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Inverse constant-Q transform using nonstationary Gabor frames

`xrec = icqt(cfs,g,fshifts)`

`xrec = icqt(___,'SignalType',sigtype)`

`[xrec,gdual] = icqt(___)`

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`

.

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].

[1] Holighaus, N., 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.*
Vol. 21, No. 4, 2013, pp. 775–785.

[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.