ESTIMATION OF ERROR R A T E S AND F A D E ON DISTRIBUTIONS A R A Y L E I G H F A D I N G CHANNEL WITH A D D I T I V E WHITE GAUSSIAN NOISE by Jimmy B.Eng., Kyoto Hon-yuenNg 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 t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March © 1986 Ji mmy Hon-yueh Ng, 1986 In presenting this requirements British available permission scholarly or understood that financial I gain agree for for purposes Department in an advanced Columbia, freely that for thesis by degree that reference extensive may his copying shall partial be or not at the Library shall of by the publication allowed The University of British 2075 Wesbrook Place Vancouver, Canada V6T IW5 Columbia of this without make Head for of It thesis my it agree thesis representatives. ENGINEERING 1986 of this DEPARTMENT OF ELECTRICAL March The University copying per mi s s i on. Date: the I further her be the of and study. granted or fulfilment my is for written 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 estimate the block error rate of a block of N model is proposed in order to data bits transmitted over the Rayleigh fading channel in the presence of additive white Gaussian noise. This model gives an average estimation error 255, 511, and fading to 71 1023, 2047 of about 4 (bits), frequencies f p MPH at a radio % over the range of blocksizes N average = 10 carrier to signal-to-noise 90 (Hz) frequency of 7 ratios = 0 5 = 63, 127, to 35 (dB) corresponding to vehicle speeds of 8 850 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 The probability examined. Empirical function of the % over the same set of simulated data. distributions of the models are derived fade rate and the fade for probability rate and the the estimation fade of the duration are also probability density function of the mass 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 2. A Simple Model for the Estimation of Block Error Rates on the Rayleigh Fading Channel with A W G N 6 2.1 6 2.2 3. 5. 6. Review 2.1.1 The Steady Signal A W G N Channel 6 2.1.2 The Rayleigh Fading A W G N Channel 7 The Empirical Model '. 8 2.2.1 The Software Simulation 8 2.2.2 The Empirical Formula 10 A Second Model for the Estimation of Block Error Rates 17 3.1 Basic Model 17 3.1.1 18 3.2 4. 1 Extending the Range of the Basic Model The Second Model (Model II) 20 Estimation of the Probability Distributions of the Fade Rate & the Fade Duration 25 4.1 Review 25 4.2 The Simulation 4.3 Probability Mass Function of Fade Rate 26 4.4 Probability Density Function of Fade Duration 33 „ 26 Estimation of Probability of m-bit Errors in an N-bit Block 40 5.1 Review Conclusion 5.2 An Approach to Estimating Q(ni,N) 40 52 41 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 1.2 Rayleigh Fading Signal Envelope at f^ = 2.1 Effective vs Average N = Bit Error 127 Rate p ^ £ bits 3 10 Hz 4 SNR y ; 0 , 12 2.2 Effective Bit Error Rate p - vs Ratio L; 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 gfi Mass Function of Fade Rate N n = & 127 its bits 13 Empirical Normal Function 4.5 31 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) 5.1 Variation f D = of Ratio Q(m,63) / 40 Hz, N = Q(0,63) with Average SNR 63 bits & 38 y; 0 42 5.2 Variation of F i with m; f D = 40 Hz 43 5.3 Variation of F f D = 40 Hz 44 2 with m; iv 5.4 Cumulative Mass Function of Q(m,N) & the unsealed C M F of the Empirical Function (5.2.6); f 5.5 D = 45 Hz, Cumulative N = 511 bits Mass Function of Q(m,N) 47 & the scaled C M F of the Empirical Function (5.2.6); f 5.6 D = 45 Hz, Cumulative N = 511 bits Mass Function of Q(m,N) 48 & the scaled C M F of the Empirical Function (5.2.6); f 5.7 D = 20 Hz, Cumulative N = 127 bits Mass Function of Q(m,N) 50 & the scaled C M F of the Empirical Function (5.2.6); f~ = 70 Hz, N = 255 bits v 51 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 <t> = fade T • f duration in millisec*Hz; 35 D 7 Signal-to-Noise power ratio (SNR) 6 7o average SNR in dB 8 7, 10 i fade length in bits/second L fade ratio in bits/cycle-of-fade; X R F carrier wavelength in meters/cycle 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 /2 two sided power spectral density of Gaussian noise 6 Np average fade rate in fades/second (at signal level p) v vehicle speed in meters/second p(7) bit 0 7 o / 1 ° error 18 rate (BER) at vi 33 SNR L = R/f 11 D 2 25 2 7 in Random Error Channel 6 effective BER in Rayleigh fading channel Pf BER in Rayleigh fading channel P (0,x) block correct rate (BKCR) of a x-bit block in Rayleigh c fading channel Pj(0,N) 7 17 block error rate (BKER) of an N-bit block in Rayleigh fading channel Q(m,N) 10 7 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 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 8 Acknowledgement I whose would like to express my grateful thanks continuous guidance and encouragement to my supervisor Dr. throughout the research C. Leung, work of this thesis are sincerely appreciated. The financial supports received by me including a Research Assistantship from NSERC Grant Electrical A1731, Engineering and at a Teaching University Assistantship of British from Columbia the are Department also of gratefully acknowledged. I would also like to thank Mr. Brian Rayleigh fading simulator program. viii H. Maranda for the use of his 1. In the mobile transmitted radio radio arrives received signal at environment, the the via receiver i s made up of a number waves s c a t t e r e d by o b s t r u c t i o n s such as b u i l d i n g s i n the vicinity of VHF/UHF signal m u l t i p l e paths. The of INTRODUCTION of the r e c e i v e r . The amplitude, phases arrival resultant local of these s i g n a l amplitude geographical amplitude [6,8,16]. waves is area, slow completely i s Rayleigh log-normally The are and angles random. distributed of the over larger local The over the mean value of the distributed change and a signal areas mean i s due to shadowing by i n t e r v e n i n g f e a t u r e s . Due to the o c c a s i o n a l deep fade i n s i g n a l s t r e n g t h , the errors in a block of received data tend to occur in c l u s t e r s . I n t e r l e a v i n g t r a n s m i s s i o n techniques together with Forward Error Repeat-request Correction (ARQ) e r r o r s [9,15,21]. proposed In (FEC) and/or Automatic can be used to combat these A number of other schemes have f o r use on the mobile t h i s t h e s i s , we r a d i o channel clustering also [7,8,9,13]. focus on the r a p i d R a y l e i g h fades of the s i g n a l envelope. Our main o b j e c t i v e i s t o develop empirical (BKER's) models of data for been estimating transmitted the over block the channel with A d d i t i v e White Gaussian Noise 1 error Rayleigh (AWGN). simple rates fading 2 A R a y l e i g h f a d i n g s i m u l a t o r [2] i s used to generate the f a d i n g envelope. The theoretical the complex s i g n a l envelope antenna power s p e c t r a l d e n s i t y r e c e i v e d by an omnidirectional i s as f o l l o w s [16,19] : , { 1 - (|-) ,2 E -1/2 } 2 , f^f, S(f) = < (LD 0 , where E f £>£ = RMS value of the s i g n a l = Doppler Q simulated spectrum spectrum S(f) [3]. and time Fourier domain Transform quadrature to Fig. 1.2 shows 1.1 discrete (IDFT), shows envelopes by they a Rayleigh are fading t y p i c a l Rayleigh 1 i n Hz. (discrete) the spectrum spectra and ; to c r e a t e a simulated signal create a Fig. the independently generated filtered D envelope (fading) frequency A Gaussian n o i s e source i s into of theoretical S(f). are Two transformed Inverse then Discrete added signal in envelope. fading s i g n a l envelope generated by t h i s s i m u l a t o r . In Chapter 2, a model based channel as an e q u i v a l e n t random on t r e a t i n g the e r r o r channel i s fading examined. T h i s model a l l o w s the e s t i m a t i o n of the BKER through the use of an e f f e c t i v e b i t e r r o r r a t e P ff e The Doppler ( f a d i n g ) frequency i s d e f i n e d as f = v/\ (Hz), where v i s the v e h i c l e speed i n meters/second, and X i s the RF c a r r i e r wavelength 1 D i n meters. 