% Pregled formula - DOS
% Copyright (C) 2007 Tomislav Petkovic (tomislav.petkovic.ml@fer.hr)
% Copyright (C) 2007 University of Zagreb - FER - ZESOI
% May be freely distributed.
% This file is intended to be processed by plain TeX.
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\naslov={Pregled formula iz \it Digitalne obradbe signala}
\upozorenje={\mali Na pismenom ispitu  iz {\maliit Digitalne obradbe signala}
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://dos.zesoi.fer.hr/}.
\copyright\ Sveu\v cili\v ste u Zagrebu-FER-ZESOI, 2007.
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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\def\rect{\mathop{\rm rect}\nolimits}
\def\sinc{\mathop{\rm sinc}\nolimits}
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\ln{\mathop{\rm ln}\nolimits}
\def\tg{\mathop{\rm tg}\nolimits}
\def\ctg{\mathop{\rm ctg}\nolimits}
\def\dirac{\mathop{\delta}\nolimits}
\def\real{\mathop{\rm Re}\nolimits}
\def\erf{\mathop{\rm erf}\nolimits}
\def\erfc{\mathop{\rm erfc}\nolimits}
\def\imag{\mathop{\rm Im}\nolimits}
\def\step{\mathop{\rm s}\nolimits}
\def\tri{\mathop{\rm tri}\nolimits}
\def\dft{\mathop{\rm DFT}\nolimits}
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\def\Dj {\hbox{\Di }} 

\dio Osnovne trigonometrijske jednakosti

$$\sin x={1\over2j}(e^{jx}-e^{-jx})$$
$$\cos x={1\over2}(e^{jx}+e^{-jx})$$
$$\sin\Bigl(x\pm{\pi\over2}\Bigr)=\pm\cos x$$
$$\cos\Bigl(x\pm{\pi\over2}\Bigr)=\mp\sin x$$

$$\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y$$
$$\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y$$

$$\sin x+\sin y=2\sin{x+y\over2}\cos{x-y\over2}$$
$$\sin x-\sin y=2\sin{x-y\over2}\cos{x+y\over2}$$
$$\cos x+\cos y=2\cos{x+y\over2}\cos{x-y\over2}$$
$$\cos x-\cos y=2\sin{x+y\over2}\sin{y-x\over2}$$

$$\sin x\sin y={1\over2}\bigl(\cos(x-y)-\cos(x+y)\bigr)$$
$$\cos x\cos y={1\over2}\bigl(\cos(x-y)+\cos(x+y)\bigr)$$
$$\sin x\cos y={1\over2}\bigl(\sin(x-y)+\sin(x+y)\bigr)$$

$$\sin(2x)=2\sin x\cos x$$
$$\cos(2x)=\cos^2x-\sin^2x$$
$$\sin(3x)=3\sin x-4\sin^3x$$
$$\cos(3x)=4\cos^3x-3\cos x$$

$$2\sin^2x=1-\cos(2x)$$
$$2\cos^2x=1+\cos(2x)$$
$$4\sin^3x=3\sin x-\sin(3x)$$
$$4\cos^3x=3\cos x+\cos(3x)$$
$$8\sin^4x=3-4\cos(2x)+\cos(4x)$$
$$8\cos^4x=3+4\cos(2x)+\cos(4x)$$

$$a\cos x-b\sin x=r\cos(x+\phi)$$
$$r=\sqrt{a^2+b^2}$$
$$\tg\phi={b/a}$$
$$a=r\cos\phi$$
$$b=r\sin\phi$$

\dio Tablice suma i integrala

\poddio Kona\v cne sume

$$\sum_{i=1}^{n}i={n(n+1)\over2}$$
$$\sum_{i=1}^{n}i^2={n(n+1)(2n+1)\over6}$$
$$\sum_{i=1}^{n}i^3={n^2(n+1)^2\over4}$$
$$\sum_{i=0}^{n}x^i={x^{n+1}-1\over x-1}$$
$$\sum_{i=0}^{n}e^{j(\theta+n\phi)}={
\sin\bigl({(n+1)\phi/2}\bigr)\over\sin(\phi/2)}e^{j(\theta+n\phi/2)}$$
$$\sum_{i=0}^{n}{n\choose i}=\sum_{i=1}^{n}{n!\over i!(n-i)!}=2^n$$


