hard limiting of two signals in random noise

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34 IEEE TRANSACTIONS ON INFORMATION THEORY January

Hard-Limiting of Two Signals in Random Noise*J. J. JONESt, MEMBER, IRE

Summary--Two sinusoidal signals and Gaussian noise lying ina narrow band are passed through an ideal band-pass limiter thatconfines the output spectrum to the vicinity of the input frequencies.The output spectrum, consisting of both discrete and continuouscomponents, is studied in terms of its corresponding autocorrelationfunction. The discrete output components are identified with theoutput signals and intermodulation products due to interferencebetween the two input signals. The continuous part of the spectrumis associated with the output noise. The effects of limiting areexpressed by ratios among the average powers of the output spectralcomponents. Performance curves are given that show signalsuppression, the ratio of output to input SNR’s, and the relativestrength of the intermodulation terms.

I. INTRODUCTION

-u

SING THE characteristic function method ofRice,l DavenportZV3 studied a power-law, band-pass limiter fed by a sinusoid and a narrow band

of Gaussian noise. In particular, he gave numerical resultsfor the ratio of the output SNR to the input SNR thatis produced by an ideal or hard limiter and showed thatthis ratio lies between 7r/4 and 2. B1achman,4 using aFourier expansion approach, studied this same problemand obtained the result for the hard limiter in a moreconcise form than the formulation of Davenport. Thenoiseless case of two sinusoids passed through an idealband-pass limiter was investigated by Granlund’ andlater, Baghdady.’ Again using a Fourier representation,they obtained the amplitude relations among the outputsignals and the intermodulation terms that result frominterference between the two input sine waves. In thispaper, ’ we study the effects of ideal band-pass limitingupon two sinusoids lying in a narrow band of Gaussiannoise. This problem becomes important, for example, inthe study of the effects of limiters in long-pulse (e.g.,pulse compression) radars, where close lying echoing

* Received March 30, 1962.i Applied Research Laboratory, Sylvania Electronic Systems,

Waltham, Mass. (A Division of Sylvania Electric Products, Inc.)1 S. 0. Rice, “Mathematical analysis of random norset” Bell

i3.w. Tech. J., vol. 23, pp. 282-332, Julv, 1944: vol. 24, pp. 46-156,January, 1945. ^ ̂

“. _-

2 W. B. Davennort. Jr.. “Sianal-to-noise ratios in band-nasslimiters,” J. Appl. ‘Phyk., vol. 24;~~. 720-727. June, 1953. )

3 W. B. Davenport, Jr. and W. L. Root, “An Introduction tot,he Theory of Random Signals and Noise,” McGraw-Hill BookCo., Inc., New York, N. Y., pp. 277311; 1958.

4 N. M. Blachman, “The output signal-to-noise ratio of a power-law device,” J. Appl. Phys., vol. 24, pp. 783-785; June, 1953.

5 J. Granlund. “Interference in Freauencv-Modulation Recen-tion,” M.I.T. Res. Lab. of Elect,ronics,& Cambridge, Mass., Tech.Rept. No. 42; January, 1949.

6 E. J. Baghdady, “Interference Rejection in FM Receivers,”M.I.T. Res. Lab. of Electronics, Cambridge, Mass., Tech. Rept.No. 252; September, 1956.

‘i J. J. Jones. “Hard-Limiting of Two Signals in Noise.” Svlvania

Electronic Systems Appl. R&. Lab., Watham, Mass.,” ARMNo. 277; November 30, 1961.

objects result in substantial pulse overlaps, giving riseto an interference problem.

II. STATEMENT OFTHE PROBLEM

The essentials of the system to be studied are shownin Fig. 1. The input process to the limiter

d4 = s,(t) + s,(t) + 77(t) (1)

consists of two signals and a narrow band of randomnoise. Each signal is of the form

s,(t) = x4& cos (CL&t 4,) (P = 172) (2)

whereas o(t) is a sample function of zero-mean, stationary,Gaussian noise. The random phase angles 9, are consideredto be statistically independent with a uniform distributionover the interval (0, 2~). It is assumed that the inputsignals and noise, prior to limiting, pass through a sym-metrical band-pass filter (center frequency w,) thatdetermines a symmetrical narrow-band spectrum for thenoise. As indicated in Fig. 1, the ideal limiter is describedby its amplitude characteristic g(z), with which the limiteroutput y(t) can be expressed uniquely in terms of theinput as

y(t) = g[x(t)] ={‘!I if x(L)[:i. (

The limiter is followed by a band-pass filter (also centeredon w,) that confines the output spectrum to the funda-mental band.

