icqt

Inverse constant-Q transform using nonstationary Gabor frames

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.

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

Perfect Reconstruction of Constant-Q Transform

Load and plot the Handel signal.

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

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

cfs — Constant-Q coefficients

matrix | cell array | structure array

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.

g — Nonstationary Gabor constant-Q analysis filters

cell array

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

fshifts — Frequency bin shifts

real-valued vector

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

sigtype — Signal type

'real' (default) | 'complex'

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

xrec — Inverse constant-Q transform

vector | matrix

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.

gdual — Dual frames

cell array

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 (NSG) frames for frequency-adaptive analysis and efficient algorithms for analysis and synthesis using NSG frames are due to Dörfler, Holighaus, Grill, and Velasco [1],[2]. The algorithms used in cqt and icqt were developed by Dörfler, Holighaus, Grill, and Velasco and are described in [1],[2]. In [3], 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 [4].

References

[1] 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.

[2] 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.

[3] 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.

[4] 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

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• To minimize compilation time, define the nonstationary Gabor constant-Q analysis filters g as variable size using coder.typeof (MATLAB Coder). If you define the filters g as fixed size, compilation time is significant with minimal gain in execution efficiency.

Version History

Introduced in R2018a