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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