BAND-PASS FILTER

Fig. I-An ideal band-pass limiter showing an input composedof two signals in random noise.

Using Davenport’s technique derived from Rice, it isshown in Appendix I that the autocorrelation functionR,(T) of the limiter output is given by

where co = 1 and E, = 2 (m > 0). The coefficients hkmnin (4) are defined by the relation



Jones: Hard-Limiting of Two Signals in Random Noise 35m(k+m+n-l)/zs vk-l Jm(v xf2S,) J&J 42X,)

0

hkmn= dv for k + m + n odd (5)

for k + m + n even

where J,( ) is the mth-order Bessel function of the firstkind. The condition that the hkmn are nonzero only forodd integral values of the sum k + m + n is a conse-quence of the odd symmetry of the limiter’s amplitudecharacteristic. This forces the limiter output to havespectral components lying only in the vicinity of oddharmonics of the input frequencies. However, besidestheir functional dependence on the indices k, m, and n,these coefficients are also functions of the input powerratios through their dependence on the input powers S,,X,, and N. Also in (4), the factor R,,(T), which is theautocorrelation function of the input noise, may bewritten’ as

R,(T) = NP(T) cosw,r (6)

because of the assumed symmetry of the narrow-bandnoise spectrum about w,. The envelope function p(7) in (6)is considered to be normalized to unit power; i.e., p(O) = 1.

III. AUTOCORRELATION FUNCTION OFTHE FILTERED OUTPUT

The first step to be taken in determining the auto-correlation function of the filtered output is to furtherexpand (4). Upon introduction of (6) into (4)) the resultingexpression contains the factor cask W,T for which it isappropriate to employ the identities9

COSkW,T

- cos (k - 2i)w,7 + (7)

g Ikg’* (k J)! i! cos (k - 22]w,r

for k odd

Following the insertion of (6) and subsequently (7) into(4), the autocorrelation function of the unfiltered outputbecomes

(k-1)/2

’ ~ (k _li)! i! ‘OS (k - 2i)w,~

cos mwlr cos nu2r @a>

8 Davenport and Root, op. cit., p. 169.9 G. Chrystal, “Algebra, Vol. II,” Chelsea Publishi ng Co.,

New Yor k, N. Y., p. 278; 1959. See also Davenport and Root,op. cit., p. 298.

for k odd and m + n even, and is given bym m m

R,(T) c c c %&ALPk(T)k=O.2. m=O n=O

n+TL=1.t3..*.

*cos m&r cosnw~r @b)for k even and m + n odd where we have introducedthe relation

By reduction of the cosine products in (8), a completeexpansion of R,(T) is achieved, but the resulting expressionis too ponderous to be given here. It remains for one toseparate out of this expansion those terms with fre-quencies lying within the first harmonic band. Examinationof (8), under the assumption that both w1 and wa differby only a small amount from w,, shows that the onlycombinations of Ic, m, and n which shift spectral com-ponents into the vicinity of w, are the following:

(m&n 1 =

1

072J4J . .. k + 1 for k odd(10)

1,3,5, f+* k + 1 for k even

Values of k, m, and n which satisfy (10) also satisfy therequirement that their sum is an odd positive integer.For each value of j m f n 1 given in (lo), it is possibleto write down as a subtotal the sum, over even or oddvalues of k, of all terms that contribute to the filteredoutput for those particular values of m and n. For example,for j m f n j = 0, these terms are given by

,=$ ,g 2 * P”(T) cos [(Z - 1)WZ ZwJ]r. .

