Performance of quadratic time-frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic timefrequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed. I. I Instantaneous frequency (IF) estimation is an important research topic in signal analysis [2], [3], [14]-[22], [28]-[29]. There are several approaches to this problem. Time-frequency distribution (TFD)-based approach is one of them [14]-[16], [18], [28], [29]. The basis for using TFDs in the IF estimation is their first moment property, [3], [4], [12]. The first-order TFD moment, with respect to frequency, provides an acceptable IF definition for a timevarying signal. The TFD, which is used to IEEE Transactions on Signal Processing, Vol.51, No.1, Jan. 2003. recover the IF as its first moment, provides an unbiased estimate. The presence of noise, however, leads to a serious degradation of the first moment estimate due to the absence of any averaging in its definition. In other words, the first moment may have a high statistical variance, even for high values of input signalto-noise ratio (SNR) [22]. TFDs concentrate the energy of a signal at and around the IF in the time-frequency plane, [3], [18], [22], [23]. Consequently, the peak detection of the TFDs is used as an IF estimator, as a natural alternative to the first moment. The IF estimation based on TFDs maxima is analyzed in [3], [5], [14]-[17], [21], [22], [28], and [29]. Out of the quadratic class of TFDs, only the most frequently used ones are considered there: the Wigner distribution (WD) for linear frequency-modulated (FM) signal and the spectrogram for signals with constant frequency. It has been shown that in the case of noisy signals, this estimate highly depends on the SNR, as well as on the window width. In this paper, we present a general analysis of an arbitrary shift covariant quadratic TFD as an IF estimator for any FM signal. Expressions for the bias and variance of this IF estimator are derived. When the IF is a nonlinear function of time, then its estimate is biased for all TFDs from this class (Cohen class of TFDs-CD), whereas they behave as unbiased estimators in the case of monocomponent signals with linear IF. The exact expressions for the IF estimator variance in the cases of white stationary and white nonstationary noises are derived. The corresponding expressions for some frequently used TFDs from the CD are obtained as special cases as well. We have presented the well-known rePERFORMANCE OF QUADRATIC TIME-FREQUENCY DISTRIBUTIONS AS IF ESTIMATORS 577 sults for the pseudo WD and linear FM signal and for the spectrogram of signals whose IF may be considered as a constant. In addition, we have derived the variance expression for the spectrogram of a linear FM signal. This signal is considered in the cases of other commonly used TFDs, such as Born-Jordan (BJD) and Choi-Williams distribution (CWD). It has been shown that the reduced interference distributions (RID) outperform the pseudo WD but only in the case when the IF is constant or its variations are small. For highly nonstationary signals, the pseudo WD can produce better results. The analysis is extended to the multicomponent signals. It has been shown that the results obtained for monocomponent signals remain valid for the multicomponent ones when TFDs from RID class are used and signal components are well separated. For the pseudo WD-based IF estimation, the variance of each component depends on the total power of all signal components. The paper is organized as follows. After this introduction, the IF estimator is defined, and the problem is described. In Section III, analysis of the estimation error is performed. The bias and variance of the estimation error in the cases of commonly used quadratic TFDs are derived in Section IV. The IF estimation of multicomponent signals is considered in Section V. The obtained results are checked numerically and statistically in Section VI. II. B T Consider discrete-time observations x(nT ) = f(nT ) + ε(nT ), f(t) = A(t) exp(jφ(t)) (1) where n is an integer, T is a sampling interval, ε(nT ) is a white noise, and A(t) is a slowvarying amplitude of the analyzed signal. By definition, [6], [16], [18], the IF is a first derivative of the signal phase ω(t) = φ(t) ≡ dφ(t) dt . (2) Assume that ω(t) is an arbitrary smooth differentiable function of time with bounded derivatives ∣ω(r)(t) ∣∣ = ∣∣φ(r+1)(t) ∣∣ ≤ Sr(t), r ≥ 1. The general form of the quadratic shiftcovariant TFDs (CD) is, [4], [6], [12], [13], [30] Cx(t, ω; c) = 1 2π ∞ ∫