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Estimation of error rates and fade distributions on a Rayleigh fading channel with additive white… Ng, Jimmy Hon-yuen 1986

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EST IMATION OF ERROR RATES AND FADE D ISTRIBUTIONS ON A RAYLEIGH FADING CHANNEL WITH ADD IT IVE WHITE GAUSSIAN NOISE B.Eng., Kyoto University (Japan), 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept th i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA by Jimmy Hon-yuenNg March 1986 © Ji mmy Hon-yueh Ng, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written per mi s s i on. DEPARTMENT OF ELECTRICAL ENGINEERING The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T IW5 Date: March 1986 Ahstraci Several characteristics of the Rayleigh fading channel are examined. A digital Rayleigh fading simulator is used to generate the (fading) signal envelope from which various statistics are derived. Based on the simulation results, a simple model is proposed in order to estimate the block error rate of a block of N data bits transmitted over the Rayleigh fading channel in the presence of additive white Gaussian noise. This model gives an average estimation error of about 4 % over the range of blocksizes N = 63, 127, 255, 511, 1023, 2047 (bits), average signal-to-noise ratios 7 0 = 5 to 35 (dB) and fading frequencies f p = 10 to 90 (Hz) corresponding to vehicle speeds of 8 to 71 MPH at a radio carrier frequency of 850 MHz. A second somewhat more complex model for estimating the block error rate is found to yield a lower average estimation error of 2.4 % over the same set of simulated data. The probability distributions of the fade rate and the fade duration are also examined. Empirical models are derived for the estimation of the probability mass function of the fade rate and the probability density function of the fade duration. These empirical models allow fairly accurate estimates without the need for cosdy and time-consuming simulations. The probability of m-bit errors in an N-bit block is an important parameter in the design of erroT-correcting codes for use on the mobile radio channel. However, such probabilities are difficult to determine without performing extensive simulation or field trials. An approach to estimate them empirically is proposed. i Table of Contents Abstract i List of Figures iv List of Symbols vi Acknowledgement viii 1. Introduction 1 2. A Simple Model for the Estimation of Block Error Rates on the Rayleigh Fading Channel with A W G N 6 2.1 Review 6 2.1.1 The Steady Signal A W G N Channel 6 2.1.2 The Rayleigh Fading A W G N Channel 7 2.2 The Empirical Model '. 8 2.2.1 The Software Simulation 8 2.2.2 The Empirical Formula 10 3. A Second Model for the Estimation of Block Error Rates 17 3.1 Basic Model 17 3.1.1 Extending the Range of the Basic Model 18 3.2 The Second Model (Model II) 20 4. Estimation of the Probability Distributions of the Fade Rate & the Fade Duration 25 4.1 Review 25 4.2 The Simulation „ 26 4.3 Probability Mass Function of Fade Rate 26 4.4 Probability Density Function of Fade Duration 33 5. Estimation of Probability of m-bit Errors in an N-bit Block 40 5.1 Review 40 5.2 An Approach to Estimating Q(ni,N) 41 6. Conclusion 52 ii 6.1 Concluding Remarks 52 6.2 Future Work 53 REFERENCES 54 APPENDIX A 56 iii List of Figures Figure Page 1.1 Power Spectral Density of Received Signal 3 1.2 Rayleigh Fading Signal Envelope at f^ = 10 Hz 4 2.1 Effective Bit Error Rate p £ ^ vs Average SNR y 0 ; N = 127 bits , 12 2.2 Effective Bit Error Rate p g f i - vs Ratio L; N = 127 bits 13 4.1 Probability Mass Function of Fade Rate n 27 4.2 Variation of Mean v with 29 4.3 Variation of Standard Deviation o with 30 4.4 Probability Mass Function of Fade Rate n & its Empirical Normal Function 31 4.5 Cumulative Mass Function of Fade Rate n & the C M F of its Empirical Normal Function 32 4.6 Probability Mass Function of Fade Length i 34 4.7 Probability Density Function of Normalized Fade Duration <f> 36 4.8 Cumulative Density Function of Normalized Fade Duration <l> & the C D F of its Empirical Function (4.4.4) 38 5.1 Variation of Ratio Q(m,63) / Q(0,63) with Average SNR y0; f D = 40 Hz, N = 63 bits 42 5.2 Variation of F i with m; f D = 40 Hz 43 5.3 Variation of F 2 with m; f D = 40 Hz 44 iv 5.4 Cumulative Mass Function of Q(m,N) & the unsealed C M F of the Empirical Function (5.2.6); f D = 45 Hz, N = 511 bits 47 5.5 Cumulative Mass Function of Q(m,N) & the scaled C M F of the Empirical Function (5.2.6); f D = 45 Hz, N = 511 bits 48 5.6 Cumulative Mass Function of Q(m,N) & the scaled C M F of the Empirical Function (5.2.6); f D = 20 Hz, N = 127 bits 50 5.7 Cumulative Mass Function of Q(m,N) & the scaled C M F of the Empirical Function (5.2.6); f~ = 70 Hz, N = 255 bits 51 v List of Symbols Symbol Definition Reference Page d divisor 22 E RMS value of the signal envelope 2 fp fading frequency in Hz 2 4> normalized fade duration in millisec*Hz; <t> = T • f D 35 7 Signal-to-Noise power ratio (SNR) 6 7 o average SNR in dB 8 7 , 1 0 7 o / 1 ° 18 i fade length in bits/second 33 L fade ratio in bits/cycle-of-fade; L = R / f D 11 X RF carrier wavelength in meters/cycle 2 m number of errors in a block (bits) 40 u mean of fade rate in fades/second 26 n fade rate in fades/second 26 N blocksize of transmitted binary data in bits/block 8 N 0 /2 two sided power spectral density of Gaussian noise 6 Np average fade rate in fades/second (at signal level p) 25 v vehicle speed in meters/second 2 p(7) bit error rate (BER) at SNR 7 in Random Error vi Channel 6 effective BER in Rayleigh fading channel 10 Pf BER in Rayleigh fading channel 7 Pc(0,x) block correct rate (BKCR) of a x-bit block in Rayleigh fading channel 17 Pj(0,N) block error rate (BKER) of an N-bit block in Rayleigh fading channel 7 Q(m,N) probability of m-bit errors in an N-bit block in Rayleigh fading channel 40 r 2 / 7 , 18 (r) Pochhammer's symbol 40 R transmission bit rate in bits/second 8 p signal envelope amplitude to amplitude RMS value ratio 25 a standard deviation of fade rate in fades/second 26 T fade duration in milliseconds 35 T average fade duration in seconds (at signal level p) 25 x sub-blocksize of an N-bit block (bits) 17 vii Acknowledgement I would like to express my grateful thanks to my supervisor Dr. C. Leung, whose continuous guidance and encouragement throughout the research work of this thesis are sincerely appreciated. The financial supports received by me including a Research Assistantship from NSERC Grant A1731, and a Teaching Assistantship from the Department of Electrical Engineering at University of British Columbia are also gratefully acknowledged. I would also like to thank Mr. Brian H. Maranda for the use of his Rayleigh fading simulator program. viii 1. INTRODUCTION In the mobile VHF/UHF radio environment, the transmitted radio signal a r r i v e s at the receiver via multiple paths. The received signal i s made up of a number of waves scattered by obstructions such as buildings in the v i c i n i t y of the receiver. The amplitude, phases and angles of a r r i v a l of these waves are completely random. The resultant signal amplitude i s Rayleigh d i s t r i b u t e d over a l o c a l geographical area, and the mean value of the signal amplitude i s log-normally d i s t r i b u t e d over larger areas [ 6 , 8 , 1 6 ] . The slow change of the l o c a l mean i s due to shadowing by intervening features. Due to the occasional deep fade in signal strength, the errors in a block of received data tend to occur in c l u s t e r s . Interleaving transmission techniques together with Forward Error Correction (FEC) and/or Automatic Repeat-request (ARQ) can be used to combat these cl u s t e r i n g errors [9,15,21]. A number of other schemes have also been proposed for use on the mobile radio channel [7,8,9,13]. In t h i s t h esis, we focus on the rapid Rayleigh fades of the signal envelope. Our main objective i s to develop simple empirical models for estimating the block error rates (BKER's) of data transmitted over the Rayleigh fading channel with Additive White Gaussian Noise (AWGN). 