( >!

and for 1 m f n 1 = i (i > 0) we have

qL II-ii-11os Liwc + (I - i + 1) ~~ - ZWJT

+ Gz, IZ-i-11 cos [iw + (I - i - 1)~~ - lwJ7}

(11)

(159The totality of all such terms may be written in a singlecompact form as the autocorrelation function R,(7) of theJiltered output z(t). This form is given by

R,(T) 2 5 2 2 ~~IZIIZ-li+llli=-m z=-~ ~=I~I.I~I+z. (k ; Ii!>! (k ; Iii)! ‘ “(‘)

.cos[Iijw, - I i + 1 I % + Z(wz - Wl)]T. (13)From (13) the spectrum of the filtered output is seen

to contain both discrete components (due to periodicterms in R,(r) for which k = 0) and continuous com-



36 IEEE TRANSACTIOhTS OhT INFORMATION THEORY January

ponents (corresponding to nonzero values of k). Perhapsthe clearest way to indicate the spectral content of z(t)

is to write in the notation of Davenport

R,(T) = %,xs,(d + %x,(-r) + %,X,(T)

+ %,X,(T) + %xszxd7?). (14)

In this notation, all of the periodic terms contained in(13) are represented by the factor R,,X,,(~). These com-ponents include terms at both of the signal frequenciesw1 and w2 and at all of the interference frequencies

mw, - nw, 1 where m - n = f 1. The complete set ofperiodic terms is obtained from (13) by setting k = 0.This set is given by

where, in particular, the terms at the signal frequencies are

and2&, COS WIT

(16)2?,& COS WQT.

direct feed-through noise. This set of components is ob-

The remaining four sets of terms in (14) for which k > 0correspond to the continuous part of R,(T) and are asso-ciated with the output noise. For example, the factorR,,,(T) represents the autocorrelation function of the

IV. FUNCTIONAL EVALUATION OFTHE bkrnn

Since the coefficients hkmn appear squared in theautocorrelation function, the sign controlling factorC-1) (k+nz+n-1)‘2 n (5) is of no consequence; therefore, itwill be dropped. Furthermore, the hkmn were redefinedby (9) in terms of the bkmn. Thus, it is appropriate tostudy the latter quantities defined by

b mn =

vk-’ J,(v v”%$ J,(v w%S,,

dv for k + m + n odd 0%

for k + m + n even

A solution to this integral has been given” as

bkmn

.,F,

functionll**’ defined by

-i, -i -m;n + 1; 2 for k + m + n odd (20)1>

where 2Fl(a, b; c; x) is the Gaussian hypergeometric

tained from (13) under the condition m = n = 0 andis given by

The factor R,,,, (T) denotes the autocorrelation func-tion of those noise components that result from inter-action of signal No. 1 with the input noise and a cor-responding remark applies to the factor R,2X11(~)or signalNo. 2. These terms are obtained from (13) by settingn = 0 for the first set (m = 0 for the second set) and thesum of the remaining two indices takes on all odd positiveintegers except 1 (i.e., neither index may be 0). Thefirst set is given by

&,x&d 2 22---m k-lil.li+Z!, k + 1 1 1

i#O,-1;’k,s+uoPkb) ’

( 2 z )!(c-2 z )!

*COS[IilW, - Ii+ 1 jWl]T (18)

and a similar expression holds for the second set.The factor RslXsnXI(~)is the autocorrelation function

of those noise components that are produced by bothsignals interacting together with the input noise. Forthese terms the sum k + m + n takes on all odd-positiveintegers except 1 and all three indices are nonzero. Since(13) with appropriate conditions is the most convenientform of expressing this set of components, it will not be

repeated here.

. . . (21)

the solutionnd I?( ) is the Gamma function. This form ofis useful for numerical computations because the hyper-geometric function in (20) terminates aft’er the i + 1stterm due to the parameter a = -i.

Another form of the solution to (19) can be obtainedby replacing J,(v 6) by its power series expansionand interchanging the order of summation and integration.This results in the form

bm, =

(+l)i+m’2i! (i jf m)!

2[S-- v2i+k+m-1

a 0J,,(vv’%%$ exp (-: u’) dv] (22)

where the integral in brackets has been attributed to

10 Staff of the Bateman Manuscript Project, “Higher Transcen-dental Fund tions, Vo!. II,”York, N. Y., p. 49; 1953.

McGraw-Hill Book Co., Inc., New

11Staff of the Bateman Manuscrint Project, “Higher Tran-ons, Vol. I,” M&raw-Hill Book Co.. Inc..cendental Functi

New York, N. Y., p. 56; 1953.12W. Magnus and F. Oberhettinger, “Formulas and Theorems

for the Functions of Mathematical Physics,” Chelsea PublishingCo., New York, N. Y., p. 7; 1954.



