ECE 301: Signals and Systems Course Notes Prof. Shreyas Sundaram PDF

Types of Signal Systems and their Properties

Even signals are symmetric around vertical axis and. Odd signals are symmetric about origin. Even Signal: A signal is referred to as an even if it is

Even and Odd Functions

signal is even only cosines are involved whereas if the signal is odd then only sines are involved. We determine if a function is even or odd or neither.

Even and Odd Functions

signal is even only cosines are involved whereas if the signal is odd then only sines are involved. We determine if a function is even or odd or neither.

Chameli Devi Group of Institutions Indore Department of Electronics

II. Even and Odd Signals : A signal is said to be even signal if it is symmetrical about the amplitude axis. The even signal amplitude is.

ECE 301: Signals and Systems Course Notes Prof. Shreyas Sundaram

A signal is even if f(t) = f(?t) for all t ? R (in continuous-time) It is easy to verify that o(t) is an odd signal and e(t) is an even signal.

ECEN 314: Signals and Systems - Homework 2 Solutions I Reading

(1.3) Determine the value of P? and E? for each of the following signals (b) Show that if x1[n] is an odd signal and x2[n] is an even signal ...

Signals in Time Domain

Any signal can be represented by a sum of even and odd signals: (2.10) where. (2.11) or. (2.12). Example 2.3. Determine if the following signals are even or

Solution Manual for Additional Problems for SIGNALS AND

Determine for which of the given values of Ts the discrete-time signal has lost If that is not the case even a stable system would provide an unbounded.

UNIT-I

Even and odd signals: a signal is even if x(-t) = x(t) (i.e. it can be reflected in the axis at zero). A signal is odd if x(-t) = -x(t).

Even and Odd Functions

as if the signal is even only cosines are involved whereas if the signal is odd then only sines are involved. determine if a function is even or odd or.

[PDF] Types of Signal Systems and their Properties

Even and Odd Signal One of characteristics of signal is symmetry that may be useful for signal analysis Even signals are symmetric around vertical axis

Signals and Systems Even and Odd Signals - Tutorialspoint

11 nov 2021 · Therefore the even signals are also called the symmetrical signals Cosine wave is an example Find whether the signals are even or odd

[PDF] 2 Signals and Systems: Part I - MIT OpenCourseWare

The signal is therefore neither even nor odd (e) In similar manner to part (a) we deduce that x[n] is even (f) x

[PDF] Even and Odd Signalspdf

6 oct 2012 · Even and Odd Signals A signal x(t) is said to be Even if x(t) odd if can be written as the sum of an even signal and an odd Signal

[PDF] Even and Odd Functions

In this Section we examine how to obtain Fourier series of periodic functions which are either even or odd We show that the Fourier series for such

[PDF] Even and Odd Functions

Even and Odd Functions A function f is even (or symmetric) when f(x) = f(?x) A function f is odd (or antisymmetric) when f(x) = ?f(?x)

[PDF] ECE 314 – Signals and Systems Fall 2012

Find the even and odd components of each of the following signals: Solution: We may solve this by inspection if we consider the following prop- erties:

[PDF] Signals and Systems

6552111 Signals and Systems Even and Odd Signals: Example 4 Find the even and odd components of the signals shown in figure below 30 Sopapun Suwansawang

Even and Odd Signal MCQ [Free PDF] - Objective Question Answer

9 fév 2023 · If x(t) is an odd function then x(t) = -x(-t) Analysis: Looking at the graph given in the question we can see that f(t) = f(-t) The graph is

How do you check if a signal is even or odd?
Even signals are symmetric around vertical axis, and Odd signals are symmetric about origin. Even Signal: A signal is referred to as an even if it is identical to its time-reversed counterparts; x(t) = x(-t).
Explanation: Signals are classified as even if it has symmetry about its vertical axis. It is given by the equation x (-t) = x (t). Explanation: Signals is said to be odd if it is anti- symmetry over the time origin. And it is given by the equation x (-t) = -x (t).

