NUSC Technical Report 8887 i 21 May 1991 . IJ AD-A237 388 Spectra and Covariances for "Classical" Nonlinear Signal Processing Problems Involving Class A Non-Gaussian Noise Albert H. Nuttall Surface ASW Directorate David Middleton Consultant/Contractor Naval Underwater Systems Center Newport, Rhode Island / New London, Connecticut Approved for public release; distribution Is unlimited. I I (1111 llii l i; llI I 11(1 Iill l! ~Ii 11l
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NUSC Technical Report 8887 i21 May 1991
. IJ
AD-A237 388
Spectra and Covariances for"Classical" Nonlinear SignalProcessing Problems InvolvingClass A Non-Gaussian Noise
Albert H. NuttallSurface ASW Directorate
David MiddletonConsultant/Contractor
Naval Underwater Systems CenterNewport, Rhode Island / New London, Connecticut
Approved for public release; distribution Is unlimited.
I I (1111 llii l i; llI I 11(1 Iill l! ~Ii 11l
PREFACE
This research was conducted under NUSC Project No. A70272,
Subproject No. RROOOO-NO1, Selected Statistical Problems in
Acoustic Signal Processing, Principal Investigator Dr. Albert H.
Nuttall (Code 304). This technical repozt was prepared with
funds provided by the NUSC In-House Independent Research and
Independent Exploratory Development Program, sponsored by the
Office of the Chief of Naval Research. Also, this research is
based in part on work by Dr. David Middleton, supported
originally by Code 10 under ONR Contract N00014-84-C-0417 with
Code 1111.
The techiiical reviewer for this report was Roy L. Deavenport
(Code 3112).
REVIEWED AND APPROVED: 21 MAY 1991
DAVID DENCE
ATD for Surface Antisubmarine Warfare Directorate
S p e c t r a n dT C o v a r a n c e f o r CF o r m A p p r o v e d
i R E Signal PO C ME N gA I PPBP No. OmsE-or1 5PUINI rlD~ltl Jrff frthis collec'tion of intformaiion is estitled to average t hour Def feOnt. including the time for rebviewing instr'ucti ons, Watlching 9-iStttg data source.githefil~l nd ml~llm at a i l nc i.e and monitmg and refviewing th e 4colle< ton of informaio. If d€m e ¢~rhgti tretL~lieo n ~ e'i4LC ft¢otl 'tO ofinf~r ltu ctdn iugt on$ , o red.ciA9 this burden.,to WAshorqtOn headquarters nev~e S wen doe tegfardn Ino tioun tmt o an y Otr so Of efrthis
DaisP Highway. StJhe 1204. Arl tn.v 22202-4302. and to the Office of Manage"In and budget. Paperwork fleduction Project (0704-0 166). Washington, DC 2050]
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
21 May 1991 Progress4. TJITLE AND SUBTITLE S. FUNDING NUMBERSSpectra and Covariances for "Classical"Nonlinear Signal Processing Problems PE 61152NInvolving Class A Non-Gaussian Noise
6. AUTHOR(S)
Albert H. NuttallDavid Middleton
7. PERFORMING ORGANIZATION NAME(S) AND ADORESS(ES) B. PERFORMING ORGANIZATIONREPORT NUMBER
Naval Underwater Systems Center TR 8887New London LaboratoryNew London, CT 06320
9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING / MONITORINGAGENCY REPORT NUMBER
Chief of Naval ResearchOffice of the Chief of Naval ResearchArlington, VA 22217-5000
11. SUPPLEMENTARY NOTES
12a. DISTRIBUTION / AVAIL B; .ITY STATEMENT 12b. DISTRIBUTION CODE
Approved for public release;distribution is unlimited.
13. ABSTRACT (Maximum 200 words)
Because of the critical r6le of non-Gaussian noise processesin modern signal processing, which usually involves nonlinearoperations, it is important to examine the effects of the latteron such noise and the extension to added signal inputs. Here,only non-Gaussian (specifically Class A) noise inputs, with anadditive Gaussian component, are considered.
The "classical" problems of zero-memory nonlinear (ZMNL)devices serve to illustrate the approach and to provide avariety of useful output statistical quantities, e.g., mean or"dc" values, mean intensities, covariances, and their associatedspectra. Here, Gaussian and non-Gaussian noise fields areintroduced, and their respective temporal and spatial outputsare described and numerically evaluated for representativeparameters of the noise and the ZMNL devices. Similar
14. SUBJECT TERMS IS. NUMBER OF PAGEScovariance spectranonlinear Clae_ A noise 16. PRICE CODEnon-Gaussian signal processing
17. SECURiTY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT
UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED SARNSN 7S401-280-5S00 Standard Form 298 (Rev 2-89)
PVCr*" by Ai Stil IWISM IN~
UNCLASSIFIEDSECURITY CLASSIFICATIONOF THIS PAGE
13. ABSTRACT (continued)
statistics for carriers which are phase or frequencymodulated by Class A (and Gaussian) noise are alsopresented, numerically evaluated, and illustrated in thefigures. A series of appendices and programs provide thetechnical support for the numerical analysis.
14. SUBJECT TERMS (continued)
zero acrary nonlinearity "- law .. tificationfrequency modulation phase modulation
UNCLASSIFIEDSECURITY CLASSIFICATIONOF THIS PAGE
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TABLE OF CONTENTS
Page
LIST OF FIGURES iii
LIST OF PRINCIPAL SYMBOLS v
SUMMARY OF NORMALIZING AND NORMALIZED PARAMETEJS vii
PART I. ANALYTIC RESULTS AND NUMERICAL EXAMPLES
1. INTRODUCTION 1
2. ANALYTIC RESULTS: A SUMMARY 2
2.1 THE SECOND-ORDER CLASS A CHARACTERISTIC FUNCTION 3
2.2 PROBLEM I: HALF-WAVE v-TH LAW RECTIFICATION 5
(STATIONARY HOMOGENEOUS FIELDS)
2.2-1 Gauss Processes Alone (A=0) 8
2.2-2 Case B, Figure 2.1 29I. The Intensity E(y 2 ) 10
II. The Mean Value, E(y) 2 2 12
III. The Continuum Intensity, E(y )-E(y) 12
2.2-3 Case A, Figure 2.1 13
2.2-4 Remarks 15
2.2-5 Spectra 16
I. Wavenumber Spectrum 16
II. Frequency Spectrum 18
III. Wavenumber-Frequency Spectrum 19
2.2-6 Frequency and Phase Modulation 20
by Class A and Gaussian NoiseI. Frequency Modulation 22
II. Phase Modulation 23
3. NUMERICAL ILLUSTRATIONS AND DISCUSSION 25
I. Gauss Noise Alone 25
(Figures 3.1 - 3.4)
II. Class A and Gauss Noise 27
(Figures 3.5 - 3.10)
EXTENSIONS , o " 29
-q' I. A0I- I
tr n
-%"l a i F [S !,
| | | | m| n- - -
TR 88R7
Page
PART II. MATHEMATICAL AND COMPUTATIONAL PROCEDURES
4. SOME PROPERTIES OF THE COVARIANCE FUNCTION 43
4.1 Simplification and Evaluation of B (Y) 43
4.2 Limiting Values of the Covariance Function 45
4.3 Value at Infinity 46
4.4 Value at the Origin 48
PART III. APPENDICES AND PROGRAMS
A.1 - EVALUATION OF COVARIANCi FUNCTIONS FOR ZERO SEPARATION 49
(AR=O)
A.2 - EVALUATION OF COVARIANCE FUNCTIONS FOR ZERO DELAY 53
(f,f '0)A.3 - EVALUATION OF TEMPORAL INTENSITY SPECTRUM FOR ZERO 55
SEPARATI'fl (AR=O)
A.4 - EVALUATION OF WAVENUMBER INTENSITY SPECTRUM FOR ZERO 59
DELAY (ff'=O)
A.5 - EVALUATION OF PHASE MODULATION INTENSITY SPECTRUM 63
A.6 - EVALUATION OF FREQUENCY MODULATION INTENSITY SPECTRUM 69
REFERENCES 75
ii
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LIST OF FIGURES
Figure Page
2.1 A. Two-point sensor array (R2) , giving sampled field 6
at two space-time points.