1 2 A Rayleigh fading simulator [2] i s used to generate the fading envelope. The t h e o r e t i c a l power spectral density of the complex signal envelope received by an omnidirectional antenna i s as follows [16,19] : ,2 , -1/2 E { 1 - (|-) 2 } , f ^ f , S(f) = < ( L D 0 , £>£ D ; where E = RMS value of the signal envelope f Q = Doppler (fading) frequency 1 in Hz. A Gaussian noise source i s f i l t e r e d to create a (discrete) simulated spectrum [3]. F i g . 1.1 shows the t h e o r e t i c a l spectrum S(f) and the simulated spectrum S ( f ) . Two independently generated discrete spectra are transformed into time domain signal envelopes by Inverse Discrete Fourier Transform (IDFT), and they are then added in quadrature to create a Rayleigh fading signal envelope. F i g . 1.2 shows a t y p i c a l Rayleigh fading signal envelope generated by t h i s simulator. In Chapter 2, a model based on treating the fading channel as an equivalent random error channel i s examined. This model allows the estimation of the BKER through the use of an e f f e c t i v e b i t error rate P e f f 1 The Doppler (fading) frequency i s defined as f D = v/\ (Hz), where v i s the vehicle speed in meters/second, and X i s the RF c a r r i e r wavelength in meters. 3 7.5 - i 1-< 0.5-0 -\ i i i i i i i i i i > i i i i i i i " | " i 0 2 4 6 8 10 12 14 16 18 20 22 Frequency (Hz) Fig. 1.1 : Power Spectral Density of Received Signal I n C h a p t e r 3 , a m o d e l o r i g i n a l l y p r o p o s e d i n [ 1 ] i s s t u d i e d . T h e r a n g e o f t h e m o d e l i s i n c r e a s e d t o c o v e r t h e v e h i c l e s p e e d s f r o m 8 MPH t o 71 MPH a t a n R F c a r r i e r f r e q u e n c y o f 8 5 0 M H z ( c o r r e s p o n d i n g t o a D o p p l e r f r e q u e n c y , f D , r a n g e o f 10 H z t o 9 0 H z ) . A r e f i n e d m o d e l ( M o d e l I I ) w h i c h i m p r o v e s t h e e s t i m a t i o n o f t h e B K E R i n r a p i d f a d i n g s i t u a t i o n s i s a l s o c o n s i d e r e d . I n C h a p t e r 4 , t h e s t a t i s t i c s p r o d u c e d b y t h e R a y l e i g h f a d i n g s i m u l a t o r a r e u s e d t o o b t a i n e m p i r i c a l f o r m u l a s f o r e s t i m a t i n g t h e p r o b a b i l i t y d i s t r i b u t i o n s o f t h e f a d e r a t e 4 (gp) UD9 ^ o[ paziipiujou adoiaALrj IDU6JS Fig. 1.2 : Rayleigh Fading Signal Envelope at / n = 10 Hz 5 and the fade duration. In Chapter 5, the problem of computing the pro b a b i l i t y Q(m,N) of m-bit errors in an N-bit block i s considered. The values of Q(m,N) obtained from the simulation in Chapter 2 are used to derive an empirical formula for estimating Q(m,N). 2. A S IMPLE MODEL FOR THE ESTIMATION OF BLOCK ERROR RATES ON THE RAYLEIGH FADING CHANNEL WITH AWGN 2.1 REVIEW 2.1.1 THE STEADY SIGNAL AWGN CHANNEL In an Additive White Gaussian Noise (AWGN) Channel, signals are corrupted by the addition of Gaussian noise with two sided power spectral density (PSD) N 0 / 2 . For binary s i g n a l l i n g schemes l i k e NCFSK, CFSK and PSK which modulate every b i t independently, the Bit Error Rate (BER) p of every b i t in a block of N bi t s i s dependent only on the Signal-to-Noise power r a t i o (SNR) 7. The BER's of these three modulation schemes are as follows [4,5] : NCFSK p ( 7 ) = (1/2) exp ( - 7 / 2 ) CFSK p ( 7 ) = (1/2) e r t c l y f y / l ) ( 2 . 1 . 1 ) PSK p ( 7 ) = (1/2) erfcd/^y") where SNR 7 = E^/N0 and = energy per b i t . In the AWGN channel, b i t errors occur randomly. The Block Error Rate (BKER) P(0,N) of the received N-bit block w i l l be : 6 7 P(0,N) = 1 - (1 - p ) N (2.1.2) where p = pW) . 2.1.2 THE RAYLEIGH FADING AWGN CHANNEL In the Rayleigh Fading channel with AWGN, the b i t errors are no longer randomly d i s t r i b u t e d . Therefore, the fading BKER P f(0,N) i s not related to the BER p as in (2.1.2). Rather, P f(0,N) i s related to the Block Length N ( b i t s ) , the average Signal-to-Noise power r a t i o y0 (dB) and the Doppler (fading) Frequency f D (Hz). On a slow fading channel, the SNR i s assumed to be constant over the duration of a b i t , and may vary from b i t to b i t . Therefore, the in d i v i d u a l fading BER P f ( i ) can be calculated from (2.1.1) for the three d i f f e r e n t modulation schemes. Since the SNR 7 is time varying, i t is d i f f i c u l t to calculate the fading BKER P f(0,N) a n a l y t i c a l l y . In t h i s chapter, software simulation of Rayleigh fading s t a t i s t i c s are used to est a b l i s h a simple empirical model for the estimation of the fading BKER P f(0,N). The motivation i s to fi n d a simple model analogous to (2.1.2) for the slow Rayleigh fading AWGN channel; i . e . P f(0,N) = 1 - (1 - p f ) N . (2.1.3) The simulation and the empirical model are described in the following sections. Because the BER p(y) varies 8 according to the modulation scheme used, the NCFSK was used in t h i s simulation. Unless otherwise stated, the results derived w i l l be applicable to NCFSK only. 2.2 THE EMPIRICAL MODEL The parameters a f f e c t i n g the BKER P f(0,N) in the Rayleigh fading channel include the average SNR y0 (dB), the fading frequency f D (Hz), the b i t rate R (bps) and the block length N ( b i t s ) . 2.2.1 THE SOFTWARE SIMULATION The Rayleigh fading simulator based on [2, 3] was used to generate fading sequences of d i f f e r e n t fading frequencies f D (Hz). These sequences were normalized to various average SNR 7 0 l e v e l s , and the BER p(7) for every b i t was calculated. The decision of whether a b i t was erroneous or not was made by comparing i t s BER with a number u^ generated by a uniform pseudo-random number generator. The b i t was declared to be in error i f p(7) > u^. The e f f e c t of random FM noise on the BER was neglected in t h i s simulation because i t has been determined that the random FM noise has a s i g n i f i c a n t e f f e c t on the BER only when the average SNR exceeds about 35 dB [6]. The average BER p f ( 7 0 ) calculated from t h i s simulation and error decision method was confirmed to be close to the t h e o r e t i c a l average BER on a Rayleigh 9 fading channel [1]. In the case of NCFSK, t h i s t h e o r e t i c a l average BER i s given by : Pf(7o> = 1 / (2+7o) (2.2.1) The b i t error pattern created above was then segmented into d i f f e r e n t block sizes N (bits) and the fading BKER P f(0,N) was calculated. By using 30 t r i a l s each of 200 seconds duration for each fading frequency f Q (Hz), the simulated P^(0,N) was found to have a 95% confidence int e r v a l of at most ± 4.5 % over the whole range of the simulation. This simulation covered the following ranges of parameters. f D : from 10 Hz to 90 Hz (5 Hz interval) 7o : from 5 dB to 35 dB (5 dB interval) N : 63, 127, 255, 511, 1023, 2047 (bits per block) : 4096 (bps) The above range for f Q covers vehicle speeds from 13 KMH to 115 KMH (8 MPH to 71 MPH) at an RF c a r r i e r frequency of 850 MHz. There were 17(f D) x 7(SNR y0) x 6(blocksize N) = 714 sets of ( f D , 7o, N) which gave 714 corresponding BKER1s P f(0,N). Omitting those P f(0,N) with value 1, there were 572 10 sets of values for use to derive the empirical formula. 2.2.2 THE EMPIRICAL FORMULA It was confirmed in [1] that the BKER P f(0,N) i s related not only to the average SNR y0r but also the fading frequency f D , the b i t rate R, and the block size N. Therefore, the BKER P f(0,N) calculated in t h i s simulation did not relate to the BER p f by the simple r e l a t i o n of (2.1.3). Referring again to the simple formula (2.1.3), one can ea s i l y see that the fading BER p^ has to incorporate the effe c t of 7o, f D , R and possibly N in order to maintain the form of (2.1.3). We thus define an e f f e c t i v e BER P ej£ as a function of y0l f D , R and N; i . e . P e f f = F ( 7 0 r f D , R, N) (2.2.2) so that (2.1.3) can be re-written as : P f(0,N) = 1 - (1 - P e f f ) N . (2.2.3) It was hoped that there would be some simple dependence of p e f f o n 7 0 ' fD' R a n d N * Substituting the 572 values of P f(0,N) obtained from the simulation into (2.2.3), the corresponding values of P e££ were calculated. One c h a r a c t e r i s t i c of the Rayleigh 11 fading envelope i s that when the fading frequency f^ i s increased, the fading envelope w i l l be proportionally compressed. If the b i t rate R i s increased so that the r a t i o R/f D remains unchanged, every b i t in the fading envelope w i l l have exactly the same SNR as before. Therefore, the r a t i o L = R/fpj incorporates the e f f e c t of the b i t rate R and the fading frequency f n . It was also found that P e j f i s quite i n s e n s i t i v e to changes in N. Therefore, (2.2.2) can be reduced to : p e f f = F ( 7 ° ' L ) * (2.2.4) Plots of P e f f a s a function of 7 0 and L with N = 127 b i t s are shown in F i g . 