FFT vs DFT: Difference and Comparison

Technologies are getting ahead of everything; the developments in the technology sector are enabling the digital world to be more efficient daily.

Whatever is visible on the computer or laptop screen is not just directly connected to what a person types; it includes several units that help 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, and there are different tools that help in converting the frequency and signals.

Key Takeaways

  1. FFT (Fast Fourier Transform) is an algorithm designed to compute a sequence’s Discrete Fourier Transform (DFT) faster and more efficiently, reducing the complexity of the calculations and improving processing time.
  2. DFT (Discrete Fourier Transform) is a mathematical technique that converts a time-domain signal into its frequency-domain representation, allowing for analyzing the frequencies present in the original signal.
  3. The main difference between FFT and DFT is that FFT is an efficient algorithm used to compute the DFT. In contrast, DFT is the mathematical technique for transforming a time-domain signal into its frequency-domain representation.

FFT vs. DFT

FFTs convert signals from the time domain to the frequency domain to improve signal processing. FFT is an algorithm that can perform the transformation in much less time. DFT converts a simple sequence of numbers into complex ones that FFT can calculate.

FFT VS DFT

Comparison Table

Parameters of ComparisonFFTDFT
Full-formFast Fourier transformDiscrete Fourier transform
DefinitionThe amalgamation of several computing techniques including DFT.  The mathematical algorithm transforms the 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, an abbreviation of Fast Fourier transform, is a mathematical algorithm in computers which enables the speeding up of conversions made by DFT (discrete Fourier transform).

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FFT is widely used in processing signals. It reduces the computations needed for N points 2N2 to N log N, wherein LG is a base-two algorithm.

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

In computer science lingo, fast Fourier transforms (FFT) reduce the computations needed for problem size N. A fast Fourier transform is a mathematical algorithm 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 that helps in processing digital signals by calculating the spectrum of a finite-duration signal. 

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

There are other properties of DFT, including 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 their frequency component is important in the frequency domain.

Main Differences Between FFT and DFT

  1. FFT implements DFT, whereas DFT establishes a relationship between the time domain and the frequency domain representation.
  2. DFT is a mathematical algorithm that transforms time-domain signals into frequency-domain components. On the other hand, the FFT algorithm consists of several computation techniques, including DFT.
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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Last Updated : 11 June, 2023

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25 thoughts on “FFT vs DFT: Difference and Comparison”

  1. An enriching read that provides valuable insights into technologies and digital signal processing. A well-researched article.

    Reply
  2. This insightful article adds depth to the understanding of digital signal processing, emphasizing the importance and working of FFT and DFT.

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  3. Informative content. The article provides a clear and concise comparison of FFT and DFT, making their differences easier to comprehend.

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  4. It’s great to see such a detailed explanation. The references cited make this a reliable resource for understanding FFT and DFT.

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  5. The article simplifies complex concepts, making it an enriching read. I appreciate the analytical breakdown of FFT and DFT.

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  6. An exquisite elucidation of FFT and DFT with comprehensive information. A testament to the refinement of the knowledge presented.

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  7. The emphasis on FFT and DFT further enriches the understanding of digital signal processing and their technical significance.

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