Editing Discrete-time Fourier transform (section)

== Convolution ==
{{Main|Convolution theorem#Functions of a discrete variable (sequences)}}
The [[convolution theorem]] for sequences is''':'''

:<math>s * y\ =\ \scriptstyle{\rm DTFT}^{-1} \displaystyle \left[\scriptstyle{\rm DTFT} \displaystyle \{s\}\cdot \scriptstyle{\rm DTFT} \displaystyle \{y\}\right].</math><ref name=Proakis/>{{rp|p.297}}{{efn-la
|Oppenheim and Schafer,<ref name=Oppenheim/> p 60, (2.169), and Prandoni and Vetterli,<ref name=Prandoni/> p 122, (5.21)
}}
An important special case is the [[circular convolution]] of sequences {{mvar|s}} and {{mvar|y}} defined by <math>s_{_N}*y,</math> where <math>s_{_N}</math> is a periodic summation.  The discrete-frequency nature of <math>\scriptstyle{\rm DTFT} \displaystyle \{s_{_N}\}</math> means that the product with the continuous function <math>\scriptstyle{\rm DTFT} \displaystyle \{y\}</math> is also discrete, which results in considerable simplification of the inverse transform''':'''

:<math>s_{_N} * y\ =\ \scriptstyle{\rm DTFT}^{-1} \displaystyle \left[\scriptstyle{\rm DTFT} \displaystyle \{s_{_N}\}\cdot \scriptstyle{\rm DTFT} \displaystyle \{y\}\right]\ =\ \scriptstyle{\rm DFT}^{-1} \displaystyle \left[\scriptstyle{\rm DFT} \displaystyle \{s_{_N}\}\cdot \scriptstyle{\rm DFT} \displaystyle \{y_{_N}\}\right].</math><ref name=Rabiner/><ref name=Oppenheim/>{{rp|p.548}}

For {{mvar|s}} and {{mvar|y}} sequences whose non-zero duration is less than or equal to {{mvar|N}}, a final simplification is''':'''

:<math>s_{_N} * y\ =\ \scriptstyle{\rm DFT}^{-1} \displaystyle  \left[\scriptstyle{\rm DFT} \displaystyle \{s\}\cdot \scriptstyle{\rm DFT} \displaystyle \{y\}\right].</math>

The significance of this result is explained at [[Circular convolution#Example|Circular convolution]] and [[Convolution#Fast convolution algorithms|Fast convolution algorithms]].