2015 Fall ECE 438 Boutin Constant Q Transform Robert Schwieterman - Rhea

1. Introduction

Text of first section goes here. Here is an example of an equation. $ f(x)= \frac{1}{5} \sin x \int_{-\infty}^\alpha \pi^y dy $

2. Background

Text of second section goes here. Here is an example of a list

  The Constant Q Transform (CQT) is closely related to the Discrete fourier Transform (DFT).  But where the DFT has 

linearly spaced frequency “bins” the CQT’s are logarithmically spaced. This is motivated by the fact that human hearing is similarly inclined. Music and instruments are built around producing frequencies that are spaced, not of constant difference, but of constant ratio. The notes on a piano are related in that the every 12 notes (half-steps) you double frequency. The CQT is built to be more sensitive to this than its predecessor.

3. Theory

   We can model the  DFT as a discrete sampling of the DTFT convolved with a sinc function.  The sampling stems from the
fact that we assume our window of N samples is periodic with period N.  The sinc convolution stems from the fact that in 

the DFT we are windowing our once infinitely long sequence. By multiplying by a rectangular window in the index (time) domain, we convolve with a sinc in the frequency domain. This leads to the phenomenon of frequency bleeding. If we have a sequence x[n]=cos((2π/2.2)n) and take a 4 point DFT with frequency bins corresponding to 0, π, π/2, and 3 π/2, then we will have non zero values in the frequency bins, despite the DTFT having non zero components ONLY where w=(2 π/2.2). It is this

sinc convolution and frequency bleeding that allows us to view each frequency bin as a band-pass filter.

   It is in this filter model of the DFT that we can begin to understand the Constant Q transform.  The Q for which it gets
its name is from the “Quality factor” of a filter, defined as the ratio of center frequency to bandwidth. (fk/BW)  The width 

of our DFT “filter” is dependent on the number of samples N, the higher the N, the smaller the bandwidth. For a DFT, the number of samples is independent of the frequency bin, leading to a unchanging bandwidth for each filter. This means that bins in the higher frequencies have a higher quality index than those in the lower frequencies. By changing the number of samples used (window length) we can develop such filters that the Quality index is constant, Constant Q Transform!

4. CQT

  [Brown], who first described the CQT in 1991, sets the window length (N) by N=Q(fs/fk).  Because our sampling frequency and Q are constant,
we can say that N is inversely proportional to our bin frequency.  Just as humans take longer times to distinguish lower frequency sounds accurately,
the CQT must devote more samples, (and thus more operations) to such lower frequencies.  This makes the the CQT apt for musical applications, 

where the signal will be composed of primarily logarithmically spaced frequencies instead of linearly-spaced ones.

5. References

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

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