OVERLAP ADD and OVERLAP SAVE methods

 Overlap-Add and Overlap-Save method is used to perform linear convolution and circular convolution of long data sequences respectively. They are called as block processing techniques as the input signal is divided into blocks and then analysed.  The output signal is the combination of the convolved input signals by removing first few values. The code is then executed by entering the input values and thus checking the result. 

FAST FOURIER TRANSFORM

 The Fast Fourier Transform(FFT) is used to reduce calculations and thus the process becomes fast. In this method the input sequence is broken down into two equal parts. This breaking down of the signal was continued till no further decomposition was possible. Radix-2 algorithm is used to perform the experiment. The counter was then added in the code to check the total number of calculations. Thus we conclude that FFT is computationally fast as the number of calculations are reduced.

DISCRETE FOURIER TRANSFORM

In the second lab we studied Discrete Fourier Transform(DFT) and its practical implementation. Three different cases were taken into consideration. In the first case DFT of four point signal is studied then in the second case DFT of zero padded signal is studied and in the last case DFT of expanded spectrum is studied and the magnitude of all the three is plotted. Thus we conclude that DFT produces periodic results. The spectrum is discrete and is in the range of [0,2π). As the length of signal increases due to zero padding the frequency spacing and approximation error decreases also the resolution of spectrum increases.   

CONVOLUTION and CORRELATION

The practical approach towards Convolution and Correlation is studied during the lab sessions. Convolution is a mathematical way of combining two signals {x(n) and h(n)}to form the third one{y(n)}.  Practicals on both linear and circular convolutions are performed. Circular convolution gives aliased output. In correlation we learned to find the degree of  similarities between the two signals. If the input is delayed the auto-correlation of delayed input is same as that of the original input signal. Thus all this is verified using different values if input signals.