2.1 and F i g . 2.2 respectively. These suggest that (2.2.4) can be expressed in a simple form. After a few d i f f e r e n t relationships were t r i e d , the following empirical form was found to be suitable L ° 9 1 0 p e f f = a i + A 2 7 ° + A 3 L A " (2.2.5) where y0 = SNR in dB L = R/f D ( b i t s / c y c l e ) . The MTS Curve F i t t i n g Library UBCrCURVE subroutine NL2SN0 was used to calculate the c o e f f i c i e n t s ( a , f a 2 , a 3 , a 4 ) which would minimize the Sum of Square % Errors between the 12 10-n Average Signal to Noise Ratio yQ ( dB ) Fig. 2.1 : E f f e c t i v e Bit Error Rate p f f vs Average SNR y0 N = 127 b i t s e J J 13 1 0 " n L (bit/cycle) Fig. 2.2 : Effective Bit Error Rate p , , vs Ratio L N = 127 bits e J J 14 estimated BKER EP f(0,N) calculated from (2.2.3) with the substitution of p g f f from (2.2.5), and the simulated BKER P f(0,N) from the Rayleigh Fading Simulator; i . e . L ° 9 1 0 p e f f = a i + a 2 7 o + a 3 L a 4 EP f(0,N) = 1 - (1 - p e f f ) N (2.2.6) S.S.%E. = I { E P f ~ P f x 100}2 . P f Substituting the 572 sets of ( P e f f r To, L) in (2.2.5) for the curve f i t t i n g subroutine, the following set of c o e f f i c i e n t s (a , ,a 2 ,a 3 ,a„ ) were found: a, = -0.457 a 2 = -0.081 a 3 = -0.017 a« = 0.583 These c o e f f i c i e n t s were obtained by minimizing the Sum of Square % Errors in (2.2.6) over the whole range of f D from 10 Hz to 90 Hz. The % error between the estimated BKER EP f(0,N) and the simulated BKER P f(0,N) for each of the 572 points was computed, and the average % error was obtained; i . e . % Error at Any One Point = E P f ~ P f x 100 (%) P f Average % Error = - i - Z { E P f ~ P f x 100} (%) 15 The average % error over the whole range of f D (from 10 Hz to 90 Hz) was found to be 3.9 %. There were altogether 64 points (out of 572 points) with double d i g i t % er r o r s . A l l of these except one had an error of less than 20 %. The exception was a % error of approximately 21 % at f D = 10 Hz, 7 0 = 15 dB and N = 127 b i t s . The other 142 P^(0,N) simulation points with value 1 were not included in the evaluation of average % error. It was found that the empirical formula gave very good estimates for these 142 points ( i . e . estimated P f(0,N) = 1 for these points). The average % error including these 142 points was found to be a r t i f i c i a l l y reduced to 3.1 %. 16 We refer to th i s model as Model I and summarize i t s performance as follows : Empirical Formula : L o g l Q p e f f = -0.457 - O . O 8 I 7 0 " 0.017 L 0 , 5 8 3 P f(0,N) = 1 - (1 - p e f f ) N For the ranges of : f Q = 10 Hz to 90 Hz 7 0 = 5 dB to 35 dB N = 63, 127, 255, 511, 1023, 2047 bits/block R = 4096 bps Average % Error = 3.9 % (over 572 points) Highest % Error = 2 1 % (at 10 Hz, 15 dB, 127 bits) 3. A SECOND MODEL FOR T H E E S T I M A T I O N OF B LOCK ERROR R A T E S 3.1 BASIC MODEL A model developed in [1] i s based on the assumption that i f an N-bit block i s suitably divided into sub-blocks of size x b i t s each, each x-bit sub-block can then be assumed to have steady SNR over the whole sub-block, and the N/x sub-blocks can be assumed to be independent. Therefore, the BKER Pf(0,N) can be expressed in a form similar to (2.1.3) with each x-bit sub-block being subject to independent error ; i . e . P f(0,N) = 1 - P c ( 0 , x ) N / x (3.1.1) where P (0,x) i s the Block Correct Rate (BKCR) of an x-bit c sub-block. In [1], an empirical formula was derived to compute a d i v i s o r d 2, namely d = a, + a 2 e a 3 7 ° (3.1.2) where a, = 2.85 a 2 = 0.643 a 3 = 0.125 7o = average SNR in dB This d i v i s o r was then used to compute P (0,x). 2 The d i v i s o r d i s defined as a d i v i s o r of L to y i e l d x; i . e . x = L / d, where L = R / f D . 17 18 x = L/d (3.1.3) 7i - 1 0 ^ / 1 ° r = 2/ 7, for P c(0,x) x » 1 (3.1.4) The values x calculated from (3.1.3) and P c(0,x) calculated from (3.1.4) were then substituted into (3.1.1) to estimate P f(0,N). This empirical formula was best f i t t e d for the range of f D from 10.5 Hz to 25.5 Hz. It gave an average of 2.1 % estimation error (over 138 points, excluding 30 points with P f(0,N) = 1) from the simulated P f(0,N). When a l l the 168 points were included, the average % error was reduced to 1.7 %. There were 3 points with double d i g i t % errors, and the highest % error was 17.3 % at f Q = 10.5 Hz, 7 0 = 10 dB and N = 63 b i t s . 3.1.1 EXTENDING THE RANGE OF THE BASIC MODEL When the basic model described in Section 3.1 was used to estimate the BKER P f(0,N) over the range of f D from 10 Hz to 90 Hz, large % errors were found over a wide f D range. These large errors resulted from the fact that the empirical formula (3.1.2) was derived using a lim i t e d range of f D from 10.5 Hz to 25.5 Hz. The large errors were also suspected to be pa r t l y the resu l t of using the c r i t e r i o n of 19 minimizing the Sum of Square Errors of the d i v i s o r d. The average % error was 4.3 % over 572 points. There were 67 points (out of 572 points) with double d i g i t % errors. The estimation error exceeds 20 % for a l l blocksizes N = 63, 127, 255, 511, 1023, 2047 b i t s when f D £ 60 Hz and 7 0 = 35 dB. The highest % error was approximately 50 % at f D = 90 Hz, y0 = 35 dB and N = 63 b i t s . The UBCrCURVE subroutine NL2SNO was used to minimize the Sum of % Errors between the estimated BKER EP f(0,N) and the simulated BKER P f(0,N) over the whole range of f D from 10 Hz to 90 Hz. The r e s u l t i n g average % error was 4 % and the highest % error was 41 %. In order to reduce the highest % error, the empirical formula was optimized by minimizing the Sum of Square % Errors between the estimated BKER EP^(0,N) and the simulated BKER P f(0,N) over the whole range of f Q from 10 Hz to 90 Hz; i . e . S.S.%E * x 100 } 2 The following empirical formula was found, d = -0.12 + 2.43 e 0.087O (3.1.5) This empirical formula gave an average error of 4.5 % over 20 the whole range of 10 Hz to 90 Hz. There were 86 points (out of 572 points) with double d i g i t % errors, and the highest % error was reduced to 27 % at f Q = 90 Hz, y0 = 35 dB and N = 63 b i t s . Using (3.1.5), i t was found that 18 points had a % error exceeding 20 % instead of 37 points when (3.1.2) was used. These figures indicate that t h i s model, even with the extended range of the d i v i s o r given by (3.1.5), gives a poorer BKER estimate than Model I described in Chapter 2. Ways for improving th i s model were examined and led to the formulation of Model I I . 3.2 THE SECOND MODEL (MODEL II) It was found in [1] that the model of Section 3.1 yielded large errors in estimating P^(0,N) when the sub-block size x was small. With the empirical formula changed to (3.1.5), the errors were averaged over the whole range of 10 Hz to 90 Hz, but generally higher estimation errors s t i l l showed up at small sub-block s i z e s . This led to the search for a model which could reduce the estimation errors at small sub-block sizes x. The problem was traced to the use of (3.1.4) to approximate P c(0,x). The analysis in [1] shows that the error in the approximation i s upperbounded by r/2 (1+x). For values of y0 < 10 dB, the approximation errors for 21 x < 10 b i t s are no longer n e g l i g i b l e , and hence ( 3 . 1 . 4 ) i s less accurate for small values of x. Rather, for small x, P c ( 0,x) was calculated by d i r e c t integration [ l ] ; i . e . OD P „ ( 0 f x ) = 1 - / e" y [ l - { l - p ( 7 i y ) ) X ] dy ( 3 . 2 . 1 ) c 0 where p ( 7 i y ) = 1 / 2 e ~ 7 l Y / ' 2 for NCFSK. ( 3 . 2 . 1 ) can be used to cal c u l a t e P c ( 0,x) for any value of x, but there are several advantages in using the approximation ( 3 . 1 . 4 ) when x i s large (> 1 0 ) . F i r s t of a l l , ( 3 . 1 . 4 ) i s very accurate for large values of x. Secondly, ( 3 . 1 . 4 ) can be evaluated with just a calculator and a GAMMA function table whereas ( 3 . 2 . 