1. Convolution:
In mathematics convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other. The term refers to both the result function and to the process of computing it.
There are linear convolution and circular convolution. In linear convolution signal may not be periodic but in circular convolution signal is assumed to be periodic . Linear convolution is used in real life applications.
2. Correlation:
Correlation is a measure of similarity between two signals.
There are two types of correlation:
· Auto correlation: It is defined as correlation of a signal with itself. Auto correlation function is a measure of similarity between a signal & its time delayed version.
· Cross correlation: Cross correlation is the measure of similarity between two different signals.
3. Discrete Fourier Transform:
Discrete Fourier transform or DFT is done to do frequency analysis of a signal . DTFT is not used in real life applications because it involves integration and hence DFT is used . Processes like convolution and correlation can be obtained by simply multiplying the signals in frequency domain. However DFT is a slow process and hence alternate methods like fat Fourier transform is used.
4. Sampling and Reconstruction:
Sampling is the process of recording the values of a signal at given points in time. A continuous signal is converted into discrete signal by passing pulse train with regular spacing between it. For sampling to take place, the signal must satisfy Nyquist criteria. Reconstruction is basically obtaining the original continuous signal for the discrete signal. Ideal Reconstruction is not possible.
5. Linear FIR using FFT:
In this experiment we studied about the different ways to perform Linear FIR filtering. We studied the 2 algorithms named Overlap Add method and Overlap Save method. These 2 methods are used to obtain the linear convolution of the input data and the impulse . In this experiment the input which we gave was long as compared to the previous inputs for normal convolution or DFT or FFT. We studied both the codes and verified that the output which we get in both the cases is indeed same. This was a long experiment as it required analysis of 2 relatively algorithm.
6. Fast Fourier Transform:
A fast Fourier transform is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.These components are single sinusoidal oscillations at distinct frequencies each with their own amplitude and phase. The Fast Fourier Transform is an efficient algorithm for computing the Discrete Fourier Transform. The amount of calculations required to implement this algorithm is lesser.
7. Frequency Response of Discrete Time Signal:
In this experiment we got the frequency response of infinite and finite impulse response signals, we obtained the conditions for which they behave as different kinds of filters.For IIR filter , for pole on right it is low pass filter while for pole on left it is high pass filter.For FIR filter ,for input a = [1 6 5 2 5 6 1] it is band reject filter . For input b = [1 6 5 2 -5 -6 -1] it is band pass filter.
8. Application of Signal Processing:
This experiment was based on Simulink. We were made to pass noise through filter and observe the result. We were able to do it easily and we had used simulink before. The experiment was completed in college itself and we wrote the theory and conclusion part at home.
Learning Experience from Signal & Systems Lab:-
At the end of this course, I was able to:
1. Understand and apply the concepts about linear time-invariant (LTI) systems.
2. Understand and apply Fourier Series representation of periodic continuous-time signals.
3. Understand and apply Fourier Transform representation of periodic and aperiodic continuous-time signals 4. Apply Laplace transforms to analyze LTI Systems.
4. To Calculate and Plot the Frequency Response of Discrete Time Signals.
