Fourier Transform

Fourier Series

Given a signal $s(t)$ with period $P$, as mentioned earlier, it can be expanded into a Fourier series:

$$s(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n\cos(\frac{2\pi n t}{P}) + b_n\sin(\frac{2\pi n t}{P}))$$

where

$$a_n = \frac{2}{P}\int_{-P/2}^{P/2} s(t)\cos(\frac{2\pi n t}{P})\ dt$$

$$b_n = \frac{2}{P}\int_{-P/2}^{P/2} s(t)\sin(\frac{2\pi n t}{P})\ dt$$

Concept of Coordinates

In fact, $\cos(2\pi n t / P)$ and $\sin(2\pi n t / P)$ can be considered as orthogonal vectors in the interval $[0,P]$, with all non-zero vectors forming a basis. The coefficients $a_n$ and $b_n$ can be considered as a set of coordinates of the function $s(t)$ under this basis, and the values of $a_n$ and $b_n$ are the projections of $s(t)$ on each basis vector, or the results of the inner product.

Complex Form

Using Euler's formula

$$e^{j\omega t} = \cos(\omega t) + j \sin(\omega t)$$

and converting the $\cos$ and $\sin$ functions into complex form, we get the complex form of the Fourier series:

$$s(t) = \sum_{n=-\infty}^{\infty} c_n e^{j 2 \pi n f_0 t}$$

where

$$f_0 = \frac{1}{P}, \ \ c_n = \frac{1}{P} \int_{-P/2}^{P/2} s(t) e^{-j 2 \pi n f_0 t} \, dt$$

Concept of Coordinates

In fact, $\{e^{-j 2 \pi n f_0 t}, n\in Z\}$ can be considered as orthogonal vectors in the interval $[0,P]$, forming a basis. The coefficient $c_n$ can be considered as a set of coordinates of the function $s(t)$ under this basis, and the value of $c_n$ is the projection of $s(t)$ on each basis vector, or the result of the inner product.

Fourier Transform

Letting the period $P$ of $s(t)$ go to infinity, $s(t)$ can be seen as an aperiodic function. The frequency interval between Fourier series terms $1/P$ approaches 0, becoming a continuous spectrum. This leads to the derivation of the Fourier transform pair for $s(t)$:

$$s(t) = \int_{-\infty}^{\infty} S(f) e^{j 2 \pi f t} \, df$$

$$S(f) = \int_{-\infty}^{\infty} s(t) e^{-j 2 \pi f t} \, dt$$

where $s(t)$ can be seen as the time domain form of the signal, and $S(f)$ as the frequency domain form. The Fourier transform pair is a tool for converting between time and frequency domains. Typically, the Fourier transform of $s(t)$ is written as $\mathcal{F}\{s(t)\} = S(f)$, and the inverse Fourier transform of $S(f)$ is written as $\mathcal{F}^{-1}\{S(f)\} = s(t)$.

Convolution

The convolution of $s_1(t)$ and $s_2(t)$ is defined as

$$s_1(t) * s_2(t) = \int_{-\infty}^{\infty} s_1(\tau) s_2(t - \tau) \, d\tau$$

Then

$$\begin{align*} \mathcal{F}\{s_1(t)*s_2(t)\} &= \mathcal{F}\{s_1(t)\}\mathcal{F}\{s_2(t)\} \\ &= S_1(f)S_2(f) \end{align*}$$

$$\begin{align*} \mathcal{F}\{s_1(t)s_2(t)\} &= \mathcal{F}\{s_1(t)\}*\mathcal{F}\{s_2(t)\} \\ &= S_1(f)*S_2(f) \end{align*}$$

Exercise 1

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