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<center><font size= 4>Constant Q Transform</font size> | <center><font size= 4>Constant Q Transform</font size> | ||
| − | A [http://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Robert Schwieterman | + | A [http://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Robert Schwieterman |
| − | Partly based on the [[ECE438|ECE438 Fall 2015]] lecture material of Boutin. | + | Partly based on the [[ECE438|ECE438 Fall 2015]] lecture material of Prof Boutin. |
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| − | ==1. Introduction | + | ==1. Introduction == |
Text of first section goes here. Here is an example of an equation. | Text of first section goes here. Here is an example of an equation. | ||
<math> f(x)= \frac{1}{5} \sin x \int_{-\infty}^\alpha \pi^y dy</math> | <math> f(x)= \frac{1}{5} \sin x \int_{-\infty}^\alpha \pi^y dy</math> | ||
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| − | ==2. Background | + | ==2. Background == |
Text of second section goes here. Here is an example of a list | Text of second section goes here. Here is an example of a list | ||
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| − | ==3. Theory | + | ==3. Theory == |
We can model the DFT as a discrete sampling of the DTFT convolved with a sinc function. The sampling stems from the | We can model the DFT as a discrete sampling of the DTFT convolved with a sinc function. The sampling stems from the | ||
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| − | ==4. CQT | + | ==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. | ||
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==5. References== | ==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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Revision as of 22:54, 22 November 2015
A slecture by ECE student Robert Schwieterman
Partly based on the ECE438 Fall 2015 lecture material of Prof Boutin.
Contents
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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