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math:transforms
Table of Contents
List of Integral / Space Transforms
Laplace Transforms
$$F(s) = \int_{-\infty}^{\infty}\,f(x)e^{st}\,dt$$
| $f(t)$ | $F(s)$ |
|---|---|
| $$1$$ | $$\frac{1}{s}$$ |
| $$t^n\,,(n=0,1,2,...)$$ | $$\frac{n!}{s^{n+1}}$$ |
| $$t^nf(t)$$ | $$(-1)\stackrel{n}{F}(s)$$ |
| $$e^{at}$$ | $$\frac{1}{s-a}$$ |
| $$e^{at}f(t)$$ | $$F(s-a)$$ |
| $$te^{at}$$ | $$\frac{1}{(s-a)^2}$$ |
| $$t^ne^{at}$$ | $$\frac{n!}{(s-a)^{n+1}}$$ |
| $$\frac{e^{at}-e^{bt}}{a-b}$$ | $$\frac{1}{(s-a)(s-b)}$$ |
| $$\frac{ae^{at}-e^{bt}}{a-b}$$ | $$\frac{s}{(s-a)(s-b)}$$ |
| $$\sin(kt)$$ | $$\frac{k}{s^2+k^2}$$ |
| $$\cos(kt)$$ | $$\frac{s}{s^2+k^2}$$ |
| $$t\sin(kt)$$ | $$\frac{2ks}{(s^2+k^2)^2}$$ |
| $$t\cos(kt)$$ | $$\frac{s^2-k^2}{(s^2+k^2)^2}$$ |
| $$e^{at}\sin(kt)$$ | $$\frac{k}{(s-a)^2-k^2}$$ |
| $$e^{at}\cos(kt)$$ | $$\frac{s-a}{(s-a)^2+k^2}$$ |
| $$\frac{\sin{at}}{t}$$ | $$\text{atan}\left(\frac{a}{s}\right)$$ |
| $$\dot{f}(t)$$ | $$sF(s) -f(0)$$ |
| $$\int_0^t{\,f(t)\,dt}$$ | $$\frac{1}{s}\,F(s)$$ |
| $$f(t) * g(t)$$ | $$F(s)G(s)$$ |
| $$\delta(t)$$ | $$1$$ |
| $$\delta(t-t_0)$$ | $$e^{-st_0}$$ |
| $$u(t-a)$$ | $$\frac{e^{-as}}{s}$$ |
| $$u(t-a)f(t-a)$$ | $$e^{a-s}F(s)$$ |
Z Transforms
$$F[z] = \sum_{n=-\infty}^{\infty}{f[n]z^{-n}}$$
- Geometric Series:
$$\sum_{n=0}^{\infty}{a^n} = \frac{1}{1-a}$$
For most transforms, taking the inversion of the coefficients and arguments leads to the same transform, but the ROC is inverted. e.g.
$$\mathcal{Z}\{u[n]\}=\frac{z}{z-1}\quad ROC\,|z|>1$$ $$\mathcal{Z}\{-u[-n-1]\}=\frac{z}{z-1}\quad ROC\,|z|<1$$
| $$f[n]$$ | $$F[z]$$ | $$ROC$$ |
|---|---|---|
| $$x[n]$$ | $$X[z]$$ | $$r_2 < |z| < r_1$$ |
| $$x[n-k]$$ | $$z^{-k}X[z]$$ | $$|z|\neq 0$$ |
| $$x[n+k]$$ | $$z^{k}X[z]$$ | $$|z|\neq \infty$$ |
| $$a^nx[n]$$ | $$X\left[\frac{z}{a}\right]$$ | $$|a|r_2<|z|<|a|r_1$$ |
| $$\delta [n]$$ | $$1$$ | $$\text{All } z$$ |
| $$u[n]$$ | $$\frac{z}{z-1}=\frac{1}{1-z^{-1}}$$ | $$|z| > 1$$ |
| $$a^nu[n]$$ | $$\frac{z}{z-a}=\frac{1}{1-az^{-1}}$$ | $$|z| > |a|$$ |
| $$nu[n]$$1) | $$\frac{z}{(z-1)^2}=\frac{z^{-1}}{(1-z^{-1})^2}$$ | $$|z| > 1$$ |
| $$n^2u[n]$$ | $$\frac{z(z+1)}{(z-1)^3}=\frac{1+z^{-1}}{z(1-z^{-1})^2}$$ | $$|z| > 1$$ |
| $$na^nu[n]$$ | $$\frac{za}{(z-a)^2}=\frac{az^{-1}}{(1-az^{-1})^2}$$ | $$|z| > |a|$$ |
| $$\cos(\omega_0 n)u[n]$$ | $$\frac{z(z-\cos(\omega_0))}{z^2-2z\cos(\omega_0)+1}=\frac{1-z^1\cos(\omega_0)}{1-2z^{-1}\cos(\omega_0)+z^{-2}}$$ | $$|z| > 1$$ |
| $$\sin(\omega_0 n)u[n]$$ | $$\frac{z\sin(\omega_0)}{z^2-2z\cos(\omega_0)+1}=\frac{z^{-1}\sin(\omega_0)}{1-2z^{-1}\cos(\omega_0)+z^{-2}}$$ | $$|z| > 1$$ |
| $$a^n\cos(\omega_0 n)u[n]$$ | $$\frac{z(z-a\cos(\omega_0))}{a^2-2az\cos(\omega_0)+1}=\frac{1-az^1\cos(\omega_0)}{1-2az^{-1}\cos(\omega_0)+a^2z^{-2}}$$ | $$|z| > 1$$ |
| $$a^n\sin(\omega_0 n)u[n]$$ | $$\frac{az\sin(\omega_0)}{a^2-2az\cos(\omega_0)+z^2}=\frac{az^{-1}\sin(\omega_0)}{1-2az^{-1}\cos(\omega_0)+a^2z^{-2}}$$ | $$|z| > 1$$ |
| $$e^{at}$$ | $$\frac{z}{z-e^{at}}$$ | $$|z| > 1$$ |
1)
$ramp(t)$
math/transforms.txt · Last modified: by theorytoe
