Hard-Limiter Intermodulation with Low Input Signal-To-Noise Ratio

IEEE TR.\NSACTIONS ON COMMUNICATION TECHNOLOGY

AUGUST

653

1967

Concise Papers

Hard-Limiter Intermodulation with Low Input Signal-To-Noise Ratio IDEAL LIMITER

R. LYNN KIRLIN Fig. 1. Sum of

REFERENCE : Kirlin, R. L. : HARD-LIMITER INTERMODULATION WITH LOW INPUT SIGNAL-TO-NOISE RATIO, Utah State University, Logan, Utah, on leave of absence from the Boeing Company, Seattle, Wash. R e d d 9/19/66; revised 12/28/66 and 3/23/67. Paper 19TP67-945. IEEETRANS. ON COMMUNICATION TECHNOLOGY, 15-4 August 1967, pp. 653-654.

1~

BAND-PASS FILTER

limiter.

signals and Gaussian noise through an ideal bandpass

W

0

- 20 ABSTRACT: A simple, closed form expression is derived for the intermodulation (IM) normalized power magnitude of major products at the output of an ideal bandpass hard-limiter with low input signal-to-noise ratio, (S/N)i.This expressiontogether with no-noise results obtained by a previous author allows quick plotting of the no-noise and high-noise asymptotes of theIM product magnitude vs. (S/N)i, for any number of equal-power, input CW signals. Lastly, it is shown that for manysignalsitmay be advantageous to operate the limiter a t a lower (S/N)i inorder to suppress the intermodulation distortion.

-40

-60

- 80 -100

AdE+C*D'E-F-G-H-I

- I40 5 -160

KEYWORDS: Communication Satellites, Communication Theory, Crosstalk, Detection, Signal-to-Noise Ratio.

.mn =

I2

-where n

exp(-u2x2/2)

n

i=l

J,, (A,x)

t

I

I

5

10

6 NUMBEROFSIGNALS,

Fig. 2.

Several papershave treatedthe subjects of signals, noise, and intermodulation (IM) in hard-limiters[11-[81with the confignration :shown in Fig. 1. I n 1953, Davenport121was first to completely analyze the inpnt-output relationships of a single sinusoid pllm narrowband Gaussian noise through a hard-limiter. Jones,L31 in 1961, followed Davenport's technique for the 2-signal case, giving results for t.he normalized power out of each signal, and the third-order product .as funct,ions of (S/N); (input totalsignal-to-noise ratio). Shaft['] continued on to the n-signal problem in 1965 and derived a closed form ,expression for the magnitude of any signal or odd order IM product. Using a compllter, he plotted most of the important products as a function of n for the no-noise case (that figure is given here as Fig. 2). He also presented the ontput power magnitudes of four signals for -variolls input amplitude arrangements as a function of (Si/N)i, where SI is the st,rongest signal, and further plotted many IM products as a function of (SI/N)i for the 4-signal equal-power case. This paper extends Shaft's results by finding the asymptotic form (S/N)i of the power of major odd order IN1 of theplotagainst prod1lct.s a t low (S/N); for any number (n)of equal-amplitude, unmodulated, CW signals. These results, which do not involve a computer solution, show how the I M distortion resulting from passing several equal-amplitude signals in adjacent channels through a hardlimiter or another nonlinear device may be reduced. Shaft,['] in his equation (5a), has shown that the normalized magnitude of the output power of any IM produet in the frnldamental zone (the frequency range of the fundamental of the square-wave output) is

A+WC+D-E-F-G

- 120

50

IOC

n

Cross productsasafunction of a numbcr of signals n under conditions (from Fig. 6 of Shaft).l'l

0-noise

Jmi--mtth order Bessel function of the first kind n -number of signals A f -amplitude of the ithsinusoidal signal u 2 -mean square noise input power nzl m2 . . . m, = odd.

