1. Introduction

The effect of hard limiting of an AM or CW signal setplus noise has been analyzed by Davenport.[1] Usually itis customary to limit the amplitude of a stationary angle-modulated signal set (PM or FM) plus stationary noise

The Effect of I&Iard so that the in-phase noise will not affect the demodulator.A l .In the past, treatments of the hard limiting of a sta-

Li iting an Angle- tionary angle-modulated signal set plus stationary noiseSignal have appealed to quasi-stationary angle-modulation argu-ouulateeu gna ments, i.e., that the stationary angle-modulated signal set

is a stationary CW signal set whose frequency is slowlyPlus Noise changing, and that, therefore, Davenport's results will

approximately apply. It is the purpose of this analysis todetermine the exact effect of the hard limiting of a sta-

JESS L. SEVY, Member, IEEE tionary angle-modulated signal set plus stationary noise.TRW Space Technology Labs.Los Angeles, Calif.

II. The Theory of the Transform Methodfor Nonlinear Devices

The representation of a nonlinear device by its FourierAbstract transform was first introduced by Bennett and Rice, [2] and

The effect of hard limiting an angle-modulated signal plus narrow- was first applied to the study of noise in nonlinear devices,band Gaussian noise is analyzed. Several examples are considered- more or less simultaneously, by Bennett, 3] Rice,[4] andsinusoidal angle modulation, Gaussian angle modulation, and biphase Middleton.[5] The material covered here is given in more

detail in Rice[41 and Davenport and Root.[6]angle modulation. The general conclusion is that when a zonal band-. Let y=f(x) be the transfer function of the nonlinearpass filter is used, which rejects dc and second harmonics, an angle- device. If the nonlinear device is a hard limiter, thenf(x)modulated signal plus Gaussian noise provides the same output signal- becomesto-noise ratio as shown by Davenport for a CW signal plus Gaussian

noise. However, when a narrow bandpass filter is used, which has a f1(x) a, x > O

bandwidth approximately equal to the input angle-modulated signal, A 0 x = 0 (1)an angle-modulated signal plus Gaussian noise has a better output A - a x < 0.signal-to-noise ratio than a CW signal plus Gaussian noise. This transfer function iS shown in Fig. 1. The Fourier

transform' is

Fi(jv) A_- f(x)e-jvxdx. (2)2r -00

Due to the discontinuity at the origin and convergencedifficulties, the Fourier transform is replaced by the bi-lateral Laplace transform:[9]

Fi(w) A F1_(w) + Fi+(w),where

w u + ivand

aFI(w) AJ f(x)e-wxdx =

C0 aFi(w)_A f(-x)e-wxdx =-. (3)

5 It is noted that the transform of the nonlinear device exists in aManuscript received November 30, 1966; revised February 13, 1967, and May 31, different domain than the usual linear-filter-theory transform1967. domain.

24 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-4, NO. 1 JANUARY 1968

y(t) The output autocorrelation function becomes

+0 1 +r 0C+i c+XRy(tl, t2) = (27rj) 2 jc fc joo

a, ~~~~~x(t *F1(w1)F1(W2)M_(W1, W2)dw,dW2. (9)

If the input to the nonlinear device consists of two inde-pendent sets of signals

x(t) A a(t) + b(t), (10a)x(t) HARD LIMITER y(t) then the joint complex characteristic function for xl and

Fig. 1. Transfer function of hard limiter. X2 iS

Mx(Wl, W2) = l¢a(Wl, W2)Mb(Wl, W2), (10b)

The bilateral Laplace transform becomes where

2a Ma(Wi, W2)Fi(w) = (4) is the joint complex characteristic function of a, and a2 and

and the Fourier transform is Mlb(wl, w2)

F2(ja,) = _ (5) is the joint complex characteristic function of b1 and b2.jv The autocorrelation function for the output then is

The inverse Fourier transform 1 c+joo c+jFxI ccRy 2

(27rj)2- ooFl(w)F(W2)Ma(WWj2)

y (t) = f 1(x) '_L F i(jv)e j-vdv (6)27r 0 J1lb(W1W2)dXw1dW2.1lrepresents the output of the nonlinear device. The auto- It is assumed that the input to the limiter x(t) consists ofcorrelation function of the stationary sets Jtf(x1) and an angle stationary signal set plus stationary narrow-band{fi(x2) } is given by Gaussian noise with zero mean, i.e.,

