Signal Processing and Integrated Circuits E-Book

Hussein Baher

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 466

Veröffentlichungsjahr: 2012

Beschreibung

This book provides a balanced account of analog, digital and mixed-mode signal processing with applications in telecommunications. Part I Perspective gives an overview of the areas of Systems on a Chip (Soc) and mobile communication which are used to demonstrate the complementary relationship between analog and digital systems. Part II Analog (continuous-time) and Digital Signal Processing contains both fundamental and advanced analysis, and design techniques, of analog and digital systems. This includes analog and digital filter design; fast Fourier transform (FFT) algorithms; stochastic signals; linear estimation and adaptive filters. Part III Analog MOS Integrated Circuits for Signal Processing covers basic MOS transistor operation and fabrication through to the design of complex integrated circuits such as high performance Op Amps, Operational Transconductance Amplifiers (OTA's) and Gm-C circuits. Part IV Switched-capacitor and Mixed-mode Signal Processing outlines the design of switched-capacitor filters, and concludes with sigma-delta data converters as an extensive application of analog and digital signal processing * Contains the fundamentals and advanced techniques of continuous-time and discrete-time signal processing. * Presents in detail the design of analog MOS integrated circuits for signal processing, with application to the design of switched-capacitor filters. * Uses the comprehensive design of integrated sigma-delta data converters to illustrate and unify the techniques of signal processing. * Includes solved examples, end of chapter problems and MATLAB® throughout the book, to help readers understand the mathematical complexities of signal processing. The treatment of the topic is at the senior undergraduate to graduate and professional levels, with sufficient introductory material for the book to be used as a self-contained reference.

Leseprobe

Table of Contents

Quotes

Title Page

Copyright

Dedication

About the Author

Preface

Part I: Perspective

Chapter 1: Analog, Digital and Mixed-mode Signal Processing

1.1 Digital Signal Processing

1.2 Moore's Law and the “Cleverness” Factor

1.3 System on a Chip

1.4 Analog and Mixed-mode Signal Processing

1.5 Scope

Part II: Analog (Continuous-time) and Digital Signal Processing

Chapter 2: Analog Continuous-time Signals and Systems

2.1 Introduction

2.2 The Fourier Series in Signal Analysis and Function Approximation

2.3 The Fourier Transformation and Generalized Signals

2.4 The Laplace Transform and Analog Systems

2.5 Elementary Signal Processing Building Blocks

2.6 Realization of Analog System Functions

2.7 Conclusion

Chapter 3: Design of Analog Filters

3.1 Introduction

3.2 Ideal Filters

3.3 Amplitude-oriented Design

3.4 Frequency Transformations

3.5 Examples

3.6 Phase-oriented Design

3.7 Passive Filters

3.8 Active Filters

3.9 Use of MATLAB® for the Design of Analog Filters

3.10 Examples of the use of MATLAB®

3.11 A Comprehensive Application: Pulse Shaping for Data Transmission

3.12 Conclusion

Chapter 4: Discrete Signals and Systems

4.1 Introduction

4.2 Digitization of Analog Signals

4.3 Discrete Signals and Systems

4.4 Digital Filters

4.5 Conclusion

Chapter 5: Design of Digital Filters

5.1 Introduction

5.2 General Considerations

5.3 Amplitude-oriented Design of IIR Filters

5.4 Phase-oriented Design of IIR Filters

5.5 FIR Filters

5.6 Comparison Between IIR and FIR Filters

5.7 Use of MATLAB® for the Design of Digital Filters

5.8 A Comprehensive Application: Pulse Shaping for Data Transmission

5.9 Conclusion

Chapter 6: The Fast Fourier Transform and its Applications

6.1 Introduction

6.2 Periodic Signals

6.3 Non-periodic Signals

6.4 The Discrete Fourier Transform

6.5 The Fast Fourier Transform Algorithms

6.6 Properties of the Discrete Fourier Transform

6.7 Spectral Analysis Using the FFT

6.8 Spectral Windows

6.9 Fast Convolution, Filtering and Correlation Using the FFT

6.10 Use of MATLAB®

6.11 Conclusion

Chapter 7: Stochastic Signals and Power Spectra

7.1 Introduction

7.2 Random Variables

7.3 Analog Stochastic Processes

7.4 Discrete-time Stochastic Processes

7.5 Power Spectrum Estimation

7.6 Conclusion

Chapter 8: Finite Word-length Effects in Digital Signal Processors

8.1 Introduction

8.2 Input Signal Quantization Errors

8.3 Coefficient Quantization Effects

8.4 Effect of Round-off Accumulation

8.5 Auto-oscillations: Overflow and Limit Cycles

8.6 Conclusion

Chapter 9: Linear Estimation, System Modelling and Adaptive Filters

9.1 Introduction

9.2 Mean-square Approximation

9.3 Linear Estimation, Modelling and Optimum Filters

9.4 Optimum Minimum Mean-square Error Analog Estimation

9.5 The Matched Filter

9.6 Discrete-time Linear Estimation

9.7 Adaptive IIR Filtering and System Modelling

9.8 An Application of Adaptive Filters: Echo Cancellers for Satellite Transmission of Speech Signals

9.9 Conclusion

Part III: Analog MOS Integrated Circuits for Signal Processing

Chapter 10: MOS Transistor Operation and Integrated Circuit Fabrication

10.1 Introduction

10.2 The MOS Transistor

10.3 Integrated Circuit Fabrication

10.4 Layout and Area Considerations for IC MOSFETs

10.5 Noise In MOSFETs

Chapter 11: Basic Integrated Circuits Building Blocks

11.1 Introduction

11.2 MOS Active Resistors and Load Devices

11.3 MOS Amplifiers

11.4 High Frequency Considerations

11.5 The Current Mirror

11.6 The CMOS Amplifier

11.7 Conclusion

Chapter 12: Two-stage CMOS Operational Amplifiers

12.1 Introduction

12.2 Op Amp Performance Parameters

12.3 Feedback Amplifier Fundamentals

12.4 The CMOS Differential Amplifier

12.5 The Two-stage CMOS Op Amp

12.6 A Complete Design Example

12.7 Practical Considerations and Other Non-ideal Effects in Operational Amplifier Design

12.8 Conclusion

Chapter 13: High Performance CMOS Operational Amplifiers and Operational Transconductance Amplifiers

13.1 Introduction

13.2 Cascode CMOS Op Amps

13.3 The Folded Cascode Op Amp

13.4 Low-noise Operational Amplifiers

13.5 High-frequency Operational Amplifiers

13.6 Fully Differential Balanced Topology

13.7 Operational Transconductance Amplifiers

13.8 Conclusion

Chapter 14: Capacitors, Switches and the Occasional Passive Resistor

14.1 Introduction

14.2 MOS Capacitors

14.3 The MOS Switch

14.4 MOS Passive Resistors

14.5 Conclusion

Part IV: Switched-capacitor and Mixed-mode Signal Processing

Chapter 15: Design of Microelectronic Switched-capacitor Filters

15.1 Introduction

15.2 Sampled and Held Signals

15.3 Amplitude-oriented Filters of the Lossless Discrete Integrator Type

15.4 Filters Derived from Passive Lumped Prototypes

15.5 Cascade Design

15.6 Applications in Telecommunications: Speech Codecs and Data Modems

15.7 Conclusion

Chapter 16: Non-ideal Effects and Practical Considerations in Microelectronic Switched-capacitor Filters

16.1 Introduction

16.2 Effect of Finite Op Amp Gain

16.3 Effect of Finite Bandwidth and Slew Rate of Op Amps

16.4 Effect of Finite Op Amp Output Resistance

16.5 Scaling for Maximum Dynamic Range

16.6 Scaling for Minimum Capacitance

16.7 Fully Differential Balanced Designs

16.8 More on Parasitic Capacitances and Switch Noise

16.9 Pre-filtering and Post-filtering Requirements

16.10 Programmable Filters

16.11 Layout Considerations

16.12 Conclusion

Chapter 17: Integrated Sigma-Delta Data Converters: Extension and Comprehensive Application of Analog and Digital Signal Processing

17.1 Motivation and General Considerations

17.2 The First-order Converter

17.3 The Second-order Converter

17.4 Decimation and Digital Filtering

17.5 Simulation and Performance Evaluation

17.6 A Case Study: Fourth-order Converter

17.7 Conclusion

Answers to Selected Problems

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 15

References

Index

‘The scribes full of wisdom,

Their names will last forever.

They leave for an inheritance,

Their teachings and their books.

Their teachings are their Pyramids,

Their magical power touches all those who read their writings.’