1 ) w i l l almost c e r t a i n l y require a computer with a well proven numerical integration subroutine. Thirdly, there are some, a l b e i t small, savings in computation time i f ( 3 . 1 . 4 ) instead of ( 3 . 2 . 1 ) i s used for large values of x. These trade-offs were considered in a r r i v i n g at a model which could improve the accuracy for small values of x while retaining the s i m p l i c i t y and accuracy of ( 3 . 1 . 4 ) for large values of x. In order to evaluate the i n t e g r a l ( 3 . 2 . 1 ) numerically, i t has to be approximated by a f i n i t e i n t e g r a l . Referring to APPENDIX A of [ 1 ] , ( 3 . 2 . 1 ) can be approximated as follows : P„(0,x) = 1 - / b e" y [ l - { l - p ( 7 , y ) } X ] dy c 0 22 where 0 < J e~ y [1-{1-p(7 1y)} x] dy < e, b 00 Since J e" y [1-{1-p(7iy) ) X 1 dy b < e~ b [1-{1-P( 7ib)} x], -b x therefore, e [1-{1-p(7,b)} ] < e was used to evaluate the smallest b which would meet the error constraint of — Q e = 10 for a l l sizes of x. It was found that b = 10. Thus, (3.2.1) was approximated as : P r(0,x) = 1 - ; 1 0 e y [l-{1-p(7iY)) X] dy. (3.2.2) c 0 The MTS Library UBC:INTEGRATION subroutine DCADRE was used to evaluate (3.2.2). A few sub-block sizes were t r i e d and the r e s u l t i n g improvements were compared. The following model gave the best r e s u l t s : d = -0.12 + 2.43 e ° - 0 8 7 ° (3.2.3) x = L/d 7, = 10 7°/10 (1) for x < 4 1 0 -v P (0,x) = 1 - J e yM-{l-p(7,y>} ] dy (3.2.4) c 0 (2) for x > 4 r = 2/ 7, 23 r r d + x ) T(l+r) T( 1+x+r) (3.2.5) N/x (3.2.6) This model gave an average error of 2.4 % over the whole range of 10 Hz to 90 Hz. There were 23 points (out of 572 points) with double d i g i t % erro r s . The highest % error was approximately 18 % at £ D =10 Hz, 7 0 =30 dB and N = 511 b i t s . This model also gave very good estimates for the other 142 points of P f(0,N) with value 1 ( i . e . estimated P f(0,N) = 1 for these points). The average % error including these 142 points was found to be reduced to 1.9%. 24 We refer to t h i s model as Model II and summarize i t s performance as follows : Empirical Formula : d = -0.12 + 2.43 e 0 , 0 8 7 0 x = L/d where L = R/f. (1) i f x < 4 10 Pc<0,x) = 1 - J e y [ l - p ( 7 l y ) ) X ] dy c 0 where 7 l = i 0 7 o / / 1 0 (2) i f x > 4 P<O,X> = 2 r r ( ^ ) r ( ; + r ) c Td+x+r) where r = 2/y, P f(0 fN) = 1 - P c ( 0 , x ) N / x Average % Error = 2.4 % (over 572 points) Highest % Error = 18 % (at 10 Hz, 30 dB, 511 bits) 4. EST IMATION OF THE PROBAB IL ITY D ISTRIBUTIONS OF THE FADE RATE & THE FADE DURATION 4.1 REVIEW Two important s t a t i s t i c s of the Rayleigh fading envelope are the average fade duration 7^ and the average fade rate at signal l e v e l p. It i s known [1,2,19] that these two parameters are given by : ep2 - 1 Tn = e _ . c 1 (4.1.1) P /21F f D p N p = /21F f D p e " p 2 (4.1.2) where p = amplitude of the signal envelope RMS value of the amplitude Various authors [6,7] have simulated the cumulative d i s t r i b u t i o n function of the fade duration. So far no attempt has been made to model the pro b a b i l i t y d i s t r i b u t i o n s of the fade rate and the fade duration at signal l e v e l p. In th i s Chapter, empirical expressions for the p r o b a b i l i t y d i s t r i b u t i o n s of the fade rate and the fade duration w i l l be considered. 25 26 4 . 2 THE SIMULATION The simulation was ca r r i e d out with the Rayleigh fading simulator described in Chapter 1 over the f Q range from 5 Hz to 9 0 Hz. Thirty sequences, each of duration 2 0 0 seconds were used. Each sequence was segmented into 2 0 0 1-second windows. The fade rate and the fade duration were recorded at a certa i n threshold l e v e l . In t h i s study, the threshold l e v e l was set at 1 5 dB below the average SNR; i . e . the threshold l e v e l p = S Q R T ( 1 0 ~ 1 5 ^ 1 0 ) . This threshold was chosen based on acceptable mobile radio telecommunications service requirements [ 7 ] , Fades which occurred at a window boundary were considered as separate fades in the two consecutive windows. The means and variances of the fade rate and the fade duration were also calculated. 4 . 3 PROBABILITY MASS FUNCTION OF FADE RATE A t y p i c a l p r o b a b i l i t y mass function (pmf) of the fade rate n (fades/sec) i s shown in F i g . 4 . 1 . This suggested that a Normal d i s t r i b u t i o n function would f i t the d i s t r i b u t i o n well; i . e . P ( „ , = _ ± _ e - ( n - M ) V 2 c ^ { 4 > 3 < 1 ) y/2% o The mean n i s equivalent to n - I 5 5 B a s determined from ( 4 . 1 . 2 ) . 27 0.16 0.12-0.08-0.04-0.00 X I f D = 20 Hz R = 4096 bps v \ 1 ! T 1 1 T ' f T — T ¥ % % 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fade Rate n (fades/sec) Fig. 4.1 : P r o b a b i l i t y Mass Function of Fade Rate 28 The variations of n and a in (4.3.1) with f D are plotted in Figures 4.2 and 4.3. It can be seen that n increases l i n e a r l y with f D < The curve f i t t i n g routine NL2SN0 was used to find the c o e f f i c i e n t s of the functions AI (f ^ ) and a ( f Q ) . The minimization of the Error Sum ERRS : ERRS = Z (estimated data - simulated data) leads to the following functions : u = 0.435 f D (4.3.2) a = j/0.323 f D . (4.3.3) From (4.1.2), N _ 1 5 d B = 0.432 f D ; therefore, (4.3.2) derived from the simulated data i s close to the th e o r e t i c a l value obtained using (4.1.2). The average % estimation error A V E R R : A V E R R = — Z restimated data - simulated data x 1 Q O % } 18 simulated data over 18 points of f D from 5 Hz to 90 Hz was 1.7 % for (4.3.2), and was 1.3 % for (4.3.3). F i g . 4.4 shows the pmf of F i g . 4.1 with the empirical Normal function obtained from (4.3.1) with u and a calculated from (4.3.2) and (4.3.3). F i g . 4.5 shows the corresponding cumulative mass function (cmf) of F i g . 4.4. Fading Frequency f D (Hz) Fading Frequency f D (Hz) 31 0.16-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fade Rate n (fades/sec) Fig. 4.4 : Probability Mass Function of Fade Rate n & its Empirical Normal Function 32 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fade Rate n (fades/sec) Fig. 4.5 : Cumulative Mass Function of Fade Rate n & the CMF of its Empirical Normal Function 33 Both figures show that the simulated data are well represented by the empirical Normal function. 4.4 PROBABILITY DENSITY FUNCTION OF FADE DURATION A t y p i c a l p r o b a b i l i t y mass function of the fade length i (bits) i s shown in F i g . 4.6. There are a number of curves which may f i t the shape of t h i s d i s t r i b u t i o n . The following three functions were t r i e d . Rayleigh d i s t r i b u t i o n : P(i) = i - e ~ i 2 / 2 h ° (4.4.1) bo for i = 0, 1, 2, where b 0 = mean GAMMA d i s t r i b u t i o n : P(i) = -=$-^ ( X i ) r _ 1 e " X i (4.4.2) T(r) for i = 0, 1, 2, where r = 1, 2, Arbitrar y d i s t r i b u t i o n : P(i) = C l i C 2 e " C 3 i C 4 for i = 0, 1,2, (4.4.3) 34 N I o CM II in CL J Q CO CD O X-X-X -X-X-X-X-x-X-X-X-X-X-x-X-X-X-tO to CM ro O tO o> CN 00 CM fx CN CO CN in CN CN to CN CN CN y \ CN V) O CN o> 00 * — CO — If) C *— V _ 1 to CD "D CN ,D X-X-X-X-X-X -X-m q d m o " T ~ if) O d i o CO o o to o o o o Fi g. 4. 6 u o i p u n j SSDJ^ XjniqDqojd P r o b a b i l i t y Mass Function of Fade Length 35 After a detailed examination, (4.4.3) was selected for the empirical f i t with the additional constraint that 2 P(i) = 1 . Since the t o t a l mass of the pmf i s always one, the mass of the d i s t r i b u t i o n s h i f t s towards shorter fade lengths as the fading frequency increases. The average fade length decreases from 54 b i t s for 5 Hz to 3 b i t s for 90 Hz. In order to generalize the fade length for any transmission rate R, the fade length was transformed into a fade duration r (milliseconds), where T = (fade length (bits) / b i t rate R) * 1000. Also, the pmf was transformed into a p r o b a b i l i t y density function (pdf) so that the t o t a l mass was represented by the area of the pdf. Since the fading frequency f ^ (= v/\) i s d i r e c t l y proportional to the vehicle speed v, doubling f D by doubling the vehicle speed w i l l reduce the fade duration by h a l f . Therefore, the fade duration i s inversely proportional to the fading frequency f D . Based on t h i s argument, we define the normalized fade duration 4> = r * f D (ms*Hz). The pdf for any fading frequency could then be represented by a single pdf of the normalized fade duration 0. F i g . 