ChapterZero » simple signal compression

simple signal compression

Mathematics — Alex @ 1:04 am

Question

Suppose you have a (periodic) function $f : [0, 2\pi) \rightarrow \C$ whose support is limited to the interval $[0,\frac{2\pi}{m}]$ , where m>2 . Is there a more compact representation than the entire fourier series? It seems like a good possibility, since the information in the function is actually restricted to a smaller region.

Answer

Yes. Let $\tilde f : [0,2\pi] \rightarrow \C$ be defined as $f : \theta \rightarrow f(\tilde \theta)$ where $\tilde \theta = \theta + \frac{2\pi k}{m}$ , where $k \in \Z$ and $\tilde \theta \in [0,\frac{2\pi}{m}]$ . Then consider the fourier coefficients for $\tilde f$ :

$\displaystyle \tilde c_k = \frac{1}{2\pi} \int_0^{2\pi} \tilde f(\theta) e^{-i k \theta} d\theta = \frac{1}{2\pi} \sum_{n=0}^{m-1} \int_{n\left(\frac{2\pi}{m}\right)}^{(n+1)\left(\frac{2\pi}{m}\right)} \tilde f(\theta) e^{-i k \theta } d\theta$

$\displaystyle = \frac{1}{2\pi} \sum_{n=0}^{m-1} \int_0^{\frac{2\pi}{m}} \tilde f(\theta + \frac{n2\pi}{m}) e^{-i k \left(\theta + \frac{n 2 \pi}{m}\right)} d\theta = \left(\sum_{n=0}^{m-1} e^{- \left( i k \frac{2\pi}{m} \right) n } \right) \int_0^{\frac{2\pi}{m}} \tilde f(\theta) e^{-i k \theta} d\theta = c_k \left(\sum_{n=0}^{m-1} e^{- \left( i k \frac{2\pi}{m} \right) n } \right)$

Since the sum of exponentials is when $m \mid k$ and 0 otherwise, we can throw out all the zero coefficients and represent $\tilde f$ or using just the coefficients $\{m c_{mk}\}_{k \in\Z}$ .

Comments

This is not directly useful, but considering where I got the idea from (minus the nice proof, all thought out by me it probably ends up figuring into quadrature mirror filter theory somewhere. Maybe this has something to do with the ubiquitous downsampling I keep hearing about in relation to wavelet transforms.

Anyhow, it is an interesting result. Notice that as $m \rightarrow \infty$ , the sequence $\{m c_{mk}\}_{k\in \Z}$ approaches a delta pulse, in a certain sense. Which, in a certain sense, is as expected.