While none can deny the usefulness of the spectral analysis of stationary signals, the extension of the concept of spectrum to non-stationary or time-varying signals is fraught with difficulty. Yet most communication signals — including the most fundamental and interesting communication signal, human speech — are inherently non-stationary. Indeed the desire to analyze speech was the main motivation for the development of the short-time Fourier transform (STFT), which is still the primary spectral analysis technique for time-varying signals. In the last few years, many authors have advocated the use of time-frequency distributions (TFDs) — and in particular the Wigner-Ville distribution — for this task. TFDs attempt to describe a signal's behaviour in time and frequency in a similar manner to the way in which bivariate joint probability distributions describe the statistical behaviour of two random variables. They are a subclass of the more general class of time- frequency representations (TFRs) which includes the STFT. One interesting feature of TFRs is that the normalized first moment of some TFRs can be used to give an unbiased estimate of the instantaneous frequency (IF) of a signal— we call such estimators, TFR moment IF estimators. Several authors have suggested that these estimators may offer advantages over more conventional estimators, and some efforts have been made to evaluate their statistical performance. Researchers have also applied TFDs and closely related TFRs to speech signals and have claimed that these representations are superior to the traditional STFT in the sense that they provide greater resolution in the time-frequency domain. This thesis examines the general problem of the spectral analysis of time-varying signals and attempts to determine whether there are time-frequency representations (TFRs) which may be better suited to the task of representing the spectrum of a time-varying signal than the traditional STFT.
First, we take a close look at the concept of IF and propose a definition based on the modulating signal and the frequency modulation conversion law. While this definition may seem obvious to many communications engineers, most authors researching this field use a definition based on the derivative of the phase of the analytic signal. We then find that TFR moment IF estimators are only appropriate for estimating the IF of monocomponent signals — a task for which many satisfactory estimators already exist. Next, we examine discrete-time TFR moment IF estimators and find that the circular nature of discrete-time frequency estimators must be accounted for by introducing a periodic definition of first moment and con- volution. This is a major new result in this context, although the mathematics for handling circular data has been in existence for many years. We cite many examples of confusion in the literature regarding the circular nature of discrete-time frequency estimators. All other researchers have been using the conventional linear definition of first moment to calculate TFR moment IF estimates. Application of the linear definition of first moment results in biased estimators with larger variance. Each TFR moment IF estimators based on the periodic first moment is found to be virtually equivalent to a smoothed central finite difference (SCFD) IF estimator based on simple finite differencing of the phase of the analytic signal. The SCFD estimator is much easier to analyze than the corresponding TFR moment IF estimator and its variance is always lower. Consequently, we only need to analyse the statistical performance of the SCFD estimators to determine the best possible performance obtainable from TFR moment IF estimators.
A general expression is developed for the statistical performance of SCFD IF estimators on monocomponent signals in noise and this forces us to conclude that SCFD IF estimators, and hence TFR moment IF estimators, are usually less statistically efficient than more conventional estimators. Nevertheless, we propose the parabolic SCFD IF which is unbiased, optimal and computationally simpler than other optimal methods.
We then examine the performance of a large number of TFRs on a simple dual component signal. Many of the important properties of TFRs can be deduced by reformulating TFRs in the time-lag domain and examining the shape of the time-lag kernel function which characterizes each individual TFR. Guidelines for the design of TFRs are developed and they are used to generate a new TFR which is a blend of the STFT and the Wigner-Ville distribution.
All TFRs of multicomponent signals exhibit unwanted oscillations called cross- terms, which are caused by interactions between signal components. Despite the ability of TFDs to represent monocomponent signals with high energy concentration about their IF laws, we see that the presence of large cross-terms causes them to have lower resolution than the STFT. Attempts to reduce the cross-terms of TFDs and TFRs lead to representations equivalent to the STFT.
Many other methods for the analysis of time-varying signals have been proposed which are outside the class of TFRs. Some of these methods are based on modern parametric spectral estimation techniques using rational transfer function models of signals. These methods can perform adequately if the model is chosen well, but can be very misleading when they are applied to unknown signals.
We examine the concept of spectrum for time-varying signals and conclude that the concept may not be useful for all signals and may only have meaning for signals which are almost stationary in relation to the measurement procedures used. Such signals are often called quasi-stationary. We develop several algorithms for partitioning quasi-stationary signals into near stationary segments and then use these algorithms to perform adaptive spectral estimation. An application of adaptive techniques in oceanographic research is described.
We conclude that TFDs do not appear to offer any advantages over conventional methods and they do little to clarify the concept of spectrum for time-varying signals. Indeed, it seems that the very concept of spectral analysis of general time-varying signals is poorly defined and may be of little use in many applications. This being said, it must be admitted that while some technique may not be a panacea, it may still have a niche where it is useful. The short-time Fourier Transform has a very important niche indeed — other TFRs have yet to find theirs.