# icqt

Inverse constant-Q transform using nonstationary Gabor frames

## Description

returns the inverse constant-Q transform, `xrec`

= icqt(`cfs`

,`g`

,`fshifts`

)`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`

.

## Examples

## Input Arguments

## Output Arguments

## 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**