Difference between FFT and DFT (With Table)

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. Computers are such examples wherein the system might look easy or accessible but the internal processing is quite complex.

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

The process might sound easy but the internal complexity of the system is only known to the specialists, some different units and algorithms work in between the input and output, and the processing is so fast that it is not visible.

DSP abbreviation of digital signal processing enables this process of converting the input into readable text or clear visible picture. Every input is some of the other forms of data or information, thus DSP enables this conversion.

Within  DSP there are different components of different types which work differently in their unit, there are different tools which help in converting the frequency and signals. Some of them are Fourier transform, Laplace transform, z-transform, etc..    

The 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.

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). It helps in reducing the complexities of computing.

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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 classified into two categories those are; decimation in time and decimation of frequency. 

The FFT algorithm works differently by rearranging the input elements in bit-reversed order and then builds the output transform (time decimation). The basic working is to break up a transform of length N into two transforms of length N / 2.

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. Gauss described the factorisation step by step.

The working of FFT can be explained through example; if one operation takes 1 nanosecond, then fast Fourier transform will cut down the time to 30 seconds by computing the discrete Fourier transform for problem size N = 10*9.

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). 

Fast Fourier transform (FFT) is helpful for time reduction in computations done by DFT and the efficiency of FFT is visible in sound engineering, seismology, or in voltage measurement.

FFT

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.

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The other type of DFT is IDFT stands for inverse discrete Fourier transform although it works quite similar to that of DFT as it also transforms N discrete-frequency samples to the same number of discrete-time samples.

There are several circumstances in which the frequency content of a time-domain signal. DFT works in applications like LC oscillators to see how much noise is present in a produced sine wave. Other than spectrum estimation DFT has several other applications in DSP for example fast convolution.

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.

This is a direct examination of information encoded in the frequency phase and the amplitude of component sinusoid. For example, human speech and hearing use signals for these kinds of encoding, moreover DFT can find the system’s frequency response from the system’s impulse response and vice versa.

DFT

Main Differences Between FFT and DFT

  1. FFT stands for fast Fourier transform on the other hand DFT stands for discrete Fourier transform.
  2. FFT is a much efficient and fast version of Fourier transform whereas DFT is a discrete version of Fourier transform.
  3. FFT is useful in sound engineering, seismology, etc., on the contrary DFT is useful in spectrum estimation, convolution, etc..
  4. FFT is an implementation of DFT whereas DFT establishes a relationship between the time domain and the frequency domain representation.
  5. 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.
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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