Limiting of signals in random noise

The effect of ideal bandpass limiting on a signal lying in narrowband Gaussian noise is analyzed. General analytic expressions for the limiter output components are derived using an integral representation for the limiter characteristic. This method allows retention of the phases of all the signals and the intermodulation products at the limiter output, which are destroyed in the characteristic-function method generally used in limiter studies. Expressions for the desired signals and intermodulation product amplitudes are obtained for the case when the limiter input consists of three angle-modulated sinusoids and noise. The analysis is extended to n modulated sinusoids plus noise, and approximate expressions for the signal, intermodulation product, and noise terms are derived. Numerical results are presented for the signal suppression and the limiter output signal amplitudes for the case of three input signals, two of equal amplitude. INTRODUCTION THE calculation of the output of a limiting device when its input consists of a sum of several signals has been the subject of a great deal of theoretical analysis and has resulted in a number of widely quoted publications. Essentially, two methods of approach have been used to analyze the problem. The first approach, commonly known as the characteristic-function method of Rice, involves computing the autocorrelation function of the limiter output and then taking the Fourier transform to obtain the powerdensity spectrum. Although a general expression for the limiter output autocorrelation function can be derived, its computation becomes extremely involved when modulation of the input signals is considered. The difficulty lies in determining the characteristic function of the signals with arbitrary modulation. However, if the signals are statistically independent angle-modulated sinusoids, and only the average power or the magnitude of the signal and crossproduct terms at the limiter output is of interest, the modulation of the input signals can be ignored, which considerably simplifies the analysis. Davenport [1] was first to use this approach to investigate the effect of hard limiting a single sinusoidal signal and narrowband Gaussian noise. Jones [2] used the same method to analyze the case of two sinusoids plus noise. More recently. Shaft [3] and Gyi [4] independently extended the analyses to include n sinusoids. The magnitude of any signal or cross product is given by an untabulated infinite integral that has been numerically evaluated for a number of cases of interest. In many practical applications, the phases of the signal and cross-product terms are also of interest. For example, in an FM or PM system, the spreading of the power spectral density of the signal and cross-product terms will be a function of their phase modulation, even though the average power in any one output component is indepenManuscript received March 10, 1970; revised September 15, 1971. The author is with the Stanford Research Institute, Arlington, Va. 22209. 20040804 112 dent of its phase. Thus, modulation of the input signals must be considered in the analysis, in order to retain the phase of the signal and cross-product terms at the output of the limiter. The second method, which can be referred to as the Fourier-expansion approach, is carried out entirely in the time domain, in contrast to the autocorrelation approach, which treats the problem in the frequency domain. Analysis in the time domain does not lose the phase of the signal and cross-product terms. It also allows removal of the assumption of statistical independence between the input signals. This approach was used by Granlund [5] and later by Baghdady [6] to investigate the noiseless case of two sinusoids passed through an ideal bandpass limiter. More recently, Sollfrey [7] used the same approach to analyze the effect of hard limiting on a sum of three or four sinusoidal signals without noise. Closed-form analytic expressions for the amplitude of the desired signal terms were obtained for three input signals, two of equal amplitude, and for four signals, having two pairs of equal ampHtude. The Fourier expansion method, however, has the drawback that it is difficult to consider the effect of noise present at the input to the limiter. The above references, therefore, do not consider noise and are directed primarily to the calculation of signal amplitude at the output of the limiter. The purpose of this paper is to extend the Fourierexpansion method to include random noise in the analysis, and to derive a general analytical expression for the output of a limiting device. The approach is similar to that used by Reed [9] for calculating the amplitudes of two signals in noise. Closed-form expressions for the desired signal and cross-product amplitudes are presented for the case of three modulated sinusoids and noise. The analysis is then extended to n modulated sinusoids plus noise, and approximate expressions for the signal, cross-product, and noise terms are derived. CALCULATION OF THE LIMITER OUTPUT The specific model for the bandpass limiter to be considered is shown in Fig. 1. The input to the limiter x{t) = s{t) + n{t) (1) consists of the signal s{t) and a band of zero-mean stationary Gaussian noise n{t). It is assumed that the bandpass filter preceding the limiter is wide enough to pass the signal with negligible distortion and limits the input noise to a narrow bandwidth that is small compared to the center frequency of the filter. The limiter is followed by another bandpass filter that confines the output spectrum essentially only to the fundamental band of the signal. It is assumed that the limiter has a hard-limiting characterDISTRIBUTION STATEMEr4T A Approved for Public Release Distribution Unlimited JAIN: UMTTING SIGNALS IN RANDOM NOISE 333 BANDPASS FILTER HARD LI MITER BANDPASS FILTER as follows: z(0 = Eiyit)-] = J" sin [vsity] ■ E[Jo{vr)-] ~ . (7) Fig. 1. Model for investigation of ideal symmetric limiting. istic that limits its output to either ±1. Thus, if the limiter input is x(0, the output y{t) may be expressed in analytical form [7] as y{t) = I sin [t)x(0] — • (2) n Jo V When the expression (1) is inserted into this integral, the sine of a sum may be expanded into the sum of two products of sine and cosine. Thus, y{t) = sin [vsit)] cos [rn(0] — Tt Jo ^ + r cos [vsity] sin [vn(t)-] . (3) nJo ^ The narrowband Gaussian noise at the limiter input may be expressed as Since the noise envelope has a Rayleigh distribution ,2 p(r) = -. exp r 2?. (8) where a^ is the total noise power at the limiter input, the expected value of J^ivr) is given by [8] E\Jo{vr)1 = f" Jo{vr)p{r) dr = exp Jo L Substitution of (9) in (7) yields 2 r°° z(f) = sin [i;s(0] • exp n Jo This is identical to the expression obtained by Reed [9] for calculating the amplitudes of two signals in noise. The above integral may also be written in terms of the error function [10] as 2 . t)' ■"1 2 . dv V (9)