# Difference Between FFT And DFT With Table

We elaborate that what is the difference between fft and dft? In tabular form. Technologies are ahead of everything, developments in the technology sector are allowing the digital world to be more efficient every day. Computers are examples where the system may seem easy or accessible, but the internal processing is quite complex.

Everything that is seen on the screen of the computer or laptop is not only directly connected to what a person writes; rather, it includes several units that help process the input and convert it to human-readable output.

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

The abbreviation DSP for Digital Signal Processing allows this process of converting input into clear, visible image or readable text. Each input is some of the other forms of data or information, so DSP allows this conversion.

Inside DSP there are different components of different types that work differently in your unit, there are different tools that help convert the frequency and signals. Some of them are Fourier transform, Laplace transform, z transform, etc.

The **difference between FFT and DFT** is that FFT improves the work of DFT. Both are part of a Fourier system or are transformed but their works are different from each other.

**What is FFT?**

FFT, short for Fast Fourier transform, is a mathematical algorithm in computers that accelerates the conversions made by DFT (Discrete Fourier Transform). Helps reduce the complexities of computing.

FFT is widely used in signal processing. Reduce the number of calculations required for N points 2N2 to N log N, where LG is a base two algorithm. FFT is classified into two categories: time decimation and frequency decimation.

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

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

The operation of FFT can be explained by an example; If an operation takes 1 nanosecond, then the fast Fourier transform will reduce the time to 30 seconds when calculating the discrete Fourier transform for the problem size N = 10 * 9.

In computer jargon, the fast Fourier transform (FFT) reduces the number of calculations required for the size of problem N. Simply put, the fast Fourier transform is a mathematical algorithm used for the fast and efficient calculation of the Discrete Fourier Transform (DFT).

Fast Fourier Transform (FFT) is useful for reducing time in DFT calculations and the efficiency of FFT is visible in sound engineering, seismology or voltage measurement.

**What is DFT?**

DFT is an abbreviation of the discrete Fourier transform, it is a mathematical algorithm that helps to process digital signals by calculating the spectrum of a signal of finite duration.

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

The other type of DFT is IDFT which stands for Inverse Discrete Fourier Transform, although it works quite similarly to DFT in that it also transforms N discrete frequency samples into the same number of discrete time samples.

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

Some of the DFT properties are: –

- Linearity: according to the DFT linearity of a combination of signals it is equal to the sum of individual signals.
- Duality: a theorem is used to find the sequence of finite duration, 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 Discrete Fourier Transform works by transforming the signals in the time domain into the frequency domain components, since the representation of digital signals in terms of their frequency component is important in the frequency domain.

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

**Comparison table between FFT and DFT**

Comparison parameters |
FFT |
DFT |

Full form | Fast Fourier transform | Discrete Fourier transform |

Definition | The fusion of various computer techniques, including DFT. | The mathematical algorithm that transforms the time domain into frequency domain components. |

Job | Faster calculation | Establish the relationship between the time domain and the frequency domain |

Applications | Convolution, voltage measurement, etc. | Spectrum estimation, conviction, etc. |

Version | Fast version | Discreet version |

**Main differences between FFT and DFT**

- FFT stands for fast Fourier transform, on the other hand, DFT stands for discrete Fourier transform.
- FFT is a much more efficient and faster version of the Fourier transform, while DFT is a discrete version of the Fourier transform.
- FFT is useful in sound engineering, seismology, etc., on the contrary, DFT is useful in spectrum estimation, convolution, etc.
- FFT is an implementation of DFT, while DFT establishes a relationship between the time domain and the frequency domain representation.
- DFT is a mathematical algorithm that transforms signals in the time domain into frequency domain components, on the other hand, the FFT algorithm consists of various calculation techniques, including DFT.

## Algorithm of FFT and DFT

The most commonly used FFT algorithm is the Cooley-Tukey algorithm, which was named after J. W. Cooley and John Tukey. It’s a divide and conquer algorithm for the machine calculation of complex Fourier series. It breaks the DFT into smaller DFTs. Other FFT algorithms include the Rader’s algorithm, Winograd Fourier transform algorithm, Chirp Z-transform algorithm, etc. The DFT algorithms can be either programmed on general purpose digital computers or implemented directly by special hardware. The FFT algorithm is used to compute the DFT of a sequence or its inverse. A DFT can be performed as O(N^{2}) in time complexity, whereas FFT reduces the time complexity in the order of O (NlogN).

## Applications of FFT and DFT

DFT can be used in many digital processing systems across a variety of applications such as calculating a signal’s frequency spectrum, solving partial differential applications, detection of targets from radar echoes, correlation analysis, computing polynomial multiplication, spectral analysis, and more. FFT has been widely used for acoustic measurements in churches and concert halls. Other applications of FFT include spectral analysis in analog video measurements, large integer and polynomial multiplication, filtering algorithms, computing isotopic distributions, calculating Fourier series coefficients, calculating convolutions, generating low frequency noise, designing kinoforms, performing dense structured matrices, image processing, and more.

## Final Thought

## Both FFT and DFT are important for calculation techniques and play an important role in conversions.

FFT and DFT are part of DSP. FFT also works for DFT.

## Summary of FFT Vs. DFT

In a nutshell, the Discrete Fourier Transform plays a key role in physics as it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. It is a simple yet fairly time-consuming algorithm. However, to reduce the computing time and complexity of large transforms, a more complex but less time-consuming algorithm such as the Fast Fourier Transform can be used. FFT is an implementation of the DFT used for used for fast computation of the DFT. In short, FFT can do everything a DFT does, but more efficiently and much faster than a DFT. It’s an efficient way of computing the DFT.