\poddio Neodre\dj eni integrali

\podpoddio Racionalne funkcije

$$\int(ax+b)^n\,dx={(ax+b)^{n+1}\over a(n+1)},\quad 0<n$$
$$\int{1\over ax+b}\,dx={1\over a}\ln|ax+b|$$
$$\int{1\over ax^2+bx+c}\,dx=$$
$$\quad\quad\eqalign{
&={2\over\sqrt{4ac-b^2}}\tg^{-1}
\Bigl({2ax+b\over\sqrt{4ac-b^2}}\Bigr),\cr
&={1\over\sqrt{b^2-4ac}}\ln
\biggl|{2ax+b-\sqrt{b^2-4ac}\over2ax+b+\sqrt{b^2-4ac}}\biggr|,\cr
&={-2\over2ax+b},\cr}
\eqalign{
\strut\phantom{1\over2}b^2&<4ac\cr
\strut\phantom{1\over2}b^2&>4ac\cr
\strut\phantom{1\over2}b^2&=4ac\cr}$$
$$\int{x\,dx\over ax^2+bx+c}=$$
$$\quad\quad\eqalign{&={1\over2a}\ln|ax^2+bx+c|-{b\over2a}\int{dx\over ax^2+bx+c}\cr}$$
$$\int{dx\over a^2x^2+b^2}={1\over ab}\tg^{-1}\Bigl({ax\over b}\Bigr)$$
$$\int{x\,dx\over a^2+x^2}={1\over2}\ln(a^2+x^2)$$
$$\int{x^2\,dx\over a^2+x^2}=x-a\tg^{-1}\Bigl({x\over a}\Bigr)$$
$$\int{dx\over(a^2+x^2)^2}={x\over2a^2(a^2+x^2)}+{1\over2a^3}\tg^{-1}\Bigl({x\over a}\Bigr)$$
$$\int{x\,dx\over(a^2+x^2)^2}={-1\over2(a^2+x^2)}$$
$$\int{x^2\,dx\over(a^2+x^2)^2}={-x\over2(a^2+x^2)}+{1\over2a}\tg^{-1}\Bigl({x\over a}\Bigr)$$

\podpoddio Trigonometrijske funkcije

$$\int\cos(x)\,dx=\sin(x)$$
$$\int x\cos(x)\,dx=\cos(x)+x\sin(x)$$
$$\int x^2\cos(x)\,dx=2x\cos(x)+(x^2-2)\sin(x)$$
$$\int\sin(x)\,dx=-\cos(x)$$
$$\int x\sin(x)\,dx=\sin(x)-x\cos(x)$$
$$\int x^2\sin(x)\,dx=2x\sin(x)+(2-x^2)\cos(x)$$

\podpoddio Eksponencijalne funkcije

$$\int e^{ax}\,dx={1\over a}e^{ax}$$
$$\int xe^{ax}\,dx=\Bigl({x\over a}-{1\over a^2}\Bigr)e^{ax}$$
$$\int x^2e^{ax}\,dx=\Bigl({x^2\over a}-{2x\over a^2}+{2\over a^3}\Bigr)e^{ax}$$
$$\int x^3e^{ax}\,dx=\Bigl({x^3\over a}-{3x^2\over a^2}+{6x\over a^3}-{6\over a^4}\Bigr)e^{ax}$$
$$\int \sin(x)e^{ax}\,dx={1\over a^2+1}\bigl(a\sin(x)-\cos(x)\bigr)e^{ax}$$
$$\int \cos(x)e^{ax}\,dx={1\over a^2+1}\bigl(a\cos(x)+\sin(x)\bigr)e^{ax}$$

\poddio Odre\dj eni integrali

$$\int_{-\infty}^{+\infty}e^{-a^2x^2+bx}\,dx={\sqrt\pi\over a}e^{b^2\over4a^2},\quad a>0$$
$$\int_{0}^{+\infty}x^2e^{-x^2}\,dx={\sqrt\pi\over4}$$
$$\int_{0}^{+\infty}{\sin(x)\over x}\,dx={\pi\over2}$$
$$\int_{0}^{+\infty}{\sin^2(x)\over x^2}\,dx={\pi\over2}$$

