Difference Between FFT and DFT

Technologies are getting ahead of everything, the developments in the sector of technology are enabling the digital world to be more efficient day by day.

Whatever is visible on the computer or laptop screen is not just directly connected to what a person types; rather it includes several units that help to process the input and convert it into a readable output. 

DSP is the abbreviation of digital signal processing that enables this process of converting the input into readable text or clear visible picture.

Within DSP there are different components of different types that work differently in their unit, there are different tools that help in converting the frequency and signals.

FFT vs DFT

The main difference between FFT and DFT is that FFT enhances the work of DFT. Both of them are part of a Fourier system or transform but their works are different from each other.

FFT VS DFT

Comparison Table Between FFT and DFT

Parameters of ComparisonFFTDFT
Full-formFast Fourier transformDiscrete Fourier transform
DefinitionThe amalgamation of several computing techniques including DFT.  The mathematical algorithm which transforms time domain into frequency domain components.
WorkFaster computationEstablishing the relationship between the time domain and frequency domain
ApplicationsConvolution, voltage measurement, etc..Spectrum estimation, conviction,etc..
VersionFast versionDiscrete version

What is FFT?

FFT abbreviation of Fast Fourier transform, it is a mathematical algorithm in computers which enables the speeding up of conversions made by DFT (discrete Fourier transform).

FFT is widely used in processing signals. It reduces the number of computations needed for N points 2N2to N log N, wherein LG is a base-two algorithm.

FFT is an algorithm was discussed by Cooley and Turkey in 1965 but the critical factorization of this algorithm is described by Gauss in 1805 which is by Cooley and Tukey.

In computer science lingo, fast Fourier transform (FFT) reduces the number of computation needed for problem size N. In a nutshell, fast Fourier transform is a mathematical algorithm which is used for fast and efficient computation of discrete Fourier transform (DFT). 

What is DFT?

DFT is an abbreviation of Discrete Fourier transform, it is a mathematical algorithm which helps in processing the digital signals by calculating the spectrum of a finite-duration signal. 

DFT works by transforming N discrete-time samples to the same number of discrete frequency samples. In some applications, the shape of the time domain is not applicable for signals in which case signal frequency content becomes very useful.

Some of the properties of DFT are:-

  1. Linearity- according to linearity DFT of a combination of signals is equal to the sum of individual signals.
  2. Duality- there is theorem is used to find the finite duration sequence, the theorem used is;  X(N)⟷Nx[((−k))N]. 

There are other properties of DFT, which includes; complex conjugate properties, circular frequency shift, multiplication of two sequences, Parseval’s theorem, and symmetry.

DFT or the discrete Fourier transform works by transforming the time domain signals to the frequency domain components as the representation of digital signals in terms of its frequency component is important in the frequency domain.

Main Differences Between FFT and DFT

  1. FFT is an implementation of DFT whereas DFT establishes a relationship between the time domain and the frequency domain representation.
  2. DFT is a mathematical algorithm which transforms time-domain signals to frequency domain components on the other hand FFT algorithm consists of several computation techniques including DFT.

Conclusion

Both FFT and DFT are important for computation techniques and they play a major role in conversions.

FFT and DFT are part of DSP. FFT works for DFT as well.                     

References

  1. https://ieeexplore.ieee.org/abstract/document/115105/
  2. https://www.researchgate.net/profile/Levent_Sevgi/publication/3305825_Numerical_fourier_transforms_DFT_and_FFT/links/5ad4d519a6fdcc2935809380/Numerical-fourier-transforms-DFT-and-FFT.pdf
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