n c t i o n r ( z ) which minimizes f = / r(c) f [ c , r ( c ) ] P z ( 0 d? 2.2.5 z w h i l e m a i n t a i n i n g / r(C)P (c)d? = 1 2.2.6 z z s u b j e c t , of course, to r ( ? ) >_ 0. Although the above n o r m a l i z a t i o n w i l l be used i n the remainder of the a n a l y s i s , many of the r e s u l t s are v a l i d even when the n o r m a l i z a t i o n cannot be performed. 2.3 S o l u t i o n f o r the Continuous Rate Function I f the r a t e f u n c t i o n r ( z ) i s allowed to be continuous, then the v a r i a b l e r a t e problem of equations 2.2.5 and 2.2.6 i s a t y p i c a l v a r i a t i o n a l c a l c u l u s problem, though s i m p l i f i e d by the absence of the d e r i v a t i v e rr(z) . Equation 2.2.6 can be adjoined to 2.2.5 w i t h a Lagrange m u l t i p l i e r and the i n t e g r a n d s u b s t i t u t e d i n t o the Euler-Lagrange equation [36] to produce the necessary c o n d i t i o n f o r a minimum: -r|£ + f [ z , r ] = X 2.3.1 9r as the i m p l i c i t equation f o r r ( z ) . The Lagrange m u l t i p l i e r X i s s e l e c t e d to s a t i s f y 2.2.6. Note that although 2.3.1 does not c o n t a i n the p.d.f. P z ( 0 , there i s an i m p l i c i t dependence on p z ( ? ) through the s e l e c t i o n of X. An a l t e r n a t i v e d e r i v a t i o n i s to consider the m i n i m i z a t i o n of the integrand f o r each value of t,. D i f f e r e n t i a t i o n w i t h respect to r again leads to 2.3.1. Let us d e f i n e u = X/r so that P (X/r) = P (u). We w i l l now s t a t e e e two simple-but important lemmas p e r t a i n i n g to the s o l u t i o n r ( z ) of 2.3.1. The proofs are contained i n Appendix I I . df e Lemma 1: I f - r — • — — i s monotonically i n c r e a s i n g w i t h u, then dP du J ° e r ( z ) , the s o l u t i o n of 2.3.1 i s unique and i t i s a minimum. Proo f ; In Appendix I I . Before the next lemma i s s t a t e d , a f u r t h e r assumption must be made about the nature of the v a r i a b l e z ( t ) . With a s l i g h t l o s s of g e n e r a l i t y we w i l l consider z ( t ) to be an estimator of X ( t + T ) , i n the sense that f o r any ct, the p r o b a b i l i t y that X ( t + x)<a decreases w i t h i n c r e a s i n g z ( t ) . More p r e c i s e l y , the .cumulative d i s t r i b u t i o n f u n c t i o n (c.d.f.) of X conditioned on F x | z = " P x | z ( 3 l ^ 1 0 1 i s a decreasing f u n c t i o n of £. df e Lemma 2: I f ^ « ^ i s monotonically i n c r e a s i n g w i t h u, and i f e the c o n d i t i o n a l c.d.f. F i ( a | £,) i s a decreasing f u n c t i o n of the: X I z the s o l u t i o n r ( z ) of 2.3.1 i s a monotonically i n c r e a s i n g f u n c t i o n of z. Proof: I n Appendix I I . L e t us examine l i m i t i n g cases. I f T and A are n e g l i g i b l e compared w i t h the "time constant" of X ( t ) , then X ( t + T + A) = X ( t ) and no p r e d i c t i o n i s r e q u i r e d , a l l o w i n g z ( t ) = X ( t ) . The p r e d i c t e d p e n a l t y f [ z , r ] becomes f [ P e ( X , r ) ] and 2.3.1 becomes df 9 P e r dP~ 9 r ~ + f [ p e < x > r ) J = X 2.3. e F u r t h e r s i m p l i f i c a t i o n i s p o s s i b l e s i n c e P (X,r) has the form P (u). This e e allows 2.3.3 to be w r i t t e n as f o l l o w s : 14 rlf d P e Since the l e f t s i d e of 2 .3 .4 i s a monotonic (see Lemma 1) f u n c t i o n of u alone, and i s equated to a constant, the s o l u t i o n i s simply u = X/r = constant, which i n t u r n i m p l i e s a constant e r r o r p r o b a b i l i t y . Thus f o r x=0, A=0 and a continuously v a r i a b l e r a t e , the optimum r a t e f u n c t i o n i s the one which holds the e r r o r p r o b a b i l i t y constant. Since r = X/u, we must have u = X to make r = 1. The q u a n t i t y u i s now equal to an average energy-to-noise r a t i o per b i t , the r a t i o of the average PNR to the average r a t e R a V » ' Since the e r r o r p r o b a b i l i t y i s a f u n c t i o n only of u, the e f f e c t of adaptive r a t e c o n t r o l w i t h p e r f e c t feedback i s to r e p l a c e the time v a r y i n g channel w i t h t h i s e q u i v a l e n t constant channel, thus completely e l i m i n a t i n g the e f f e c t of the f l u c t u a t i o n s i n X ( t ) . This provides an o p t i m i s t i c bound on the performance of a r e a l feedback communication system. At the other extreme, i f x and A are much g r e a t e r than the time constant of X ( t ) , then the c o n d i t i o n a l p.d.f. p x | z ( a l ? ) o r 2 . 2 . 3 and 2 .2 .4 becomes the a p r i o r i d e n s i t y p x ( a ) > a n d f[Cj*"] = f [ r ] independent of the value assumed by z. The s o l u t i o n of 2 .3 .1 i s then r = constant, and from 2 . 2 . 6 , r = l , g i v i n g a f i x e d r a t e system. This provides a p e s s i m i s t i c bound on the performance of a r e a l feedback system. F i n a l l y we note that f o r f [ P ] = P n and P (u) = e~ U or P ( u ) = e r f c ( / u ~ ) , p e e e df g the c o n d i t i o n that -77— • — — i n c r e a s e w i t h u i s s a t i s f i e d . These are the usu a l dP du e forms of e r r o r p r o b a b i l i t y which occur. 2.4 S o l u t i o n f o r the D i s c r e t e Rate Function I t was observed i n s e c t i o n 2 .2 .2 that a r e a l i s t i c a n a l y s i s of the feedback communication system w i l l consider the e f f e c t s of a l l o w i n g only a f i n i t e number of t r a n s m i s s i o n r a t e s . The r e s u l t i s a d i s c r e t e r a t e f u n c t i o n whose behaviour might be as i l l u s t r a t e d i n F i g . 2.4.1. For M q u a n t i z a t i o n r(z) • 1 1 I 1 1 1 1 1 1 2 3 z —*-F i g . 2.4.1 A (non-optimum) 4 - i n t e r v a l r a t e f u n c t i o n i n t e r v a l s of z there are M ra t e s r ^ , and M-l s w i t c h i n g thresholds z_^ . By analogy w i t h the continuous m i n i z a t i o n problem of equations 2.2.5 and 2.2.6, Lsh to f i n d £ r- j^ a n c^ which we wis minimize M z. Z r f1 f [ c , r , ] P j ? ) d c 1 = 1 Z i - 1 w h i l e m a i n t a i n i n g i = l 1 z 2.4 M z. E r , Z 1 P z ( ? ) d ? z. 2.4 J i - 1 s u b j e c t again to r.> 0. We d e f i n e z = 0, z = °°. i - o m Before s o l v i n g these equations, l e t us consider the s l i g h t l y more general case of the m i n i m i z a t i o n w i t h respect to y(x) of 16 / F [ x , y ( x ) ] d x 2.4.3 o w i t h the c o n s t r a i n t oo / G[x,y(x)]dx = C 2.4.4 o We w i l l a l s o consider the a s s o c i a t e d M - i n t e r v a l d i s c r e t e problem of the m i n i -m i z a t i o n w i t h respect to a n <* " { x i ^ °^ M x . E V 1 F[x,y.]dx 2.4.5 i = l x. , i - i s u b j e c t to M x E / G[x,y.] = C 2.4.6 i = l x. n i - i A d j o i n i n g 2.4.3 and 2.4.4 w i t h a Lagrange m u l t i p l i e r and d i f f e r e n t i a t i n g w i t h respect to y, we o b t a i n the necessary c o n d i t i o n ^ ^ y ^ = 0 as the i m p l i c i t equation f o r a continuous s o l u t i o n , where H(x,y) = F(x,y) + uG(x,y). Equations 2.4.5 and 2.4.6 are f u n c t i o n s of the 2M-1 v a r i a b l e s y. and x.. They too can J x l J be adjoined w i t h a Lagrange m u l t i p l i e r . Subsequent d i f f e r e n t i a t i o n w i t h respect to each of the y_^  and x^ gives the f o l l o w i n g necessary c o n d i t i o n s f o r a minimum x 3H(x,y ) / 5 — — dx = 0 2.4.7 3y. X i - 1 H ( x ± , y i ) - H ( x i } y i + 1 ) = 0 2.4.8 Equation 2.4.7 must be s a t i s f i e d f o r each y_^  being considered, and 2.4.8 must be s a t i s f i e d f o r each x. being considered. I f any of the x. or y. l J l y i i s p r e v i o u s l y s p e c i f i e d , the corresponding equation does not have to be s a t i s f i e d , For example, i n Chapter 3, the v a r i a b l e corresponding to y^ w i l l be f i x e d at zero, and only y^, i=2, M and x^, i = 1, M-l w i l l be allowed to vary. I t i s important to note that '|~^X'^ must not c o n t a i n impulses i n x; i f i t d i d , equation 2.4.7 would not be continuous i n x^_^ a n d x^, and a s o l u t i o n 2.4.7 would not be continuous i n x. . and x., and a s o l u t i o n might i - i I not be p o s s i b l e . An a l t e r n a t i v e to the s o l u t i o n by d i f f e r e n t i a t i o n i s a dynamic programming s o l u t i o n of 2.4.5 and 2.4.6. F o r t u n a t e l y , i n the case of p r a c t i c a l i n t e r e s t of t h i s t h e s i s , the Rayleigh f a d i n g channel, a l l d e n s i t y f u n c t i o n s are non-impulsive. The s o l u t i o n to the s e t of equations 2.4.7 and 2.4.8 can be obtained by i t e r a t i o n on a s i n g l e parameter, say y^, by s o l v i n g a l t e r n a t e l y 2.4.7 and 2.4.8, thereby a v o i d i n g a search over the e n t i r e 2M-1 dimensional space jV.^ x { " X i ^ ' ^ e e c l u a t ^ o n s a r e s i m i l a r to those obtained by M a x [ l l ] and Larson [12], i n a d i f f e r e n t context, and i n f a c t reduce to them i f we s e t G[x,y]=0 and F[x,y] = g(x-y) where g(a) i s some even f u n c t i o n . I t would be u s e f u l to know what c o n d i t i o n s guarantee a unique s o l u t i o n to the d i s c r e t e m i n i m i z a t i o n problem of equations 2.4.5 and 2.4.6. We w i l l now s t a t e two important lemmas concerning t h i s d i s c r e t e s o l u t i o n and a c o r o l l a r y which allows us to r e l a t e the d i s c r e t e s o l u t i o n and the s o l u t i o n y Q ( x ) of the continuous m i n i m i z a t i o n problem of equations 2.4.3 and 2.4.4. Lemma 3: I f the equation ^ ^ y ^ = 0 n a s a s i n g l e root i n y f o r any given x, and a s i n g l e root i n x f o r any given y, then the i t e r a t i v e s o l u t i o n of equations 2.4.7 and 2.4.8 i s unique. Proof: I n Appendix I I . C o r o l l a r y : The continuous s o l u t i o n y (x) l i e s between y. and y. o 1 1+1 The x f o r which y (x) = y. l i e s between x. , and x.. The r e l a t i o n o 1 1-1 1 between the continuous and d i s c r e t e s o l u t i o n s i s shown i n F i g . 2.4.2. F i g . 2.4.2 R e l a t i o n Between Continuous and D i s c r e t e S o l u t i o n s 2 2 3 H 3 H Lemma 4: I f —w > 0 and — — < 0 then the i t e r a t i v e s o l u t i o n of a y 2 3x3y 2.4.7 and 2.4.8 i s unique and i t i s a minimum. Proof: In Appendix I I . Returning now to the d i s c r e t e r a t e problem of equations 2.4.1 and we can give the r e q u i r e d c o n d i t i o n s f o r a s o l u t i o n , df d P e Theorem: I f — • — — i s monotonically i n c r e a s i n g w i t h u, and i f dP du J . e the c o n d i t i o n a l c.d.f. F i (a|^) i s monotonically decreasing w i t h r, and i f the p.d.f. P z(?) i s non-impulsive, then f o r a unique mini mizing s o l u t i o n to the d i s c r e t e m i n i m i z a t i o n problem of equations 2.4.1 and 2.4.2 i t i s necessary and s u f f i c i e n t t h a t r. and z. s a t i s f y the f o l l o w i n g equations: i / { f a , r . ) + r. | 7 - ( C , r i ) - XJP (c)ds = 0 2.4 z i - l 1 r i [ f ( z 1 , r i ) - X ] - r i + 1 [ f ( z i » r i + 1 ) - ^ = 0 2.4.10 where X i s s e l e c t e d to s a t i s f y 2.4.2. Proof: I f we make the i d e n t i f i c a t i o n s x = z, y = r , and H(x,y) = r f ( z , r ) - Xr i t i s o b v i o u s l y necessary that 2.4.9 and 2.4.10 ?the analogs of 2.4.7 and 2.4.8, be s a t i s f i e d f o r each r a t e r ^ and each s w i t c h i n g t h r e s h o l d z^ under c o n s i d e r a t i o n . Next, from Lemmas 1, 2, and 3, the s o l u t i o n of 2.4.7 and 2.4.8 i s unique. F i n a l l y , i n the proof of Lemma 1 i t was shown that the second p a r t i a l d e r i v a t i v e — 9 [ r f ( z , r ) - Xr]> 0 everywhere, and i n the 3r proof of Lemma 2 i t was shown that the mixed d e r i v a t i v e - — — [ r f ( z , r ) - X ] < 0 everywhere. From Lemma 4, the s o l u t i o n of o l d Z 2.4.9 and 2.4.10 i s a minimum. R e c a l l i n g the t r a d e o f f discussed i n s e c t i o n 2.1.2 between N, the number of non-zero r a t e s , and A, the time s e p a r a t i o n between r a t e changes, we note that equation 2.4.9 and 2.4.10 allow determination of the optimum r a t e f u n c t i o n f o r any given N and A. 3. ADAPTIVE RATE CONTROL FOR THE RAYLEIGH FADING CHANNEL I t i s the purpose of t h i s chapter to examine the e f f e c t of using adaptive r a t e c o n t r o l on an important c l a s s of time v a r y i n g channels known as Rayleigh f a d i n g channels. S e c t i o n 3.1 contains a d e t a i l e d v e r s i o n of the system model developed i n Chapter 2. I n s e c t i o n 3.2, the performance of the optimum v a r i a b l e r a t e system i s analyzed and compared to e x i s t i n g t ech-niques w i t h p e r f e c t feedback c o n d i t i o n s ; and i n s e c t i o n 3.3, w i t h v a r i o u s types of imperfect feedback a c t i n g i n i s o l a t i o n . The e f f e c t of m u l t i p l e sources of imperfect feedback, and the s e l e c t i o n of the optimum s e r v i c e b i t r a t e are i l l u s t r a t e d i n s e c t i o n 3.4. F i n a l l y , s e c t i o n s 3.5 and 3.6 c o n t a i n a numerical example and a d i s c u s s i o n of the r e s u l t s , r e s p e c t i v e l y . 3.1 The System Model f o r Rayleigh Fading Channels  3.1.1 Modulation System mi s s i o n i s p o s s i b l e at VHF (30-300 MHz) and UHF (300-3000 MHz) i f low-angle r e f l e c t i o n s from the ionosphere or troposphere are used. Since at these frequencies the wavelength i s comparable to the s c a l e of i r r e g u l a r i t i e s i n the atmosphere due to turbulence, a l a r g e number of s c a t t e r e d paths i s present, causing a c h a r a c t e r i s t i c a l l y e r r a t i c r e c e i v e d s i g n a l s t r e n g t h . A s i m i l a r phenomenon occurs i n atmospheric propagation of HF (3-30 MHz) s i g n a l s . D i v e r s i t y t r a n s m i s s i o n has been the technique most w i d e l y used to improve -r e c e p t i o n on these f a d i n g s c a t t e r channels. I t w i l l be seen, however, that the v a r i a b l e r a t e scheme provides an e x c e l l e n t a l t e r n a t i v e method of coping w i t h the f a d i n g phenomenon. For s i m p l i c i t y , the complex r e p r e s e n t a t i o n of s i g n a l s w i l l be used, so that a bandpass s i g n a l y ( t ) can be w r i t t e n where the complex waveform Y ( t ) i s c a l l e d the complex envelope of y ( t ) . For narrowband s i g n a l s , the magnitude of Y ( t ) i s the conventional envelope of y ( t ) , and the angle of Y ( t ) i s the phase of y ( t ) w i t h respect to the c a r r i e r phase to t . For wideband s i g n a l s the above r e p r e s e n t a t i o n s t i l l h o l d s , but the c H i l b e r t transform i s r e q u i r e d f o r an i n t e r p r e t a t i o n of Y ( t ) [13]. In the e a r l y 1950' s i t was r e a l i z e d that beyond-the-horizon t r a n s -y ( t ) = Re Y ( t ) e j a ) C t 3.1.1 G(t) f i l t e r matched take to M ( t ) ,modulus f i l t e r matched to M ( t ) take modulus | sample • at s e l e c t l a r g e r b) COSU) t c G-(t) g a i n a ( t ) , phase s h i f t o(t) N ( t ) G*(t) f i l t e r matched to M ( t ) f i l t e r matched to M ( t ) 0 cos(o) t-0) (coherent) c , . or cosco t {.incoherent) M x ( t ) F i g . 