Overlap Add & Save Method

Experiment 4

When the data size and rate is high, we filter the long data sequence using Overlap Add Method and Overlap Save Method.In OAM, convolution of smaller groups of input with the second input is obtained. So we get convolution for each group which gets overlap depending upon the length of the input signal. In OSM, we divide the output instead of input signals and use only limited input required to calculate that output. Hence the memory requirement is reduced. Both the methods require the same amount of computations and hence the same memory.

Fast Fourier transform

Experiment 3

For finding DFT for large signal we use FFT which divides N point signal into two equal parts with each part consists of alternate values.
On comparing the arithmetic computing properties for complex addition, Complex multiplication, real addition and real multiplication of FFT with DFT, we observed that number of calculations were less for FFT, thus making FFT much faster than DFT. The reason for fast computation in FFT is parallel processing.

Discrete Fourier transform

Experiment 2

Discrete Fourier Transform (DFT) is frequency sampling of Discrete Time Fourier Transform (DTFT). 
DFT of 4 point sequence was calculated and then. we padded the 4-point signal with 4 zeroes and observed the change in magnitude spectrum from the 4-point signal. DFT results are periodic with period equal to length of the input signal.
If the input signal is expanded, its Magnitude Spectrum is compressed. The total number of multiplications and additions in DFT is higher and hence DFT is computationally slow.

Convolution and Correlation Algorithms

Experiment 1

In this experiment  Convolution (Linear Convolution, Circular Convolution and Linear Convolution using Circular Convolution) and Correlation (Auto Correlation and Cross Correlation) has been performed.
Convolution is a mathematical way of combining two signals to form a third signal. It is the single most important technique in Digital Signal Processing. Using the strategy of impulse decomposition, systems are described by a signal called the impulse response. Convolution is important because it relates the three signals of interest: the input signal, the output signal, and the impulse response. 

Linear Convolution is used for non-periodic signals and Circular Convolution is used for periodic signals. The output of Circular Convolution is aliased. Application of Convolution is to find output of the system. We observed that for Linear Convolution the length of output signal is N=L+M-1 and For the circular that length is N= MAX(l,M).

The output of Auto Correlation is the value of the output at n=0 gives the energy of the signal. Auto Correlation of delayed signal is same as that of original signal. The output was both-sided.