15 results
Frontmatter
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp i-iv
-
- Chapter
- Export citation
Preface
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp xiii-xiv
-
- Chapter
- Export citation
-
Summary
It gives me immense satisfaction in presenting this book to my students who have been eagerly waiting to see it. The difficulties encountered by students in understanding the physical significance of different concepts inspired me to write a student-friendly book, rich in technical content. The subject Signals and Systems is essential for undergraduate students of Electronics Engineering, Electrical Engineering, Computer Engineering, and Instrumentation Engineering disciplines. The subject has a diverse range of applications. A thorough knowledge of different transforms studied in Mathematics is essential for an understanding of the subject.
The subject involves a number of complex algorithms which require indepth domain knowledge. If a concept is explained with concrete examples and programs, the reader will definitely take interest in the field. The reader will experience great joy when she observes tangible outcomes of an experiment on her computer screen. A number of signals occurring in nature like speech, ECG, EEG etc. have some random components. Research in this field is somewhat difficult. Despite significant progress in an understanding of the subject, there remain many things which are not well grasped. There is room for explaining the basic concepts of signals and systems, in a better manner. Different concepts are illustrated here using MATLAB programs. The outputs of the MATLAB programs are given in the form of graphs. With extensive experience in signals and systems research, I observed, that people require significant amount of time before they can begin grasping the subject. However with proper guidance, a person can acquire considerable knowledge in this field. The motivation behind this book has been, that a new comer be provided information about where and how to start and how to proceed.
There was a request from my students as well as from well wishers that I write a book on signals and systems. It was also suggested that the book include the theory of random signals. Several concrete examples have been given to illustrate the concepts pertaining to random signals. For better understanding, we have included output from many MATLAB programs. Interpretations of the results are also explained, which will not only help the reader but also enable instructors to incorporate the examples into their classroom teaching.
6 - Fourier Transform Representation of Aperiodic Signals
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 409-512
-
- Chapter
- Export citation
-
Summary
The chapter deals with Fourier transform (FT) representation for continuous time and discrete time aperiodic signals. The FT of CT signals is called as CTFT and that for DT signals is called as DTFT. The Fourier transform is defined and is evaluated for all standard aperiodic signals such as exponential signal, rectangular pulse, triangular pulse, etc. The IFT of some standard signals such as sinc function is also discussed. The use of Dirac Delta function is explained for evaluation of FT for periodic signals. FT of DT signals are illustrated with some numerical examples. Properties of FT and DTFT are emphasized with their physical significance. The numerical examples for the calculation of FT using FT properties are illustrated. The calculation of response of LTI system to the input signal is simplified using FT.
Fourier Transform Representation of Aperiodic CT Signals
If the signal x(t) is aperiodic, a similar representation for a signal in frequency domain can be developed in terms of Fourier Transform using exponential signals as basis function. The signal xP(t) a periodic signal can be generated by repeating x(t) after a period of T0. We can now define x(t) as
The Fourier series representation can be written for the periodic signal xP(t). Equation (5.38) becomes
The coefficients can be written as
As the period tends to infinity, the spacing between the spectral lines, which is equal to 1/T0, will tend to zero and the spectrum will be a continuous spectrum. Now, the summation will turn into the integral. The coefficients will represent a frequency point on the frequency axis represented by X(f).
Index
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 767-771
-
- Chapter
- Export citation
8 - Z Transform
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 618-692
-
- Chapter
- Export citation
-
Summary
Z domain is used for DT signals. The purpose of Z transform for DT signals is similar to that of Laplace transform for analog signals. The Z transform (ZT) is used to characterize a DT system by analyzing its transfer function in the Z domain. The locations of poles and zeros of the transfer function are further used for stability analysis of the DT systems. We will first discuss the significance of a transform. To understand that Laplace transform and Z transform are parallel techniques, we will start with Laplace transform and develop its relation to Z transform. The properties of Z transform will be discussed along with the region of convergence (ROC). The overall strategy of these two transforms is as follows. Find the impulse response from the transfer function. Find poles and zeros of the transfer function and apply sinusoids and exponentials as inputs to find the response. The basic difference between the two transforms is that the s-plane used by S domain is arranged in a rectangular co-ordinate system, while the z-plane used by Z domain uses a polar format. The Z transform provides mathematical framework for designing digital filters based on the well-known theory of analog filters using a mapping technique. The reader will study this part in the course on “Digital signal processing”.
