[tgrad,fgrad,c] = gabphasegrad('dgt',f,g,a,M);
[tgrad,fgrad] = gabphasegrad('phase',cphase,a);
[tgrad,fgrad] = gabphasegrad('abs',s,g,a);
[tgrad,fgrad]=gabphasegrad(method,...) computes the relative time-frequency gradient of the phase of the dgt of a signal. The derivative in time tgrad is the relative instantaneous frequency while the frequency derivative fgrad is the negative of the local group delay.
tgrad is a measure the deviation from the current channel frequency, so a value of zero means that the instantaneous frequency is equal to the center frequency of the considered channel, a positive value means the true absolute intantaneous frequency is higher than the current channel frequency and vice versa. Similarly, fgrad is a measure of deviation from the current time positions.
fgrad is scaled such that distances are measured in samples. Similarly, tgrad is scaled such that the Nyquist frequency (the highest possible frequency) corresponds to a value of L/2. The absolute time and frequency positions can be obtained as
tgradabs = bsxfun(@plus,tgrad,fftindex(M)*L/M); fgradabs = bsxfun(@plus,fgrad,(0:L/a-1)*a);
Please note that neither tgrad and fgrad nor tgradabs and fgradabs are true derivatives of the dgt phase. To obtain the true phase derivatives, one has to explicitly pass either 'freqinv' or 'timeinv' flags and scale both tgrad and fgrad by 2*pi/L.
The computation of tgrad and fgrad is inaccurate when the absolute value of the Gabor coefficients is low. This is due to the fact the the phase of complex numbers close to the machine precision is almost random. Therefore, tgrad and fgrad may attain very large random values when abs(c) is close to zero.
The computation can be done using four different methods.
| 'dgt' | Directly from the signal using algorithm by Auger and Flandrin. |
| 'phase' | From the phase of a DGT of the signal. This is the classic method used in the phase vocoder. |
| 'abs' | From the absolute value of the DGT. Currently this method works only for Gaussian windows. |
| 'cross' | Directly from the signal using algorithm by Nelson. |
[tgrad,fgrad]=gabphasegrad('dgt',f,g,a,M) computes the time-frequency gradient using a DGT of the signal f. The DGT is computed using the window g on the lattice specified by the time shift a and the number of channels M. The algorithm used to perform this calculation computes several DGTs, and therefore this routine takes the exact same input parameters as dgt.
The window g may be specified as in dgt. If the window used is 'gauss', the computation will be done by a faster algorithm.
[tgrad,fgrad,c]=gabphasegrad('dgt',f,g,a,M) additionally returns the Gabor coefficients c, as they are always computed as a byproduct of the algorithm.
[tgrad,fgrad]=gabphasegrad('cross',f,g,a,M) does the same as above but this time using algorithm by Nelson which is based on computing several DGTs.
[tgrad,fgrad]=gabphasegrad('phase',cphase,a) computes the phase gradient from the phase cphase of a DGT of the signal. The original DGT from which the phase is obtained must have been computed using a time-shift of a using the default phase convention ('freqinv') e.g.:
[tgrad,fgrad]=gabphasegrad('phase',angle(dgt(f,g,a,M)),a)
[tgrad,fgrad]=gabphasegrad('abs',s,g,a) computes the phase gradient from the spectrogram s. The spectrogram must have been computed using the window g and time-shift a e.g.:
[tgrad,fgrad]=gabphasegrad('abs',abs(dgt(f,g,a,M)),g,a)
[tgrad,fgrad]=gabphasegrad('abs',s,g,a,difforder) uses a centered finite diffence scheme of order difforder to perform the needed numerical differentiation. Default is to use a 4th order scheme.
Currently the 'abs' method only works if the window g is a Gaussian window specified as a string or cell array.
F. Auger and P. Flandrin. Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Process., 43(5):1068--1089, 1995.
E. Chassande-Mottin, I. Daubechies, F. Auger, and P. Flandrin. Differential reassignment. Signal Processing Letters, IEEE, 4(10):293--294, 1997.
J. Flanagan, D. Meinhart, R. Golden, and M. Sondhi. Phase Vocoder. The Journal of the Acoustical Society of America, 38:939, 1965.
Z. Průša. STFT and DGT phase conventions and phase derivatives interpretation. Technical report, Acoustics Research Institute, Austrian Academy of Sciences, 2015. [ .pdf ]