IFFT

IFFT stands for inverse fast Fourier transform, an efficient algorithm to compute the inverse discrete Fourier transform (IDFT). It converts a sequence of frequency-domain samples X[k] into a time-domain sequence x[n], effectively reconstructing the original signal from its spectrum within numerical precision and the chosen scaling convention.

For an N-point transform, the IDFT is x[n] = (1/N) sum_{k=0}^{N-1} X[k] e^{i 2π kn / N}, for n = 0,...,N-1. The IFFT computes this result using the same computational structure as the FFT, but with the sign of the exponent reversed and the 1/N scaling. In practice, many software libraries implement the IFFT by applying a forward FFT to the complex-conjugated input and then taking complex-conjugates of the result, with a final 1/N normalization.

Properties of the IFFT include linearity and invertibility (assuming the input adheres to the DFT definitions). Real-valued frequency-domain data yield conjugate-symmetric time-domain sequences, a feature that can be exploited to reduce computation. The concept extends to multi-dimensional data, so 2D and higher-dimensional IFFTs are used for images and other multidimensional signals.

Applications of the IFFT are widespread in signal processing and related fields. It is used to reconstruct time-domain signals from spectra, in digital communications, audio and image processing, spectral analysis, filtering, and in procedures that manipulate data in the frequency domain and then convert back to time or spatial representations. The IFFT is a fundamental tool in both real-time and offline digital signal processing.