FFT is an algorithm used to compute the Discrete Fourier Transform (DFT) and its inverse efficiently. The DFT converts a sequence of values into components of different frequencies, which is useful in many fields such as signal processing, image analysis, and data compression.
Mathematical Formulation:
Given a sequence x[n] of length n, the DFT $X[k]$ is defined by:
where:
X[k] is the k-th frequency component.
x[n] is the n-th time-domain sample.
The FFT algorithm reduces the computational complexity of calculating the DFT from O(N2) to O(NlogN). This is achieved by decomposing the DFT into smaller DFTs recursively, exploiting symmetries and periodicities in the Fourier coefficients.
IFFT is used to convert data from the frequency domain back into the time domain. It is essentially the reverse operation of the FFT.
Mathematical Formulation:
Given the frequency domain sequence X[k], the IFFT x[n] is defined by:
where:
x[n] is the n-th time-domain sample.
X[k] is the k-th frequency component.
In practice, both FFT and IFFT are implemented efficiently in numerical libraries and software packages, making them essential tools for many digital signal processing tasks.
Numerical Recipes: The Art of Scientific Computing by William H. Press et al.
The Fast Fourier Transform and its Applications by E. Oran Brigham.
Understanding Digital Signal Processing by Richard G. Lyons.
Fourier Analysis and Its Applications by Gerald B. Folland.
These resources provide a range of perspectives on FFT and IFFT, from practical implementation to theoretical foundations.
This website uses cookies to provide you with the best browsing experience.