1. Introduction

Ideal Limiting of PeriodicSignals in Random Noise

RICHARD W. KELLYArizona State UniversityTempe, Ariz.

P. R. HARIHARANThe Magnavox CompanyFort Wayne, Ind. 46804

Abstract

A method based on Hermite polynomials is developed for analysis

of the output resulting from passing Fourier expandable signals and

noise through general zero-memory devices. New results are

developed for the output time and autocorrelation functions.

Special results are generated for an ideal clipper. Techniques for

obtaining results for practical cases are discussed. The method itself

can be modified and applied to non-Gaussian noise and general

nonlinear devices with nondeterministic signals.

Davenport [1] has determined the output autocorrela-tion function and power spectrum for sinusoidal signals andGaussian noise through an ideal clipper. Jones [2] extendedthe work of Davenport to include the case of twoindependent sinusoidal signals and Gaussian noise through asimilar clipper. Shaft [3] obtained the output power invarious spectral zones when the input to the clipperconsisted ofN independent sinusoidal signals and Gaussiannoise. In many cases the input to the clipper consists of ageneral Fourier series expandable signal and noise. For thiscase the formulas and results obtained by the authors justreferred to are not applicable. The restriction of the inputsto the clipper having to be nonharmonic is, in this paper,removed by the development of a method based onHermite polynomials. 'Formulas are developed forcalculation of the output signal and noise powers in thevarious spectral zones for general Fourier expandablefunctions and Gaussian noise. These results should help inthe solution of a wider class of problems than is presentlypossible by the transform method.

II. Fundamental Results

Let the transfer characteristic of the nonlinear devicecorresponding to a signal s(t) and noise x(t) as inputs begiven by

Y(t) = G [x(t) + s(t)]

Expand Y(t) in a Taylor series as

(1)

(2)

00

Y(t) = Z G(P) (x(t)) s(t)Pp=O

where G(P)(x(t)) denotes the pth derivative of G [x(t)] withrespect to x(t).

It is known [4] that if the real function G(x) defmed inthe infinite interval (-oo, oo) is piecewise continuous inevery finite closed interval [-a, a] and if the integral

e-x2I2G2(X)dX-Io

(3)

is finite, then the series.

G(x) = CnHn(X)n=O

with the coefficients Cn calculated from the relation

Cn= !vIf G(x)Hn(x)e-X2I2 dxWN2

ao_,

(4)

(5)

Manuscript received August 25, 1970; revised November 10, 1970. converges to G(x) at every continuity point of G(x) and to

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-7, NO. 4 JULY 1971644

the limit

I[G(x+0)+G(x -0)] (6)

at every point of discontinuity. Here G(x + 0) is used todenote the value of the function as the discontinuity isapproached from the right, and G(x - 0) to denote thesame when the discontinuity is approached from the left.Hn(x) are the Hermite polynomials defined as

Hn(x)=(1I)?ex 12~ex 2/2 (7)dxyn

which are orthogonal with weight e-x2 /2 such that

f'ex2I/2Hn(xWm(x)cdx= N!V-, forn =m=0,~ forn m. (8)

Under the assumption that the device underconsideration satisfies these requirements, we can expandthe same in a series of Hermite polynomials as

G(x) ZC Hn(x=) E[x] E[x] 0 (9)n=0

The coefficients C,n are calculated from the relation

1 (222Cfl = JN2i~u G(x)Hn(xIu)e-xI/2 dx. (10)

For the Hermite polynomials,

dx H()=nH-l(x)so that from (9) and (11) we get

G(l)(X) =on±rHn.i(x/u)

n=l

G(2) (x) = Cn ((n) (n- ) Hn -.2(x/ )n=2

C pn)!Hn(x)

(11)

The coefficients Ck+p are evaluated using the relation

uPCk+

uk p)!/~ G(P) (x)Hk(x/o) e -x2'2a2dx.

(13)It is proved in Appendix I that (13) simplifies to give

Ck+=(O)+pE'[G(k+p) (x)] (14)Ck+P=(u)k+P (k+p)!

where

E[G (k+p) (x)] fG(k +p) (x) e-x 22a2 dx.