3 7.5 - i 1-< 0.50 -\ 0 i i i 2 4 i i i i 6 i i 8 i > i 10 i i i 12 14 i i 16 i | " 18 " 20 i 22 Frequency (Hz) Fig. 1.1 In Chapter studied. vehicle : Power Spectral 3, a The range of speeds model Density o r i g i n a l l y the model from 8 MPH to i s of 850 MHz ( c o r r e s p o n d i n g f of 10 , range which improves situations In fading i s Hz the also Chapter simulator estimating to 4, 90 H z ) . A estimation of Received at to Doppler a refined in to MPH the Signal proposed increased 71 frequency D of an [1] cover RF model i s the c a r r i e r frequency, (Model BKER in rapid produced by the II) fading considered. the are s t a t i s t i c s used the probability to obtain empirical d i s t r i b u t i o n s of Rayleigh formulas the fade for rate 4 (gp) Fig. UD9^ 1.2 : o[ paziipiujou adoiaALrj IDU6JS Rayleigh at / n Fading = 10 Signal Hz Envelope 5 and the fade d u r a t i o n . In Chapter 5, the problem of computing the probability Q(m,N) of m-bit e r r o r s i n an N - b i t block i s c o n s i d e r e d . v a l u e s of Q(m,N) obtained from are used Q(m,N). to d e r i v e an The the s i m u l a t i o n i n Chapter empirical formula 2 f o r estimating 2. A S I M P L E MODEL FOR T H E E S T I M A T I O N O F B L O C K 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, s i g n a l s are c o r r u p t e d by the a d d i t i o n of Gaussian two sided signalling power spectral density (PSD) N / 2 . For b i n a r y 0 schemes l i k e NCFSK, CFSK and PSK every b i t independently, the B i t E r r o r Rate bit in a block of N bits S i g n a l - t o - N o i s e power r a t i o The follows BER's of these (BER) p of every only on the three modulation schemes a r e as [4,5] : CFSK p ( ) = (1/2) e r t c l y f y / l ) PSK p ( ) = ( 1 / 2 ) erfcd/^y") 7 7 (2.1.1) 7 7 where and will modulate (SNR) 7. p ( ) = (1/2) e x p ( - / 2 ) Block which i s dependent NCFSK In noise with SNR 7 = E^/N 0 = energy per b i t . the AWGN channel, b i t e r r o r s Error occur randomly. The Rate (BKER) P(0,N) of the r e c e i v e d N - b i t block be : 6 7 P(0,N) = 1 - (1 - p ) p = pW) . where 2.1.2 (2.1.2) N THE RAYLEIGH FADING AWGN CHANNEL In the R a y l e i g h Fading e r r o r s are no longer randomly fading BKER P (0,N) f (2.1.2). Rather, (bits), with distributed. i s not r e l a t e d AWGN, the b i t T h e r e f o r e , the t o the BER p as i n P (0,N) i s r e l a t e d t o the f Block the average S i g n a l - t o - N o i s e power r a t i o y the Doppler On channel a ( f a d i n g ) Frequency f slow fading D Length (dB) and 0 (Hz). channel, the SNR i s assumed t o be constant over the d u r a t i o n of a b i t , and may vary to bit. Therefore, calculated from N from b i t the i n d i v i d u a l f a d i n g BER P f ( i ) can be (2.1.1) f o r the three different modulation schemes. Since the SNR 7 i s time v a r y i n g , i t i s d i f f i c u l t t o c a l c u l a t e the f a d i n g chapter, software BKER analytically. f In this s i m u l a t i o n of R a y l e i g h f a d i n g s t a t i s t i c s are used t o e s t a b l i s h estimation P (0,N) a simple empirical model for the of the f a d i n g BKER P ( 0 , N ) . The m o t i v a t i o n i s t o f i n d a simple f model analogous to (2.1.2) f o r the slow R a y l e i g h f a d i n g AWGN channel; i . e . P (0,N) = 1 - (1 - p ) . (2.1.3) N f f The the s i m u l a t i o n and the e m p i r i c a l model are d e s c r i b e d i n following sections. Because the BER p(y) varies 8 according to the modulation scheme used, the NCFSK was in simulation. this derived w i l l 2.2 THE otherwise s t a t e d , the r e s u l t s be a p p l i c a b l e to NCFSK o n l y . EMPIRICAL MODEL The parameters Rayleigh length N the BKER P (0,N) y the Rayleigh fading simulator based on f to v a r i o u s SNR 7 0 These sequences were normalized levels, calculated. not was by a and The the BER p(7) uniform pseudo-random was been determined effect exceeds about 35 dB on bit was erroneous or The effect bit was of random in t h i s simulation because that the random the BER only when the average [ 6 ] . The from t h i s s i m u l a t i o n and average with a number u^ generated > u^. The neglected used frequencies every number generator. i n e r r o r i f p(7) FM n o i s e on the BER significant for d e c i s i o n of whether a b i t was made by comparing i t s BER d e c l a r e d t o be i t has block [2, 3] was fading (Hz). the SOFTWARE SIMULATION to generate f a d i n g sequences of d i f f e r e n t D the (dB), 0 (Hz), the b i t r a t e R (bps) and D in f (bits). THE The affecting f a d i n g channel i n c l u d e the average SNR f a d i n g frequency f 2.2.1 Unless used average BER FM noise p (7 ) f BER on a SNR calculated 0 e r r o r d e c i s i o n method was to be c l o s e t o the t h e o r e t i c a l average has a confirmed Rayleigh 9 f a d i n g channel average BER [ 1 ] . In the case of NCFSK, (2.2.1) = 1 / (2+7o) f b i t e r r o r p a t t e r n c r e a t e d above was i n t o d i f f e r e n t block s i z e s P (0,N) was f calculated. seconds d u r a t i o n f o r simulated theoretical i s given by : P (7o> The this P^(0,N) By each was and using fading found i n t e r v a l of at most ± 4.5 simulation. N (bits) the have the fading BKER t r i a l s each of frequency to % over 30 then segmented a whole f Q 95% (Hz), 200 the confidence range of the T h i s s i m u l a t i o n covered the f o l l o w i n g ranges of parameters. f : from D 10 Hz to 90 Hz 7o : from 5 dB t o 35 dB N : 63, 127, 255, (5 Hz (5 dB 511, interval) interval) 1023, 2047 (bits per block) : 4096 (bps) The above range f o r f 115 KMH (8 MPH covers v e h i c l e speeds Q t o 71 MPH) at an RF c a r r i e r from 13 KMH to frequency of 850 x 6 ( b l o c k s i z e N) = 714 MHz. There were s e t s of ( f , 7o, N) D x 7(SNR y ) 17(f ) 0 D which gave 714 corresponding BKER s P ( 0 , N ) . O m i t t i n g those P (0,N) with value 1, there were f f 1 572 10 s e t s of values f o r use t o d e r i v e the e m p i r i c a l formula. 