DSPP Experiment 4 : Overlap Add and Overlap Save Methods

OAM method involves decomposing the input signal and performing linear convolution on each of them individually. The decomposed outputs are then concatenated together. The value of N chosen is generally a power of 2.

Overlap Add Method

OSM method involves decomposing of input. But in this method, we divide the output signals and find the resultant as opposed to dividing the input signals in OAM and perform zero padding to modify the input signal x[n] and zero padding to h[n]. 

OAM and OSM are Block Processing Techniques and are suitable for real time signal processing.

Image source : Wikipedia

DSPP Experiment 3 : Fast Fourier Transform

In today's world, real time processing of signals is required, be it in stock markets or industrial applications, where even a difference of a seconds can lead to losses in millions. Hence, using conventional DFT algorithm is not feasible and Fast Fourier Transform was developed by two friends Cooley and Tukey in 1965. 

We studied implementing FFT using Radix-2 Decimation In Time Fast Fourier Transform(DITFFT) algorithm. The signal is decimated in time domain which helps to reduce the number of complex calculations. This, in turn increases the speed of processing. The order of the inputs and outputs are in a bit reversed manner. 

For example, when N=4; while DFT takes 240 real additions and 256 real multiplications, the numbers in FFT are reduced to 16 and 48 respectively!!

Thus, FFT is faster, efficient and more easily realizable at higher values of N.

DSPP Experiment 2 : Discrete Fourier Transform

Discrete Fourier Transform (DFT) is the sampled form of DTFT(Discrete Time Fourier Transform).

It was realized on plotting the magnitude spectrum, that on appending zeros at the end of the input signal, gives us a better spectrum, on which
1. Reduces error
2. Frequency spacing decreases 
3. Spectrum resolution increases. If we expand the original time domain signal, the DFT output gets compressed and vice versa.

To perform DFT, we need to perform [4N(^2)] real multiplications and [4N(^2) - 2N] real additions. For N=4, these values are 240 and 256 respectively. Thus, DFT requires a lot of mathematical computation which renders it to be quite slow for practical applications. A better way is to use Fast Fourier Transform which we will look into in the third experiment. 

DSPP Experiment 1 : Convolution and Correlation Algorithms

Convolution involves finding output of a Linear Time Invariant system. Correlation provides degree of similarity between  two signals.
The user inputs the signals x(n) and h(n).

To find convolution, the length of the signals are calculated to be L and M respectively. Thus, the length of output signal is given by

1. L= N + M - 1 For linear convolution

2. L= Max(N, M) For circular convolution

3. L >= N + M - 1 For linear convolution by circular convolution

To find correlation, the output y(n) is an even signal i.e. y(n)=y(-n). 
1. If the input signals are delayed, the output does not change. 

2. If we try cross correlation of a signal with its delayed self, we get a result that is an advanced signal from the auto correlation output.