IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998 2315 Instantaneous Frequency Estimation Using the Wigner Distribution with Varying and Data-Driven Window Length Vladimir Katkovnik, Member, IEEE, and LJubiˇ sa Stankovi´ c, Senior Member, IEEE Abstract—Estimation of the instantaneous frequency (IF) of a harmonic complex-valued signal with an additive noise using the Wigner distribution is considered. If the IF is a nonlinear function of time, the bias of the estimate depends on the window length. The optimal choice of the window length, based on the asymptotic formulae for the variance and bias, can be used in order to resolve the bias-variance tradeoff. However, the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the IF. The goal of this paper is to develop an adaptive IF estimator with a time-varying and data-driven window length, which is able to provide quality close to what could be achieved if the smoothness of the IF were known in advance. The algorithm uses the asymptotic formula for the variance of the estimator only. Its value may be easily obtained in the case of white noise and relatively high sampling rate. Simulation shows good accuracy for the proposed adaptive algorithm. I. INTRODUCTION A complex-valued harmonic with a time-varying phase is a key model of the instantaneous frequency (IF) concept, as well as an important model in the general theory of time–frequency distributions. It has been utilized for the study of a wide range of signals, including speech, music, biological, radar, sonar, and geophysical ones [14]. An overview of methods for the IF estimation, as well as an interpretation of the IF concept itself, is presented in [2] and [6]. Beside other efficient techniques for the IF estimation (e.g., [2], [11], [15], [16]), the time–frequency distribution approach is interesting and commonly applied [2], [6]. This approach is based on the property of time–frequency distributions to concentrate the energy of a signal, in the time–frequency plane, at and around the IF [1], [2], [4], [5], [12]. Out of the general Cohen class of time–frequency distributions with a signal- independent kernel, the Wigner distribution (WD) produces the best concentration along the linear IF [4], [5], [19], [20]. In order to improve the concentration when the IF is not a Manuscript received May 9, 1997; revised February 5, 1998. The work of LJ. Stankovi´ c was supported by the Alexander von Humboldt Foundation and the Montenegrin Academy of Science and Art. The associate editor coordinating the review of this paper and approving it for publication was Dr. Frans M. Coetzee. V. Katkovnik is with the Department of Statistics, University of South Africa, Pretoria, Republic of South Africa. LJ. Stankovi´ c is with the Signal Theory Group, Ruhr University Bochum, Bochum, Germany, on leave from the University of Montenegro, Podgorica, Montenegro (e-mail: l.stankovic@ieee.org). Publisher Item Identifier S 1053-587X(98)05954-6. linear function of time, various higher order time–frequency representations have been introduced [3], [8], [9], [19]. Here, we will focus our attention on the WD only. If the IF is a nonlinear function of time, then its estimate, using the WD, is biased. In the case of noisy signals, this estimate is highly signal and noise dependent [8], [17], [21]. Using the asymptotic formulae for the estimation variance and bias, we can, theoretically, find the optimal window length in the WD and resolve the bias-variance tradeoff. However, the optimal window length depends on the unknown smoothness of the IF, making this approach practically useless. The main goal of this paper is to develop an adaptive estimator with a time-varying and data-driven window length that is able to provide quality close to what could be achieved if the smoothness of the IF were known in advance. The idea of the approach developed in this paper originated from [7], where it was proposed and justified for the local polynomial fitting of regression. For the time-varying IF estimation, this approach was used in [10], where the algorithm with the time-varying and data- driven window length was presented for the local polynomial periodogram. This approach uses only the formula for the variance of the estimate, which does not require information about the IF to be known in advance. Simulations based on the discrete WD, with several noisy signal examples, show a good accuracy of the presented adaptive algorithm, as well as an improvement in the time–frequency representation of signals with a nonlinear IF. The structure of the paper is as follows: The WD, as an IF estimator, is considered in Section II. The asymptotic bias and variance of the IF estimate, along with the optimal window size for the IF estimation, are also presented in Section II. The adaptive estimation of the IF with a time-varying and data-driven window length is developed in Section III. A numerical implementation of the adaptive algorithm, along with simulation results, is discussed in Section IV. Proofs are given in the Appendix. II. BACKGROUND THEORY Consider the problem of the IF estimation from the discrete- time observations with (1) 1053–587X/98$10.00  1998 IEEE