2.2.2 THE EMPIRICAL FORMULA It was confirmed in [1] that the r e l a t e d not only to the average SNR y f , the D b i t rate R, and T h e r e f o r e , the BKER P (0,N) c a l c u l a t e d f did not relate to the BER p f f but a l s o the 0r frequency BKER P (0,N) i s the in block this fading s i z e N. simulation by the simple r e l a t i o n of (2.1.3). Referring easily again to the simple see that the f a d i n g BER p^ effect formula has to (2.1.3), one can incorporate the of 7o, f , R and p o s s i b l y N i n order t o maintain the D form of (2.1.3). We thus d e f i n e an e f f e c t i v e BER P j £ e f u n c t i o n of y 0l f , R and N; i.e. P f , R, N) D e f f = F( 7 0 r as a (2.2.2) D so t h a t (2.1.3) can be r e - w r i t t e n as : P (0,N) = 1 - (1 - P f It p eff was o n 7 0 ' D' f R a n d N the s i m u l a t i o n i n t o e (2.2.3) hoped t h a t t h e r e would be some simple dependence of Substituting P ££ ) . N e f f were * the 572 (2.2.3), calculated. One v a l u e s of P (0,N) o b t a i n e d from f the corresponding values of c h a r a c t e r i s t i c of the R a y l e i g h 11 fading envelope increased, i s that the fading when the f a d i n g envelope will be compressed. I f the b i t r a t e R i s i n c r e a s e d R/f D remains unchanged, every b i t will have exactly the fading quite frequency proportionally so that the r a t i o the fading envelope same SNR as b e f o r e . T h e r e f o r e , the r a t i o L = R/fpj i n c o r p o r a t e s the in frequency f ^ i s the e f f e c t of the b i t r a t e R and f . It was a l s o found that P j f i s n e i n s e n s i t i v e t o changes i n N. T h e r e f o r e , (2.2.2) can be reduced t o : p Plots eff of N = 127 b i t s = F P ff ( a 7 ° ' s L ) a e are shown * (2.2.4) function of Fig. 2.1 in r e s p e c t i v e l y . These suggest that a simple form. A f t e r tried, a few L where The MTS ° 10 9 p eff = a i + = SNR i n dB L = R/f Curve D A 27° and L with Fig. 2.2 relationships were + A 3L " A (2.2.5) (bits/cycle). Fitting Library NL2SN0 was used t o c a l c u l a t e the which and form was found t o be s u i t a b l e y 0 0 (2.2.4) can be expressed i n different the f o l l o w i n g e m p i r i c a l 7 UBCrCURVE subroutine coefficients (a, a ,a ,a ) f 2 3 4 would minimize the Sum of Square % E r r o r s between the 12 10-n Average Signal to Noise Ratio Fig. 2.1 : E f f e c t i v e Bit Error Rate N = 127 b i t s p f e J f J y Q ( dB ) vs Average SNR y 0 13 1 0 " n L Fig. 2.2 : Effective (bit/cycle) Bit Error Rate N = 127 bits p ,, e J J vs Ratio L 14 estimated BKER EP (0,N) calculated f s u b s t i t u t i o n of p g f f from from (2.2.5), and (2.2.3) with the simulated the BKER P (0,N) from the R a y l e i g h Fading S i m u l a t o r ; i . e . f L ° 10 eff 9 p = i a + + a L 2 7 o a EP (0,N) = 1 - (1 - p f S.S.%E. = I { f ~ E P P Substituting for the curve coefficients the 572 fitting P s e t s of (P subroutine, ) f (2.2.6) N 2 . e f f r the To, L) i n (2.2.5) following a a a« = 2 % Sum of 0.583 the E r r o r s i n (2.2.6) over the whole range of f 10 Hz t o 90 The of = -0.081 These c o e f f i c i e n t s were obtained by m i n i m i z i n g Square set found: 3 = -0.017 f f x 100} a, = -0.457 3 e f (a , , a ,a ,a„ ) were 2 a 4 3 D from Hz. % e r r o r between the estimated BKER EP (0,N) and f s i m u l a t e d BKER P (0,N) f for computed, and the average % E r r o r at Any One each of % e r r o r was Point = E P the 572 points obtained; i . e . f ~ f x 100 f P (%) P Average % E r r o r = - i - Z { f~ E P P f x 100} (%) the was 15 The to average 90 Hz) was points of % e r r o r over the whole range of f (out 0 %. There these except one had an e r r o r of less altogether other 142 found that e s t i m a t e s f o r these for these p o i n t s was 64 % errors. A l l than 20 %. D The = 10 Hz, bits. P^(0,N) s i m u l a t i o n p o i n t s with value 1 were not i n c l u d e d i n the e v a l u a t i o n of average was (from 10 Hz a % e r r o r of approximately 21 % at f = 15 dB and N = 127 The were of 572 p o i n t s ) with double d i g i t e x c e p t i o n was 7 found to be 3.9 D the empirical formula % error found to be a r t i f i c i a l l y error. gave 142 p o i n t s ( i . e . estimated p o i n t s ) . The average % very It good P (0,N) = 1 f i n c l u d i n g these reduced to 3.1 %. 142 16 We refer to this model as Model I and summarize i t s performance as f o l l o w s : Empirical Formula : Log p l Q e f f = -0.457 - O . O 8 I 7 0 " 0.017 P (0,N) = 1 - (1 f For p e f f ) 0 , 5 8 3 L N the ranges of : f Q = 10 Hz t o 90 Hz 7 0 = 5 dB to 35 dB N = 63, 127, 255, 511, 1023, 2047 b i t s / b l o c k R = 4096 bps Average % E r r o r = 3.9 % (over Highest % E r r o r = (at 572 points) 21% 10 Hz, 15 dB, 127 b i t s ) 3. A 3.1 MODEL FOR THE ESTIMATION BLOCK A model developed i n [1] i f an N - b i t block i s suitably divided each sub-block size x b i t s each, assumed t o have steady SNR N/x OF ERROR RATES BASIC MODEL that of SECOND i s based x-bit on the into (2.1.3) Pf(0,N) can with independent each error sub-blocks can then be over the whole sub-block, and the sub-blocks can be assumed to be independent. the BKER assumption be expressed i n a form x-bit sub-block being Therefore, similar subject to to ; i.e. P (0,N) = 1 - P ( 0 , x ) f (3.1.1) N / x c where P (0,x) i s the Block c C o r r e c t Rate (BKCR) of an x-bit sub-block. In [ 1 ] , an e m p i r i c a l formula was d e r i v e d to compute a d i v i s o r d , namely 2 d = a, + a e a 3 7 2 where a, = 2.85 a ° 2 = 0.643 7o = average SNR T h i s d i v i s o r was (3.1.2) a 3 = 0.125 i n dB then used to compute P (0,x). The d i v i s o r d i s d e f i n e d as a d i v i s o r of L to y i e l d i . e . x = L / d, where L = R / f . 2 D 17 x; 18 x = L/d 7i - (3.1.3) 10^ / 1 ° = 2/ , r 7 P (0,x) (3.1.4) c for The x » 1 v a l u e s x c a l c u l a t e d from (3.1.3) and P ( 0 , x ) c from (3.1.4) were then s u b s t i t u t e d calculated i n t o (3.1.1) t o estimate P (0,N). f This empirical f D from 10.5 estimation Hz t o error P (0,N) = 1) f formula was best 25.5 Hz. I t (over 1.7 %. There were 3 and bits. 30 p o i n t s % error at f Q % with When a l l the 168 f was reduced to = 10.5 Hz, 7 0 and = 10 dB EXTENDING THE RANGE OF THE BASIC MODEL When the b a s i c model d e s c r i b e d to estimate 10 Hz t o the BKER 90 Hz, l a r g e range. These l a r g e empirical of f of 2.1 p o i n t s with double d i g i t % e r r o r s , % e r r o r was 17.3 % 3.1.1 P (0,N). the average the highest N = 63 gave an average 138 p o i n t s , e x c l u d i n g from the simulated p o i n t s were i n c l u d e d , f i t t e d f o r the range of D formula from 10.5 P (0,N) f i n Section over the range of % e r r o r s were found e r r o r s r e s u l t e d from (3.1.2) was d e r i v e d Hz t o 25.5 Hz. 3.1 was f used from D over a wide the f a c t that using a l i m i t e d D the range The l a r g e e r r o r s were suspected t o be p a r t l y the r e s u l t of using f the c r i t e r i o n also of 19 minimizing the Sum of Square E r r o r s of the d i v i s o r The average % e r r o r were 67 points errors. The blocksizes N f D £ 60 Hz approximately N = 63 (out was of and 7 50 0 error over 572 p o i n t s . 511, = The % 35 dB. at f 1023, y Hz, for 2047 % % all bits when error was = 35 0 There digit 20 % highest = 90 D w i t h double exceeds 127, 255, dB and bits. The UBCrCURVE s u b r o u t i n e the Sum the s i m u l a t e d BKER P (0,N) over NL2SNO was used to minimize of % E r r o r s between the e s t i m a t e d BKER EP (0,N) the whole range of f Hz. The r e s u l t i n g average % h i g h e s t % e r r o r was e r r o r was from D 4 % and 41 %. In order to reduce the formula was and f f 10 Hz t o 90 the % 572 p o i n t s ) estimation = 63, 4.3 d. o p t i m i z e d by highest % e r r o r , the m i n i m i z i n g the Sum of empirical Square % E r r o r s between the e s t i m a t e d BKER EP^(0,N) and the simulated BKER P (0,N) f over the whole range of f from 10 Q Hz to 90 Hz; i . e . * x 100 } S.S.