(15)

From (2), (12'), and (14) we get

Y(t) =__ E[G (k+p) (X)Juk Hk(X/ur) 6P=O k! P!~~~~~16

for the instantaneous output of the zero-memory nonlineardevice with signal'and noise as input. It is also perfectlygeneral in that the signal itself can be periodic or randomand the noise can be Gaussian or n-on-Gaussian.

Ill. Output Correlation Function

If we let Y(t1) and Y(t2) represent the output of thezero-memory device at times t, and t:z, the outputautocorrelation function for the sum of signal andindependent noise as input is given by

Ry (r) ~j E[G (k+p) (xl)]E[G (Q+q) (x2)] a1(k + Q)k,Q,p,q=0 k!Q

<.Hk(X IU)HQ(X21)AV < s(t1ps(q2j > AVp!! (17)

Making use of the relationship between E[G(k+p) (xi)] and,the Laplace transform of the device characteristicdeveloped in Appendix II, (17) could be written as

Ry(T) =k, Q,p,q=o

(12) J71{(G(S)Sk+Pes2 a212)e-sx 4C-17'.j(G(S)SQ+q eS2 a2 /2)e-SX 21k!Q

or

G(P) (x)= C(k+p) (k+P)! H(II)k=O aP(k)! H(/) (12')

aOk+Q <Hk (X, /u1)HQ(x2 /a) >,AV < AV1)S(2)p!q! >A

(18)-which is especially useful for vth-law devices.

KEL.LY AND HARIHARAN: IDEAL LIMITING OF PERIODIC SIGNALS IN RANDOM NOISE 645

If xl, x2 are jointly Gaussian with normalized correlation coefficient p(T), then their joint density fx(xl, x2) can beexpanded in Hermite polynomials as [51

fx(x(,x2)= 2+x7T2 nEp(! Hn(xl/U)Hn(x2/u).n=O

(19)

Using the orthogonal property of the Hermite polynomial, from (8) and (18) we have

00

Ry(r)=k=O

00

= E 1 l(G(s)sk+Pes2or212)e-sx1}q=O

VPelfi'fs(\k+nes2 2 [ap(r)]k < (tS(t2 >AVxx1lGss+ eS g/2)e-SX2} k! < p!qV , (xi and x2 Gaussian)

for the output autocorrelation function.

IV. Ideal Clippers

For an ideal clipper

e1l{(G(s)sk+Pea2 2S 2)e- sx}

Ifej

e7r ,-

sk+ple- s2 /2 ds

+ SkPlesk+miea2s2I2 ds

Je -jOOwhich is reduced to give

f'l{(G(s)sk+Pea2 /2)e-sx}

j1k+p f"sk+p2e-a2s2/2 ds, fork+poddiT J

=0, for k + p even.

This correlation function has to be operated on for derivingthe various components from which the output signal andnoise powers can be obtained.

A. Output Signal Power

The signal part of the output correlation function isobtained from (24) by considering the terms with k = 0 as

Rsxs(r) =4 (-<)p(s(t, )/xf2a)2p+ 1RS)7T< / , p!(2p + 1)

(21)00

/ 2-22 l

's 5( - )qS(t2)/iS2q >XZ, ~q!(2q+ 1) >AV'q=O

(25)

The summations can be simplified [7] to give

Rsxs(T) =-< erf (s(tj )Iv'2a2) erf (s(t2)/V/292) > AV

(26)

(22) whereEvaluation of this integral [61 yields

ir1 {(G(s)sk+Per2s212)e-sx}

2(k+p)2_ , fork+p odd

(a2) (k+p)12 r(i _ k + P

=0, for (k + p) even. (23)

For Gaussian noise and arbitrary signal through an idealclipper the output autocorrelation function is obtainedfrom (20) and (22) as

erfz=z e-2 d. (27)

If s(tl) is Fourier expandable, erf (s(tj )/x/2a2 ) is Fourierexpandable and can be written as

erf (s(t1 )K/2uJ2 ) = C0 +

00

C,n cos ncost, + Dn sin nwst1n=l

(28)

