to update ：1 APR 2016

For Fourier series, see Mathematical equations ：Fourier Series

Fourier Transformation ：

For satisfying Dirichlet The function of the condition $$f(t)$$ Define... At its continuous point

$$F(\omega)=\int_{-\infty}^{+\infty}f(t)e^{-\mathrm{i}\omega t}dt$$

be $$f(t)$$ Can be transformed into

$$f(t)=\dfrac{1}{2\pi}\int_{-\infty}^{+\infty}F(\omega)e^{\mathrm{i}\omega t}d \omega$$

This is known as Fourier Transformation , It's a one-to-one mapping in a function space , Write it down as

$$F(\omega)=\mathscr{F}[f(t)],\qquad f(t)=\mathscr{F}^{-1}[F(\omega)]$$

Fourier The basic properties of transformation ：

1. linear

$$\mathscr{F}[\alpha f_1(t)+\beta f_2(t)]=\alpha \mathscr{F}[f_1(t)]+\beta \mathscr{F}[f_2(t)]$$

2. Differentiability

(1) $$\mathscr{F}[f’(t)]=\mathrm{i}\omega\mathscr{F}[f(t)]$$

(2) $$\dfrac{d}{d\omega}\mathscr{F}[f(t)]=\mathscr{F}[-\mathrm{i}tf(t)]$$

3. Integrality

Ruodong $$t \rightarrow +\infty$$ when ,$$g(t)=\int_{-\infty}^tf(a)da \rightarrow 0$$, be

$$\mathscr{F}\left[\int_{-\infty}^tf(a)da\right]=\dfrac{1}{\mathrm{i}\omega}\mathscr{F}[f(t)]$$

Convolution

Convolution is a binary operation defined in function space . For the function $$f_1(t)$$,$$f_2(t)$$, Define convolution operations $$*$$

$$f_1(t)*f_2(t)=\int_{-\infty}^{+\infty}f_1(\tau)f_2(t-\tau)d\tau$$

Convolution operation satisfies commutative law 、 Associative law 、 The distributive law of addition .

Convolution theorem

if $$f_1(t)$$,$$f_2(t)$$ Can be done Fourier Transformation , be

$$\mathscr{F}[f_1(t)*f_2(t)]=\mathscr{F}[f_1(t)]\mathscr{F}[f_2(t)]$$

Transpose convolution and multiplication .

In mathematical equation, it can be used to solve the function which is difficult to inverse transformation —— Decomposing factors to simplify transformations .

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