### Signals and Stochastic Processes

In typical applications of science and engineering, we have to processsignals, using systems. While the applications can be varied large communication systems to control systems but the basic analysis and design tools are the same. In a signals and systems course, we study these tools: convolution,Fourier analysis, z-transform, and Laplace transform. The use of thesetools in the analysis of linear time-invariant (LTI) systems with deterministic signals. For most practical systems, input and output signals are continuous and these signals can be processed using continuous systems. However,due to advances in digital systems technology and numerical algorithms,it is dvantageous to process continuous signals using digital systems byconverting the input signal into a digital signal. Therefore, the study of both continuous and digital systems is required.As most practical systems are digital and the concepts are relativelyeasier to understand, we describe discrete signals and systems _rst, immediately followed by the corresponding description of continuous signalsand systems

• Module 1 &2

Introduction to Signals :This module gives introduction to signals and gives an idea about  classification of signals and basic operations on signals.

• Module 3

System Classification and Analysis:To understand the concept of systems, classification, signal transmission through linear systems with bandwidth considerations

• Module 4

Ideal Characteristics of filters: To understand the magnitude response characteristics of ideal filters and concept of causality and physical reliazability

• Module 5

Convolution in time and frequency domain: To understand the importance of Convolution operation in LTI systems.

• Module 6

Introduction to Fourier series :To understand Fourier series representation of Periodic signals

• Module 7

Properties of Fourier Series and Complex Fourier Spectrum :To understand the change in Fourier series coefficients due to different signal operations and to plot complex Fourier spectrum

• Module 8

Introduction to Fourier Transform :

To understand how the Fourier series is extended to aperiodic signals in the form of Fourier transform

• Module 9

Fourier Transform of Standard Signals :To find the Fourier transform of standard signals like unit impulse, unit step etc. and any periodic signal

• Mod 10 & 11

Properties of continuous Fourier Tranform :These properties provides significant amount of insight into the transform and into the relationship between the time-domain and frequency domain descriptions of a signal. Many of these properties are useful in reducing the complexity Fourier transforms or inverse transforms. By using these properties we can translate many Fourier transform properties into the corresponding Fourier series properties

• Module 12 & 13

To Understand the concept of Sampling a signal its reconstruction and various methods of Sampling

• Module 14

Laplace Transforms : To apply Laplace transform for analyzing continuous time signals and to understand the relation to Fourier transforms

• Module 15

Region of Convergence (ROC)  : To understand the meaning of ROC in Laplace transforms and the need to consider it.

• Module 16

Properties of Laplace Transform :To understand the properties of Laplace Transform and associating the knowledge of properties of ROC in response to different operations on signals

• Module 17

Inverse Laplace Transform and Waveform Synthesis : To describe how to obtain inverse Laplace transform making use of the knowledge of properties of Laplace Transform and properties of ROC.

• Module 18

Discrete Time Signals and Z-Transforms :To understand representing discrete time signals, apply z transform for analyzingdiscrete time signals and to understand the relation to Fourier transform

• Module 19

Region of Convergence (ROC)  (Z-Transforms) :To understand the meaning of ROC in Z transforms and the need to consider it.

• Module 20

Properties of Z-Transform :To understand the properties of Z-Transform and associating the knowledge of properties of ROC in response to different operations on discrete signals

• Module 21

InverseZ-Transform :To describe how to obtain inverse z-transform making use of the knowledge of properties of z-Transform and properties of ROC

• Module 22

Random Processes – Temporal Characteristics : To make the student to understand the concept of Random Processes, its types and its distribution and density function

• Module 23

Independence and Stationarity :  To introduce the concepts of Statistical Independence, Stationarity and its types w.r.to random processes. This module also presents the concept of Ergodicity

• Module 24

Time based Parameters and their properties :

To analyze the characteristics of a random signal in time domain, one need to estimate certain parameters like Mean, Variance, correlation etc. This particular module deals with Autocorrelation, Cross Correlation, Covariance and its properties. These parameters play a vital role in signal estimation and analysis in various applications like RADR, SONAR, and NAVY etc.

• Module 25

Gaussian and Poisson Random Processes : This module presents the Concept of Gaussian and Poisson Random Processes

• Module 26

Analysis of LTI System Response :This particular module discusses the methods of describing the out response of a linear time invariant system (LTI) when a continuous random process is applied at the input

• Module 27

Random Processes – Spectral Characteristics : To design any LTI filter which is intended to extract or suppress the signal, it is necessary to understand how the strength of a signal is distributed in the frequency domain, relative to the strengths of other ambient signals. Similar to the deterministic signals, it turns out to be just as true in the case of random signals

• Module 28

Cross Power density spectrum :To determine the relationship between two time series as a function of frequencyusing Cross spectral analysis

• Module 29

Spectral characteristics of system response  :The quality of a communication system deals with the delivery of message  to the user, who is available after the receiver. The present module deals with the effect of an LTI system on the input random process. This in turn helps in  the computation of S/N at the output of thr receiver in a communication system