Editing Discrete-time Fourier transform (section)

==Properties==
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
* <math>*\!</math> is the [[Convolution#Discrete convolution|discrete convolution]] of two sequences
* <math>s^{*}[n]</math> is the [[complex conjugate]] of <math>s[n].</math>

{| class="wikitable"
|-
! Property
! Time domain<br/>{{math|''s''[''n'']}}
! Frequency domain<br/><math>S_{2\pi}(\omega)</math>
! Remarks
! Reference
|-
| Linearity
| <math>a\cdot s[n] + b\cdot y[n]</math>
| <math>a\cdot S_{2\pi}(\omega) + b\cdot Y_{2\pi}(\omega)</math>
| complex numbers <math>a,b</math>
| <ref name=Proakis/>{{rp|p.294}}
|-
| Time reversal / Frequency reversal
| <math>s[-n]</math>
| <math>S_{2\pi}(-\omega) \!</math>
|
| <ref name=Proakis/>{{rp|p.297}}
|-
| Time conjugation
| <math>s^*[n]</math>
| <math>S_{2\pi}^*(-\omega) \!</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Time reversal & conjugation
| <math>s^*[-n]</math>
| <math>S_{2\pi}^*(\omega) \!</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Real part in time
| <math>\operatorname{Re}{(s[n])}</math>
| <math>\frac{1}{2}(S_{2\pi}(\omega) + S_{2\pi}^*(-\omega))</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Imaginary part in time
| <math>\operatorname{Im}{(s[n])}</math>
| <math>\frac{1}{2i}(S_{2\pi}(\omega) - S_{2\pi}^*(-\omega))</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Real part in frequency
| <math>\frac{1}{2}(s[n]+s^*[-n])</math>
| <math>\operatorname{Re}{(S_{2\pi}(\omega))}</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Imaginary part in frequency
| <math>\frac{1}{2i}(s[n]-s^*[-n])</math>
| <math>\operatorname{Im}{(S_{2\pi}(\omega))}</math>
|
| <ref name=Proakis/>{{rp|p.291}}
|-
| Shift in time / Modulation in frequency
| <math>s[n-k]</math>
| <math>S_{2\pi}(\omega)\cdot e^{-i\omega k}</math>
| integer {{mvar|k}}
| <ref name=Proakis/>{{rp|p.296}}
|-
| Shift in frequency / Modulation in time
| <math>s[n]\cdot e^{ian} \!</math>
| <math>S_{2\pi}(\omega-a) \!</math>
| real number <math>a</math>
| <ref name=Proakis/>{{rp|p.300}}
|-
| Decimation
| <math>s[nM]</math>
| <math>\frac{1}{M}\sum_{m=0}^{M-1} S_{2\pi}\left(\tfrac{\omega - 2\pi m}{M}\right) \!</math>&nbsp; {{efn-ua
|This expression is derived as follows:<ref name=Oppenheim/>{{rp|p.168}}

:<math>
\begin{align}
\sum_{n=-\infty}^{\infty} s(nMT)\ e^{-i\omega n}
&= \frac{1}{MT}\sum_{k=-\infty}^{\infty} S\left(\tfrac{\omega}{2\pi MT} - \tfrac{k}{MT}\right)\\
&= \frac{1}{MT}\sum_{m=0}^{M-1} \quad \sum_{n=-\infty}^{\infty} S\left(\tfrac{\omega}{2\pi MT} - \tfrac{m}{MT} - \tfrac{n}{T}\right),
\quad \text{where} \quad k \rightarrow m + nM\\
&=\frac{1}{M}\sum_{m=0}^{M-1} \quad \frac{1}{T}\sum_{n=-\infty}^{\infty}S\left(\tfrac{(\omega - 2\pi m)/M}{2\pi T} - \tfrac{n}{T}\right)\\
&= \frac{1}{M}\sum_{m=0}^{M-1} \quad S_{2\pi}\left(\tfrac{\omega - 2\pi m}{M}\right)
\end{align}
</math>
}}
| integer <math>M</math>
|
|-
| Time Expansion
| <math> \scriptstyle \begin{cases}
s[n/M] & n=\text{multiple of M} \\
0 & \text{otherwise}
\end{cases}</math>
| <math>S_{2\pi}(M \omega) \!</math>
| integer <math>M</math>
|<ref name=Oppenheim/>{{rp|p.172}}
|-
| Derivative in frequency
| <math>\frac{n}{i} s[n] \!</math>
| <math>\frac{d S_{2\pi}(\omega)}{d \omega} \!</math>
|
| <ref name=Proakis/>{{rp|p.303}}
|-
| Integration in frequency
| <math> \!</math>
| <math> \!</math> 
|
|
|-
| Differencing in time
| <math> s[n]-s[n-1] \!</math> 
| <math> \left( 1-e^{-i \omega} \right) S_{2\pi}(\omega)  \!</math>
|
|
|-
| Summation in time
| <math> \sum_{m=-\infty}^{n} s[m] \!</math> 
| <math> \frac{1}{\left( 1-e^{-i \omega} \right)} S_{2\pi}(\omega) + \pi S(0) \sum_{k=-\infty}^{\infty} \delta(\omega-2\pi k) \!</math>
|
|
|-
| Convolution in time / Multiplication in frequency
| <math>s[n] * y[n] \!</math>
| <math>S_{2\pi}(\omega) \cdot Y_{2\pi}(\omega) \!</math>
|
| <ref name=Proakis/>{{rp|p.297}}
|-
| Multiplication in time / Convolution in frequency
| <math>s[n] \cdot y[n] \!</math>
| <math>\frac{1}{2\pi}\int_{-\pi}^{\pi}S_{2\pi}(\nu) \cdot Y_{2\pi}(\omega-\nu) d\nu \!</math>
| [[Periodic convolution]]
| <ref name=Proakis/>{{rp|p.302}}
|-
| [[Cross correlation]]
| <math>\rho_{sy} [n] = s^{*}[-n] * y[n] \!</math>
| <math>R_{sy} (\omega) = S_{2\pi}^*(\omega) \cdot Y_{2\pi}(\omega) \!</math> 
|
|
|-
| [[Parseval's theorem]]  
| <math>E_{sy} = \sum_{n=-\infty}^{\infty} {s[n] \cdot y^*[n]} \!</math>
| <math>E_{sy} = \frac{1}{2\pi}\int_{-\pi}^{\pi}{S_{2\pi}(\omega) \cdot Y_{2\pi}^*(\omega) d\omega} \!</math>
|
| <ref name=Proakis/>{{rp|p.302}}
|}