Fourier变换与逆变换
Fourier变换（纯数学形式）：
F [ f ( t ) ] = ∫ − ∞ ∞ f ( t ) e − i ω t d t F\left[ f\left( t \right) \right]=\int_{-\infty }^{\infty }{f\left( t \right)}{ {e}^{-i\omega t}}dt F[f(t)]=∫−∞∞f(t)e−iωtdt
Fourier逆变换（纯数学形式）：
f ( t ) = 1 2 π ∫ − ∞ ∞ F ( ω ) e i ω t d ω f\left( t \right)=\frac{1}{2\pi }\int_{-\infty }^{\infty }{F\left( \omega \right)}{ {e}^{i\omega t}}d\omega f(t)=2π1∫−∞∞F(ω)eiωtdω
Fourier余弦变换与逆变换
Fourier余弦变换（纯数学形式）：
F [ f ( t ) ] = ∫ − ∞ ∞ f ( t ) e − i ω t d t = 2 ∫ 0 ∞ f ( t ) cos ( ω t ) d t F\left[ f\left( t \right) \right]=\int_{-\infty }^{\infty }{f\left( t \right)}{
{e}^{-i\omega t}}dt=2\int_{0}^{\infty }{f\left( t \right)}\cos \left( \omega t \right)dt F[f(t)]=∫−∞∞f(t)e−iωtdt=2∫0∞f(t)cos(ωt)dt
令 F c [ f ( t ) ] = 2 ∫ 0 ∞ f ( t ) cos ( ω t ) d t {
{F}_{c}}\left[ f\left( t \right) \right]=2\int_{0}^{\infty }{f\left( t \right)}\cos \left( \omega t \right)dt Fc[f(t)]=2∫0∞f(t)cos(ωt)dt
Fourier余弦逆变换（纯数学形式）：
f ( t ) = 1 2 π ∫ − ∞ ∞ F ( ω ) e i ω t d ω = 1 2 π ∫ − ∞ ∞ ( 2 ∫ 0 ∞ f ( t ) cos ( ω t ) d t ) e i ω t d ω = 1 2 π ∫ − ∞ ∞ ( 2 ∫ 0 ∞ f ( t ) cos ( ω t ) d t ) cos ( ω t ) d ω = 1 π ∫ 0 ∞ ( 2 ∫ 0 ∞ f ( t ) cos ( ω t ) d t ) cos ( ω t ) d ω = 1 π ∫ 0 ∞ F c ( ω ) cos ( ω t ) d ω \begin{align} & f\left( t \right)=\frac{1}{2\pi }\int_{-\infty }^{\infty }{F\left( \omega \right)}{ {e}^{i\omega t}}d\omega \\ & \text{ }=\frac{1}{2\pi }\int_{-\infty }^{\infty }{\left( 2\int_{0}^{\infty }{f\left( t \right)}\cos \left( \omega t \right)dt \right){ {e}^{i\omega t}}d\omega } \\ & \text{ }=\frac{1}{2\pi }\int_{-\infty }^{\infty }{\left( 2\int_{0}^{\infty }{f\left( t \right)}\cos \left( \omega t \right)dt \right)\cos \left( \omega t \right)d\omega } \\ & \text{ }=\frac{1}{\pi }\int_{0}^{\infty }{\left( 2\int_{0}^{\infty }{f\left( t \right)}\cos \left( \omega t \right)dt \right)\cos \left( \omega t \right)d\omega } \\ & \text{ }=\frac{1}{\pi }\int_{0}^{\infty }{ { {F}_{c}}\left( \omega \right)}\cos \left( \omega t \right)d\omega \\ \end{align} f(t)=2π1∫−∞∞F(ω)eiωtdω =2π1∫−∞∞(2∫0∞f(t)cos(ωt)dt)eiωtdω =2π1∫−∞∞(2∫0∞f(t)cos(ωt)dt)cos(ωt)dω =π1∫0∞(2∫0∞f(t)cos(ωt)dt)cos(ωt)dω =π1∫0∞Fc(ω)cos(ωt)dω
