Discrete-time Fourier Transform - Inverse 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}

Read more about this topic:  Discrete-time Fourier Transform

Famous quotes containing the words inverse and/or transform:

    The quality of moral behaviour varies in inverse ratio to the number of human beings involved.
    Aldous Huxley (1894–1963)

    But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.
    Walter Pater (1839–1894)