Cross-spectral methods for processing speech.

We present time-frequency methods which are well suited to the analysis of nonstationary multicomponent FM signals, such as speech. These methods are based on group delay, instantaneous frequency, and higher-order phase derivative surfaces computed from the short time Fourier transform (STFT). Unlike more conventional approaches, these methods do not assume a locally stationary approximation of the signal model. We describe the computation of the phase derivatives, the physical interpretation of these derivatives, and a re-mapping algorithm based on these phase derivatives. We show analytically, and by example, the convergence of the re-mapping to the FM representation of the signal. The methods are applied to speech to estimate signal parameters, such as the group delay of a transmission channel and speech formant frequencies. Our goal is to develop a unified method which can accurately estimate speech components in both time and frequency and to apply these methods to the estimation of instantaneous formant frequencies, effective excitation time, vocal tract group delay, and channel group delay. The proposed method has several interesting properties, the most important of which is the ability to simultaneously resolve all FM components of a multicomponent signal, as long as the STFT of the composite signal satisfies a simple separability condition. The method can provide super-resolution in both time and frequency in the sense that it can simultaneously provide time and frequency estimates of FM components, which have much better accuracy than the Heisenberg uncertainty of the STFT. Super-resolution provides the capability to accurately "re-map" each component of the STFT surface to the time and frequency of the FM signal component it represents. To attain high resolution and accuracy, the signal must be jointly estimated simultaneously in time and frequency. This is accomplished by estimating two surfaces, which are essentially the derivatives of the STFT phase with respect to time and frequency. To avoid phase ambiguities, the differentiation is performed as a cross-spectral product.