B. A general array (R) (preformed beam), converting
the field a(R,t) into a single time process x(t1 ).
Thus, the rather formidable expression, above, for B (Y) can be
evaluated by the use of just one square root and one arcsin when
S= 0, 1, 2.
The following limiting values, which are obvious, are needed
for various special cases:
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B0 (0) = i B0 (1) = 2n
B,(0) = 1 , BI(1) = Rt
B2 (0) = n/4 , B2 (1) = 3n/2
B3 (0) = 1 , B3 (1) = 15n/4
B4 (0) = 9n/16 , B4 (1) = 105R/8 • (4.6)
These are special cases of
B (0) = r2(, + 1) (4.7)
B (1) = 2 n+ + (4.8)
the latter following from [10; (15.1.20)J.
4.2 LIMITING VALUES OF THE COVARIANCE FUNCTION
The covariance function at normalized separation AR and
delay f is given by (2.7) as
W D A(l - PH m 1+M 2M (tR,f) = exp[-A(2-p)] [ mT ! m2
mI=0 m2=0
-n v/2 v/2
X n + rl [ A + B (Y) , (4.9)
n=0
where
p = p(f) = max{0, 1 - Ili , (4.10)
kL + r, kA L + A n (4.11)
Y - Y(ml'm2'nl)" ' n + m + rj n + m2 + r 4
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-'2 _1 1Lj 2 f2kL =kL(R,fl = exp - L2_1 , (4.12)
k G kG(ERf) = exp- [(GI a2- 2 i2 . (4.13)
The functions p, kL, kG can be replaced by other functional
dependencies, if desired. The function B (Y) has been
considered earlier and considerably simplified for v = 0, 1, 2.
4.3 VALUE AT INFINITY
As LR or f - ±a, then
p 4 0 , kL 4 0, kG 4 0, Y 4 0 (4.14)
(If IfI remains less than 1 as SR tends to infinity, then p does
not approach zero; this nuance has been discussed elsewhere in
this report.) Then, it follows that
W m 1 +m2 rI v/m2v + r}v/2
M exp(-2A) Lk m+r+ B (0)My T T ml1! m 2! AA
m1=0 m2=0
=B (0) expi(--A) m + r' ) ,/2 (4.15)v )tx~IL m! (EA + AJ
m=0
because the sum on n can be terminated with the n = 0 term.
The sum on m can be effected in closed form, for v = 0, 2, ,
etc., by using the following results:
AmZ ! = exp(A) , (4.16)
m=O
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- =m 1) A exp(A) , (4.17)
m=O m=l
. AM m2 An (m -i+ 1)m=O m=l
= Am Amnm( - 2)! + - m 1)! (A2 + A) exp(A) . (4.18)
m=2 m=i
There follows
It for v = O'
A 2M(o) = (0+r) for v = 2 (4.19)
9n 4 1 +'2 2)H [- + i + rAJ for v = 4
The case for v = 1 requires a numerical summation, once A and
r' are specified. When these limiting values are subtracted from
the correlation function, we obtain the covariance function.
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4.4 VALUE AT THE ORIGIN
For AR = 0, f = 0, then
p = 1, kL = 1, kG = 1 , (4.20)
and
A- n ,AA)
M (0,0) = exp(-A) ' + F B (1) (4.21)
2 . n! LA Ak BVl (.1n=0
because the sums on mI and m2 can be terminated with the zero
terms, thereby also leading to Y = 1.
The sum on n can be accomplished in closed form, for
V= 0, 1, 2, etc., by using results given earlier. There follows
2n for v = 0
A
My(0,0) =nj + q for v = 1 (4.22)
3 + (l+r) 2 1 forv=2
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PART III. APPENDICES AND PROGRAMS
APPENDIX A.1 - EVALUATION OF COVARIANCE FUNCTION
FOR ZERO SEPARATION (AR = 0)
A program for the numerical evaluation of covariance
M y( R,) for AR = 0 is contained in this appendix. Inputsrequired of the user are A, r', (awL/) 2 (AWG/) 2 , 6(f), N(f),
in lines 20 - 70. Since we are generally interested in values of
A less than 1, the series for Ay in (4.9) will not have to be
taken to very large values of mi1, m2 , n; accordingly, the values
of (A /k!] are tabulated once in lines 260 - 300 with a tolerance
of 1E-10 set in line 80.
The values of the covariance at infinity, as given by (4.19),
are computed and subtracted in lines 220 - 240 and 400 - 420;
this is in anticipation of taking a Fourier transform of a
covariance function which decays to zero for large arguments AR.A '
The functions BV(Y) and My (R,f) are available in the two
subroutines stdrting at lines 1010 and 1120, respectively. The
latter subroutine actually calculates the covariance at general
nonzero values of both AR and f, although we only employ it for
AR - 0 in this appendix; see lines 10 and 380. Also, for AR - 0,
the parameter Lg2 - (AL/AG)2 is not relevant and, hence, is
entered as zero in line 380.
The exponential Gaussian forms for kL and kG are used in
lines 1200 and 1210, while the triangular form for p is entered
in line 1240. Any of these can be replaced, if desired, by forms
more appropriate to the user.
The program is written in BASIC for the Hewlett Packard 9000
Computer Model 520. The designation DOUBLE denotes integer
variables, not double precision. The output from the program is
stored in data files AOTO, AOT1, AOT2, for v - 0, 1, 2,
respectively.