Ry(t1, t2) A E[fi(xDfi(x2)] z(t) = s(t) + n (t), (12)

1 (wC+JoUC+jOe where

(27rj)2 cJo -Joo s(t) = A cos [cw0t + f + 0(t)]

*exp (wlx, + w2x2)dw,dw2l and the set 0 is uniformly distributed between 0 and 2r.It is further assumed that the stationary signal set and the

1 f c~e+ C+YJOC noise are independent. It therefore follows that

(2I) C-Jo Mx(wl, w2) = M2(wl, w2)Mn(w1, W2), (13)-E{exp (wlxl +w2x2)}dw1dw2, (7) where

where it is assumed that M8(wi, W2)

E[f(x)] = E[f(X2)] = O, C > u is the joint characteristic function of the input signal setsand { s(t1) and s(t2) } and

Xk =X(tk), k = 1, 2. Mn1(w1, W2)

The joint complex characteristic function of xi and x2 is is the joint characteristic function of the input noise{ n(t.) and n(t2) }.

Mx(w,l W2) The joint characteristic function of a stationary Gaus-

A E; exp [w1x1 + w2x2] } sian random process with zero mean is

= rU p(X1,X2 exp [w1x1 w2x2]dx1dx2. (8)

J__OJx2) = exp {24Rn(O)w12 ± 2Rn(T-)w1w2 + Rn(O)w22] }, (14)

SEVY: HARD LIMITING OF ANGLE-MODULATED SIGNAL PLUS NOISE 25

where is the autocorrelation function of the stationary input sig-

R,,,(-r)=E[n(tl)n(t2)nal set. Also the calculation of the expectation of the sig-RZ,(T)= E[n(tD)n(t2)1. nal at T=O shows that

Now Rsm(0) = (23)

exp {IR.A(T)wiw2} = E (15) The joint characteristic function for the stationary inputk=O k signal set is therefore

Therefore, oc

Mln(wl, w2) = exp i2[Rn (O)wl2 + R, (O)w22]} M,(w 1, w2) = EEm2Im(Awl)Im(AW2)Rsm (T). (24)

00 Rnk(r)WilkW2k*k* (16) Hence the output autocorrelation function isk=O k!

x o e22

The joint characteristic function for the stationary input Ry(T) = hmk2Rnk(T)Rsm(T)7 (25)signal set is m=O k=O k!

M8,(W2, W2) = E{exp [w1A cos (,t1 + + 01) where

+ w2A COS (Wci2 + +± 02]}. (17) hmk =- f] exp-2 -w2

h.k2 7j0

w [wl ep 2-dw. (26)Using the Jacobi-Anger formula[72

00 The coefficient hmk is the same coefficient that Daven-exp [z Cos 4] E EmIm(z) cos mq5, (18) portR1] evaluates for the CW plus noise signal into a hard

m=0 limiter. Davenport shows that

the joint characteristic function for the stationary input [m + k 1signal set becomes a2k12(S/N)imI2lFL 2 ;m + 1; -(SIN)ijJIM(wi, w2) hm,k / m +k\

Rnk/2(0)F(m + 1)FK1 2 2

= > { EmEnlm(Awi)In(AW2)m=O n=O m + A odd

*E[cosm(cw,t1+ +0i) cosn(w(Wt2+4)+02)]}, (19) = 0, m + k even, (27)

where where

Ei = 1, Em = 2, iFl[b; c; -z] is the confluent hypergeometric function

(m = 1, 2, 3 * ) are the Neumann numbers. F(x) is the gamma function.

