Fast Fourier Transfrom

An algorithm for fast computation of DFT is FFT(Fast Fourier Transform).The FFT algorithm divides the N-point DFT input signal into 2 parts-even and odd signal

We performed 4-point and 8-point FFT. Then we verified the answer by performing Inverse FFT for both.

Later, number of calculations involved in FFT and DFT were compared and we found that very fewer calculations are required for FFT. Hence, FFT is used for parallel processing as it improves operational efficiency. 

  

Filtering of Data sequence

In order to reduce the delay and the data latency obtained during long data sequences overlap add and overlap save method is employed. These methods do not require the entire data to be present before carrying out the convolution. In case of overlap add method, the entire signal is divided into small length signals and the convolution of the signals are carried out with the response of the system. After carrying out the convolution the overlap portion is added to get the output. In case of overlap save method, the overlapped portion is saved to get the result. The result is same as that of time domain convolution. This methods are used to obtain the output of the system. 

Discrete Fourier Transform

DFT (Discrete Fourier Transform) is carried out to overcome the limitation of Fourier series as well as DTFT.
The programming language used in this experiment was C. For DFT(Discrete Fourier Transform) experiment we had considered 2 cases. The first case included calculations of 4 point DFT of given input signal. The second case had calculation of 8 point DFT of the same input signal x[n]. Magnitude spectrum was plotted for both the cases. The magnitude spectrum of 8 point signal has more resolution as compared to that of 4 point signal. We also performed the inverse Discrete Fourier transform of the same signal.

Linear and Circular Convolution and Correlation

The aim of this experiment was to study linear convolution, circular convolution and linear convolution using circular convolution and correlation of signals.
Convolution is an mathematical expression used to express the relation between input and output of a system.
Mathematically, Linear convolution is expressed as y[n]=x[n]*h[n]
Where x[n] is the input to the system and h[n] is the impulse response of the system.
Linear convolution is use to find the output of the system.
The length of the output signal obtained was the one less than the addition of the length of the 2 input signals. 
In case of Circular convolution the length of the output signal is chosen as maximum length of both the signals.
Circular convolution gives alliased output.
Correlation is use to find the degree of similarity of two signals.
Auto-correlation is an Even signal.