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10 Rc=O. 1DeiR-20 A=.2 I A(subA)30 Cf,. 00 1 IGAMMA' (subA)40 WlIb2=1 I <(DeIW (SUbL )/Beta)A^250 14gb2=25. I(De IW (subC) /Bet a)A260 Dtc=.01 INCREMENT IN TauJA70 Ntc=2Ou NUMBER OF Tau^ VALUESso Tolerance=1.E-109e Cciii AfVO: 40), C(0: SO), ScO: 80)
100 CON DOUBLE J1 INTEGER110 DIM K'ag(200),Tc(0:200),FO(0:200') ,F1 (0:200), F2(l0:20 0)120 DOUBLE Ntc,K IINTEGERS130 FOR K=O TO Ntc140 Tc=K*Dtc T TaUA150 Rho=MAX(0.,1.-ABS(Tc)) IRho160 T2=.5*Tc*Tc170 K1=EXP(-Wlb2*T2)180 'gFVP(-W-gb2*T2)190 K1ag(K')=(Rhio*K1+Gp)*Kg)/(1.+Gp) IINPUT COVARIANCE200 NEXT K210 A=.AA>0 REQUIRED220 F0i nCf=P I230 Fliif=FtFXInC(A,Gp)240 F2inf=.25*PI*( 1.+Gp)*( 1. Gp)250 Af'(0)=1.260 FOR K=1 TO 40270 J=K280 AK)TRK-)AKIA-K./KI290 IF T<Tolerance THEN 320300 NEXT K310 PRINT "40 TERMS IN Af(*)"320 FOR K=0 TO 3*2330 C(K)=TK*A1+Gp340 Sq(K")1.'SQR(T)350 NEXT K360 FOR K=O TO Ntc370 Tc(K)=Tc=K*Dtc ITauA380 CALL Myc(Rc,Tc,A,Gp,'lb2,Wgb2,0. ,FO(K),F1(K),F2(K))390 NEXT K400 MAT FO=FO-(FOinf)410 NAT F1=F1-(Flinf)420 MAT F2=F2-(F2inf)430 MAT FO=FO'(FO(0))440 MAT F1=F1'(F1(0))450 MAT F2=F2/(F2(0))460 PRINT "IN.FINITY: ";F~inf;Flinf;F2inf470 PRINT "MINIMA: " ;MIN(FO(*));MIN(F1 (*));MIN(F2(*) )480 PRINT "AT Ntc: ".;FO(Ntc);F1(Ntc);F2(Ntc)490 CREATE DATA "AIT1",8500 ASSIGN #1 TO "AITI"510 PRINT #1;Kag(*)520 CREATE DATA "AOTO",8530 ASSIGN #1 TO "ACTO"540 PRINT *1;FO(*)550 CREATE DATA "AOTI18
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560 ASSIGN #1 TO "ROTI"570 PRINT #I;F1(*)580 CREATE DATA "ROT2",8590 ASSIGN #1 TO "AOT2"600 PRINT #1;F2(*)610 ASSIGN #1 TO *620 Tcrax=Dtc*Ntc630 GINIT 200/260640 PLOTTER IS 505,"HPGL"650 PRINTER IS 505660 LIMIT PLOTTER 505,0,200,0,260670 VIEJPORT 22,85,19,122680 WINDOW 0.1.,0.,1.690 PRINT "'VS5"708 GRID .25,.25710 PRINT "VS36"720 PLOT Tc(*),Kag(*)730 PENUP740 PLOT Tc(*),FO(*)758 PENUP760 PLOT Tc(*),F1(*)770 PENUP780 PLOT Tc(*),F,.'(*)790 PENUP800 PAUSE810 PRINTER IS CRT820 PLOTTER 505 IS TERMINATED838 END84e850 DEF FNFlinf(A,Gp) I for v(=nu) = 1860 Tol=I.E-18870 Ag=A*Gp880 T=I.890 S=SQR(I.+Ag)900 FOR M=2 TO 108910 T=T*A/M920 P=T*SQR(M+Rg)938 S=S+P940 IF P<S*Tol THEN 970958 NEXT M960 PRINT "100 TERMS IN FNF1inC"970 T=Gp+A*S*S+2.*SQR(Ag)*S980 RETURN EXP(-2.*A)*T990 FNEND
18081010 SUB Bnu(Y,BO,BI,B2) I Bv(Y) for v=8,1,21820 IF Y>I. THEN PRINT "Y = I +";Y-1.1030 IF '>1. THEN Y=I.1040 Y2=Y*Y1050 Sq=SQR(1.-Y2)1060 T=ASN(Y)+1.57079632679489661070 BO=T+T1080 BI=Y*T+Sq1090 B2=(.5+Y2)*T+I.5*Y*Sq1100 SUBEND1118
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1120 SUB fp1kc (Pc, Tc ,A,G :r, wllb2,1 Ngb2 ,,Lg2, SO, Si, S2)1130 CON AfA*),C(*),Sq(*)1140 CON DOUBLE J INTEGER
1150 ALLOCATE Ap(0:J),Apl(0:J)
1160 DOUBLE K,t11,N2,N,K1,K2 INTEGERS
1170 A1=1.'A ! >O REQUIRED
1180 T2=.5*Tc*Tc1190 R2=Rc*Rc1200 K1=EXP(-R2-W1b2*T2)1210 Kg=EXP(-Lg2*R2-W.gb2*T2)1220 Ak=1A1*K111230 Gk=Gp*Kg1240 RhoMPAY,-(0.,.-ABS(Tc)) Rho1250 Rho1=1.-Rho1260 Ap(0)ARpl(0)=Pk=PklIl.1270 FOR K=1 TO J1280 Pk=Pk*Rho1290 PklI=P1klI*Rlen1300 T=AC(s'1310 Ap(K)=T*Pk1320 Apl(K)=T*Pkl1330 NEXT K1340 S0rn151m15S2m10O.1350 FOR 111=0 TO J1360 66r2=1n2=S2r2=0.1370 FOR M2=0 TO J1380 80n5lnS2n0O.1390 FOR N-=0 TO J1400 K1=N+M111410 K2=1NtM21420 T=Rp(N)1430 P=C(K1)*C(K2)1440 Y=(N*Ak+Gk)*Sq(K1)*Sq(K2)1450 CALL Bnu(Y,BO,D1,B2)1460 SOn=SOn+T*BO1470 Sln=SlntT*SQR(P)*BI1480 S2n=S2n+T*P*B21490 NEXT N1500 T2=Apl(M2)1510 SO2SOm2+T2*SOn1520 Slr,21r2+T2*Sln)1530 52,2=S2rn2tT2*52l1540 NEXT M21550 T1=Apl(Mi)1560 SOnl=SOrltTl*SOf21570 Slui,=Slmnl+T1*S'r21580 S2ra1 =S2rl+T1*S2m21590 NEXT Ill1600 T=EXP(-A*(2.-Rho))1610 S0=T*S~nm11620 51=T*Slnl1630 S2=T*S2,11640 SUBEND
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APPENDIX A.2 - EVALUATION OF COVARIANCE FUNCTION
FOR ZERO DELAY (f,f' = 0)
A program for the numerical evaluation of covarianceAM (AR,f) for f,f' 0 is contained in this appendix. Inputs
y 2required of the user are A, r,, (AL/AG) , 6(AR), N(AR), in lines
20 - 60. The tolerance for terminating the triple infinite sums
is set at 1E-15 in line 70. The output from the program is
stored in data files AORO, AOR1, AOR2, for v = 0, 1, 2,
respectively. Other relevant comments are made in appendix A.1.
The limit of M at AR = (when f = 0) is given by the closedyform results
HK for v = 0
y( 0) = A + ' for v = 1 (A.2-1)MyA
+ fi + r,)21 for v = 2A
These values have been subtracted from My so that we can Bessel
transform a function which tends to zero as AR 4 =.
