Overlap Add and Overlap save Method

In this session we studied and practically implemented the commonly used block processing techniques namely OAM(Overlap Add method) and OSM(Overlap save method).The general idea here is that when the input to a digital FIR filter is a very long sequence, performing convolution of the entire signal could prove to be an arduous task.Thus a very efficient way of finding the filter output is by using the linear filtering methods viz. OAM ,OSM wherein we decompose the original input signal and perform linear convolution on each of them individually. OAM and OSM are thus most suitable for processing the real time signals.

Fast Fourier Transform

As the name suggested FFT went on to prove itself to be computationally faster than its discrete counterpart. This was the very result we verified in our third lab session.Two cases were considered, , one with a four point input and other with an eight point input.The number of real additions and multiplications (Computations) were considerably lesser compared to the DFT computations.This very well justified the computational efficiency of the Fast Fourier Transform.   

Discrete Fourier Transform

Initially a four point and later an eight point signal was fed as the input to implement the Discrete Fourier Transform which basically converted any signal from time to frequency domain.Thus in simple words DFT was nothing but the frequency sampled version of DTFT(Discrete time fourier transform).The results produced in both cases were found to be periodic in nature.Magnitude spectrums were plotted for each case which highlighted the effect of increasing the input signal length on frequency spacing , approximation error and spectrum resolution.The most revealing observation however was that as we expanded the input in time domain we achieved a compressed spectra in frequency domain.

Convolution and Correlation algorithms

The basic mathematical operations of convolution (Linear, Circular) and correlation(Auto, Cross) were studied and practically implemented in this session.It was observed that linear convolution produced a causal output for a causal input. Circular convolution however produced an aliased output.That means first few values of the output got overlapped with the values beyond 'N' where N acted as the length of input and output signals.Correlation was used to test the degree of similarity between two signals.It was observed that auto correlation signal always turned out to be an even signal.C programming language was used to construct the test codes.