%E The f o l l o w i n g e m p i r i c a l 2 formula was d = -0.12 + 2.43 e found, 0.087 T h i s e m p i r i c a l formula gave an average O (3.1.5) e r r o r of 4.5 % over 20 the whole range of 10 Hz t o 90 Hz. There were 86 p o i n t s (out of 572 p o i n t s ) with double d i g i t e r r o r was reduced to N = 63 b i t s . Using had a % error 27 % at % e r r o r s , and the h i g h e s t % f Q Hz, y = 90 0 (3.1.5), i t was found exceeding 20 % instead = 35 dB that of 37 18 and points points when (3.1.2) was used. These extended f i g u r e s i n d i c a t e that t h i s range of the poorer BKER estimate model, even with d i v i s o r given than Model I Ways f o r improving t h i s model by (3.1.5), the gives d e s c r i b e d i n Chapter a 2. were examined and l e d to the f o r m u l a t i o n of Model I I . 3.2 THE SECOND MODEL (MODEL I I ) I t was yielded found i n large sub-block s i z e [1] errors x t h a t the in model of estimating was s m a l l . With Section P^(0,N) the when empirical whole range of 10 Hz to errors s t i l l showed up a t small sub-block s i z e s . T h i s l e d t o the search f o r a model generally higher the formula changed t o (3.1.5), the e r r o r s were averaged over the 90 Hz, but 3.1 which c o u l d estimation reduce the e s t i m a t i o n e r r o r s a t s m a l l sub-block s i z e s x. The problem approximate error values was P (0,x). c traced to the The a n a l y s i s in use of (3.1.4) [1] shows that to the i n t h e approximation i s upperbounded by r/2 (1+x). For of y 0 < 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 ) l e s s a c c u r a t e f o r small P ( 0 , x ) was c v a l u e s of x. c a l c u l a t e d by d i r e c t Rather, integration is f o r small x, [ l ] ; i.e. OD P„(0 x) = where p(7iy) f 1 - c (3.2.1) of x, but approximation / e" 0 = [l-{l-p(7iy)) ] 1/2 ~ e 7 l Y / ' (3.2.1) dy X y f o r NCFSK. 2 can be used to c a l c u l a t e P ( 0 , x ) f o r any there are several advantages when x i s l a r g e (> (3.1.4) accurate for large in using First 10). i s very (3.1.4) can be e v a l u a t e d with j u s t a c a l c u l a t o r and a f u n c t i o n t a b l e whereas ( 3 . 2 . 1 ) computer with a well will values of x. almost proven numerical some, a l b e i t in computation time instead for l a r g e values of x. model which small of accuracy of ( 3 . 1 . 4 ) x has while retaining P„(0,x) = c (3.2.1) 1 - integration small, (3.2.1) savings is / b 0 used in the accuracy for simplicity and the f o r l a r g e v a l u e s of to be approximated by a f i n i t e APPENDIX A of [ 1 ] , of c o u l d improve In order to e v a l u a t e the i n t e g r a l it GAMMA These t r a d e - o f f s were c o n s i d e r e d a r r i v i n g at a values Secondly, c e r t a i n l y require s u b r o u t i n e . T h i r d l y , there are if (3.1.4) the of a l l , (3.1.4) a value c x. numerically, (3.2.1) i n t e g r a l . R e f e r r i n g to can be approximated as f o l l o w s : e" y [l-{l-p(7,y)} ] X dy 22 where 0 < J e~ y e" y [ 1 - { 1 - p ( 7 y ) } ] dy < e, x 1 b 00 Since J < therefore, e smallest b -b b e~ [ 1 - { 1 - p ( 7 i y ) ) 1 dy X [1-{1-P( ib)} ], b x 7 x [1-{1-p(7,b)} ] < e was used to evaluate which would meet the error constraint the of —Q e = 10 for a l l s i z e s of x. I t was found that b = 10. Thus, (3.2.1) was approximated as : P (0,x) = 1 - ; e 1 0 r c The MTS L i b r a r y 0 [ l - { 1 - p ( 7 i Y ) ) ] dy. y X UBC:INTEGRATION subroutine DCADRE used t o e v a l u a t e (3.2.2). A and the r e s u l t i n g (3.2.2) few sub-block s i z e s were improvements were compared. The was tried 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 °/10 7 (1) for x < 4 1 P (0,x) = 1 - J c (2) 0 2/ , 7 -v e M-{l-p(7,y>} ] dy y for x > 4 r = 0 (3.2.4) 23 r r d + x ) T(l+r) T( 1+x+r) (3.2.5) N/x T h i s model gave an average error whole range of 10 Hz t o 90 Hz. There 572 p o i n t s ) with double d i g i t was approximately N = 511 18 % (3.2.6) of 2.4 % over were 23 p o i n t s (out of % e r r o r s . The h i g h e s t % at £ D =10 Hz, 7 =30 0 P (0,N) = 1 of P (0,N) f for i n c l u d i n g these 1.9%. and bits. points f error dB T h i s model a l s o gave very good e s t i m a t e s f o r the 142 the these 142 with value points). points The was found 1 (i.e. average to be other estimated % error reduced to 24 We r e f e r t o t h i s model as Model II and summarize i t s performance as f o l l o w s : E m p i r i c a l Formula : (1) d = -0.12 + 2.43 e x = L/d where L = R/f. 0 , 0 8 7 0 if x < 4 10 P <0,x) = 1 - J c where (2) e 0 c = 7 l i0 y [l-p( y)) ] X 7 l dy 7 o / / 1 0 if x > 4 P<O,X> = 2 r r ( c ^ ) r ( ; r ) Td+x+r) where r = 2/y, P (0 N) = 1 - P ( 0 , x ) f + f N / x c Average % E r r o r = 2.4 % (over 572 p o i n t s ) Highest % E r r o r = (at 18 % 10 Hz, 30 dB, 511 bits) 4. E S T I M A T I O N OF T H E P R O B A B I L I T Y OF THE FADE RATE & THE FADE DISTRIBUTIONS DURATION 4.1 REVIEW Two important envelope are fade r a t e statistics the average of the Rayleigh fading fade d u r a t i o n 7^ and the average at s i g n a l l e v e l p. I t i s known [1,2,19] that these two parameters are given by : e - 1 = _. P /21F f p p2 T e (4.1.1) 1 n c D N where p = /21F f p e " p = V a r i o u s authors distribution attempt (4.1.2) p 2 D amplitude of the s i g n a l envelope RMS v a l u e of the amplitude [6,7] have function of the simulated fade the duration. has been made to model the p r o b a b i l i t y cumulative So f a r no distributions of the fade r a t e and the fade d u r a t i o n a t s i g n a l l e v e l p. In t h i s Chapter, empirical expressions f o r the probability d i s t r i b u t i o n s of the fade r a t e and the fade d u r a t i o n w i l l be considered. 25 26 4.2 THE SIMULATION The s i m u l a t i o n was c a r r i e d out with the R a y l e i g h f a d i n g simulator described to 9 0 Hz. T h i r t y were used. i n Chapter 1 over the f sequences, each Each of Q range from 5 Hz a certain chosen level below the p = SQRT(10~ 1 5 ^ based on a c c e p t a b l e mobile service requirements boundary were consecutive average 1 0 ). SNR; This radio recorded i . e . the threshold was telecommunications [ 7 ] , Fades which o c c u r r e d at a window considered windows. as The separate means fades in the two and v a r i a n c e s of the fade r a t e and the fade d u r a t i o n were a l s o 4.3 were t h r e s h o l d l e v e l . In t h i s study, the t h r e s h o l d l e v e l was s e t at 1 5 dB threshold seconds was segmented i n t o 2 0 0 1-second sequence windows. The fade r a t e and the fade d u r a t i o n at 200 duration calculated. 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 f u n c t i o n (pmf) of the fade r a t e n ( f a d e s / s e c ) i s shown i n F i g . 4 . 1 . T h i s suggested that a Normal d i s t r i b u t i o n f u n c t i o n would f i t the distribution well; i . e . P The mean ( 4 . 1 . 