On subspace-based methods for frequency estimation of random amplitude sinusoidal signals

Sinusoidal signals with random time-varying amplitude show up in many signal processing applications. Amplitude modulation results in degeneracy of the signal subspace, i.e. the signal subspace corresponding to one amplitude modulated sinusoid is no longer spanned by one vector. In this paper, we propose modi cations of two subspace-based techniques, namely ESPRIT and MODE for estimating the center frequency of a sinusoidal signal with random time-varying ARMA amplitude. Numerical simulations illustrate the good performance of the methods. Finally, a robusti ed scheme of the proposed methods is described and succesfully applied to real radar data. 1. PROBLEM STATEMENT Analysis of sinusoidal signals with random or time varying amplitude appears to be of importance in many elds of signal processing. In the radar application, as soon as the target scintillates or with targets that cannot be considered as points, a random amplitude model is more appropriate than a constant amplitude one[1]. This problem is also encountered in the array processing context when the source signals impinge on the array over a spread of angles [2]. Most of the time, the amplitude may be viewed as a perturbation term (a "multiplicative noise") and a key problem consists of estimating the center frequency of such processes. From a spectral point of view, multiplicative noise is more di cult to cope with, random modulation causing a spread of the peak. Subspace-based techniques, which have proved to be powerful in the constant amplitude case, deteriorate in the case of a random amplitude. As a matter of fact, unless speci c assumptions are made on the envelope, the signal subspace is degenerate, i.e. it has spread into a higher than one dimensional subspace. Consequently, subspace-based methods, which rely on a clear discrimination between signal and noise subspaces are no longer suitable. Therefore, there is an interest in proposing subspace-based methods that properly handle the case of random amplitude signals. This is the aim of this paper. More precisely, we propose to use modi cations of ESPRIT[3] and MODE[4] algorithms in order to estimate the frequency of an exponential signal with random ARMA amplitude. Moreover, as we focus on the frequency estimation (and not on the ARMA amplitude parameter estimation), we describe simpli cations of these two methods, which reduce the associated computational burden. 2. MODEL AND FREQUENCY ESTIMATION Let us consider the problem of estimating the frequency !0 from N samples of the following signal x(t) = (t)e0 +w(t); t = 1; :::; N (1) where ' is a deterministic phase, w(t) is a sequence of zero-mean i.i.d. random variables with variance 2 w . In (1), (t) is assumed to be a real-valued stationary ARMA(p; q) process, independent of w(t). Let S (z) = B(z)B(z )= A(z)A(z ) denote the spectrum of (t) and de ne by yk = ke j k k=1::p the zeroes of A(z). It can be readily veri ed that the spectrum Sx(z) has the following ARMA(p; p) form: Sx(z) = C(z)C(z ) A(ze 0)A(z e0) (2) from which the following set of Yule-Walker equations can be written: