Exp - 4. Filtering of Long Data Sequence

For implementing fast processing algorithms like FFT we need all data present at input at the same time. In practical implementation it is not possible to have all data together at same time. Hence it is necessary to decompose the incoming signal into multiple finite length signals to perform operations.  This is done by using - Overlap Add Method (OAM) and Overlap Save Method ( OSM)

In Overlap Add Method x[n] is decomposed into non overlapping sequence of
length L and is performed using Linear Convolution.

In overlap save method, input data sequence x[n] is decomposed into number of
L point sequences, where each data block begins with the last (N-L) values in the previous data block, except the first data block which begins with (N–L) zeros and is performed using Circular Convolution.

Expt. 3 Fast Fourier Transform

The FFT algorithm divides the N-point input signal into 2 parts-even and odd signal values of the input. Computation is carried out only for N/2 signal values, this increases the speed of computations.

The input & output sequence are in a bit reversed manner.

In this lab session we studied and executed FFT of 4 point and 8 point sequence using DITFFT.

We also calculated no. of computations required to find out FFT of a sequence. Comparing it with that of DFT, we can conclude that FFT is much faster.

Expt - 2. Discrete Fourier Transform

Discrete Fourier Transform (DFT) is frequency sampled version  of Discrete Time Fourier Transform (DTFT).It works on the assumption that input is always periodic, to produce periodic results. In this experiment we had to perform  4 point DFT  & 8 point DFT of same input signal.

The magnitude spectrum of both the cases was plotted and compared which helped us draw following conclusion - 
As Length of Input SIgnal (N) increases - 
1. Frequency Spacing increases.
2. Approximation error decreases.
3. Resolution of spectrum increases.

We also calculated the no. of computations required to gt DFT of a signal and concluded that is DFT computationally slow 

Expt - 1. Convolution & Correlation

The aim of the experiment was to study mathematical operations like Convolution and Correlation.
Convolution is the process of computing the output, y[n], of a Linear Time Invariant system to a given arbitrary input x[n] via its impulse response , h[n], summarized in the notation of y[n]=x[n]⋆h[n]:

y[n]=∑k=−∞∞x[k]h[n−k]

It is further classified into two - 
1. Linear Convolution 
    The length of the output signal is N where N=L+M-1 . It is used to determine response of a system when a non - periodic input is applied.

2. Circular Convolution
    The length of the output signal is N=max(L,M). It is used to determine response of a system when a periodic input is applied
(L - Length of input signal i.e x[n] , M - Length of impulse response i.e h[n] ).

Correlation is a process of determining similarity between two input signals.