Today, everything involves data. That is why technology and science continue to work in tandem as times continue to evolve. We need extremely powerful technology that will help convert mathematical theories giving physical implementations. DSP started as just a graduate course but has over time developed as a potential game-changer in the fields of Science and Engineering.

Technology has seemed to take over everything happening today and certainly remains ahead. The digital world is gaining more efficiency due to the availability and the help of new emerging technologies. Computers are an example of a system looking easy and accessible but the programs involved get very complicated.

Whatever the computer or laptop screen displays are not the only thing that happens. There are programs integrally involved to make everything operate as it is.

Digital Signal Processing for example is what converts input into readable text or pictures that are clear and can be interpreted. Input is usually informed of data or information that is later converted by the Digital Signal Processing tool.

There are components of different types that work within the DSP units. These tools help in the conversion of frequency and signals. Some of these components include Fourier transform, Laplace transform, and z-transform, among others.

## FFT Vs. DFT

The main difference between the FFT and DFT is that the FFT enhances the work done by the DFT. They are both part of the Fourier transform systems but work interchangeably. Both are important but the FFT is a more sophisticated process. It makes computations easier and helps to complement tasks done by the DFT. As a result, FFT increases efficiency while handling tasks, and results that would have otherwise taken a long period to achieve are achieved within a short time.

### Comparison between FFT and DFT

- FFT stands for Full Fourier Transform while DFT stands for Discrete Fourier Transform.
- FFT is the resulting process of computing techniques while DFT is the algorithm that transforms the time domain into frequency domains mathematically.
- FFT has a faster computation rate. DFT on the other hand needs to establish a relationship between the time and frequency domains before computing.
- The FFT is a fast version while the DFT is a discrete version.

## What is FFT?

The mathematical algorithm in computers that enables the speeding up of conversions made by the discrete Fourier transform is called the fast Fourier transform. Computing comes with many complexities but these are reduced with the help of FFT.

FFT is used to process signals and helps to reduce the number of computations needed. The two categories under which the FFT is classified are time decimation and frequency decimation.

The work of this FFT algorithm is to rearrange the input elements. The order is a bit reversed and reduced to build the output transform which is the time decimation. The work needed is to break the form of length N into two transforms N/2.

The FFT algorithm was a discussion in 1965 by Cooley and Tukey. The critical factors of this algorithm were later discovered and discussed in 1805 by Gauss. He described the factorization step by step.

In computer science, the FFT algorithm is supposed to reduce the complexities by minimizing the number of computations. In summary, the fast Fourier transform is a mathematical algorithm whose purpose is to produce fast and efficient computation of the discrete Fourier transform.

It helps to reduce the time used up during computations done by DFT as well as increase efficiency in sound engineering, seismology, and voltage measurements.

## What is DFT?

Discrete Fourier Transform is an algorithm used to process digital signals by calculating the finite duration signal.

The N discrete-time samples are transformed into the same number of discrete frequency samples with the help of DFT. Some applications’ time-domain signal is not applicable, making the signal frequency content useful.

Another type of discrete Fourier transform is the IDFT. That is the Inverse Discrete Fourier Transform. It works in an almost similar way to the DFT since it reduces the N frequency to the same number of discrete-time samples.

Applications like the LC oscillators are compatible with the DFT. They allow the DFT to see how much noise is present in a produced sine wave. Some properties of the DFT include:

- Linearity- A combination of signals is equal to the sum of individual signals, according to the DFT linearity.
- The theorem used to find the finite duration is; X(N)⟷Nx[((−k))N].
- Other properties of DFT include; complex conjugate properties, circular frequency shift, multiplication of two sequences, symmetry, and Parseval’s Theorem.
- DFT, discrete Fourier transform transforms the time domain signals to frequency domain components.

## Difference Between FFT and DFT

- FFT stands for fast Fourier transform while DFT stands for discrete Fourier transform.
- FFT compared to DFT is a faster version of the Fourier Transform. DFT on the other hand as its name states is a more discrete version.
- FFT is mainly used in sound engineering and seismology while DFT is used in spectrum estimation and convolution.
- FFT helps to implement the DFT while DFT helps to build a relationship between the time domain and the frequency domain.
- DFT deals more with numbers. It is used to transform time-domain signals to frequency domain signals. FFT on the other hand does more computation techniques some of which include the DFT.
- DFT is only useful in signals with a discrete and finite length. FFT on the other hand helps to implement the DFT processes.

## Conclusion

In conclusion, FFT and DFT play a huge role in conversions and are very much needed during computation processes. Both are a part of Digital Signal Processing. Processes evolve with time and FFT and DFT come in handy to reduce the complexities that come with these processes.

Since the FFT works like an updated version of the DFT, its main role is to reduce any computation difficulties. Reduction of these difficulties eventually results in efficiency allowing processes to take less time. Results are achieved much faster.