60 Nrc=900 N NUMBER OF DeiR ^ VALUES70 Tolerance=1.E-1580 COM Af(0:40),C(0:80),Sq(0:80)90 COM DOUBLE J ! INTEGER
100 DIM Rc(0:900),Kag(0:900),FO(0:900),F1(0:900),F2(0:900)110 DOUBLE Nrc,K I INTEGERS120 81=1./A 8 R>0 REQUIRED130 FOinf=PI LIMITS FOR140 Flinf=l.+Gp I Rc=infinity150 F2irf=.25*PI*((l.+Gp)*(1.+p)+A1) 1 AND Tc=O160 Af(B)=1.170 FOR K=1 TO 40180 J=K190 Af(K)=T=Af(K-1)*A/K I A^K/K!200 IF T<Tolerance THEN 230210 NEXT K220 PRINT "40 TERMS IN Af(*)"230 FOR K=O TO J*2240 C(K)=T=K*AI+Gp250 Sq(K)=I./SQR(T)260 NEXT K
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270 FOR K=O TO Nrc280 Rc(K)=Rc=K*Drc DeJRA290 R2=Rc*Rc3uf KI=EXP(-R2)310 Kg=EXP(-Lg2*R2)320 Rho=t-A,(0., 1.-ABS(Tc)) Rho330 Kag(K)=(Rho*K 1 +Gpt*Kg)/( 1. +Gp) I INPUT COVARIANCE340 CALL Myc(Rc,Tc,A,Gp,6.,6.,Lg2,F6(K),FI(K),F2(K)W356 NEXT K360 MAT F6=F-(Fginf)370 MAT FI=Ft-(Flinrf)386 NAT F2=F2-(F2inf)390 MAT F6=F6/(F6(6))460 MAT FI=F1/(FI(6))410 MAT F2=F2/(F2(6))420 PRINT " INFINITY: ";FOinf;Flinf;F2in430 PRINT "MINIMA: " ;MIN(F(*));MIN(F1(*));MIN(F2(*))440 PRINT "AT Nrc: ";FO(t-c);Fl(Nh-rc);F2(Nrc)456 CREATE DATA "AIR",33460 ASSIGN #1 TO "AIRI"476 PRINT #1;Kag(*)480 CREATE DATA "AOR",33490 ASSIGN #1 TO "AOR6"566 PRINT #1;F6(*)516 CREATE DATA "AORI",33526 ASSIGN #1 TO "AORI"536 PRINT #1;F1(t)540 CREATE DATA "AOR2",33556 ASSIGN #1 TO "AOP2"566 PRINT #i;F2(*)576 ASSIGN #1 TO *586 Rcmax=Drc*Nrc590 GINIT 200/260666 PLOTTER IS 565,"HPGL"616 PRINTER IS 565626 LIMIT PLOTTER 565,0,266,6,266636 VIEWPORT 22,85,19,122640 WINDOW 0.,3.,B.,1.656 PRINT "VS5"660 GRID .5,.25676 PRINT "VS36"680 PLOT Rc(*),Kag(*)696 PENUP760 PLOT Rc(*),FO(*)710 PENUP726 PLOT Rc(*),F1(*)736 PENUP746 PLOT Rc(*),F2(*)756 PENUP766 PAUSE776 PRINTER IS CRT786 PLOTTER 505 IS TERMINATED796 END80 I810 SUB Bnu(Y,B,BI,B2) I Bv(Y) for v=r,1,2820 ! SEE APPENDIX A.1906 SUBEtID910 I920 SUB Myc (Rc, Tc, A, Gp, Wl b2, Wgb2, Lg2, $6, Si,52)930 ! SEE APPENDIX A.11440 SUBEND
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APPENDIX A.3 - EVALUATION OF TEMPORAL INTENSITY
SPECTRUM FOR ZERO SEPARATION (AR = 0)
A program for the numerical evaluation of the Fourier
transform of covariance , (0f) - My () is contained in this
appendix. Inputs required of the user are listed in lines10 - 30. The data input, AOTO or AOT1 or AOT2, as generated bymeans of the program in appendix A.1, is injected by means of
lines 410, 600, and 790.
In order to keep the FFT (fast Fourier transform) size, N inlines 30 and 320, at reasonable values, the data sequence is
collapsed, without any loss of accuracy, according to the method
given in [8; pages 7 - 8] and [9; pages 13 - 16]. The
integration rule documented here is the trapezoidal rule; this
procedure is very accurate and efficient and is recommended for
numerical Fourier transforms.
18 Nt c=280 ! NUMBER OF Tau- VALUES28 Dtc=.81 I ICREMENT IN Tau-38 N=1824 SIZE OF FFT; N > Ntc REQUIRED48 DOUBLE Ntc,N,N4,N2,Ns INTEGERS58 N4=/468 N2=N/278 REDIM Cos(0:N4),X(O:N-1),Y(e:N-I)80 DIM Cos(256),X(123),Y(I023),A(200)90 T=2.,*P I /N
180 FOR Ns=8 TO N4110 Cos(Ns)=COS(T*Ns) QUARTER-COSINE TABLE IN Cos(*)120 NEXT Hs130 GINIT 200/260140 PLOTTER IS 505,"HPGL"158 PRINTER IS 505160 LIMIT PLOTTER 505,0,200,0,260170 VIE1PORT 22,85, 19, 122180 WINDOW 0,N2,-5,1190 PRINT "VS5"200 GRID N/10,1210 PRINT "VS36"220 ASSIGN #1 TO "AITI"230 READ #1;A(*)240 MAT X=(0.)250 MAT Y=(0.)260 X(8)=.5*A(O)270 FOR Ns=1 TO Htc-1280 X(Ns)=A(tls)2:00 11 EXT .1300 X(Ntc)=.5*A(Ntc)
55
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310 MAT X=X*(Dtc*4.)320 CALL Fft14(tCo (*,'(*),Y(*))330 FOR Ns=0 TO 12340 Ar=X(Ns)350 IF Ar>0. THEN 380360 PENUP370 GOTO 390380 PLOT tNls,LGT(Ar)390 NEXT Ns400 PENUP410 ASSIGN #1 TO "ROTO"420 READ #1;A(*)430 MAT X=(O.)440 MAT Y=(0.)450 X(O)=.5*A(O)460 FOR Ns=l TO Ntc-1470 X(Nts)=A(Ns)480 NEXT Ns49e X(Ntc)=.S*A(Ntc)500 MAT X=X*(Dtc*4.)510 CALL Fft14(N,Cos(*),X(*),Y(*))520 FOR Ns=O TO N2530 Ar=X(ts)540 IF Ar>0. THEN 570550 PENUP560 GOTO 580570 PLOT Ns,LGT(Ar)580 NEXT Ns590 PENUP600 ASSIGN #1 TO "AOTI"610 READ #1;A(*)620 MAT X=(O.)630 MAT Y=(O.)640 X(O)=.5*A(0)650 FOR Ns=l TO Ntc-1660 X(Ns)=(Ns)670 NEXT Ns680 X(Htc)=.5*A(Ntc)690 MAT X=X*(Dtc*4.)700 CALL FftI4(N,Cos(*),X(*),Y(*))710 FOR Ns'0 TO N2720 Ar=X(Ns)730 IF Ar>0. THEN 760740 PENUP750 GOTO 770760 PLOT Ns,LGT(Ar)770 NEXT Ns780 PENUP790 ASSIGN #1 TO "AOT2"800 READ #1;A(*)810 MAT A=A/(A(0))820 MAT X=(O.)830 MAT Y=(0.)
