Digital Signal Processing E-Book

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• Herausgeber: Packt Publishing
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• Sprache: Englisch

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Veröffentlichungsjahr: 2024

Beschreibung

This book covers the fundamental concepts of digital signal processing (DSP), introducing discrete-time signals, the z-transform, frequency analysis, discrete and fast Fourier transforms, digital filters, FIR, and statistical DSP. With applications ranging from engineering to telecommunications and medical imaging, this text focuses on representing signals with mathematical models and processing them using discrete-time systems.
Understanding DSP is crucial for professionals in various fields as it allows for effective signal representation and manipulation. The book starts with an introduction to DSP and a review of discrete-time signals and systems, followed by a detailed exploration of the z-transform and frequency analysis using DTFT. It further delves into discrete Fourier transforms, fast Fourier transform algorithms, and the implementation of discrete-time systems.
The journey continues with in-depth discussions on digital filters, including the design and analysis of infinite impulse response (IIR) and finite impulse response (FIR) filters. The book also covers statistical digital signal processing, multirate DSP, and applications in speech and radar. Through a variety of exercises and a comprehensive glossary, readers will gain a solid understanding of DSP fundamentals and applications.

Leseprobe

DIGITALSIGNALPROCESSING

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DIGITALSIGNALPROCESSING

An Introduction

R. Anand, Ph.D.

MERCURY LEARNING AND INFORMATIONDulles, VirginiaBoston, MassachusettsNew Delhi

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CONTENTS

Preface

Chapter 1:Introduction to Digital Signal Processing (DSP)

1.1Introduction

1.2Applications of Digital Signal Processing

1.3Signals

1.4Classification of Signals

1.5Signal Processing Systems

1.6Signal Processing

1.7Advantages of Digital Signal Processing over Analog Signal Processing

1.8Elements of Digital Signal Processing System

Exercises

Chapter 2:Review of Discrete—Time Signals and Systems

2.1Introduction

2.2Discrete-Time Signals

2.3Discrete-Time Systems

2.4Convolution of Two Discrete-Time Signals

2.5Inverse Systems

2.6Correlation of Two Discrete-Time Signals

2.7Signals and Vectors

2.8Representation of Signals on Orthogonal Basis

2.9Sampling of Continuous-Time Signals

2.10Reconstruction of a Signal from its Sample Values

Exercises

Numerical Exercises

Chapter 3:The z-Transform

3.1Introduction

3.2Definition of the z-Transform

3.3Region of Convergence (ROC)

3.4Properties of z-Transform

3.5Some Common z-Transform Pairs

3.6The Inverse z-Transform

3.7System Function

3.8Poles and Zeros of Rational z-Transforms

3.9Solution of Difference Equations Using z-Transform

3.10Analysis of Linear Time-Invariant (LTI) Systems in the z-Domain

Exercises

Numerical Exercises

Chapter 4:Frequency Analysis Using DTFT

4.1Introduction to Discrete-Time Fourier Transform (DTFT)

4.2Development of the Discrete-Time Fourier Transform (DTFT)

4.3Convergence of the DTFT

4.4Fourier Transform of Discrete-Time Periodic Signals

4.5Properties of the DTFT

4.6Tabulation of Properties of DTFT

4.7Tabulation of DTFT Pairs

4.8Duality

4.9Discrete-Time LTI Systems Characterized by Linear Constant-Coefficient Difference Equations

Exercises

Numerical Exercises

Chapter 5:Discrete Fourier Transforms (DFTs)