The expectation ThusEm

E[cos m(wjti + 4)+ 01) cos n(W,t2 + 4) + 02)] Ry(r) = > hmkiRnk(T)Rsm(T), m + Io odd (28)

= 1 cos [mt, - nt2)co }El cos (m01 - n02)} m=i k=O 1!-E { sin [(mt, - nt2)CO] } E{ sin (m01- nO2) } is the general form of the output autocorrelation func-

+ E{t 2cos [(mti ± nt2)cv -+- 24] } E{ sin (mOi + nO2) }tion when the input is a stationary angle-modulated signalplus stationary noise. The output autocorrelation may be

-Et 2 sin [(mtl + nt2)w0 + 24] }E{ sin (m01 + nO2) } separated into signal, noise, and signal-times-noise termsCos mwcrE{ COS m (1 - 02) } as follows:

= 2sin mw,rE{ sin m(1 - 02) } form = n 00

(O form # n R,(r) = E Em2hhm2Rsm() (output signal)m=1

= E cos [mW,T +m (01 - 02)]} (20) m odd00 ho2

Defining + E k Rok(Q) (output noise)k=1 k!

Rsm(T) = E[eos m(0ti ± 4) ± 01) k odd

*eosm(w0et2 + 4) + 02)] (21) + fE-mk2RskTRmit is seen that for m= 1 m=l_A=lk!

no-j-k odd

Rsi(T) = E[cos (A,Ctl + 4) + 01) cos (w0t2 + 4) ± 02)] (22) (output signal times noise). (29)

26 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JANUARY 1968

The output signal power and noise power for the zonal 111. Angle-Modulated Output Autocorrelation Functionsbandpass filter are given by In the previous section it was shown that the outputSO 2h,o2Rs1(O) autocorrelation function for a stationary angle-modulated

x x Em2 signal set plus stationary Gaussian noise into a hardNo ho02R.(O) + ZE hEmk2Rnk(O)Rsm(O). (29a) limiter is given by

m=1 k=1 k!m+k odd 00 EmJm-kj==1 RY(T) L= E hmk2Rnk(T)R,m(T).

m=O k=O k!If the output of the hard limiter is followed by a narrow To analyze specific cases, the autocorrelation functionbandpass IF filter of bandwidth approximately equal tothe signal, then the output signal power and noise power Rsm(T) = Elcos m [w,ti + q5 + 01]are given by *cos m[cvt2 + 4) + 02]} (33)

So 2h,02 a)C(0)C(/3)Rj(}3- a)dad/ will have to be evaluated. No fundamental distinction00cc will be made between phase and frequency modulation

00 c since they differ only by a linear operation (differentiationNo = J_ f C(a)C()4ho02R,(3 - a) or integration) on the modulating signal.

-0 -00

A. Sinusoidal Modulation

+ o EM 2 a The stationary input signal set is assumed to be+ E E k! hmk2Rk(_- a)RsmI(3 - a) dado, (29b) s(t) = A cos [wet + 4 + 0(t)] (34)

m+k oddIm-kh=1j where

where C(a) is the impulse response of the narrow band- 0(t) = D sin (w,t + a),pass filter. D is the modulation index for pulse modulation, and theThe output spectral density is given by deviation ratio for frequency modulation and the sets 4)

00 00xEm2 and a are uniformly distributed over 0 to 27r.Gy(w) = E - hmk2 The signal autocorrelation function is

2Wrm~Ok=o k!Rsm(T) E{ cosm [wti + 4 + 01] cosm [cOt2 + 4) + 02j }

f Gnk(W/)Gsm(W - w')dw', (30) =Jo2(mD) 00 J 2(mD)G00 -,,( 4 c 3)C'OSmw0T +Z

Em r==1 Emwhere * [cos (mw, + rc0)r + cos (mw, - rw,)7]. (35)

, x

G,k(w) = Rnk(r)e-i-rdr The result may be obtained by using the Bessel function-00 expansion of s(t1) and s(t2) and calculating the expectationr 0 of each product.

Gsm(Wi) = | Rsm(r)e-jtdr. (31)J_ B. Gaussian Modulation

The expanded spectral density is given by The stationary input signal set is assumed to be

s(t) = A cos [w,t + 4) + 0(t)], (36)

Gy,(w)= E Em2hmo2Gsm(W) (output signal) where 0(t) is the Gaussian process for phase modulationm=l

?n odd and is the integral of a Gaussian process for frequency00 hok2 modulation, which is a new Gaussian process, and the set

+ Z G0k (c) (output noise) 4) is uniformly distributed over 0 to 2r.k=l k! The signal autocorrelation function is given in Middle-k odd ton8 by

100 00 2

+-E E- h.k2 RIsr(T) = E{ cosm[w,t++ + 01] cosm[w0t2 + 4 + 02]}n=1 k=1 1

m+k~~~~~~~~~~ -d_ cOsmw0rexp {mIIft(r) - R(0()] }, (3'7)00 G(@)s(- c')dwt' where E

-00

(output signal times noise). (32) Ro(r) =E[0(t1)0(t2)] .