4.7 shows the pdf's of the normalized fade duration for 10 Hz, 45 Hz and 90 Hz. These 3 curves have a t o t a l area of 0.95 each. Even though i d e a l l y they are expected to be i d e n t i c a l , F i g . 4.7 **1 00 A. C #o I M ind «•» >i -. © i f N CJ- >» «» Q ft. O-c ft -• Q> o. - O c <* • — t a _Q J3 *> O JQ a * * i Pro c a o A 10 Hz X 45 Hz • 90 Hz i 1 1 1 1 1 1 1 90 100 110 120 130 140 150 160 Normalized Fade Duration cp (ms * Hz) CO 37 shows some differences e s p e c i a l l y near the peak. These differences can be explained as follows : A l l 3 simulations were run for 90 t r i a l s of 200 seconds/trial. The fade lengths are between 1 and 63 b i t s for f D = 10 Hz, between 1 and 14 b i t s for f D = 45 Hz and between 1 and 7 b i t s for f D = 90 Hz. The "fluctuations" in the f D = 10 Hz pdf are due to s t a t i s t i c a l v a r i a t i o n s . The UBC graphic routine *TELLAGRAF used to interpolate the sample points would also tend to further "smooth out" the pdf's for high f Q values. The cumulative density function (cdf) of the fade duration can be used in designing an error correcting code. Fi g . 4.8 shows the cdf's of the 3 curves in F i g . 4.7. The empirical formula was obtained by f i t t i n g a curve of the form of (4.4.3) to the simulated data for f D = 45 Hz. The r e s u l t i n g curve i s : 0 63 pdf(0) = 5 . 1 8 X 1 0 " 8 0 5* 6 1 e ' ° ' 8 < t > ' (4.4.4) where <t> = T * fp. The cdf calculated from (4.4.4) i s also shown in F i g . 4.8. It can be seen that i t f i t s the simulated data quite c l o s e l y . 38 uoipunj XjisusQ 3Aj4D|nujno Fig. 4.8 : Cumulative Density Function of Normal ized Fade Durat i on 0 & the CDF of its Empirical Function (4.4.4) 39 We summarize the two empirical models as follows : ( 1 ) Empirical model for estimation of p r o b a b i l i t y mass function of fade rate n : pm£(n) = e - < n - « > 2 / 2 c r * where n = 0 . 4 3 5 f D a = v^0.323 f D Average % estimation error for u = 1 .7 % Average % estimation error for o = 1 .3 %. ( 2 ) Empirical model for estimation of p r o b a b i l i t y density and cumulative density functions of normalized fade duration <t> : pdf(*) = 5 . 1 8 X 1 0 " 8 e - O - 8 0 ° ' 6 3 a cdf(tf>) = j pdf(tf>) d<*> 0 where a i s a real number. 5. ESTIMATION OF PROBABILITY OF m-BIT ERRORS  IN AN N-BIT BLOCK 5 . 1 REVIEW The p r o b a b i l i t y Q(m,N) of m-bit errors in an N-bit block in a mobile radio fading channel i s usually determined by simulations or f i e l d t r i a l s . Unfortunately, these methods tend to be quite c o s t l y . It i s shown in [ 1 , 2 2 ] that for very slow Rayleigh fading channels, in which the received signal strength can be assumed constant over the duration of the block, Q(m,N) can be approximated as : Q(m,N) = Q(0,N) -21 ( 5 . 1 . 1 ) m! where r = 2 / 7 and ( r ) m = r(r+1)(r+ 2)....(r+m-1). The error in t h i s approximation, e(m,N), is upperbounded by : c(m,N) ) N-m/ T \ m / 2 W m(1+N-m) for 0 < r < 1 . When th i s condition of very slow fading i s relaxed, ( 5 . 1 . 1 ) i s no longer accurate in estimating the pro b a b i l i t y Q(m,N). 40 41 5.2 AN APPROACH TO ESTIMATING Q(m,N) The values of Q(m,N) obtained from the simulation in Chapter 2 were analysed to see how Q(m,N) varies with the average SNR 7 0, the fading frequency f D , the blocksize N and the number of errors m. Thirty t r i a l s each of duration 200 seconds were generated for each ( f D , N, 7 0) at a transmission rate of 4096 bps. Only those values of m with a minimum of 50 blocks in the simulation were used. A plot of loQio Q(m,N)/Q(0,N) as a function of the average SNR y0 for dif f e r e n t values of f D , N and m showed that the rel a t i o n s h i p was approximately l i n e a r . F i g . 5.1 shows a t y p i c a l plot f o r f D = 40 Hz and N = 63 b i t s . A lin e a r equation of the following form was considered : L o g 1 0 IJ^ll = F , + F 2 7o, (5.2.1) where F , and F 2 are functions of f D , N and m. The simulated data for f Q = 20, 40, 60 and 80 Hz and blocksizes N = 63, 127, 255, 511, 1023 and 2047 b i t s were used with the MTS curve f i t t i n g routine NL2SNO to derive the values of F , and F 2 . Optimization was obtained by minimizing the sum of error between the estimated r a t i o and the simulated r a t i o Q(m,N)/Q(0,N). The range of m considered was from 1 to 19 b i t s . Typical v a r i a t i o n s of and F 2 with m for f D = 40 Hz and N = 63, 255, 1023 b i t s are shown in Figures 5.2 and 5.3 respectively. One possible form for the 42 10-3 O o fD = 40 Hz N = 63 bits o.oooi 0.01-0.001-15 20 25 Average SNR y0 (dB) Fig. 5.1 : Variation of Q(m,N) / Q(0,N) with Average SNR y0 fn = 40 Hz, N = 63 bits 44 45. functions F , ( f D , N, m) and F 2 ( f D , N, m) i s as follows : F, = a i f D a z N a 3 m a« (5.2.2) F 2 = b , f D b 2 N b a mba. (5.2.3) The curve f i t t i n g routine was applied to fi n d the best sets of (a , ,a 2,a 3,a 4) and (b,,b 2,b 3,b„) for (5.2.2) and (5.2.3) respectively. Optimization was performed by minimizing the sum of error between the estimated F, and F 2, and t h e i r corresponding simulated values. The following formulas were obtained : F, = 5 . 5 X 1 0 " 3 f D 0 ' 7 3 N 0 ' 4 7 m 0' 5 4 F 2 = -4.5X10" 3 f D 0 ' 5 2 N 0 ' 1 5 m0'6. (5.2.4) and (5.2.5) were used to calculate F, and F 2 for f D = 20, 40, 60 and 80 Hz, N = 127 and 511 b i t s and 7o = 15, 20 and 25 dB. These F / s and F 2's were substituted into (5.2.1) to estimate the p r o b a b i l i t y of m-bit errors Q(m,N) . The cmf of Q(m,N) i s important in the design of an error-correcting code. From (5.2.1), the cmf of up to M-bit errors cmf(M,N) can be calculated as follows : (5.2.4) (5.2.5) M cmf(M,N) = Z Q(m,N), m=0 (5.2.6) 46 where Q(m,N) can be obtained from (5.2.1) as : Q(m,N) = Q(0,N) 10 (F, + F 2 7 o ) (5.2.7) F, and F 2 are calculated from (5.2.4) and (5.2.5). F i g . 5.4 shows a t y p i c a l plot of the estimated cmf 1s with f D = 45 Hz, N = 511 b i t s and 7 0 = 15, 20 and 25 dB. Although the simulated and the empirical cmf's are d i f f e r e n t , they do exhibit similar shapes. In an attempt to get a better estimate of the simulated cmf, the empirical cmf was multiplied by a scaling factor. This factor i s applied to a l l Q(m,N) for m * 0. Denoting the la s t predicted and simulated cmf values by cmfp and cmf g respectively, the scaling factor k can be calculated as follows : where L i s the number of the l a s t predicted cmf value. Then, Applying t h i s scaling factor k to the cmf's in F i g . 5.4, the res u l t i n g scaled cmf's are shown in Fi g . 5.5. The predicted and simulated cmf's for other f D , N and 7 0 combinations in the range of consideration were checked. Figures 5.6 and 5.7 L Q(0,N) + Z Q(m,N) = cmf m=1 cmf - Q(0,N) k = (5.2.8) show the scaled cmf' s for (20 Hz, 127 bi t s ) and 47 u o j p u n j SSD^ a A i j D i n u j n Q Fig. 5.4 : Cumulative Mass Function of Q(m,N) & the unsealed CMF of the Empirical Function (5.2.7) fn = 45 Hz, N = 511 bits 48 uoipunj S S D ^ aAip|nujnQ Fig. 5.5 : Cumulative Mass Function of Q(m,N) & the scaled CMF of the Empirical Function (5.2.7) fD = 45 Hz, N = 511 bits 49 (70 Hz, 255 bits) respectively. This approach for estimating the pro b a b i l i t y Q(m,N) and i t s cmf i s quite e f f e c t i v e even though the procedure i s somewhat lengthy. The c o e f f i c i e n t s in the empirical formulas (5.2.2) and (5.2.3) may change i f the range of f D , N and 7 0 are extended. We summarize the results as follows : Estimation of Q(m,N) : Q(m,N) = k Q ( 0 , N ) 1 0 ( F i + F 2 7 o ) where m= 1, 2, 3, k = scaling factor calculated from (5.2.8) F, = 5 . 5 X 1 0 ' 3 f D 0 ' 7 3 N 0 ' 4 7 m 0' 5 4 F 2 = -4.5X10' 3 f D 0 ' 5 2 N 0 ' 1 5 m0'6. 