Egyptian Hieratic Papyrus in the British Museum

‘The instinct of constructiveness, which is one of the chief incentives to artistic creation, can find in scientific systems a satisfaction more massive than any epic poem. Disinterested curiosity, which is the essence of almost all intellectual effort, finds with astonished delight that science can unveil secrets which might well have seemed for ever undiscoverable…A life devoted to science is therefore a happy life, and its happiness is derived from the very best sources that are open to dwellers on this troubled and passionate planet.’

Bertrand Russell

‘The Place of Science in a Liberal Education’

This edition first published 2012

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Library of Congress Cataloging-in-Publication Data

Baher, H.

Signal processing and integrated circuits / Hussein Baher.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-71026-5 (cloth)

1. Signal processing—Equipment and supplies. 2. Signal processing—Mathematics. 3. Integrated circuits—Design and construction.

4. Electric filters. I. Title.

TK5102.9.B353 2012

621.382′2 – dc23

2011053421

A catalogue record for this book is available from the British Library.

ISBN: 9780470710265

About the Author

Professor Hussein Baher was born in Alexandria and received his early education at the University of Alexandria and the American University in Cairo. He obtained his Ph.D. in Electronic Engineering from University College Dublin and has held academic positions at universities worldwide, including University College Dublin, Virginia Polytechnic Institute and State University, the Professorship of Electronic Engineering at Dublin City University, and the prestigious Analog Devices Professorship of Microelectronics in Massachusetts, United States. He has also been Visiting Professor at the Technische Universitaet Wien (TUW), Vienna, Austria.

He has published extensively on circuit design and signal processing including five books: Synthesis of Electrical Networks (John Wiley & Sons, Ltd, 1984), Selective Linear-phase Switched-capacitor and Digital Filters (Kluwer, 1993), Microelectronic Switched-capacitor Filters: with ISICAP: a Computer-aided Design Package (John Wiley & Sons, Ltd, 1996) and Analog and Digital Signal Processing (John Wiley & Sons, Ltd, 1990; 2nd Edition, 2001).

Prof. Baher spends his time in Dublin, Vienna and Alexandria as Professor Emeritus of Electronic Engineering. He is also a Member of the Archaeological Society of Alexandria and lectures in Dublin and Vienna on Ancient Egyptian civilization.

Preface

‘Exact Calculation: The Gateway to Everything.’

Ahmes

‘An Egyptian Mathematical Papyrus1, 1850 BC’

In 2006, Austria was celebrating the 250th anniversary of Mozart's birth. While enjoying the festivities, I gave two graduate-level courses at the Technical University of Vienna (TUW). One course was on Digital Signal Processing and the other dealt with Analog Integrated Circuits for Signal Processing with application to the design of Switched-capacitor Filters and Sigma-Delta Data Converters. The two courses complemented each other to such an extent that the idea of writing a book combining the material of both courses was quite attractive. As the idea became more compelling, the material was updated and the result is this book.

The objective of this book is to provide a coherent and harmonious account of both analog and digital signal processing. In the case of digital systems, the design is at the relatively high level of adders, multipliers and delays. In the case of analog systems, the emphasis is laid on integrated circuit implementations of both continuous-time and sampled-data (discrete) circuits and systems, reaching all the way to the transistor level. This provides a comprehensive treatment of analog MOS integrated circuits for signal processing, with application to the design of microelectronic switched-capacitor circuits and extension to the design of mixed-mode processors in the form of integrated sigma-delta data converters. In this context, integrated circuit realizations which have been used successfully in submicron and deep submicron implementations for ultra high frequency applications are also discussed. Finally, MATLAB®2 is used throughout as a useful aid to the analysis and design problems.

The level of treatment is at the senior to first-year graduate and professional levels while providing enough coverage of fundamental junior-level material to make the book self-contained. The book is divided into four parts.

Part I contains one chapter, which is a general introduction. Chapter 1 gives a general overview and perspective of the general area of signal processing and the related disciplines, mentioning several applications. The growing areas of Systems on a Chip (SoC) and mobile communications are used for illustration of the wealth of knowledge required to design a complex signal processing system and to demonstrate the complementary relationship between analog and digital systems.

