105 Abstract—We propose a two-step procedure to estimate the frequency of a deterministic sinusoid, with unknown parameters, corrupted by additive, white, zero-mean noise, based on the Pisarenko Harmonic Decomposition. A rough PHD estimation is performed in the first step, and a multiple of the unknown frequency is estimated in the second step. The variance of the PHD estimator is significantly reduced. Keywords—Frequency estimation, Noisy real sinusoid, Pisarenko’s method, Variance analysis. I. INTRODUCTION E consider the problem of the frequency estimation of a signal that consists of a sine wave with additive, white noise, from a finite number of consecutive samples. This problem has a long history [1], and it is relevant for a wide field of applications such as communications, radar, sonar, speech processing, measurement, etc [2]. In many applications, complex valued exponentials are prefered to real valued sinusoids, especially when both in- phase and quadrature components of real signals are available, so that complex valued signals can be readily constructed inside computers [2]. The interest in complex versions of signals in frequency estimation originated in the discovery of the fact that the maximum likelihood (ML) frequency estimator for a complex exponential is the maximum of the signal's periodogram [3]. As ML estimators are not computationally efficient, other more efficient suboptimal estimators have been proposed and studied [4−9]. The problem of the ML estimator for a real sinusoid has also been addressed [10], and the solution turned out to be less simple than in the complex case. The ML estimator suffers again of the problem of computational complexity, so that suboptimal but more computationally efficient estimators exist in this case too. Some of these estimators can be derived from the more general spectrum estimators presented in the review [1]. The PHD is such a suboptimal estimator that relies on the eigenvalues of the covariance matrix of the signal. Its Manuscript received February 15, 2007; Revised May 19, 2007. Aldo De Sabata is with the Faculty of Electronics and Teleco- mmunications, Politehnica University of Timisoara, Romania (corresponding author to provide phone: +40-256-403370; fax: +40-256-403362; e-mail: aldo.desabata@ etc.upt.ro). Liviu Toma is with the Faculty of Electronics and Telecommunications, Politehnica University of Timisoara, Romania (e-mail: liviu.toma@etc.upt.ro). Septimiu Mischie is with the Faculty of Electronics and Telecommunications, Politehnica University of Timisoara, Romania (e-mail: septimiu.mischie@etc.upt.ro). statistical properties has been extensively studied, and implementation in the case of a real sinusoid turned out to be very simple (see [11], [12] and the references cited therein). Also for the case of a single real sinusoid, a Reformed PHD (RPHD) estimator, which exhibits better statistical properties than the PHD, has been introduced recently [13], [14]. In order to decrease the estimator variance, we propose in this paper a new frequency estimation algorithm. Namely, we propose to extend the PHD to a k-PHD, through which we compute the signal's (digital) frequency from the estimated cosine of its k'th multiple. In order to raise the inherent ambiguity, a rough estimate of the frequency must be known a priori, or it must be estimated by other, less computationally complex means. Here we will use the PHD for the initial estimate too, in order to show that the variance of the newly introduced estimator is significantly reduced with respect to the variance of the known one. We will also compare the variance of our estimator to the Cramer-Rao lower bound (CRLB). II. DESCRIPTION OF THE METHOD We consider the following discrete-time signal model () () () cos( ) ( ), 1.. xn sn qn n qn n N α ω ϕ = + = + + = (1) where the amplitude 0 α > , (angular) frequency (0, ) ω π ∈ and phase ϕ are deterministic but unknown quantities, and q(n) is a Gaussian, white noise, uncorrelated with the signal, with zero-mean and variance 2 σ . The signal-to-noise ratio is 2 2 /(2 ) SNR α σ = . Following [1], we review here briefly the derivation of the PHD estimator in order to introduce its extension to the estimation of a multiple of ω. The sinusoid obeys the following k’th order linear prediction equation for any fixed integer k: () 2cos( )( ) ( 2) 0 sn k sn k sn k ω − − + − = (2) (n denotes the discrete time). It is possible to derive from (1) and (2) the following vector equation: T T n n = xa qa (3) where [() ( ) ( 2 )] T n xn xn k xn k = − − x , [() ( ) ( 2 )] T n qn qn k qn k = − − q and A New Two-Step Single Tone Frequency Estimation Algorithm Aldo De Sabata, Liviu Toma, and Septimiu Mischie W INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Issue 2, Vol. 1, 2007