3.1.1 Modulation System f o r R a y l e i g h Fading Channels a) Complex Model — Incoherent D e t e c t i o n b) Complex Model — Coherent D e t e c t i o n c) Real Model envelope d e t e c t o r — I J sample I at s e l e c t l a r g e r envelope d e t e c t o r 1 I sample I at -4? T -3* s e l e c t l a r g e r (envelope d e t e c t o r s f o r i n coherent d e t e c t i o n only) of the channel i s to m u l t i p l y M^(t) by a complex gain G ( t ) , which i s a narrow band complex Gaussian process (see s e c t i o n 3.1.2) and i s wide sense s t a t i o n a r y . This gain changes very s l o w l y compared w i t h the pul s e r e p e t i t i o n r a t e r R At the r e c e i v e r there i s added white Gaussian noise w i t h complex av r envelope N(t) and double-ended power d e n s i t y N^/2 on both the r e a l and imaginary p a r t s . The r e c e i v e r consists, of matched f i l t e r s or c o r r e l a t o r s matched to M^Cr.) cosu^t, which are followed by envelope detectors i n the case of incoherent r e c e p t i o n . The r e s u l t s are sampled a f t e r T seconds and a d e c i s i o n device accepts the index 1 or 0 which has the l a r g e r c o r r e l a t i o n . The pulses M^CO w i l l be con s t r a i n e d to be orthogonal, s i n c e the channel introduces a random phase s h i f t . Binary phase s h i f t keyed (PSK) and b i n a r y frequency s h i f t keyed (FSK) waveforms are the u s u a l b i n a r y orthogonal waveforms employed over random phase channels. In each case, there i s a c a r r i e r component present; i f the phase d r i f t on the channel i s slow enough, t h i s c a r r i e r component can be e x t r a c t e d w i t h a very narrow band f i l t e r and used as a phase r e f e r e n c e , e n a b l i n g the r e c e i v e r to d i s c a r d the envelope detectors and use coherent d e t e c t i o n . I f the phase d r i f t i s too f a s t , or i f the a d d i t i o n a l complexity of phase t r a c k i n g i s u n d e s i r a b l e , the r e c e i v e r r e t a i n s the envelope detectors and uses incoherent d e t e c t i o n , w i t h a conse-quent decrease i n e f f e c t i v e energy or i n c r e a s e i n e r r o r p r o b a b i l i t y . The l o s s i n e f f e c t i v e energy i s not s i g n i f i c a n t [18], only about ^ db, at normal values (10-15 db) of energy-to-noise r a t i o . At lower values of energy-to-noise r a t i o i t i s more important. I t should be noted that i n the s e l e c t i o n of v a r i a b l e d u r a t i o n orthogonal waveforms i t i s impossible to f i n d non zero waveforms which s a t i s f y T C k j = ; \ ( t ) M j ( t ) d t = 2 P T 6 k j 3 ' 1 , 2 23 f o r a l l T, where P i s the t r a n s m i t t e r power. Never t h e l e s s , i n binaryPSK and bi n a r y FSK, 3.1.2 i s s a t i s f i e d i f T i s an i n t e g e r m u l t i p l e of the h a l f p e r i o d of the modulating waveform M_^(t). Since the frequency of the modulating waveforms can be much g r e a t e r than the average data r a t e when one of the r e s u l t i n g sidebands i s removed before t r a n s m i s s i o n , t h i s r e s t r i c t i o n on T i s not s i g n i f i c a n t . We can f u r t h e r note that i f c ^ C l l decays f a i r l y r a p i d l y w i t h T as i t does w i t h b i n a r y PSK and FSK, then a good approximation to 3.1.2 i s obtained even i f T i s not an i n t e g e r m u l t i p l e of the modulator h a l f - p e r i o d . 3.1.2 The S c a t t e r Channel Assuming a l i n e a r channel, and a mu l t i t u d e of s c a t t e r e r s which cause a range of s m a l l delays and doppler s h i f t s UK , we can w r i t e the complex envelope of the r e c e i v e d s i g n a l i n the absence of noise as f o l l o w s [33] where g^ i s the complex gain of the i 1 " * 1 s c a t t e r e r : \(t) = E g ± M ^ t - T ± ) e ^ V 3.1.3 i The magnitude of g i s the s t r e n g t h of the 1th s c a t t e r e d s i g n a l and the phase of i s U)c'r^' Equation 3.1.3 i n c l u d e s both time and frequency d i s p e r s i o n which cause frequency s e l e c t i v e and time s e l e c t i v e f a d i n g , r e s p e c t i v e l y . Now i f the maximum s i g n i f i c a n t incremental path delay i s much l e s s than the r e c i p r o c a l bandwidth of M^(t) , then ^ ( t ) can be approximated as R ^ t ) = M ^ t ) I g. e ~ j V 3.1.4 i or R ^ t ) = Mj c(t) • G(t) 3.1.5 24 Note that there i s no frequency s e l e c t i v e f a d i n g i n 3.1.5 and we are l e f t w i t h t i m e - s e l e c t i v e f a d i n g , the focus of t h i s t h e s i s . B e l l o [15] has used a combination of experimental and t h e o r e t i c a l evidence to o b t a i n an estimate of the mean power c o n t r i b u t e d by s c a t t e r e r s as a f u n c t i o n of the incremental delay on a t r o p o s c a t t e r l i n k . The maximum s i g n i f i c a n t delays thus found were about 0.5 usee. Since the data r a t e on a s i n g l e frequency m u l t i p l e x e d sub-channel i s u s u a l l y l e s s than 200 k b i t s / s e c , the approximation made i n 3.1.4 i s w e l l j u s t i f i e d . Intersymbol i n t e r f e r e n c e can be e l i m i n a t e d when the channel has the form of 3.1.5 simply by l e a v i n g a guard time between p u l s e s . This guard time w i l l be s m a l l compared to p u l s e d u r a t i o n f o r the same reason the approximation 3.1.4 was made. At times a phenomenon known as m u l t i p a t h r e c e p t i o n [14] occurs, i n which there are one or more s c a t t e r i n g l a y e r s each p r o v i d i n g a s u c c e s s i v e l y delayed r e f l e c t e d s i g n a l having the form of 3.1.5 w i t h s t a t i s t i c a l l y indepen-dent complex gains G ( t ) . Although m u l t i p a t h r e c e p t i o n can c o n t r i b u t e to i n t e r s y m b o l i n t e r f e r e n c e , methods e x i s t to compensate f o r i t [30,34]. The Rake system [30] i s a c l a s s i c a l example. The complex gain and p o s i t i o n of each m u l t i p a t h i s tracked and the separate s i g n a l s brought i n t o coincidence before d e t e c t i o n . I f the cross c o r r e l a t i o n f u n c t i o n of the pulses I s approximately zero f o r a l l time s h i f t s (as i n FSK) then the system i s sub-s t a n t i a l l y e q u i v a l e n t to a s i n g l e m u l t i p a t h system, although w i t h a somewhat hig h e r noise l e v e l . In view of t h i s e quivalence, and since-our i n t e r e s t i s p r i m a r i l y i n time s e l e c t i v e f a d i n g , the a n a l y s i s w i l l consider only a s i n g l e m u l t i p a t h having the simple form of 3.1.5. I f a l a r g e number of s c a t t e r e r s w i t h randout.phase are present,-and are s t a t i s t i c a l l y independent of each other, the c e n t r a l l i m i t theorem can be used [33] to show that G(t) i s a complex Gaussian process. I n a d d i t i o n , experimental evidence quoted i n . references [14, 16, 19] i n d i c a t e s t h a t , apart from d i u r n a l and seasonal v a r i a t i o n s , the envelope a and phase 9 of G are indeed Ra y l e i g h and uniform l y d i s t r i b u t e d (equation 3.1.6) as r e q u i r e d by the complex Gaussian assumption; 2 a Pa,e ( a> B ) = S ^ 3' 1' 6 In 3.1.6, B i s the second moment of a, and i s equal to twice the v a r i a n c e of the r e a l and imaginary processes u ( t ) and v ( t ) , where u ( t ) = Re[G(t)] and v ( t ) = Im[G(t)]. I t i s easy to show from 3.1.4 that the a u t o c o r r e l a t i o n f u n c t i o n G(t) G(t-ty) =0 which i m p l i e s that the a u t o c o r r e l a t i o n f u n c t i o n s R u ( Y ) = R V ^ Y ) and that R (y) = -R (Y)• In f a c t , i t can be shown that i f the a u t o c o r r e l a -vu uv t i o n f u n c t i o n G(t) G(t+y) =0 then the process Re[G(t) e ^ c ' ] , the channel response to a c a r r i e r at w , need not be wide-sense s t a t i o n a r y . I f the u s u a l c a u t o c o r r e l a t i o n f u n c t i o n G(t) G* (t+y) = ~Rn(y) i s c a l c u l a t e d from 3.1.4 the f o l l o w i n g expression i s obtained: R G(Y) = E |g.| 2 e ^ V .. i i 12 D e f i n i n g o(w), the frequency s c a t t e r i n g f u n c t i o n , as the sum of the \g^\ w i t h doppler s h i f t between w and co + dco , we can w r i t e the above equation as 00 R G(Y) = / O ( O J ) e j W Y du> - 3.1.7 —OO Noting that R_(y) = R (y) + R (Y) + j [ R (Y) - R ( Y ) ] , we see that i f R„(y) 1* u v vu uv G i s r e a l , then R (y) = R (Y) = 0, s i n c e we already have R (v) = -R (Y). vu uv vu uv F i n a l l y RQ(Y) i s r e a l i f a (to) i n the F o u r i e r transform r e l a t i o n 3.1.7 i s even i n w, a s i t u a t i o n o c c u r r i n g i n tropo s p h e r i c and moonbounce t r a n s m i s s i o n , and approximately i n the o r b i t i n g d i p o l e b e l t of p r o j e c t West Ford [33]. Since u ( t ) and v ( t ) are u n c o r r e l a t e d and Gaussian, they are independent. No d e r i v a t i o n s or experimental measurements of the a u t o c o r r e l a t i o n f u n c t i o n r u ( Y ) f o r the t r o p o s c a t t e r channel seem to have been made, but experimental evidence quoted i n [19] i n d i c a t e s a fading r a t e of 0.2 to 5Hz at a c a r r i e r frequency of 108 MH Z» Measurements made on an HF channel [20] gave a fading bandwidth of from 0.1 to 0.3 Hz. The assumption that G(t) does not change s i g n i f i c a n t l y over the d u r a t i o n of a s i n g l e pulse i s c e r t a i n l y j u s t i f i e d . The above fading r a t e s are f o r t u i t o u s . They are low enough that the combined processing time and round t r i p delay time i s u n l i k e l y to be more than a s m a l l f r a c t i o n of the f a d i n g p e r i o d . Yet they are h i g h enough that a b u f f e r s e p a r a t i n g a f i x e d r a t e source from the v a r i a b l e r a t e modulator would probably be of reasonable s i z e . 3.1.3 P r o b a b i l i t y of E r r o r When the angle of G(t) changes s l o w l y , as the above d i s c u s s i o n on fading r a t e s shows i s l i k e l y , the r e c e i v e r can e x t r a c t the c a r r i e r component from the orthogonal s i g n a l s w i t h a very narrowband f i l t e r and use i t as a phase reference. Coherent d e t e c t i o n of the orthogonal s i g n a l s can then be used, r e s u l t i n g i n the w e l l known [18] b i t e r r o r p r o b a b i l i t y P = ^ e r f c vTa 2T/(2N ) 3.1.8 e 2 o where P i s the t r a n s m i t t e r power, and a i s the instantaneous channel g a i n . I f , however, no phase reference i s employed, incoherent (envelope) d e t e c t i o n must be used, g i v i n g another w e l l known r e s u l t [18] f o r the b i t e r r o r p r o b a b i l i t y : 27 As mentioned i n s e c t i o n 3.1.1, coherent d e t e c t i o n i s not s i g n i f i c a n t l y b e t t e r 2 than incoherent d e t e c t i o n at normal values of P a T/N . o 3.1.4 P r e d i c t i o n In order to o b t a i n the p r e d i c t e d penalty of equations 2.2.3 and 2.2.4 we need the c o n d i t i o n a l p.d.f. of a ( t + t ) based on a l l the past h i s t o r y of G($) , g<t, which i s a v a i l a b l e to the r e c e i v e r . L e t us assume f o r the moment that the r e c e i v e r t r a c k s 9 ( t ) , the angle of G ( t ) , as w e l l as the mag-nit u d e a ( t ) . Then the two independent Gaussian processes u ( t ) and v ( t ) , which are the r e a l and imaginary p a r t s of G ( t ) , r e s p e c t i v e l y , can be obtained from u ( t ) = Re[G(t)] and v ( t ) = I > [ G ( t ) ] . We know that a Gaussian d i s t r i b u t i o n i s c h a r a c t e r i z e d by i t s mean and v a r i a n c e alone, and that l i n e a r p r e d i c t i o n of a Gaussian process can produce the c o n d i t i o n a l mean w i t h some f i x e d e r r o r v a r i a n c e . A c c o r d i n g l y , consider u ( t ) and v ( t ) as being l i n e a r l y f i l t e r e d s e p a r a t e l y , producing the 2 p r e d i c t i o n s or c o n d i t i o n a l means y and y w i t h mean squared e r r o r a . u v Denoting the f u t u r e value of a q u a n t i t y by the s u b s c r i p t 2, we have the p.d.f. x - y u 2 y _ y v 2 / N i - i/2 ( — - ) - i/2 e - — r P„ „ (x,y) = ~ e a e a U 2 V 2 2TTO 2iTCT 2a Changing to the p o l a r coordinates a and 9 y i e l d s 1 1 2 2 2 2 2 exp [ ^ <x + y + u u +UV ~2xy u.-2yu v) ] P 0 o ( a , 3 ) = — e x p [ \r (a2+\i 2+y . 2-2a(y . cosB + y s i n g ) ) ] 2 2 2ira 2a 1 a K — exp [ — (y^ cosg s i n g ) ] a 28 where a r 1 , 2 2 2... K = — exp .[ r- (a + y + y )] a Z 2a U V To o b t a i n the p.d.f. of a, we i n t e g r a t e over 9. , TT p (a) = K — / exp [ — (y cosS + y s i n g )]dB a„ 2TT 2 u v 2 - T T a a where I (y) i s the zero order modified B e s s e l f u n c t i o n of the f i r s t k i n d , o Hence ? 