50 uoipunj SSD^ aAi jDinujnQ Fig. 5.6 : Cumulative Mass Function of Q(mJ^) & the scaled CMF of the Empirical Function (5.2.7) fn = 20 Hz, N = 127 bits 51 O > 0 0 r ^ l O l T ) ' ^ ' K ) C M » 7 0 0 0 0 0 0 0 0 ° uoijounj SSD|^ a A j j D i n i u n Q Fig. 5.7 : Cumulative Mass Function of Q(mJV) & the scaled CMF of the Empirical Function (5.2.7) fn = 70 Hz, N = 255 bits 6 . CONCLUSION 6.1 CONCLUDING REMARKS The block error rate i s an important parameter in the design of an e f f i c i e n t d i g i t a l radio transmission system. For example, the e f f i c i e n c y of an ARQ scheme which uses only error detection i s highly dependent on the retransmission p r o b a b i l i t y . In t h i s thesis, a simple and f a i r l y accurate empirical model for estimating the block error rate P f (0,N) on a Rayleigh fading channel with additive white Gaussian noise was described. A previously studied model [ 1 , 2 2 ] was shown to be v a l i d for a wide range of vehicle speeds. The res u l t i n g model parameters were derived. The estimation of the pr o b a b i l i t y d i s t r i b u t i o n s of the fade rate and the fade duration were also examined. Empirical models which y i e l d reasonably good results were proposed. These quantities are useful in characterizing the nature of the errors in the mobile radio channel. The estimation of the p r o b a b i l i t y Q(m,N) of m-bit errors in an N-bit block was also considered. An approach to estimating Q(m,N) and i t s cmf as a function of the fading frequency f D , blocksize N, SNR y0 and number of errors m gives quite encouraging r e s u l t s . Further studies would be required to determine how good t h i s approach i s . 52 53 6.2 FUTURE WORK The simulation in this thesis was ca r r i e d out exclusively for a non-coherent FSK demodulation scheme. This i s mainly due to the popular use of t h i s scheme. A similar approach can be used in further studies for other modulation schemes. The attempt in Chapter 5 to estimate the pro b a b i l i t y Q(m,N) i s by no means f i n a l . This study suggests an empirical form for the rel a t i o n s h i p between Q(m,N) and the various parameters. Further work could involve the examination of a broader range of the independent parameters and the improvement of the accuracy in predicting Q(m,N). REFERENCES [I] B. H. Maranda, "The Computation of the Block Error Rate on a Rayleigh Fading Channel in the presence of Additive White Gaussian Noise," M.A.Sc. Thesis, Dept. of Electrical Engineering, University of British Columbia, Nov. 1982. [2] G. A. Arredondo, W. H. Chriss and E. H. Walker, "A Multipath Fading Simulator for Mobile Radio," IEEE Trans. Commun., Vol. COM-21, No. 11, pp. 1325-1328, Nov. 1973. [3] J. I. Smith, "A Computer generated Multipath Fading Simulation for Mobile Radio," IEEE Trans. Veh. Technol., Vol. VT-24, No. 3, pp. 39-40, Aug. 1975. [4] M. Schwartz, W. R. Bennett and S. Stein, Communication Systems and  Techniques. New York : McGraw-Hill. 1965. [5] J . M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York : John Wiley, 1965. [6] G. A. Arredondo and J . I. Smith, "Voice and Data Transmission in a Mobile Radio Channel at 850 MHz," IEEE Trans. Veh. Technol., Vol. VT-26, No. 1, pp. 88-93, Feb. 1977. [7] M. R. Karim. "Transmission of Digital Data Over a Rayleigh Fading Channel," IEEE Trans. Veh. Technol., Vol. VT-31, No." 1, pp. 1-6, Feb. 1982. [8] G. A. Arredondo, J. C. Feggeler and J . I. Smith, "Advanced Mobile Phone Service : Voice and Data Transmission," Bell System Technical Journal, Vol. 58, pp. 97-122, Jan 1979. [9] K. Otani, K. Daikoku and H. Omori "Burst Error Performance Encountered in Digital Land Mobile Radio Channel," IEEE Trans. Veh. Technol., Vol. VT-30, No. 4, pp. 156-160, Nov. 1981. [10] F. H. Blecher, "Advanced Mobile Phone Service," IEEE Trans. Veh. Technol., Vol. VT-29, No. 2, pp. 238-244, May 1980. [II] D. O. Reudink, "Properties of Mobile Radio Propagation above 400 MHz" IEEE Trans. Veh. Technol., Vol. VT-23, No. 4, pp. 143-158, Nov. 1974. [12] R. E. Eaves and A. H. Levesque, "Probability of Block Error for Very Slow Rayleigh Fading in Gaussian Noise," IEEE Trans. Commun., Vol. COM-25, No. 3, pp. 368-373, Mar. 1977. 54 55 [13] W. C. Jakes, Jr., "A Comparison of Specific Space Diversity Technigues for Reduction of Fast Fading in UHF Mobile Radio Systems," IEEE Trans. Veh. Technol., Vol. VT-20, No. 4, pp. 81-92, Nov. 1971. [14] R. C. French, "Error Rate Predictions and Measurements in the Mobile Radio Data Channel," IEEE Trans. Veh. Technol., Vol. VT-27, No. 3, pp. 110-116, Aug. 1978. [15] P. J . Mabey, "Mobile Radio Data Transmission - Coding for Error Control," IEEE Trans. Veh. Technol., Vol. VT-27, No. 3, pp. 99-109, Aug. 1978. [16] R. H. Clarke, "A Statistical Theory of Mobile-Radio Reception," Bell System Technical Journal, pp. 957-1000, Jul.-Aug. 1968. [17] R. P. Brent, "A Gaussian Pseudo-Random Number Generator [G5]," Communications of the ACM, Vol. 17, No. 12, pp. 704-706, Dec. 1974. [18] P. G. Moore, D. E. Edwards and E. A. C. Shirley, "Standard Statistical Calculations." New York : John Wiley, 1972. [19] W. C. Jakes, Jr., Microwave Mobile Communications. New York : John Wiley, 1974. [20] S. Lin, An Introduction to Error-Correcting Codes. New Jersey : Prentice-Hall, 1970. [21] R. A. Comroe and D. J. Costello, Jr., "ARQ Schemes for Data Transmission in Mobile Radio Systems," IEEE Trans. Veh. Technol., Vol. VT-33, No. 3, pp. 88-96, Aug. 1984. [22] B. Maranda and C. Leung, "Block Error Performance of Noncoherent FSK Modulation on Rayleigh Fading Channels," IEEE Trans. Commun., Vol. COM-32, No. 2, pp. 206-209, Feb. 1984. [23] C. Leung, B. Maranda and J . Ng, "Empirical Models for Evaluating Block Error Rates on Rayleigh Fading Channels," IEEE International Communications and Energy Conference, Montreal, Oct. 1984. APPENDIX A T h e s e FORTRAN p r o g r a m s c o m p i l e on t h e EE D e p t . VAX/VMS FORTRAN77 C o m p i l e r . To c o m p i l e them on t h e S t a n d a r d FORTRAN77 C o m p i l e r , a l l DO Loops h a v e t o be m o d i f i e d t o c o n v e n t i o n a l I n d e x e d DO Loops b e f o r e c o m p i l a t i o n . M a i n P rog ram SIM.FOR c c ** A S i m u l a t i o n P r o g r a m t o g e n e r a t e a RAYLEIGH FADING SEQUENCE c ** o f N ( b p s ) f o r MXSEC ( s e c ) c ** and c a l c u l a t e BLOCK ERROR RATE P s ( M , N B I T ) where c ** P s ( M , N B I T ) = BKER o f more t h a n M - b i t e r r o r i n a NBIT b l o c k c * * NTRYS = # o f T r i a l s c * * DBMIN = M i n . SNR i n dB c ** DBINC = I nc remen t o f SNR i n dB r * * NLEVS = # o f L e v e l s o f S e q u e n c e L e n g t h c ** NSEC = Sequence L e n g t h i n Seconds c ** NSIZE = # o f B l o c k s i z e s c ** NBIT = B l o c k s i z e i n b i t s / b l o c k c ** FD = F a d i n g F r e q u e n c y i n Hz c ** 1 1 , 12 = 2 s t a r t i n g Seeds f o r t h e G a u s s i a n Random # G e n e r a t o r c ** SUBROUTINES : SEQU.FOR, F A D E . F O R , F S S T . F O R , RANDN.FOR c R E A L * 8 S U M O ,30,6), SUMSQ(7, 30,6) , CDFSQ(7 ,30,6) REAL Y(819200), B T E R R(7 , 6 ) , F(100), ANORM(7) INTEGER NSEC(7), N B I T ( 6 ) , ICHAN(6) INTEGER*2 I I , 12 COMMON DBMIN, DBINC, N , MEM, N L E V S , NOUT, NTRYS COMMON / S E E D S / I I , 12 DATA SUM / 1 2 6 0 * 0 . D 0 / , SUMSQ / 1 2 6 0 * 0 . D 0 / , CDFSQ / 1 2 6 0 * 0 . D 0 / , BTERR / 4 2 * 0 . D 0 / DATA ICHAN / l l , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 / MEM = 819200 MAXBIT = 2048 N = 4096 NOUT = 30 56 OPEN (10, FILE='INPUT', STATUS='OLD', ERR=990) READ (10, 80) NTRYS READ (10, 90) DBMIN READ (10, 90) DBINC READ (10, 80) NLEVS IF (NLEVS .GT. 7) CALL GOOF(l) READ (10, 80) (NSEC(I), 1 = 1 , NLEVS) READ (10, 80) NSIZE IF (NSIZE .GT. 6) CALL GOOF(2) READ (10, 80) (NBIT(I), 1 = 1 , NSIZE) READ (10, 90) FD READ (10, 80) II READ (10, 80) 12 CLOSE (10) DO I = 1, NSIZE CALL HEADER(NSEC, NBIT(I), FD, ICHAN(I)) ENDDO MXSEC = 0 DO I = 1, NLEVS MXSEC = MAXO(MXSEC, NSEC(I)) ENDDO L = N * MXSEC IF (L .GT. MEM) CALL GOOF(3) IF (IFIX(FD) .GE. 100) CALL GOOF(4) IF ((NTRYS .LT. 2) .OR. (NTRYS .GT. 30)) CALL GOOF(5) DO I = 1, NSIZE IF (NBIT(I) .GE. MAXBIT) CALL GOOF(6) ENDDO 58 Z = RANDN(O) CALL FILTER(F, FD) DO LOOP = 1, NTRYS CALL SEQU(Y, F, FD, MXSEC) DO K = 1, NSIZE CALL NORML(Y, ANORM, NBIT(K), NSEC, MXSEC, * ICHAN(K)) CALL ERROR(SUM(l,l,K), SUMSQ(1,1,K), CDFSQ(1,1,K), * Y, BTERR(1,K), NBIT(K), NSEC, ANORM, ICHAN(K)) ENDDO ENDDO DO K = 1, NSIZE CALL STATS(SUM(1,1,K), SUMSQd, 1,K), CDFSQ(1,1,K), * BTERR(1,K), ICHAN(K)) ENDDO STOP 80 FORMAT (16) 90 FORMAT (F6.3) 990 WRITE (6,*) '*** Error in opening the INPUT Fil e ***' STOP END S u b p r o g r a m SEQU.FOR SUBROUTINE SEQU ( Y , F , F D , NSEC) COMMON DBMIN, DBINC, N , MEM, N L E V S , NOUT, NTRYS REAL Y (MEM) , F ( 1 0 0 ) , X ( 4 0 9 6 ) I F (N . G T . 