Editing Discrete-time Fourier transform (section)

== Table of discrete-time Fourier transforms ==
Some common transform pairs are shown in the table below.  The following notation applies''':'''

*<math>\omega=2 \pi f T</math> is a real number representing continuous angular frequency (in radians per sample). (<math>f</math> is in cycles/sec, and <math>T</math> is in sec/sample.)  In all cases in the table, the DTFT is 2π-periodic (in <math>\omega</math>).
*<math>S_{2\pi}(\omega)</math> designates a function defined on <math>-\infty < \omega < \infty </math>.
*<math>S_o(\omega)</math> designates a function defined on <math>-\pi < \omega \le \pi</math>, and zero elsewhere. Then: <math display="block">S_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} S_o(\omega - 2\pi k).</math>
* <math>\delta ( \omega )</math> is the [[Dirac delta function]]
* <math>\operatorname{sinc} (t)</math>  is the normalized [[sinc function]]
* <math>\operatorname{rect}\left[{n \over L}\right] \triangleq \begin{cases}
1 & |n|\leq L/2 \\
0 & |n| > L/2
\end{cases}</math>
* <math>\operatorname{tri} (t)</math> is the [[triangle function]]
* {{mvar|n}} is an integer representing the discrete-time domain (in samples)
* <math>u[n]</math> is the discrete-time [[Heaviside step function#Discrete form|unit step function]]
* <math>\delta[n]</math> is the [[Kronecker delta]] <math>\delta_{n,0}</math>

{| class="wikitable"
|-
! Time domain <br /> ''s''[''n'']
! Frequency domain <br />''S''<sub>2''π''</sub>(''ω'')
! Remarks
! Reference
|-
| <math>\delta[n]</math>
| <math>S_{2\pi}(\omega) = 1</math>
| 
| <ref name=Proakis/>{{rp|p.305}}
|-
| <math>\delta[n-M]</math>
| <math>S_{2\pi}(\omega) = e^{-i\omega M}</math>
| integer <math>M</math>
| 
|-
|<math>\sum_{m = -\infty}^{\infty} \delta[n - M m] \!</math>
|<math>S_{2\pi}(\omega) = \sum_{m = -\infty}^{\infty} e^{-i \omega M m} = \frac{2\pi}{M}\sum_{k = -\infty}^{\infty} \delta \left( \omega - \frac{2\pi k}{M} \right) \,</math><br>
<math>S_o(\omega) = \frac{2\pi}{M}\sum_{k = -(M-1)/2}^{(M-1)/2} \delta \left(\omega - \frac{2\pi k}{M} \right) \,</math> &nbsp; &nbsp; odd ''M''<br>
<math>S_o(\omega) = \frac{2\pi}{M}\sum_{k = -M/2+1}^{M/2} \delta \left(\omega - \frac{2\pi k}{M} \right) \,</math> &nbsp; &nbsp; even ''M''<br>
| integer <math>M > 0 </math>
| 
|-
| <math>u[n]</math>
| <math>S_{2\pi}(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \sum_{k=-\infty}^{\infty} \delta (\omega - 2\pi k)\!</math><br>
<math>S_o(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \cdot \delta (\omega)\!</math>
|The <math>1/(1-e^{-i \omega})</math> term must be interpreted as a [[distribution (mathematics)|distribution]] in the sense of a [[Cauchy principal value]] around its [[pole (complex analysis)|poles]] at <math>\omega=2 \pi k</math>.
| 
|-
| <math>a^n u[n]</math>
| <math>S_{2\pi}(\omega) = \frac{1}{1-a e^{-i \omega}}\!</math>
| <math>0 < |a| < 1 </math>
| <ref name=Proakis/>{{rp|p.305}}
|-
| <math>e^{-i a n}</math>
| <math>S_o(\omega) = 2\pi\cdot \delta (\omega +a),</math> &nbsp; &nbsp; -π < a < π<br>
<math>S_{2\pi}(\omega) = 2\pi \sum_{k=-\infty}^{\infty} \delta (\omega +a -2\pi k)</math>
| real number <math>a</math>
| 
|-
| <math>\cos(a\cdot n)</math>
| <math>S_o(\omega) = \pi \left[\delta \left(\omega -a\right)+\delta \left(\omega +a\right)\right],</math><br>
<math>S_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} S_o(\omega - 2\pi k)</math>
| real number <math>a</math> with <math>-\pi < a < \pi</math>
| 
|-
| <math>\sin(a\cdot n)</math>
| <math>S_o(\omega) = \frac{\pi}{i} \left[\delta \left(\omega -a\right)-\delta \left(\omega +a\right)\right]</math>
| real number <math>a</math> with <math>-\pi < a < \pi</math>
| 
|-
| <math>\operatorname{rect} \left[ { n - M \over N  } \right] \equiv \operatorname{rect} \left[ { n - M \over N-1  } \right]</math>
| <math>S_o(\omega) = { \sin( N \omega / 2 ) \over \sin( \omega / 2 ) } \,  e^{ -i \omega M } \!</math>
| integer <math>M,</math> and <u>odd</u> integer <math>N</math>
|
|-
| <math>\operatorname{sinc} ( W (n+a))</math>
| <math>S_o(\omega) =  \frac{1}{W} \operatorname{rect} \left( { \omega \over 2\pi W } \right) e^{ia\omega}</math>
| real numbers <math>W,a</math> with <math>0 < W < 1</math>
| 
|-
| <math>\operatorname{sinc}^2(W n)\,</math>
| <math>S_o(\omega) = \frac{1}{W} \operatorname{tri} \left( { \omega \over 2\pi W } \right)</math>
| real number <math>W</math>, <math>0 < W < 0.5</math>
| 
|-
| <math> \begin{cases}
0 & n=0 \\
\frac{(-1)^n}{n} & \text{elsewhere}
\end{cases}</math>
| <math>S_o(\omega) = j \omega</math>
|it works as a [[differentiator]] filter
|-
| <math>\frac{1}{(n + a)} \left\{ \cos [ \pi W (n+a)] - \operatorname{sinc} [ W (n+a)] \right\}</math>
| <math>S_o(\omega) = \frac{j \omega}{W} \cdot \operatorname{rect} \left( { \omega \over \pi W } \right) e^{j a \omega}</math>
| real numbers <math>W,a</math> with <math>0 < W < 1 </math>
| 
|-
| <math>\begin{cases}
\frac{\pi}{2}  & n = 0 \\
\frac{(-1)^n - 1}{\pi n^2} & \text{ otherwise}
\end{cases}</math>
| <math>S_o(\omega) = |\omega|</math>
| 
|
|-
| <math>\begin{cases}
0; & n \text{ even} \\
\frac{2}{\pi n} ; & n \text{ odd}
\end{cases}</math>
| <math>S_o(\omega) = \begin{cases}
j & \omega < 0 \\
0 & \omega = 0 \\
-j & \omega > 0
\end{cases}</math>
|[[Hilbert transform]]
| 
|-
| <math>\frac{C (A + B)}{2 \pi} \cdot \operatorname{sinc} \left[ \frac{A - B}{2\pi} n \right] \cdot \operatorname{sinc} \left[ \frac{A + B}{2\pi} n \right]</math>
| <math>S_o(\omega) = </math>[[Image:Trapezoid signal.svg|250px]]
| real numbers <math>A,B</math> <br /> complex <math>C</math>
| 
|}