Difference between DFT and FFT

In the field of Digital Signal Processing (DSP), Fourier analysis is used to decompose the signals. The mathematical tool Discrete Fourier transform (DFT) is used to digitize the signals. The collection of various fast DFT computation techniques are known as the Fast Fourier transform (FFT). In simpler words, FFT is just an implementation of the DFT. In this article, we see the exact difference between DFT and FFT.

Difference between DFT and FFT – Comparison Table

The following table summarises the comparison between DFT and FFT.

DFT FFT
The DFT stands for Discrete Fourier Transform. The FFT stands for Fast Fourier Transform.
The DFT is only applicable for discrete and finite-length signals. Discrete time-domain signals are transformed into discrete frequency domain signals using DFT. It is an implementation of DFT.
X(k)=\sum_{n=0}^{N-1}x(n)e^{-\frac{j2\pi kn}{N}} FFT mainly works with computational algorithms for the fast execution of DFT.
The time complexity required for a DFT to perform is equal to the order of N2 or o(N2). The time complexity reduces in the case of FFT and becomes equal to o(NlogN).
The DFT has less speed than the FFT. It is the faster version of DFT.
Some applications of the DFT are spectral analysis, solution of partial differential equations, correlation analysis, etc. Filtering algorithms, multiplication of integer and polynomials, etc. are some applications of the FFT.

What is DFT?

The signals found in nature are basically analog type of signals. But the digital computers that are used for the analysis of the signals can work only with the information that is discrete in nature and finite in length. Hence, the digitization of signals is performed. The Fourier transform of a signal within a finite range is called Discrete Fourier Transform. The mathematics and the algorithms of the Fourier transform are the heart of the DFT.

The Discrete Fourier Transform of a signal x(n) is mathematically expressed as:

    \[X(k)=\sum_{n=0}^{N-1}x(n)e^{-\frac{j2\pi kn}{N}}\]

where, k = 0, 1, 2,…,N-1

The signals obtained in the discrete Fourier transform are discrete and periodic in nature. The tool DFT is able to calculate the frequency spectrum of a signal. The frequency response of a signal from its impulse response can be obtained from the DFT of the required signal. The DFT allows the frequency domain analysis of the signal and examines the information encoded in the frequency, phase and amplitude of the signal.

What is FFT?

The Fast Fourier Transform FFT is nothing but an implementation of a very common tool of DSP called the DFT. The FFT provides a more efficient result than DFT. The computational time required for a signal in the case of FFT is much lesser than that of DFT. Hence, it is called Fast Fourier Transform which is a collection of various fast DFT computation techniques. The FFT works with some algorithms that are used for computation.

Summary

The transformation of time-domain signals to frequency domain signals are the key part of Digital Signal Processing. This process of transformation includes various tools such as DTFT, DFT, FFT etc. As time passes, the evolution of processes takes place. The FFT is the updated version or way of implementation of the DFT that takes less computational time and more efficient results than that of ordinary DFTs.

Author

Sunmoni Gohain
NIT Silchar

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