00

Ry(T)== 2kp()kk=O

00 2(p+q)/2 <(s(tl)) p(S(t2)yq. a a AV ,fork+p,k+qodd.

p,q=O p!!(l k2P)r IQk+q)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JULY 1971

(20)

(24)

646

with

Cn= T erf (s(t1)/Iv'2U2) cos nco,tj dt,

rT

D= T2f erf (s(tl)/V/22) sin nwst, dt, (29)

and with T the period of the signal s(t). A similar expansionexists for erf (S(t2)/V2a2). Substituting (28) into (26) andperforming the indicated average gives

Rsxs('r) C Dn cosn)CT,n=0 en

en = 1, for n = 0

= 2, for n > 1. (30)

The output signal power at the ith spectral zone is

It is readily seen that

to 0(t, ) = ez

t1 O(tl) = ez(l + 2z)

t20( )= ez(3 + 12z +4z2)

tko(ti ) = ez ( ak2oz2)Q2=O

(36)

where

akQo = (2k + 22 - 1) a(k-l)Qo + 2a(k-1) (Q-l)o

aooo= ,aloo = ,aklo =0, forQ>k. (37)

Since s(tl) is periodic, tko(tl) is periodic also and can beexpanded in a Fourier series as

Pi = (C 2 +D12)/e1. (31)

The required Fourier series expansion is performed on adigital computer using fast Fourier transform (FFT)techniques. Also, this expansion need be performed onlyonce for obtaining the output signal power which savescomputation time.

B. Output NGise Correlation Function

The correlation function for the continuous part of theoutput is obtained from (24) as

tko(ti) = CkoO + CkmO Cos m&)tlm=l

+Dkmo sin mwct,and

co oo

RNo(T Y= Hkmo p(Tr)2k+ 1 cos mWcok=O m=O

whereCkmo2 +Dkmo2

Hkmoko em(2k+ I)!1

2kp('r)k 2PI2(S(t1)kJYk! 0p P!( 2p)

s2q "(S(t2)I)q.> , for k + p,k + q odd.I=tr -k+g AV

For convenience this is split up into two parts by considering even and odd values of k separately as

RNe(T) = E p((r)2kk=l p=O

and

(-l)P(s(ti)Ia)2p+1(2k + 2p)!2k+P(k + p)!(2p + 1)!

*N-rn (- lI(s(t2 )/a)2q+ 1 (2k + 2q)!>h=. 2k+q(k + q)!(2q + 1)!q=O

2 p(,r)2k+1 (- I)P(S(ti)/a)2p(2k+2p)!k-0 +p0o

*'K' (-l)q (s(t2 )/a)2q(2k + 2q)>!2k+q(k + q)!(2q)! AV

q=O

Consider the summation over p in (34) so that

L 00

Pk(s= E 2k(2p)!(k+p)! ,Z=-(' 222

In a similar ma-.ier if we define tke(ti) as

tke(ti) = s(ti) ' zP(2k + 2p)!a*et) () 2k(k+p)!(2p+ 1)!

P=O

KELLY AND HARIHARAN: IDEAL LIMITING OF PERIODIC SIGNALS IN RANDOM NOISE

00

RN(T)=k=l

(38)

(39)

(40)

(32)

(33)

(34)

(41)

647

then

ti e(ti ) = (s(ti )/a) eZ

t2 e(ti ) = (S(t1 )/a) ez(3 + 2z)

k-1

4ke(tl) (S(t )a) ez akQeZQ (42)Q=O

where

akQe (2k + 2Q-l)a(k-l)Qe + 2a(k - 1) (Q - l)e * (43)

Again Fourier series expansion of the various t functions ispossible so that

4ke(ti ) = EkOe +

00

i Ekme cosmxctm=l

+Fkmesinmc,ct1 (44)

with

2 f

Ekme = J tke(t1) cosmcoct1 dt,0

and thebecomes

T

km~T 4ke(t ) sinmcct, dt,0

output correlation function for even values of k

Go oo

RNe(T)=-2

Ikme p(T)2k cosMcTC

k=l m=l

where

(45)

Ekme2 + FkmeImke - em(2k)!

