The DFT and IDFT

• The Discrete Fourier Transform (DFT) is given by:
    $$X[k] \ensuremath{\stackrel{\Delta}{=}}\sum^{N-1}_{n=0}x[n] e^{-j 2 \pi k n/N}, \hspace{0.2in} k = 0, 1, 2, \ldots, N - 1.$$ (4)

• The operation of the DFT can be reversed to recover a time-domain signal from its frequency representation. This is done with the Inverse Discrete Fourier Transform (IDFT).

• The IDFT is given by:
    $$x[n] = \frac{1}{N} \sum^{N-1}_{k=0}X[k] e^{j 2 \pi k n/N}, \hspace{0.2in} n = 0, 1, 2, \ldots, N - 1.$$ (5)

• In practice, we use the more efficient FFT and IFFT algorithms to perform these operations.