2 ) . n is ( „ , = _ ± _ y/2% o equivalent e -(n-M)V2c^ to n - I 5 5 B { 4 > 3 < 1 ) a s determined from 27 0.16 f D R = 2 0 Hz = 4 0 9 6 bps 0.12- v 0.08- 0.04- 0.00 X 2 I 3 4 5 6 \ 1 7 8 Fade Rate Fig. 4.1 : P r o b a b i l i t y Mass ! 9 T 1 1 T ' f T—T¥ % % 10 11 12 13 14 15 16 17 18 19 20 n (fades/sec) Function of Fade Rate 28 The v a r i a t i o n s of n plotted in Figures 4.2 i n c r e a s e s l i n e a r l y with f a and and D < in 4.3. (4.3.1) It with f D are can be seen that n The curve f i t t i n g r o u t i n e NL2SN0 to f i n d the c o e f f i c i e n t s of the f u n c t i o n s AI (f ^) and was used a ( f ) . The m i n i m i z a t i o n of the E r r o r Sum ERRS : Q ERRS = Z (estimated data - simulated data) leads t o the f o l l o w i n g f u n c t i o n s : u = 0.435 f (4.3.2) D a = j/0.323 f . (4.3.3) D From (4.1.2), N_ = 0.432 f ; t h e r e f o r e , 1 5 d B d e r i v e d from the simulated data i s c l o s e to the value o b t a i n e d using (4.1.2). The average A V E R R A theoretical % estimation error : V E R R = — Z 18 over (4.3.2) D restimated data - simulated data s i m u l a t e d data 18 p o i n t s of f D from 5 Hz to 90 Hz was x 1 Q O % } 1.7 % f o r (4.3.2), and was 1.3 % f o r (4.3.3). F i g . 4.4 shows the pmf of F i g . 4.1 with Normal function c a l c u l a t e d from corresponding obtained (4.3.2) and cumulative from (4.3.1) (4.3.3). mass the e m p i r i c a l with F i g . 4.5 u and a shows the f u n c t i o n (cmf) of F i g . 4.4. Fading F r e q u e n c y f D (Hz) Fading F r e q u e n c y f D (Hz) 31 0.16-1 0 1 2 3 4 5 6 7 8 Fade Rate Fig. 4.4 9 10 11 12 13 14 15 16 17 18 19 20 n (fades/sec) : Probability Mass Function of Fade & its Empirical Normal Function Rate n 32 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fade Rate Fig. 4.5 : Cumulative & the CMF of its n (fades/sec) Mass Function of Empirical Normal Fade Rate Function n 33 Both figures show that the simulated r e p r e s e n t e d by the e m p i r i c a l Normal data are well function. 4.4 PROBABILITY DENSITY FUNCTION OF FADE DURATION A typical probability i (bits) mass f u n c t i o n of the fade length i s shown i n 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 f u n c t i o n s 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 bo (4.4.1) for i = 0, 1, 2, where b = mean 0 GAMMA d i s t r i b u t i o n : P(i) = -=$-^ T(r) (Xi) r _ 1 for i = 0, 1, 2, where r = 1, 2, Arbitrary e" X i (4.4.2) distribution : P(i) = for C l i C 2 e " i = 0, C 3 i C 4 1,2, (4.4.3) 34 tO to CM in ro CL N I o JQ O CO CD O CN tO o> 00 CM CM fx CN CO CN X- II in X- CN X- CN to X- CN CN CN X- CN O CN X- o> X- 00 xX- * XX- — If) C to CD CN xXXXXXXXXXXi o CO o o to o m o "T~ o o o if) O d u o i p u n j SSDJ^ XjniqDqojd 4. 6 V _1 X- Fi g. — CO *— X- m q d y \ V) Probability Mass Function of Fade Length "D ,D 35 After the a d e t a i l e d examination, empirical f i t with the (4.4.3) was s e l e c t e d f o r 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 distribution the f a d i n g frequency s h i f t s towards s h o r t e r fade l e n g t h s as increases. The average fade length decreases from 54 b i t s f o r 5 Hz t o 3 b i t s f o r 90 Hz. In order transmission to generalize the fade length for any r a t e R, the fade l e n g t h was transformed i n t o a fade d u r a t i o n r ( m i l l i s e c o n d s ) , where T = (fade l e n g t h ( b i t s ) / b i t r a t e R) * 1000. A l s o , the pmf was transformed i n t o a 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 (pdf) so that the t o t a l mass was r e p r e s e n t e d by the area of the p d f . Since the fading frequency p r o p o r t i o n a l t o the v e h i c l e speed (= v/\) f^ v, doubling f D directly by d o u b l i n g the v e h i c l e speed w i l l reduce the Therefore, duration i s inversely proportional to the fade fade is duration the f a d i n g frequency f . Based on t h i s argument, D the n o r m a l i z e d fade d u r a t i o n 4> = r * f any pdf D the we half. define (ms*Hz). The pdf f o r f a d i n g frequency c o u l d then be r e p r e s e n t e d by of by a single normalized fade d u r a t i o n 0. F i g . 4.7 shows the pdf's of the normalized fade d u r a t i o n f o r 10 Hz, 45 Hz and 90 Hz. These 3 curves have a t o t a l area though ideally of 0.95 each. Even they are expected t o be i d e n t i c a l , F i g . 4.7 **1 00 A. C IM o ind # «•» >i i f -. © N CJ«» Q ft. O- ft o. -• - A X 10 Hz • 90 Hz 45 Hz >» c Q> O c <* • — t a _Q O JQ a **i Pro J3 *> c a o 1 i 90 Normalized Fade Duration 100 cp 1 110 1 120 (ms * Hz) 1 130 1 140 1 150 1 160 CO 37 shows some differences especially near the peak. These d i f f e r e n c e s can be e x p l a i n e d as f o l l o w s : All 3 simulations seconds/trial. bits f for f = 45 Hz D were run fade lengths The = 10 Hz, D between and between 1 and "fluctuations" statistical in for the f 90 variations. 1 and 14 7 bits for f The pdf UBC D tend to f u r t h e r 200 63 bits for = 90 Hz. The are due graphic *TELLAGRAF used to i n t e r p o l a t e the sample also of are between 1 and = 10 Hz D trials to routine points would "smooth out" the pdf's f o r high f Q values. The cumulative d u r a t i o n can be used Fig. 4.8 shows the empirical formula was form (4.4.3) of density function (cdf) of the fade i n d e s i g n i n g an e r r o r c o r r e c t i n g code. c d f ' s of the 3 curves i n F i g . 4.7. The obtained by f i t t i n g the a to the simulated data f o r f curve D of = 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 " where can closely. be 0 * 5 6 1 e '°' 8 < t > ' (4.4.4) <t> = T * f p . The cdf c a l c u l a t e d from It 8 seen that (4.4.4) i s a l s o shown it fits the in F i g . 4.8. s i m u l a t e d data q u i t e 38 uoipunj XjisusQ 3Aj4D|nujno Fig. & the 4.8 : Cumulative Density Function of Normal ized Fade Durat i on 0 CDF of its Empirical Function (4.4.4) 39 We (1) summarize the two e m p i r i c a l models as f o l l o w s : E m p i r i c a l model f o r e s t i m a t i o n of f u n c t i o n of fade n = -< -«> /2cr* n e f 0.435 a = v^0.323 (2) 2 D f D Average % e s t i m a t i o n error for u = 1.7 % Average % e s t i m a t i o n error for o = 1.3 %. Empirical density model and normalized pdf(*) for estimation cumulative fade d u r a t i o n = 5.18X10" density <t> : 8 e - O - 8 0 ° ' a cdf(tf>) = j where 0 mass rate n : pm£(n) = where probability pdf(tf>) d<*> a i s a real number. 6 3 of probability functions of 5. ESTIMATION OF PROBABILITY OF m-BIT ERRORS IN AN N-BIT BLOCK 5.1 REVIEW The block probability Q(m,N) of m-bit i n a mobile r a d i o f a d i n g channel by s i m u l a t i o n s or f i e l d trials. e r r o r s i n an N - b i t i s u s u a l l y determined Unfortunately, tend t o be q u i t e c o s t l y . I t i s shown i n [ 1 , 2 2 ] slow Rayleigh f a d i n g channels, these methods that f o r very i n which the r e c e i v e d s t r e n g t h can be assumed constant over the d u r a t i o n signal of the block, Q(m,N) can be approximated as : Q(m,N) = Q(0,N) (5.1.1) -21 m! where in r = 2 / 7 and ( r ) t h i s approximation, c(m,N) m = r ( r + 1 ) ( r + 2 ) . . . . ( r + m - 1 ) . The e r r o r e(m,N), i s upperbounded by : ) N-m/ T \ m/ 2 (1+N-m) W m for 0 < r < 1. When t h i s c o n d i t i o n of very (5.1.1) i s no longer a c c u r a t e Q(m,N). 