Editing Discrete-time Fourier transform (section)

== Inverse transform ==
An operation that recovers the discrete data sequence from the DTFT function is called an ''inverse DTFT''.  For instance, the inverse continuous Fourier transform of both sides of {{EquationNote|Eq.3}} produces the sequence in the form of a modulated Dirac comb function''':'''

:<math>\sum_{n=-\infty}^{\infty} s[n]\cdot \delta(t-n T) = \mathcal{F}^{-1}\left \{S_{1/T}(f)\right\} \ \triangleq \int_{-\infty}^\infty S_{1/T}(f)\cdot e^{i 2 \pi f t} df.</math>

However, noting that <math>S_{1/T}(f)</math> is periodic, all the necessary information is contained within any interval of length <math>1/T.</math>&nbsp; In both {{EquationNote|Eq.1}} and {{EquationNote|Eq.2}}, the summations over <math>n</math> are a [[Fourier series]], with coefficients <math>s[n].</math>&nbsp; The standard formulas for the Fourier coefficients are also the inverse transforms''':'''

{{Equation box 1
|title=
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation = {{NumBlk||<math>\begin{align}
s[n] &= T \int_{\frac{1}{T}} S_{1/T}(f)\cdot e^{i 2 \pi f nT} df \quad \scriptstyle{\text{(integral over any interval of length }1/T\textrm{)}} \\
\displaystyle &= \frac{1}{2 \pi}\int_{2\pi} S_{2\pi}(\omega)\cdot e^{i \omega n} d\omega \quad \scriptstyle{\text{(integral over any interval of length }2\pi\textrm{)}}
\end{align}</math> &nbsp; &nbsp; |{{EquationRef|Eq.4}}}}
}}