56
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840 X(O)=.5*A(0)850 FOR N=1 TO Ntc-1860 X(t:E )=A 1s)878 NEXT Ns880 X(Nt c)=.5*A(Ntc)898 MAT X=X*(Dtc*4.)900 CALL Fft14(N,Cos(*),X(*),Y(*))910 FOR Ns=O TO N2926 Ar=X(Ns)930 IF Ar>0. THEN 966940 PENUP950 GOTO 970960 PLOT NsLGT(Ar)97A NEXT Ns980 PENUP990 PAUSE1000 END10101020 SUE Fftl4(DOUBLE N,RERL Cos(*),X(*),Y(*)) N t<=2-14=16384; 0 SUBS1030 DOUBLE Log2n,NI,N2,N3,N4,J,K ! INTEGERS < 2'31 = 2,147,483,6481040 DOUBLE 11,12,13,14,15,16, I7,18,19,11 ,II1,112, I13, I14,L(0:13)1050 IF N=1 THEN SUBEXIT1060 IF N>2 THEN 11401070 A=X(0)+X(1)loes X(1)=X(0)-X(1)1090 X()=A1100 R=Y(O)+Y(1)1110 Y(1)=Y(O)-Y(1)1120 Y(O)=R1130 SUBEXIT1140 A=LOG(N)/LOG(2.)1150 Log2n=A1160 IF ABS(A-Log2n)<I.E-8 THEN 11901178 PRINT "N =";N;"IS NOT A PO4ER OF 2; DISALLOWED."1180 PAUSE1190 NI=N/41200 12=N1+11210 N3=N2+11220 N4=N3+N11230 FOR I1=1 TO Log2n1240 12=2A<Log2n-I1)1250 13=2*121260 14=N/131270 FOR 15=1 TO 121280 16=(15-1)*14+11290 IF 16<=N2 THEN 13301300 Al=-Cos(N4-16-1)1310 R2=-Cos(I6-NI-1)1320 GOTO 13501330 AI=Cos(16-1)1340 A2=-Cos(N3-16-1)1350 FOR 17=0 TO N-13 STEP 131360 18=17+I5-11370 19=18+121380 TI=X(I8)1390 T2=X(19)
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1400 T3=Y(I8)1410 T4=Y(19)1420 A3=TI-T21430 R4=T3-T41440 X(I8)=TI+T21450 Y(18)=T3+T41460 X(19)=AI*A3-A2*A41470 Y(19)=AI*A4+R2*R31480 NEXT 171490 NEXT !51500 NEXT 111510 l=Log2n,+1720 FOR 12=1 TO 141530 L(12-1)=11540 IF 12>LWg2n THEN 15681550 L(12-I)r2"(11-12)1560 NEXT 121578 K=O1580 FOR 11=1 TO L(13)1590 FOR 12=11 TO L(12) STEP L(13)1600 FOR 13=I2 TO L(11) STEP L(12)1610 FOR 14=13 TO L(1G) STEP L(11)1620 FOR 15=14 TO L(9) STEP L(10)1630 FOR 16=15 T L3) STEP L(9)1640 FOR 17=I6 TO L(7) STEP L(8)1650 FOR 18=17 TO L(6) STEP L(7)1660 FOR 19=18 TO L(5) STEP L(6)1678 FOR I10=T? TO L(4) STEP L(5)1680 FOR III=Ii TO L(3) STEP L(4)1698 FOR 112=111 TO L(2) STEP L(3)1700 FOR 113=112 TO L(1) STEP L(2)1710 FOR 114=113 TO L(0) STEP L(1)1720 J=114-11738 IF K>J THEN 18081740 F=X(K)1750 X(K)=X(J)1760 X(J)=A1770 A=Y(K)1780 Y(K)=Y(J)1790 Y(J)=R1800 K=K+11810 NEXT 1141820 NEXT 1131830 NEXT 1121840 NEXT III1850 NEXT 1101860 NEXT 191870 NEXT 181880 NEXT 171890 NEXT 161900 NEXT 151910 NEXT 141920 NEXT 131930 NIEXT 121940 NEXT Il1950 SUBEND
58
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APPENDIX A.4 - EVALUATION OF WAVENUMBER INTENSITY
SPECTRUM FOR ZERO DELAY (f,f' = 0)
A program for the numerical evaluation of the zeroth-order
Bessel transform of covariance M y(AR,0) - (-o) is contained in
this appendix. Inputs required of the user are listed in lines
10 - 40 and are coupled to appendix A.2, where the data input,
AORO or AOR1 or AOR2, was generated. The numerical Bessel
transform is accomplished by means of Simpson's rule with end
correction [11; pages 414 - 418], and is exceedingly accurate for
the small increment, .005, in AR employed in line 30.
18 Dkc=.4 I INCREMENT IN k"A20 rlkc=208 NUMBER OF k' VALUES38 Drc=.885 INCREMENT IN DelR.40 Nrc=908 NUMBER OF DelR ^ VALUES58 DOUBLE Nrc,-Okc, I,Ns I INTEGERS68 REDIri C(0:Nrc)70 REDIr Ni (8:Nkc), W080:Nk c),1(0:Nkc),W2(8: Nkc)80 DIM C(900),Wi (200 ),,0(200), WI 200),W2(200)90 ASSIGN #1 TO "AIRI"100 READ #I;C(*)118 FOR I=8 TO Nkc120 Kc=I*Dkc 1 k-130 T=Kc*Drc140 Se=So=8.158 FOR Ns=1 TO Nhrc-1 STEP 2160 So=So+tIl.*FN Jo(T*t.ls)*C tls)178 NEXT Ns180 FOR Ns=2 TO Nrc-2 STEP 2190 Se=Se+Ns*FNJo(T*Hs)*C(Ns)200 NEXT Ns210 Wi(I)=C(0)+16.*So+14.*Se220 NEXT I230 MAT Wi=Wi*(Drc*Drc*2.*PI/15.)248 ASSIGN #1 TO "AOR8"250 READ #1;C(*)260 FOR 1=8 TO Nkc270 Kc=I*Dkc280 T=Kc*Drc290 Se=So=.308 FOR Ns=1 TO Nrc-1 STEP 2
310 So=So+N*FNJo(T*N-)*C(Ns)320 NEXT Hs330 FOR Ns=2 TO Nrc-2 STEP 2340 Se=Se+N*FNJo(T*Ns)*C(Ns)350 NEXT Ns360 8( I )=C(8)+ 16.*So+ 14.*Se370 NEXT I380 MAT W0=W0*(Drc*Drc*2.*PI/15.)
