Two versions of Constant Q Transform (CQT) doesn't match each other?

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.

• 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. – tjwrona1992 Aug 21 at 22:36