\dio Fourierova transformacija

Fourierova transformacija funkcije $x(t)$ je:
$${\cal F}\bigl[x(t)\bigr]=X(\omega)=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}\,dt$$
Inverzna transformacija je:
$${\cal F}^{-1}\bigl[X(\omega)\bigr]=x(t)={1\over2\pi}\int_{-\infty}^{+\infty}X(\omega)e^{j\omega t}\,d\omega$$
Funkcija $x(t)$ i njen spektar $X(\omega)$ \v cine transformacijski par:
$$x(t)\ufrek X(\omega)$$
Dovoljni (ali ne i nu\v zni) uvjeti za postojanje transformacije funkcije $x(t)$ su:

1. Funkcija x(t) zadovoljava Dirichletove uvjete (funkcija je ograni\v cena s kona\v cnim brojem
maksimuma i minimuma te kona\v cnim brojem diskontinuiteta u bilo kojem kona\v cnom vremenskom
intervalu).

2. $\strut\displaystyle\int_{-\infty}^{+\infty}|x(t)|dt<\infty$

\poddio Svojstva Fourierove transformacije

Neka je $x(t)\ufrek X(\omega)$ i neka su $\alpha_i$, $t_0$ i $\omega_0$ konstante.
Fourierova transformacija tada zadovoljava sljede\' ca svojstva:

\podpoddio Linearnost

$$x(t)=\sum_{i=1}^n\alpha_ix_i(t)\ufrek\sum_{i=1}^n\alpha_iX_i(\omega)=X(\omega)$$

\podpoddio Dualnost

$$X(t)\ufrek2\pi x(-\omega)$$

\podpoddio Pomak u vremenu i frekvenciji

$$x(t-t_0)\ufrek X(\omega)e^{-j\omega t_0}$$
$$x(t)e^{j\omega_0t}\ufrek X(\omega-\omega_0)$$

\podpoddio Skaliranje

$$x(\alpha t)\ufrek{1\over|\alpha|}X\Bigl({\omega\over\alpha}\Bigr)$$

\podpoddio Deriviranje

$${d^nx(t)\over dt^n}\ufrek(j\omega)^nX(\omega)$$
$$(-jt)^nx(t)\ufrek{d^nX(\omega)\over d\omega^n}$$

\podpoddio Integriranje

$$\int_{-\infty}^tx(\tau)\,d\tau\ufrek\pi X(0)\dirac(\omega)+{X(\omega)\over j\omega}$$
$$\pi x(0)\dirac(t)-{x(t)\over jt}\ufrek\int_{-\infty}^\omega X(\xi)\,d\xi$$

\podpoddio Konjugacija

$$x^*(t)\ufrek X^*(-\omega)$$
$$x^*(-t)\ufrek X^*(\omega)$$

\podpoddio Konvolucija

$$\int_{-\infty}^{+\infty}x_1(\tau)x_2(t-\tau)\,d\tau\ufrek
X_1(\omega)X_2(\omega)$$
$$x_1(t)x_2(t)\ufrek
{1\over2\pi}\int_{-\infty}^{+\infty}X_1(\xi)X_2(\omega-\xi)\,d\xi$$

\podpoddio Korelacija

$$\int_{-\infty}^{+\infty}x_1^*(\tau)x_2(t+\tau)\,d\tau\ufrek
X_1^*(\omega)X_2(\omega)$$
$$x_1^*(t)x_2(t)\ufrek
{1\over2\pi}\int_{-\infty}^{+\infty}X_1^*(\xi)X_2(\omega+\xi)\,d\xi$$

\podpoddio Parsevalov teorem

$$\int_{-\infty}^{+\infty}x_1^*(t)x_2(t)\,dt=
{1\over2\pi}\int_{-\infty}^{+\infty}X_1^*(\omega)X_2(\omega)\,d\omega$$
$$\int_{-\infty}^{+\infty}|x(t)|^2\,dt=
{1\over2\pi}\int_{-\infty}^{+\infty}|X(\omega)|^2\,d\omega$$

