UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

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

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  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

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>
                        
                    
IIIF logo 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

Comment

Related Items