2 ^ 2 2 l i + y a + y + y / x _ a ,q/ u. v N r u v -, o i i n P a 0 ( a ) " ~1 T o ( 2 } 6 X P [ 71 ] 3 ' 1 , 1 0 , 2 a a 2a Equation 3.1.10 i s a general e x p r e s s i o n f o r the p.d.f. of a ( t + T) c o n d i t i o n e d on G(y) up to y =t. There has been no r e s t r i c t i o n so f a r on the a u t o c o r r e l a t i o n f u n c t i o n R^Cy) = R ( y ) , o t h e r than that of a fading r a t e much l e s s than the b i t r a t e . Here we w i l l suppose the a u t o c o r r e l a t i o n f u n c t i o n to be R u ( y ) = R v ( y ) = | E _ V I Y ' f o r two reasons. F i r s t , i t i s known [21] that f o r t h i s a u t o c o r r e l a t i o n f u n c t i o n (which corresponds to the s o - c a l l e d f i r s t - o r d e r Butterworth power spectrum [37])knowledge of u ( t ) i s s u f f i c i e n t f o r the p r e d i c t i o n of u ( t + r ) . Equation 3.1.10 i s s i m p l i f i e d c o n s i d e r a b l y , and i n f a c t i s dependent only on a ( t ) and not 9 ( t ) . Second, and more important, recent measurements of the time-frequency c o r r e l a t i o n f u n c t i o n on an HF l i n k between Hawaii and New Jersey [20] show an a u t o c o r r e l a t i o n f u n c t i o n R (Y) which can be c l o s e l y approximated by exp (-v|y|). Figure 3.1.2 shows one such auto-c o r r e l a t i o n f u n c t i o n taken from F i g . 3 of [20] f o r a c a r r i e r frequency of 20.4 MHz. 29 F i g . 3.1.2 A u t o c o r r e l a t i o n f u n c t i o n of u ( t ) Denoting the present value of a q u a n t i t y by the s u b s c r i p t 1, the form of the p r e d i c t o r f o r the above form of R (y) i s [21] y u = k U l ' y v = k V l 2 B t-1 1 2 ^ 0 = — (1-k ) - V T where k = e . S u b s t i t u t i o n i n t o 3.1.10 gives the c o n d i t i o n a l p.d.f. a 2 | a l (a | B ) = 2 2 2 2a T , 2kag ^ r a +k 3 o i n j - I ( ^-)exp[ 5—] 3.1.11 B ( l - l c ) ° B ( l - k ) B ( l - k ) Note that f o r t h i s Butterworth spectrum the phase 9^ does not appear i n the p r e d i c t i o n of a^j so that the r e c e i v e r need t r a c k only a ( t ) . I f the spectrum i s not Butterworth, the r e c e i v e r can s t i l l avoid t r a c k i n g 9 ( t ) , provided p r e d i c t i o n of a(t+t) i s based only on G ( t ) . 3.2 C a l c u l a t i o n s f o r P e r f e c t Feedback In a l l the c a l c u l a t i o n s of t h i s chapter, the average e r r o r pro-b a b i l i t y c r i t e r i o n , f[P ] = P £ w i l l be used because of i t s simple i n t e r p r e t a -t i o n , although c r i t e r i a such as the n ^ - moment c r i t e r i o n , f [P ] = P present no insurmountable d i f f i c u l t i e s . Let us define the normalized power-to-noise r a t i o x = P a 2 / (N R ) o av which from 3.1.6 i s e x p o n e n t i a l l y d i s t r i b u t e d : 1 _ « p (a) = — e b a>o 3.2.1 x b — where the average energy-to-noise r a t i o b = PB/N R . Then, r e c a l l i n g that O clV T = l / ( r R ) , equations 3.1.8 and 3.1.9 can be r e w r i t t e n av P = -J- e r f c /x/(2r) 3.2.2 e 2 and P = =r e 2r 3.2.2 e 2 r e s p e c t i v e l y . By p e r f e c t feedback we mean that the r a t e f u n c t i o n r ( z ) i s continuous, that there are no bounds on the s i z e of r ( z ) (other than r ( z ) >_ 0), and that the round t r i p delay T and r a t e change p e r i o d A are n e g l i g i b l e w i t h respect to the f a d i n g time constant v ^. More s u c c i n c t l y , N=°°, VT<<1, vA<<l and 0 <r(z)<°°. For the optimum v a r i a b l e r a t e system w i t h p e r f e c t feedback we can use 2.3.4 to get r=x/b which from 3.2.1 has r = l . The r e s u l t i n g e r r o r p r o b a b i l -1 - 1 - -i t i e s p e = "2 e r f c Vb/2 (coherent) and P g = — e 2 (incoherent) are p l o t t e d on Figures 3.2.1 and 3.2.2. Comparison of the f i x e d r a t e and optimum v a r i a b l e r a t e curves show a s t r i k i n g d i f f e r e n c e ; f o r t y p i c a l values of average e r r o r p r o b a b i l i t y —2 —6 (10 - 10 ) the re q u i r e d value of b = PB/N^R can be reduced by a f a c t o r of 10 to 10^, or ten to f i f t y db, by the use of the v a r i a b l e r a t e s t r a t e g y . 31 The savings are even g r e a t e r f o r lower values of e r r o r p r o b a b i l i t y . A l s o p l o t t e d on Figures 3.2.1 and 3.2.2 are curves r e p r e s e n t i n g the performance of a communication system which adapts the instantaneous power, r a t h e r than the instantaneous r a t e , i n order to minimize the average e r r o r p r o b a b i l i t y f o r a s p e c i f i e d average power. These curves were taken from Figures 2 and 4 of Hayes [8] f o r a s i n g l e m u l t i p a t h , where the q u a n t i t y 2 2k R of Hayes has been i d e n t i f i e d w i t h b of t h i s work. The minor change has been made that h i s coherent curve has been s h i f t e d 3 db to the r i g h t to represent our use of orthogonal, rather than a n t i p o d a l , s i g n a l s . The same assumptions of n o i s e l e s s d e l a y l e s s feedback were made. I t i s c l e a r t h a t , e f f e c t i v e as adaptive power c o n t r o l i s , adaptive r a t e c o n t r o l i s even more powerful. Comparison w i t h the standard technique of d i v e r s i t y t r a n s m i s s i o n i s a l s o i n t e r e s t i n g . D i v e r s i t y c o n s i s t s of t r a n s m i s s i o n of the s i g n a l s over L s t a t i s t i c a l l y independent channels which may be s p a t i a l l y , temporally o r frequency d i s j o i n t . Although they are mathematically e q u i v a l e n t , we focus on time d i v e r s i t y . I t i s evident that there i s a t r a d e o f f i n L; as L i n c r e a s e s , the e r r o r p r o b a b i l i t y decreases, but so does the data r a t e . Optimum d i v e r -s i t y c o n s i s t s of using the optimum number of re t r a n s m i s s i o n s L to minimize the e r r o r p r o b a b i l i t y f o r a given energy-to-noise r a t i o per b i t . I f the r e c e i v e r t r a cks n e i t h e r the gain a or the phase 9 on the separate t r a n s m i s s i o n s , the optimum r e c e i v e r s t r u c t u r e c o n s i s t s of s e l e c t i o n of the l a r g e r of the two sums of L sampled envelope d e t e c t o r outputs [18]. This i s known as equal gain post d e t e c t i o n combining. A d e r i v a t i o n of optimum d i v e r s i t y f o r b i n a r y orthogonal s i g n a l s i n t h i s s i t u a t i o n i s per-formed i n reference [18] which shows that the optimum L i s approximately b. The r e s u l t i n g e r r o r p r o b a b i l i t y i s shown i n F i g . 3.2.2. I t can be 32 10 l o g 1 0 b 55 10 15 20 H , 1 1 1 h 1 1 1 1 f 1 1 -i 1 j " F i g . 3.2.1 Comparison of Modulation Techniques f o r Coherent Reception of R a y l e i g h Faded B i n a r y Orthogonal Pulses w i t h P e r f e c t Feedback. 33 10 log b 5 10 i U 15 20 Fig. 3«2.2 Comparison of Modulation Techniques for Incoherent Reception of Rayleigh Faded Binary Orthogonal Pulses with Perfect Feedback. seen that adaptive r a t e c o n t r o l i s co n s i d e r a b l y more e f f e c t i v e than t h i s form of d i v e r s i t y r e c e p t i o n . Now i f the r e c e i v e r tracks a and 9 on the separate t r a n s m i s s i o n s , the samples can be brought i n t o phase coincidence and a weighted sum formed before d e t e c t i o n . Brennan [34] has shown that the optimum weight f o r each-t r a n s m i s s i o n i s the channel gain a. This form of d i v e r s i t y combination i s known as maximal r a t i o p r e - d e t e c t i o n combining (MRPDC). An a n a l y s i s of optimum MRPDC d i v e r s i t y i s contained i n Appendix I I I . I t i s shown that optimum MRPDC can achieve the same e r r o r p r o b a b i l i t y as adaptive r a t e c o n t r o l , but i t req u i r e s an i n f i n i t e number of s t a t i s t i c a l l y independent transmissions to do so. The e r r o r p r o b a b i l i t y f o r the more t y p i c a l f i g u r e of L=4 d i v e r s i t y transmissions obtained from equation A.3.4 i s shown on F i g . 3.2.2 and i s considerably worse than that o b t a i n a b l e w i t h r a t e c o n t r o l . A l s o shown i s MRPDC d i v e r s i t y w i t h the r a t h e r l a r g e value of L=10 d i v e r s i t y t r a n s m i s s i o n s . Although the MRPDC d i v e r s i t y scheme improves w i t h i n c r e a s i n g L and can be considered a competitor of the v a r i a b l e r a t e scheme, the d i s c u s s i o n i n s e c t i o n 3.5 demonstrates t h a t , f o r other reasons, the v a r i a b l e r a t e scheme i s pre-f e r a b l e . In f a c t , no scheme which uses b i n a r y pulses on a fa d i n g channel can giv e a lower e r r o r p r o b a b i l i t y than the adaptive r a t e scheme w i t h p e r f e c t feedback. I f b i s i n t e r p r e t e d as the average energy-to-noise r a t i o per b i t , then f o r a constant channel w i t h power equal to the average r e c e i v e r power PB, and r a t e equal to the average r a t e R , 3.1.8 and 3.1.9 can be w r i t t e n I 2 _ b_ P e = Y e r f c A>/2 and P^ = — e 2, r e s p e c t i v e l y , which are e x a c t l y the e r r o r p r o b a b i l i t i e s d erived f o r the adaptive r a t e scheme. At the expense of some incr e a s e i n complexity, adaptive r a t e c o n t r o l has completely e l i m i n a t e d the e f f e c t of t i m e - s e l e c t i v e f a d i n g . 35 3.3 I n d i v i d u a l E f f e c t of the Feedback Parameters In t h i s s e c t i o n we w i l l i n v e s t i g a t e the i n d i v i d u a l e f f e c t of bounds on the instantaneous r a t e , of feedback delay, of r a t e change spacing, and of q u a n t i z a t i o n of the r a t e f u n c t i o n . In s e c t i o n 3.3.4 and Appendix IV a method of c a l c u l a t i n g the optimum r a t e f u n c t i o n f o r any s e l e c t i o n of the feedback parameters i s presented. This method i s c e n t r a l to the i n v e s t i g a t i o n of t r a d e o f f s among these parameters i n s e c t i o n 3.4. 3.3.1 Bounds on the Instantaneous Rate The e f f e c t of a c o n s t r a i n t on the range of the a l l o w a b l e i n s t a n t a n -eous r a t e , that i s , e r < r < r , was considered. Note that s i n c e we must o — — o have ,r=l, there i s a r e s t r i c t i o n on the maximum r a t e r ^ f o r a given r a t e r a t i o e, namely, 1 < r < :e — o — -1 Assuming that the other feedback parameters are p e r f e c t (N = ° ° , V T << 1, vA << 1) then from 2.3.4 the optimum r a t e f u n c t i o n i s ( F i g . 3.3.1) I O — — O 3.3.1 r ( x ) = \ x/u er u < x < r u o — — o : r u < x < oo o o — where u i s chosen to make r= l (equation 2.2.6) F i g . 3.3.1 Rate Function w i t h Upper and Lower Bounds 36 I f the r e s u l t i n g p r o b a b i l i t y i s now minimized w i t h respect to r o i n the range 1 <_ r ^ <_ e \ then only the rat.e r a t i o e, the r a t i o of minimum to maximum r a t e s , a f f e c t s the performance. The e r r o r p r o b a b i l i t y , shown on F i g s . 3.3.3 and 3.3.4, i s only about two orders of magnitude lower than that f o r f i x e d r a t e n o n - d i v e r s i t y t r a n s m i s s i o n . As e incre a s e s to e=l, the system becomes i d e n t i c a l to the f i x e d r a t e n o n - d i v e r s i t y system. However, i f the t r a n s m i t t e r i s allowed to shut down, that i s , to transmit at r a t e r=0, more encouraging r e s u l t s are obtained. The r a t e f u n c t i o n now has the form ( F i g . 3.3.2) 0 x < x r ( x ) =^ o x < x < er u o — — o x/u er u < x < r u o — — o r r u < x < <» o o — where x s a t i s f i e s P (x ,er ) = X" (see 2.4.10) and u, which i s r e l a t e d to X o e o o ' through 2.3.4, i s s e l e c t e d to make r = l . Both r Q and e a f f e c t the e r r o r per-formance . F i g . 3.3.2 Rate Function w i t h r=o Allowed I f we again minimize the e r r o r p r o b a b i l i t y w i t h respect to the maximum r a t e r , then only the r a t e r a t i o e i n f l u e n c e s the e r r o r curve. This e r r o r curve i s di s p l a y e d i n F i g s . 3.3.3 and 3.3.4, again f o r the l a r g e value e=0.1, and i t can be seen to be only 0.2db worse than that o b t a i n a b l e w i t h no r a t e c o n s t r a i n t , As e increases to e=1.0, the system becomes i d e n t i c a l to the s i n g l e non-zero 37 10 10 l o g 1 ( ) b 15 20 F i g * 3«3«3 E f f e c t of a Rate Range C o n s t r a i n t on the Performance of a V a r i a b l e Rate System Using Incoherent D e t e c t i o n o f R a y l e i g h Faded P u l s e s . 3b Fig. 3.3.^ Effect of a Rate Range Constraint on the Performance of a Variable Rate System Using Coherent Detection of Rayleigh Faded Pulses. r a t e system of s e c t i o n 3.3.4, which i s s t i l l a s i g n i f i c a n t improvement over the f i x e d r a t e n o n - d i v e r s i t y system. For e <^0.01, there was no p e r c e p t i b l e degradation from the v a r i a b l e r a t e system w i t h no r a t e c o n s t r a i n t s . The importance o f a l l o w i n g the system to cease t r a n s m i s s i o n i n bad channel c o n d i -t i o n s has been demonstrated. Henceforth no c o n s t r a i n t s w i l l be placed on the al l o w a b l e r a t i o of r a t e s , s i n c e i t w i l l be assumed from the above d i s c u s s i o n that such a con-s t r a i n t , f o r reasonable values of e, w i l l not a p p r e c i a b l y a f f e c t the r e s u l t s . The maximum r a t e r ^ i s s t i l l a r a t h e r important parameter. Since the system uses a v a r i a b l e t r a n s m i s s i o n r a t e , the bandwidth used f l u c t u a t e s , and i s greater than the average bandwidth R some f r a c t i o n of the time. The parameter r Q i s the maximum bandwidth expansion, and as such i t determines the frequency spacing of frequency m u l t i p l e x e d s u b c a r r i e r s . To i n v e s t i g a t e the e f f e c t of r Q alone, we use the r a t e f u n c t i o n 3.3.1 w i t h e=0. Again u i s chosen to make r = 1 i n equation 2.2.6 and the e r r o r p r o b a b i l i t y evaluated from 2.2.5. Displayed on F i g . 3.3.5 and 3.3.6 are the e r r o r curves w i t h r =2 and r =4. Although the bandwidth expansion r has a very minor e f f e c t o o r o on performance, t h i s upper L>ound should be i n c l u d e d i n the c a l c u l a t i o n of the optimum r a t e f u n c t i o n i n any p r a c t i c a l s i t u a t i o n . 3.3.2 E f f e c t of Feedback Time Delay In t h i s s e c t i o n we wish to evaluate the e f f e c t on e r r o r performance of a delay t between the r a t e request and the time i t takes e f f e c t , which i s s i g n i f i c a n t compared to the f a d i n g time cons tant, w h i l e the other feedback parameters are p e r f e c t (N= °°, vA =0, r Q= «) . In order to o b t a i n the optimum ra t e f u n c t i o n from equation 2.3.1, we need the p r e d i c t e d penalty f [ z , r ] of 2.2.3, which f o r the average e r r o r p r o b a b i l i t y c r i t e r i o n i s f[z,r]=P [ z , r ] . 40 5 10 1 0 l o g i o b 15 20 ^ — , — I — I — I — I — , — , — I — I — I 1 — I — I — I — I — Fig, 3«3»5 Effect of Bandwidth Expansion Constraint r Q on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses. 41 F i g . 3.3.6 E f f e c t of Bandwidth Expansion Constraint r Q on the Performance of a V a r i a b l e Rate System Using Coherent Detection of Rayleigh Faded Pulses. 42 I d e n t i f y i n g z w i t h the channel gain a^ at time t , we have 00 P (f3,r) = / P (a,r) p , (a|g)da 3.3.2 a2 l a l which, f o r incoherent r e c e p t i o n , can be w r i t t e n : co 2 2 2 2 P (B,r) = / x- I ( r-) expf : r - ]da o B ( l - k ) ° B ( l - k ) - B ( l - k ) 2r R N av o from equations 3.1.11 and 3.2.3. This complicated i n t e g r a l i s of a form evaluated i n Abromowitz and Stegun [22] as t h e i r equation 11.4.29. The i n t e g r a l i s 2 2 A r R N k P a, T> t \ av o r 1 , P (a ,r) = : — exp [ =-] 2r R N +PB(l-k ) 2r R N +PB(l-k ) av o av o 2 Upon n o r m a l i z a t i o n w i t h x = Pa, /(N R ) we o b t a i n 1 o av 2 P (x,r) = j- • exp[ 7 ] 3.3.3 2r+b(l-k ) 2r+b(l-k ) Note that now x takes the place of z i n Chapter 2. The i m p l i c i t equation f o r r ( x ) can be obtained by s u b s t i t u t i o n of 3.3.3 i n t o 2.3.1, r e s u l t i n g i n r - k 2 x ., exp [ =-] 2r+b(l-k ) ( r [ 2 r b ( l - k 2 ) + b 2 ( l - k 2 ) 2 + 2 r k 2 x ] 2 r + b ( l - k 2 ) [ 2 r + b ( l - k 2 ) ] 3 3.3.4 = ^ w i t h A s e l e c t e d to make r i n 2.2.6 equal to 1. The corresponding equation f o r coherent r e c e p t i o n was not obtained, s i n c e the author could not evaluate 3.3.2 i n a c l o s e d form. A numerical i n -t e g r a t i o n would have been p o s s i b l e , but f a r too c o s t l y i n computer time. Since coherent d e t e c t i o n i s not s i g n i f i c a n t l y b e t t e r than incoherent d e t e c t i o n at normal energy-to-noise r a t i o s , the e f f e c t of delay on coherent d e t e c t i o n was not i n v e s t i g a t e d . 43 10 1 0 l o g 1 0 b 15 20 — I 1 1 1 1 + i i 1 1 r F i g . 3.3.7 E f f e c t of Feedback Delay o f Fading Time Constants on the Performance of a V a r i a b l e Rate System Using Incoherent D e t e c t i o n of R a y l e i g h Faded P u l s e s . I t i s i n t e r e s t i n g to note that f o r k=e < 1, the optimum r a t e f u n c t i o n computed from 3.3.4 does not go to zero at x=0, s i n c e even when the current channel gain i s zero, there i s s t i l l a f i n i t e p r o b a b i l i t y of a non-zero gain T seconds i n the f u t u r e . As one would expect, e v a l u a t i o n of the e r r o r p r o b a b i l i t y r e s u l t i n g from the use of the optimum r a t e f u n c t i o n of 3.3.4 had to be performed numer-i c a l l y . I t was done, i n f a c t , by using the d i s c r e t e r a t e s o l u t i o n i n s e c t i o n 3.3.4, s i n c e any numerical s o l u t i o n would quantize the ra t e s i n some f a s h i o n , and the d i s c r e t e r a t e s o l u t i o n quantizes them o p t i m a l l y . I n s p e c t i o n of F i g . 3.3.9 shows th a t f o r 16 non-zero ra t e s the e r r o r p r o b a b i l i t y i s a c l o s e approximation to the continuous r a t e f u n c t i o n e r r o r p r o b a b i l i t y , so the curves presented i n F i g . 3.3.7 f o r x~v ^  are the d i s c r e t e r a t e s o l u t i o n s f o r N=16, and various x. Figure 3.3.7 shows c l e a r l y that the presence of delay s e r i o u s l y degrades the performance of the adaptive r a t e system. Comparison w i t h F i g . 3.2.2 shows that a delay of 0.05 fa d i n g time constants causes an e r r o r p r o b a b i l i t y only s l i g h t l y lower than the MRPDC d i v e r s i t y system w i t h 4 d i v e r -s i t y t r a n s m i s s i o n s . However, t a k i n g the t y p i c a l f i g u r e of v=0.2Hz f o r a 5000 m i l e HF l i n k between Hawaii and New Jersey [20], we can c a l c u l a t e vx=0.01, f o r which the e r r o r curve i s shown on F i g . 3.3.7. I f t h i s curve i s compared to those of F i g . 3.2.2, i t can be seen that the performance of an adaptive r a t e c o n t r o l system on t h i s l i n k i s p o t e n t i a l l y somewhat b e t t e r than t h a t of an MRPDC system w i t h 10 d i v e r s i t y t r a n s m i s s i o n s . 3.3.3 E f f e c t of Rate Change Spacing In t h i s s e c t i o n , the e f f e c t of a.rate change p e r i o d A which i s com-parable to the f a d i n g time constant v ^ w i l l be i n v e s t i g a t e d w i t h otherwise F i g . 3.3.8 E f f e c t of Rate Change Spacing of 2<d Fading Time Constants on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses. p e r f e c t feedback (N=°°, vx~0, ro=°°) . We must f i r s t o b t a i n the p r e d i c t e d penalty of equation 2.2.4. This p r e d i c t e d penalty i s j u s t the p r e d i c t e d penalty of 2.2.3 ( f o r A=0) averaged over r R A values of delay i n the range [x,x+A]. We can then w r i t e the p r e d i c t e d 3 . V penalty from 2.2.4 and 3.3.3 as rR A . 2 , av k. x P e ( x ' r ) = r ¥ ~ T 2 ~ r e x p C J _ 2~ ] av j = l 2r+b(l-k. ) 2r+b(l-k. ) where k. = exp [-v(x + j / ( r R ) ] . The feedback delay x has been r e t a i n e d j av here f o r g e n e r a l i t y , although the curves presented i n t h i s s e c t i o n assume vx << 1. These curves were evaluated by d i v i d i n g the i n t e r v a l [x,x +A] i n t o 5, r a t h e r than r & a vA s u b i n t e r v a l s , to make the computer program more eco-nomical. Hence the p r e d i c t e d penalty used was , 5 k. 2x P (x,r) = — E - ry exp [ J r-] 3.3.5 j=0 2r+b(l-k. ' 2r+b(l-k. ) 3 J w i t h k^ = exp [-v(x + j A / 5 ) ] . As i n s e c t i o n 3.3.2, the e r r o r performance was c a l c u l a t e d n u m e r i c a l l y using the d i s c r e t e r a t e s o l u t i o n of s e c t i o n 3.3.4 w i t h N=16 non-zero r a t e s . The r e s u l t s f o r some values of vA w i t h vx~0.are d i s p l a y e d on F i g . 3.3.8. I t can be seen that the r a t e change spacing has a s i m i l a r e f f e c t to that of the feedback delay, although not as severe. There i s the p o s s i b i l i t y of a t r a d e o f f here. For a given feedback channel c a p a c i t y , one could use a s m a l l number of non-zero r a t e s , but change the r a t e o f t e n , or a l a r g e number of non-zero rates changed i n f r e q u e n t l y . This t r a d e o f f and others w i l l be i n v e s t i g a t e d i n s e c t i o n 3.4. 3.3.4 E f f e c t of Rate Q u a n t i z a t i o n In t h i s s e c t i o n the e f f e c t of a l l o w i n g only a f i n i t e number N of 46 non-zero t r a n s m i s s i o n rates i s i n v e s t i g a t e d . The equations presented a l l o w the feedback delay x and the r a t e change p e r i o d A to be comparable to the -1 fa d i n g time constant v , and r Q to be f i n i t e , to i n d i c a t e how the optimal r a t e f u n c t i o n i s obtained when two or more feedback parameters are i m p e r f e c t . The graphs presented here, however, assume vx~0, ' vA = 0 i n order to d i s p l a y the i s o l a t e d e f f e c t of a f i n i t e N. We know that the optimum d i s c r e t e r a t e f u n c t i o n must s a t i s f y equations 2.4.9 and 2.4.10. Note that s i n c e the p r e d i c t e d p e n a l t y f i s used, the e f f e c t s of delay and the r a t e change spacing can be i n c l u d e d . We s u b s t i t u t e the p r e d i c t e d e r r o r p r o b a b i l i t y of 3.3.5, which contains both x and A, and the p.d.f. of 3.2.1, i n t o equations 2.4.9 and 2.4.10 and perform some s t r a i g h t - f o r w a r d i n t e g r a t i o n s to o b t a i n the c o n d i t i o n s to be s a t i s f i e d by the optimum d i s c r e t e rate f u n c t i o n . I f we de f i n e _ x 5 _ £x_ 2 r b k . 2 . u.+b(l-k. 2) 2 r k . 2 $(x,r,A) = Ae b - -7- E e bu. [ 1-+ —1 3 + jL- x] 6q j u.q - u. 2 J-0 1^ J u and , 2 c k. x r r I 1 iKr,x,A) = r<X - 7- E — e u 1=0 1 6 . „ u j 2 where k. = exp[-v(x +JA/5)], q = 2r+b, u. = q - b k. , then the c o n d i t i o n s 1 1 1 2.4.9 and 2.4.10 can be w r i t t e n as $ ( x i , r 1 , X ) = l ( x 1 _ 1 , r i , A ) 3.3.6 ty(r±+1,x±,\) = i|/(r 1,x 1,X) 3.3.7 where X q=0, =co, ^=0, and A i s s e l e c t e d to make r = l . Equation 3.3.6 must be s a t i s f i e d f o r each r_^ allowed to vary, and equation 3.3.7 f o r each x. allowed to vary. I f any r. or x. i s a constant, then the corresponding equation does not have to be s a t i s f i e d . 10 10 iog 1 0b 15 20 Fig. 3.3«9 Effect of Rate Quantization to N Non-zero Rates on the Performance of a Variable Rate System Using Incoherent Detection of Rayleigh Faded Pulses. 48 5 10 1 0 l o g10* 15 20 H 1 1 1 1—^—-j 1 1 1 1 1 1 1 1 r Pig. 3.3.10 Effect of Rate Quantization to N Non-zero Rates on the Performance of a Variable Rate System Using Coherent Detection of Rayleigh Faded Pulses. The lowest r a t e , r ^ , has been s e t to zero, f o r reasons o u t l i n e d i n s e c t i o n 3.3.1. The i t e r a t i o n was performed on r a t e and the t h r e s h o l d x^ was c a l c u l a t e d from 3.3.7 w i t h i = l . The r e s t of the r . and x. are obtained 1 l by s o l v i n g 3.3.6 and 3.3.7 a l t e r n a t e l y , and a d j u s t i n g r ^ so that 3.3.6 w i t h i=N+l i s s a t i s f i e d . A somewhat more d e t a i l e d d e s c r i p t i o n of the s o l u t i o n , i n c l u d i n g the upper bound r , can be found i n Appendix IV. The e f f e c t of the d i s c r e t e nature of the r a t e f u n c t i o n alone i s shown when vx=0 and vA=0. Equations 3.3.6 and 3.3.7 s i m p l i f y c o n s i d e r a b l y and t h e i r s o l u t i o n gives the r e s u l t s shown i n F i g . 3.3.9 f o r incoherent r e -c e p t i o n . I t can be seen that even an i n t e r m i t t e n t (N=l) system provides a very s u b s t a n t i a l improvement over a f i x e d r a t e system. A d d i t i o n of more rates f u r t h e r improves performance, and at N=16 there i s only a 0.1 db de-gradation from the continuous r a t e f u n c t i o n . Results f o r coherent d e t e c t i o n can be obtained f o r vx=0 and vA=0 i n an analogous manner, by d e f i n i n g i n equations 3.3.6 and 3.3.7 the f u n c t i o n s — 1 7 - — 1 / — $(r,x,A) = Xe b - e r f c (/- ) e b + -=- ^  e r f c ( / — ) x I r 2 rs + 2tT/7 q ^ t- e ~ q X - ^ • e r f c (/^x" ) ll>(x,r,X) = r [ | e r f c ( / | ~ ) -X] where s=b/(b+r), and q= l / ( r s ) . Figure 3.3.10 d i s p l a y s the performance of a system using b i n a r y orthogonal s i g n a l s w i t h coherent d e t e c t i o n and va r i o u s numbers of non-zero r a t e s . The same general features a s . i n incoherent d e t e c t i o n are e x h i b i t e d , s i n c e the two forms of d e t e c t i o n are a s y m p t o t i c a l l y e q u i v a l e n t . 50 3.4 Adaptive Rate C o n t r o l i n a Two-Way S i t u a t i o n At t h i s p o i n t , we have the a b i l i t y to c a l c u l a t e the optimum r a t e f u n c t i o n r ( x ) and the minimum e r r o r p r o b a b i l i t y P^ as a f u n c t i o n of the average energy-to-noise r a t i o per b i t b f o r any choice of the feedback parameters r ^ , N, x, A , and the e f f e c t of each of these has been examined i n d i v i d u a l l y . Now i f more than one of these parameters i s n o n - i d e a l , then we might ask what t r a d e o f f s we can expect i n those p a r t i c u l a r v a r i a b l e s we are f r e e to choose i n a system design. The t r a d e o f f s have been mentioned before, i n s e c t i o n s 2.1.2 and 3.3.3; i t i s our i n t e n t i o n here to di s c u s s them i n d e t a i l . Let us consider a s i t u a t i o n i n which s t a t i o n A and s t a t i o n B transmit to each other over f a d i n g channels, so that A's forward channel can act as B's feedback channel, and v i c e v e r s a (see F i g . 3.4.1). Every A seconds the message b i t stream i s i n t e r r u p t e d and the s e r v i c e b i t s ( r a t e request) i n s e r t e d at the current r a t e . S e r v i c e i n f o r m a t i o n reduces the message data r a t e , so the problem i s to determine the optimum amount of s e r v i c e i n f o r m a t i o n , that i s , to p i c k N and A which y i e l d the most e f f i c i e n t system. Trans-mitter A message A B N . — s e r v i c e BA U — A A » Re-ceiver B Channel AB Channel BA Re- . ceiver A w - A — & — A *-message B A ^ — ^ s e r v i c e AB ^ F i g . 3.4.1 Two-Way Communication System Trans-mitter B 51 We assume that channel AB and channel BA are s t a t i s t i c a l l y indepen-dent, s i n c e even i f they occupy the same s c a t t e r volume, the centre frequen-c i e s w i l l u s u a l l y be s u f f i c i e n t l y f a r o f f s e t that the fades w i l l be uncorre-l a t e d . Thus we can consider the channels s e p a r a t e l y . Each time frame of d u r a t i o n A contains T seconds of s e r v i c e i n f o r -s mation and T seconds of message i n f o r m a t i o n ( F i g . 2.3.11). I f PCM t r a n s m i s s i o n m 1 T °< s message s e r v i c e ^  £ . «-F i g . 3.4.2 Time Frame A l l o c a t i o n of the r a t e request i s used, that i s , the request i s d i g i t i z e d w i t h no redun-dancy removal, then the s e r v i c e i n f o r m a t i o n b i t r a t e i s RS=M log2(N+l)/A , where each s e r v i c e b i t i s repeated M times f o r p r o t e c t i o n . This elementary "coding" i s meant to ensure that s e r v i c e i n f o r m a t i o n i s r e c e i v e d c o r r e c t l y . We have T = A-T = A-M log„(N+l)/(R r) m s z av so that the message b i t r a t e i s R = T~R r/A = R - M log„(N+l)/A m m av av °2 which gives the f o l l o w i n g expected r e s u l t R = R + R 3.2.8 av m s 52 In the design of a communication system we might reasonably expect to be able to choose the number of non-zero r a t e s N, the r a t e change p e r i o d A, the maximum bandwidth expansion r Q and the power P . The parameters more l i k e l y to be s p e c i f i e d i n advance are the b i t e r r o r p r o b a b i l i t y P , the message r a t e R and the feedback delay x. I n such circumstances a graph d i s -m p l a y i n g a l l combinations of the feedback parameters b, vA, and N which g i v e a s p e c i f i e d e r r o r p r o b a b i l i t y P £ f o r a s p e c i f i e d delay vx, and bandwidth expansion r , i s l i k e l y to prove u s e f u l . F i g . 