LECTURE NOTES On SIGNALS & SYSTEMS (PCBM4302) 5th Semester ETC Engineering Prepared by, Kodanda Dhar Sa Paresh Kumar Pasayat Prasanta Kumar Sahu INDIRA GANDHI INSTITUTE OF TECHNOLOGY, SARANG
SYLLABUS Module – I Discrete-Time Signals and Systems: Discrete-Time Signals: Some Elementary Discrete-Time signals, Classification of Discrete-Time Signals, Simple Manipulation; Discrete-Time Systems : Input-Output Description, Block Diagram Representation, Classification, Interconnection; Analysis of DiscreteTime LTI Systems: Techniques, Response of LTI Systems, Properties of Convolution, Causal LTI Systems, Stability of LTI Systems; Discrete-Time Systems Described by Difference Equations; Implementation of Discrete-Time Systems; Correlation of Discrete-Time Signals: Crosscorrelation and Autocorrelation Sequences, Properties. Properties of Continuous-Time Systems: Block Diagram and System Terminology, System Properties: Homogeneity, Time Invariance, Additivity, Linearity and Superposition, Stability, Causality. Module – II The Continuous-Time Fourier Series: Basic Concepts and Development of the Fourier Series, Calculation of the Fourier Series, Properties of the Fourier Series. The Continuous-Time Fourier Transform: Basic Concepts and Development of the Fourier Transform, Properties of the Continuous-Time Fourier Transform. Module- III The Z-Transform and Its Application to the Analysis of LTI Systems: The ZTransform: The Direct Z-Transform, The Inverse Z-Transform; Properties of the Z-Transform; Rational Z-Transforms: Poles and Zeros, Pole Location and Time-Domain Behaviour for Causal Signals, The System Function of a Linear Time-Invariant System; Inversion of the Z-Transforms: The Inversion of the ZTransform by Power Series Expansion, The Inversion of the Z-Transform by
Partial-Fraction Expansion; The One-sided Z-Transform: Definition and Properties, Solution of Difference Equations. The Discrete Fourier Transform: Its Properties and Applications: Frequency Domain Sampling: The Discrete Fourier Transform; Properties of the DFT: Periodicity, Linearity, and Symmetry Properties, Multiplication of Two DFTs and Circular Convolution, Additional DFT Properties.
Module – I 1.1 WHAT IS A SIGNAL We are all immersed in a sea of signals. All of us from the smallest living unit, a cell, to the most complex living organism (humans) are all time receiving signals and are processing them. Survival of any living organism depends upon processing the signals appropriately. What is signal? To define this precisely is a difficult task. Anything which carries information is a signal. In this course we will learn some of the mathematical representations of the signals, which has been found very useful in making information processing systems. Examples of signals are human voice, chirping of birds, smoke signals, gestures (sign language), fragrances of the flowers. Many of our body functions are regulated by chemical signals, blind people use sense of touch. Bees communicate by their dancing pattern. Some examples of modern high speed signals are the voltage charger in a telephone wire, the electromagnetic field emanating from a transmitting antenna, variation of light intensity in an optical fiber. Thus we see that there is an almost endless variety of signals and a large number of ways in which signals are carried from on place to another place. In this course we will adopt the following definition for the signal: A signal is a real (or complex) valued function of one or more real variable(s).When the function depends on a single variable, the signal is said to be one dimensional. A speech signal, daily maximum temperature, annual rainfall at a place, are all examples of a one dimensional signal. When the function depends on two or more variables, the signal is said to be multidimensional. An image is representing the two dimensional signal, vertical and horizontal coordinates representing the two dimensions. Our physical world is four dimensional (three spatial and one temporal). 1.2 CLASSIFICATION OF SIGNALS As mentioned earlier, we will use the term signal to mean a real or complex valued function of real variable(s). Let us denote the signal by x(t). The variable t is called independent variable and the value x of t as dependent variable. We say a signal is continuous time signal if the independent variable t takes values in an interval. For example t ϵ (−∞, ∞), or tϵ [0, ∞] or t ϵ[T0, T1]. The independent variable t is referred to as time, even though it may not be actually time. For example in variation if pressure with height t refers above