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Constant-Q transform

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.

Constant-Q transform applied to the waveform of a C major piano chord. The x-axis is frequency, mapped to standard musical pitches, from low (left) to high (right). The y-axis is time, starting from pressing the piano chord at the bottom, and releasing the piano chord at the top, 8 seconds later. Darker pixels correspond to higher values of the Constant-Q transform. The peaks correspond closely to the precise frequencies of the vibrating piano strings. Thus the peaks can be used to detect the notes played on the piano. The lowest 3 peaks are the fundamental frequencies of the C major chord (C, E, G). Each string also vibrates at multiples of the fundamental, known as overtones, which correspond to the remaining smaller peaks to the right of the fundamental pitches. The overtones are smaller in intensity than the fundamental pitch.
Audio of the C Major piano chord used to generate the Constant-Q transform above.
Its waveform does not visually communicate pitch information like the Constant-Q transform is able to do.

The transform can be thought of as a series of filters fk, logarithmically spaced in frequency, with the k-th filter having a spectral width δfk 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.

  1. ^ Judith C. Brown, Calculation of a constant Q spectral transform, J. Acoust. Soc. Am., 89(1):425–434, 1991.
  2. ^ 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."

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