59
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390 ASSIGN #1 TO "AORi'400 READ #I;Ct*)410 FOR I=O TO Nkc420 Kc=I*Dkc430 T=Kc*Drc440 Se=So=O.450 FOR tNs=l TO Nrc-i STEP 2460 So=SorN-*FNJo(T*Ns)*C(Ns)470 NEXT Ns480 FOR Ns=2 TO Nrc-2 STEP 2490 Se=Se+tls*FtlJo(T*Ns)*C(Ns)500 NEXT Ns510 Wl(I )=C(0)+16. *So+14.*Se520 tEXT I530 MAT W1=1*(Drc*Drc*2*PI/15.)540 ASSIGN #1 TO "AOR2"550 READ #I;C(*)560 ASSIGN #1 TO *570 FOR I=O TO Nkc580 Kc=I*Dkc590 T=Kc*Irc600 Se=So=0.610 FOR Nsl= TO Nrc-I STEP 2620 So=So+ts*FNJo(T*Ns)*C(tls)630 NEXT Ns640 FOR Ns=2 TO Nrc-2 STEP 2
650 Se=Se+Ns-t*:l ;Jo(T*tls)*C (Ns)660 NEXT Ns670 W2I )=C(0)+16.*So+14.*Se680 NEXT I690 MAT W2=W2*(Drc*Drc*2.*PI/15.)700 GINIT 200/260710 PLOTTER IS 505,'HPGL"720 PRINTER IS 505730 LIMIT PLOTTER 505,0,200,0,260740 VIEWPORT 22,85,19,122750 WINDOW 0,Nkc,-9,1760 PRINT "VS5'770 GRID 25,1780 PRINT "VS36"790 FOR 1=0 TO Nkc800 W=Wi(I)810 IF W>O. THEN 840820 PENUP830 GOTO 850840 PLOT I,LGT(W)850 NEXT I860 PENUP
60
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870 FOR I=0 TO NI'c880 WWo (I )890 IF 11>0. THEN 920900 PENUP910 COTO 9:30920 PLOT I,LGT(W)930 NEXT I
940 PENLIP950 FOR 1=0 TO IHkc960 WW1,1(I)970 IF 14>0. THEN 1000980 PENUP990 COTO 1010
i000 PLOT 1,LGT(W)1010 NEXT I1020 PENUP1030 FOR 1=0 TO Nkc1040 W=W2(I)1050 IF W>0. THEN 10801060 PENUP1070 COTO 10901080 PLOT I,LCT(W)1090 NEXT I1100 PENUP1110 PAUSE1120 PRINTER IS CRT1130 PLOTTER 505 IS TERMINATED1140 END11501160 DEF FNJo(X) Jo(X) FOR ALL X1170 Y=ABS(X)1180 IF Y>8. THEN 12801190 T=Y*Y HART, #58451200 P=2271490439.5536033-T*(5513584.56477o07522-T*5292.61713o3845574)1210 P=233448917187?869. 7-T*(47?65559442673. 588-T*(4621?2225031 .7180:3-T*P,))1220 P=18596231762189?804.-T*(44145829391815982.-T*P)1230 Q=204251483.52134357+T*(494030.79491813972+T*(884.720367-56175504+T,,1240 Q=2344?50813658996.84T*(15015462449769.7524T*(64398674535. 1332-56+T*Q)J.,1250 Q=18596231762189?733. +T*Q1260 J0 ~P'o1270 RETURN Jo1280 2=8./"? HART, #6546 & 69461290 T=2*Z1300 Pn)2204.5010439651804+T*(128.67758574871419+T*.90047934748026,-*803,:'1310 Pn)8554.8225415066617+T*C8894.4375329606194tT*Pn)1320 Pd=22114.0488519147104+T*(130.88490049992388+T)1330 Pd=8554.8225415066628+T*(8903.8361417095954tT*Pd)1340 0n.13 .990976865960680+T*(1.8497327982345548+T*.009352 59532940319)1350 On=-37.510534954957112-T*(46.093826814625175+T*r-,)1360 Od=921.56697552653090+T*(74.428389741411179+T)1370 Od=2400.6742371172675+T*(2971.9837452084920+T*Od)1380 T='(-. 785398163397448281390 Jo=.28209479177387820*SQR(Z)*(COS(T)*Pti/Pd-SIN.(T )*Z*Onlr/Qd,)1400 RETURN Jo1410 FJElJD
6 1/62Reverse Blank
TR 8887
APPENDIX A.5 - EVALUATION OF PHASE MODULATION INTENSITY SPECTRUM
The normalized covariance function for phase modulation is
given by (2.50) in the main text, namely
k 0 exp[- rA p [1 - exp(- )] - A[2 - p()] + (A5-1)
This is a fairly rapid decay with 4 and will not lead to
numerical difficulty when rA p2 << 1.2For large bp /A, the term
2
exp(- L- [1 - exp(-b4)]) (A.5-7)
is very sharp near 4 = 0; in fact, it is given approximately by
2
exp- -L bt) for 4 near 0 . (A.5-8)
Therefore, we define the sharp component of covariance k (4) as
2
ks (4) = exp[- A + A exp[- - b )J - exp(-A) for all 4 . (A.5-9)
Then
k s(0) = 1 - exp(-A) , k S(-) = 0 . (A.5-10)
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Now we let
k o() = [k (() - k ()] + k s(0 =
= kf(t) + k s() , (A.5-11)
where kf(t) is a flat function near t - 0. Then we can express
the desired difference as
ko(Z) - k() = [kf(t) - ko(-)] + k s (Z) -
kj j + ks( ) , (A.5-12)
where functions kl(<) and ks (Z) both decay to 0 at Z = . We now
employ two separate FFTs on each of the functions in (A.5-12).
The sharp component, ks (), must be sampled with a very small
increment, A , when bp2/A is large. On the other hand, the flat
component
kl( ) - kf(C) - ko(-) (A.5-13)
can be sampled in a coarser fashion. Finally, if bp2/A is
moderate, we work directly with k0 ) - k0 (-) without breaking
it into any components.
Two programs are furnished in this appendix, one for moderate
bp/A, and the other for the flat component (A.5-13) when b2 /A
is large. For sake of brevity, the Fourier transform of the
sharp -7omponent (A.5-9) is straightforward and is not presented.
The particular covariance p(t) adopted is triangular,
p(K) - 1 - L for IZI < rc 0 otherwise , (A.5-14)
but can easily be replaced. The parameter c is the cutoff value
of covariance p(C).
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The number of samples, N, taken of the covariance, in order
to perform the FFT of (A.5-4), is rather large, so as to
guarantee a very small value of truncation error at the upper end
of the integral, despite the small increment At. In order to
keep the FFT size, Mf, at reasonable values, the data sequence is
collapsed without any loss of accuracy according to the method
given in [8; pages 7 - 8] and [9; pages 13 - 16]. The
trapezoidal rule is used to approximate the integral in (A.5-4),
for reasons given in [8; appendix A].
10 SPECTRUM FOR PHASE MODULATION - MODER.9TE20 MUI=1. MUSubP30 Gp=. 001 I Gamma'40 BE=1. b50 A=.2 A60 Zc=2.*PI I Rh:(Z) = 0 for IZI>Zc; Z=zeta70 Delz.005 Zeta increment80 H=60000 M axirmum number of samples of Lkokzeta)90 Mf=16:384 Size of FFT100 DOUBLE N, Mf, Ms, Hs INTEGERS110 DIN X< 163,c-_"),Y( 16384),Cos(4096)120 REDIi X(0: Mf"-1), Y(O: Mf-1 ), C:os (0: Mf'4)130 fiAT X=(0.)140 MAT Y=(8.)150 T=2.*PI/Mf160 FOR Ms=0 TO MI/4170 Co-(Ms)=COS(T*Ils) QUARTER-COSINE TABLE180 NEXT Ms190 Ta=Gp*Mup*Mup200 IF A=O. THEN 220210 Tb=Mup*Mup/A220 Tc=2.*A*FHExp(Tb)230 Kirf=FNEx,(Ta+2.*A-Tc) CORRELATION AT INFINITY240 COM A,Bs,ZcTa,Tb,Tc,Kinf250 T=I.-Kinf260 PRINT 0,T270 X(O0=T*.5 TRAPEZOIDAL RULE280 FOR Ns= TO N290 Corr.=FNKo(_1s*DeIz) CORRELATION ko(zeta)300 IF Ns<6 THEN PRINT Ns,Corr310 IF ABS(Corr.)<I.E-30 THEN 350320 'ls=Ns MODULO Mf COLLAPSING330 X(Ms)=X(Ms>+Corr340 NEXT Ns350 PRINT "Final value of Corr =";Corr;" N =";H .
