Overalp ADD and Overlap SAVE.

Whenever we come across larger sequences that are required to be processed, say a large valued sequence is given to a digital FIR filter, Overlap add method or Overlap save method is used, where faster processing of the signals is needed.

This can be either done by overlapping values after decomposing the input, adding zeros to make it equivalent to L (where L=N-M+1) and linear convolution of these decomposed inputs with h(n) ie OAM

or by decomposing the input, adding preceding values to each of the decomposed input to make it equivalent to L, circular convolution with h(n) and then eliminating padded number of values from each convoluted output to obtain the final output. ie OSM.

OAM and OSM are block processing techniques as seen above and computationally occupy similar memory space.

Fast Fourier Transform

FFT is a faster computational method compared to DFT as the signal is divided into even and odd parts and computations for both are done alongside. We see that the first value is always the sum of input signal values. 

We have taken two sequences 4pt and 8 pt. and found their Fast Fourier Transforms to study these radix 2 FFT algorithms. The computation is carried out only for N/2 values and makes it the fastest algorithm computationally.

 

 

Discrete fourier transform

In this experiment we have verified the Discrete Fourier Transform of a signal by executing a C- program for the same wherein arrays have been used to store real and imaginary parts of the signal.
This signal is manipulated (such as zero padded or expanded form of the same signal) in different ways through this experiment and for each situation we have studied the result.

DFT is the frequency sampled version of DTFT hence giving periodic results.
We also plot its magnitude spectrum by approximation for both 4pt and 8pt (zeros appended to 4pt sequence) signals as well as expanded signal by adding zeros between the for 4pt sequence.
We observe that expansion in time domain gives a compressed spectrum in frequency domain and by finding number of additions and multiplications required we concluded that DFT is computationally slow.

Discrete Convolution and Correlation

Through this experiment, we have aimed to comprehend the Convolution and Correlation of Discrete signals by programming for the same in C-language and tallying them with their mathematically verified outputs.
Results obtained for the following were analyzed to derive legitimate conclusions:
1. Linear Convolution
2. Circular Convolution
3. Linear using Circular Convolution
4. Auto-correlation
5. Cross-correlation
In convolution, we have taken varied lengths of the two input signals to get different results for linear and circular convolution and have observed an aliasing effect in circular convolution. 
In Correlation, we have found auto-correlation and cross-correlation in combinations of original, delayed and scaled values of the same signal in order to find degree of similarity between two signals.