5.1Introduction

5.2Definition of DFT

5.3The DFT as a Linear Transformation Tool

5.4Properties of DFT

5.5Tabulation of Properties of DFT

5.6Relationship Between DFT and z-Transform

5.7Linear Convolution Using DFT

5.8Pitfalls in Using DFT

Exercises

Numerical Exercises

Chapter 6:Fast Fourier Transform (FFT) Algorithms

6.1Introduction

6.2Goertzel Algorithm

6.3Fast Fourier Transform Algorithms

Exercises

Chapter 7:Implementation of Discrete-Time Systems

7.1Introduction

7.2Major Factors Influencing Our Choice of Specific Realization

7.3Network Structures for IIR Systems

7.4Network Structure for FIR Systems

Exercises

Numerical Exercises

Chapter 8:Digital Filters

8.1Introduction

8.2Major Considerations in Using Digital Filters

8.3Comparison Between Digital and Analog Filters

8.4Comparison Between IIR and FIR Digital Filters

8.5Realization Procedures for Digital Filters

8.6Notch Filters

8.7Comb Filters

8.8All-Pass Filters

8.9Digital Sinusoidal Oscillators

8.10Digital Resonators

Exercises

Chapter 9:Design and Analysis of Infinite Impulse Response (IIR) Digital Filters

9.1Introduction

9.2Approximation of IIR Digital Filters from Analog Filters

9.3Frequency Transformation

Exercises

Numerical Exercises

Chapter 10:Design and Analysis of Finite Impulse Response (FIR) Digital Filters

10.1Introduction

10.2Properties of FIR Digital Filters

10.3Design of FIR Digital Filters Using Fourier Series Method

10.4Design of FIR Digital Filter Based on Numerical-Analysis Formulae

10.5Design of Optimal Linear-Phase FIR Digital Filters Using M-Clellan–Parks Method

10.6Finite Word Length Effects in Digital Filters

Exercises

Numerical Exercises

Chapter 11:Statistical Digital Signal Processing

11.1Introduction to Spectral Estimation

11.2Energy Density Spectrum

11.3Estimation of the Autocorrelation and Power Spectrum of Random Signals

11.4DFT in Spectral Estimation

11.5Non-Parametric Methods of Power Spectrum Estimation

11.6Parametric Methods of Power Spectrum Estimation

Exercises

Numerical Exercises

Chapter 12:Multirate Digital Signal Processing (MDSP)

12.1Introduction

12.2Sampling Rate Conversion

12.3Interpolation of Sampling Rate by a Integer Factor (I)

12.4Sampling Rate Alternation or Conversion by a Rational Factor

12.5Filter Design and Implementation for Sampling Rate Alternation or Conversion

12.6Sampling Rate Conversion by an Arbitrary Factor

12.7Application of Multirate Digital Signal Processing

Solved Examples

Exercises

Chapter 13:Applications of Digital Signal Processing to Speech

13.1Introduction

13.2Model of Speech Production

13.3Short-Time Fourier Transform (STFT)

13.4Speech Analysis-Synthesis Using STFT

13.5Analysis Considerations

13.6Overall Analysis-Synthesis System

13.7Channel Vocoder

13.8Pitch Detection and Voiced–Unvoiced Decisions

13.9Voiced–Unvoiced (Buzz-Hiss) Detection

13.10Voiced–Unvoiced (Buzz-Hiss) Detection

13.11Voiced Fricative Excitation Network

13.12Random Number Generator

13.13Principles of Digital Operation

13.14Linear Prediction of Speech

13.15A Computer Voice Response System

Exercises

Chapter 14:Applications of Digital Signal Processing to Radar

14.1Introduction

14.2Applications and Advantages of Radars

14.3Limitations of Using Radar

14.4Chirp z-Transform (CZT) Algorithm

14.5Radar System and Radar Parameters

14.6Radar Signal Design and Ambiguity Functions

14.7Ambiguity Functions of Chirps and Sinusoidal Pulses

14.8Ambiguity Function of a CW Pulse

14.9Ambiguity Functions of a Burst

14.10Other Signals

14.11Airborne Surveillance Radar for Air Traffic Control (ATC)

14.12Long-Range Demonstration Radar (LRDR)

14.13Digital Matched Filter for a High-Performance Radar (HPR)

Exercises

Glossary

Index

PREFACE

The great advancements in the design of microchips, digital systems, and computer hardware over the past 40 years have given birth to digital signal processing (DSP) which has grown over the years into a ubiquitous, multifaceted, and indispensable subject of study. As such, DSP has been applied in most disciplines ranging from engineering to economics and from astronomy to molecular biology. Consequently, it would take a multivolume encyclopedia to cover all the facets, aspects, and ramifications of DSP, and such a treatise would require many authors. This book focuses instead on the fundamentals of DSP, namely, on the representation of signals by mathematical models and on the processing of signals by discrete-time systems. Various types of processing are possible for signals, but the processing of interest in this volume is almost always linear. It typically involves reshaping, transforming, or manipulating the frequency spectrum of the signal of interest.

The author considers the processing of continuous- and discrete-time signals to be different facets of the same subject of study without a clear demarcation where the processing of continuous-time signals by analog systems ends, and the processing of discrete-time signals by digital systems begins. Discrete-time signals sometimes exist as distinct entities that are not derived from or related to corresponding continuous-time signals. The processing of such a signal would result in a transformed discrete-time signal, which would be, presumably, an enhanced, or in some way, more desirable version of the original signal. Obviously, reference to an underlying continuous time signal would be irrelevant in such a case. However, more often than not discrete-time signals are derived from corresponding continuous-time signals and, as a result, they inherit the spectral characteristics of the latter. Discrete-time signals of this type are often processed by digital systems, and after that, they are converted back to continuous-time signals. A case in point can be found in the recording industry where music is first sampled to generate a discrete-time signal, which is then recorded on a disc. When the disc is played back, the discrete-time signal is converted into a continuous-time signal. In order to preserve the spectrum of the underlying continuous-time signal, e.g., that delightful piece of music, through this series of signal manipulations, special attention must be paid to the spectral relationships that exist between continuous- and discrete-time signals.

In the past, signal processing appeared in various concepts in more traditional courses like telecommunications, control, circuit theory, and in instrumentation. The signal processing done was analog, and discrete components were used to achieve the various objectives. However, in the later part of the 20th century we saw the introduction of computers and their fast and tremendous growth. In the late 1960s and early 1970s, a number of researchers resorted to modeling and simulation of various concepts in their research endeavors, using digital computers, in order to determine performance and optimize their designs. It is these endeavors that led to the development of many digital signal processing algorithms which we know today. With the rapid growth of computing power in terms of speed and memory capacity, a number of researchers wanted to obtain their results from near real-time to real time. This saw the development of processors and I/O devices that were dedicated to real-time data processing; though initially at lower speeds, they are currently capable of processing high speed data including video signals. The many algorithms that were developed in the research activities, combined with software and hardware that was developed for processing by industry, ushered in a new course into the university curriculum – Digital Signal Processing.

For many years, the course titled Digital Signal Processing was offered as a postgraduate course with students required to have a background in telecommunications (spectral analysis), circuit theory and of course mathematics. The course provided the foundation to do more advanced research in the field. Though this was useful, it did not provide all the necessary background that many industries required; to write efficient programs and to develop applications. In many institutions a simplified version of the postgraduate course has filtered into the undergraduate programs. This book is an attempt to bridge the gap. It can serve as a text for undergraduate or graduate courses and various scenarios are possible depending on the background preparation of the class and the curriculum of the institution.

CHAPTER 1

INTRODUCTION TO DIGITAL SIGNAL PROCESSING (DSP)

1.1 INTRODUCTION

Digital signal processing (DSP) is an area of science and technology that has developed rapidly over the past few decades. The techniques and applications of DSP are as old as Newton and Gauss and as new as digital computers and integrated circuits (ICs). The rapid development of DSP is a result of the significant advances in digital computer technology and IC fabrication.

DSP is concerned with the representation of signals by sequences of numbers or symbols and the processing of these sequences. Processing means the modification of sequences into a form that is in some sense more desirable.

In another words, DSP is a mathematical manipulation of discrete-time signals to get more desirable properties of the signal, such as less noise or distortion.

The classical numerical analysis formulas such as those used for interpolation, differentiation, and integration are also DSP algorithms.

DSP finds application in various fields such as speech communication, data communication, image processing, radar engineering, seismology, sonar engineering, biomedical engineering, acoustics, nuclear science, and many others.