\poddio Tablica $\cal F$ transformacije

Neka je:

$$\step(x)=\left\{\eqalign{&1,\cr&0,\cr}
\eqalign{x&>0\cr x&<0\cr}\right.$$
$$\rect(x)=\left\{
\eqalign{&1,\strut\phantom{1\over2}\cr&0,\strut\phantom{1\over2}\cr}
\eqalign{-{1\over2}&<\phantom{|}x\phantom{|}<{1\over2}\cr{1\over2}&<|x|\cr}\right.$$
$$\tri(x)=\left\{\eqalign{&1-|x|,\cr&0,\cr}
\eqalign{|x|&<1\cr|x|&>1\cr}\right.$$
$$\sinc(x)={\sin(\pi x)\over\pi x}$$

Uz te oznake va\v znije transformacije su:

$$1\ufrek2\pi\dirac(\omega)$$
$$\dirac(t)\ufrek1$$
$$\step(t)\ufrek\pi\dirac(\omega)+{1\over j\omega}$$
$${1\over2}\dirac(t)-{1\over2\pi jt}\ufrek\step(\omega)$$
$$\sgn(t)\ufrek{2\over j\omega}$$
$$\rect\Bigl({t\over T}\Bigr)\ufrek T\sinc\Bigl({\omega T\over2\pi}\Bigr)$$
$$\sinc(at)\ufrek{1\over a}\rect\Bigl({\omega\over2\pi a}\Bigr)$$
$$\tri\Bigl({t\over T}\Bigr)\ufrek T\sinc^2\Bigl({\omega T\over2\pi}\Bigr)$$
$$\sinc^2(at)\ufrek{1\over a}\tri\Bigl({\omega\over2\pi a}\Bigr)$$
$$e^{j\omega_0t}\ufrek2\pi\dirac(\omega-\omega_0)$$
$$\delta(t-t_0)\ufrek e^{-j\omega t_0}$$
$$\sin(\omega_0t)\ufrek-j\pi\bigr(\delta(\omega-\omega_0)-\delta(\omega+\omega_0)\bigl)$$
$$\cos(\omega_0t)\ufrek\pi\bigr(\delta(\omega-\omega_0)+\delta(\omega+\omega_0)\bigl)$$
$$\sum_{i=-\infty}^{+\infty}\delta(t-iT_0)\ufrek{2\pi\over T_0}\sum_{i=-\infty}^{+\infty}
\delta\Bigl({\omega\over2\pi}-{i\over T_0}\Bigr)$$
$$\sin(\omega_0t)\step(t)\ufrek-{j\pi\over2}\bigr(\delta(\omega-\omega_0)-\delta(\omega+\omega_0)\bigl)+
{j\omega\over\omega_0^2-\omega^2}$$
$$\cos(\omega_0t)\step(t)\ufrek{\pi\over2}\bigr(\delta(\omega-\omega_0)+\delta(\omega+\omega_0)\bigl)+
{j\omega\over\omega_0^2-\omega^2}$$
$$e^{-at}\step(t)\ufrek{1\over a+j\omega},\quad a>0$$
$$te^{-at}\step(t)\ufrek{1\over (a+j\omega)^2},\quad a>0$$
$$t^2e^{-at}\step(t)\ufrek{2\over (a+j\omega)^3},\quad a>0$$
$$t^3e^{-at}\step(t)\ufrek{6\over (a+j\omega)^4},\quad a>0$$
$$e^{-a|t|}\ufrek{2a\over a^2+\omega^2}$$
$$e^{-{t^2\over2a^2}}\ufrek a\sqrt{2\pi}e^{-a^2\omega^2\!/2}$$


\dio Vremenski diskretna Fourierova\hfil\break transformacija

Vremenski diskretna Fourierova transformacija (DTFT -- {\sl Discrete-Time Fourier Transform}) niza $x[n]$ je:
$${\cal F}_{\rm vd}\bigl[x[n]\bigr]=X(\omega)=\sum_{n=-\infty}^{+\infty}x[n]e^{-j\omega n}$$
Inverzna transformacija je:
$${\cal F}^{-1}_{\rm vd}\bigl[X(\omega)\bigr]=x[n]={1\over2\pi}\int_{-\pi}^{+\pi}X(\omega)e^{j\omega n}\,d\omega$$
Niz $x[n]$ i njegov spektar $X(\omega)$ \v cine transformacijski par:
$$x[n]\ufrek X(\omega)$$
Dovoljan (ali ne i nu\v zni) uvjet za postojanje transformacije niza $x[n]$ je
apsolutna sumabilnost:
$$\strut\displaystyle\sum_{n=-\infty}^{+\infty}\bigl|x[n]\bigr|<\infty$$