SEVY: HARD LIMITING OF ANGLE!-MODULATED SIGNAL PLUS NOISE 27

C. Biphase Random Pulse Code Modulation IV. Angle-Modulated Signal-Plus-Noise

The stationary input signal set is assumed to be of the Output Spectral Density Functionform In Section II it was shown that the output spectral

A cos [co0t + + L(t)], (38) density for a stationary angle-modulated signal set plusstationary noise into a hard limiter is given by

where 0(t) is a random switching function with values+±r/2 and the set 4 is uniformly distributed between 0 G,(w) = > Em2hmo2Gsm(W) (output signal)and 27r. m=lA useful property of the biphase random PCM signal m odd

is that Co hok'is~~~~~~~~~~~~~~~~~~~~thaE G,k( (output noise)[cos t + 4 + 6(t)] = p(t) cos [wct + 0]2 (39) k= Gk!

k odd

where p(t) is a random switching function with values 1 00 00 Em2+1. +-E E- hmkConsider the function 27rm=l k=l= k!

m+k odd

cos m[wt + ) + 06(t)]. (40) Xf Gnk(W')Gsm(co CD')dwo'

Now the values that mo(t) will assume will be

± m7r/2. (output signal times noise). (47)

For The output signal spectral density function is found by

m = 4r r = (0, 1, 2, ) taking the Fourier transforms of the output autocorrela-y ' ' ' tion functions given in Section III. For sinusoidal modu-cos m [Wct + ) + 6(t)] = cosm[W0t + 4)]. (41) lation the unfiltered output signal spectral density is

For irFor ~~~~~~~~~~~~~G,m(w)-1Jo2(iD)[3(w - mw0) -1 6(w + mwc)]

m = 4r + 1, r = (O, 1, 21. . .sm(co) =-JO2(mD) [6(@-muc) + b(cs + ]Emcosim[W0t + 4 + 0(t)] = p(t) cosim[W0t + 4]. (42) 7r JX

+ -E1 Jr2(mD) [b (w-mw - rw,)For Em r=1

m =4r+2, r =(0,1,2, *) +8(w+mwc+rw,) +3(w-mw,+rcov)cos m[cojt + 4) + (t)] = -cos m[wog + 4]. (43) + (w -+ mco - rw,)]. (48)

For Examination of the spectral density for sinusoidal mod-

m = 4r + 3, r= (0,1,2,*) ulation shows that the modulation index or deviation

cos m[0 + 4) + 6(t)] -p(t) cos m[wt ± 4]. (44) ratio is mD where D is the original modulation index or

deviation ratio and m is the order of the spectral densityThe signal autocorrelation function is term. If the bandwidth of each spectral density term is

Rs,m(T) E{ COS m[w0ti + 4 + 01] COS M[COt2 ± 4) + 62] } proportional to twice the modulation index or deviationratio, which is approximately true for a modulation index

(1 or a deviation ratio greater than five, then the bandwidth

emcos mw0r, for m evenof each spectral density term increases in direct propor-=m

^(45) tion to the order of that term. The first three terms are

I R,() cos JcfW0T for m odd shown in Fig. 2.( Em For the biphase random pulse code modulation the

where unfiltered output signal spectral density is

RP(T) = E[p(t1)p(t2)] Gsm(w) =- [ (co - mc) + 8(±+ mwj)] for m evenEm

The autocorrelation function of a random switching func- rjFT 'sin mw0] T/2 2tion with equally likely values of ±1 and constant bit GA(() = (time T iS given in Rice [4] by Em L2 \ Lw- mw]Tj2 i(9

RP(T)=1(l T )'-T<rT<T + (si [an + mwc]T/'2 21]

O, ~~-T> r> T form odd.