1963 Jones: Hard-Limiting of Two Signals in Random Noise 37

Weber and Sonine.‘3 The solution to (22) may now bewritten as

. 2 (-l,i($gri+lc+m+n

i=o z! (i + m)! ( 2 )

;n+l;-$

for Ic + m + n odd (23)

where lFl(a; c; x) is the confluent hypergeometric func-tion14 defined by

a(a + 1) 2’= 1+;++ c(C+l) y+ ***. (24)

This form of the solution is convenient for theoreticalmanipulations as in the computation of limiting con-ditions. Notice that the only difference between the twosolutions (20) and (23) is in the type and the parametersof the hypergeometric function that is involved.

V. SOME NUMERICAL RESULTS

The spectrum of the filtered output was shown to con-tain discrete components corresponding to the outputsignals and intermodulation terms, and to have a host ofcontinuous components that collectively make up theoutput noise spectrum. The total power contained inthese discrete components is given by (15) evaluated atT = 0. Thus, the noiseless output power is

and, by application of (20), the output signal powers aregiven by

and12

(261

)I. (27)

13Bateman Manuscript Project, op. cit., Vol. I, p. 56. See alsoMagnus and Oberhettinger, op. cit., p. 35.

14 Bateman Manuscript Project, op. cit., Vol. I, p. 248. See alsoMagnus and Oberhettinger, op. cit., pp. 86-87.

Another important term in (25) is the spectral contri-bution at the frequency 2w, - w2. The power containedin this term is

,(i + i),,(-i,-i - 2;2;g)/

‘. (28)

When the ratio (LQ’S,)~ I 1, this quantity is the powercontained in the strongest intermodulation product.

The output noise power (N)O is the total power con-tained in the continuous part of the spectrum and isgiven by

0% = R,,,(O) + %x&O) + &x,(O) + Rs,x,,x&X. (29)

However, the output noise power may also be expressedas the difference between the total power in the funda-

mental band and the noiseless power as given by (25).In general, the total power contained in the kth harmonicband of the limiter output is a constant regardless ofhow the power is distributed within the band. This poweris equal to the power of the kth harmonic in a Fourierseries expansion of a rectangular waveform which has itspeak-to-peak amplitude determined by the limiter (vix., 2).Thus, the total power in the l&h-odd harmonic band is8/(&)‘. In place Iof (29) we have

WI, = $ - R,,x,,(O). (30)

The output signal and noise powers including the inter-modulation term of (28) were evaluated for a wide rangeof input power ratios by computation on a Recomp IIdigital computer. The data thus obtained was used toplot the performance curves given in Figs. 2, 3, and 4.

The first set of curves, Fig. 2, shows the effect on signal-to-noise power ratio produced by the ideal band-passlimiter. The ratio of the output SNR (&IN), to the inputSNR (L&/N); is plotted as a function of the latter. Theinput signal-to-signal power ratio (S,/SJi is a parameterfor these curves. Although the curves were plotted forsignal No. 1, they apply to signal No. 2 as well. To obtainthe ratio (X,/N),/(&/N>, for given values of the inputratios (S1/N)i and (SZ/S1)i, multiply these two numbersto obtain the ratio (X2/N)i and then enter Fig. 2 at thisabcissa value. The normalized SNR for signal No. 2 liesat the intersection of this abcissa value with the curveplotted for the reciprocal of the given value of (X,/X,)i.

From Fig. 2 it is seen that the ratio of output SNRto input SNR lies in the range 0 to 2 for two signals innoise as compared with the range rr/4 to 2 for one signalin noise. The uppermost curve in Fig. 2, which is plottedfor the condition that signal No. 2 is much weaker than




Fig. 2-The ratio of the output SNR to the input SNR as a functionof the latter for the ideal band-pass limiter.