4096) CALL GOOF(7) DO INDEX = 1 , NSEC CALL FADE ( X , F , F D , N) MARK = N * ( INDEX - 1) DO 1 = 1 , N Y(MARK + I ) = X ( I ) ENDDO ENDDO RETURN END SUBROUTINE NORML ( Y , ANORM, N B I T , NSEC, MXSEC, ICHAN) COMMON DBMIN, DBINC, N , MEM, N L E V S , NOUT, NTRYS REAL Y(MEM) , ANORM(NLEVS) INTEGER NSEC(NLEVS) R E A L * 8 D B L E , D L , DFLOAT, S l , S2 NBL = N / NBIT LEN = NBIT * NBL DL = DFLOAT(LEN) S l = O.ODO DO INDEX = 1, MXSEC MARK = N * ( INDEX - 1) S2 = O.ODO DO 1 = 1 , LEN S2 = S2 + DBLE(Y( I+MARK) ) ENDDO S l = S l + S2 / DL DO J = 1 , NLEVS I F ( INDEX . E Q . N S E C ( J ) ) ANORM(J) = S N G L ( S 1 / D F L 0 A T ( I N D E X ) ) ENDDO ENDDO RETURN END SUBROUTINE ERROR (SUM, SUMSQ, CDFSQ, Y, BTERR, NBIT, * NSEC, ANORM, ICHAN) INTEGER*2 I I , 12 COMMON DBMIN, DBINC, N, MEM, NLEVS, NOUT, NTRYS COMMON /SEEDS/ I I , 12 REAL*8 SUM(7,30), SUMSQ(7,30), CDFSQ(7,30), * DFLOAT, DNBLT, SIGMA, TEMP REAL Y(MEM), BTERR(NLEVS), ANORM(NLEVS) INTEGER NSEC(NLEVS), ERR(2048) NBL = N / NBIT DO LEVEL = 1, NLEVS NSECS = NSEC(LEVEL) NBLT = NBL * NSECS DNBLT = DFLOAT(NBLT) DB = DBMIN + DBINC * FLOAT(LEVEL - 1) SCALE = (10.0 ** (DB/10.0)) / ANORM(LEVEL) MAXERR = 0 NBIT1 = NBIT + 1 D O 1 = 1 , NBIT1 E R R ( I ) = 0 E N D D O D O 1 = 1 , NSECS DO INDEX = 1, NBL NERR = 1 MARK = NBIT*(INDEX-1) + N*(I-1) DO J = 1, NBIT SNR = SCALE * Y(MARK+J) IF (SNR .GT. 50.0) GO TO 20 BIT = 0.5 * EXP(-0.5*SNR) TEST = R A N C H , 12) IF (TEST .LT. BIT) NERR=NERR+1 0 ENDDO ERR(NERR) = ERR(NERR) + 1 IF (NERR .GT. MAXERR) MAXERR = NERR ENDDO ENDDO 61 L = NBIT * NBLT ACTL = BITERR (ERR, MAXERR, L) BTERR(LEVEL) = BTERR(LEVEL) + ACTL SIGMA = O.DO NUM = MINO(MAXERR, NOUT) DO K = 1, NUM TEMP = DFLOAT(ERR(K)) / DNBLT SUM(LEVEL,K) = SUM(LEVEL,K) + TEMP SUMSQ(LEVEL,K) = SUMSQ(LEVEL,K) + TEMP * TEMP SIGMA = SIGMA + TEMP TEMP = 1.D0 - SIGMA CDFSQ(LEVEL,K) = CDFSQ(LEVEL,K) + TEMP * TEMP ENDDO ENDDO RETURN END SUBROUTINE STATS (SUM, SUMSQ, CDFSQ, BTERR, ICHAN) COMMON DBMIN, DBINC, N, MEM, NLEVS, NOUT, NTRYS REAL*8 SUM(7,30), SUMSQ(7,30), CDFSQ(7,30), DBLE, DFN, MEAN, S, SIGMA, T REAL BTERR(NLEVS) WRITE (ICHAN, 30) FN = FLOAT(NTRYS) DFN = DBLE(FN) DO LEVEL = 1, NLEVS DB = DBMIN + DBINC * FLOAT(LEVEL - 1) ACTL = BTERR(LEVEL) / FN CALL TITLE (DB, ACTL, ICHAN) WRITE (ICHAN, 40) SIGMA = 0.D0 DO J = 1, NOUT S = SUM(LEVEL,J) T = SUMSQ(LEVEL,J) MEAN = S / DFN VAR1 = ZERO((DFN*T-S*£) / (DFN*(DFN-1.0D0))) SIGMA = SIGMA + MEAN S = 1.D0 - SIGMA T = CDFSQ(LEVEL,J) VAR2 = ZERO((T-DFN*S*S) / (DFN-1.0D0)) NERR = J - 1 SMEAN = SNGL(MEAN) CDF = ZERO(S) WRITE (ICHAN, 50) NERR, SMEAN, VAR1, CDF, VAR2 ENDDO ENDDO RETURN FORMAT ( ' i * * * * * * * * * * FINAL STATISTICS **********•/) FORMAT ('0# OF ERRORS', T21, 'PROB', T39, 'VARIANCE', T60, 'MORE THAN ? ERRORS', T85, 'VARIANCE'/) FORMAT (T5, 13, T15, 1PE15.6, T34, E15.6, T60, E15.6, T80, E15.6) END 63 SUBROUTINE HEADER (NSEC, NBIT, FD, ICHAN) COMMON DBMIN, DBINC, N, MEM, NLEVS, NOUT, NTRYS INTEGER NSEC(NLEVS) WRITE (ICHAN, 20) NTRYS, NBIT, N WRITE (ICHAN, 30) NBL = N / NBIT DO LEVEL = 1, NLEVS DB = DBMIN + DBINC * FLOAT(LEVEL - 1) NBLT = NBL * NSEC(LEVEL) FL = FLOAT(NBIT) * FLOAT(NBLT) WRITE (ICHAN, 40) DB, NSEC(LEVEL), NBLT, FL ENDDO WRITE (ICHAN, 50) FD RETURN 20 FORMAT ('OFADING SIMULATION'/'OTHE NUMBER OF TRIALS IS', * 13, '.'/'OBITS PER BLOCK :', 14, '.'/' BIT RATE * 15, ' B I T S / S E C ) 30 FORMAT ('OPARAMETERS (PER TRIAL) :'//T10, 'DB', T19, 'SECONDS', T31, 'NUMBER OF', T46, "NUMBER OF'/ T33, 'BLOCKS', T47, 'SAMPLES'/) 40 FORMAT (T8, F5.1, T20, 13, T33, 15, T46, F9.1) 50 FORMAT ('OTHE DOPPLER FREQUENCY IS', 2X, F6.3, * 2X, 'HZ.'/) END C =============================================== SUBROUTINE TITLE (DB, ACTL, ICHAN) SNR = 10.0 ** (DB/10.0) THEOR = 1.0 / (SNR + 2.0) WRITE (ICHAN, 10) WRITE (ICHAN, 20) DB, ACTL, THEOR WRITE (ICHAN, 10) RETURN 10 FORMAT (' ', 26(' ')) 20 FORMAT (T2, F5.1, ' DB', 4X, •***', T44, * 'BIT ERROR PROB :', 1PE15.6, 5X, * 'THEORETICAL', E15.6) END 6 4 SUBROUTINE GOOF(N) WRITE (6,5) FORMAT C * * * * * * * * * * ERROR IF (N .EQ. 1) WRITE (6, 10) IF (N .EQ. 2) WRITE (6, 20) IF (N .EQ. 3) WRITE (6, 30) IF (N .EQ. 4) WRITE (6, 40) IF (N .EQ. 5) WRITE (6, 50) IF (N .EQ. 6) WRITE (6, 60) IF (N .EQ. 7) WRITE (6, 70) * * * * * * * * * * 7) STOP 10 FORMAT C TOO MANY SNR''S.') 20 FORMAT C TOO MANY BLOCK SIZES.') 30 FORMAT C TOO MANY SAMPLES.') 40 FORMAT (' DOPPLER FREQUENCY TOO HIGH.') 50 FORMAT C TOO FEW OR TOO MANY TRIALS.*) 60 FORMAT ( ' TOO MANY BITS PER BLOCK.') 70 FORMAT (' BIT RATE IS TOO HIGH.') r rr — END FUNCTION BITERR (ERR, MAXERR, L) INTEGER ERR(2048) NSUM = 0 DO K = 2, MAXERR NSUM = NSUM + (K - 1) * ERR(K) ENDDO BITERR = FLOAT(NSUM) / FLOAT(L) RETURN END FUNCTION ZERO(A) REAL*8 A, DABS ZERO = SNGL(A) IF (DABS(A) .GT. l.D-15) RETURN ZERO = 0. RETURN END Subprogram FADE.FOR C **============================================ C ** FADE.FOR generates a Rayleigh Fading Sequence X(I) based on C ** a Model proposed i n IEEE Trans. COM.21 v o l . 11 C ** p.1325 Nov. 1973 SUBROUTINE FADE (X, F, FD, N) REAL X(N), F(100), Y(5002), Z(5002) IM = IFIX(FD) IN = IM + 1 DO I = 1, N+2 Y(I) = 0.0 Z(I) = 0.0 ENDDO Y( l ) = RANDN(O) DO I = 2, IN Y(2*I-1) = Y(2*I) ENDDO Z( l ) = RANDN(O) DO I = 2, IN Z(2*I-1) = F(I) * RANDN(O) Z(2*I) = F(I) * RANDN(O) ENDDO C ** Subroutine FSST i s i n the f i l e SYSSUSERDISK:[IEEESIG]FAST.FOR CALL FSST (Y, N) CALL FSST (Z, N) DO I = 1, N X(I) = Y(I) * Y(I) + Z(I) * Z(I) ENDDO F ( I ) * RANDN(O) F ( I ) * RANDN(O) RETURN END SUBROUTINE FILTER (F, FD) REAL F(100) PI = 3.1415927 IM = IFIX(FD) IN = IM + 1 F ( l ) = 1.0 DO I = 2, IM FF = FLOAT(I - 1) P = SQRTd.O - (FF/FD)**2) F(I) = 1.0 / SQRT(P) ENDDO P = (FLOAT(IM - 1)) / SQRT(FLOAT(2*IM - 1)) F(IN) = SQRT(FLOAT(IM)*(PI/2.0 - ATAN(P))) RETURN END 67 Subprogram RANDN.FOR C C ** Gaussian random number generator. C ** This program was written by R.P. Brent. It appears i n C ** Communications of the ACM, v o l . 17, no. 12, pp. 704-706, C ** December 1974. C ** The program uses the function RAN, which i s a uniform C ** random number generator implemented on the VAX. The C ** seeds II and 12 for the generator must be passed by C ** a COMMON statement. C **==================^ ^ FUNCTION RANDN(IDUMMY) INTEGER*2 I I , 12 COMMON /SEEDS/ I I , 12 DIMENSION D(32) DATA D ( l ) , D(2), D(3), D(4), D(5), D(6), D(7), D(8), D(9), D(10), D ( l l ) , D(12), D(13), D(14), D(15), D(16), D(17), D(18), D(19), D(20), D(21), D(22), D(23), D(24), D(25), D(26), D(27), D(28), D(29), D(30), D(31), D(32) /0.674489750,0.475859630,0.383771164, 0.328611323,0.291142827,0.263684322, 0.242508452,0.225567444,0.211634166, 0.199924267,0.189910758,0.181225181, 0.173601400,0.166841909,0.160796729, 0.155349717,0.150409384,0.145902577, 0.141770033,0.137963174,0.134441762, 0.131172150,0.128125965,0.125279090, 0.122610883,0.120103560,0.117741707, 0.115511892,0.113402349,0.111402720, 0.109503852,0.107697617/ DATA U /0.0/ 68 A = 0.0 1 = 0 10 U = U + U IF (U .LT. 1.0) GO TO 20 U = U - 1.0 1 = 1 + 1 A = A - D(I) GO TO 10 20 W = D(I+1) * U V = W * (0.5 * W - A) 30 U = RAN(II, 12) IF (V .LE. U) GO TO 40 V = RAN(II, 12) IF (U .GT. V) GO TO 30 U = RAN(II, 12) GO TO 20 40 U = RAN(II, 12) U = U + U IF (U .LT. 1.0) GO TO 50 U = U - 1.0 RANDN = W - A RETURN 50 RANDN = A - W RETURN END 69 Main Program FDSTAT.FOR c c ** A program to c a l c u l a t e ( i n Rayleigh Fading) c *r* (1) P r o b a b i l i t y Mass D i s t r i b u t i o n of Fade # c ** (2) P r o b a b i l i t y Mass D i s t r i b u t i o n of Fade Length c ** (3) Confident Limits of P r o b a b i l i t i e s c ** (4) Conditional P r o b a b i l i t y D i s t r i b u t i o n of c ** Fade Length (Assuming N fades i n a Window) c ** NTRY = # of T r i a l s (must be MULTIPLE of 10) c ** N = B i t Rate (must be a power of 2) c ** NSEC = Fading Sequence Length i n seconds c ** NSIZE = # of Windowsizes c ** NBIT = Windowsize i n b i t s c ** MAXCNT = = Highest Fade # c ** MAXLEN = • Longest Fade Length c ** KFAD = Highest Fade # to be pri n t e d c ** KLEN = Longest Fade Length to be printed c ** DB = Threshold i n dB c ** FD = Fading Frequency c * * 11, 12 = S t a r t i n g Seeds for RAN c * * TVAL = t-value i n Student-t d i s t r i b u t i o n c * * IDF = Degree of Freedom i n Student-t d i s t r i b u t i o n c ICL = % Confidence Limit c n ** SUBROUTINES : FADE.FOR, FSST.FOR, RANDN.FOR