CORRESPO~ENCE 109 REFERENCES [l] W. T. Cochran et al., “What is the fast Fourier transform,” Proc. IEEE, vol. 5!, pp. 1664-1674, Oct. 1967. [2] A. J. Viterbl, Principles of Coherent Communication. New York: McGraw-Hill, 1966, pp. 272-274. [3) D. Slepian, “Estimation of signal parameters in the presence of noise,” IRE Trans. Inform. Theory, vol. PGIT-3, pp. 68-89, Mar. 1954. [4] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Part I. New York: Wiley, 1968?p. 278. [5] L. P. Seidman, “Performance hmitations and error calculations for parameter estimation,” Proc. IEEE, vol. 58, pp. 644-652, May 1970. [6] G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications. San Francisco: Holden-Day, 1968, p. 238. Asymptotic Statistics of Periodogram-Type Spectral Estimates for the Output of Nonlinear Devices MICHAEL M. FITELSON Abstracl-The high resolution statistics of a periodogram-type spectral estimate for a periodic deterministic signal plus Gaussian noise are well understood. If, however, the signal plus noise is first passed through a nonlinear memoryless device (such as a hard limiter), the situation becomes much more complicated. The main result of thii correspondenceis the derivation of the high resolution limit of the joint statistics of the I and Q components of a periodogram for a certain class of nonlinear devices with a deterministic periodic signal and Gaussian noise as input. This result permits the calculation of the asymptotic central moments of the spectral estimate formed from the foregoing periodogram. It then becomes possible to subject a device employing this technique for spectral estimation to a detailed performance analysis. I. INTRODUCTION One of the most commonly used approaches to real-time spectral analysis is the periodogram technique. If z(t) is the time waveform whose spectrum is to be estimated, the spectral esti- mate F&a) obtained by this technique is given by where FTC4 = ZT2(4 + Q&4 (1) IT(o) = & s z2 dt cos wtz(t) QT(a) = - .& s T’2 JT -T/Z dt sin wtz(t). (2) When z(t) consists of a periodic waveform plus stationary gaussian noise, the central moments of FT(o) are readily obtained. However, if z(t) is the result of passing a periodic waveform plus Gaussian noise through a nonlinear memoryless device (such as a hard limiter) it becomes very difficult to describe the statistics of IT(W), QT(o), and FT(w) unless T is large compared to l/W, where W is the effective bandwidth of the noise. It is the purpose of this correspondence to present a derivation of the large-Tjoint statistics of IT(w) and Q&w). With this result, in which Z=(o) and Q&z) are shown to be asymptotically normal and statistically independent with equal variances, the asymp- totic central moments of FT(w) are expressible in terms of the first and second moments of IT(w) and Q,(o). Manuscript received April 17, 1973; revised June 26, 1973. The author is with the Electronic Systems Division, General Electric Company, Syracuse, N.Y. 13201. The proof hinges on a central limit theorem proved by Sun [l ] that can be stated as follows. A. Sun’s Theorem Let x(t) (- co < t < 03) be a real stationary Gaussian pro- cesswhich satisfies i) lo,..+,, E(x(t) - ~(t~))~ = 0, for all to; ii) Ex(t) = 0, for all t; iii) p(z) = E(x(t + 7)x(t)) = J?Fm da exp (2ni;lz)f(l), for all t, z, and withf(1)6L2(-a,co). Condition i) implies that p(z) is continuous [2]. Let G(x(t),t) be a timedependent memoryless device that satisfies iv) EG(x(t),t) = 0, for all t; v) EG2(x(t),t) < M < co, for all t; vi) lim,,,, E(G(x(t),t) - G(x(to),to))2 = 0, for all to; vii) where there exists a p > 0 such that G(x(t + np), t + np) = G(x(t + np), t), Let forn = O,fl,rt2,.-.. f’(a) = 2 f A + : . n=-m ( 1 (3) Suppose viii) f*(;1)EL2 -;, ; ( 1 ix) lim --A- s rrlp sin2 (N/2)11 N+oo 2nN f*(a) da < 03 -n,p sin2 ap and exists, then zT (= ($) s_T:z2 dtG(x(t),t)) is asymptotically normal with lim E(exp(ia2,)) = exp ,02= lim EzT2< co. T-03 T-rC.3 (4) In the next section, Sun’s theorem will be used to establish the joint asymptotic normality of Z,(w) - El,(o) and Q,(w) - EQT(w). II. THE ASYMPTOTIC STATISTICS OF IT(w) AND Q,(O) The following lemma establishes the asymptotic normality of fT(d = ITk) - El,(m) QT(~) = Q,(m) - EQT(~). (5) Lemma 2.1: Let x(t) be a real stationary Gaussian process that satisfies conditions it-iii), viii), and ix) of Sun’s theorem. Let p(0) = 1. Let H be a real nonlinear memoryless time-independent device that satisfies i) ii) iii) Let EH(x(t)) = 0, for all t; IH( I cedlxl, for some c, d > 0; H(x) is continuous almost everywhere. At) = x(t) + s(t) (6)