NSIGT_REAL - Nonstationary Gabor synthesis for real signals

Usage

fr = nsigt_real(c,g,shift,M,Ls)

Input parameters

c Cell array of nonstationary Gabor coefficients
g Cell array of synthesis windows
shift Vector of time shifts
M Number of frequency channels (vector/scalar)
Ls Length of the analyzed signal

Output parameters

fr Synthesized real-valued signal (Channels are stored in the columns)

Description

Given the cell array c of nonstationary Gabor coefficients, a set of windows g and time shifts shift, this function computes the corresponding real-valued nonstationary Gabor synthesis. Let N=numel(g) and \(P(n)=\sum_{l=1}^{n} shift(l)\) , then the complex valued synthesis formula reads:

\begin{equation*} fr[l] = \sum_{n=0}^{N-1}\sum_{m} c\{n\}(m)g\{n\}[l-P(n)] e^{2\pi i(l-P(n))m/M(n)}, \end{equation*}

for \(l=0,\cdots,Ls-1\) . In practice, the synthesis formula is realized by ifft and overlap-add. In the real valued case, ifftreal provides the missing frequency content normally given by the coefficients c{n}(m) for \(m \geq floor(M(n)/2)\) .

If a nonstationary Gabor frame was used to produce the coefficients and g is a corresponding dual frame, this function should perfectly reconstruct the originally analyzed signal to numerical precision.

Note that nsigt_real requires the input parameter M to guarantee that the vectors used in the overlap-add process are of the correct length.

Multichannel output will save each channel in a column of fr.

References:

P. Balazs, M. Dörfler, F. Jaillet, N. Holighaus, and G. A. Velasco. Theory, implementation and applications of nonstationary Gabor Frames. J. Comput. Appl. Math., 236(6):1481-1496, 2011.