GABPHASEDERIV DGT phase derivatives
Usage: [phased,c] = gabphasederiv(dflag,'dgt',f,g,a,M);
phased = gabphasederiv(dflag,'cross',f,g,a,M)
phased = gabphasederiv(dflag,'phase',cphase,a,difforder);
phased = gabphasederiv(dflag,'abs',s,g,a);
[{phased1,phased2,...}] = gabphasederiv({dflag1,dflag2,...},...);
[{phased1,phased2,...},c] = gabphasederiv({dflag1,dflag2,...},'dgt',...);
phased=GABPHASEDERIV(dflag,method,...) computes the time-frequency
derivative dflag of the phase of the DGT of a signal using algorithm
method.
The following strings can be used in place of dflag:
't' First phase derivative in time.
'f' First phase derivative in frequency.
'tt' Second phase derivative in time.
'ff' Second phase derivative in frequency.
'tf' or 'ft' Second order mixed phase derivative.
phased is scaled such that (possibly non-integer) distances are measured
in samples. Similarly, the frequencies are scaled such that the Nyquist
frequency (the highest possible frequency) corresponds to a value of L/2.
The computation of phased 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, phased attain very large random values when abs(c)
is close to zero.
The phase derivative computation can be done using four different methods
depending on the string method:
'dgt' Directly from the signal using algorithm by Auger and
Flandrin.
'cross' Directly from the signal using algorithm by Nelson.
'phase' From the unwrapped phase of a DGT of the signal using a
finite differences scheme. This is the classic method used
in the phase vocoder.
'abs' From the absolute value of the DGT exploiting explicit
dependency between partial derivatives of log-magnitudes and
phase.
Currently this method works only for Gaussian windows.
phased=GABPHASEDERIV(dflag,'dgt',f,g,a,M) computes the time-frequency
derivative 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.
[phased,c]=GABPHASEDERIV(dflag,'dgt',f,g,a,M) additionally returns
the Gabor coefficients c, as they are always computed as a byproduct
of the algorithm.
phased=GABPHASEDERIV(dflag,'cross',f,g,a,M) does the same as above
but this time using algorithm by Nelson which is based on computing
several DGTs.
phased=GABPHASEDERIV(dflag,'phase',cphase,a) computes the phase
derivative 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.:
phased=gabphasederiv(dflag,'phase',angle(dgt(f,g,a,M)),a)
phased=GABPHASEDERIV(dflag,'abs',s,g,a) computes the phase derivative
from the absolute values of DGT coefficients s. The spectrogram must have
been computed using the window g and time-shift a e.g.:
phased=gabphasederiv(dflag,'abs',abs(dgt(f,g,a,M)),g,a)
Currently the 'abs' method only works if the window g is a Gaussian
window specified as a string or a cell array.
phased=GABPHASEDERIV(dflag,'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.
Phase conventions
-----------------
First derivatives in either direction are subject to phase convention.
The following additional flags define the phase convention the original
phase would have had:
'freqinv' Derivatives reflect the frequency-invariant phase of dgt.
This is the default.
'timeinv' Derivatives reflect the time-invariant phase of dgt.
'symphase' Derivatives reflect the symmetric phase of dgt.
'relative' This is a combination of 'freqinv' and 'timeinv'.
It uses 'timeinv' for derivatives along frequency and
and 'freqinv' for derivatives along time and for the
mixed derivative.
This is usefull for the reassignment functions.
Please see ltfatnote042 for the description of relations between the
phase derivatives with different phase conventions. Note that for the
'relative' convention, the following holds:
gabphasederiv('t',...,'relative') == gabphasederiv('t',...,'freqinv')
gabphasederiv('f',...,'relative') == -gabphasederiv('f',...,'timeinv')
gabphasederiv('tt',...,'relative') == gabphasederiv('tt',...)
gabphasederiv('ff',...,'relative') == -gabphasederiv('ff',...)
gabphasederiv('tf',...,'relative') == gabphasederiv('tf',...,'freqinv')
Several derivatives at once
---------------------------
phasedcell=GABPHASEDERIV({dflag1,dflag2,...},...) computes several
phase derivatives at once while reusing some temporary computations thus
saving computation time.
{dflag1,dflag2,...} is a cell array of the derivative flags and
cell elements of the returned phasedcell contain the corresponding
derivatives i.e.:
[pderiv1,pderiv2,...] = deal(phasedcell{:});
[phasedcell,c]=GABPHASEDERIV({dflag1,dflag2,...},'dgt',...) works the
same as above but in addition returns coefficients c which are the
byproduct of the 'dgt' method.
Other flags and parameters work as before.
References:
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.
K. R. Fitz and S. A. Fulop. A unified theory of time-frequency
reassignment. CoRR, abs/0903.3080, 2009.
D. J. Nelson. Instantaneous higher order phase derivatives. Digital
Signal Processing, 12(2-3):416--428, 2002. [1]http ]
F. Auger, E. Chassande-Mottin, and P. Flandrin. On phase-magnitude
relationships in the short-time fourier transform. Signal Processing
Letters, IEEE, 19(5):267--270, May 2012.
Z. Průša. STFT and DGT phase conventions and phase derivatives
interpretation. Technical report, Acoustics Research Institute,
Austrian Academy of Sciences, 2015.
References
1. http://dx.doi.org/10.1006/dspr.2002.0456
Url: http://ltfat.github.io/doc/gabor/gabphasederiv.html
See also: resgram, gabreassign, dgt, pderiv, gabphasegrad.
Package: ltfat