I Il- I

1 .o - I' Il.0

0.81 \

I- I 1-1 .- ,- --- 3

(Sl/N)iFig. a--The ratio of the output signal-to-si gnal power ratio to

the input signal-to-signal power ratio as a function of the largerinput SNR.

s2/Sl) = INPUT SIGNAL-

(S1/N)Fig. 4-The power ratio of the strongest intermodulation product

to weakest output signal as a function of the larger input SNR.

signal No. 1, is the same performance curve as given byDavenport for one signal in noise. Under the conditionof strong noise dominating the limiter, the SNR’s ofboth signals are decreased by the factor ?r/4 which isa loss of about 1 db. But, when one signal is much strongerthan both the noise and the other signal, the SNR ofthe strong signal increases by the factor 2 for a gainof 3 db whereas the SNR of the weak signal decreases

by the factor l/2 for a loss of 3 db. However, when the twosignals are of equal strength and the noise is weak, thenormalized SNR tends to zero for both signals. That is,the equal output SNR’s are large compared with unity,but are much smaller than the large and equal inputSNR’s such that the output over input ratio is a smallnumber. This point is considered in more detail in Sec-tion VI.

The second set of performance curves, given in Fig. 3,shows the relative suppression of one signal by the otheras influenced by the noise level. The ratio of the outputsignal-to-signal power ratio (S,/S,)O to the input signal-

to-signal power ratio (AS’,/S,)~ is plotted as a function ofthe input SNR (AS’,/N)~ with (L‘$/X,)~ as a parameter.It can be seen from these curves that when the signalsare buried in noise, the limiter is essentially a lineardevice in the sense that little or no suppression of onesignal by another takes place (i.e., the relative signalstrengths are preserved). This condition is approximatelyin effect when the larger of the two input SNR’s is - 10 dbor smaller. On the other hand, the maximum degree ofsuppression of a weak signal by a strong signal occursat large input SNR’s and is seen to be 6 db. Thus, thesmallest possible signal-to-signal power ratio at the output

of an ideal band-pass limiter is l/4 of the correspondinginput ratio. The asymptotic values approached by thecurves of Fig. 3 at large input SNR’s are equivalent to aportion of the Granlund-Baghdady results.

The set of curves given in Fig. 4 expresses in power therelationship of the strongest intermodulation product tothe weakest output signal. The output power ratio(L!&‘&),, is plotted against the input SNR (S,/N)i withthe ratio (X,/AS’,)~ as a parameter. Under the condition(S,/S,)i 5 1, the quantity (S,,)0 is the power of thestrongest intermodulation term, which occurs at thefrequency 2wl - w2,and (A’,), is output power of theweakest signal. These two output components are ofequal strength when the strong signal dominates both thenoise and the weak signal, but when the input signalstrengths are equal and much stronger than the noise,the strongest intermodulation term is only l/9 of eitherof the equal output signal powers. However, the strengthsof all intermodulation products fall off rapidly with de-creasing SNR. For example, when the larger of the twoinput SNR’s is unity, the strongest intermodulation termis more than 12 db below the weakest signal and decreasesat a rate of about 2 to 1 db for smaller SNR’s.



196.3 Jones: Hard-Limiting of Two Xignals in Random Noise 39

VI. ASYMPTOTIC FORMS FOR Two SIGNALS As an interesting application of the above results,IN WEAK NOISE consider the situation which occurs when both input

When one or both of t,he input signals rise substan- signals are of equal strength and are much stronger than

tially above the input noise level, computation of the the input noise. Eq. (33) applies when X, = X, -+ 0~

bkmn oefficients based on either (20) or (23) breaks down such that (LS’,/N)~ = (&IN)< -+ m . From (33) it is seen

and it becomes necessary to employ asymptotic forms. that the equal output signal powers t’ake on the limiting

In Appendix II, asymptotic forms are derived for the valuebkmn or use in the presence of weak noise. The first of

(SJO = (SJO = 2G,, = 5 & = $these results is given as

for ($ = ($ -+ m (34)bkmn

rk+m+n( 2 )

m-j-n k-i-m-n’ 2

;m+l;$2

Ic=O,l

and a complementary result is

b

F k-l-m-l-n k-i-n-m‘2 1( 2 ’ 2 ;n + l;$

1>

and the equal powers of the strongest intermodulationcomponents approach the value

(S,,), = (S,,), = 2b2,,1 = -$ ’ = $

(31)r2 g r2 3

0 0

for (J$); = ($)i -+ co. (35)