\poddio Svojstva vremenski diskretne Fourierove\hfil\break transformacije

Neka je $x[n]\ufrek X(\omega)$ i neka su $\alpha_i$, $n_0$ i $\omega_0$ konstante.
Vremenski diskretna Fourierova transformacija tada zadovoljava sljede\' ca svojstva:

\podpoddio Linearnost

$$x[n]=\sum_{i=1}^n\alpha_ix_i[n]\ufrek\sum_{i=1}^n\alpha_iX_i(\omega)=X(\omega)$$

\podpoddio Pomak u vremenu i frekvenciji

$$x[n-n_0]\ufrek X(\omega)e^{-j\omega n_0}$$
$$x[n]e^{j\omega_0 n}\ufrek X(\omega-\omega_0)$$

\podpoddio Deriviranje i diferenciranje

$$\Delta x[n]\ufrek(e^{j\omega}-1)X(\omega)$$
$$n^ix[n]\ufrek j^i{d^iX(\omega)\over d\omega^i}$$

\podpoddio Sumiranje

$$\sum_{i=-\infty}^nx[i]\ufrek{1\over1-e^{-j\omega}}X(\omega)$$

\podpoddio Konjugacija

$$x^*[n]\ufrek X^*(-\omega)$$
$$x^*[-n]\ufrek X^*(\omega)$$

\podpoddio Konvolucija

$$\sum_{i=-\infty}^{+\infty}x_1[i]x_2[n-i]\ufrek
X_1(\omega)X_2(\omega)$$
$$x_1[n]x_2[n]\ufrek
{1\over2\pi}\int_{-\pi}^{+\pi}X_1(\xi)X_2(\omega-\xi)\,d\xi$$

\podpoddio Parsevalov teorem

$$\sum_{n=-\infty}^{+\infty}x_1^*[n]x_2[n]=
{1\over2\pi}\int_{-\pi}^{+\pi}X_1^*(\omega)X_2(\omega)\,d\omega$$
$$\sum_{n=-\infty}^{+\infty}\bigl|x[n]\bigr|^2=
{1\over2\pi}\int_{-\pi}^{+\pi}\bigl|X(\omega)\bigr|^2\,d\omega$$

\poddio Relacije simetri\v cnosti

Neka je $x[n]$ \v cisto realan niz i neka je $x[n]\ufrek X(\omega)$. Tada je:
$${1\over2}\bigl(x[n]+x[-n]\bigr)\ufrek\real\bigl[X(\omega)\bigr]$$
$${1\over2}\bigl(x[n]-x[-n]\bigr)\ufrek j\imag\bigl[X(\omega)\bigr]$$

Tako\dj er vrijedi:
$$X(\omega)=X^*(-\omega)$$
$$\real\bigl[X(\omega)\bigr]=\real\bigl[X(-\omega)\bigr]$$
$$\imag\bigl[X(-\omega)\bigr]=-\imag\bigl[X(\omega)\bigr]$$