DSP can be applied to one-dimensional signals as well as multidimensional signals. Example of the one-dimensional signal is speech and an example of the two-dimensional signal is an image. Many picture processing applications require the use of two-dimensional signal processing techniques. Two-dimensional signal processing includes X-ray enhancement, analysis of aerial photographs (these photographs are necessary for detection of a forest fire or crop damage), analysis of satellite weather photographs, etc. Analysis of seismic data is required in oil exploration, earthquake measurements, and monitoring of nuclear tests. These utilize multidimensional signal processing techniques. The impact of DSP techniques will undoubtedly promote revolutionary advances in many fields of application. A notable example is telephony where digital techniques dramatically increased economy and flexibility in implementing switching and transmission systems.

1.2 APPLICATIONS OF DIGITAL SIGNAL PROCESSING

There are a variety of application areas of DSP because of the availability of high-resolution spectral analysis. It requires high-speed processor to implement the Fast Fourier transform (FFT). Some of these areas are

1.Speech processing,

2.Image processing,

3.Radar signal processing,

4.Digital communications,

5.Spectral analysis, and

6.Sonar signal processing.

Many of the above applications are discussed in Chapter 13.

Some of the other applications of DSP are in

a.Transmission lines,

b.Advanced optical fiber communication,

c.Analysis of sound and vibration signals,

d.Implementation of speech recognition algorithms,

e.Very Large-Scale Integration technology,

f.Telecommunication networks,

g.Microprocessor systems,

h.Satellite communications,

i.Telephony transmission,

j.Aviation,

k.Astronomy,

l.Industrial noise control, and

m.New DSP algorithms and many more.

Speech Processing: Speech is a one-dimensional signal. Digital processing of speech is applied to a wide range of speech problems such as speech spectrum analysis and channel vocoders (voice coders). DSP is applied to speech coding, speech enhancement, speech analysis and synthesis, speech recognition, and speaker recognition.

Image Processing: Any two-dimensional pattern is called an image. Digital processing of images requires two-dimensional DSP tools such as discrete Fourier transform, fast Fourier transform (FFT) algorithms, and z-transforms. Processing of electrical signals extracted from images by digital techniques includes image formation and recording, image compression, image restoration, image reconstruction, and image enhancement.

Radar Signal Processing: Radar stands for “radio detection and ranging.” Improvement in signal processing is possible by digital technology. The development of DSP has led to greater sophistication of radar tracking algorithms. Radar systems consist of transmitting–receiving antenna, digital processing system, and control unit.

Digital Communications: Application of DSP in digital communication especially telecommunications comprises digital transmission using PCM, digital switching using time-division multiplexing, echo control, and digital tape recorders. DSP in telecommunication systems is found to be cost-effective due to the availability of medium- and large-scale digital ICs. These ICs have desirable properties such as small size, low cost, low power, immunity to noise, and reliability.

Spectral Analysis: Frequency-domain analysis is easily and effectively possible in DSP using fast Fourier transform (FFT) algorithms. These algorithms reduce computational complexity and also reduce the computational time.

Sonar Signal Processing: Sonar stands for “sound navigation and ranging.” Sonar is used to determine the range, velocity, and direction of targets that are remote from the observer. Sonar uses sound waves at lower frequencies to detect objects underwater.

DSP can be used to process sonar signals, for the purpose of navigation and ranging.

1.3 SIGNALS

A signal can be defined as a function of one or more independent variable(s) which conveys information. Independent variables may be time, space, etc., and depend on the type of signals.

Examples of signals are speech signals, pictures, electrocardiogram (ECG) signals, etc. A speech signal is represented mathematically as a function of time and a picture signal is represented as a brightness function of two spatial variables.

FIGURE 1.1  Speech signals.

1.4 CLASSIFICATION OF SIGNALS

Any investigation in signal processing is started with a classification of signals involved in the specific application. Signals can be classified in the following classes:

1.Multichannel and multidimensional signals,

2.Continuous-time and discrete-time signals,

3.Analog and digital signals,

4.Deterministic and random signals,

5.Energy and power signals, and

6.Periodic and non-periodic signals.

