Reassignment Method - Separability

Separability

The short-time Fourier transform can often be used to estimate the amplitudes and phases of the individual components in a multi-component signal, such as a quasi-harmonic musical instrument tone. Moreover, the time and frequency reassignment operations can be used to sharpen the representation by attributing the spectral energy reported by the short-time Fourier transform to the point that is the local center of gravity of the complex energy distribution.

For a signal consisting of a single component, the instantaneous frequency can be estimated from the partial derivatives of phase of any short-time Fourier transform channel that passes the component. If the signal is to be decomposed into many components,

and the instantaneous frequency of each component is defined as the derivative of its phase with respect to time, that is,

then the instantaneous frequency of each individual component can be computed from the phase of the response of a filter that passes that component, provided that no more than one component lies in the passband of the filter.

This is the property, in the frequency domain, that Nelson called separability and is required of all signals so analyzed. If this property is not met, then the desired multi-component decomposition cannot be achieved, because the parameters of individual components cannot be estimated from the short-time Fourier transform. In such cases, a different analysis window must be chosen so that the separability criterion is satisfied.

If the components of a signal are separable in frequency with respect to a particular short-time spectral analysis window, then the output of each short-time Fourier transform filter is a filtered version of, at most, a single dominant (having significant energy) component, and so the derivative, with respect to time, of the phase of the is equal to the derivative with respect to time, of the phase of the dominant component at . Therefore, if a component, having instantaneous frequency is the dominant component in the vicinity of, then the instantaneous frequency of that component can be computed from the phase of the short-time Fourier transform evaluated at . That is,

Just as each bandpass filter in the short-time Fourier transform filterbank may pass at most a single complex exponential component, two temporal events must be sufficiently separated in time that they do not lie in the same windowed segment of the input signal. This is the property of separability in the time domain, and is equivalent to requiring that the time between two events be greater than the length of the impulse response of the short-time Fourier transform filters, the span of non-zero samples in .

In general, there is an infinite number of equally valid decompositions for a multi-component signal. The separability property must be considered in the context of the desired decomposition. For example, in the analysis of a speech signal, an analysis window that is long relative to the time between glottal pulses is sufficient to separate harmonics, but the individual glottal pulses will be smeared, because many pulses are covered by each window (that is, the individual pulses are not separable, in time, by the chosen analysis window). An analysis window that is much shorter than the time between glottal pulses may resolve the glottal pulses, because no window spans more than one pulse, but the harmonic frequencies are smeared together, because the main lobe of the analysis window spectrum is wider than the spacing between the harmonics (that is, the harmonics are not separable, in frequency, by the chosen analysis window).