40 slow fading is relaxed, i n e s t i m a t i n g the p r o b a b i l i t y 41 5.2 AN APPROACH TO ESTIMATING Q(m,N) The v a l u e s of Q(m,N) o b t a i n e d from Chapter 2 average the were analysed t o see how Q(m,N) v a r i e s with the SNR 7 , the f a d i n g frequency 0 number seconds the s i m u l a t i o n i n f , the b l o c k s i z e N and D of e r r o r s m. T h i r t y t r i a l s each of d u r a t i o n 200 were generated for each ( f , N, 7 ) D 0 at a t r a n s m i s s i o n r a t e of 4096 bps. Only those v a l u e s of m with a minimum of 50 b l o c k s i n the s i m u l a t i o n were used. A p l o t loQio Q(m,N)/Q(0,N) of SNR y f o r as a f u n c t i o n of the average 0 d i f f e r e n t v a l u e s of f , N and m showed that the r e l a t i o n s h i p D was f D l i n e a r . F i g . 5.1 shows a t y p i c a l p l o t f o r approximately = 40 Hz and N = 63 b i t s . A linear equation of the f o l l o w i n g form was c o n s i d e r e d : Log where F , and F The 2 1 0 IJ^ll = F, N = 63, v a l u e s of F , and F of 7o, (5.2.1) D 127, Q error 2 = 20, 40, 60 and 80 Hz and 255, 511, 1023 and 2047 b i t s were used with the MTS curve f i t t i n g the sum 2 a r e f u n c t i o n s of f , N and m. s i m u l a t e d data f o r f blocksizes + F r o u t i n e NL2SNO t o d e r i v e the . O p t i m i z a t i o n was o b t a i n e d by m i n i m i z i n g between the estimated ratio and the simulated r a t i o Q(m,N)/Q(0,N). The range of m c o n s i d e r e d was from 1 t o 19 b i t s . T y p i c a l v a r i a t i o n s of for f D = 40 Hz and N = 63, 255, 1023 and F 2 with m b i t s a r e shown i n F i g u r e s 5.2 and 5.3 r e s p e c t i v e l y . One p o s s i b l e form f o r the 42 10-3 f = 40 Hz N = 63 bits D O 0.01- o 0.001- o.oooi 15 20 25 A v e r a g e SNR y 0 Fig. 5.1 (dB) : Variation of Q(m,N) / with Average SNR y = 40 Hz, N = 63 bits 0 f n Q(0,N) 44 45. functions F , ( f , N, m) and F ( f , N, m) D 2 F, = F The 2 a = curve f i t t i n g 2 3 a 4 N z D b, f s e t s of (a , , a , a , a ) (5.2.3) f i D b respectively. m . (5.2.3) ba was a p p l i e d to f i n d the best (b,,b ,b ,b„) 2 : (5.2.2) a b a routine and m « a 3 N 2 D i s as f o l l o w s for 3 Optimization was (5.2.2) performed and by m i n i m i z i n g the sum of e r r o r between the estimated F, and F , 2 and their corresponding simulated values. The following formulas were obtained : 5.5X10" F, = F (5.2.4) for f D = -4.5X10" and 3 f D f D (5.2.5) = 20, 40, 60 and 7o = 15, into 2 3 0 ' 7 3 N ' 0 ' 5 2 N ' 0 0 4 7 m ' 54 (5.2.4) m'. (5.2.5) 0 1 5 0 6 were used to c a l c u l a t e F, and F 80 Hz, N = 127 and 511 b i t s 20 and 25 dB. These F / s and F ' s were 2 (5.2.1) to estimate the probability of 2 and substituted m-bit errors Q(m,N) . The cmf of Q(m,N) i s important error-correcting code. From in the of an (5.2.1), the cmf of up t o M-bit e r r o r s cmf(M,N) can be c a l c u l a t e d as f o l l o w s cmf(M,N) = design M Z m=0 Q(m,N), : (5.2.6) 46 where Q(m,N) can be o b t a i n e d from (5.2.1) as : Q(m,N) = Q(0,N) 10 ( F , + F F, and F 2 are c a l c u l a t e d 2 ) 7 o (5.2.7) from (5.2.4) and (5.2.5). F i g . 5.4 shows a t y p i c a l p l o t of the estimated c m f s with f 1 N = 511 b i t s and 7 = 15, 0 simulated and the e m p i r i c a l exhibit similar 20 and cmf's last respectively, follows and by a s c a l i n g simulated the s c a l i n g cmf v a l u e s by cmfp and cmf factor k can be k = as L Z Q(m,N) = cmf m=1 cmf r e s u l t i n g scaled simulated (5.2.8) k t o the cmf's i n F i g . 5.4, the cmf's a r e shown i n F i g . 5.5. The cmf's f o r other f , N and 7 D range of c o n s i d e r a t i o n the cmf v a l u e . Then, - Q(0,N) Applying t h i s s c a l i n g factor show calculated g : where L i s the number of the l a s t p r e d i c t e d the factor. t o a l l Q(m,N) for m * 0. Denoting the Q(0,N) + and do estimate of the simulated cmf was m u l t i p l i e d i s applied predicted they the shapes. the e m p i r i c a l This factor = 45 Hz, Although are d i f f e r e n t , In an attempt t o get a b e t t e r cmf, 25 dB. D scaled 0 predicted combinations i n were checked. F i g u r e s 5.6 and 5.7 cmf' s for (20 Hz, 127 b i t s ) and 47 u o j p u n j SSD^ aAijDinujnQ Fig. 5.4 : Cumulative Mass Function of Q(m,N) & the unsealed CMF of the Empirical Function (5.2.7) f = 45 Hz, N = 511 bits n 48 uoipunj S S D ^ aAip|nujnQ Fig. 5.5 & the scaled : Cumulative Mass Function of CMF of the Empirical Function f = 45 Hz, N = 511 bits D Q(m,N) (5.2.7) 49 (70 Hz, 255 b i t s ) respectively. T h i s approach f o r e s t i m a t i n g the p r o b a b i l i t y Q(m,N) and its cmf i s quite effective even though the procedure i s somewhat l e n g t h y . The c o e f f i c i e n t s i n the e m p i r i c a l (5.2.2) are and (5.2.3) may change formulas i f the range of f , N and 7 D extended. We summarize the r e s u l t s as f o l l o w s : E s t i m a t i o n of Q(m,N) : Q(m,N) = k Q ( 0 , N ) where m= F, = 2 ( F i + F 2 7 o ) 1, 2, 3, k = scaling F 1 0 factor calculated 5.5X10' 3 = -4.5X10' 3 f D f D 0 ' 7 3 N ' 0 4 7 0 ' 5 2 N ' 1 5 0 from m ' 0 54 m'. 0 6 (5.2.8) 0 50 uoipunj SSD^ & aAijDinujnQ Fig. 5.6 : Cumulative Mass Function of Q(mJ^) the scaled CMF of the Empirical Function (5.2.7) f = 20 Hz, N = 127 bits n 51 O 0 > 0 0 0 ^ r 0 l O 0 l T 0 ) ' 0 ^ ' 0 K ) 0 C M » 7 ° u o i j o u n j 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) f = 70 Hz, N = 255 bits n 6 . CONCLUSION 6.1 CONCLUDING REMARKS The design For block of an e r r o r r a t e i s an important parameter i n the efficient digital radio transmission system. example, the e f f i c i e n c y of an ARQ scheme which uses only error detection i s h i g h l y dependent p r o b a b i l i t y . In t h i s t h e s i s , a simple e m p i r i c a l model f o r e s t i m a t i n g on a R a y l e i g h noise was d e s c r i b e d . A previously retransmission and f a i r l y the block fading channel with shown t o be v a l i d on the error rate a d d i t i v e white accurate P (0,N) f Gaussian s t u d i e d model [ 1 , 2 2 ] was of v e h i c l e speeds. The f o r a wide range r e s u l t i n g model parameters were d e r i v e d . The fade estimation rate and of 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 the fade E m p i r i c a l models which duration were y i e l d reasonably also examined. good r e s u l t s proposed. These q u a n t i t i e s a r e u s e f u l i n c h a r a c t e r i z i n g were the nature of the e r r o r s i n the mobile r a d i o c h a n n e l . The estimation of the p r o b a b i l i t y Q(m,N) e r r o r s i n an N-bit block was a l s o c o n s i d e r e d . estimating i t s cmf fading Q(m,N) and frequency f , e r r o r s m gives D b l o c k s i z e N, q u i t e encouraging as a 0 m-bit An approach t o function SNR y of and of the number r e s u l t s . Further studies would be r e q u i r e d t o determine how good t h i s approach i s . 