Now, we will discuss these in detail in subsequent sections.

1.4.1 Multichannel and Multidimensional Signals

Multichannel Signals: Signals which are generated by multiple sources or multiple sensors are called multichannel signals. These signals are represented by a vector:

.

The signal represents a 3-channel signal. In electrocardiography, 3-lead and 12-lead electrocardiographs are often used in practice, which results in 3-channel and 12-channel signals, respectively.

Multidimensional Signal: A signal is called a multidimensional signal if it is a function of M independent variables. For example, Speech signal is a one-dimensional signal because the amplitude of the signal depends upon a single independent variable, namely, time. TV Picture Signal: A B/W picture signal is an example of a two-dimensional signal because the brightness of the signal at each point is a function of two spatial independent variables, namely, x and y. Variables x and y are width and height of the picture element.

A colored picture signal is an example of three-dimensional signal because brightness of the signal at each point is a function of three independent variables, namely, x, y, and time (t).

1.4.2 Continuous-time and Discrete-time Signals

Continuous-time Signals: A signal that varies continuously with time is called a continuous-time signal. These are defined for every value of the independent variable, namely, time. For example, speech signal and temperature of the room are continuous-time signals. The continuous-time signal is shown in Figure 1.2.

FIGURE 1.2  Continuous-time signal.

Discrete-time Signal: Discrete-time signals are signals which are defined at discrete times (Figure 1.3). These are represented by sequences of numbers. For example, the rail traffic signal is a discrete-time signal.

FIGURE 1.3  Discrete-time signal.

Discrete-time signals can be recovered by periodic sampling of continuous-time signals. Figure 1.3 illustrates the discrete-time signal.

1.4.3 Analog and Digital Signals

Analog Signals: Analog signals are signals of which both the dependent variable and the independent variable(s) are continuous in nature. Analog signals arise when a physical waveform is converted into an electrical signal. This conversion is performed by means of a transducer. For example, telephone speech signals, TV signals, etc., are very common types of the analog signal.

Telephone Speech Signals. A telephone message comprises speech sounds having vowels and consonants. These sounds produce an audio signal. These sound waves are converted into analog electrical signals by means of a transducer (microphone). The transducer is a device that converts non-electrical quantities into electrical signals, for example, a microphone. Continuous-amplitude, continuous-time signals are called analog signals. The Analog signal is shown in Figure 1.1.

Digital Signals: Digital signals are signals of which both the dependent variable and the independent variables are discrete in nature. Digital signals comprise pulses occurring at discrete intervals of time. Telegraph and teleprinter signals are the examples of digital signals. Figure 1.4 illustrates a telegraph signal.

FIGURE 1.4  Telegraph signal (Digital signal).

1.4.4 Deterministic and Random Signals

Deterministic Signals. A deterministic signal is one that has no uncertainty with respect to its value at any value of an independent variable, namely, time. For example, the rectangular pulse given by Eq. (1.1) is a deterministic signal. Figures 1.5 and. 1.6 illustrate rectangular pulse and cosine signal, respectively; both are an example of the deterministic signal.

FIGURE 1.5  Rectangular pulse.

FIGURE 1.6  Cosine signal.

.(1.1)

Another example of the deterministic signal is sinusoidal signals such as sine waves and cosine waves as given in Eq. (1.2):

s(t) A cos wt, − ∞ <t<∞.(1.2)

Random signal: A random signal is a signal which has some degree of uncertainty with respect to its value at any value of independent variable namely, time. For example, thermal agitation noise in conductors is a random signal.

FIGURE 1.7  Random signal.

1.4.5 Energy and Power Signals

Energy signal: A signal is called an energy signal if and only if its total energy is finite. For example, the rectangular pulse is an energy signal.

Power signal: A signal is called a power signal if and only if its average power is finite. For example, sinusoidal waves are power signals.

The energy signals have zero average power and power signals have infinite energy. It means that both signals are mutually exclusive.

1.4.6 Periodic and Non-periodic Signals

Periodic Signal: A signal which repeats its waveform after a fixed period of time is called as a periodic signal. This fixed time is called Time period (T0).