Fourier正弦变换与逆变换
Fourier正弦变换（纯数学形式）：
F [ f ( t ) ] = ∫ − ∞ ∞ f ( t ) e − i ω t d t = − 2 i ∫ 0 ∞ f ( t ) sin ( ω t ) d t F\left[ f\left( t \right) \right]=\int_{-\infty }^{\infty }{f\left( t \right)}{
{e}^{-i\omega t}}dt=-2i\int_{0}^{\infty }{f\left( t \right)}\sin \left( \omega t \right)dt F[f(t)]=∫−∞∞f(t)e−iωtdt=−2i∫0∞f(t)sin(ωt)dt
令 F s [ f ( t ) ] = 2 ∫ 0 ∞ f ( t ) sin ( ω t ) d t {
{F}_{s}}\left[ f\left( t \right) \right]=2\int_{0}^{\infty }{f\left( t \right)}\sin \left( \omega t \right)dt Fs[f(t)]=2∫0∞f(t)sin(ωt)dt
Fourier正弦逆变换（纯数学形式）：
f ( t ) = 1 2 π ∫ − ∞ ∞ F ( ω ) e i ω t d ω = 1 2 π ∫ − ∞ ∞ ( − 2 i ∫ 0 ∞ f ( t ) sin ( ω t ) d t ) e i ω t d ω = 1 2 π i ∫ − ∞ ∞ ( − 2 i ∫ 0 ∞ f ( t ) sin ( ω t ) d t ) sin ( ω t ) d ω = 1 π ∫ 0 ∞ ( 2 ∫ 0 ∞ f ( t ) sin ( ω t ) d t ) cos ( ω t ) d ω = 1 π ∫ 0 ∞ F s ( ω ) sin ( ω t ) d ω \begin{align} & f\left( t \right)=\frac{1}{2\pi }\int_{-\infty }^{\infty }{F\left( \omega \right)}{
{e}^{i\omega t}}d\omega \\ & \text{ }=\frac{1}{2\pi }\int_{-\infty }^{\infty }{\left( -2i\int_{0}^{\infty }{f\left( t \right)}\sin\left( \omega t \right)dt \right){
{e}^{i\omega t}}d\omega } \\ & \text{ }=\frac{1}{2\pi }i\int_{-\infty }^{\infty }{\left( -2i\int_{0}^{\infty }{f\left( t \right)}\sin\left( \omega t \right)dt \right)\sin \left( \omega t \right)d\omega } \\ & \text{ }=\frac{1}{\pi }\int_{0}^{\infty }{\left( 2\int_{0}^{\infty }{f\left( t \right)}\sin\left( \omega t \right)dt \right)\cos \left( \omega t \right)d\omega } \\ & \text{ }=\frac{1}{\pi }\int_{0}^{\infty }{
{
{F}_{s}}\left( \omega \right)}\sin\left( \omega t \right)d\omega \\ \end{align} f(t)=2π1∫−∞∞F(ω)eiωtdω =2π1∫−∞∞(−2i∫0∞f(t)sin(ωt)dt)eiωtdω =2π1i∫−∞∞(−2i∫0∞f(t)sin(ωt)dt)sin(ωt)dω =π1∫0∞(2∫0∞f(t)sin(ωt)dt)cos(ωt)dω =π1∫0∞Fs(ω)sin(ωt)dω
若 f ( t ) f(t) f(t)是偶函数，则
F c [ f ( t ) ] = F [ f ( t ) ] {
{F}_{c}}\left[ f\left( t \right) \right]=F\left[ f\left( t \right) \right] Fc[f(t)]=F[f(t)]
若 f ( t ) f(t) f(t)是奇函数，则
F s [ f ( t ) ] = i F [ f ( t ) ] {
{F}_{s}}\left[ f\left( t \right) \right]=iF\left[ f\left( t \right) \right] Fs[f(t)]=iF[f(t)]
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