ECE 301: Signals and Systems

Course Notes

Prof. Shreyas Sundaram

School of Electrical and Computer Engineering

Purdue University

Acknowledgments

These notes very closely follow the book:Signals and Systems, 2nd edition, by Alan V. Oppenheim, Alan S. Willsky with S. Hamid Nawab. Parts of the notes are also drawn from

Linear Systems and Signalsby B. P. Lathi

A Course in Digital Signal Processingby Boaz Porat

Calculus for Engineersby Donald Trim

I claim credit for all typos and mistakes in the notes. The L ATEX template forThe Not So Short Introduction to LATEX2"by T. Oetiker et al. was used to typeset portions of these notes.

Shreyas Sundaram

Purdue University

1 Introduction 1

1.1 Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Outline of This Course . . . . . . . . . . . . . . . . . . . . . . . .

2 Properties of Signals and Systems 5

2.1 Signal Energy and Power . . . . . . . . . . . . . . . . . . . . . .

2.2 Transformations of Signals . . . . . . . . . . . . . . . . . . . . . .

2.3 Periodic, Even and Odd Signals . . . . . . . . . . . . . . . . . . .

2.4 Exponential and Sinusoidal Signals . . . . . . . . . . . . . . . . .

2.4.1 Continuous-Time Complex Exponential Signals . . . . . .

2.4.2 Discrete-Time Complex Exponential Signals . . . . . . . .

2.5 Impulse and Step Functions . . . . . . . . . . . . . . . . . . . . .

2.5.1 Discrete-Time . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.2 Continuous-Time . . . . . . . . . . . . . . . . . . . . . . .

2.6 Properties of Systems . . . . . . . . . . . . . . . . . . . . . . . .

2.6.1 Interconnections of Systems . . . . . . . . . . . . . . . . .

2.6.2 Properties of Systems . . . . . . . . . . . . . . . . . . . .

3 Analysis of Linear Time-Invariant Systems 21

3.1 Discrete-Time LTI Systems . . . . . . . . . . . . . . . . . . . . .

3.2 Continuous-Time LTI Systems . . . . . . . . . . . . . . . . . . .

3.3 Properties of Linear Time-Invariant Systems . . . . . . . . . . . .

3.3.1 The Commutative Property . . . . . . . . . . . . . . . . .

3.3.2 The Distributive Property . . . . . . . . . . . . . . . . . .

3.3.3 The Associative Property . . . . . . . . . . . . . . . . . .

25
vi CONTENTS

3.3.4 Memoryless LTI Systems . . . . . . . . . . . . . . . . . .

3.3.5 Invertibility of LTI Systems . . . . . . . . . . . . . . . . .

3.3.6 Causality of LTI Systems . . . . . . . . . . . . . . . . . .

3.3.7 Stability of LTI Systems . . . . . . . . . . . . . . . . . . .

3.3.8 Step Response of LTI Systems . . . . . . . . . . . . . . .

3.4 Dierential and Dierence Equation Models for Causal LTI Systems

3.4.1 Linear Constant-Coecient Dierential Equations . . . .

3.4.2 Linear Constant Coecient Dierence Equations . . . . .

3.5 Block Diagram Representations of Linear Dierential and Dier-

ence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Fourier Series Representation of Periodic Signals 37

4.1 Applying Complex Exponentials to LTI Systems . . . . . . . . .

4.2 Fourier Series Representation of Continuous-Time Periodic Signals

4.3 Calculating the Fourier Series Coecients . . . . . . . . . . . . .

4.3.1 A Vector Analogy for the Fourier Series . . . . . . . . . .

4.4 Properties of Continuous-Time Fourier Series . . . . . . . . . . .

4.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . .

4.4.4 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.5 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.6 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . . .

4.5 Fourier Series for Discrete-Time Periodic Signals . . . . . . . . .

4.5.1 Finding the Discrete-Time Fourier Series Coecients . . .

4.5.2 Properties of the Discrete-Time Fourier Series . . . . . . .

5 The Continuous-Time Fourier Transform 59

5.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 Existence of Fourier Transform . . . . . . . . . . . . . . .

5.2 Fourier Transform of Periodic Signals . . . . . . . . . . . . . . . .

5.3 Properties of the Continuous-Time Fourier Transform . . . . . .

5.3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 Time-Shifting . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS vii

5.3.3 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . .

5.3.5 Time and Frequency Scaling . . . . . . . . . . . . . . . .

5.3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.7 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . . .

5.3.8 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.9 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .

6 The Discrete-Time Fourier Transform 75

6.1 The Discrete-Time Fourier Transform . . . . . . . . . . . . . . .

6.2 The Fourier Transform of Discrete-Time Periodic Signals . . . . .

6.3 Properties of the Discrete-Time Fourier Transform . . . . . . . .

6.3.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.3 Time and Frequency Shifting . . . . . . . . . . . . . . . .

6.3.4 First Order Dierences . . . . . . . . . . . . . . . . . . . .

6.3.5 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.6 Time-Reversal . . . . . . . . . . . . . . . . . . . . . . . .

6.3.7 Time Expansion . . . . . . . . . . . . . . . . . . . . . . .

6.3.8 Dierentiation in Frequency . . . . . . . . . . . . . . . . .

6.3.9 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . . .

6.3.10 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.11 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .

7 Sampling 87

7.1 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . .

7.2 Reconstruction of a Signal From Its Samples . . . . . . . . . . .

7.2.1 Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . .

7.2.2 First-Order Hold . . . . . . . . . . . . . . . . . . . . . . .

7.3 Undersampling and Aliasing . . . . . . . . . . . . . . . . . . . . .

7.4 Discrete-Time Processing of Continuous-Time Signals . . . . . .

91
viii CONTENTS

8 The Laplace Transform 95

8.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . .

8.2 The Region of Convergence . . . . . . . . . . . . . . . . . . . . .

8.3 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . .

101

8.3.1 Partial Fraction Expansion . . . . . . . . . . . . . . . . .

102

8.4 Some Properties of the Laplace Transform . . . . . . . . . . . . .

103

8.4.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . .

104

8.4.2 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . .

104

8.4.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . .

104

8.5 Finding the Ouput of an LTI System via Laplace Transforms . .

105

8.6 Finding the Impulse Response of a Dierential Equation via Laplace

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Chapter 1 Introduction

1.1 Signals and Systems

Loosely speaking,signalsrepresent information or data about some phenomenon of interest. This is a very broad denition, and accordingly, signals can be found in every aspect of the world around us. For the purposes of this course, asystemis an abstract object that acceptsinput signalsand producesoutput signalsin response.SystemInputOutput Figure 1.1: An abstract representation of a system.

Examples of systems and associated signals:

Electrical circuits: voltages, currents, temperature,... Mechanical systems: speeds, displacement, pressure, temperature, vol- ume, ... Chemical and biological systems: concentrations of cells and reactants, neuronal activity, cardiac signals, ... Environmental systems: chemical composition of atmosphere, wind pat- terns, surface and atmospheric temperatures, pollution levels, ... Economic systems: stock prices, unemployment rate, tax rate, interest rate, GDP, ... Social systems: opinions, gossip, online sentiment, political polls,... Audio/visual systems: music, speech recordings, images, video, ...

2 Introduction

Computer systems: Internet trac, user input, ...