Hkmo and Ikme can be generated using the FFT algorithm,and the output correlation function for noise powercalculation is obtained as

series expansions called for in previous sections could beobtained analytically. With the aid of the digital computer,however, this method becomes a powerful tool for theevaluation of the output signal powers and output noisespectrum in many practical problems.

The output signal powers are given by (31) once the Cand D coefficients are known. These coefficients can beobtained numerically using a digital computeryconstructing a sampled version of the signal erf (s(t)/v/2a2)over one period, and then performing an FFT. The FFToutputs will be the Fourier coefficients for erf (s(t)/V/2&').The accuracy resulting from this method is dependent uponthe number of sample points used for representing thesignal under consideration. For the example considered inthe next section, 2048 sample points were used. Thesubroutine DRHARM from the IBM Scientific SubroutinePackage [9] was used to perform the FFT.

Similarly, the Fourier coefficients needed to evaluate theHmko and Ikme coefficients in (40) and (45), respectively,can be obtained using sampled approximations for thetko(t) and tke(t) functions of (36) and (42), respectively,and the FFT algorithm. The noise spectrum can also beapproximated using the FFT algorithm. From thewell-known Wiener-Khintchine theorem the output noisespectrum is obtained as

SN(A ) = f RN(r) exp (- j&r) dr

TN= 2 ,RN(T) cos (ncor) dr

0

(47)

where TN is chosen to be large enough so that RN(T) 0for r > TN.

Let w0 = 27r/TN and let RN*(T) be the periodicexpansion of the portion of RN(r) for values of r between0 and TN:

RN(T), 0<T < TNRN*(r) =

RN*(T+ TN), allr.

RN*(r) has a Fourier series expansion

(48)

C+

RN*(Tr) = CO + 2 (Cn cos nwo0r + Dn sin nwor). (49)n=l

RN(T) = RNO(T) + RNe(T). (46)

The output spectrum is obtained from (46) by using theWiener-Khintchine theorem and the result is used forcalculation of output signal-to-noise ratios.

V. Numerical Techniques

Without the aid of digital computers, the resultsdeveloped so far would be of little practical use exceptpossibly in very special cases where the various Fourier

The value of the noise spectrum at the points X = nwc0 canthen be approximated by

SN(nfO) = 2f RN*(r) cos ncor d'r

0

= TNCn. (50)

The CN coefficients can also be obtained by samplingRN*and using FFT. Thus with the help of the digital computer

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JULY 1971648

J4l Li Li I LIFig. 2. Input signal waveform.

and the corresponding autocorrelation function is

and FFT algorithm, one can closely approximate the signalpowers and output noise spectrum of the ideal clipper.

VI. Example

A block diagram of the system under consideration isshown in Fig. 1. The signal generator is assumed to be aparticular periodic rectangular wave and is shown in Fig. 2.The fundamental frequency of the rectangular wave isassumed to be 20 Hz. This signal can be expanded in aFourier series as

co

s(t) = Pn cos nwct. (51)n=1

The noise generator was assumed to generate whiteGaussian noise with a power level of ain' per hertz.

The input signal-to-noise ratio with respect to a 1-Hzband was defined for each harmonic as

(SNR)in = 10 log (yn2/40in2) dB. (52)

The transfer function of the bandpass filter was taken tohave the form

S22

H(s) (S2 +265ow,s+co,2) S.2+262C2S+ W22

s +2633S+c3253)

With f =f2 = I 0.0, f3 = f4 =300, 6 1 =0.75, 6 2 = 0.40,and 63 = 0.45, the filter's amplitude response is flat from11 Hz to 230 Hz, down 3 dB at 9 Hz and 320 Hz, anddown 20 dB at 6.5 Hz and 660 Hz. The phase response islinear between 10 and 300 Hz.

The signal portion of the input to the clipper is

Ys(t) = 2 nH(fnc)I cos (ncct+ 0(ncic)) (54)n=1

where

H(fw) = IH(1C) eX¢O(w)is obtained from (53) by replacing s by j&.