Part II contains eight chapters dealing with the techniques of signal processing in the analog and digital domains at the system and circuit levels while not reaching the transistor level. Chapter 2 is a review of the fundamental concepts and mathematical tools of analog signal and system analyses. This review can be regarded as a comprehensive summary of the fundamentals of analog signals and systems. It is the distillation of courses on these topics which are usually covered at the junior undergraduate level. Therefore, the discussion is quite compact, and the material can be used as an easy reference for later chapters and as a short revision course. Chapter 3 discusses the general theory and techniques of analog continuous-time filter design. These are important in themselves and are also of direct relevance to the design of all types of filter, including those which are of the sampled-data type such as digital and switched-capacitor filters. This is because the filtering operation is based on the same principles and, very often, analog continuous-time models are used as starting points for the design of other types. The chapter concludes with a guide to the use of MATLAB® in the design of analog filters. Extensive use of this material will also be made in later chapters. Chapter 4 gives a brief and concise review of the process of analog to digital conversion and the representation of discrete signals and systems. This should serve as a revision of the fundamentals of discrete signal and system analyses. In Chapter 5, the design techniques of digital filters are discussed in detail. Emphasis is, at first, laid on the conceptual organization and analytical methods of design. Then the chapter concludes with the details of how to use MATLAB® as a computer-aided deign tool. Numerous examples are given throughout the chapter of both analytical and computer-aided design methods. Chapter 6 provides a discussion of the computational algorithms that have come to be known collectively as the fast Fourier transform (FFT). The discrete Fourier transform is introduced and its properties are examined. The applications of the FFT are discussed in relation to spectral analysis, convolution, correlation and filtering of signals. Chapter 7 introduces the concepts and techniques suitable for the description of stochastic (random) signals. The discussion encompasses both analog and digital signals. However the systems which perform the processing of these signals are themselves deterministic. Chapter 8 deals with the effects of using binary words with finite lengths in representing the various quantities in digital signal processors. The degradation caused by these effects is examined and the resulting errors are assessed quantitatively. In Chapter 9, a central problem in signal processing is addressed, namely: the estimation of some signal of interest from a set of received noisy data signals. This leads to the area of adaptive filtering. A closely related area is that of the modelling or simulation of the behaviour of an unknown system (or process) by a linear system. Initially, the principles of linear estimation and modelling are discussed, then it is shown how these can be implemented using adaptive algorithms.

Part III is devoted to the design of analog MOS integrated circuits for signal processing. In Chapter 10 a brief review of MOS transistor fundamentals and integrated circuit fabrication techniques are given. Chapter 11 provides a discussion of the basic integrated circuit building blocks such as amplifiers, current mirrors, and load devices. In Chapter 12 the two-stage CMOS operational amplifier is introduced and complete design examples are given. Chapter 13 deals with high performance operational amplifiers and operational transconductance amplifiers which are used in Gm-C circuits. Integrated circuit realizations which have been used successfully in submicron and deep submicron implementations for ultra high frequency applications are also discussed in this chapter. Chapter 14 deals with integrated resistors, capacitors and switches which are building blocks in analog signal processing systems.

Part IV is devoted to the design of signal processing systems using switched-capacitor and mixed-mode (i.e. both analog and digital) circuits. Chapter 15 is a detailed account of the design techniques of microelectronic switched-capacitor filters. These are analog sampled-data circuits which have established themselves as viable alternatives to digital circuits in many applications. Furthermore, they are particularly amenable to implementation using the same CMOS integrated circuit technology which is used in digital processing, and consequently they can be easily integrated on the same chip with the digital circuits. Both the theoretical foundation and practical considerations are discussed in detail. Of particular importance in analog systems is the non-ideal behaviour of the building blocks, since this can lead to deteriorated performance if not understood and taken into account early in the design. These are treated in detail in Chapter 16 together with many practical considerations in the design of analog integrated circuits. Chapter 17 gives a detailed discussion of a highly instructive class of signal processor: the Σ-Δ converter. Its analysis and design require knowledge of both analog and digital signal processing, as well as most of the analytical and computational techniques discussed in this book. Therefore, it is ideal for inclusion in this book, which attempts to unify the two fields in one volume and should serve as a good illustration of the validity of the adopted approach.

Numerous applications in the electronic communications field are given throughout the book at the appropriate points in the chapters. These include: pulse shaping networks for data transmission, switched-capacitor filters for speech CODECs, full duplex data MODEMs, adaptive echo cancellation in the satellite transmission of speech signals, linear estimation, system modelling and adaptive filtering. In addition, the final chapter on sigma-delta data converters constitutes a comprehensive application, bringing together all the signal processing techniques in the book (switched-capacitor techniques, digital filters, decimators, FFT, and analog COMS integrated circuits) to design a mixed-mode processor with a wide range of applications as an analog to digital converter.

