Wednesday, August 14, 2013

A Short Recap on Fourier's Toys

Fourier's building blocks were sine and cosine waves. All the majesty of periodic functions can be constructed by them.

Fourier series

Just a glimpse. A periodic function $f(x)$, with $(-\pi, \pi)$ as its basic periodic unit, can be approached by the sum of an infinite series.

$$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)dx \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx)dx \\ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx)dx \\ a_0 + \sum_{n=1}^{\infty}(a_n \cos (nx) + b_n \sin (nx))$$

Fourier Transform

Also a quick glance,

$$\mathcal{F}(f(t))(s) = \int_{-\infty}^{+\infty} f(t)e^{-2\pi i s t} dt$$

Discrete Fourier Transform (DFT)

A bit into the point. The input of DFT is a finite series of equally-spaced (in time) samples. e.g., f(0), f(0.1), f(0.2), ..., f(2π), and the output is a series of complex numbers encoding the amplitude and phase information, say $\mathcal{F}(0)$, $\mathcal{F}(F_0)$, ..., $\mathcal{F}(-F_0)$. And $F_0$ is the fundamental frequency, which is calculated as

$$F_0 = \frac{1}{N \Delta}$$

$\Delta$ is the sample period, in our example it is $0.1$. The x-axis of these numbers are the frequency spaced as in the figure below,

However, the discrete transform is thus performed, $$\mathcal{F}(n) = \sum_{k=0}^{N} f(k) e^{-2\pi i n k/N}$$

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