REAL Y(819200), F(100), X(4096), * PCFDdO, 5,200), PCLEN(10,5,1200), CDF(5,1200), * AVELEN(5,1200), AVEFD(5,200), * VARLEN(5,1200), VARFD(5,200), * CONFD(5,200), C0NLEN(5,1200), * AFD(5), VFD(5), ALEN(5), VLEN(5), * PCLN2(4,5,2000), CDF2(4,5,2000) REAL*8 TVAL, DNY, DM INTEGER FDLEN(5,1200), NFD(5,200), NBIT(5), NBLK(5), FDLN2(4,5,2000), SUMI, SUM2 INTEGER*2 I I , 12 COMMON /SEEDS/ II , 12 DATA FDLEN /6000*0/, NFD /1000*0/, AVEFD /1000*0./, AVELEN /6000*0./, VARFD /1000*0./, VARLEN /6000*0./, AFD /5*0./, ALEN /5*0./, VFD /5*0./, VLEN /5*0./, FDLN2 /40000*0/ OPEN ( i o , : FILE='FDIN*, STATUS' READ (10,*) NTRY, N, NSEC READ (10,*) NSIZE, FD READ (10,*) (NBIT(K), K = READ (10,*) KFAD, KLEN, KLN2 READ (10,*) 11, 12, DB READ (10,*) TVAL, IDF, ICL CLOSE (10) MEM = N * NSEC NCNT = 1 NONE = 0 NTWO = • 0 YMAX = 0. YMIN = 0. MAXLEN = 0 MAXCNT = 0 MAXTWO = 0 NLEN = 0 NFADE = 0 NTRY10 = NTRY / 10 FLN10 = FLOAT(10) I1C = 11 I2C = 12 DO K = 1, NSIZE NBLK(K) = MEM/NBIT(K) ENDDO Z = RANDN(0) CALL FILTER(F, FD) DO LY = 1, 10 DO LZ = 1, NTRY10 S = 0. DO J = 1, NSEC CALL FADE(X, F, FD, N) MK = N * ( J - l ) DO I = 1, N Y(MK + I) = X(I) S = S + X(I) ENDDO ENDDO S = S / FLOAT(MEM) Normalize Y(MEM) to 0 (dB) MEAN Fading Envelop DO 1=1, MEM Y(I) = 10.*ALOG10(Y(I)/S) YMAX = AMAX1(Y(I), YMAX) YMIN = AMIN1(Y(I), YMIN) ENDDO Calculate Prob. D i s t r . of Fade # & Fade length ** DO K = 1, NSIZE DO NBN = 1, NBLK(K) MK = (NBN-1) * NBIT(K) DO I =1, NBIT(K) IF (Y(MK+I) .GT. DB) GOTO 11 NONE = NONE + 1 IF (I .EQ. NBIT(K)) GOTO 13 GOTO 12 IF (NONE .EQ. 0) GOTO 12 FDLEN(K,NONE) = FDLEN(K,NONE)+1 NLEN = MAXCKNLEN, NONE) NTWO = NTWO + NONE NCNT = NCNT + 1 NONE = 0 ENDDO NFD(K,NCNT) = NFD(K,NCNT) + 1 NFADE = MAX0(NFADE, NCNT) IF (NCNT .EQ. 1) GOTO 14 IF (NCNT .GT. 5) GOTO 15 IF (NTWO .GT. 2000) GOTO 15 IFD = NCNT - 1 FDLN2(IFD,K,NTWO)=FDLN2(IFD,K,NTWO)+1 MAXTWO = MAXO(MAXTWO, NTWO) NCNT = 1 NTWO = 0 ENDDO ENDDO ENDDO DO K = 1, NSIZE SUM1 = 0 DO 1 = 1 , NFADE SUM1 = SUM1 + NFD(K,I) ENDDO DO 1 = 1 , NFADE PCFD(LY, K, I)=FLOAT(NFD(K,I))/FLOAT(SUM1) AFDK = AFDK + FLOAT(I-l)*PCFD(LY,K,I) VFDK = VFDK+(FLOAT(I-l))**2*PCFD(LY,K,I) NFD(K,I) = 0 ENDDO VFDK = VFDK - AFDK*AFDK AFD(K) = AFD(K) + AFDK VFD(K) = VFD(K) + VFDK AFDK = 0. VFDK =0. 72 SUM2 = 0 DO 1 = 1 , NLEN SUM2 = SUM2 + FDLEN(K,I) ENDDO DO 1 = 1 , NLEN PCLEN(LY,K,I)=FLOAT(FDLEN(K,I))/FLOAT(SUM2) ALENK = ALENK + FLOAT(I)*PCLEN(LY,K,I) VLENK = VLENK + (FLOAT(I))**2*PCLEN(LY,K,I) FDLEN(K, I) = 0 ENDDO VLENK = VLENK - ALENK*ALENK ALEN(K) = ALEN(K) + ALENK VLEN(K) = VLEN(K) + VLENK ALENK = 0. VLENK = 0. ENDDO MAXLEN = MAXO(MAXLEN, NLEN) MAXCNT = MAXCKMAXCNT, NFADE) NLEN = 0 NFADE = 0 ENDDO WRITE (6,601) MAXLEN, MAXCNT-1, MAXTWO 601 FORMAT ('OMAXLEN =', 16, 15X, 'MAXFD =*, 16/' MAXTWO =', 16/) DO K = 1, NSIZE C ** Calculate MEANS & VARIANCES of Fade # & Fade Length ** AFD(K) = AFD(K) / FLN10 VFD(K) = VFD(K) / FLN10 ALEN(K) = ALEN(K) / FLN10 VLEN(K) = VLEN(K) / FLN10 C ** Calcul a t e MEANS & VARIANCES of P r o b a b i l i t y D i s t r i b u t i o n s ** DO 1 = 1 , MAXCNT DO LY = 1, 10 AVEFD(K,I) = AVEFD(K,I) + PCFD(LY,K,I) VARFD(K,I) =VARFD(K,I)+(PCFD(LY,K,I))**2 ENDDO VARFD(K,I) = FLN10*VARFD(K,I) - (AVEFD(K,I))**2 VARFD(K,I) = VARFD(K,I) / (FLN10*(FLN10 - 1.)) AVEFD(K,I) = AVEFD(K,I) / FLN10 ENDDO DO 1 = 1 , MAXLEN DO LY = 1, 10 AVELEN(K, I )=AVELEN(K, I ) + P C L E N ( L Y , K , I ) VARLEN(K , I )=VARLEN(K , I )+ (PCLEN(LY ,K , I ) ) * *2 ENDDO VARLEN(K , I ) =FLN10*VARLEN(K , I ) - (AVELEN(K , I ) ) * *2 VARLEN(K, I )=VARLEN(K, I ) / (FLN10*(FLN10 - 1.) ) AVELEN(K, I )=AVELEN(K, I ) / FLN10 ENDDO CDF(K ,1 ) = AVELEN(K,1) DO 1 = 2 , MAXLEN CDF (K , I ) = CDF (K , I - 1 ) + AVELEN(K, I ) ENDDO ENDDO KFAD = MIN0(MAXCNT, KFAD) KLEN = MINO(MAXLEN, KLEN) KLN2 = MINO(MAXTWO, KLN2) OPEN ( 1 , F I L E = ' F D S T A T ' , STATUS='NEW') OPEN ( 7 , F I L E = ' C O N F \ STATUS='NEW' ) WRITE ( 1 , 1 0 0 ) NTRY, N, NSEC, NSIZE, FD, YMAX, YMIN, DE I1C, I2C, MAXCNT-1, MAXLEN WRITE (1 ,110 ) (NB IT (K ) , K = 1, 5 ) , (AFD(K ) , K = 1, 5 ) , (VFD(K) , K = 1, 5) WRITE (1 ,111) ( 1 - 1 , ( A V E F D ( K , I ) , K = l , 5 ) , 1=1,KFAD) WRITE (1 ,120) (NB IT (K ) , K = 1, 5 ) , ( A L E N ( K ) , K = 1, 5 ) , ( V L E N ( K ) , K = 1 , 5) DO I = 1, KLEN WRITE (1 ,121) I, ( AVELEN(K , I ) , C D F ( K , I ) , K= l , 5 ) ENDDO Calculate Confidence Limits of P r o b a b i l i t i e s ** DNY = DSQRT(DBLE(FLN10)) DO K = 1, NSIZE DO 1 = 1 , MAXCNT IF (AVEFD(K,I) .EQ. 0.) GOTO 21 DM = DSQRT(DBLE(VARFD(K,I))) DM = TVAL * DM / DNY CONFD(K,I) = SNGL(DM) * 100. / AVEFD(K,I) ENDDO DO 1 = 1 , MAXLEN IF (AVELEN(K,I) .EQ. 0.) GOTO 22 DM = DSQRT(DBLE(VARLEN(K,I))) DM = TVAL * DM / DNY CONLEN(K,I) = SNGL(DM) * 100. / AVELEN(K,I) ENDDO ENDDO WRITE (7,700) NTRY, N, NSEC, NSIZE, FD, DB, TVAL, IDF, ICL WRITE (7,710) (NBIT(K), K=l,3) DO I = 1, KFAD WRITE (7,711) 1-1, (AVEFD (K,I), VARFD(K,I), CONFD(K,I), K=l,3) ENDDO WRITE (7,710) (NBIT (K ) , K=3,5) DO I = 1, KFAD WRITE (7,711) 1-1, (AVEFD(K,I), VARFD(K,I), CONFD(K,I), K=3,5) ENDDO WRITE (7,720) (NBIT(K), K=l,3) DO I = 1, KLEN WRITE (7,711) I, (AVELEN(K.I), VARLEN (K,I), CONLEN(K,I), K=l,3) ENDDO WRITE (7,720) (NBIT(K), K=3,5) DO I = 1, KLEN WRITE (7,711) I, (AVELEN(K,I), VARLEN(K,I), CONLEN(K,I), K=3,5) ENDDO CLOSE (7) CLOSE (1) Calculate Conditional (on Fade #) P r o b a b i l i t y D i s t r i b u t i o n s DO K = 1, NSIZE SUM2 = 0 DO IFD =1, 4 DO 1 = 1 , MAXTWO SUM2 = SUM2 + FDLN2(IFD,K,I) ENDDO DO 1 = 1 , MAXTWO FLEN2 = FLOAT(FDLN2(IFD,K,I)) PCLN2(IFD,K,I) = FLEN2 / FLOAT(SUM2) ENDDO CDF2(IFD,K,1) = PCLN2(IFD,K,1) DO 1 = 2, MAXTWO CDF2(IFD,K,I) = CDF2(IFD,K,I-1) * + PCLN2(IFD,K,I) ENDDO ENDDO OPEN (1, FILE='WSTAT', STATUS='NEW') WRITE (1,770) NBIT(K), MAXTWO, (IFD, IFD=1,4) DO 1 = 1 , KLN2 WRITE (1,771) I, (PCLN2(IFD,K,I), * CDF2(IFD,K,I), IFD=1,4) ENDDO CLOSE (1) ENDDO 76 100 FORMAT ('OPROBABILITY DISTRIBUTIONS OF FADE # & FADE LENGTH' * ////' Parameters :'//' # of T r i a l s ' , T21,'=',I8/ * ' B i t Rate', T21,'=',I8,' bps'/ * ' Sequence Length', T21,'=',I8,' sec'/ * ' # of Windowsize', T21,'=*,I8/ ' Fading Frequency', T21,*=',F8.2,* Hz*// * ' Envelop (Max)', T21,'=',F8.2,' dB'/ ' Envelop (Min)', T21,'=',F8.2,' dB'// 'Threshold', T21,'=*,F8.2,' dB'// ' S t a r t i n g Seeds', T21,'=',218// * ' Max # of Fade', T21,'=',I8/ ' Max Fade Length', T21,'=',I8,' b i t s ' ) 110 FORMAT ( ' l P r o b a b i l i t y D i s t r i b u t i o n of Fade #*//// * ' # OF FADES',T31,'PROBABILITY'/ T10,5(10X,I5)/' \10('-'),T16,75(*-')/ * ' AVERAGE',5F15.2/' VARIANCE',5F15.2/) 111 FORMAT (16, 4X, 5F15.7) 120 FORMAT ( ' l P r o b a b i l i t y D i s t r i b u t i o n of Fade Length'/ * ' (CDF of P r o b a b i l i t y D i s t r i b u t i o n i n Parentheses)', * ////' FADE LENGTH',T31,'PROBABILITY'/ T4,' bits',T3,5(17X,15)/' ',12('-'),T16,105('-')/ * ' AVE LENGTH',F15.3,4F22.3/ VARIANCE ',F15.3,4F22.3/) 121 FORMAT (16, 4X, 5(F13.6,' (',F6.4,')')) 77 700 FORMAT ('OCONFIDENCE LIMITS OF PROBABILITIES'//// * * Parameters :'//' # of T r i a l s ' , T23 f'=',I8/ * ' B i t Rate', T23 #*=',I8,' bps'/ * ' Sequence Length', T23,'=',I8,' sec'/ * ' # of Windowsize', T23,'=',I8/ * ' Fading Frequency' T23,'=',F8.2,' Hz'// ' Threshold', T23,'=',F8.2,' dB'// * ' t-value', T23,'=',F8.4/ * ' Degree of Freedom', T23,'=',I8/ * ' Confidence Limits', T23,'=',I8,' %') 710 FORMAT ('IConfidence Limits of Fade # P r o b a b i l i t i e s ' / / / / * ' # OF FADES', T31, ' PROBABILITY' / * T16, 'AVERAGE—VARIANCE—CONFIDENCE LIMITS (AVE+-%) ' , * 10X,'WINDOWSIZE =', 318,' ( b i t s ) ' / * * *,10(*-'),T16,105('-')/) 711 FORMAT (17, 3(5X, 2F12.7, F8.3)) 720 FORMAT ('IConfidence Limits of Fade Length P r o b a b i l i t i e s ' / / / / * ' FADE LENGTH',T31,'PROBABILITY'/T4,' b i t s ' , * T16,'AVERAGE—VARIANCE—CONFIDENCE LIMITS (AVE+-%)\ * 10X,'WINDOWSIZE =', 318,' ( b i t s ) ' / ' \12('-'),T16,105('-')/) 7 7 0 FORMAT (' Conditional P r o b a b i l i t y D i s t r i b u t i o n s * of Fade Length', * ' (Assuming N fades i n a Window)'/ * ' (CDF i n Parentheses)'/// * ' Windowsize', T18,'=',I6,' b i t s ' / * ' Max Fade Length', T18, ,=',I6,' b i t s ' / / * ' FADE LENGTH', T31,'Conditional Probabilty', * ' (N fades per window)'/T4,' b i t s ' , T25,I1, 3(26X,I1)/ ' ',12('-'),T16, 105C-')/) 7 7 1 FORMAT (16,5X, 4(1PE18.7, * (',0PF6.4,')')) STOP END 

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