360 MAT X=X*(Delz*4.)370 CALL Fft14(Mf,Cos(*),X(*),Y(*))
66
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380 GINIT390 PLOTTER IS "GRAPHICS"400 GRAPHICS ON410 WINDOW -2,2,-60,0420 LINE TYPE 3430 GRID 1,10440 LINE TYPE 1450 Del f=l./(Nf*Delz)460 FOR t1;=i TO Mf-'2470 F=is*Delf FREQUENCY480 PLOT LGT(F),10.*LGT(X(Ms))490 NEXT lMs500 PENUP510 PAUSE520 END530 I540 DEF FNExp(Xminus) I EXP(-X) WITHOUT UNDERFLOW550 IF Xrninus>708.3 THEN RETURN 0.560 RETURN EXP(-Xminus)570 FNEND580590 DEF FNKo(Zeta) I CORRELATION ko(zeta)600 COM A,Bs,Zc,Ta,Tb,Tc,Kinf610 Rho=MAX(0., 1.-Zeta/Zc) TRIANGULAR RHO620 El=Ta*(I.-FNExp(Zeta))630 E2=Tb*(I.-F.Exp(Bs*Zeta))640 E3=A*Rho*FNExp(E2)650 RETURN FNExp(E1+A*(2.-Rho)-Tc*(I.-Rho)-E3)-Kirnf660 FNEND670680 SUB Fft14(DOUBLE N,REAL Cos(*),X(*),Y(*)) ! N<=2"14=16384; 0 SUBS690 ! SEE APPENDIX A.3
10 SPECTRUM FOR PHASE MODULATION - FLAT COMPONENT20 Mup=I. MUsubP30 Gp=.001 I Gamma'40 Bs=l. I b50 A=0. I A60 Zc=2.*PI Rho(Z) = 0 for iZI>Zc; Z=zeta70 Delz=.005 Zeta increment8 N.=60000 Maximum number of sampl-Es of kl(zeta)90 Mf=16384 I Size of FFT
100 DOUBLE N,Mf,Ms,Ns I INTEGERS110 DIM X(16384),Y(16384),Cos(4096)120 REDIM X(O:MC-I1),Y(O: Nf-1),Cos(OaMf/4)130 MAT X=(0.)140 MAT Y=(.)150 T -2. x : -,160 FOR Ms=O TO Mf/4170 Cos(Ms)=COS(T*Ms> I QUARTER-COSINE TABLE
180 NEXT Ms190 Ta=Gp*Mup*Mup200 IF 8=0. THEN 220
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210 Tb=Miup*l'lUp/'A220 Tc=2. *A*FIIExp( Tb)230 Tb=5.E5524C Ki n=FHExp( Ta+2. *A-Tc) CORRELATION1 AT INFINITY250 Ea=FMExp(A)260 Tbb=Tb*Bs270 COM A, E.s,Zc,Ta, Tb,Tc,KinflEa, Tbb280 T1l-Kinf-(1.-Ea) SUBTRACT SHARP COMPONENT290 PRINT 0,T300 X(O)=T*.5 TRAPEZOIDAL RULE310 FOR Ns=1 TO N320 Corr=FKh1(Ns*Delz) CORRELATIOM k1(zeta)330 IF Ms<6 THEM PRIMT tMs,Corr340 IF ABS(Corr)<1.E-30 THEM 380350 M==s MODULO Mf COLLAPSING360 X(Ms)=XM's)+Corr370 NEXT Ms380 PRINT "Final valJE of Corr =";Corr;" I- ="0N
390 MAT X=X*(Delz*4.)400 CALL Fft 14(M,Cos(*),X,(*),Y(*)>410 GINIT420 PLOTTER IS "GRAPHICS"430 GRAPHICS ON440 WINDOW -2,2,-60,0450 LINE TYPE 3460 GRID 1,10470 LINE TYPE 1480 Derlf:I./(NIf*Delz)490 FOR Ms=1 T'1 MC/2500 F=Ms*Delf FREQUENCY510 T=X(Ms)520 IF T>0. THEN 550530 PENUP540 GOTO 560550 PLOT LGT(F),10.*LGT(T)560 NEXT Ms570 PENUP580 PAUSE590 END600610 DEF FNExp(Xminus) EXP(-X) WITHOUT UNDERFLOW620 IF Xrfitius>708.3 THEN RETURN 0.630 RETURN EXP(-Xminus)640 FNEND650660 DEF FNIK1(Zeta) I CORRELATION kl(zeta)670 COM A, Bs, Zc ,Ta, Tb, Tc, Kinf, Ea, Tbb680 Rho=MAX(0. , 1.-Zeta/Zc) TRIANGULAR RHO690 E1=Ta*(1.-FNExp(Zeta))700 E2=Tb*(1.-FHExp(Bs*Zeta))710 E3=A*Rho*FtlExp(E2)720 E4=FNExp(Tbb*Zeta)730 Sharp=FHExp(A*(1.-E4))-Ea 1 ks(zeta)740 RETURN FNExp(E1+A*(2.-Rho)-Tc*(l.-Rho)-E3)-Kinf-Sharp750 FNEND760770 SUE Fftl4(DOUBLE N,REAL Cos(*),X(*),Y(*)) N .<-214=16384; 0 SUBS780 I SEE APPENDIX A.3
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APPENDIX A.6 - EVALUATION OF FREQUENCY MODULATION
INTENSITY SPECTRUM
The normalized covariance function for frequency modulation
+ A p(<) exp(_ 2 [exp(-b7) + b - 1] for Z 0 , (A.6-1)Ab
where t is the time delay and p(Z) is the temporal normalized2 2 ncovariance of the field process. Also, pF = FG and
b = A /w . Since (A.6-1) involves an exponential of an
exponential of an exponential, and because a wide range of
parameter values are of interest, care must be taken in numerical
evaluation of this covariance and its transform.
Observe that
k (0) = 1 , because p(O) = 1 . (A.6-2)
Also, as delay i 4 +-, then p 4 0, giving
k0 () - exp(- r 4 (< - 1) - 2A) S kl() for C > 0 . (A.6-3)
This term, kl(), decays slowly with C if rA P << 1
The spectrum of interest is given by
W0 (w) - 4 f dC cos(wC) k0 () for w Z 0 ; w = 2nf . (A.6-4)
0
The spectrum corresponding to the limiting component, kl(C) in
(A.6-3), is directly available in closed form as
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TR 8887
WI(w ) = 4 f d cos(wL) kl(t)
0
2 2'~ , 2~4 rA 2
expi ( F - 2 2 A * (A.6-5)
if q p 2«< 1, this latter quantity is large and very sharply
peaked at w = 0; hence, this term has been subtracted out and
handled separately when FA P << 1. The residual covariance,F
ko(<) - kl(t), then decays very rapidly with r and is easily
handled directly by means of an FFT. This breakdown is not
necessary when rA PF _ 1 and is avoided, then, by handling
k o() directly in one FFT.0 2For pF/A >> 1, the term
2
expA( 2 [exp(-b ) + bi, - 1]) (A.6-6)Ab
inside the exponential in (A.6-1) behaves like
2
exp A 2 near < = 0 ,(A.6-7)
where its major sharp contribution arises. For example, if
PF = 50, A = 1, then increment AZ - .005 leads to values for
(A.6-7) of
exp(-0.156 n2 ) at t - n AC , (A.6-8)
which is adequately sampled in order to track its dominant
contribution; the actual sequence of values is 1, .856, .536,2.247, .083. For smaller p F/A, this sampling interval is more
than adequate.
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TR 8887
Two programs are furnished in this appendix, one each for the2cases of large and small rA pF The particular covariance p(C)
adopted is triangular,
p(C) = 1 - 10' for I < Cc ' 0 otherwise , (A.6-9)c
but can easily be replaced. The parameter Cc is the cutoff
covariance value and is specified numerically in the examples in
figures 3.9a through 3.10b.
