Hoán chuyển tích phân

Phép toán giải tích của một tích phân xác định dùng trong việc Hoán chuyển hệ tóan .

Hệ thời gian	Hệ Laplace	Hệ Fourier	Hệ Góc độ
${\frac {d}{dt}}$	$s$	$j\omega$	$\omega \angle 90^{o}$
${\frac {d^{n}}{dt^{n}}}f(x)$	$s^{n}f(x)$	$j\omega ^{n}f(x)$	${\frac {\omega }{n}}\angle 90^{o}$
$\int dt$	${\frac {1}{s}}=-s$	${\frac {1}{j\omega }}=-j\omega$	${\frac {1}{\omega }}\angle -90^{o}$

Hoán Chuyển Laplace

Hoán Chuyển Laplace được dùng để biến đổi hệ số thời gian sang hệ số phức

$f(t)->F(s)$ .

Cong thuc

${\mathcal {L}}\{f(t)\}=F(s)=\int \limits _{0^{-}}^{\infty }f(t)e^{-st}dt$

Thí dụ

${\frac {d}{dt}}$	$\int dx$
${\frac {d}{dt}}f(t)$	$sf(t)$

Luật toán hoán chuyển Laplace

Property	Definition
Linearity	${\mathcal {L}}\left\{af(t)+bg(t)\right\}=aF(s)+bG(s)$
Differentiation	${\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0^{-})$ ${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})$ ${\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$
Frequency Division	${\mathcal {L}}\{tf(t)\}=-F'(s)$ ${\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)$
Frequency Integration	${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma$
Time Integration	${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{u(t)*f(t)\right\}={1 \over s}F(s)$
Scaling	${\mathcal {L}}\left\{f(at)\right\}={1 \over a}F\left({s \over a}\right)$
Initial value theorem	$f(0^{+})=\lim _{s\to \infty }{sF(s)}$
Final value theorem	$f(\infty )=\lim _{s\to 0}{sF(s)}$
Frequency Shifts	${\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)$ ${\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)$
Time Shifts	${\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)$ ${\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)$
Convolution Theorem	${\mathcal {L}}\{f(t)*g(t)\}=F(s)G(s)$

Where:

$f(t)={\mathcal {L}}^{-1}\{F(s)\}$

$g(t)={\mathcal {L}}^{-1}\{G(s)\}$

$s=\sigma +j\omega$

Công thức hoán chuyển Laplace

	$x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$	$X(s)={\mathcal {L}}\left\{x(t)\right\}$
1	${\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds$	$\int _{-\infty }^{\infty }x(t)e^{-st}dt$
2	$\delta (t)\,$	$1\,$
3	$\delta (t-a)\,$	$e^{-as}\,$
4	$u(t)\,$	${\frac {1}{s}}$
5	$u(t-a)\,$	${\frac {e^{-as}}{s}}$
6	$tu(t)\,$	${\frac {1}{s^{2}}}$
7	$t^{n}u(t)\,$	${\frac {n!}{s^{n+1}}}$
8	${\frac {1}{\sqrt {\pi t}}}u(t)$	${\frac {1}{\sqrt {s}}}$
9	$e^{at}u(t)\,$	${\frac {1}{s-a}}$
10	$t^{n}e^{at}u(t)\,$	${\frac {n!}{(s-a)^{n+1}}}$
11	$\cos(\omega t)u(t)\,$	${\frac {s}{s^{2}+\omega ^{2}}}$
12	$\sin(\omega t)u(t)\,$	${\frac {\omega }{s^{2}+\omega ^{2}}}$
13	$\cosh(\omega t)u(t)\,$	${\frac {s}{s^{2}-\omega ^{2}}}$
14	$\sinh(\omega t)u(t)\,$	${\frac {\omega }{s^{2}-\omega ^{2}}}$
15	$e^{at}\cos(\omega t)u(t)\,$	${\frac {s-a}{(s-a)^{2}+\omega ^{2}}}$
16	$e^{at}\sin(\omega t)u(t)\,$	${\frac {\omega }{(s-a)^{2}+\omega ^{2}}}$
17	${\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)$	${\frac {1}{(s^{2}+\omega ^{2})^{2}}}$
18	${\frac {t}{2\omega }}\sin \omega t$	${\frac {s}{(s^{2}+\omega ^{2})^{2}}}$
19	${\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)$	${\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}$

Hoán Chuyển Fourier

Hoán Chuyển Fourier , chuyển hệ từ hệ thời gian thực sang hệ số ảo

$f(t)->F(j\omega )$ .

Cong thuc

${\mathcal {L}}\{f(t)\}=F(s)=\int \limits _{0^{-}}^{\infty }f(t)e^{-j\omega t}dt$

Thí dụ

$f(x)$	${\frac {d}{dt}}$	$\int dx$
$F(j\omega )=\int f(x)e^{j\omega x}dx$	$j\omega$	${\frac {1}{j\omega }}$

Luật hoán chuyển Fourier

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$	$G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$	$G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$
1	$a\cdot g(t)+b\cdot h(t)\,$	$a\cdot G(\omega )+b\cdot H(\omega )\,$	$a\cdot G(f)+b\cdot H(f)\,$	Linearity
2	$g(t-a)\,$	$e^{-ia\omega }G(\omega )\,$	$e^{-i2\pi af}G(f)\,$	Shift in time domain
3	$e^{iat}g(t)\,$	$G(\omega -a)\,$	$G\left(f-{\frac {a}{2\pi }}\right)\,$	Shift in frequency domain, dual of 2
4	$g(at)\,$	${\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,$	${\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,$	If $|a|\,$ is large, then $g(at)\,$ is concentrated around 0 and ${\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,$ spreads out and flattens
5	$G(t)\,$	$g(-\omega )\,$	$g(-f)\,$	Duality property of the Fourier transform. Results from swapping "dummy" variables of $t\,$ and $\omega \,$.
6	${\frac {d^{n}g(t)}{dt^{n}}}\,$	$(i\omega )^{n}G(\omega )\,$	$(i2\pi f)^{n}G(f)\,$	Generalized derivative property of the Fourier transform
7	$t^{n}g(t)\,$	$i^{n}{\frac {d^{n}G(\omega )}{d\omega ^{n}}}\,$	$\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}G(f)}{df^{n}}}\,$	This is the dual to 6
8	$(g*h)(t)\,$	${\sqrt {2\pi }}G(\omega )H(\omega )\,$	$G(f)H(f)\,$	$g*h\,$ denotes the convolution of $g\,$ and $h\,$ — this rule is the convolution theorem
9	$g(t)h(t)\,$	$(G*H)(\omega ) \over {\sqrt {2\pi }}\,$	$(G*H)(f)\,$	This is the dual of 8
10	For a purely real even function $g(t)\,$	$G(\omega )\,$ is a purely real even function	$G(f)\,$ is a purely real even function
11	For a purely real odd function $g(t)\,$	$G(\omega )\,$ is a purely imaginary odd function	$G(f)\,$ is a purely imaginary odd function