Physical Significance of a Transform
When any natural signal is to be processed, the signal parameters or features are required to be extracted. A transform helps in three ways.
The spectral properties of the signal are better revealed in a transform domain.
The system transfer function directly indicates the locations of poles and zeros that help in analyzing the stability of the system.
A signal, while transmission over a communication channel, needs to be compressed. Energy compaction can be achieved in a transform domain. The signal containing, for example, 128 sample points when transformed in Fourier domain will result in Fourier coefficients 128 in number. However, only a small number of them will have significant values and the others will be tending to zero.
3 - CT and DT Systems
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 181-245
-
- Chapter
- Export citation
-
Summary
This chapter concentrates on system definition and properties. The properties of systems such as linearity, time invariance, causality, invertibiity, memory and stability are discussed in detail. We have discussed sampling in chapter 1. The reader is already familiar with CT and DT signals. We will define properties of CT and DT systems simultaneously. Finally, we will see how to represent a system as an interconnection of operators and series/parallel; connection of systems.
Properties of CT and DT Systems – Linearity and Shift Invariance
We will discuss and explain properties of CT and DT systems in the following sections. We will study the property of linearity and Shift/time invariance in this section.
Linearity property
Any CT or DT system is said to be linear if it obeys two important properties. The first is homogeneity and the second is additivity.
Homogeneity property
The system is said to obey the property of homogeneity if the following condition holds:
• For CT and DT input signal of x(t) or x[n], respectively, if the output is given by y(t) or y[n], then if the input signal is scaled by a factor of k to get input signal of kx(t) or kx[n], then the output is also scaled by the same factor k i.e., output is ky(t) or ky[n], where k is any scaling factor. Let H represent the system operator, then
• if y(t) = H[x(t)] then ky(t) = H[kx(t)] and
• if y(n) = H[x(n)] then ky(n) = H[kx(n)]
Additivity Property (Superposition Property)
The system is said to obey additivity property if the following condition holds good.
• Let y1(t) and y1[n] be the output for the CT and DT input signal x1(t) and x1[n], respectively, and y2(t) and y2[n] be the output for input signal x2(t) and x2[n], respectively, for a CT and DT system. The property of additivity states the following:
If the input to a system is the addition of two signals, then the output of the system is the addition of the respective outputs. Let H represent the system operator, then
• If y1(t) = H[x1(t)] and y2(t) = H[x2(t)], then
y1(t) + y2(t) = H[x(t) + x2(t)]
If y1(t) = H[x1(n)] and y2(n) = H[x2(n)], then
y1(n) + y2(n) = H[x(n) + x2(n)]
7 - Laplace Transform
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 513-617
-
- Chapter
- Export citation
-
Summary
We will discuss the unilateral and bilateral Laplace Transform (LT) and its significance in analyzing the systems. The significance of complex frequency will be discussed. The relation between LT and FT will be explained. Different examples based on calculation of LT and inverse LT will be solved. LT of some standard signals will be evaluated. LT is very useful for analyzing the stability of the system. Stability of the system in Laplace domain will be explained.
Definition of Laplace Transform
Laplace transform was first proposed by Laplace (year 1980). This is the operator that transforms the signal in time domain in to a signal in a complex frequency domain called as ‘S’ domain. The complex frequency domain will be denoted by S and the complex frequency variable will be denoted by ‘s’. Let us understand the significance of Laplace transform. The reader must be familiar with complex numbers. The complex frequency S can be likewise defined as s = σ + jω, where σ is the real part of s and jω is the imaginary part of s. The complex numbers are defined by mathematicians and are the mathematical abstractions useful for the analysis of signals and systems. It simplifies the mathematics. Similarly, the complex frequency plane is also the mathematical abstraction useful for the simplification of mathematics. Laplace transform does not have any physical significance. We can just say that ω stands for the real frequency and Laplace transform transforms the signal from time domain to some kind of frequency domain.
Physical significance of Laplace transform Laplace transform has no physical significance except that it transforms the time domain signal to a complex frequency domain. It is useful to simply the mathematical computations and it can be used for the easy analysis of signals and systems. The stability of the system is directly revealed when the transfer function of the system is known in Laplace domain. LT is used for solving differential equations.