Finally, the enthusiasm and professionalism of Alexandra King and Liz Wingett of John Wiley and Sons, Ltd (Chichester, UK) have been of great help in the completion of this book.

H. Baher

Vienna and Dublin, 2012

2 MATLAB® is a registered trademark of the Mathworks Inc.

Part I

Perspective

‘Science as it exists at present is partly agreeable, partly disagreeable. It is agreeable through the power which it gives us of manipulating our environment, and to a small but important minority, it is agreeable because it affords intellectual satisfaction. It is disagreeable because, however we may seek to disguise the fact, it assumes a determinism which involves, theoretically, the power of predicting human actions; in this respect, it seems to lessen human power.’

Bertrand Russell‘Is Science Superstitious?’ (in ‘Sceptical Essays’)

Chapter 1

Analog, Digital and Mixed-mode Signal Processing

1.1 Digital Signal Processing

The widespread use of digital signal processing systems is due to many factors including reliability, reproducibility, high precision, freedom from aging and temperature effects, low cost and efficient computational algorithms. Furthermore, the revolution in the microelectronics field [1–3] has been characterized by a continuous increase in the level of integration leading to complete systems being integrated on a single chip, that is, systems on a chip (SoC) [3–5].

1.2 Moore's Law and the “Cleverness” Factor

The integrated circuit dates back to around 1960. Since then, the number of devices on a chip has increased dramatically in line with an observation [1, 2] predicting a doubling every year. Now, millions of transistors can be manufactured on a single chip allowing phenomenal processing capability. If we define a pixel as the smallest spot on a chip that can be controlled in the fabrication process, then this will determine the contribution of device miniaturization and chip area to the content of the chip. This contribution can be measured by the quantity A/S where A is the chip area and S is the pixel area. As progress continued, it was found that the number of devices on a chip was actually increasing faster than A/S. This additional growth was a result of “clever” techniques of exploiting the space on the chip. These include forming thin-film capacitors on the side holes etched into a chip instead of on the surface, and self-aligned structures where part of the device is used as the mask in the fabrication process. Next came the effect of the wiring on limiting the size of the chip. This, again, has been tackled [1] by the “cleverness” of increasing the number of wire layers.

1.3 System on a Chip

Such a system comprises application specific integrated circuits (ASICs). Examples are the single-chip TV or the single chip camera, and the ever-emerging new generations of integrated telecommunication systems particularly in the mobile communication area. Such systems include analog and digital sections on the same chip where the technology of choice has been CMOS and possibly BiCMOS. Most functions on these chips are implemented using digital signal processing circuits. However, analog circuits are needed as an interface between the system and the real world which is, of course, analog in nature. Figure 1.1 shows a typical SoC containing embedded digital signal processors, embedded memory, reconfigurable logic, and analog circuits to interface with the analog continuous-time world.

Figure 1.1 System on a chip (SoC)

The design of signal processing systems with low-power requirements is one of the most important areas of research [6, 7] which together with the need for high speed and density of integration have led to great advances in technology and clever circuit design methods [8].

1.4 Analog and Mixed-mode Signal Processing

The trend to replace, for example, analog filters by digital filters is understandable in view of the advantages of digital filters. However, there are some functions on the processor which will always remain analog [4]. These are the following:

a. At the input of the system, signals from a sensor, microphone, antenna or cable must be received, amplified and filtered, that is processed and brought to a level that permits digitization with acceptable signal to noise ratio and low distortion. Here, we need low-noise amplifiers (LNAs), variable gain amplifiers (VGAs), filters, oscillators and mixers. Applications are:

Data and biomedical instrumentation.Sensor interfaces such as airbags and accelerometers.Telecommunications receivers such as telephone or cable modems and wireless telephones.

b. At the output of the system the signal is reconverted from digital to analog form and strengthened so that it can drive an external load such as an antenna or a loudspeaker with low distortion. Here we also need buffers, filters, oscillators and mixers. Applications are the following

Telecommunications transmittersAudio and video, such CD, SACD, DVD and BluerayLoudspeakersTVPC monitorsHearing aids

c. Mixed-mode circuits are also needed for the interface between the analog and digital parts. These include sample and hold circuits for the sampling of signals, analog to digital (A/D) converters as well as digital to analog converters for signal reconstruction. These are mixed-mode circuits.

d. The integrated circuits discussed above need stable references for their operation which are analog voltage and current sources and crystal oscillators.