10 I SPECTRUM FOR FREOIUENCY MODULATION - LARGE Gp*Muf-220 Muf=50. MUsubF30 Gp=.001 G am rra'40 Bs=I. b50 A=.2 A60 Zc=2.*PI Rho(Z) = 0 for IZI>Zc Z=zeta70 t'elz=.005 Zeta increment80 N=60000 Max i mum number of s amp e s c k o z t aL90 Mf=16384 Size of FFT
100 DOUBLE N1Pf.,MsNs INTEGERS110 DIM X(16d4),Y(16384),Cos(4096)120 REDIM X(O:MfI-I),Y(O:Mf-1),Cos(O:Mf/4)130 MAT X=(O.)140 MAT Y=(O.)150 T=2.*PI/Mf160 FOR Ms=O TO Mf/4170 Cos(Mc)=COS(TkMs) QUARTER-COSINE TABLE180 NEXT Ms190 Ta=Gp;Muf*Muf200 IF A=0. THEN 220210 Tb=Muf*Muf/(A*Bs*Bs)220 Tc=FNExp(2.*A-Ta)*Ta230 Td=Ta*Ta240 COM H,Bs,Zc,Ta,Tb250 X(0)=.5 I TRAPEZOIDAL RULE260 FOR Ns=1 TO N270 Corr=FNKo(Ns*Delz) I CORRELATION ko(zeta)280 IF Corr<1.E-20 THEN 320290 Ms=Ns MODULO ME COLLAPSING300 X(Ms)=X(1-1s)+Corr310 NEXT NS320 PRINT "Final value of Corr =";Corr;" Ns =';Ns330 MAT X=X*(Delz)340 CALL Ff't 14<Mf,Cos(*))<(*),W*))
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350 GINIT360 PLOTTER IS "GRAPHICS'"
370 GRAPHICS ON380 WINDOW -4,2,-70,30390 LINE TYPE 3400 GRID 1,10410 LINE TYPE I420 DE1C=l."(Mf*De]z)430 FOR M1= TO Mf/2440 F=Ms-*De 1 f FREQUENCY450 T=X(N)460 IF T>O. THEN 490470 PENUP480 GOTO 500490 PLOT LGTtF),1O.*LGT(T)500 NE XT 1MIs510 PENUP520 Add=X(0)-Tc/Td ORIGIN CORRECTION530 F=1.E-4540 FOR Ns=l TO 100550 1=2.*PI*F560 WL=Tc/(Td+W*W)570 T=WI+Add580 IF T>0. THEN 610590 PENUP600 GOTO 620610 PLOT LGT(F),10.*LGT(WI1+Rdd)620 F=F*I. I FREQUENCY630 IF F>Delf THEN 650640 NEXT Ns650 PENUP
660 PAUSE670 END680690 DEF FNExp(Xminus) EXP(-X) WITHOUT UNDERFLOW700 IF Xrninus>708.3 THEN RETURN 0.710 RETURN EXP(-Xminus)720 FNEND730740 DEF FNKo(Zeta) CORRELATION ko(zeta)750 COM A,B s,Zc,Ta,Tb760 EI=FHExp(Zeta)+Zeta-1.770 T=Bs*Zeta780 E2=FNExp(T)+T-1.790 Rho=MAX(0., 1-Zeta/Zc) TRIANGULAR RHO
10 SPECTRUM FOR FFEQLIEHCY MOTILLATIONt-i SMALL Gp*Hlui "220 MU f 1IIlU HULbF30 Gp=.O0cl1 G am A)a40 B=1 b50 A=. A60 2c=2.*PI Rho(Z) =0 for, jZI >Zc; Zz.eta70 DelIz=.005 Zeta incr-eiments0 H= I 10000 MaSXI muml turjibe r of sarip 1 E s of ko< zeta90 Mf=6192 ISize of FFT
100 DOLIBLE N, Hlf, Hs , tN INTEGERS110 DIM X(8192), Y(8192) ,Cos(2048)120 REIM X( 0: f - I, Y (0: tlt -1), Cos (0: lf'4)130 MATX=0)140 MAT Y=(0.)150 T =2. * P I/Hlf,160 FOR Hs=0 TO Hf/4170 C's (Mi_) COS( I*Hsc) IQLIARTER-COSINE TABLE180 NEXT M-s190 Ta=Gp*Huf*HUf200 IF A=0. THENA 220210 TbHtLJC *HUf, kA*Bs*BS)220 T=FNExp(2.*A-TaL)230 Tc=T*TaL240 Td=Ta*TaL250 Delf=.1*Ta/(2.*PI) INCREMENT IN FREQUENCY260 COM A,Bs,Zc,Ta,Tb270 Y<(0 =.5*(1.-T) TRAPEZOIDAL RULE280 FOR NslI TO N290 Corrp-FNKol (Ns*tjel z) ICORRELATION ko(zeta)-k I (zet a)300 IF ABS(Corr)<1.E-30 THEN4 340310 Hs=Ns HODULO Hf ICOLLAPSING320 Y"(Ms)=X(Ms)+Corr330 NEXT Ns340 PR INT ' F inalI val ue of Corr ;Corr-; Ns '";Ns350 MAT X=X*(Delz)360 CALL Fft14(H', Cos(*) ,X(*) ,Y(
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370 GINIT380 PLOTTER IS "GRAPHICS"390 GRAPHICS ON400 WIN1DOW -4,2',-70,30410 LINE TYPE 3420 GRID 1,10
430 LINE TYPE I440 FOR M= TO 2000450 F=M_.-*De 1 f FREQUENCY460 N=2.*PI*F470 I=Tc /(Td+*W) SHARP SPECTRAL COMPONENT480 T=i"f*Delz*F490 Ns=INT(T)500 Fr=T-Ns510 W2=Frt::(l4+1)+(1 -Fr) ,. ( Nl BROAD SPECTRAL COMPONENT520 PLOT LGT\F),10.*LGT(W1+W2)530 NEXT ls540 N MA ..=M ( t.s , I550 FOR tMts=Ns TO rlf/2560 F=M=/ (MCf*De l z) FREQUENCY570 1.=2. *PI*F580 W1 =Tc, (Td+H*N)590 W2=X(Ms600 T= W1 +W2610 IF T>O. THEN 640620 PENUP630 GOTO 650640 PLOT LGT(F),l0.*LGT(T)650 NEXT M'ls660 PENUP670 PAUSE680 ENID6907LO DEF FtNExp( >minu- ) E P(-X) WITHOUT UNDERFLOW710 IF Xrirus>708.3 THEN RETURN 0.720 RETURN EXP(-Xi nus)730 FNEND7407'fl DEF FN4,ol(2ta) CORPELATION ko(zeta)-k1(zeta,760 COM A,Bs,Z,:,Ta, Tb770 E1=FNExo(Zeta) +Zeta-1.780 =Bs*Zcta
790 E2=FtlExp(T. +f-1.900 R ,=MA.(0. , 1-Zet a-Zc ) TRIANGULAR RHO810 T=Ta*EI+A*2.-RV c)-A*Pho*FllExp(Tb*E2)820 RETURN FNE..p T)-FOF xp(Ta*(2eta-1. >+2.*A)630 FNEM'a48850 1UB Fft I'D UBLE H,RE AL CoE(*),X(*,Y(* ) N(2-1416384; i SUBS
SEE AFFEIII!X h. 3
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REFERENCES
[1) D. Middleton, An Introduction to Statistical Communication
Theory, McGraw-Hill Book Company, Inc., New York, NY, 1960.