Reassignment method
The method of reassignment is a technique for sharpening a time-frequency representation (e.g. spectrogram or the short-time Fourier transform) by mapping the data to time-frequency coordinates that are nearer to the true region of support of the analyzed signal.
This mapping to reassigned time-frequency coordinates is very precise for signals that are separable in time and frequency with respect to the analysis window.
They map the one-dimensional time-domain signal into a two-dimensional function of time and frequency.
But the windowing operation required in spectrogram computation introduces an unsavory tradeoff between time resolution and frequency resolution, so spectrograms provide a time-frequency representation that is blurred in time, in frequency, or in both dimensions.
[2] One of the best-known time-frequency representations is the spectrogram, defined as the squared magnitude of the short-time Fourier transform.
[2] A time-frequency representation having improved resolution, relative to the spectrogram, is the Wigner–Ville distribution, which may be interpreted as a short-time Fourier transform with a window function that is perfectly matched to the signal.
The method has been shown to reduce time and frequency smearing of any member of Cohen's class.
[2][3] In the case of the reassigned spectrogram, the short-time phase spectrum is used to correct the nominal time and frequency coordinates of the spectral data, and map it back nearer to the true regions of support of the analyzed signal.
[4] Their technique enhances the resolution in time and frequency of the classical Moving Window Method (equivalent to the spectrogram) by assigning to each data point a new time-frequency coordinate that better-reflects the distribution of energy in the analyzed signal.
defined by[4]: 74 This phenomenon is known in such fields as optics as the principle of stationary phase, which states that for periodic or quasi-periodic signals, the variation of the Fourier phase spectrum not attributable to periodic oscillation is slow with respect to time in the vicinity of the frequency of oscillation, and in surrounding regions the variation is relatively rapid.
Analogously, for impulsive signals, that are concentrated in time, the variation of the phase spectrum is slow with respect to frequency near the time of the impulse, and in surrounding regions the variation is relatively rapid.
[4]: 73 In reconstruction, positive and negative contributions to the synthesized waveform cancel, due to destructive interference, in frequency regions of rapid phase variation.
This analogy is a useful reminder that the attribution of spectral energy to the center of gravity of its distribution only makes sense when there is energy to attribute, so the method of reassignment has no meaning at points where the spectrogram is zero-valued.
[2] In digital signal processing, it is most common to sample the time and frequency domains.
The reassignment operations proposed by Kodera et al. cannot be applied directly to the discrete short-time Fourier transform data, because partial derivatives cannot be computed directly on data that is discrete in time and frequency, and it has been suggested that this difficulty has been the primary barrier to wider use of the method of reassignment.
[6] It is easily shown that Nelson's cross spectral surfaces compute an approximation of the derivatives that is equivalent to the finite differences method.
Since these algorithms operate only on short-time spectral data evaluated at a single time and frequency, and do not explicitly compute any derivatives, this gives an efficient method of computing the reassigned discrete short-time Fourier transform.
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.
This is the property, in the frequency domain, that Nelson called separability[6] 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
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.
[6]: 2585 Gardner and Magnasco (2006) argues that the auditory nerves may use a form of the reassignment method to process sounds.
The authors come up with a variation of reassignment with complex values (i.e. both phase and magnitude) and show that it produces sparse outputs like auditory nerves do.
By running this reassignment with windows of different bandwidths (see discussion in the section above), a "consensus" that captures multiple kinds of signals is found, again like the auditory system.