Since the noise generator is assumed to be white,, the

spectrum of the input noise to the clipper is

Sn(&) = Oin 2 jH(W) (55)

I

~~~~RAOT= 27 Sn(wo) exp (cwr) dcj_ 00

ai 2 i'802irico

H(s) H(- s) exp (rs) ds

= uin2 residues {H(s) H(- s) exp (hT I s)} (56)LHP

where the summation is taken over all left-half-plane polesof H(s). After simplification, (56) can be written for thefilter under consideration as

4

RN(T) = Gin2 (Ai cos Pi3rI - Bi sin ,3ilr 1) exp (rT la,)i=li (57)

where Ai and Bi are the real and imaginary parts of theresidue of H(s) H(- s) at the ith left-half-plane poles withreal part ai and imaginary part pi, respectively. From (57)the total input noise power to the clipper is obtained as

4

a2 = ain2 Ai.

i=l

(58)

Computer programs were written to perform thecomputations described in the last section for erf(Ys(t)/ViJ/) andRN *(r) obtained from (57). The programsperform all the computations required to obtain the outputsignal powers, noise powers, and signal-to-noise ratios for a

given set of input signal and filter parameters. Since theprograms are long and run to several pages they are notshown here, but the output results of these are shown inTable I.

VII. Summary

This theoretical paper has developed a new mathematicaltechnique for the analysis of periodic signals and noisethrough general nonlinear devices. The results are thenspecialized for the case of the nonlinear device being an

ideal clipper. A practical example for which presentlyavailable techniques are not applicable explains the actualnumerical techniques employed for obtaining physicallymeaningful results.

KELLY AND HARIHARAN: IDEAL LIMITING OF PERIODIC SIGNALS IN RANDOM NOISE 649

DEAL

Fig. 1. System block diagram.

t

T/3

r I

TABLE IClipper Output with Rectangular Wave and Gaussian Noise as Input

Frequency Input SNR* Output Signal Power* Output Noise Power* Output (SNR)* Clipping Loss(Hz) (dB) (dB) (dB) (dB) (dB)

20 18.49 - 9.1 -26.5 17.4 1.140 12.469 -16.1 -26.8 10.7 1.860 - -46.9 -26.9 -20.0 -

80 6.449 -23.3 -26.9 3.6 2.9100 4.510 -25.1 -26.9 1.8 2.7120 - -52.1 -26.9 -25.3 -140 1.588 -28.3 -26.9 - 1.4 3.0160 0.428 -29.5 -27.0 - 2.5 2.9180 - -56.3 -27.2 -29.1 -

200 - 1.510 -32.2 -27.5 - 4.7 3.2220 - 2.338 -33.7 -28.0 - 5.7 3.4240 - -59.8 -28.7 -31.1 -

*All powers are with respect to a 1-Hz band.

Appendix I

Proof of Relation Between Cn+p and G(n+P)(x)

It is known [8] that under some generally fulfilledconditions

G(x) d?P%(X) dx = (- i)P dPG(x)(dx.

(59)

From (12) we also have the result

_ _ _ f 2 2

Ck+p (k=+p;/ ] G(P) (x)Hk(x/u) eX /2ag dx.

(6

Evaluate Ck+p for various values of k:

CP = j G(P) (x) ex2/2U2 dxPE/27 1()

oaPE[G(P) (x)].

00aPP+=(p+1)~A G(P) (x) d)e7X /2a2 dxup2+1)! 2d d

q(P+1) G(P1)J(x) e-x2/2u2 dx(p + 1)!V/27 G(

oaP+l1 2/2 2G(Pl)(x)e-X7 /2 dx

C -=f(p+2)!(x2)x/d

*J G(P)(X)(I I)X2 12o2 dx00

((p+2) j

(p + 2)!-\27ro'

g(p+2) f

(p + 2)! L

0) Cp+2= f(p+2) (P+2)!

G(P) (x)-2X2/2u dX

G(P+2) (x) e-X2/2a2 dX

(62)

Continuing this process, the general result is obtained as

Cktp = Ok+p EIG(k+P) (x)](63)

Appendix 11

Relation Between E[G(k+P)(x)] and Inverse BidirectionalLaplace Transform of G(x)

Consider the relations

E[G(k+P) (x)] = G(k+P) (x) e-x2/2a2 dx

(64)and

(65)

C + = U(P+1) E(G(P+l) (x)](p + 1)!

(61) where G(s) is the bidirectional Laplace transform of G(x)and c is the Bromwich contour.