52 of 53 6.2 FUTURE WORK The simulation in this thesis was carried out e x c l u s i v e l y f o r a non-coherent FSK demodulation scheme. T h i s i s mainly due t o the popular use of t h i s scheme. A similar approach can be used i n f u r t h e r s t u d i e s f o r other modulation schemes. The attempt i n Chapter 5 t o Q(m,N) i s by empirical various no means final. e s t i m a t e the This form f o r the r e l a t i o n s h i p parameters. Further work study probability suggests an between Q(m,N) and the could involve examination of a broader range of the independent the parameters and the improvement of the accuracy i n p r e d i c t i n g 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, Techniques. New York : McGraw-Hill. 1965. [5] J . M. Wozencraft and I. M. Jacobs, Principles New York : John Wiley, 1965. Communication Systems and of Communication Engineering. [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. Channel," IEEE Feb. 1982. "Transmission Trans. of Digital Veh. Technol., Data Over Vol. VT-31, a Rayleigh No." 1, Fading pp. 1-6, [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," Technical Journal, pp. 957-1000, Jul.-Aug. 1968. Bell System [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, Calculations." New York : John Wiley, 1972. [19] W. C. Jakes, Jr., Microwave Mobile New York : John Wiley, 1974. "Standard Statistical Communications. [20] S. Lin, An Introduction to Error-Correcting New Jersey : Prentice-Hall, 1970. Codes. [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 o n t h e E E D e p t . V A X / V M S FORTRAN77 C o m p i l e r . To c o m p i l e t h e m o n t h e S t a n d a r d FORTRAN77 C o m p i l e r , a l l DO L o o p s 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 Indexed DO L o o p s before compilation. Main Program c c c c c ** ** ** ** ** * c * c ** r ** c ** c c c c c c c ** ** ** ** ** SIM.FOR A Simulation P r o g r a m t o g e n e r a t e a R A Y L E I G H FADING SEQUENCE of N (bps) for MXSEC (sec) and c a l c u l a t e BLOCK ERROR RATE Ps(M,NBIT) where 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 NTRYS = # of T r i a l s DBMIN = M i n . SNR i n dB DBINC = I n c r e m e n t o f SNR i n dB NLEVS = # of L e v e l s of Sequence Length NSEC = Sequence Length i n Seconds NSIZE = # of B l o c k s i z e s NBIT = Blocksize in bits/block FD = Fading F r e q u e n c y i n Hz 1 1 , 12 = 2 s t a r t i n g S e e d s f o r t h e G a u s s i a n Random # G e n e r a t o r SUBROUTINES : SEQU.FOR, FADE.FOR, F S S T . F O R , RANDN.FOR REAL* 8 REAL INTEGER INTEGER*2 COMMON COMMON DATA DATA S U M O , 3 0 , 6 ) , SUMSQ(7, 30,6), CDFSQ(7 ,30,6) Y(819200), B T E R R ( 7 , 6 ) , F ( 1 0 0 ) , ANORM(7) NSEC(7), N B I T ( 6 ) , ICHAN(6) II, 12 DBMIN, D B I N C , N , MEM, N L E V S , NOUT, NTRYS /SEEDS/ II, 12 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 / ICHAN / l l , 1 2 , 1 3 , 14, 15, 16/ MEM = 819200 MAXBIT = 2 0 4 8 N = 4096 NOUT = 30 56 OPEN READ READ READ READ (10, (10, (10, (10, (10, FILE='INPUT', STATUS='OLD', ERR=990) 80) NTRYS 90) DBMIN 90) DBINC 80) NLEVS IF (NLEVS .GT. 7) READ READ (10, 80) (NSEC(I), (10, 80) NSIZE IF (NSIZE .GT. 6) READ READ READ READ CLOSE CALL GOOF(l) 1 = 1 , NLEVS) CALL GOOF(2) (10, 80) ( N B I T ( I ) , (10, 90) FD (10, 80) I I (10, 80) 12 (10) 1 = 1 , NSIZE) DO I = 1, NSIZE CALL HEADER(NSEC, N B I T ( 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 DO I = 1, NSIZE IF (NBIT(I) .GE. MAXBIT) CALL GOOF(6) ENDDO GOOF(5) 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 90 990 FORMAT FORMAT WRITE STOP END (16) (F6.3) (6,*) '*** Error i n opening the INPUT F i l e ***' Subprogram SEQU.FOR SUBROUTINE SEQU ( Y , F , COMMON DBMIN, D B I N C , N , REAL Y(MEM), F(100), F D , NSEC) MEM, N L E V S , NOUT, X(4096) NTRYS I F (N . G T . 4 0 9 6 ) CALL GOOF(7) DO INDEX = 1 , NSEC CALL FADE ( X , F , F D , N) MARK = N * ( I N D E X - 1) DO 1 = 1 , N Y(MARK + I) = X ( I ) ENDDO ENDDO RETURN END SUBROUTINE NORML ( Y , ANORM, N B I T , N S E C , MXSEC, ICHAN) COMMON DBMIN, D B I N C , N , MEM, N L E V S , NOUT, NTRYS REAL Y ( M E M ) , ANORM(NLEVS) INTEGER NSEC(NLEVS) REAL*8 DBLE, D L , DFLOAT, S l , S2 NBL = N / NBIT LEN = N B I T * 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 + D B L E ( Y ( I + M A R K ) ) ENDDO S l = S l + S2 / DL DO J = 1 , NLEVS IF (INDEX . E Q . N S E C ( J ) ) ANORM(J) = SNGL(S1/DFL0AT(INDEX)) 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 DO 1=1, NBIT1 ERR(I) = 0 ENDDO DO 1 = 1 , DO 0 NSECS 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 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, ENDDO ENDDO RETURN CDF, VAR2 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, WRITE (ICHAN, 30) NBIT, N 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 ) FORMAT ('OPARAMETERS (PER TRIAL) :'//T10, 'DB', T19, 'SECONDS', T31, 'NUMBER OF', T46, "NUMBER OF'/ T33, 'BLOCKS', T47, 'SAMPLES'/) FORMAT (T8, F5.1, T20, 13, T33, 15, T46, F9.1) FORMAT ('OTHE DOPPLER FREQUENCY IS', 2X, F6.3, * 2X, 'HZ.'/) * * 30 40 50 END C =============================================== SUBROUTINE TITLE (DB, ACTL, 10 20 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 FORMAT (' ', 2 6 ( ' ')) 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 * * * * * * * * * * IF (N .EQ. 1) WRITE IF (N .EQ. 2) WRITE IF (N .EQ. 3) WRITE IF (N .EQ. 4) WRITE IF (N .EQ. 5) WRITE IF (N .EQ. 6) WRITE IF (N .EQ. 7) WRITE STOP FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT END 10 20 30 40 50 60 70 r C C C (' C (' (' ERROR (6, 10) (6, 20) (6, 30) (6, 40) (6, 50) (6, 60) (6, 70) * * * * * * * * * * 7) TOO MANY SNR''S.') TOO MANY BLOCK SIZES.') TOO MANY SAMPLES.') DOPPLER FREQUENCY TOO HIGH.') TOO FEW OR TOO MANY TRIALS.*) TOO MANY BITS PER BLOCK.') BIT RATE IS TOO HIGH.') rr — FUNCTION BITERR (ERR, MAXERR, INTEGER ERR(2048) L) 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) ZERO = 0. RETURN END RETURN Subprogram FADE.FOR C **============================================ C ** FADE.FOR g e n e r a t e s a R a y l e i g h F a d i n g Sequence X ( I ) based on C ** a Model proposed i n IEEE T r a n s . COM.21 v o l . 11 C ** p.1325 Nov. 1973 SUBROUTINE FADE (X, F, FD, N) REAL X(N), F ( 1 0 0 ) , 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) = F ( I ) * RANDN(O) F(I) * RANDN(O) 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 ** S u b r o u t i n e FSST i s i n t h e f i l e CALL FSST (Y, N) CALL FSST (Z, N) I = 1, N X(I) = Y(I) * Y(I) ENDDO SYSSUSERDISK:[IEEESIG]FAST.FOR DO RETURN END + Z(I) * Z(I) 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 C C C C C C C C C C RANDN.FOR ** G a u s s i a n random number g e n e r a t o r . ** T h i s program was w r i t t e n by R.P. B r e n t . I t appears i n ** Communications o f t h e ACM, v o l . 