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Constant-Q nonstationary Gabor transform

`cfs = cqt(x)`

`[cfs,f] = cqt(x)`

`[cfs,f,g,fshifts] = cqt(x)`

`[cfs,f,g,fshifts,fintervals] = cqt(x)`

`[cfs,f,g,fshifts,fintervals,bw] = cqt(x)`

`[___] = cqt(___,Name,Value)`

`cqt(___)`

returns the constant-Q transform (CQT), `cfs`

= cqt(`x`

)`cfs`

, of input signal
`x`

.

If

`x`

is a vector, then`cqt`

returns a matrix corresponding to the CQT.If

`x`

is a matrix, then`cqt`

obtains the CQT for each column (independent channel) of`x`

. The function returns a multidimensional array corresponding to the maximally redundant version of the CQT.

`[`

returns the frequency intervals, `cfs`

,`f`

,`g`

,`fshifts`

,`fintervals`

] = cqt(`x`

)`fintervals`

, corresponding the
rows of `cfs`

. The `k`

th element of
`fshifts`

is the frequency shift in DFT bins between the
`((k-1) mod N)`

and` (k mod N)`

element of
`fintervals`

with `k = 0,1,2,...,N-1`

where
`N`

is the number of frequency shifts. Because MATLAB^{®} indexes from 1, `fshifts(1)`

contains the frequency
shift between `fintervals{end}`

and
`fintervals{1}`

, `fshifts(2)`

contains the
frequency shift between `fintervals{1}`

and
`fintervals{2}`

, and so on.

`[___] = cqt(___,`

returns the CQT with additional options specified by one or more
`Name,Value`

)`Name,Value`

pair arguments, using any of the preceding
syntaxes.

`cqt(___)`

with no output arguments plots the CQT
in the current figure. Plotting is supported for vector inputs only. If the input
signal is real and `Fs`

is the sampling frequency, the CQT is
plotted over the range `[0,Fs/2]`

. If the signal is complex, the
CQT is plotted over the range [`0,Fs`

).

[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.