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icqt

Inverse constant-Q transform using nonstationary Gabor frames

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Syntax

xrec = icqt(cfs,g,fshifts)
xrec = icqt(___,'SignalType',sigtype)
[xrec,gdual] = icqt(___)

Description

xrec = icqt(cfs,g,fshifts) returns the inverse constant-Q transform, xrec, of the coefficients cfs. cfs is a matrix, cell array, or structure array. g is the cell array of nonstationary Gabor constant-Q analysis filters used to obtain the coefficients cfs. fshifts is a vector of frequency bin shifts for the constant-Q bandpass filters in g. icqt assumes by default that the original signal was real-valued. To indicate the original input signal was complex-valued, use the 'SignalType' name-value pair. If the input to cqt was a single signal, then xrec is a vector. If the input to cqt was a multichannel signal, then xrec is a matrix. cfs, g, and fshifts must be outputs of cqt.

example

xrec = icqt(___,'SignalType',sigtype) designates whether the signal was real-valued or complex-valued. Valid options for sigtype are 'real' or 'complex'. If unspecified, sigtype defaults to 'real'.

[xrec,gdual] = icqt(___) returns the dual frames of xrec as a cell array the same size as g. The dual frames are the canonical dual frames derived from the analysis filters.

Examples

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Open Live Script

Load and plot the Handel signal.

load handel
t = (0:length(y)-1)/Fs;
plot(t,y)
title('Handel')
xlabel('Time (s)')

Figure contains an axes object. The axes object with title Handel, xlabel Time (s) contains an object of type line.

Obtain the constant-Q transform of the signal using the sparse transform option. Because the transform will be inverted, you must also return the Gabor frames and frequency shifts used in the analysis.

[cfs,~,g,fshifts] = cqt(y,'SamplingFrequency',Fs, ...
    'TransformType','sparse');

Invert the constant-Q transform and demonstrate perfect reconstruction by showing the maximum absolute reconstruction error and the relative energy error in dB.

xrec = icqt(cfs,g,fshifts);
maxAbsError = max(abs(xrec-y))
maxAbsError = 
6.9389e-16

relEnergyError = 20*log10(norm(xrec-y)/norm(y))
relEnergyError = 
-301.4160

Input Arguments

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Constant-Q coefficients of a signal or multichannel signal, specified as a matrix, cell array, or structure array. cfs must be the output of cqt.

Nonstationary Gabor constant-Q analysis filters used to obtain the coefficients cfs, specified as a cell array. cfs must be the output of cqt.

Frequency bin shifts for the constant-Q bandpass filters in g, specified as a real-valued vector. fshifts must be the output of cqt.

Signal type of the original signal, specified as 'real' or 'complex'. Use sigtype to designate whether the original signal was real-valued or complex-valued. If unspecified, sigtype defaults to 'real'.

Output Arguments

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Inverse constant-Q transform, returned as a vector or matrix. If the input to cqt was a single signal, then xrec is a vector. If the input to cqt was a multichannel signal, then xrec is a matrix.

Dual frames used in the synthesis of xrec, returned as a cell array the same size as g. The dual frames are the canonical dual frames derived from the analysis filters.

Algorithms

The theory of nonstationary Gabor transforms (NSGTs) was introduced by Jaillet [1] and Balazs, Dörfler, Jaillet, Holighaus, and Velasco [2]. Dörfler, Holighaus, Grill, and Velasco [3], [4] develop a framework for an efficient, perfectly invertible CQT. The algorithms used in cqt and icqt were developed by Dörfler, Holighaus, Grill, and Velasco and are described in [3], [4]. In [5], Schörkhuber, Klapuri, Holighaus, and Dörfler develop and provide algorithms for a phase-corrected CQT transform which matches the CQT coefficients that would be obtained by naïve convolution. The Large Time-Frequency Analysis Toolbox (https://github.com/ltfat) provides an extensive suite of algorithms for nonstationary Gabor frames [6].

References

[1] Jaillet, Florent. “Représentation et traitement temps-fréquence des signaux audionumériques pour des applications de design sonore.” Ph.D. dissertation, Université de la Méditerranée, Aix-Marseille II, 2005.

[2] Balazs, P., M. Dörfler, F. Jaillet, N. Holighaus, and G. Velasco. “Theory, Implementation and Applications of Nonstationary Gabor Frames.” Journal of Computational and Applied Mathematics 236, no. 6 (October 2011): 1481–96. https://doi.org/10.1016/j.cam.2011.09.011.

[3] Holighaus, Nicki, M. Dörfler, G. A. Velasco, and T. Grill. “A Framework for Invertible, Real-Time Constant-Q Transforms.” IEEE Transactions on Audio, Speech, and Language Processing 21, no. 4 (April 2013): 775–85. https://doi.org/10.1109/TASL.2012.2234114.

[4] Velasco, G. A., N. Holighaus, M. Dörfler, and T. Grill. "Constructing an invertible constant-Q transform with nonstationary Gabor frames." In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx-11). Paris, France: 2011.

[5] Schörkhuber, C., A. Klapuri, N. Holighaus, and M. Dörfler. "A MATLAB® Toolbox for Efficient Perfect Reconstruction Time-Frequency Transforms with Log-Frequency Resolution." Submitted to the AES 53rd International Conference on Semantic Audio. London, UK: 2014.

[6] Průša, Z., P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs. The Large Time-Frequency Analysis Toolbox 2.0. Sound, Music, and Motion, Lecture Notes in Computer Science 2014, pp 419-442.

Extended Capabilities

Version History

Introduced in R2018a

See Also

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