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At a glance, the constant-Q fourier transform and the complex Gabor-Morlet wavelet transform seem the same. Both are time-frequency representations, based on constant-Q filters, windowed sinusoids, etc. But maybe there's a difference that I'm missing?

Constant-Q Transform Toolbox for Music Processing says:

CQT refers to a time-frequency representation where the frequency bins are geometrically spaced and the Q-factors (ratios of the center frequencies to bandwidths) of all bins are equal.

Time-scale analysis says:

That is, computing the CWT of a signal using the Morlet wavelet is the same as passing the signal through a series of bandpass filters centered at $f = \frac{5/2\pi}{a}$ with constant Q of $5/2\pi$.

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3 Answers

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Simply speaking both the const-Q-transform and the Gabor-Morlet wavelet-transform are just continuous wavelet transforms. Or, more precisely, approximations thereof, as there will always be discretization issues in real applications.

A property of wavelet transforms is that they have build in the constant Q-factor property, or in other words logarithmic scaling. Gabor and Morlet are just two names of a particular wavelet function (complex exponentials with a gaussian window) which is used most commonly. The CQ-transform just uses another basis function/wavelet and has a special name attached to it, probably to some historical reason.

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It is important to note that the various wavelets that have been developed offer different decompositions of the signals they are used to study. Specific wavelets are chosen to reveal specific signal features in a particular way. When you compute wavelet coefficients, you perform a correlation of the chosen wavelet with the signal of interest; thus the shape of the wavelet determines the shape of signal features that are revealed.

Some wavelet functions have been "designed" to provide decompositions that can relate to Fourier decompositions (actually more in line with short term Fourier decompositions used to produce spectrograms of signals). The Morlet wavelet is a good example of such a wavelet function. Other wavelets have been "designed" to identify discontinuities or edges of signals. I've see papers that use Daubechies wevelet functions for this.

It may be helpful to do some research to see how each of the wavelet functions you've mentioned are being used in practice. I think this will give you a better understanding of how various wavelets differ.

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The question is specifically about the Morlet wavelet only, though, and how it relates to the constant-Q transform, which is also a type of Fourier decomposition. Is there any difference between them, or are they re-inventions of the same thing? I've also found the "Fixed-Point per Octave (FPPO) algorithm" which "utilizes a measurement time window that varies as a function of frequency, utilizing a long time window at low frequencies (for narrow frequency resolution) and a successively shorter time window at high frequencies" rationalacoustics.com/files/FFT_Fundamentals.pdf –  endolith Jan 12 '13 at 20:22
    
I posted a specific comment regarding the question. My other post was intended to encourage the poster to understand how wavelet transforms are unique and why it makes sense to develop transforms based on different wavelet functions. –  user2718 Jan 14 '13 at 14:47
    
"Is there any difference between them, or are they re-inventions of the same thing?" They are different. The foundation of Fourier methods is based in sinus functions and has no time scale resolution. Windowed versions of the Fourier transform approach what is done with wavelets. Wavelet transforms are founded on compactly supported basis functions and the transform is a time/scale representation rather than time/frequency representation. Some wavelet functions mimic Fourier methods by design, but this is not a requirement. –  user2718 Jan 15 '13 at 15:47
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The constant Q transform is not a wavelet transform. The constant Q transform is a particular variation on the short term Fourier transform in which the frequency bins are exponentially spaced instead of linearly spaced as is the case with the discrete Fourier transform.

See: http://en.wikipedia.org/wiki/Constant_Q_transform for details.

Some wavelet transforms are also considered to be constant Q transforms because in the discrete versions of the transforms, the scale of the wavelet is varied exponentially (base being 2 in this case). According to the following paper from Stanford university ( https://ccrma.stanford.edu/~jos/sasp/Continuous_Wavelet_Transform.html ):

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.12.5Before 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. See Appendix E for related discussion.

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"The constant Q transform is not a wavelet transform." How so? –  endolith Jan 15 '13 at 4:10
    
This is probably a bit of a semantics problem, but the "constant Q transform" developed from the short term Fourier transform, so no wavelet function is used in the analysis. It is similar to wavelet analysis in that the frequency bins are spaced exponentially. Wavelet transforms specifically do not deal with frequency. Wavelet transforms only deal with scale. The combination of scale and wavelet function can be related back to frequency, but the two things are not the same. –  user2718 Jan 15 '13 at 15:15
    
From what I've read, the Gabor-Morlet wavelet was the first continuous wavelet transform, and was focused on frequency, not scale, since it was derived from the Gabor transform, which is a windowed Fourier transform. Ignoring semantic differences, is there a difference in the way the the CQT and Morlet WT are calculated? –  endolith Jan 15 '13 at 21:38
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Aren't those mathematically equivalent, assuming the window function is the same and the wavelet is made from a complex exponential? –  endolith Jan 15 '13 at 22:23
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I think you can arrange for a windowed Fourier transform that is equivalent to a wavelet transform. Typically in the application of the constant Q transform, the window function isn't chosen to enforce the admissibility conditions required of wavelets, so in general the constant Q transform is not the same as a wavelet transform. The admissibility conditions for wavelets ensure that the analysis is reversible (i.e. you can reconstruct your time signal from the transform results) which is not true in general for the constant Q transform. –  user2718 Jan 15 '13 at 23:13
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