9 - Random Signals and Processes
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 693-766
-
- Chapter
- Export citation
-
Summary
The theory of signals and systems is applied for the processing of signals like speech signal, RADAR signal, SONAR signal, earth quake signal, ECG and EEG signals. All these signals are naturally occurring signals and hence have some random component. The primary goal of this chapter is to introduce the principles of random signals and processes for in-depth understanding of the processing methods. We will introduce the concept of probability and will discuss different standard distribution functions used for the analysis of random signals. Different operations on the random variables namely expectation, variance and moments will be introduced. We will then discuss central limit theorem. The classification of random processes namely wide sense stationary, ergodic and strict sense stationary processes will be defined.
Probability
Consider the experiment of throwing a dice. The result of the throw is random in nature in the sense that we do not know the result i.e., the outcome of the experiment until the outcome is actually available. The randomness in the outcome is lost once the outcome is evident. The result of the experiment is either a 1dot, 2dots, 3dots and so on. The outcome is a discrete random variable. Here, there are only 6 possibilities and so one can easily find the probability of getting 1 dot is 1/6 evaluated as one of total possible outcomes. If the experiment is performed a large number of times, the ratio of frequency of occurrence of a 1dot and the total number of times the experiment is performed approaches 1/6 (Refer to Eq. (9.1)).
We will first define the term event. When any experiment is performed to generate some output, it is called as an event. The output obtained is called as the outcome of the experiment. The possible set of outcomes of the experiment is called as a space or as event space or sample space. Many times, there is the uncertainty or randomness involved in the outcome of the experiment. The term probability is closely related to uncertainty. In case of the experiment of throwing of a die, the uncertainty is involved in the outcome. When a die is thrown, we do not know what will be the outcome. Using the notion we may define a relative frequency of occurrence of some event. Consider the example of a fair coin. Let it be tossed n times. Let nH times Head appear.
1 - Introduction to Signals
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 1-23
-
- Chapter
- Export citation
-
Summary
We discuss the basic definitions of analog/continuous time (CT) and discrete time (DT) signals in this chapter. We need to understand the basics of discrete time signals, i.e., the sampled signals. The theory of sampled signals is introduced in this chapter. The analog signal is first interfaced to a digital computer via analog to digital converter (ADC). ADC consists of a sampler and a quantizer. We will mainly discuss the sampler in this chapter. The analog signal, when sampled, gets converted to discrete time (DT) signal. Here, the time axis is digitized with a constant sampling interval T. The inverse of T is the sampling frequency. The sampling frequency must be properly selected for faithful reconstruction of the analog signal.
Introduction to Signals
Any physical quantity that carries some information can be called a signal. The physical quantities like temperature, pressure, humidity, etc. are continuously monitored in a process. Usually, the information carried by a signal is a function of some independent variable, for example, time. The actual value of the signal at any instant of time is called its amplitude. These signals are normally plotted as amplitude vs. time graph. This graph is termed as the waveform of the signal. The signal can be a function of one or more independent variables. Let us now define a signal.
Definition of a signal A signal can be defined as any physical quantity that varies with one or more independent variables.
Let us consider temperature measurement in a plant. The measured value of temperature will be its amplitude. This temperature changes from one instant of time to another. Hence, it is a function of time, which is an independent variable, as it does not depend on anything else. Temperature can be measured at two different locations in a plant. The values of temperature at two different locations may be different. Hence, the temperature measured depends on the time instant and also on the location. We can say that temperature is a function of two independent variables, namely time and location in a plant. This temperature can take on any continuous value, like 30.1, 30.001, 31.3212, etc. The time axis is also continuous i.e. the temperature is noted at each time value. The signal is then called continuous time continuous amplitude (CTCA) signal.
2 - Signals and Operations on Signals
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 24-180
-
- Chapter
- Export citation
-
Summary
This chapter aims to introduce basic concepts related to CT and DT signals and operations on the signals such as translation, scaling, addition, multiplication, etc.
Natural signals like speech, seismic signal, Electro Cardio Gram (ECG) and Electro Encephalogram (EEG) signals are processed using sophisticated digital systems. All these information-carrying signals are functions of the independent variables, namely, time and space. We first deal with basic definition and classification of signals.