\poddio Tablica $\cal F_{\rm vd}$ transformacije

$$\delta[n]\ufrek1$$
$$1\ufrek\sum_{i=-\infty}^{+\infty}2\pi\delta(\omega+2\pi i)$$
$$e^{j\omega_0 n}\ufrek\sum_{i=-\infty}^{+\infty}2\pi\delta(\omega-\omega_0+2\pi i)$$
$$\step[n]\ufrek{1\over1-e^{-j\omega}}+\sum_{i=-\infty}^{+\infty}\pi\delta(\omega+2\pi i)$$
$$a^n\step[n]\ufrek{1\over1-ae^{-j\omega}},\quad|a|<1$$
$$na^n\step[n]\ufrek{ae^{j\omega}\over(e^{-j\omega}-a)^2},\quad|a|<1$$
$$\sin(\omega_0 n)\ufrek\!\!\sum_{i=-\infty}^{+\infty}\!
j\pi\bigl(\delta(\omega+\omega_0+2\pi i)-\delta(\omega-\omega_0+2\pi i)\bigr)$$
$$\cos(\omega_0 n)\ufrek\!\!\sum_{i=-\infty}^{+\infty}
\pi\bigl(\delta(\omega+\omega_0+2\pi i)+\delta(\omega-\omega_0+2\pi i)\bigr)$$
$$a^n\sin(\omega_0 n)\step[n]\ufrek{ae^{j\omega}\sin(\omega_0)\over
e^{2j\omega}-2ae^{j\omega}\cos(\omega_0)+a^2},\quad|a|<1$$
$$a^n\cos(\omega_0 n)\step[n]\ufrek{e^{j\omega}\bigl(e^{j\omega}-a\cos(\omega_0)\bigr)\over
e^{2j\omega}-2ae^{j\omega}\cos(\omega_0)+a^2},\quad|a|<1$$

\dio Diskretna Fourierova transformacija

Diskretna Fourierova transformacija kona\v cnog niza $x[n]$ duljine $N$ je:
$$X[k]=\sum_{n=0}^{N-1}x[n]W^{nk}_N,\quad 0\le k\le N-1$$
Inverzna transformacija je:
$$x[n]={1\over N}\sum_{k=0}^{N-1}X[k]W^{-nk}_N,\quad 0\le n\le N-1$$
Niz $x[n]$ i njegov spektar $X[k]$ \v cine transformacijski par:
$$x[n]\ufrek X[k]$$
Pri tome je $W^{nk}_N=e^{-2\pi jnk/N}$.

\poddio Svojstva diskretne Fourierove transformacije

Neka je $x[n]\ufrek X[k]$ i neka su $\alpha_i$, $n_0$ i $k_0$
konstante. DFT tada zadovoljava
sljede\' ca svojstva:

\podpoddio Linearnost

$$x[n]=\sum_{i=1}^n\alpha_ix_i[n]\ufrek\sum_{i=1}^n\alpha_iX_i[k]=X[k]$$

\podpoddio Dualnost

$$X[n]\ufrek Nx\bigl[\langle-k\rangle_N\bigr]$$

\podpoddio Cirkularni pomak u vremenu i frekvenciji

$$x\bigl[\langle n-n_0\rangle_N\bigr]\ufrek X[k]W^{kn_0}_N$$
$$x[n]W^{k_0n}_N\ufrek X\bigl[\langle k-k_0\rangle_N\bigr]$$

\podpoddio Cirkularna konvolucija

$$\sum_{i=0}^{N-1}x_1[i]x_2\bigl[\langle n-i\rangle_N\bigr]\ufrek
X_1[k]X_2[k]$$
$$x_1[n]x_2[n]\ufrek
{1\over N}\sum_{i=0}^{N-1}X_1[i]X_2\bigl[\langle k-i\rangle_N\bigr]$$

\podpoddio Parsevalova relacija

$$\sum_{n=0}^{N-1}x^*_1[n]x_2[n]\ufrek{1\over N}\sum_{k=0}^{N-1}X^*_1[k]X_2[k]$$
$$\sum_{n=0}^{N-1}\bigl|x[n]\bigr|^2\ufrek{1\over N}\sum_{k=0}^{N-1}\bigl|X[k]\bigr|^2$$

\dio $\cal Z$-transformacija

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

\poddio 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:

\podpoddio Linearnost

$$f[n]=\sum_{i=1}^n\alpha_if_i[n]\ufrek\sum_{i=1}^n\alpha_iF_i(z)=F(z)$$

\podpoddio Pomak

$$ {f[n+1]}\ufrek {zF(z)- zf[0]} $$
$$ {f[n+m]}\ufrek {z^mF(z) - \sum _{i=0}^{m-1} f[i] z^{m-i}} $$
$$ {f[n-1]}\ufrek {{1\over z} F(z) + f[-1]} $$
$$ {f[n-m]}\ufrek {z^{-m}F(z) + \sum _{i=0}^{m-1} f[i-m] z^{-i}} $$

