215 Detection of the Peak of an Arbitrary Spectrum LEONARD KLEINROCK, MEMBER, IEEE Summary-A new procedure is described for determining that frequency at which the spectrum of a signal has its absolute peak. The salient feature of the procedure is that it does not explicitly involve the estimation of the spectrum of the signal itself. Specifi- cally, it is shown that the limit of the iterated normalized auto- correlation [see (8) and (9)J of a function, f(t), is a pure cosine wave whose frequency corresponds to the location of the peak of the spectrum of f(f). Furthermore, if one is willing to accept an estimated peak fre- quency of maximum energy to within a given finite spectral resolu- tion, then the procedure terminates after a specified finite number of iterations. Results from a computer simulation of the procedure are described. The areas of application of this procedure are dis- cussed, and the results indicate that this method of detecting a signal (Le., by the peak of its spectrum) merits further consideration. It is important to note that the consideration of random processes has not been undertaken in this initial study; the results apply to the spectral peak of a deterministic signal only. I. INTRODUCTION I N COMMUNICATIONS engineering, it is often useful to be able to extract, from an incoming signal, that frequency which contains a greater (power or energy) density’ than any other frequency. In this study, a new procedure is described for determining that fre- quency; the significant aspect of the procedure is that the spectrum of the signal need not be calculated. The fundamental theorem of Section III describes the mathematically interesting result in the case where we carry our procedure to the limit. The more useful theorem in Section V, however, describes a realizable procedure for determining the frequency of maximum power or energy to within a finite spectral resolution. We begin by defining the quantities basic to the procedure. II. DEFINITIONS Consider that class of real functions or signals f(t) whose autocorrelation function, g(t) has the following properties, 0 < g(0) < c-3 s m -m IdOl dt < a (1) (2) g(t) is continuous in t. (3) Since g(t) is an autocorrelation, it is therefore an even function of its argument. In Section IV, we modify these Manuscript received August 28,1963; revised December 31,1963. This work was performed while the author was employed at the Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Mass., which is operated with support from the U. S. Army, Navy, and Air Force. The author is at the Department of Engineering, University of California, Los Angeles, Calif. 1 Whether this is the energy density spectrum or power density spectrum depends upon the class that the signal falls in. See Sec- tion II. restrictions to allow periodic and discrete time functions. For the above class, we define functions f(t) to be of finite energy if s m o< _m If(t dt < ~0 (4) in which case g(t) = I- f(df(t + 4 dT. -m (5) Further, we define functions f(t) to be of finite average power if 0 < lim 1 s T T-m 2T -T tf(Ol” dt < ~0 (f-3) in which case g(t) = ;+t & s_: f(df(t + 4 d7. (7) Functions of finite energy or of finite average power therefore satisfy (a), and if their autocorrelation function satisfies (1) and (3), then our results hold.’ We now define a set of normalized autocorrelation functions, R,(t) as R (t) 0 = !@m g(O) s m Rn-1(4Rn-l(t + T) dr R,(t) = -m s m n = 1,2,3, ... . (9) R:-,(T) dr -m We recognize that R,(t) is the normalized autocorrelation function of the normalized autocorrelation function R,,(t), etc, We may thus consider R,,(t) to be the nth iterated normalized autocorrelation function of f(t). Consider the Fourier transform, X,(w), defined by X,(w) = j-- R,(t)e-‘“’ dt for all n (10) --m and its inverse R,(t) = & lrn X,(w)e’“’ da. (11) m Defining as usual, 8: (w) to be the energy density spectrum of R,(t), we note that X,(w) is proportional to the energy density spectrum of R,‘-,(t) for n 2 1; in particular, So(w) is proportional to the spectral density3 of f(t). We s The properties expressed in (1) and (3) are most easily stated in terms of g(t) and will be left in that form. 3 That is, for signals of finite energy, SO(w) represents their normalized energy density spectrum, whereas for signals of finite average power, SO(W) represents their normalized pourer density spectrum.