In other words, a signal which satisfies the condition s(t) s(t+T0) for all t is called a periodic signal. For example, sinusoidal signals are the example of a periodic signal.

Non-periodic Signal: A signal which does not satisfy the above condition is called non-periodic signal.

Unit rectangular pulse is an example of a non-periodic signal.

Usually, periodic signals and random signals are power signals and deterministic signals, and non-periodic signals are energy signals.

1.5 SIGNAL PROCESSING SYSTEMS

A system responds to particular signals by producing other signals having some desired behavior.

Signal processing systems are of two types depending on the type of signal to be processed.

1.Continuous-time systems.

2.Discrete-time systems.

1.5.1 Continuous-time Systems

Continuous-time systems are the systems for which both input and output are continuous-time signals. H(s) is the transfer function of a continuous-time system. Figure 1.8 illustrates the block diagram of a continuous-time system.

FIGURE 1.8  Block diagram of continuous-time system.

An example of continuous-time system is an analog filter which is used to reduce the noise corrupting a message signal.

1.5.2 Discrete-time Systems

Discrete-time systems are systems for which both the input and output are discrete-time signals. H(z) is the transfer function of a discrete-time system. Figure 1.9 illustrates the block diagram of a discrete-time system.

FIGURE 1.9  Block diagram of discrete-time system.

An example of a discrete-time system is a digital computer.

1.6 SIGNAL PROCESSING

Changing the basic nature of signal to obtain the desired shaping of the input signal is called signal processing. Signal processing is concerned with the representation, transformation, and manipulation of signals and the information they contain.

Signal processing is of two types depending upon the type of signal to be processed.

1.Analog signal processing (ASP), and

2.Digital signal processing (DSP).

1.6.1 Analog Signal Processing

In ASP, continuous-amplitude continuous-time signals are processed. Various types of analog signals are processed through low-pass filters, high-pass filters, band-pass filters, and band-reject filters to obtain the desired shaping of the input signal. Another example of ASP is the production of the modulated carrier using a high-frequency oscillator, and the modulating audio signal and a modulator. Figure 1.10 illustrates the block diagram of an ASP system.

FIGURE 1.10  Block diagram of ASP system.

1.6.2 Digital Signal Processing

Digital signal processing (DSP) is a numerical processing of signals on a digital computer or some other data processing machine. Figure 1.11 illustrates the block diagram of DSP system.

FIGURE 1.11  Block diagram of DSP system.

A digital system such as digital computer takes input signal in discrete-time sequence form and converts it in discrete-time output sequence.

1.7 ADVANTAGES OF DIGITAL SIGNAL PROCESSING OVER ANALOG SIGNAL PROCESSING

Digital signal processing has the following advantages:

1.Digital signal processing operations can be changed by changing the program in a digital programmable system, that is, these are flexible systems.

2.Better control of accuracy in digital systems is compared to analog systems.

3.Digital signals are easily stored on magnetic media such as magnetic tape without loss of quality of reproduction of the signal.

4.Digital signals can be processed offline, that is, these are easily transported.

5.Sophisticated signal processing algorithms can be implemented by DSP method.

6.Digital circuits are less sensitive to tolerances of component values.

7.Digital systems are independent of temperature, aging, and other external parameters.

8.Digital circuits can be reproduced easily in large quantities at a comparatively lower cost.

9.Cost of processing per signal in DSP is reduced by time-sharing of given processor among a number of signals.

10.Processor characteristics during processing, as in adaptive filters can be easily adjusted in digital implementation.

11.Digital system can be cascaded without any loading problems.

1.8 ELEMENTS OF DIGITAL SIGNAL PROCESSING SYSTEM

A majority of the signals encountered in science are analog in nature. In analog signals, both the dependent variable and independent variable(s) are continuous. Such signals may be processed directly by analog systems (i.e., analog filters) for the purpose of changing their characteristics or extracting some desired information.