17, no. 12, pp. 704-706, ** December 1974. ** The program uses t h e f u n c t i o n RAN, which i s a uniform ** random number g e n e r a t o r implemented on t h e VAX. The ** seeds I I and 12 f o r t h e g e n e r a t o r must be p a s s e d by ** a COMMON statement. **==================^^ 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 10 20 30 40 50 A = 0.0 1 = 0 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 W = D(I+1) * U V = W * (0.5 * W - A) 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 U = RAN(II, 12) U = U + U IF (U .LT. 1.0) GO TO 50 U = U - 1.0 RANDN = W - A RETURN RANDN = A - W RETURN END 69 Main Program c ** c c *r* c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c ** c c c c ** ** ** FDSTAT.FOR A program t o c a l c u l a t e ( i n Rayleigh Fading) (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 o f Fade # (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 o f Fade Length (3) C o n f i d e n t L i m i t s o f P r o b a b i l i t i e s (4) C o n d i t i o n a l P r o b a b i l i t y D i s t r i b u t i o n o f Fade Length (Assuming N fades i n a Window) NTRY = # of T r i a l s (must be MULTIPLE o f 10) = B i t Rate (must be a power of 2) N = F a d i n g Sequence Length i n seconds NSEC = # o f Windowsizes NSIZE = Windowsize i n b i t s NBIT MAXCNT == H i g h e s t Fade # MAXLEN =• Longest Fade Length KFAD = H i g h e s t Fade # t o be p r i n t e d KLEN = Longest Fade Length t o be p r i n t e d DB = Threshold i n dB FD = F a d i n g Frequency = S t a r t i n g Seeds f o r RAN 1 1 , 12 = t - v a l u e i n Student-t distribution TVAL IDF = Degree of Freedom i n Student-t d i s t r i b u t i o n ICL = % Confidence Limit SUBROUTINES : FADE.FOR, FSST.FOR, RANDN.FOR n * * * * * * 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/ I I , 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 READ READ READ READ READ READ CLOSE ( i o , :FILE='FDIN*, STATUS' (10,*) NTRY, N, NSEC (10,*) NSIZE, FD K = (10,*) (NBIT(K), (10,*) KFAD, KLEN, KLN2 (10,*) 11, 12, DB (10,*) TVAL, IDF, ICL (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) = 11 I1C = 12 I2C 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) t o 0 (dB) MEAN F a d i n g DO 1 = 1 , MEM Y ( I ) = 10.*ALOG10(Y(I)/S) YMAX = AMAX1(Y(I), YMAX) YMIN = AMIN1(Y(I), YMIN) ENDDO Envelop C a l c u l a t e Prob. D i s t r . o f Fade # & Fade l e n g t h ** 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) VLEN(K) ALENK = VLENK = ENDDO MAXLEN MAXCNT NLEN NFADE ENDDO 601 C ** C ** = = = = = ALEN(K) + ALENK = VLEN(K) + VLENK 0. 0. MAXO(MAXLEN, NLEN) MAXCKMAXCNT, NFADE) 0 0 WRITE (6,601) MAXLEN, MAXCNT-1, MAXTWO FORMAT ('OMAXLEN =', 16, 15X, 'MAXFD =*, 16/' MAXTWO =', 16/) DO K = 1, NSIZE C a l c u l a t e MEANS & VARIANCES o f Fade # AFD(K) = AFD(K) / FLN10 VFD(K) = VFD(K) / FLN10 ALEN(K) = ALEN(K) / FLN10 VLEN(K) = VLEN(K) / FLN10 & Fade Length ** C a l c u l a t e MEANS & VARIANCES o f P r o b a b i l i t y Distributions 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) + PCLEN(LY,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 - AVELEN(K,I)=AVELEN(K,I) / FLN10 1.)) ENDDO CDF(K,1) DO = AVELEN(K,1) 1 = 2 , MAXLEN CDF(K,I) = CDF(K,I-1) + AVELEN(K,I) ENDDO ENDDO KFAD = MIN0(MAXCNT, KLEN = MINO(MAXLEN, KLN2 = MINO(MAXTWO, OPEN OPEN (1, (7, WRITE ( 1 , F I L E = ' F D S T A T ' , STATUS='NEW') F I L E = ' C O N F \ STATUS='NEW' ) 1 0 0 ) WRITE (1,110) WRITE WRITE (1,111) (1,120) DO I = 1, WRITE ENDDO KFAD) KLEN) KLN2) NTRY, N, NSEC, NSIZE, F D , YMAX, YMIN, I 1 C , I 2 C , M A X C N T - 1 , MAXLEN (NBIT(K), K = 1, 5 ) , (AFD(K), K = 1, 5 ) , (VFD(K), K = 1, 5) (1-1, (AVEFD(K,I), K=l,5), (NBIT(K), K = 1, 5 ) , (ALEN(K), K = 1, 5), (VLEN(K), K = 1 , 5) 1=1,KFAD) KLEN (1,121) I, (AVELEN(K,I), CDF(K,I), K=l,5) DE 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, ( A V E F D ( K , I ) , VARFD(K,I), CONFD(K,I), K=l,3) ENDDO WRITE (7,710) ( N B I T ( K ) , K=3,5) DO I = 1, KFAD WRITE (7,711) 1-1, (AVEFD(K,I), VARFD(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), V A R L E N ( 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), K=3,5) ENDDO CLOSE CLOSE (7) (1) CONFD(K,I), CONLEN(K,I), CONLEN(K,I), 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', WRITE (1,770) NBIT(K), MAXTWO, DO 1 = 1, WRITE * ENDDO CLOSE ENDDO (1) STATUS='NEW') (IFD, IFD=1,4) KLN2 (1,771) I, (PCLN2(IFD,K,I), CDF2(IFD,K,I), IFD=1,4) 76 100 FORMAT * * * * * * 110 FORMAT * * 111 120 FORMAT FORMAT * * * 121 FORMAT ('OPROBABILITY DISTRIBUTIONS OF FADE # & FADE LENGTH' ////' Parameters :'//' # o f T r i a l s ' , T21,'=',I8/ ' B i t Rate', T21,'=',I8,' b p s ' / ' Sequence Length', T21,'=',I8,' s e c ' / ' # o f Windowsize', T21,'=*,I8/ ' F a d i n g Frequency', T21,*=',F8.2,* Hz*// ' Envelop (Max)', T21,'=',F8.2,' dB'/ ' Envelop ( M i n ) ' , T21,'=',F8.2,' dB'// 'Threshold', T21,'=*,F8.2,' dB'// ' Starting Seeds', T21,'=',218// ' Max # of Fade', T21,'=',I8/ ' Max Fade Length', T21,'=',I8,' b i t s ' ) ( ' l P r o b a b i l i t y D i s t r i b u t i o n o f Fade #*//// ' # OF FADES',T31,'PROBABILITY'/ T10,5(10X,I5)/' \ 1 0 ( ' - ' ) , T 1 6 , 7 5 ( * - ' ) / ' AVERAGE',5F15.2/' VARIANCE',5F15.2/) (16, 4X, 5F15.7) ( ' l P r o b a b i l i t y D i s t r i b u t i o n o f Fade L e n g t h ' / ' (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 P a r e n t h e s e s ) ' , ////' FADE LENGTH',T31,'PROBABILITY'/ T4,' b i t s ' , T 3 , 5 ( 1 7 X , 1 5 ) / ' ',12('-'),T16,105('-')/ ' AVE LENGTH',F15.3,4F22.3/ VARIANCE ',F15.3,4F22.3/) (16, 4X, 5(F13.6,' (',F6.4,')')) 77 700 FORMAT * * * * * f # * * * 710 FORMAT * * * * 711 720 FORMAT FORMAT * * * 770 FORMAT * * * * * * * 771 ('OCONFIDENCE LIMITS OF PROBABILITIES'//// * Parameters :'//' # o f T r i a l s ' , T23 '=',I8/ ' B i t Rate', T23 *=',I8,' b p s ' / ' Sequence Length', T23,'=',I8,' s e c ' / ' # o f Windowsize', T23,'=',I8/ ' Fading Frequency' T23,'=',F8.2,' Hz'// ' Threshold', T23,'=',F8.2,' dB'// ' t-value', T23,'=',F8.4/ ' Degree o f Freedom', T23,'=',I8/ ' Confidence L i m i t s ' , T23,'=',I8,' %') ('IConfidence L i m i t s o f 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('-')/) (17, 3(5X, 2F12.7, F8.3)) ('IConfidence L i m i t s 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 ( A V E + - % ) \ 10X,'WINDOWSIZE =', 318,' ( b i t s ) ' / ' \12('-'),T16,105('-')/) (' C o n d i t i o n a l 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 Fade Length', ' (Assuming N fades i n a Window)'/ ' (CDF i n P a r e n t h e s e s ) ' / / / ' Windowsize', T18,'=',I6,' b i t s ' / ' Max Fade Length', T18, =',I6,' b i t s ' / / ' FADE LENGTH', T 3 1 , ' C o n d i t i o n a l P r o b a b i l t y ' , ' (N fades p e r window)'/T4,' b i t s ' , T25,I1, 3(26X,I1)/ ' ',12('-'),T16, 105C-')/) (16,5X, 4(1PE18.7, * (',0PF6.4,')')) , FORMAT STOP END
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Estimation of error rates and fade distributions...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Estimation of error rates and fade distributions on a Rayleigh fading channel with additive white… Ng, Jimmy Hon-yuen 1986
pdf
Page Metadata
Item Metadata
Title | Estimation of error rates and fade distributions on a Rayleigh fading channel with additive white Gaussian noise |
Creator |
Ng, Jimmy Hon-yuen |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | 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 70 = 5 to 35 (dB) and fading frequencies f[sub D] = 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 error-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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096917 |
URI | http://hdl.handle.net/2429/26318 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1986_A7 N53.pdf [ 3.31MB ]
- Metadata
- JSON: 831-1.0096917.json
- JSON-LD: 831-1.0096917-ld.json
- RDF/XML (Pretty): 831-1.0096917-rdf.xml
- RDF/JSON: 831-1.0096917-rdf.json
- Turtle: 831-1.0096917-turtle.txt
- N-Triples: 831-1.0096917-rdf-ntriples.txt
- Original Record: 831-1.0096917-source.json
- Full Text
- 831-1.0096917-fulltext.txt
- Citation
- 831-1.0096917.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0096917/manifest