Short-time Fourier transform with variable resolution
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform[1] and very closely related to the complex Morlet wavelet transform.[2] Its design is suited for musical representation.
The transform can be thought of as a series of filters fk, logarithmically spaced in frequency, with the k-th filter having a spectral widthδfk equal to a multiple of the previous filter's width:
where δfk is the bandwidth of the k-th filter, fmin is the central frequency of the lowest filter, and n is the number of filters per octave.
^Continuous Wavelet Transform "When the mother wavelet can be interpreted as a windowed sinusoid (such as the Morlet wavelet), the wavelet transform can be interpreted as a constant-Q Fourier transform. Before the theory of wavelets, constant-Q Fourier transforms (such as obtained from a classic third-octave filter bank) were not easy to invert, because the basis signals were not orthogonal."