3.4.3 i s such a graph f o r P^ = 10~ 6, VT=0.01, and r = 2. o As an example of the use of F i g . 3.4.3, suppose that a t r o p o s c a t t e r system must be designed to handle a data r a t e R at an average b i t e r r o r m p r o b a b i l i t y of 10 ^. The r o u n d - t r i p propagation delay i s 0.01 f a d i n g time constants and the maximum bandwidth expansion r i s 2. We wish to s e l e c t o q - log„(N+l) and A so that the r e q u i r e d power-to-noise r a t i o PB/N i s 2. o minimized. From the d e f i n i t i o n b = PB/N R and from 3.4.1: o av PR b(q,A) (R m + Mq/A) = ~^ 3.4.2 o where b(q,A) i s the f u n c t i o n shown on F i g . 3.4.3. To c a l c u l a t e the optimum q and A, we f i r s t p i c k q = 1, then f i n d the A which minimizes the l e f t s i d e of 3.4.2. Next, the procedure i s repeated f o r q = 2, q = 3 and so on to q~ 4 a f t e r which the e f f e c t on e r r o r p r o b a b i l i t y of i n c r e a s i n g " ^ i s n e g l i g i b l e . PB The q, A combination which gives the lowest value f o r — i s optimum. o An a l t e r n a t i v e use of F i g . 3.4.3 occurs i n the s i t u a t i o n where, • . -t r . . . . PB , _ , , _ _ . PB i n s t e a d of minimizing — f o r a given R , we maximize R f o r a given — . I n N ° m m N o o t h i s case, we rearrange 3.4.2 and f i n d the q, A combination which maximizes _ PB . i _ "Ma m N q b(q,A) A Fig. 3A.3 Two-Way System Paramenters b , H , A which Produce an Error Probability of 10"^ on a Rayleigh Fading Channel with Feedback Delay t = 0.01 Fading Time Constants, and Maximum Bandwidth Expansion r Q = 2. P i g . jA.k Two-Way System Parameters b , N , A which Produce an Error P r o b a b i l i t y of 10 on a Rayleigh Fading Channel with Feedback Delay c = 0.02 Fading Time Constants, and Maximum Bandwidth Expansion r Q = 2 . 55 A t h i r d way i n which F i g . 3.4.3 can be used i s the s i t u a t i o n i n which, as w e l l as a c o n s t r a i n t on the maximum bandwidth expansion, there i s a con-s t r a i n t on the maximum bandwidth, r R . This i m p l i e s a s p e c i f i e d R . I f o av av PB/N i s f i x e d (thus f i x i n g b) then we wish to maximize R = R - N q/A. o o t m av We simply c a l c u l a t e the value of b and then, on F i g . 3.4.3, c a l c u l a t e q/A f o r each of the fou r values of q at that b. The q, A combination thus found which minimizes q/A i s optimum. A l t e r n a t i v e l y , i f P B / N q i s not f i x e d (so n e i t h e r i s b ) , and R = M q/A (and thus R ) i s f i x e d , we can minimize the s m r e q u i r e d value of P B / N q by f i n d i n g the q, A (A = Mq/Rg) p o i n t on F i g . 3.4.3 which has the minimum value of b(q,A). F i g . 3.4.4 i s another graph l i k e that of F i g . 3.4.3, but f o r vx = 0.02. 3.5 A Numerical Example Suppose a modulation system i s to be designed f o r a 300-mile t r o p o s c a t t e r l i n k which has a fading bandwidth v = 3 Hz. Each of s e v e r a l frequency m u l t i p l e x e d subchannels i s to have a message data r a t e R^ = 100 — —6 k b i t s / s e c , an e r r o r p r o b a b i l i t y P ^ = 10 , and a maximum bandwidth expansion r Q = 2. What choice of N and A gives the minimum r e q u i r e d r e c e i v e r power-to-noise r a t i o P B / N ? o F i r s t we p i c k M = 3 so that a double e r r o r i s r e q u i r e d f o r a mis-—6 take i n a s e r v i c e b i t . The b i t e r r o r p r o b a b i l i t y i s already 10 , so that we can n e g l e c t the p o s s i b i l i t y of i n c o r r e c t r e c e p t i o n of the r a t e request. Now s i n c e vx = 0.01, F i g . 3.4.3 summarizes the necessary i n f o r m a t i o n . We w i l l t r y to minimize P B / N q = b(q,A) (10~* + 3q/A) . On the q=l curve, t r i a l and e r r o r gives a minimum value f o r P B / N of 9.1 x 10 , a t t a i n e d a t A = .75 -3 x 10 . S i m i l a r l y , on the q = 2 curve the minimum value of P B / N i s 6.5 x o —6 —3 10 , al s o at A =.75 x 10 . Continuing i n t h i s f a s h i o n , we f i n d the minimum 6 ~*3 value of PB/N i s 6.03 x 10 , a t t a i n e d a t q = 4, and A = 1.25 x 10 . o Let us compare t h i s r e s u l t w i t h the re q u i r e d PB/N^ f o r an MRPDC d i v e r s i t y system w i t h an eq u i v a l e n t bandwidth expansion; that i s , a d u a l -frequency d i v e r s i t y system. From equation A.3.3, the e r r o r p r o b a b i l i t y i s : P = I ( 1 + Ll PA-2 e 2 U 4 N o —1 —6 S u b s t i t u t i n g T = 100 k b i t s / s e c , and P = 10 , the necessary PNR, PB/N , c e o g i s obtained as 2.8 x 10 . Thus at the same data r a t e and e r r o r p r o b a b i l i t y , and f o r only 9.6% more bandwidth (caused by the s e r v i c e i n f o r m a t i o n ) , the v a r i a b l e r a t e scheme e f f e c t s the remarkable saving of 16.7 db i n r e q u i r e d t r a n s m i t t e r power. Now, as a second example, l e t us suppose that bandwidth c o n s t r a i n t s are very t i g h t and we can al l o w only a 1% i n c r e a s e i n bandwidth f o r the -3 s e r v i c e i n f o r m a t i o n . Thus R = 0.01-R = 1 k b i t / s e c , and A = 3q x 10 s m Examining F i g . 3.4.3 w i t h these q,A p a i r s , we see that the minimum value f o r -3 b i s 105, and occurs at q = 1, A = 3 x 10 . Then s i n c e R =R +R =101 k b i t s / av m s sec, we f i n d that PB/N = 1.06 x 10 7. This i s s t i l l a saving of 14.2 db o ° from a dual-frequency MRPDC d i v e r s i t y system. I t should be noted that the above values are somewhat c o n s e r v a t i v e , s i n c e the fa d i n g bandwidths encountered on t r o p o s c a t t e r channels are very o f t e n l e s s than 3 Hz. 3.6 D i s c u s s i o n We have seen that when feedback c o n d i t i o n s are p e r f e c t , adaptive r a t e c o n t r o l can completely e l i m i n a t e the e f f e c t of f a d i n g , thereby saving 10 - 50 db i n energy-to-noise r a t i o per b i t over a f i x e d r a t e n o n - d i v e r s i t y system f o r t y p i c a l values of e r r o r p r o b a b i l i t y . In view of the f a c t that 57 t r o p o s c a t t e r systems are c h a r a c t e r i z e d by very h i g h t r a n s m i t t e r power, t y p i c a l l y 10-50 kwatt, t h i s saving i s most a t t r a c t i v e . The disadvantage of the v a r i a b l e r a t e scheme i s that i t expands the bandwidth; f i r s t , because i t i s a v a r i a b l e r a t e scheme, and second, because s e r v i c e i n f o r m a t i o n must be t r a n s -m i t t e d . However, t h i s disadvantage i s minor, s i n c e a l l o w i n g a maximum bandwidth expansion of only 2 r e s u l t s i n a mere 0.9 db degradation from the i n f i n i t e bandwidth case. The c l o s e s t competitor i s maximal-ratio p r e d e t e c t i o n combined (MRPDC) d i v e r s i t y t r a n s m i s s i o n , which r e q u i r e s both the gain and the phase to be tracked on each t r a n s m i s s i o n . D i v e r s i t y t r a n s m i s s i o n here means frequency d i v e r s i t y . Space d i v e r s i t y would r e q u i r e s e v e r a l p h y s i c a l l y separated antennas, a consi d e r a b l e expense. Time d i v e r s i t y not only r e q u i r e s b u f f e r storage at the t r a n s m i t t e r (as does r a t e c o n t r o l ) , but f o r these s l o w l y f a d i n g channels r e q u i r e s storage of very many analog values at the r e c e i v e r . We are l e f t w i t h frequency d i v e r s i t y , which a l s o i n c r e a s e s the bandwidth. A comparison of a v a r i a b l e r a t e system to an MRPDC system, i n a r e a l - l i f e s i t u a t i o n , both w i t h the same values of e r r o r p r o b a b i l i t y , data r a t e , and bandwidth ( s e c t i o n 3.5) has shown that adaptive r a t e c o n t r o l achieves an impressive 14-17 db saving over MRPDC, i t s c l o s e s t competitor. Yet a f u r t h e r advantage a r i s e s from the f a c t t h a t , i n order to be s t a t i s t i c a l l y independent, the separate d i v e r s i t y transmissions must be widely separated i n frequency, at l e a s t g r e a t e r than the c o r r e l a t i o n bandwidth [15] of the channel. Since a t y p i c a l t r o p o s c a t t e r l i n k has a t o t a l bandwidth of about 10 MHz imposed by the k l y s t r o n c a v i t y and a c o r r e l a t i o n bandwidth of about 4 MHz [15], there i s obvious d i f f i c u l t y i n using more than dual-frequency d i v e r s i t y . Channels w i t h l a r g e r c o r r e l a t i o n bandwidths present even more problems s i n c e not a l l subchannels can use frequency d i v e r s i t y . Adaptive r a t e c o n t r o l does not s u f f e r from t h i s d i s -advantage, s i n c e the bandwidth r e q u i r e d i s not i n d i s j o i n t regions of the spectrum. The s e r v i c e i n f o r m a t i o n b i t r a t e used i n s e c t i o n s 3.4 and 3.5 i s a c t u a l l y more than r e q u i r e d , s i n c e f o r A << v \ the su c c e s s i v e r a t e requests are h i g h l y c o r r e l a t e d , and some form of redundancy removal could be used. Also the f a c t that the channel gains w i l l be s t r o n g l y c o r r e l a t e d between adjacent frequency-multiplexed subchannels again means that redundancy removal could be a p p l i e d . 4. ADAPTIVE RATE CONTROL FOR THE MULTI-USER CHANNEL In Chapter 4, we s h a l l study the improvement to. be made when the v a r i a b l e r a t e scheme i s a p p l i e d to a second important c l a s s of time-varying channels, the m u l t i - u s e r channels. Sections 4.1 and 4.2 develop a d e t a i l e d system model f o r these channels, and the model i s used i n s e c t i o n 4.3 to o b t a i n the optimum r a t e f u n c t i o n and the e f f e c t on e r r o r p r o b a b i l i t y of i t s use. Some numerical examples are performed i n s e c t i o n 4.4, and f i n a l l y , s e c t i o n 4.5 contains a d i s c u s s i o n of the r e s u l t s . 4.1 System Model f o r M u l t i - U s e r Channels  4.1.1 Forward Channel Model When users share the same time-frequency channel, i n t e r f e r e n c e r e s u l t s , causing a degradation i n q u a l i t y of the r e c e i v e d s i g n a l . However, i n some cases, an u n c o n t r o l l e d s h a r i n g s i t u a t i o n i s d e s i r a b l e , as i n m u l t i p l e access to a s a t e l l i t e repeater o r , more g e n e r a l l y , a s i t u a t i o n where a l a r g e number of p o t e n t i a l users each transmit over a s i n g l e channel only a s m a l l f r a c t i o n of the time. The r e s u l t i s a time-varying noise l e v e l . One method of m u l t i p l e x i n g f o r these channels, shown on F i g . 4.1.1, known as pseudo-noise modulation [23], c o n s i s t s of m u l t i p l y i n g the message b i t stream m(t) w i t h a high speed pseudo-random b i t stream, pn(t) which i s unique to that user, and b i n a r y phase coding a c a r r i e r w i t h the product. The r e c e i v e r c o r r e l a t e s the r e c e i v e r s i g n a l a gainst a r e p l i c a of the pseudo-noise stream and compares the r e s u l t to a t h r e s h o l d to determine the s i g n of the t r a n s m i t t e d d i g i t . The e f f e c t of pseudo-noise modulation i s to spread the s i g n a l s over a l a r g e bandwidth. A f t e r c o r r e l a t i o n , the l^1 t i l r e c e i v e r has brought the i s i g n a l back to i t s o r i g i n a l bandwidth, but s i n c e the other s i g n a l s are u n c o r r e l a t e d w i t h pn^(t) they remain spread over the pseudo-noise bandwidth, and are mostly excluded by the narrow band f i l t e r i n g a c t i o n of the c o r r e l a t o r . Two models of the channel i t s e l f w i l l be considered. A l i n e a r channel merely sums a l l the i n p u t s i g n a l s and adds white n o i s e . This model i s a p p l i c a b l e i n a s i t u a t i o n where s e v e r a l users share the same wide-band ca b l e . A hard l i m i t i n g channel, on the other hand, i s a good model of a s a t e l l i t e repeater c o n t a i n i n g a t r a v e l l i n g wave tube (TWT). In t h i s model the s i g n a l s are summed and passed through a hard l i m i t e r ( i n f i n i t e c l i p p e r ) which i s i n s t a l l e d to keep the TWT o p e r a t i n g near s a t u r a t i o n i n order to achieve e f f i c i e n t conversion of DC power to RF power. A zonal f i l t e r f o l l o w s the TWT to remove harmonics not centered on the c a r r i e r frequency and, f i n a l l y white n o i s e i s added before the c o r r e l a t o r at the r e c e i v e r . Both models w i l l be analyzed i n d e t a i l i n s e c t i o n 4.2. In the a n a l y s i s the performance of channel Q w i l l be considered. A l l the users indexed from 1 to M w i l l be c a l l e d "other users". F i g . 4.1.1. Pseudonoise Modulation System 4.1.2 Channel Usage S t a t i s t i c s Since the number of users change w i t h time, some s t a t i s t i c a l mod of the channel must be obtained. To t h i s end, the usage f u n c t i o n u_^(t) of th the i t r a n s m i t t e r i s def i n e d to be 1 when i t i s on the a i r , and 0 when i i s not. This usage f u n c t i o n i s modelled as an inhomogeneous Poisson squar wave ( F i g . 4.1.2) where i n any sm a l l time i n t e r v a l At, the p r o b a b i l i t y of a s t a t e t r a n s i t i o n i s p At i f the user i s i n s t a t e 0, and P-^At i f the user i s i n s t a t e 1. u.(t) (state l ) 1 (state 0) 0 F i g . 4.1.2 T y p i c a l usage f u n c t i o n We define P Q ( t ) as the p r o b a b i l i t y that u ( t ) = 0, P ^ t ) as the p r o b a b i l i t y that u ( t ) = 1. Then or P 1 ( t + A t) = P 0 ( t ) p 0 1 A t + P 1 ( t ) [ l - p 1 ( ) A t ] P Q ( t + A t ) = P 1 ( t ) p 1 Q A t + P Q ( t ) [ l - p 0 1 A t ] p i ( t ) ° p o ( t ) , p 0 r p i ( t ) p i o p 0 ( t ) = V ^ ' P i o - V ^ o i These equations can be so l v e d using Pg(t) + P^( t ) = 1, to o b t a i n P Q ( t ) = P Q ( 0 ) e _ ! 