Definition: f 1 ( x ) f_1(x)f1Both ( x ) and f_2(x) can be Fourier transformed:
f 1 ( x ) ∗ f 2 ( x ) = ∫ − ∞ + ∞ f 1 ( x − ξ ) f 2 ( ξ ) d ξ = ∫ − ∞ + ∞ f 1 ( ξ ) f 2 ( x − ξ ) d ξ f_1(x)*f_2(x)=\int_{-\infty}^{+\infty}f_1(x-\xi)f_2(\xi)d\xi = \int_{-\infty}^{+ \infty}f_1(\xi)f_2(x-\xi)d\xif1(x)∗f2(x)=∫− ∞+ ∞f1(x−ξ ) f2( ξ ) d ξ=∫− ∞+ ∞f1( ξ ) f2(x−ξ ) d ξ
f 1 ( t ) ∗ f 2 ( t ) = ∫ 0 t f 1 ( t − τ ) f 2 ( τ ) d τ = ∫ 0 t f 1 ( τ ) f 2 ( t − τ ) d τ f_1(t)*f_2(t)=\int_{0}^{t}f_1(t-\tau)f_2(\tau)d\tau = \int_{0}^{t}f_1(\tau)f_2(t-\tau)d\tau f1(t)∗f2(t)=∫0tf1(t−t ) f2( t ) d t=∫0tf1( t ) f2(t−t ) d t
Fourier transform
Definition: function f ( x ) f(x)f ( x ) is piecewise smooth, and at( − ∞ , + ∞ ) (-\infty,+\infty)( − ∞ ,+ ∞ ) is absolutely integrable (∫ − ∞ + ∞ ∣ f ( x ) ∣ dx < + ∞ \int_{-\infty}^{+\infty}|f(x)|dx<+\infty∫− ∞+ ∞∣f(x)∣dx<+ ∞ ), Fourier transform:
F [ f ( x ) ] ( w ) = ∫ − ∞ ∞ f ( x ) e i w x d x F[f(x)](w)=\int_{-\infty}^{\infty}f(x)e^{iwx}dx F[f(x)](w)=∫− ∞∞f(x)eiwxdx
Inverse transform
F − 1 [ g ( w ) ] ( x ) = 1 2 π ∫ − ∞ ∞ g ( w ) eiwxdw F^{-1}[g(w)](x)=\frac{1}{2\pi }\int_{-\infty}^{\infty}g(w)e^{iwx}dwF−1[g(w)](x)=2 p.m1∫− ∞∞g ( w ) ei w x dw
nature
Linearity
F [ α f 1 ( x ) + β f 2 ( x ) ] ( w ) = α F [ f 1 ( x ) ] ( w ) + β F [ f 2 ( x ) ] ( w ) F[\alpha f_1(x)+\beta f_2(x)](w)=\alpha F[f_1(x)](w)+\beta F[f_2(x)](w) F[αf1(x)+βf2(x)](w)=αF[f1(x)](w)+βF[f2(x)](w)
Convolution Theorem
F [ f 1 ( x ) ∗ f 2 ( x ) ] ( w ) = F [ f 1 ( x ) ] ( w ) ⋅ F [ f 2 ( x ) ] ( w ) F[f_1(x)*f_2(x)](w)=F[f_1(x)](w)\cdot F[f_2(x)](w) F[f1(x)∗f2(x)](w)=F[f1(x)](w)⋅F[f2(x)](w)
Differential properties
F [ f ( n ) ( x ) ] ( w ) = ( iw ) n F [ f ( x ) ] ( w ) F[f^{(n)}(x)](w)=(iw)^nF [f(x)](w)F[f(n)(x)](w)=(iw)nF[f(x)](w)
Integral properties
F [ ∫ − ∞ x f ( ξ ) d ξ ] = 1 i w F [ f ( x ) ] F[\int_{-\infty}^xf(\xi)d\xi]=\frac{1}{iw}F[f(x)] F[∫− ∞xf ( ξ ) d ξ ]=iw1F[f(x)]
Laplace Transform
Definition: the function f ( t ) f(t)f ( t ) satisfies the condition
{ 0 t < 0 f ( t ) t ≥ 0 \left\{\begin{matrix}0 & t<0 \\f(t) & t\geq 0 \end{matrix}\right. {
0f(t)t<0t≥0
t → + ∞ t\rightarrow+\infty t→+ ∞ , there are constantsM > 0 , S 0 > 0 M>0, S_0>0M>0 ,S0>0 ( S 0 S_0 S0is f ( t ) f(t)Growth index of f ( t )) ∣ f ( t ) ∣ ≤ M e S 0 t , ( 0 < t < + ∞ ) |f(t)|\leq Me^{S_0t}, (0<t<+\ infty)∣f(t)∣≤MeS0t,( 0<t<+ ∞ )
f(t) of f(t)Laplace transform of f ( t ) :
L [ f ( t ) ] ( p ) = ∫ 0 + ∞ f ( t ) e − p t d t L[f(t)](p)=\int_0^{+\infty}f(t)e^{-pt}dt L[f(t)](p)=∫0+ ∞f(t)e- pt dt
Inverse transform
f ( t ) = 1 2 π i ∫ β − i ∞ β + i ∞ g ( p ) e p t d p f(t)=\frac{1}{2\pi i}\int_{\beta-i\infty}^{\beta+i\infty}g(p)e^{pt}dp f(t)=2πi1∫β − i ∞β + i ∞g(p)eptdp
nature
Linearity
L [ α f 1 ( t ) + β f 2 ( t ) ] ( p ) = α L [ f 1 ( t ) ] ( p ) + β L [ f 2 ( t ) ] ( p ) L[\alpha f_1(t)+\beta f_2(t)](p)=\alpha L[f_1(t)](p)+\beta L[f_2(t)](p) L[αf1(t)+βf2(t)](p)=αL[f1(t)](p)+βL[f2(t)](p)
Convolution Theorem
L [ f 1 ( t ) ∗ f 2 ( t ) ] = L [ f 1 ( t ) ] ⋅ L [ f 2 ( t ) ] L[f_1(t)*f_2(t)]=L[f_1(t)]\cdot L[f_2(t)] L[f1(t)∗f2(t)]=L[f1(t)]⋅L[f2(t)]
Differential properties
L [ f ′ ( t ) ] ( p ) = p L [ f ( t ) ] ( p ) − f ( 0 ) L[f'(t)](p)=pL[f(t)](p)-f(0) L[f′(t)](p)=pL[f(t)](p)−f(0)
L [ f ′ ′ ( t ) ] ( p ) = p 2 L [ f ( t ) ] ( p ) − pf ( 0 ) − f ′ ( 0 ) L[f''(t)](p)=p ^2L[f(t)](p)-pf(0)-f'(0)L[f′′(t)](p)=p2L[f(t)](p)−pf(0)−f′ (0)
Integral properties
L [ ∫ 0 tf ( τ ) d τ ] = 1 p L [ f ( t ) ] L[\int_0^tf(\tau)d\tau]=\frac{1}{p}L[f(t) ]L[∫0tf ( τ ) d τ ]=p1L[f(t)]