Under these conditions, the entire discrete spectrum of

the limiter output is symmetrical about the frequency(a1 + wZ)/2 and the power of each intermodulationcomponent is proportional to the reciprocal of a squaredodd interger.16 Thus, the total power contained in allthe a1 X s2 terms is given by

for

Note the restrictions stated in (31) and (32) on the inputsignal-to-signal ratio and on the index k. These conditionsare imposed by the convergence propert ies of the zFIfunction. Finally, for the special case of equal input signalstrengths we have

b ’‘\ 2 J

kmn= -52 T + l)r(y +l)rtT + 1)

for ($); = ($), + QJ (36)

as expected.Unfortunately, the behavior of the output noise power

for the case of equal input signal strengths and weaknoise cannot be obtained directly, since (33) does nothold for k = 1. However, a computer study of (32) forinput signal-to-signal ratios close to unity shows that theoutput noise power behaves as

,,J(i!Ji = (& co (33) o,ia($)i-o for (+($Ji* me (37)

Ik=OThe further restriction on k stated in (33) is due to thespecial conditions under which (31) and (32) may beevaluated for S, = S,. In essence, the above three resultswith k = 0 were given by Davenport and Root? for theinterference study of two signals in the absence of noise.For k = 0 these results are also equivalent to the Gran-lund-Baghdady results. However, (31) and (32) are shownhere to hold also for k = 1 and as such are useful in de-termining the behavior of the output noise when one orboth strong signals control the limiter.

15 Davenport and Root, op. cit., pp. 308-309.

The proportionality quantity a! is a number which growslarge as (S2/S1)i -+ 1, but is always considerably lessthan (S,/N)i, so that the output noise power tends tozero. Under these conditions, the ratio of the output SNRto the input SNR tends to zero; i.e., from (34) and (37)we have

because a! is a large number.Some simple physical reasoning may serve to explain

16 Note that this result w as not given by either Granlund orBaghdady since it was unattainable through their analyses.




this phenomenon. When the input contains two strongsignals of equal strength that are close in frequency, thecombination may be thought of as a double-sideband,suppressed carrier signal with sine wave modulation atone half of the difference f requency, and a carrier fre-quency at one half of the sum frequency. The first dele-terous effect is due to the sine wave envelope functionof the combined signals which crosses the time axis at

the difference frequency. In the vicinity of each zerocrossing of the envelope the noise, which is momentarilystronger that the signals, controls the limiter and con-tributes heavily to the output noise. Thus, while theinput signals are beating destructively toegther, the noisehas an opportunity to slip t’hrough the limiter. Thisexplains why at large input SNR’s the output noise,though small in value, is much larger for signals of equalstrength than for a situation where a considerable im-balance exists between the signals. The second effect isthat the limiter fixes the equal output signal powers atabout 40 per cent of the total output power with theremainder distributed among the intermodulation pro-ducts. The net result under these conditions is that theoutput SNR’s are constrained to nonzero equal valueswhich are large relative to unity, but are necessarilymuch smaller than the large input SNR’s. Consequently,the normalized SNR tends to zero for this situat’ion.

VII. CONCLUSION

The effect of ideal band-pass limiting upon two signalsin random noise has been demonstrated in terms ofpower relations among the output spectral components.One important conclusion to be noted from these resultsis that under certain conditions the SNR with two signals

present may be degraded well below that which is en-countered with only one signal. The worst case in whichsuch degradation takes place is when two equally strongsignals are present in relative weak noise. For this situationthe output SNR tends to a very small fraction of theinput SNR. A secondary point to not’e is the absence ofsignal suppression at small input SNR’s and the maximumsuppression of 6 db (in power) that occurs at large SNR’swhen a substantial imbalance exists between the signals.