3 t + a 0 [ l - e " e t ] P 1 ( t ) = P^O) e " B t + a ; L [ l - e ~ B t ] 4.1 where A . A P 1 0 A P 0 1 = p i o + p o i > ao = T " 5 a i = T " and ct^, the a p r i o r i p r o b a b i l i t y of s t a t e 1, equals 1 - a^, where i s the a p r i o r i p r o b a b i l i t y of s t a t e 0. The a u t o - c o r r e l a t i o n f u n c t i o n of u ( t ) i s r e a d i l y obtained. which y i e l d s a f i r s t order Butterworth power spectrum* 2 2 a l a O ! 3 S ( f ) = a / 6(f) + 1 U U 1 D 2_l_/0 C\Z 3 +(2irf) The number of other users on the a i r at time t i s M(t) = E u . ( t ) i = l 1 where M^ , i s the maximum number of p o t e n t i a l other users. I f a l l users have the same s t a t i s t i c s the a p r i o r i p.d.f. of M i s b i n o m i a l P M ( x ) = E 1-0 i V 1 V i/ct aQ 6(x-i) where [ j / i s the usu a l b i n o m i a l c o e f f i c i e n t . For c l a r i t y we w i l l r e s t r i c t i to i n t e g e r values and w r i t e P M ( i ) =1 , / a, a„"T 4.1 *The author was shown t h i s economical d e r i v a t i o n of the spectrum of the inhomogeneous Poisson process by Dr. R.E. Burgess of the U.B.C. Physics Department. Like u ( t ) , M(t) has a f i r s t order Butterworth spectrum because i t i s a sum of independent Butterworth processes. For purposes of p r e d i c t i o n we require the co n d i t i o n a l density function P., . (J K) where K i s the value of M(t) and J i s the value of M M T.! M(t+x). Note that, because the processes are Poisson, i t i s s u f f i c i e n t to base the p r e d i c t i o n on only the most recent value of M. Now M(t) = K implies that at time t, K users are i n state 1 and K'=M^-K users are i n state 0. We define S as the sum at time t+x of the usage functions of those users i n state 1 at time t, and S' as the corresponding sum for the users i n state 0 at time t . Both S and S' are binomially d i s t r i b u t e d p s ( l ) = ( i j q n qio V j ) =(j;qoi qoo where from equations 4.1.1 q n = a i + ao 6 qio _ a o ( 1 - e } -Bx. , -Bx qoi = a i ( 1 " e } qoo = ao + a i e Since M(t+x) = S + S 1, and S and S' are s t a t i s t i c a l l y independent, r e f e r r i n g to d i s j o i n t sets of users, the co n d i t i o n a l p.d.f. of M(t+x) i s the convolution of the p.d.f.'s of S and S'. PM | M ( J I K ) ° \ n P 8<i)P 8(J-i) x1 i=0 \ / K ] / r | i K - i J - i K'-J+i . . , = ±^\tl\J-±) q n qio qoi qoo. 4 - 1 - 3 This c o n d i t i o n a l density function has the correc t behaviour at 64 where <S. . i s the Kronecker d e l t a . Also f o r t -> °°, we get PM |M<JIK> " " I " ° 0 K + K ' " J T 1 • which i s the a p r i o r i p.d.f. of M. Now that we have the r e q u i r e d s t a t i s t i c a l model of the channel usage, the form of the e r r o r p r o b a b i l i t y must be examined. 4.2 P r o b a b i l i t y of E r r o r 4.2.1 The L i n e a r Channel M (t) n (t) pn (t)«sinu) t (t) .(t) c / ( • ) at 0 c(t) F i g . 4.2.1 L i n e a r Channel The l i n e a r channel i s shown i n F i g . 4.2.1, where channel 0 has been th s i n g l e d out f o r a n a l y s i s . The i b i n a r y phase coded s i g n a l i s S! (t) = /2P? m ( t ) p n . ( t - e . ) s i n ( w t+$.) 1 1 3- 1 C 1 where m(t) = E IIL r e c t ( t - k I ) , the message b i t stream. k=l k 1 = +1, the s i g n of which-i s the i n f o r m a t i o n t r a n s m i t t e d . CO pn.(t) = E b r e c t ( t - k T .)> the pseudorandom b i t stream. k=l o b_^ = +1» the pseudonoise b i t s , s e l e c t e d equiprobably and independently of a l l other b. . r e c t , ( t ) = -A f 1 0 <_ t <_ A 0 elsewhere T = pseudorandom b i t d u r a t i o n , o T = message b i t d u r a t i o n , T = n T q w i t h n an i n t e g e r , and n>>l. 9^ = time displacement of pn_^(t) from some re f e r e n c e , u n i f o r m l y d i s t r i b u t e d on [ 0 , T ] f o r i ^ 0. For i = 0, 9 = 0 . o o. (J^ = phase s h i f t of i t h c a r r i e r , u n i f o r m l y d i s t r i b u t e d on [ — r r , i r ] f o r 1 ^ 0. For i = 0, <j> = 0 . o w = c a r r i e r frequency, w > > T ^ c ^ J ' c o The channel adds v h i t e Gaussian noise of double-ended power d e n s i t y N q / 2 , and r e c e i v e r 0 c o r r e l a t e s the re c e i v e d waveform a g a i n s t p n Q ( t ) . The c o r r e l a t o r output Q(T) i s given by M Q(T) = E C. + v i=0 1 where v, caused by the channel n o i s e , i s a zero mean Gaussian random v a r i a b l e w i t h v a r i a n c e T N /4, and o T C. = / s . ( t ) pn (t) s i n w t dt 4.2.1 1 l o c o We assume coherent detection of the c a r r i e r and pseudorandom sequence syn-chronization so that (j> = 0, 9 =0. Therefore, o o ' C = + T/P 12 o — o since wc>>>"|'> t y p i c a l l y z 1Q6/T. For i ^ 0, we have C. = /2P. / pn.(t-9.)pn (t)sin(w t+d>.)sinw t dt 1 1 1 X O c x c o n = /2~P~ Z d (9 (j, ) 1 k=l k 1 1 4.2.2 where n = T/T , and o kT o d, (9.,cj>.) = / pn.(t-0.)pn (t)sin(w t+<j).)sinw t dt (k—X) "JJ 1 1 O C 1 c The random v a r i a b l e d, (9.,d>.) i s a function of the three random processes k x l p n ^ ( t ) , Q^'^i" Since the b ^ are s t a t i s t i c a l l y independent, the l i m i t s i n the above i n t e g r a l can be changed and the i n t e r v a l broken i n t o the s t a t i s t i c a l l y independent subintervals [0,9.] and [9.,T ], as shown on F i g . 4.2.2. Then d, x x o ° k can be wr i t t e n as 0 pn.(t-0.) 1 x F i g . 4.2.2 Microscopic View of I n t e r f e r i n g Waveforms 0. = b i k - l b o k ' L s i n ( w c t ^ . ) s i n w^ dt ' o T + b.,b / sin(w t+(j).)sin w t dt 4.2.3 IK. OlC Q c i c i We wish to c a l c u l a t e the mean and v a r i a n c e of C. f o r i 4 0. From l 4.2.2, the mean of C. i s the sum of the means of the d, (9.,d>.), and from 4.2.3 the average of ^ ( B ^ , ^ ) over (J). i s zero. Hence each of the i s a zero mean random v a r i a b l e , and the mean of the c o r r e l a t o r output Q(T) = C . o I n s p e c t i o n of 4.2.3 shows that the d, (9.,d>.)» when con d i t i o n e d on k i I 9^ and <fr, are independent and i d e n t i c a l l y d i s t r i b u t e d . Thus the v a r i a n c e of C. i s p r o p o r t i o n a l to n times the vari a n c e of d„ (9. ,<b.) : l 1 i i C.2 = 2nP. d 2 ( 0 . ,d>.) 4.2.4 l i 1 l T i 2 In order to evaluate d^ (9^><j>^)j we perform the i n t e g r a t i o n i n 4.2.3, and note that over the ensemble b^^j , d^(9^,<j>^) takes on the values cosd). l 1 T — T . — w i t h p r o b a b i l i t y -r o 2 r J 4 cos<j)^ ,• -T — 7 - ^ — w i t h p r o b a b i l i t y -r o 2 r 3 4 cosd). i . . . . . . . 1 (T Q-29 i) — ^ — w i t h p r o b a b i l i t y ~r COS(}). (29 i-T Q) — ^ w i t h p r o b a b i l i t y | where, f o r s i m p l i c i t y , the f a c t that w >>T has already been used. E v a l u a t i o n c o J 2 of d^ (9.,<j>.) over the three ensembles ^ ^ Q > ^ Q Q » a n ^ $ • r e s u l t s i n d. 2 (9. ,<)>,) = T 2/48 i l l o 2 S u b s t i t u t i o n i n t o 4.2.4 y i e l d s the vari a n c e C^ . 2 The variance of the c o r r e l a t o r output, a q ( x ) » ^ s t n e s u m ° ^ t ^ i e other-user variances and the channel noise variance: N T T 2 M CTQ(T) = T " + "24 n \ = 1 P i F i n a l l y , i f we assume P^ = P for a l l i £ 0, then N T ' MPT T 2 _ o , o 0 Q(T) _ 4 24 Now i f T/T^ = n » l , then each C i s a sum of a large number of independent, i d e n t i c a l l y d i s t r i b u t e d random va r i a b l e s (namely, the d^(9^,$/fi-lls e of the c e n t r a l l i m i t theorem enables the err o r p r o b a b i l i t y to be written as follows: P = ^ e r f c /6TP /MPT e 2 o o For c l a r i t y , the e r r o r p r o b a b i l i t y w i l l be w r i t t e n as a function of the number of other users M, rather than of the power to noise r a t i o . S i m p l i f i c a t i o n y i e l d s P e(M,r) = \ e r f c i ^ ^ ) where T R _ T R. r = ° a v — = o. av 1 12 ' P 12y 4.2.5 o Y = P /P , the power r a t i o o N R 6N y L2 2P P T 1 o o o The quantity [r(C^M+ C^)] ^ i s the energy-to-noise r a t i o per message 4.2.2 The Hard L i m i t i n g C h a n n e l An a n a l y s i s o f t h e h a r d - l i m i t i n g c h a n n e l has been p e r f o r m e d by A e i n [ 2 4 ] , among o t h e r s [ 2 5 - 2 8 ] . The c h a n n e l i s m o d e l l e d as i n F i g . 4.2.3. The i s i g n a l s S ^ ( t ) a r e n 2 ( t ) Bandpass F i l t e r B.W. T 0 Hard Lim-i t e r - Zonal F i l t e r "*0^  Corre-l a t o r Deci-sion F i g . 4.2.3 The Hard L i m i t i n g C h a n n e l t h o s e o f s e c t i o n 4.2.1 e x c e p t t h a t f o r ease o f a n a l y s i s P^=P, f o r a l l i , so t h a t no s i g n a l c a p t u r e s the l i m i t e r [ 2 9 ] . The s i g n a l s a r e added, a l o n g w i t h N l w h i t e G a u s s i a n n o i s e o f power d e n s i t y — j and bandpass f i l t e r e d . I t i s assumed t h a t the f i l t e r m e r e l y band l i m i t s t h e n o i s e and has n e g l i g i b l e e f f e c t on the s i g n a l s . A f t e r t he h a r d l i m i t e r , a z o n a l f i l t e r removes a l l f r e q u e n c y bands n o t c e n t e r e d on w . The r e s u l t i n g waveform c o n s i s t s o f a c a r r i e r a t w , c c phase m o d u l a t e d w i t h the phase o f the l i m i t e r i n p u t s i g n a l , and h a v i n g power N 2 P r- A f t e r w h i t e G a u s s i a n n o i s e o f power d e n s i t y —^ i s added, the r e c e i v e r i a t t e m p t s t o r e c o v e r t h e o r i g i n a l message by c o r r e l a t i o n w i t h S ( t ) . A e i n o showed t h a t t h e r e s u l t i n g e r r o r p r o b a b i l i t y can be w r i t t e n V M i r ) = | e r f c / 2 r ( C j M + C 2 ) 4.2.6 where 9 P T +N 1 TT P av C = -, N.(P T +N„)R 2 1 r o 2 av TT PP T r o PT Some a s s u m p t i o n s i m p o r t a n t t o t h i s t h e s i s were made i n t h e d e r i v a t i o n . F i r s t , the t i m e - b a n d w i d t h p r o d u c t ( r R T ) ^ = T/T must be l a r g e enough t o s a t i s f y av o o J the requirements of the c e n t r a l l i m i t theorem. Next, Aein's i s e s s e n t i a l l y a s m a l l s i g n a l a n a l y s i s so that the s i g n a l at the l i m i t e r i n p u t must be b u r i e d PT 0 c 2 deeply i n n o i s e , s p e c i f i c a l l y , ^ T + N <<— . The f i r s t assumption a l s o o 1 appears i n s e c t i o n 4.2.1 and can be s a t i s f i e d by a c o n s t r a i n t on the maximum r a t e . As f o r the second, only a very s m a l l value of M w i l l cause the approximation to be v i o l a t e d . For a f a i r l y l a r g e average number of other users, a-jHj,» t h i s event occurs w i t h n e g l i g i b l e p r o b a b i l i t y , so the e r r o r introduced can be s a f e l y neglected. 4.3 C a l c u l a t i o n s Now that we have a model of the channel usage s t a t i s t i c s and the form of the e r r o r p r o b a b i l i t y , we wish to determine the optimum r a t e f u n c t i o n and the e r r o r performance r e s u l t i n g from i t s use. The normalized power-to-noise r a t i o X of Chapter 2 becomes (C^ K + C^) ^ f o r m u l t i - u s e r channels, where K i s the number of other users (at time t+x). The c o n d i t i o n a l p.d.f. of K depends on M, the current number of other users (at time t ) , which w i l l be i d e n t i f i e d w i t h z of Chapter 2. For s i m p l i c i t y , then, the r a t e f u n c t i o n w i l l be w r i t t e n as a f u n c t i o n of M, r a t h e r than of z. 4.3.1 O p t i m i z a t i o n Equations I f the rates can be s e l e c t e d from a continuum, and i f the feedback delay T and the r a t e change p e r i o d A are n e g l i g i b l e compared w i t h the usage time constant 3 \ then we have the " p e r f e c t feedback" s i t u a t i o n . From equation 2.3.4 the optimum r a t e f u n c t i o n r(M) i s the one which holds the e r r o r p r o b a b i l i t y constant. Thus 71 r ( M ) " u ( C ; L M + C 2) where u i s a constant r e l a t e d to the Lagrange m u l t i p l i e r y through 2.3.4. The r e s u l t i n g e r r o r p r o b a b i l i t y i s given by P = i e r f c /u/2 e 2 where u i s s e l e c t e d to make the average rat e - i ? 0 ft) = 1 i=0 1 2 I f the feedback l i n k does conta i n a delay, the p r e d i c t e d p e n a l t y of equation 2.2.3 i s r e q u i r e d before the optimum r a t e f u n c t i o n can be deter-mined. The p r e d i c t e d penalty w i l l a l s o be w r i t t e n as a f u n c t i o n of the number of current other users M, r a t h e r than of the PNR. S u b s t i t u t i o n of the c o n d i t i o n a l p.d.f. of M(equation 4.1.3) i n t o 2.2.3, and use of the average e r r o r p r o b a b i l i t y c r i t e r i o n f [ P e ( K , r ) ] = P e ( K , r ) y i e l d s the p r e d i c t e d e r r o r p r o b a b i l i t y M r p ' Z fx, \ 1 v v (Ml/M* \ i M-i J - i M ' - J+i , 1  P e ( M ' r ) = 2 J = 0 • J = 0 ( i / ( j - i h l l q l Q q 0 1 q00 6 r f c V 2 r ( C ; L i + C 2 ) where M' • M - M. The i m p l i c i t equation f o r the optimum r a t e f u n c t i o n r(M) can be obtained from 2.3.1 as 1 „ T t /MUM 1 \ i M-i J - i M'-J+i r _ % — , 2 J = 0 l=Q q H q l 0 q 0 1 q00 t e 2 Atfto + e r f c ^721 = A 4.3.2 where u^ = r ( C ^ i + C^) ^ and i s no longer constant. Equation 4.3.2 must be s a t i s f i e d f o r each M, and X then s e l e c t e d to make the average r a t e M=0 1 ° The r e s u l t i n g e r r o r p r o b a b i l i t y i s obtained from 2.2.5 as C r(M) P e(M,r(M)) a* aV 6 = M= The r a t e f u n c t i o n r(M) i s , of course, d i s c r e t e w i t h + 1 d i s t i n c t v a l u e s . I f however, the r a t e s are r e s t r i c t e d to N < M^ , + 1 d i s t i n c t v a l u e s , then a problem a r i s e s . The i t e r a t i v e s o l u t i o n o u t l i n e d i n s e c t i o n 2.4 i s not a p p l i c a b l e , s i n c e the p.d.f. of the SNR i s i m p u l s i v e . The technique of dynamic programming w i l l c e r t a i n l y apply, but the computation time r e q u i r e d would be e x o r b i t a n t , and as s e c t i o n 4.3.2 demonstrates, the r e s u l t s would not be p a r t i c u l a r l y rewarding. For these reasons, the d i s c r e t e s o l u t i o n to the m u l t i - u s e r channel was not i n v e s t i g a t e d . In order to s a t i s f y the requirements of the c e n t r a l l i m i t theorem so that the noise can be considered Gaussian, there must be a lower bound on the time-bandwidth product ( r R T ) \ as noted above. Although the e f f e c t av c of the r e s u l t a n t upper bound r on r , should not be c r i t i c a l , s i n c e R T max av c -3 i s t y p i c a l l y on the order of 10 , i t was i n c l u d e d i n the computations. 