From a mathematical perspective, signals can be regarded as functions of one or more independent variables. For example, the voltage across a capacitor in an electrical circuit is a function of time. A static monochromatic image can be viewed as a function of two variables: anx-coordinate and ay-coordinate, where the value of the function indicates the brightness of the pixel at that (x;y) coordinate. A video is a sequence of images, and thus can be viewed as a function of three variables: anx-coordinate, ay-coordinate and a time- instant. Chemical concentrations in the earth's atmosphere can also be viewed as functions of space and time. In this course, we will primarily be focusing on signals that are functions of a single independent variable (typically taken to be time). Based on the examples above, we see that this class of signals can be further decomposed into two subclasses: Acontinuous-time signalis a function of the formf(t), wheretranges over all real numbers (i.e.,t2R). Adiscrete-time signalis a function of the formf[n], wherentakes on only a discrete set of values (e.g.,n2Z). Note that we use square brackets to denote discrete-time signals, and round brackets to denote continuous-time signals. Examples of continuous-time sig- nals often include physical quantities, such as electrical currents, atmospheric concentrations and phenomena, vehicle movements, etc. Examples of discrete- time signals include the closing prices of stocks at the end of each day, population demographics as measured by census studies, and the sequence of frames in a digital video. One can obtain discrete-time signals bysamplingcontinuous-time signals (i.e., by selecting only the values of the continuous-time signal at certain intervals). Just as with signals, we can consider continuous-time systems and discrete- time systems. Examples of the former include atmospheric, physical, electrical and biological systems, where the quantities of interest change continuously over time. Examples of discrete-time systems include communication and computing systems, where transmissions or operations are performed in scheduled time- slots. With the advent of ubiquitous sensors and computing technology, the last few decades have seen a move towardshybridsystems consisting of both continuous-time and discrete-time subsystems { for example, digital controllers and actuators interacting with physical processes and infrastructure. We will not delve into such hybrid systems in this course, but will instead focus on systems that are entirely either in the continuous-time or discrete-time domain. The termdynamical systemloosely refers to any system that has an internal state and some dynamics (i.e., a rule specifying how the state evolves in time).

1.1 Signals and Systems 3

This description applies to a very large class of systems, including individual ve- hicles, biological, economic and social systems, industrial manufacturing plants, electrical power grid, the state of a computer system, etc. The presence of dy- namics implies that the behavior of the system cannot be entirely arbitrary; the temporal behavior of the system's state and outputs can be predicted to some extent by an appropriatemodelof the system. Example 1.1.Consider a simple model of a car in motion. Let the speed of the car at any timetbe given byv(t). One of the inputs to the system is the accelerationa(t), applied by the throttle. From basic physics, the evolution of the speed is given bydvdt =a(t):(1.1) The quantityv(t) is the state of the system, and equation (1.1) species the dynamics. There is a speedometer on the car, which is a sensor that measures the speed. The value provided by the sensor is denoted bys(t) =v(t), and this

is taken to be the output of the system.Much of scientic and engineering endeavor relies on gathering, manipulating

and understanding signals and systems across various domains. For example, in communication systems, the signal represents voice or data that must be transmitted from one location to another. These information signals are often corrupted en route by other noise signals, and thus the received signal must be processed in order to recover the original transmission. Similarly, social, physical and economic signals are of great value in trying to predict the current and future state of the underlying systems. The eld of signal processing studies how to take given signals and extract desirable features from them, often via the design of systems known aslters. The eld of control systems focuses on designing certain systems (known as controllers) that measure the signals coming from a given system and apply other input signals in order to make the given system behave in an desirable manner. Typically, this is done via a feedback loopof the formControllerSystemDesiredquotesdbs_dbs9.pdfusesText_15

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[PDF] ECE 301: Signals and Systems Course Notes Prof. Shreyas Sundaram

How do you check if a signal is even or odd?

ECE 301: Signals and Systems

Course Notes

Prof. Shreyas Sundaram

School of Electrical and Computer Engineering

Purdue University

Acknowledgments

Linear Systems and Signalsby B. P. Lathi

Calculus for Engineersby Donald Trim

Shreyas Sundaram

Purdue University

Contents

1 Introduction 1

1.1 Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Outline of This Course . . . . . . . . . . . . . . . . . . . . . . . .

