To my knowledge, there's two major CQT papers, the one by Brown in 1991, and the one by Schorkhuber in 2010.

The 2010 paper claims to be a more computationally efficient implementation of the 1992 and 1991 paper by Brown. (The 1992 paper is also an efficient version of the 1991 paper.)

I discovered that the mathematics behind the two papers are not exactly the same.

Definition of $Q$ and $N_k$ by Brown in 1991 and 1992

$Q=\frac{f_k}{\Delta f_k}=\frac{f_k}{f_{k+1}-f_k}=\frac{1}{2^{\frac{1}{b}}-1}$ ----(1A), where $b$ is the number of bins per octave.

$N_k=\frac{f_s}{\Delta f_k}=\frac{f_s}{f_k}Q$ ----(2A), where $f_s$ is the sampling rate.

Definition of $Q$ and $N_k$ by Schorkhuber in 2010

$Q = \frac{f_k}{\Delta f_k} = \frac{N_k f_k}{\Delta \omega f_s}$ ----(1B)

$Q = \frac{q}{\Delta \omega (2^{\frac{1}{b}}-1)}$ ----(2B)

By combining equation (1B) and (2B), you can get the window length $N_k$ as below

$N_k = \frac{qf_s}{f_k(2^{\frac{1}{b}}-1)}$ ----(3B)

where $\Delta \omega$ is the equivalent noise bandwidth, $q$ is the scaling factor.

Here's the problem that I have

If you try to equate (2A) and (3B) together, you will get another expression for Q.

$\frac{f_s}{f_k}Q = \frac{qf_s}{f_k(2^{\frac{1}{b}}-1)}$

After simplifying, you will get the follow expression.

$Q = \frac{q}{2^{\frac{1}{b}}-1}$. Which looks very similar to equation (2B) with a missing $\Delta \omega$ factor.

So, there's 3 different expressions for $Q$ now? To me, it seems a contradiction. Either of the papers must be wrong in their definition on $Q$.

Another mystery is where does the term $\Delta \omega$ come from? I don't see any necessity in adding this term. It seems to me the source of the contradiction.

  • $\begingroup$ While I don't know the answer to this question thank you very much for posting something that lead me to these articles on the constant Q transform! I've been trying to find good info on it for a little while now. $\endgroup$ – tjwrona1992 Aug 21 '19 at 22:36

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