Analog signals can also be processed digitally using DSP techniques. To process analog signals digitally, an interface between the analog signal and digital processor is needed. This interface is termed an analog-to-digital converter. The output of the analog-to-digital converter is a digital signal. This digital signal is appropriate for the digital processor.

The digital signal processor may be a large programmable digital computer or a small microprocessor.

In some applications such as in speech communication, we require digital signal in analog form at the receiver end. Here, we need another interface, called digital-to-analog converter. Figure 1.12 illustrates the block diagram of a DSP system.

FIGURE 1.12  Block diagram of a digital signal processing system.

EXERCISES

1.Define a signal. Give some examples of signals.

2. Give the classification of signals.

3. What is signal processing? Differentiate between ASP and DSP.

4. What are the basic elements of the DSP system?

5. What are the advantages of DSP over ASP?

6. Differentiate multichannel and multidimensional signals. Give some examples of these signals.

7. What is the importance of DSP in various fields of engineering and technology? Give a brief account of its applications.

CHAPTER 2

REVIEW OF DISCRETE—TIME SIGNALS AND SYSTEMS

2.1INTRODUCTION

In Chapter 1, we have introduced the concept of digital signal processing. In this chapter, we will study discrete-time signals and systems. Discrete-time signals are obtained either by periodical sampling of continuous-time signals or by a recursion formula. Discrete-time signals are represented by discrete-time sequences.

If both input and output for a system are discrete, then this system is termed a discrete-time system. An example of a discrete-time system is a digital computer.

In this chapter, we first study discrete-time signals: various ways of representing discrete-time signals, different methods of obtaining discrete-time signals, elementary discrete-time signals, and manipulation of discrete-time signals.

After studying discrete-time signals, we will study discrete-time systems and their classification. In this chapter, we will also study LTI discrete-time systems, convolution and correlation operations for LTI discrete-time systems, inverse systems, and deconvolution operations.

Finally, we will study sampling of continuous-time signals, Nyquist rate, sampling theorem, aliasing, and reconstruction of the sampled version of continuous-time signals.

2.2DISCRETE-TIME SIGNALS

Discrete-time signals are defined for discrete values of an independent variable (time). Discrete-time signal is not defined at instants between two successive samples.

Discrete-time signals are represented in two ways:

s(n), N1≤n≤N2(2.1)

where N1 and N2 are the first and the last sample points, respectively, in a given discrete-time signal.

It represents non-uniformly spaced samples, and these are shown in Figure 2.1(a):

s(nTs), N1≤n≤N2(2.2)

It represents uniformly spaced samples, and these are shown in Figure 2.1(b).

FIGURE 2.1  (a) Discrete-time signal showing non-uniformly spaced samples (there is no sampling period Ts) and (b) Discrete-time signal showing uniformly spaced samples.

2.2.1 Representation of Discrete-Time Signals

Discrete-time signal sequences can be represented in the following four ways:

1.Graphical Representation

2.Functional Representation

3.Tabular Representation

4.Sequence Representation.

Graphical Representation: Discrete-time signals can be represented by a graph when the signal is defined for every integer value of n for −∞<n<∞. This is illustrated in Figure 2.2.

FIGURE 2.2  Graphical representation of a discrete-time signal.

Functional Representation: Discrete-time signals can be represented functionally as given below:

(2.3)

Tabular Representation: Discrete-time signals can also be represented by a table as follows:

Sequence Representation: An infinite-duration (−∞≤n≤∞) signal with the time as origin (n 0) and indicated by the symbol ↑.

(2.4)

2.2.2 Methods of Obtaining a Signal Sequence

2.2.3 Some Elementary Discrete-Time Signals

2.2.4 Manipulation of Discrete-Time Signals

2.3DISCRETE-TIME SYSTEMS

2.3.1Classification of Discrete-time Systems

2.3.2 Linear-Time-Invariant (LTI) Systems

2.4CONVOLUTION OF TWO DISCRETE-TIME SIGNALS

2.4.1 Procedure for Computing Convolution Sum

2.4.2 Linear Convolution

2.4.3 Properties of Convolution Sum

Convolution is a mathematical operation between two signal sequences s(n) and h(n). This operation satisfies following properties:

1.Commutative law

2.Associative law

3.Distributive law.

