Discrete-time Fourier Transform

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 Eq.2 produces the sequence in the form of a modulated Dirac comb function:

$\sum_{n=-\infty}^{\infty} x\cdot \delta(t-n T) = \mathcal{F}^{-1}\left \{X_{1/T}(f)\right\} \ \stackrel{\mathrm{def}}{=} \int_{-\infty}^\infty X_{1/T}(f)\cdot e^{i 2 \pi f t} df.$

However, noting that is periodic, all the necessary information is contained within any interval of length 1/T. In both Eq.1 and Eq.2, the summations over n are a Fourier series, with coefficients x. The standard formulas for the Fourier coefficients are also the inverse transforms:

$\begin{align} x &= T \int_{\frac{1}{T}} X_{1/T}(f)\cdot e^{i 2 \pi f nT} df \quad \scriptstyle {(integral\ over\ any\ interval\ of\ length\ 1/T)} \\ \displaystyle &= \frac{1}{2 \pi}\int_{2\pi} X(\omega)\cdot e^{i \omega n} d\omega. \quad \scriptstyle {(integral\ over\ any\ interval\ of\ length\ 2\pi)} \end{align}$