Figure 1.2 illustrates the above points with the block diagram of a mobile telephone/Bluetooth receiver section [9]. This highlights the fact that both analog and digital circuits coexist on the same chip employing CMOS technology, and also the interrelationship between analog and digital signal processing.

Figure 1.2 The analog and digital parts of a mobile telephone/Bluetooth receiver section

1.5 Scope

Now, what do we need to know in order to be able to design a system on a chip? Our knowledge must include the following:

1. Methods of description of both analog and digital signals in the time and frequency domains.

2. Methods of description of the systems which process the signals. We need to do this for both analog and digital systems.

3. Design techniques for analog circuits such as amplifiers, integrators, differentiators, and most importantly: filters taking into account the non-ideal effects.

4. Integrated circuit implementations of analog circuits using CMOS technology.

5. Design of digital filters taking into account the finite word-length effects inherent in all digital processors.

6. Random signals require special methods for their description and processing, leading to the subject of adaptive filters. These, together with the related topics of linear prediction, estimation, and system modelling are essential.

7. Modern design techniques of discrete-time filtering using switched-capacitor techniques, since these are particularly amenable to implementation using VLSI techniques.

8. Design of A/D and D/A converters since these act as the interfaces between the digital and analog parts of the system.

Detailed treatment of the above topics is the aim of this book. To facilitate the numerical calculations, and to be able to study the responses of systems and evaluate their performances, we need a powerful software package. MATLAB® is a good choice, and it is used throughout the book.

Part II

Analog (Continuous-time) and Digital Signal Processing

‘It is very desirable in instruction, not merely to persuade the student of the accuracy of important theorems, but to persuade him in the way which itself has, of all possible ways, the most beauty.’

Bertrand Russell‘The Study of Mathematics’

Chapter 2

Analog Continuous-time Signals and Systems

2.1 Introduction

In this chapter the fundamental concepts and mathematical tools of analog signal and system analyses are reviewed. This review can be regarded as a comprehensive summary of the fundamentals of analog signals and systems. It is the distillation of courses on these topics which are usually covered at the junior to senior undergraduate levels. Therefore, the discussion is quite compact, and the material can be used as an easy reference for later chapters and as a short revision course.

2.2 The Fourier Series in Signal Analysis and Function Approximation

2.2.1 Definitions

A signal f(x) defined over an interval [−l, l] and satisfying conditions of considerable generality, can be represented as an infinite series of pure sine and cosine signals as

2.1

where the coefficients are given by

2.2a

2.2b

A simplified notation results by letting

2.3

so that the signal is defined over the range [−π, π] and the series becomes

2.4

with

2.5a

2.5b

or the alternative versions in

2.6

or

2.7

with

2.8

2.9

and

2.10

If the signal is periodic, the representation is valid for all values of the independent variable. If the signal is non-periodic, then the representation is valid only over the fundamental range [−l, l] or [−π, π].

2.2.2 The Time and Discrete Frequency Domains

If the signal is a function of time t, examples of which are shown in Figure 2.1, then the representation is given by

2.11

2.12a

2.12b

where

2.13

with T being the period and ω0 is the fundamental radian frequency.

Figure 2.1 Examples of signals as functions of time

A more compact (complex) Fourier series uses the exponential signal and is given by

2.14

with

2.15

For a function of time, the series is given by

2.16

with

2.17

or

2.18

The coefficients of the series give a representation in the frequency domain. The amplitudes of the coefficients give the amplitude spectrum whereas their phases give the phase spectrum. For a periodic signal these are line spectra, and give the frequency domain representation of the signal.

2.2.3 Convolution

The convolution of two signals f1(θ) and f2(θ) is given by

2.19

2.20

and the complex Fourier series of the convolution has as its coefficients the products of the corresponding ones in the series of the individual signals.

2.2.4 Parseval's Theorem and Power Spectrum

Parseval's theorem relates the average power of a signal to the sum of the squares of the amplitudes of the complex Fourier coefficients as expressed by

2.21

The squared amplitudes of the complex Fourier coefficients are called the and a plot of these versus frequency is called the .