\podpoddio Skaliranje
$$ {a^n f[n]}\ufrek {F({z\over a} )} $$

\podpoddio Diferenciranje i deriviranje

$$ \Delta f[n]\ufrek (z-1)F(z)$$
$$ {n f[n] }\ufrek {-z {{dF(z)}\over {dz}}} $$

\podpoddio Konvolucija

$$ {\sum _{i=0}^{+\infty} f[i]g[n-i]}\ufrek {F(z)G(z)} $$

\poddio Tablica $\cal Z$ transformacije

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

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\dio Pregled Fourierovih transformacija

\poddio Fourierova transformacija

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\vskip -10pt
Fourierova transformacija se uobi\v cajeno koristi za prikaz kontinuiranih signala.
Transformacijski par je $x(t)\ufrek X(\omega)$ i vrijedi:
$$X(\omega)=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}\,dt$$
$$x(t)={1\over2\pi}\int_{-\infty}^{+\infty}X(\omega)e^{j\omega t}\,d\omega$$

\poddio Fourierov red

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\vskip -10pt
Fourierov red (FS -- {\sl Fourier Sum}) se uobi\v cajeno koristi samo za prikaz periodi\v ckih kontinuiranih signala.
Transformacijski par je $x(t)\ufrek X[k]$ i vrijedi:
$$X[k]={1\over T_P}\int_{T_P}x(t)e^{-j\omega_Pkt}\,dt$$
$$x(t)=\sum_{k=-\infty}^{+\infty}X[k]e^{j\omega_Pkt}$$

\poddio Vremenski diskretna Fourierova transformacija

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\vskip -10pt
Vremenski diskretna Fourierova transformacija (DTFT -- {\sl Discrete-Time Fourier Transform}) se koristi
za prikaz nizova. Transformacijski par je $x[n]\ufrek X(\omega)$ i vrijedi:
$$X(\omega)=\sum_{n=-\infty}^{+\infty}x[n]e^{-j\omega n}$$
$$x[n]={1\over 2\pi}\int_{-\pi}^{\pi}X(\omega)e^{j\omega n}\,d\omega$$

\poddio Diskretna Fourierova transformacija

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\vskip -10pt
Diskretna Fourierova transformacija 
se koristi za prikaz kona\v cnog niza duljine $N$ (DFT -- {\sl Discrete Fourier Transform}).
Transformacijski par je $x[n]\ufrek X[k]$ i vrijedi:
$$X[k]=\sum_{n=0}^{N-1}x[n]W^{kn}_N,\quad 0\le k\le N-1$$
$$x[n]={1\over N}\sum_{k=0}^{N-1}X[k]W^{-kn}_N,\quad 0\le n\le N-1$$

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    \special{pa 2700 1400}%
    \special{pa 2700 1400}%
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    \special{ar 2700 1400 20 20 0 6.28319}%
    \graphtemp=\baselineskip\multiply\graphtemp by 1\divide\graphtemp by 2
    \advance\graphtemp by .5ex\advance\graphtemp by 1.450in
    \rlap{\kern 2.300in\lower\graphtemp\hbox to 0pt{\hss $N$\hss}}%
    \graphtemp=\baselineskip\multiply\graphtemp by 1\divide\graphtemp by 2
    \advance\graphtemp by .5ex\advance\graphtemp by 1.450in
    \rlap{\kern 0.500in\lower\graphtemp\hbox to 0pt{\hss $-N$\hss}}%
    \hbox{\vrule depth1.450in width0pt height 0pt}%
    \kern 2.900in
  }%
}%
    \box\graph
    \vss
  }
  \hss
}
\vskip -10pt
Transformacija je usko povezana s diskretnim Fourierovim redom (DFS -- {\sl Discrete Fourier Series})
koji se koristi za prikaz periodi\v ckog niza. Izrazi su gotovo jednaki:
$$X[k]={1\over N}\sum_{n=0}^{N-1}x[n]e^{-2\pi jkn/N}$$
$$x[n]=\sum_{k=0}^{N-1}X[k]e^{2\pi jkn/N}$$

\bye