4.3.2 Results There are a l a r g e number of parameters to c o n s i d e r , namely, C^, M_, a. , P , r ; as w e l l as the f i x e d r a t e and v a r i a b l e r a t e cases. Probably T 1 e max J the most convenient way of d i s p l a y i n g the data i s a graph of C^ vs w i t h curves of constant P , f o r d i f f e r e n t s e l e c t i o n s of M_, a, , T , and r e T' 1 max Figures 4.3.1 to 4.3.4 are such graphs f o r v a r i o u s values of the average number of other u s e r s , as c a l c u l a t e d by an IBM 360/67 computer. Note that the s c a l e i s not the same on a l l graphs. F o r t u n a t e l y , some of the parameters M ', CL , and r have l i t t l e T ± max e f f e c t i n d i v i d u a l l y . Before we d i s c u s s the improvement i n performance caused by using adaptive r a t e c o n t r o l , l e t us see what changes these para-meters can make i n the curves presented. F i r s t , we can see by comparing F i g . 4.3.1 to 4.3.4 that M = ct-jMrjo the average number of other users , a f f e c t s the curves s trongly..However, fo a constant a-jHr' ct^ and had l i t t l e i n d i v i d u a l e f f e c t . For example, 1 -3 Figure 4.3.1, which i s f o r M^lOO and = shows that = 2.02 x 10 , -3 -3 = 2.5 x 10 , gives an e r r o r p r o b a b i l i t y P g = 10 . I f i s doubled -3 and h a l v e d , again g i v i n g ct-^ M^  = 50, then must be 2.00 x 10 to give -3 -3 P = 1 0 at C„ = 2.5 x 10 . This was the l a r g e s t v a r i a t i o n noted f o r a e 2 & two-to-one change i n (or a^). A more s u b s t a n t i a l v a r i a t i o n can be ob-served f o r a f i v e - t o - o n e change i n M^, as i l l u s t r a t e d on Figures 4.3.3 and 4.3.4, which both have aj^j< = 10. However, i t i s c l e a r t h a t changes i n the average number of other users, a r e ^ a r m o r e important than i n d i v i d u a l changes i n and M^, so the graphs presented can be taken as r e p r e s e n t a t i v e of many other systems, each w i t h the same value of ct-^Hp' A l s o i n v e s t i g a t e d was the e f f e c t on the curves of the upper bound on the r a t e . No p e r c e p t i b l e change was found, even f o r r as low as 6. max Now l e t us consider what improvement adaptive r a t e c o n t r o l can pro duce on a m u l t i - u s e r system. I n s p e c t i o n of Figures 4.3.1 to 4.3.4 r e v e a l s a g e n e r a l l y d i s a p p o i n t i n g performance f o r the v a r i a b l e r a t e system. Even under s t r o n g l y i n t e r f e r e n c e - l i m i t e d c o n d i t i o n s (C^ > C^), a v a r i a b l e r a t e system w i t h the same e r r o r p r o b a b i l i t y as a f i x e d r a t e system allowed a maximum i n c r e a s e of only 50% i n the other users' noise c o n t r i b u t i o n , , shown on F i g . 4.3.3. This represents only 1.8 db i n r e c e i v e r SNR saved 0 . 0 0 1 . 0 0 2 " ° 1 F i g . 4 . 3 . 1 Combinations of Thermal Noise C o e f f i c i e n t C^ and Interference Noise C o e f f i c i e n t C^ which Produce a Given E r r o r P r o b a b i l i t y on a Multi-User Channel. Maximum Number of Other Users M = 100 , Duty Cycle of Users a, = 0 . 5 . F i g . 4 . 3 . 2 Combinations of Thermal Noise C o e f f i c i e n t C and I n t e r f e r e n c e Noise C o e f f i c i e n t C which Produce a Given E r r o r P r o b a b i l i t y on a Mu l t i - U s e r Channel. Maximum Number of Other Users MT=100 , ^ Duty Cycle of Users CC-, =0.25 F i g . 4.3-3 Combinations of Thermal Noise C o e f f i c i e n t C 2 and I n t e r f e r e n c e Noise C o e f f i c i e n t C which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users = 100 , Duty Cycle of Users a, =0.1 . ON 0 .002 .004 .006 .008 .01 .012 °1 F i g . 4-3.4 E f f e c t of Feedback Delay on Combinations of Thermal Noise C o e f f i c i e n t C„ and I n t e r f e r e n c e Noise C o e f f i c i e n t C which Produce a Given E r r o r P r o b a b i l i t y on a M u l t i - U s e r Channel. Maximum Number of Other Users M_ = 20, Duty Cycle of Users a = 0.5 . (see equations 4.2.5 and 4.2.6) or a 50% i n c r e a s e i n data r a t e . The r e l a t i v e l y s m a l l improvement of the v a r i a b l e r a t e system i s a t t r i b u t a b l e to the f a c t t h a t , i n c o n t r a s t to the Rayleigh f a d i n g channel, very low values of SNR do not occur f o r a s u b s t a n t i a l f r a c t i o n of the time. The e f f e c t of feedback delay i s i l l u s t r a t e d i n F i g . 4.3.4 f o r BT=.1 and BT=.5 at M^  = 20, = -J- A l a r g e r value of M^, would have been chosen, but the program was already expensive, and the computer time r e q u i r e d i n -2 creased approximately as M^, . As one would expect from the r e s u l t s of the study of the R a y l e i g h f a d i n g channel, a delay of only a f r a c t i o n of a time constant caused a severe d e t e r i o r a t i o n i n the already modest improvement due to the v a r i a b l e r a t e system. One would expect that the r a t i o of the standard d e v i a t i o n to the mean of M, J— ±, would p r e d i c t , to some ex t e n t , the r e l a t i v e improvement V l due "to the v a r i a b l e r a t e scheme. This turns out to be the case, because the improvement i s much grea t e r at lower values of ct^M^ (compare F i g s . 4.3.1 and 4.3.3), and f o r the same value of a^M^ the improvement i s g r e a t e r f o r s m a l l e r values of (compare F i g s . 4.3.3 and 4.3.4). Thus an i d e a l c o n d i t i o n f o r the v a r i a b l e r a t e system i s the r a t h e r a t y p i c a l s i t u a t i o n of an enormous number of p o t e n t i a l other users, M^ , each using the channel a t i n y f r a c t i o n of the time. 4.4 Two Numerical Examples 4.4.1 M u l t i - U s e r S a t e l l i t e Channel The m u l t i - u s e r s a t e l l i t e channel i s an example of the h a r d - l i m i t i n g channel of s e c t i o n 4.2.2, f o r which 2 N 2 C = £ R (T + 1 TT av o P r I f there are p o t e n t i a l l y 100 other u s e r s , each on the a i r JTJ of the time, the r e q u i r e d values of and to gi v e .an e r r o r p r o b a b i l i t y of 10 ^ are shown on F i g . 4.3.3. The u p - l i n k c o n s i s t s of a powerful ground t r a n s m i t t e r w i t h the r e c e i v e r i n the o r b i t i n g repeater. A f t e r frequency s h i f t i n g , hard l i m i t i n g and a m p l i f i c a t i o n at IF i n the s a t e l l i t e , the s i g n a l i s s h i f t e d again i n frequency and tr a n s m i t t e d back to ground. The down-link power c o n s t r a i n t can be severe, w i t h a t y p i c a l s a t e l l i t e t r a n s m i t t e r power of 2 - 4 wa t t s . We w i l l use t y p i c a l f i g u r e s from the Appendix of reference [23] f o r a s a t e l l i t e r epeater at an a l t i t u d e of 6000nmi. The u p - l i n k power i s not cons t r a i n e d i t i s on the down-link, so s a t e l l i t e i n p u t s i g n a l - t o - t h e r m a l noise r a t i o s PT ° of the order of 20 db are o b t a i n a b l e . Hence N l c 2 = .01 c± With a s a t e l l i t e t r a n s m i t t e r power of 4 watts (6 db), a down l i n k a t t e n u a t i o n ( i n c l u d i n g antenna gains) of 142 db, and a r e c e i v e r thermal n o i s e d e n s i t y of -206 db, we o b t a i n N„/P = -70 db or N 0/P = 10~ 7. A repeater bandwidth 2 r 2 r of at l e a s t 10 MHz w i l l be a v a i l a b l e , so that we can assume T =0.1 sec. o For the f i x e d r a t e system, F i g . 4.3.3 shows that must equal 3.3 -3 x 10 . From the d e f i n i t i o n of C. and the above values f o r T and N„/P , 1 o 2 r we o b t a i n an average data r a t e R =26 k b i t s / s e c . av -3 In c o n t r a s t , the v a r i a b l e r a t e system can t o l e r a t e = 4.9 x 10 , which i m p l i e s an average data r a t e R = 39 k b i t s / s e c . This i s c e r t a i n l y not a s p e c t a c u l a r improvement, but i n some cases may be worthwhile. A l t e r n a t i v e l y , we may consider the saving as a r e d u c t i o n i n repeater power. For a data r a t e of 25 k b i t s , the f i x e d r a t e repeater power i s , of course, 4 watts. When the v a r i a b l e r a t e scheme i s used, the -3 value of repeater power which makes C^ = 4.9 x 10 i s 2.04 w a t t s , a saving of 3 db. I t i s evident from the form of C. that i f the system bandwidth T 1 o were s m a l l e r , say about 1MHz, then a s m a l l i n c r e a s e i n a l l o w a b l e C^, i n going from f i x e d to v a r i a b l e r a t e , could produce a much l a r g e r change i n r e q u i r e d repeater power. However, i t i s very u n l i k e l y that anyone would launch a s a t e l l i t e w i t h such a s m a l l bandwidth. Most have a bandwidth of hundreds of megahertz, which, i n the above example, r e s u l t s i n only a 1.7 db decrease i n repeater power. 4.4.2 M u l t i - U s e r Cable I f many users share a c o a x i a l c a b l e , we have a l i n e a r channel ( s e c t i o n 4.2.1) w i t h T R _ ° a v 1 " 12 Y N R 6N Y r = ° a v = °_ r 2 2P P T 1 o o o I n a cable s i t u a t i o n i t i s l i k e l y to be bandwidth, r a t h e r than power, which determines cost. We can again assume a hig h s i g n a l - t o - t h e r m a l n o i s e r a t i o .is the r e c e i v e r so that C~ < C.. We w i l l take the t y p i c a l value f o r R of 60 2 1 J * av k b i t s / s e c f o r PCM v o i c e , and assume the power r a t i o Y=l- Note that f o r a given C^ and T , Y can a f f e c t only R a v> not the r e l a t i v e improvement i n performance. We w i l l assume M^ , = 200, and = .05 so that ct^M^ = 10. F i g u r e 4.3.3 i l l u s t r a t e s the behaviour of systems w i t h M^ = 10, and f o r P = 10 ^ -3 shows that a f i x e d r a t e system r e q u i r e s C^ = 3.3 x 10 . Thus the r e q u i r e d T = 6.6 x 10 which- i m p l i e s a system bandwidth of 1.5 MHz. The v a r i a b l e o -3 r a t e scheme, on the other hand, can have = 4.9 x 10 , and hence a system bandwidth of 1 MHz, which i s not a s i g n i f i c a n t r e d u c t i o n . 4.5 D i s c u s s i o n A p p l i c a t i o n of the v a r i a b l e r a t e scheme to m u l t i - u s e r channels produced g e n e r a l l y poor r e s u l t s , even under p e r f e c t feedback c o n d i t i o n s . There was some improvement, but i t was no grea t e r than 3 db i n a l l cases examined. Since feedback delay and r a t e q u a n t i z a t i o n reduce t h i s already meagre sav i n g , i t must be concluded that adaptive r a t e c o n t r o l on a m u l t i -user channel i s not worth the expense of the e x t r a complexity i n v o l v e d . I t would appear that the preceding r e s u l t s are a p p l i c a b l e p r i -m a r i l y to data, r a t h e r than v o i c e t r a n s m i s s i o n , as the f o l l o w i n g d i s c u s s i o n w i l l demonstrate. Since we should assume that the usage f u n c t i o n of the user under c o n s i d e r a t i o n has the same bandwidth as that of the other i n t e r -f e r i n g users, "average r a t e " must mean r a t e averaged over s e v e r a l times on the a i r . However, i f v o i c e i s a s u b s t a n t i a l f r a c t i o n of a v o i c e plus data mixture then the bandwidth of the usage f u n c t i o n s must be l a r g e i n order to keep b u f f e r s i z e and delay reasonable. T h i s , however, i s i n c o n s i s t e n t w i t h the l a r g e a c q u i s i t i o n time of pseudonoise codes, so that v o i c e , i f present, must be only a s m a l l f r a c t i o n of a s t a t i o n ' s t r a f f i c . 5. CONCLUSIONS A scheme f o r a d a p t i v e l y changing t r a n s m i s s i o n r a t e to compensate f o r a time-varying power-to-noise r a t i o has been described and analyzed. I m p l i c i t equations f o r the optimum r a t e request as a f u n c t i o n of past and current instantaneous PNR has been derived f o r an a r b i t r a r y p r o b a b i l i t y d e n s i t y f u n c t i o n of the PNR. The e f f e c t s of a bandwidth c o n s t r a i n t , of time delay i n the feedback l i n k and of time and amplitude d i s c r e t e r a t e requests have been i n c l u d e d i n the a n a l y s i s . For the Rayleigh f a d i n g channel w i t h p e r f e c t feedback c o n d i t i o n s there i s an enormous r e d u c t i o n i n r e q u i r e d power over a f i x e d r a t e non-d i v e r s i t y system, up to 50 db f o r t y p i c a l values of e r r o r p r o b a b i l i t y . The best a l t e r n a t i v e to adaptive r a t e c o n t r o l i s maximal-ratio p r e d e t e c t i o n combined (MRPDC) frequency d i v e r s i t y . For the same values of bandwidth, data r a t e , and e r r o r p r o b a b i l i t y , the v a r i a b l e r a t e scheme can save many db (16 db i n the example of s e c t i o n 3.5) over an MRPDC frequency d i v e r s i t y system. While i t i s tru e that the round t r i p delay must be l e s s than a s i g n i f i c a n t f r a c t i o n of the fa d i n g time constant or t h i s advantage i s l o s t , n e vertheless the fa d i n g rates encountered on s c a t t e r channels are low enough that t h i s c o n d i t i o n i s very o f t e n met. A method has been given which allows t r a d e o f f s among power, bandwidth and data r a t e on the Rayleigh f a d i n g channel to be examined graphically-. On m u l t i - u s e r channels the improvement due to the v a r i a b l e r a t e method i s d i s a p p o i n t i n g . For the range of parameters considered l i k e l y , there i s a maximum of about 50% or 0.7 db improvement over f i x e d r a t e systems, even when feedback c o n d i t i o n s are p e r f e c t . The presence of feedback delay makes the v a r i a b l e r a t e method even l e s s a t t r a c t i v e . The f a c t that the improvement i s s m a l l i s a t t r i b u t e d to the low p r o b a b i l i t y of very h i g h noise l e v e l s . There are s e v e r a l r e l a t e d areas which could be f r u i t f u l l y i n v e s t i -gated. One f a i r l y obvious one i s refinement of the channel model. In