28 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JANUARY 1968

Bl lG lI 1 G 1 1;)y ___~~~WC_ _ _ __ _

wc-W w o c

-+. 2B .F Gs2(w) -+- 2B 1-. C C

- G 0O 22wl- 3B 3G3(w) c 3B -w - WC c

c c c WC c c~~~~~~~~~~~~~~~~~~~~C C- | l Gs3 (w)--3w-ww -2uo -°0Xc2wC3wc_3,\ 2 [ 0_1\ S_

Fig. 2. First three terms of output signal spectral density for -3w -2wC w0wJ 2c 3wsinusoidal modulation input. Gs4 (w)

-4w -3w -2wl -w 2 3wl4wlThe first four terms of the output signal spectral density 2 c c c

for random biphase PCM input are shown in Fig. 3. For Fig. 3. First four terms of output signal spectral density for random bi-the Gaussian modulation function it is assumed that the phase PCM input.noise is mutually shaped by an RC filter, i.e.,

4Ro(0)/A N Fig. 4. First three terms of the output noise spectralGN+(f) Co (50) density for Gaussian noise input.

1 + c0'/AWN-i B 1- Gn, I (wt) B1

Then the baseband autocorrelation function is nA 2 WGG2 (w) --- 2B I-

Rsm(T) = exp [-mRo(0)(1 - e Tr)] cos (51) _____C __N_______2B_2 '~~~ ~ ~~~AWN-2w 0

and the corresponding spectral density is G 3 (w) - 3B

-3w -% 0 w 3wA2e-mRo(0) oo(Re(0))n n C

Gsmn(co) - E n! 2+12 (52)more as wide as the noise output spectrums. When the

where biphase random PCM spectrum is convolved with the

c)-c output noise spectrums, the resulting spectrums will haved =

WN

the same bandwidth for m even and will be twice the band-width for m odd. When the CW spectrum is convolved

The output noise spectral density is with the output noise spectrums, the resulting bandwidth( w will be the same bandwidth as the original output noise

Gnk(X) RJ (r)-jT dr spectrums.-00

1 r 0oo o=. . . Gn(k-) V. Angle-Modulated Signal Power and Noise Power(2w)k1S°° Output of the Bandpass Limiter

X Gn^((Wk-2 -( 'k-1) ' ' ' The bandpass filter following the limiter may be eitherG0(W - Wl)dWkl dw2, (53) a zonal bandpass filter, which rejects the dc and the har-

where monics of the signal while passing a very wide zone of fre-quencies about the fundamental of the signal, or a nar-

G0(w£) = C Rf(T)ewXrdrc row bandpass filter similar to the IF bandpass character-00

istic.

Davenport's analysis of the bandpass limiter for a CWThe first three terms of the output noise spectral density signal plus noise uses a zonal bandpass filter to filter the

for Gaussian noise input are shown in Fig. 4. The output output signal and noise terms. Consider the comparisonsignal times noise spectral density is given by between the CW signal and noise output and the angle-

1 °° x EM2 oo modulated signal and noise output for a zonal bandpassZE E--hk2 Gnk(W-)Gsm(W-W'/)dw'. (54) filter following the hard limiter. The filtered signal-times-

2rm-lk-odd_0 signal output power for the CW signal and the angle-modulated signal are equal, since the average power in

When the sinusoidal angle-modulated output spectrums both signals is the same. The noise-times-noise outputare convolved with the noise output spectrums, the result- power for the CW signal and angle-modulated signal areing spectrums will have bandwidths that will be twice or equal, since the noise-times-noise terms are identical. The

