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 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. 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 that work differently in their unit, there are different tools that help in converting the frequency and signals. Some of them are Fourier transform, Laplace transform, z-transform, etc..
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.
Comparison Table Between FFT and DFT
| Parameters of Comparison | FFT | DFT |
| Full-form | Fast Fourier transform | Discrete Fourier transform |
| Definition | The amalgamation of several computing techniques including DFT. | The mathematical algorithm which transforms time domain into frequency domain components. |
| Work | Faster computation | Establishing the relationship between the time domain and frequency domain |
| Applications | Convolution, voltage measurement, etc.. | Spectrum estimation, conviction,etc.. |
| Version | Fast version | Discrete 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.
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.
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.
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:-
- Linearity- according to linearity DFT of a combination of signals is equal to the sum of individual signals.
- 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.
References
- https://ieeexplore.ieee.org/abstract/document/115105/
- https://www.researchgate.net/profile/Levent_Sevgi/publication/3305825_Numerical_fourier_transforms_DFT_and_FFT/links/5ad4d519a6fdcc2935809380/Numerical-fourier-transforms-DFT-and-FFT.pdf