~~~~~~~~~~~~~~~IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JULY 1971

G(x) = 2l JG(s) esx dsc

650

Differentiating both sides of (65) (k + p) times we get

G(k+P) (x) = f G(s)sk+P esx ds.cir

(66)

Substitution of (66) in (64) and interchanging the order ofintegration gives

E[G(k+P) (x)] [11

[2]

=2ii G(S)Sk+P ds | e-x /2a esX dx

(67)

The integral in the { } brackets of (67) is evaluated as

| esx e-x 2/2a2 dx= eS2a2/2 (68) [6]/27ro12 [7]

to give [81

E[G(k+p) (X)1 ssk+pG(s) es2ao2/2 ds (69) [9]

and

E[G(k+P) (x)] =.C'F {(s(k+P)G(s) es2o212) e-sx} (70)

which is the desired result.

References

W.B. Davenport, Jr., "Signal-to-noise ratios in bandpass limit-ers," J. Appl. Phys., vol. 24, pp. 720-727, June 1953.J.J. Jones, "Hard limiting of two signals in random noise,"IEEE Trans. Information Theory, vol. IT-9, pp. 34-42, January1963.P.D. Shaft, "Limiting of several signals and its effect oncommunication system performance," IEEE Trans. Communi-cation Technology, vol. COM-13, pp. 504-512, December1965.N.N. Lebedev, Special Functions and Their Applications,Englewood Cliffs, N.J.: Prentice-Hall, 1965, p. 71.R.L. Stratanovich, Topics in the Theory ofRandom Noise, vol.I. New York: Gordon and Breach, 1963, p. 42.D. Middleton, Introduction to Statistical CommunicationTheory. New York: McGraw-Hill, 1960, p. 1079.L.B.W. Jolley, Summation of Series. New York: Dover, 1961,p. 181.S. Nabeya, "Absolute and incomplete moments of the multi-variate normal distribution," Biometrica, vol. 48, pp. 77-84,February 1961."System/360 Scientific Subroutine Package, (360A-CM-03X),version III," Programmer's Manual H20-020503, IBM Corp.,White Plains, N.Y.

Richard W. Kelly was born in Iowa City, Iowa, on September 6, 1935. He received theB.S.E.E., M.S.E.E., and Ph. D. degrees from the University of Iowa, Iowa City, in 1958,1962, and 1965, respectively.

From 1958 to 1965 he was with the electrical engineering faculty at the University ofIowa. In 1965 he joined the electrical engineering faculty at Arizona State University,Tempe, where he is currently a Professor of Engineering Science. Since 1966 he has beena consultant to the Government Electronics Division of Motorola, Scottsdale, Ariz. Hehas done work in the areas of modulation theory, detection and estimation theory, andsignal representation.

Dr. Kelly is a member of Tau Beta Pi, Eta Kappa Nu, Sigma Xi, and the AmericanSociety for Engineering Education.

P. R. Hariharan was born in Palghat, India, on October 26, 1938. He received the B.S.E.E.degree from Madras Institute of Technology, Madras, India, in 1960, the M.S.E.E. degreefrom Kansas State University, Manhattan, in 1963, and the Ph. D. degree from ArizonaState University, Tempe, in 1970.

From 1960 to 1962 he was a member of the research staff at the Central ElectronicsEngineering Research Institute, Pilani, Rajasthan, India. From 1964 to 1967 he was aSenior Engineer in the Research and Development Department at AMECO, Inc., Phoenix,Ariz. Since 1967 he has been with The Magnavox Company, Fort Wayne, Ind., where heis presently a Staff Engineer on the Technical Staff for ASW Operations. His primaryresearch areas include general network synthesis and detection of signals after nonlineartransformations.

KELLY AND HARIHARAN: IDEAL LIMITING OF PERIODIC SIGNALS IN RANDOM NOISE 651