The emphasis placed here on SNR may or may not bejustified depending on the intended application of thelimiter. It has been pointed outI that SNR is importantwhere fidelity of transmission is of concern, but whenthe presence of the signal is not a foregone conclusion,then signal detectability is of primary concern. The factthat an SNR enhancement by a factor as large as 2 occursat high SNR’s does not imply that a band-pass limitercan improve signal detectability. A loss in signal detect-ability is always experienced with a limiter but the losscan be made to approach a small value at low input SNR’s.Manasse et aZ.17have shown that it is possible to operate

I7 R. Manasse, R. Price,, and R. Lerner, “Loss of signal detect-ability in band-pass liml ters,” IRE TRANS. ON INFORMATIONTHEORY,vol. IT-4, pp. 34-38; March , 1958.

a limiter such that the detectability of a signal is essentiallyunchanged even though the SNR decreases.

APPENDIX I

AUTOCORRELATI ON FUNCTION OF THE LIMITER OUTPUT

The autocorrelation function of the limiter output y(t)is defined as the statistical average

G/(4 t’> = Ez~d~(~)lsL4Ul) (39)where we have written t’ for t + T. It has been shown”that the amplitude characteristic as described in (3) forthe ideal limiter may be represented as a sum of contourintegrals in the complex w plane (w = u + jv). Thisrepresentation is given by

d4=& s,,xp (WX) g + 1 s7v c-xp (wx) g. (40)

The contour of integration c, always encloses the originof the w plane to the left whereas the contour c- always

encloses the origin to the right. Both contours includethe entire w = jv axis except for appropriate avoidancesof the origin due to a pole there. For brevity and sincethe integrands of (40) are identical, let g(x) be momen-tarily represented by a single contour integral over thecontour c. Introducing this representation for g(x) into(39), the autocorrelation function of the limiter outputmay be writt#en as

dwR,(t,t’) = & c $s

-s%Ez expvi[w,x(t) + wt4t')l]. (41)

The statistical average indicated in (41) is definedIgas a two-dimensional joint characteristic function andis written

M,(w,w~) = E,iexp [wdt> + w,x(t'>l). (42)

Since x(t)s the sum of zero-mean, statistically independentrandom processes, the joint characteristic function Mz maybe written” as the product of the individual joint charac-teristic functions M,<; i.e.,

MAW, w,,> = fi Mz<(wt, wt,>i=l

= a E,,{exp [wx,(t> + wt4'II. (43)

The xi may be identified with the individual terms in (1).It can be shown” that the joint characteristic functionfor stationary Gaussian noise is

M,,(wt, w,,) = exp[;(w: + ~9,) + R~(dw~wr.] (44)

I8 Davenport and Root, op. cit., pp. 277-282.I9 Davenport and Root, op. cit., pp. 52-54.2o Davenport and Root, op. cit., pp. 149, 289.



1963 Jones: Hard-Limiting of Two Signals in Random Noise 41

where R,(T) is the autocorrelation function of the input reduces to a single integral. Passing to a real integral,noise and r = t’ - t. Further, it can be shown21 that the definition of the hkmnbecomesfor each of the sine waves the joint characteristic functionhas the form hkmn= i C-1)

(k+m+n-l)/z

M,,(wt, w,,) = 2 GJ,(wt~~~n=O.I,(w,,/2xi) cos muiT. (45)

Here I,( ) is the mth-order modified Bessel function ofthe first kind and the E, are the Neumann numbersdefined above.

Using the series expansion for the exponential

exp Ru(4wtwt,l= g $ [%(dwtwt~lk

in (44) and inserting the result into (41), the autocorrela-tion function of the limiter output takes the form

In order to simplify (46) after insertion of the expressionfor MS;, we make the following definition. Let

h1- --

kmn 27d

. ws k-l~,(w~)II,(~~) exp (47)c

vkel J,(vv’%f$J,(v&!?$ exp (49)

It may be of interest to note that had a smooth limiter

been used with its amplitude characteristic described byan error function curve

exp (-x2) dx for x > 0

-9+(-x) for x < 0(50)

the only effect this change would have on the end resultwould be to augment N in (49) by a’. The quantity a isa number which determines how fast saturation is ap-

proached in the smooth limiter and, when a = 0, thesmooth limiter reduces to the hard limiter.

