傅立葉轉換

Fourier Transform
1-Dimension
F(u) = ∫ f(x)*exp(-i*2Pi*u*x) dx — continuous
= (1/N)*Σ( f(x)*exp(-i*2Pi*u*x/N) ) — discrete
f(x,y) = ∫ F(u)*exp(i*2Pi*u*x) du
= (1/N)*Σ( F(u)*exp(i*2Pi*u*x/N) )
2-Dimension
F(u,v) = ∫ f(x,y)*exp(-i*2Pi*(ux+vy)) dxdy
= (1/MN) * Σ( f(x,y)*exp(-i*2Pi*(ux/M+vy/N) )
f(x,y) = ∫( F(u,v)*exp(i*2Pi*(ux+vy) ) dudv
(1/MN) * Σ( F(u,v)*exp(i*2Pi*(ux/M+vy/N) )
Convolution:
f(x)*g(x) = ∫(f(t)g(x-t)) dt
Convolution Theorem :
f(x)*g(x) <=> F(u)*G(u)
Coorelation:
f(x).g(x) = ∫( f"(t)g(x+t) ) dt — f"(t) means the complex conjugate(共軛複數) of f(t)
Coorelation Theorem :
f(x).g(x) = F"(u)*G(u)
f"(x)*g(x)= F(u).G(u)

拉普拉斯轉換

Laplace Transform
F(s) = L[ f(t) ] = ∫ f(t)*exp(-st) dt , 0 <= t <= infinite, s = a + bi is a complex number
f(t) = L-[ F(s) ] = ∫ F(s)*exp(st) dt / 2*Pi*i , c - i * infinite < s < c + i * infinite
Laplace transform table
original function f(t)
Laplace transform F(s)
u ( t ) 1 / s
exp ( a * t ) 1 / s - a
sin ( b * t ) b / ( s ^ 2 + b ^ 2 )
cos ( b * t ) s / ( s ^ 2 + b ^ 2 )
sinh ( b * t ) b / ( s ^ 2 - b ^ 2 )
cosh ( b * t ) s / ( s ^ 2 - b ^ 2 )
exp ( - a * t ) * f ( t ) F ( s + a )
t^n n! / s ^ ( n+1 )
註:事實上, Fourier Transform 是 Laplace Transform 的一個 Special case

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