Ponimban Fourier

Ponimban Fourier nopo nga' iso ponimban donsompuu (integral transform) it ogumu kopio ampos toi kounalan id mogikaakawo gana' lobi po id pongumbangan pandu dontuntuu (digital signal processing).

Ponimban diti poposimban do pandu mantad raung hiza (time domain) kumaa raung sinaru (frequency domain).

Pongomoi

Montok iso pampos $s(t)$ ii kotuluk do nunung Dirichlet (Dirichlet conditions), ii sinimban Fourier nopo dau nga' ${\hat {s}}(f)$ om ponimban Fourier do ${\mathcal {F}}:s\rightarrow {\hat {s}}$ tu' osimban no raung hingkaa $t\rightarrow f$. Id siriba no ii govit pongomoi do ponimban Fourier:

${\mathcal {F}}(s(t)):={\hat {s}}(f)=\int _{-\infty }^{+\infty }s(t)e^{-2\pi jt}dt$

om ponimban sambalik dau nopo, ${\mathcal {F}}^{-1}$

${\mathcal {F}}^{-1}({\hat {s}}(f)):=s(t)=\int _{-\infty }^{+\infty }{\hat {s}}(f)e^{2\pi jf}df$

It ponimban Fourier nopo okito saagal do koromigan do rayat Fourier, id saau nopo it govit do koponutunan (representation) isoiso pampos id ngaan di tongo pamagat Fourier, $c_{n}$.

$f(x)=\sum _{n=-\infty }^{\infty }c_{n}\exp {(2\pi inx/p)}$ $\quad {\textrm {ontok}}\quad x\in \left[-{\frac {P}{2}},{\frac {P}{2}}\right]$

Soroho andasan gia di vaza pongomoi ii pampos $f(x)$ nopo nga' pampos osompuu don Riemann (Riemann-integrable) om pampos osikap kumuri. Id sinalom diti komoon do pampos osikap kumuri nopo nga' nunu $f(x):\mathbb {R} \rightarrow \mathbb {R}$ ii kotuluk:

$\lim _{n\rightarrow \infty }f^{(n)}(x)=0$ $\quad {\textrm {ngai}}\quad n\in \mathbb {N}$

Bahazan

Ingkaa no bahazan timpou do ponimban Fourier kaanu popouhan do pongihiman ponimban Fourier do nunu pampos tosudong:

Tinulid (Linearity)

$$a\cdot s(t)+b\cdot m(t)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\cdot {\widehat {s}}(\omega )+b\cdot {\widehat {m}}(\omega );\quad a,b\in \mathbb {C}$$

Insir hiza (Time shifting)

$$s(t-t_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi t_{0}\omega }\ {\widehat {s}}(\omega );\quad \ t_{0}\in \mathbb {R}$$

Insir sinaru (Frequency shifting)

$$e^{i2\pi \omega _{0}t}s(t)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {s}}(\omega -\omega _{0});\quad \ \omega _{0}\in \mathbb {R}$$

Panganatan hiza (Time scaling)

$$s(at)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {s}}\left({\frac {\omega }{a}}\right);\quad \ a\neq 0$$ Pinili $a=-1$ pakayaan doid bahazan pomolik hiza: $$s(-t)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {s}}(-\omega )$$