SEVY: HARD LIMITING OF ANGLE-MODULATED SIGNAL PLUS NOISE 29

signal-times-noise output power for the CW signal and put of the zonal bandpass, since again the noise onlythe angle-modulated signal are equal since average power partially passes through the narrow-band bandpass filter.of the terms of each case is equal. The signal-times-noise The narrow-band bandpass limiter is not analyzed forspectral densities have a much wider bandwidth for the the CW or angle-modulated signal since the output signal-angle-modulated signal than for the CW signal; however, to-noise power is dependent on a specific system. Thethe very wide zonal bandpass filter will pass these terms. specific system parameters are the type of signal involved,Even though the autocorrelation and spectral density the modulation index or deviation ratio, and the band-functions for the CW signal and angle-modulated signal widths of the input signal and noise and output bandpassdiffer in appearance, if a wide enough bandpass filter fol- filter.lows the limiter, the average signal and noise powers willbe equal. Therefore, for a zonal bandpass limiter, the out- Conclusionsput signal-to-noise ratio versus input signal-to-noise The output signal-to-noise ratio of a zonal bandpassratio for the angle-modulated signal and noise will be the filter is the same for an angle-modulated signal and asame as for a CW signal and noise, as analyzed by Daven- CW signal. The output signal-to-noise ratio of a narrow-port. When the bandpass filter has approximately the band bandpass filter will be reater than the output signal-bandwidth of the input signal, the angle-modulated signal tono bandpass filter for anal-will yield a higher output signal-to-noise ratio than the to-nolserate o of a zonal bandpass filter for an angle-CW signal, where it is assumed that the same bandpass modulated signal and a CW signal. The output signal-to-filter is used for the CW signal. noise ratio for an angle-modulated signal is greater thanWhen the bandpass filter has a narrow bandwidth, the the output signal-to-noise ratio for a CW signal if they

signal-times-signal output power is equal for the CW sig- are passed through a narrow-band bandpass filter.

nal and angle-modulated signal. The noise-times-noise The reduction of noise by using a narrow-band rathern

u wsalandthe angle-modulatedsignal.aThenoise-t -noeC than zonal-band bandpass limiter is due to the higher-output power IS equal for the angle-modulated and CW odrniesetu em pedn nbnwdh hsignals and is less than noise-times-noise output power order noise spectrum terms spreading in bandwidthu Thefor the zonal bandpass filter case, since the higher-order reductilon of nose of an angle-modulated signas over a

noise terms have their spectrum spread and, therefore, CW sgnal In a narrow-band bandpass limiter is due to

only partially pass through the narrow bandpass filter, the fact that the spread spectrum of the angle-modulatedThe signal-times-noise output power for the angle-modu- signal spreads the noise spectral terms.lated signal is less than the CW signal since the spectrum REFERENCESof the angle-modulated signal spreads2 the noise spec- t1l N. B. Davenport, Jr., "Signal to noise ratios in bandpasstrums even wider, and therefore less noise passes. The limiters," J. Appl. Phys., June 1953.signal-times-noise output power of the narrow bandpass [2] W. R. Bennett and S. 0. Rice, "Note on methods ofcomput-

ing modulation products," Phil. Mag., September 1954.filter for a CW signal is less than the corresponding out- [3] W. R. Bennett, "Response of a linear rectifier to signal and

noise," J. Acoust. Soc. Am., January 1944.I4] S. 0. Rice, "Mathematical analysis of random noise," Bell

Sys. Tech. J., vol. 23, 1944, and vol. 24, 1945.2 The nonlinear device increases the deviation ratio or modula- (5] D. Middleton, "The response of biased, saturated linear and

tion index of the angle-modulated signal in direct proportion to the quadratic rectifiers to random noise," J. Appl. Phys., October 1946.order of that product and thus spreads the spectrum. In the noise [6] W. B. Davenport, Jr., and W. L. Root, Random Signals andcase, the spectrum is spread since the nonlinear device multiplies Noise. New York: McGraw-Hill, 1958.the noise times itself in direct proportion to the order of that prod- t[' W. Magnus and F. Oberhettinger, Functionis of Mathematicaluct. For the signal-times-noise terms, the increased modulation Physics. New York: Chelsea, 1949.index of the signal (spread signal) multiplies times the noise terms (8] D. Middleton, StatisticalCommunlicationTheory. NewYork:and hence spreads the spectrum of the signal-times-noise terms. McGraw-Hill, 1960.

Jess L. Sevy (M'60) was born in Salt Lake City, Utah, on May 24, 1934. He receivedthe B.S. degree in mathematics from the University of Southern California, Los Ange-les, in 1956. He has pursued graduate studies in mathematics and engineering at theUniversity of Southern California and at the University of California at Los Angeleswhere he received the M.S. degree in engineering.

In the last eleven years, he has worked in the fields of computers, controls, struc-tures, missile and space telemetry, and radar and space communications, at the HughesAircraft Company, the Aerospace Corporation, and presently at TRW Space Tech-nology Laboratories, Los Angeles, Calif.

30 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JANUARY 1968