Commutative Law: Commutation sum satisfies commutative law. According to commutative law for a system shown in Figure 2.17,

FIGURE 2.17  LTI system

This is true only for LTI discrete-time systems.

Associative Law: Convolution sum also satisfies the associative law.

According to associative law for the systems shown in Figure 2.18,

[s(n) * h1(n)] * h2(n) s(n) * [h1(n) * h2(n)]

FIGURE 2.18  Cascading of two discrete-time LTI systems.

Distributive Law: This law is also satisfied by convolution sum of two discrete-time LTI systems. According to the distribution law for the systems shown in Figure 2.19,

s(n) * [h1(n) +h2(n)] s(n) * h1(n) +s(n) * h2(n)

FIGURE 2.19  Two discrete-time LTI systems in parallel.

2.5INVERSE SYSTEMS

2.6CORRELATION OF TWO DISCRETE-TIME SIGNALS

A mathematical operation that has close resemblance with convolution is called correlation. Correlation operation also requires two discrete-time sequences just as convolution.

The objective in computing the correlation between two signals is to measure the degree of similarity of two signals. By measuring the degree of correlation, we can extract some information that depends on the application. Here, application means the type of system where correlation operation is used for extracting some information. It is required in radar, sonar, digital communications, and other areas of engineering and technology. Resultant of correlation operation of two discrete-time sequences is a discrete-time sequence.

If the two sequences are identical, then the resultant of correlation of two discrete-time sequences is called auto-correlation sequence.

If the two sequences are different, then the resultant of correlation of two sequences is called cross-correlation sequence.

Digital communication is one of the areas where correlation operation is often used.

2.6.1 Cross-correlation and Auto-correlation Sequences

2.7 SIGNALS AND VECTORS

There is a perfect analogy between signals and vectors. Signals are not just like vectors. Signals are vectors. A vector can be represented as a sum of its components in a variety of ways, depending on the choice of coordinate system. A signal can also be represented as a sum of its components in a variety of ways.

2.7.1 Component of a Vector

A vector is specified by its magnitude and its direction. Here, vectors are represented by an alphabet over which an arrow is shown. For example, is a vector with magnitude or length . Consider two vectors and , as shown in Figure 2.23. Let the component of and be c. Geometrically the component of along is the projection of on . The component of along is obtained by drawing a perpendicular from the tip of on the vector . It is shown in Figure 2.23.

FIGURE 2.23  Illustration of two vectors v and x

Vector can be expressed in terms of as follows:

c+(2.29)

However, this is not the only way to express vector in terms of vector .Figure 2.24 shows two of the infinite other possibilities.

FIGURE 2.24  Illustration of approximation of a vector in terms of another vector.

From Figs. 2.24(a) and 2.24(b), we have

(2.30)

In each of these three representations of Figs. 2.23, 2.24(a), and 2.24(b), vector can be represented in terms of plus another vector (called the error vector).

If we approximate by c in Figure 2.23,

c(2.31)

The error in this approximation is the vector . Similarly, the errors in the approximations in Figs. 2.24(a) and 2.24(b) are and 2. But the error vector e is the smallest.

Now, we can define mathematically the component of a vector along to be c, where c is chosen to minimize the length of the error vector .

For convenience, we can define the scalar or dot or inner product of two vectors and as follows:

where θ angle between vectors and .

By using above definition, we can express magnitude of vector , i.e., as follows:

(2.33)

Magnitude of a vector is also called length of the vector.

Now, the length of the component of along is cos θ, but it is also equal to c.

On multiplying both sides of Eqn. (2.34) by , we get

From Eqns. (2.33) and (2.35), we get

c(2.36)

since .

From Figure 2.23, it is apparent that when vectors and are perpendicular, or orthogonal, then vector has a zero component along ; consequently, c 0.

Now, we can conclude from Eqn. (2.36) that if vectors and are to be orthogonal then their dot or scalar product must be zero, i.e.,

2.7.2 Component of a Signal