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\font\mali=cmr8
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\naslov={Tablice transformacija za \it Signale i sustave}
\upozorenje={\mali Na pismenom ispitu ili na kolokviju iz {\maliit Signala  i  sustava}
dozvoljeno je imati isklju\v civo pribor za pisanje, kalkulator bez
bilje\v zaka vezanih uz predmet te ovaj \v salabahter. Ovaj \v salabahter je dostupan
na {\malitt http://sis.zesoi.fer.hr/}.
\copyright\ ZESOI-FER-Sveu\v cili\v ste u Zagrebu, 2004.
Dozovljeno je umna\v zanje i distribucija ovog \v salabahtera samo ako svaka kopija sadr\v zi
gore navedenu informaciju o autorskim pravima te ovu dozvolu o umna\v zanju. }
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\dio Fourierova transformacija

$\strut\displaystyle {\cal F} \bigl[f(t)\bigr] = \int _{-\infty}^{+\infty}f(t)e^{-j\omega t}dt$

Tablica $\cal F$ transformacije
$$\rect\Bigl({t\over T}\Bigr)\ufrek T\sinc\Bigl({\omega T\over2\pi}\Bigr)$$
$$\sinc(at)\ufrek{1\over a}\rect\Bigl({\omega\over2\pi a}\Bigr)$$
$$1\ufrek2\pi\delta(\omega)$$
$$\sin(\omega_0t)\ufrek{-j\pi}\bigr(\delta(\omega-\omega_0)-\delta(\omega+\omega_0)\bigl)$$
$$\cos(\omega_0t)\ufrek{\pi}\bigr(\delta(\omega-\omega_0)+\delta(\omega+\omega_0)\bigl)$$
$$\sgn(t)\ufrek{2\over j\omega}$$
$$\sum_{i=-\infty}^{+\infty}\delta(t-iT_0)\ufrek{2\pi\over T_0}\sum_{i=-\infty}^{+\infty}
\delta\Bigl({\omega\over2\pi}-{i\over T_0}\Bigr)$$

Pri tome je:
$$\rect(x)=\biggl\{{1,-1/2<x<1/2\atop0,|x|>1/2}$$
$$\sinc(x)={\sin(\pi x)\over\pi x}$$

\dio Laplaceova transformacija

$\strut\displaystyle {\cal L} \bigl[f(t)\bigr] = \int _{0}^{+\infty}f(t)e^{-st}dt $

Tablica $\cal L$ transformacije
$$1\ufrek{1\over s}$$
$$t\ufrek{1\over s^2}$$
$$e^{-at}\ufrek{1\over s+a}$$
$${1\over b-a}(e^{-at}-e^{-bt})\ufrek{1\over (s+a)(s+b)}$$
$${1\over a-b}(ae^{-at}-be^{-bt})\ufrek{s\over (s+a)(s+b)}$$
$${1\over a}e^{-bt}\sin(at)\ufrek{1\over (s+b)^2+a^2}$$
$$e^{-bt}\bigl(\cos(at)-{b\over a}\sin(at)\bigr)\ufrek{s\over (s+b)^2+a^2}$$


\dio $\cal Z$-transformacija

$\strut\displaystyle {\cal Z} \bigl[f[n]\bigr] = \sum _{n=0}^{+\infty} f[n] z^{-n}$

Tablica $\cal Z$ transformacije
$$ \delta [n] \ufrek 1 $$
$$ \delta [n-m] \ufrek z^{-m} $$
$$ n \ufrek {z\over (z-1)^2}$$
$$ {1^n}\ufrek {{1}\over {1-z^{-1}}} = {{z}\over {z-1}}$$
$$ {a^n}\ufrek {{1}\over {1-az^{-1}}} = {{z}\over {z-a}} $$
$$ (n+1)a^n\ufrek {{z^2}\over {(z-a)^2}} $$
$$ {(n+1)(n+2)\over 2!}a^n \ufrek {z^3\over(z-a)^3}$$
$$ {{{(n+1)(n+2)\dots (n+m-1)}\over {(m-1)!}} a^n}\ufrek {{z^m}\over {(z-a)^m}} $$
$$ {a^n - \delta (n)}\ufrek{a\over {z - a}}$$
$$ {\sin {(an)}}\ufrek {{z \sin {(a)}}\over {z^2 - 2z \cos {(a)} + 1}} $$
$$ {\cos {(an)}}\ufrek {{z^2 - z \cos {(a)}}\over {z^2 - 2z \cos {(a)} + 1}} $$

Svojstva $\cal Z$ transformacije

Neka je ${\cal Z} \bigl[f[n]\bigr] = F(z)$ i ${\cal Z} \bigl[g[n]\bigr] = G(z)$. Tada vrijedi:
$$ {a f[n]}\ufrek {aF(z)} $$
$$ {a^n f[n]}\ufrek {F({z\over a} )} $$
$$ {n f[n] }\ufrek {-z {{dF(z)}\over {dz}}} $$
$$ {f[n+1]}\ufrek {zF(z)- zf[0]} $$
$$ {f[n+m]}\ufrek {z^mF(z) - \sum _{i=0}^{m-1} f[i] z^{m-i}} $$
$$ {f[n-1]}\ufrek {{1\over z} F(z) + f[-1]} $$
$$ {f[n-m]}\ufrek {z^{-m}F(z) + \sum _{i=0}^{m-1} f[i-m] z^{-i}} $$
$$ {\sum _{i=0}^{+\infty} f[i]g[n-i]}\ufrek {F(z)G(z)} $$

\bye