Signals
We have gone through the basics of signals in chapter 1. We will go through it in detail in this chapter. Let us first define a signal.
Definition of a signal A signal is defined as a single valued function of an independent variable, namely, time that conveys some information. At every instant of time there is a unique value of the function. The unique value may be real or complex.
A signal X is represented as X = F(t), where F is some function and t represents time, the independent variable. Here, time is a continuous variable and X also takes on continuous values. X is then termed as continuous time, continuous amplitude (CTCA) signal that is the analog signal. When a signal is written as some mathematical equation, it is deterministic in the sense that the signal value at any time t can be determined using the mathematical equation. A signal is called scalar-valued if its value is unique for every instant of time. Most of the naturally occurring signals are scalar-valued, real, analog and random in nature. Speech signal is an example of such naturally occurring random signals. Speech signal is generated when a person utters a word. Researchers are still trying to generate a model for speech signal and exact mathematical equation cannot be written for a speech signal. It has a random signal component of some strength. It is called as random signal. If a signal has multiple values at any instant of time, it is called vector-valued. For example, color images, where at every pixel there is a three-dimensional vector representing RGB color components. In this chapter we will refer to scalar-valued and deterministic signals only. Hence, the signals will always be represented in the form of an equation.
5 - Fourier Series Representation of Periodic Signals
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 331-408
-
- Chapter
- Export citation
-
Summary
This chapter deals with the Fourier series representation of periodic CT signals. Chapter 4 dealt with the representation of signals in terms of impulses where the impulse function is treated as a basic unit. Any signal can be represented as a linear combination of scaled and shifted impulses. This chapter uses the decomposition of a signal in terms of sines and cosines or exponential signals and represents the signal as a vector with each coefficient of the vector representing the amount of similarity of the signal with the basic functions namely sines and cosines or exponential functions. This representation is termed as Fourier series representation after the scientist named Fourier who claimed that every signal can be represented as a linear combination of sinusoidal functions.
Firstly, we will study the representation of signals in terms of vectors, which makes it easy to understand the concept of orthogonality. The use of exponential series representation is elaborated. The trigonometric series representation and exponential representations are compared. We will then classify the signals as periodic and aperiodic. Periodic signals will be represented in terms of Fourier series representation. Aperiodic signals will be represented in terms of Fourier transform representation. Fourier series representation will be illustrated for CT and DT signals.
Signal Representation in Terms as Sinusoids
We have discussed the decomposition of any CT signal in terms of impulses and decomposition of any DT signal in terms of unit impulse function in chapter 4. Unit impulse is considered as a basic unit because the LTI systems can be characterized by impulse response. We can also decompose any signal in terms of sinusoids or in terms of complex exponentials. Here, the sinusoids or exponential functions are used as basic units. The decomposition of a signal as a linear combination of scaled sinusoids or complex exponentials is discussed further.
Why we select a sinusoid or complex exponentials?
The reason is when a sinusoid or complex exponential is given as input to any LTI system, the output is the same sinusoid or the same complex exponential except the change in amplitude and/or the change of phase. No other signal can have this claim. We will first prove that the sinusoids of different frequencies are orthogonal to each other and complex exponentials of different frequencies are also orthogonal to each other.
Contents
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp vii-xii
-
- Chapter
- Export citation
4 - Time Domain Response of CT and DT LTI Systems
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp 246-330
-
- Chapter
- Export citation
Dedication
- Shaila Dinkar Apte
-
- Book:
- Signals and Systems
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016, pp v-vi
-
- Chapter
- Export citation
Signals and Systems
- Principles and Applications
- Shaila Dinkar Apte
-
- Published online:
- 05 May 2016
- Print publication:
- 09 May 2016
-
This book provides a rigorous treatment of deterministic and random signals. It offers detailed information on topics including random signals, system modelling and system analysis. System analysis in frequency domain using Fourier transform and Laplace transform is explained with theory and numerical problems. The advanced techniques used for signal processing, especially for speech and image processing, are discussed. The properties of continuous time and discrete time signals are explained with a number of numerical problems. The physical significance of different properties is explained using real-life examples. To aid understanding, concept check questions, review questions, a summary of important concepts, and frequently asked questions are included. MATLAB programs, with output plots and simulation examples, are provided for each concept. Students can execute these simulations and verify the outputs.