+ +

+

And the mi represent the harmonic of the ith signal cont,ribrItillg t,o the IM product. For example, Mag,2,00 would be the magnitude of he fifth order intermodulation product2fi - 2J. +sa for a 5 signal inprlt., where thefi aredistinct signal freqllencies. If all A t = A and theharmonic of any contributing frequency is less than or equal to 3, then upon substituting y = Ax,

where p , q, and r represent, respectively, the number of 3rd, 2nd, and fundamental harmonics of distinct signal freqllencies contributing to a given I M product. Note that anexpression similar to (1) could be given which would include even higher harmonics, and the following results made t o include more products, but additional productjs are not of much importance.

ASYMPTOTES For any n signals of equal input power, the high (S/N)i asymptotes of the normalized magnitude of the output power of the important I M products are given by the values in Fig. 3. It will now be shown that, when plotted against (S/N);, theasymptotes of the normalized magnitude of the output power of the major

ci54

IEEE TRANSACTlONS ON COMMUNICATION TECHNOLOGY

(SIN);

20

IO

0

-

DE -10

Fig. 3. Cross product asymptotes versus

-20

-30

(;3/N)i for 5 equal signals.

12(n)is valid for any (S/N)i, and accurate towithin 0.1 dB for n 2 4 signals. Z,(n) squared will reduce for small (S/N); to ( 2 )of this paper when p = 0 in ( 2 ) . The method of approximation in Gyils1 could be extended to include third harmonics as contributors to the IM products. Although Gyi’s results may be more generally useful (even though limited to n 2 4, and harmonics 2 2), this paper is concerned more with the outputsa t low (S/N)i, as thefollowing portion points out.

S I G N ~AND L TOTAL I M POWER IN CENTER CHANNEL FOR LARGE n AND Low (S/N);

+ +

Bennett141has shown that as n -* m the number of -4 B C (or f2 - f3) products in the center channels of n equally spaced signals approaches ( 3 / S ) n 2and that thenumber of 2A - B products B C type are approaches ( n - 2 ) / 2 ; therefore, only the A significant,. Thus for large n and low (S/N)i, and assuming equally spaced, equal amplitude, unmodulated signals, the ratio of one signal (8,) to intermodulationin the center channel (or one near the center) is

fl

+

+ +

where

+ +

L = 3p 2q r, the order of the I M product R = (S/N)i = nA2/(2a2) 0,

[LIP. D. Shaft, “Limiting of several signals and its effect on communication systemperformance.” I E E E Trans.Communication Technology, vol.COM-13, pp. 504-512, December 1965. 121 W. B. Davenport, Jr., “Signal-to-noise ratios in bandpass limiters,” Phrra 24. ~~. nn 72C-727. , Jltne ~.~~ 1953. ”.I . _A_n _n 1r _. _ ..i.__.“01. . ~~, [SI J. J. Jones “Hard-limiting o-f two signals in random noise,” I E E E Trans. Information Thebry, vol. IT-9, pp. 34-42, January 196:i. F.1 a m -161 w . R. Bennett,“Crosemodulationrequirementsonmultichann plifiers below overload,’‘Bell Sus. Tech. J . . vol. 19, pp. 58 7-610, October 1946.--[SI R. Manasse, R. Price, and R. M . Lerner. “Loss of signal detectability in IT-4,. DD. 34-38. band-Dass limiters.” I R E Trans.InformationTheow.vol. March 1958. 161 M. Gyi, “Some topics on limiters and FM demodulators,” Stanford ElectronicsLab.,StanfordUniversity,Stanford.Calif..Tech.Rept.,SEL-65-056. July 1965. [’I W. C.Babcock,“Intermodulationinterferenceinradiosystems,” Bell Sys. Tech. J . , v01.,32, pp. 63-73, January 19R3. [SI J. L.SevyTheeffectofmultiple CIV and FM signalspassedthrough a hard limiter o;TWT.” I E Z E Trans. Communication Technology, vol. COM-14, pp. 568-578, October 1966.

..

which is identical to ( 2 ) . The low (S/N)i asymptotes are plotted in Fig. 3 for the 5-signal case ( n = 5 ) . Some high (S/N); asymptotes are shown as given by Fig. 2. It shordd be noted that here the horizontal axis is (S/N)i where

8 = 581. Although the previous argument does not constitute a proof, the result (2) is valid as verified by the computer results of Shaft[*]and

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