Returning now to (49), the integration along the nega-tive portion of the jv axis may be expressed (after a changeof variable) in the form

s [ ] dv = - (-l)k+m+n /- [ ] dv. (50-m 0

Eq. (46) may now be written compactly as Applying (51) to (49), the hkmllbecome

hkmn= ; (- 1) (k+m+fi--1)/2[1 (- l)k+m+y

f cos mw,r cos m2i-. (4% mAs pointed out above, the integral in (47) is actually ’ o

vkml ,(v a) J,(v 4%) exp dv (52)

the sum of two contour integrals, where the contoursof integration avoid the origin of the w plane because of where the bracketed quantity in (52) has the valuea possible pole at that point. However, the occurrenceof a pole there may be prevented if either k, m, or n is 2 for k + m + n odd

nonzero. It is clear from (47) that the condition k > 11 - (-l)ki-m+n = - (53)

eliminates a pole at the origin. However, if 1~ = 0, the 0 for k + m + n even

condition m or n > 1 also prevents a pole there becauseof the limiting behavior of the Bessel function for smallvalues of the argument; vz’x.,

P

I,(w) -3 -?- for 11) + 0.2pp!

Thus, no pole will occur at the origin if at least one ofk, m, or n is nonzero. By this restriction, the dc or zero-frequency term is eliminated from the autocorrelationfunction, but this is no loss since the inputs to’the limiterhave been assumed to have zero mean. The advantagegained by this restriction is that both contours of inte-gration may now use the jv axis in its entirety and (47)

21Davenport and Root, op. cit., p. 290.

Thus, the final defining form for the hkmn s

Ii mvk-l J,(v a) J,(v 4%)h 0kmn=

dv for L + m + n odd* (541

for lc $ m f n even

Note that the condition for a nonzero hkmn; viz., k +mfn = 1,3,5 e-e, includes the condition under whichthe occurrence of a pole at the origin is prevented.




APPENDIX II

WEAK NOISE ASYMPTOTIC FORMS FORTHE hmn COEFFICIENTS

$

l?(m + l)r ( i + k+Y+nri+ >(lc+m-n

2 >

r(i+ mf i)r

(k-km-l-n

2 )(rlc+m-n

2 )Consider the case of two very strong signals in weak

noise. Let S, and S, -+ *, such that (S,/N)i ands i

( >-A

(S2/N); + 03, but for the moment place no restrictions2

*i! @O>on the ratio (S,/S,)i. For very large values of the variable

2, 1F, assumes the limiting form”r(+-a

but the sum in (60) defines a ,F, function. Thus, we maywrite (60) as

2’da; C; - X> -+ cc- 4for

Thus, the hypergeometric function inthe form

c;n+l;-%

>

x-+ a. (55)

(23) approaches bkmnr

-i-(k+m+n)/2

*- (56)(S./N)i-m r

In terms of (56), we may rewrite (23) as

\ Y I

k-l-m-n . 81>

[:j,2< 1; I=+O,ll! (61)

The restrictions stated in (61) on the input signal-to-signal ratio and the index k are imposed by the con-vergence properties of 2FI. By interchanging m with nand S, with S,, (61) becomes

rk+m+n

. ~ (-lji(JL)‘r(i + lc + T + ,)( >

i=o .t!(i+ m)!r c 1 - i -

k-j-m-n’)

(57)bkma= ik ($)k’2(f$T2 r(m _ i t n + 1)

Q

From (10) it can be seen that the factor k + m - nI

takes on only the values

k+m-n=

It can be further shownthe indices, the identity

. s21

TIOl,? (62)

Al, + 3, + 5, *** . (53) When the variable x is unity, ,F, is eva1uatedz3 by thethat, under these conditions on identity

r(c)r(c - a - b)d’,(a, b;c; 1) = r(~- u>rcc _ b)

for a+b-c<O. (63)

Thus, by application of (63) to either (61) or (62), we have

(-uir(i + it + T - “)=

rk+m-nrn-lc-m

( 2 >( 2 + 4

(59)

must hold also. After inserting the relation (59) into (57),the bkmnmay be expressed as for ($ji = (%ii + co. (64)

bkm,,= 5 (~)“‘@)~” m,r;r-+kl;i 1>

lc=O

The further restriction on k in (64) is because of the

2conditions under which (63) holds.

22Bateman Manuscript Project, op. cit., Vol. I, p. 278.23Bateman Manuscript Project, op. cit.,Vol. I, p. 104. See aIs0

Magnus and Oberhettinger, op. cit., p. 8.

hard limiting of two signals in random noise

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