par-t i c u l a r , p r e c i s e c a l c u l a t i o n of the e f f e c t s of intersymbol i n t e r f e r e n c e i n a s c a t t e r channel together w i t h adaptive e q u a l i z a t i o n at the r e c e i v e r , as i n the Rake system [30], w i l l give a more r e a l i s t i c estimate of the p e r f o r -mance of very high data r a t e systems. Another extension i s the a n a l y s i s of simultaneous r a t e and power a d a p t i v i t y . This could be u s e f u l i n imperfect feedback c o n d i t i o n s , s i n c e w i t h p e r f e c t feedback, adaptive r a t e c o n t r o l e l i m i n a t e s the e f f e c t s of f a d i n g There i s the p o s s i b i l i t y of changing the i n f o r m a t i o n r a t e i n coded systems w i t h e i t h e r a l g e b r a i c or p r o b a b i l i s t i c decoding at the r e c e i v e r . The former was, i n f a c t , i n v e s t i g a t e d by the author but abandoned because random coding bounds were too l o o s e . However, study of s p e c i f i c code s e t s , such as those given by S l e p i a n [31] may i n d i c a t e whether or not f u r t h e r work i s worthwhile. As f o r p r o b a b i l i s t i c decoding, Wozencraft and H o r s t e i n [32] have suggested the use of v a r i a b l e redundancy c o n v o l u t i o n a l codes w i t h r a t e changes dependent on the s e q u e n t i a l decoding t h r e s h o l d l a s t used, i n order to keep the decoder op e r a t i n g at f u l l c a p a c i t y . U n f o r t u n a t e l y t h i s would have to be a s i m u l a t i o n study, s i n c e a n a l y t i c bounds i n t h i s area do not agree c l o s e l y w i t h r e a l i t y . F i n a l l y , one should consider the e f f e c t of f i n i t e b u f f e r s i n the system. I f these b u f f e r s separate a f i x e d r a t e source from the v a r i a b l e r a t e modulator, there i s a p o s s i b i l i t y of l o s i n g b i t s on overflows and of i n e f f i c i e n t o p e r a t i o n on emptyings. Any r a t e c o n t r o l scheme i n these circumstances would have'to monitor the b u f f e r queue l e n g t h , as w e l l as the instantaneous power-to-noise r a t i o . This too would most l i k e l y i n v o l v e a s i m u l a t i o n study. APPENDIX I ; Glossary of Symbols General N number of non-zero ra t e s T round t r i p time delay (sec.) A time s e p a r a t i o n between r a t e changes (sec.) R data r a t e ( b i t s / s e c ) T b i t d u r a t i o n (sec. T = 1/R R^v average data r a t e ( b i t s / s e c ) p(.) r a t e f u n c t i o n ( b i t s / s e c ) r ( . ) normalized r a t e f u n c t i o n p(.)/R av X Lagrange m u l t i p l i e r ii)^ " c a r r i e r frequency f [ P g ] p e n a l t y weighting of the e r r o r p r o b a b i l i t y P^  Chapter 2 S ( t ) r e c e i v e d s i g n a l power to noise d e n s i t y r a t i o (PNR) v bandwidth of S(t) X ( t ) normalized PNR, S ( t ) / R av z ( t ) estimator of S(t+-r) u energy-to-noise r a t i o per b i t X/r Chapter 3 M^(t) v a r i a b l e d u r a t i o n s i g n a l p u l s e s ; k = 0,1 G(t) complex channel g a i n P t r a n s m i t t e r power a ( t ) s t r e n g t h of channel gain |G(t) | 6-(t) phase angle i n t r o d u c e d by channel, arg[G(t)] N Q s i n g l e ended noise power d e n s i t y v bandwidth of G(t) B second moment of gain a k p r e d i c t i o n f a c t o r 2 x normalized power-to-noise r a t i o P a /N R o av b average energy-to-noise r a t i o per b i t , b =x = PB/N R o av r maximum bandwidth expansion o R message b i t r a t e m R s e r v i c e b i t r a t e , R - R s av m M number of r e p e t i t i o n s of each s e r v i c e b i t Chapter 4 pn^(t) i t b pseudo-random b i t stream M(t) number of other users at'time t maximum number of other users u ^ ( t ) usage f u n c t i o n of i t b user; u ^ ( t ) = 0,1 f r a c t i o n of time users t r a n s m i t , duty c y c l e 3 bandwidth of u ( t ) T pseudorandom b i t d u r a t i o n o C^ i n t e r f e r e n c e noise c o n t r i b u t i o n per user channel noise c o n t r i b u t i o n APPENDIX I I : Proofs of the Lemmas of Chapter 2 df d P e Lemma 1: I f 1T, •-— i s monotonically i n c r e a s i n g w i t h u, then r ( z ) , dP du e the s o l u t i o n of 2. 3 . 1 j i s unique and i t i s a minimum. »• Proof: We w i l l use d e f i n i t i o n 2.2.3 of f f o r A=0, r a t h e r than 2.2.4 f o r A>0 merely to avoid the cumbersome summations. Then equation 2.3.1 can be w r i t t e n oo / { f [ P (-)] + r •— f [P (—)]] P I ( a | c ) d a = X l e r dr e r j x z ' h o The dependence of the integr a n d on r i s contained i n the f a c t o r K(u) . dP r dP •) = -u •§! "r~ + f. The d e r i v a t i v e K ' ( u ) = -u ^ - 4 • — r ^ ( < 0, by dP^ du du ^ dP g duj . hypothesis. Hence the int e g r a n d i n c r e a s e s monotonically w i t h r and there can be only one root of 2.3.1. Next, i f the d e r i v a t i v e w i t h r e -spect to r of the l e f t s i d e of 2.3.1 i s p o s i t i v e then we have a minimum Since the operator —- i s e q u i v a l e n t to - — and s i n c e K'(u)<0, we 3r r du do have minimum. df e Lemma 2: I f • i s monotonically i n c r e a s i n g w i t h u, and i f the e c o n d i t i o n a l c.d.f. F i ( a | t ; ) i s a decreasing f u n c t i o n of r then the X I 2 s o l u t i o n r ( z ) i s a monotonically i n c r e a s i n g f u n c t i o n of z, Proof: From the d e f i n i t i o n of K(u) i n Lemma 1 2.3.1 can be w r i t t e n : / K(u) P x | z ( « k ) d a = X A.2 Let J(a) = K ( a / r ) . Then i n t e g r a t i o n by pa r t s of the l e f t s i d e of A.2.1 y i e l d s OO 00 J(a) F | (a\0 | - / J' ( a ) F , (a | da = X x | z o o X | Z The f i r s t term above i s independent of t,, s i n c e a c.d.f. always has 87 upper and lower l i m i t s of 1 and 0. The integrand of the second term incr e a s e s w i t h w i t h the r e s u l t that the l e f t s i d e of A.2.1 decreases w i t h £. But i t was shown i n Lemma 1 that the l e f t s i d e of A.2.1 inc r e a s e s w i t h r . Hence r ( z ) i s a monotonically i n c r e a s i n g f u n c t i o n of z. Lemma 3: I f the equation T | ~ ^ X ' ^ = 0 has a s i n g l e root i n y f o r any given x, and a s i n g l e root i n x f o r any given y, then the i t e r a t i v e s o l u t i o n of equations 2.4.7 and 2.4.8 i s unique. 9H(x y) Proof: L e t us denote by y (x) a continuous s o l u t i o n of — ' = 0 . ; ; o ' 9y The f i r s t c o n d i t i o n above i s e q u i v a l e n t to a l l o w i n g a s i n g l e extremum y Q ( x ) and the second c o n d i t i o n i s e q u i v a l e n t to y Q ( x ) being monotonic. We w i l l take y Q ( x ) to be i n c r e a s i n g , although i t could e q u a l l y w e l l be decreasing. Consider the s o l u t i o n of 2.4.8, where y.,, must be found when y. l + l l and x^ are given. Since there i s only a s i n g l e root i n y of "|~~^X,^=0> the s i t u a t i o n i s as depicted i n F i g . A.2.1. I n s p e c t i o n shows a unique value f o r y..». Note that y (x) l i e s between y. and y.,,. 1+1 J o J1 31+1 Next consider the s o l u t i o n of 2.4.7 where x. must be found when y. l y i and are given. The integr a n d must change s i g n at l e a s t once i n x. But s i n c e there i s only a s i n g l e value of x which makes -^.(x>y)=o the 9y value of x_^  i s unique. Note again that the x f o r which y (x) = y^ l i e s between x. , and x.. i - i l 2 2 9 H 9 W Lemma 4: I f ~ v <0 and —~ >0 then the i t e r a t i v e s o l u t i o n of 2.4.7 and 9x9y . 2 9y 2.4.8 i s unique and i t i s a minimum. 2 Proof: I f \ \ <0 then the equation -!pl( x>y) _ 0 can have only a s i n g l e 9x9y 2 y root i n x f o r any given y, and i f - ^ M r >0 then - j ^ x ' y ) = o can have only a 3y^ 7 y i y i + l F i g . A.2.1 Behaviour of H(x^ , y) s i n g l e root i n y f o r any given x. From Lemma 3, the d i s c r e t e s o l u t i o n i s unique. Consider the s o l u t i o n of 2.4.7 which must be s a t i s f i e d by y . The d e r i v a t i v e of the l e f t s i d e of 2.4.7 w i t h respect to y^ i s p o s i t i v e , by h y p o t h e s i s , so tha t y^ i s a minimum. Next consider the s o l u t i o n of 2.4.8 which must be s a t i s f i e d by x.. J x For a minimum, the d e r i v a t i v e of the l e f t s i d e of 2.4.8 w i t h r e s p e c t to x_^  must be p o s i t i v e , that i s : |H(x ±, y i) _ |H_(x y 1 + 1 ) > ax. 3x. x x Since the mixed d e r i v a t i v e the s o l u t i o n i s a minimum. 3x3y <0, t h i s c o n d i t i o n i s s a t i s f i e d and 89 APPENDIX I I I : Optimum D i v e r s i t y f o r Maximal R a t i o P r e - D e t e c t i o n Combining (MRPDC) The development up to equation A.3.3 e s s e n t i a l l y paraphrases that th i n Appendix I I of B e l l o [17]. The complex envelope of the j d i v e r s i t y channel output i s u. = G. M. ( t ) + n. ( t ) J 3 K j where G^  i s the j t b complex g a i n , s t a t i s t i c a l l y independent of a l l G_^ , i ^ j , and n.(t) i s white noise w i t h double ended power d e n s i t y N /2. I f the 3 ° r e c e i v e r tracks a l l the G^  i t can b r i n g a l l L s i g n a l s i n t o phase coincidence before d e t e c t i o n and form the optimal weighted sum [34] L * , S ( t ) = E G. u. 3=1 3 3 L 2 L A.3.1 M. ( t ) E a. + E G. n ( t ) * j - 1 2 3=1 3 to maximize the s i g n a l - t o - n o i s e r a t i o . Although the s i g n a l component of S ( t ) can now be detected c o h e r e n t l y , i t i s usual to use an envelope d e t e c t o r a f t e r the matched f i l t e r or c o r r e l a t o r s i n c e the l o s s i n e f f e c t i v e energy i s not s i g n i f i c a n t at normal values of energy-to-noise r a t i o . Then sum S ( t ) L 2 L 2 2 c o n s i s t s of a s i g n a l M. ( t ) E a. w i t h power P [E a. ] embedded i n white ^ j = l 3 T L 3=21 3 noise w i t h double ended power de n s i t y — N E a. . I t f o l l o w s from 2 ° j = l 3 equation 3.1.9 that the e r r o r p r o b a b i l i t y f o r incoherent d e t e c t i o n , given the channel gains, i s 1 PT L P e "2 6 X P [" 2T 1 , a j ] A ' 3 - 2 o 3 = 1 where T c i s the d u r a t i o n of a s i n g l e t r a n s m i s s i o n . Noting that the 8..i= a_.2 are s t a t i s t i c a l l y independent, and from 3.1.6 are a l l e x p o n e n t i a l l y d i s t r i b u t e d 1 - -. P Q (a) = - e B ct > 0 j \ " we can average A.3.2 over the j o i n t ensemble of L channel gains to o b t a i n the e r r o r p r o b a b i l i t y PT B c 1 + 2N o -L A.3 I f the i d e n t i f i c a t i o n T L = 1/R i s made, i t can be seen that f o r L = l , A.3. c av agrees w i t h the f i x e d r a t e e r r o r p r o b a b i l i t y i n s e c t i o n 3.2.1. Let us d e f i n e n =PBT c / N o. Then the average energy-to-noise r a t i o b = ri L. The e r r o r p r o b a b i l i t y A. 3.3 can be w r i t t e n = | exp (-bg(n)) A. 3 where g ( n ) = n 1 l n ( l + n / 2 ) Optimum d i v e r s i t y c o n s i s t s of minim i z i n g the f u n c t i o n g ( n ) . I t turns out that g(n) a t t a i n s the maximum value of at n=0. Thus optimum MRPDC 1 d i v e r s i t y can a t t a i n the same e r r o r p r o b a b i l i t y , p E = "J E 2, as adaptive r a t e c o n t r o l w i t h p e r f e c t feedback, but i t req u i r e s an i n f i n i t e number of s t a t i s t i c a l l y independent transmissions to do so. APPENDIX IV: C a l c u l a t i o n of the D i s c r e t e Rate Function We wish to f i n d the optimum r a t e f u n c t i o n r ( x ) f o r a given delay T, r a t e change p e r i o d A, number of non-zero r a t e s N, maximum bandwidth expansion r Q , and energy-to-noise r a t i o per b i t b. The equations to be s a t i s f i e d , 3.3.6 and 3.3.7, are reproduced below $ ( x i , r i , A ) = (jj (x±_lfxitX) 3.3 ^(r i + 1,x 1»A) = K i V . x ,A) 3.3 For a given A, the equations may be solved i t e r a t i v e l y as f o l l o w s N+l 0—>m pick r 2 f i n d x^ such that ¥(r 2,x ,\)= y ( r 1 , x 1 > \ ) N:l 2 — > i i:N -f i n d x. such that 1 $ (x. ,r. ,X)= frx. ,r . ,\) 1 l i - i l no s o l u t i o n f i n d r. . such that l + l ¥ (r. ... ,x. tfr ,x ,\) l + l l i i i+1-m:0 ) ( x N ' r N + l ' A ) ~ * £ e:0 ' $ (x , r i } \ ) :0 decrease r ^ increase r^ r -r„ — * e o N+l r N + l : r o A m : ° increase vr decrease r, The process i s to terminate when only very s m a l l changes i n r are made. The Lagrange m u l t i p l i e r X i s then changed and the i t e r a t i v e process repeated. When r , evaluated from 2.2.6, equals 1 f o r some A, the s o l u t i o n i s complete. 94 REFERENCES 1. 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Clowes, et a l , "A V a r i a b l e Rate Binary Communication System f o r U n c o n t r o l l e d Sharing of a S a t e l l i t e Repeater"', T e c h n i c a l Note No. 7, Communications Laboratory, Defence Research Telecommunications E s t a b l i s h -ment, Ottawa, Canada, May 1967. 11. J . Max, "Quantizing f o r Minimum D i s t o r t i o n " , IRE Trans. Information  Theory, v o l . IT-6, pp. 7-12, March 1960. 12. R.E. Larson, "Optimum Q u a n t i z a t i o n i n Dynamic Systems", IEEE Trans.  Automatic C o n t r o l , v o l . AC-12, pp. 162-168, A p r i l 1967. 13. D.J. S a k r i s o n , Communication Theory: Transmission of Waveforms and D i g i t a l I nformation, New York: John Wiley, 1968. 14. " S c a t t e r Propagation Issue", Proc. IRE, October 1955. 15. P.A. B e l l o , "A Troposcatter Channel Model", IEEE Trans. Comm. Tech., v o l . COM-17, pp. 130-137, A p r i l 1969. 16. J.T.A.C., "Radio Transmission by Ionospheric and Tropospheric S c a t t e r , p a r t I I " , Proc. 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New York: John Wiley, 1969. 34. D.G. Brennan, " L i n e a r D i v e r s i t y Combining Techniques", Proc. IRE, v o l . 47, pp. 1075-1102, June 1959. 35. J . J . Metzner and K.C. Morgan, "Variable-Data-Rate Procedures f o r Feedback Communication Systems", Proc. S i x t h Ann. A l l e r t o n Conf. on C i r c u i t and Systems Theory, M o n t i c e l l o , 111., Oct. 1968. 36. J.T. Tou, Modern C o n t r o l Theory. New York: McGraw-Hill, 1964. 37. D.J. S a k r i s o n , Communication Theory: Transmission of Waveforms and  D i g i t a l I n formation , p. 215. New York: John Wiley, 1968. 

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