2 Properties of Signals and Systems 5

2.1 Signal Energy and Power . . . . . . . . . . . . . . . . . . . . . .

2.2 Transformations of Signals . . . . . . . . . . . . . . . . . . . . . .

2.3 Periodic, Even and Odd Signals . . . . . . . . . . . . . . . . . . .

2.4 Exponential and Sinusoidal Signals . . . . . . . . . . . . . . . . .

2.4.1 Continuous-Time Complex Exponential Signals . . . . . .

2.4.2 Discrete-Time Complex Exponential Signals . . . . . . . .

2.5 Impulse and Step Functions . . . . . . . . . . . . . . . . . . . . .

2.5.1 Discrete-Time . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.2 Continuous-Time . . . . . . . . . . . . . . . . . . . . . . .

2.6 Properties of Systems . . . . . . . . . . . . . . . . . . . . . . . .

2.6.1 Interconnections of Systems . . . . . . . . . . . . . . . . .

2.6.2 Properties of Systems . . . . . . . . . . . . . . . . . . . .

3 Analysis of Linear Time-Invariant Systems 21

3.1 Discrete-Time LTI Systems . . . . . . . . . . . . . . . . . . . . .

3.2 Continuous-Time LTI Systems . . . . . . . . . . . . . . . . . . .

3.3 Properties of Linear Time-Invariant Systems . . . . . . . . . . . .

3.3.1 The Commutative Property . . . . . . . . . . . . . . . . .

3.3.2 The Distributive Property . . . . . . . . . . . . . . . . . .

3.3.3 The Associative Property . . . . . . . . . . . . . . . . . .

3.3.4 Memoryless LTI Systems . . . . . . . . . . . . . . . . . .

3.3.5 Invertibility of LTI Systems . . . . . . . . . . . . . . . . .

3.3.6 Causality of LTI Systems . . . . . . . . . . . . . . . . . .

3.3.7 Stability of LTI Systems . . . . . . . . . . . . . . . . . . .

3.3.8 Step Response of LTI Systems . . . . . . . . . . . . . . .

3.4 Dierential and Dierence Equation Models for Causal LTI Systems

3.4.1 Linear Constant-Coecient Dierential Equations . . . .

3.4.2 Linear Constant Coecient Dierence Equations . . . . .

3.5 Block Diagram Representations of Linear Dierential and Dier-

4 Fourier Series Representation of Periodic Signals 37

4.1 Applying Complex Exponentials to LTI Systems . . . . . . . . .

4.2 Fourier Series Representation of Continuous-Time Periodic Signals

4.3 Calculating the Fourier Series Coecients . . . . . . . . . . . . .

4.3.1 A Vector Analogy for the Fourier Series . . . . . . . . . .

4.4 Properties of Continuous-Time Fourier Series . . . . . . . . . . .

4.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . .

4.4.4 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.5 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.6 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . . .

4.5 Fourier Series for Discrete-Time Periodic Signals . . . . . . . . .

4.5.1 Finding the Discrete-Time Fourier Series Coecients . . .

4.5.2 Properties of the Discrete-Time Fourier Series . . . . . . .

5 The Continuous-Time Fourier Transform 59

5.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 Existence of Fourier Transform . . . . . . . . . . . . . . .

5.2 Fourier Transform of Periodic Signals . . . . . . . . . . . . . . . .

5.3 Properties of the Continuous-Time Fourier Transform . . . . . .

5.3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 Time-Shifting . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS vii

5.3.3 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . .

5.3.5 Time and Frequency Scaling . . . . . . . . . . . . . . . .

5.3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.7 Parseval's Theorem . . . . . . . . . . . . . . . . . . . . . .

5.3.8 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.9 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .

6 The Discrete-Time Fourier Transform 75

6.1 The Discrete-Time Fourier Transform . . . . . . . . . . . . . . .

6.2 The Fourier Transform of Discrete-Time Periodic Signals . . . . .

6.3 Properties of the Discrete-Time Fourier Transform . . . . . . . .

6.3.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . .