Understanding digital signal processing

310 Pages·2018·2.375 MB·English

by  Gazi O

Springer Topics in Signal Processing
Orhan Gazi
Understanding
Digital Signal
Processing
Springer Topics in Signal Processing
Volume 13
Series editors
Jacob Benesty, Montreal, Canada
Walter Kellermann, Erlangen, Germany
More information about this series at http://www.springer.com/series/8109
Orhan Gazi
Understanding Digital
Signal Processing
123
Orhan Gazi
ElectronicsandCommunicationEngineering
Department
Çankaya University
Etimesgut/Ankara
Turkey
ISSN 1866-2609 ISSN 1866-2617 (electronic)
SpringerTopics inSignal Processing
ISBN978-981-10-4961-3 ISBN978-981-10-4962-0 (eBook)
DOI 10.1007/978-981-10-4962-0
LibraryofCongressControlNumber:2017940604
©SpringerNatureSingaporePteLtd.2018
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Preface
In this book, we tried to explain digital signal processing topics in detail. We paid
attentiontothesimplicityoftheexplanationlanguage.Andweprovidedexamples
with increasing difficulty. The reader of this book should have some background
about signals. If it is possible, the reader should learn fundamental concepts on
signals and systems since, in this book, more attention is paid on digital signal
processing concepts rather than continuous time signal processing topics. Hence,
weassumethatthereaderhasfundamentalknowledgeaboutalltypesofsignalsand
transforms.
All the topics in this book are presented in an orderly manner. We tried to
simplifythelanguageofthisbookaspossibleaswecan.Wealsoprovidedoriginal
examples explaining the aim of the subjects studied in this book. Numerical
examples are provided for the comprehension of the subjects. Unnecessary abun-
dance of mathematical details is omitted for the simplicity of the presentation
language. In addition, to indicate both continuous and digital time frequencies, we
preferredtousethesameparameter.Wethoughtthatusingtwodifferentparameters
mixes the students’ mind and it is not necessarily needed.
This book includes four different chapters. And in these chapters, sampling of
continuoustimesignals,multiratesignalprocessing,discreteFouriertransform,and
filter design concepts are covered. In sampling of continuous time signals and
multirate signal processing chapters, we provided some original practical tech-
niques to draw the spectrum of aliased signals. In discrete time Fourier transform
chapter, well-designed numerical examples are provided to illustrate the operation
of the fast Fourier transform algorithm. In filter design chapter, both analog and
digitalfilterdesigntechniquesareexplainedindetail.Fortheanalogfilters,wealso
provided analog filter circuit design methods for the designed analog filter transfer
function.
Maltepe/Ankara, Turkey Orhan Gazi
November 2016
v
Contents
1 Sampling of Continuous Time Signals.... .... .... .... ..... .... 1
1.1 Sampling Operation for Continuous Time Signals .... ..... .... 2
1.1.1 Sampling Frequency . .... .... .... .... .... ..... .... 4
1.1.2 Mathematical Characterization of the Sampling
Operation .... ..... .... .... .... .... .... ..... .... 5
1.2 Sampling Operation .. ..... .... .... .... .... .... ..... .... 6
1.2.1 The Fourier Transform of the Product Signal .. ..... .... 7
1.3 How to Draw Fourier Transforms of Product Signal
and Digital Signal.... ..... .... .... .... .... .... ..... .... 11
1.3.1 Drawing the Fourier Transform of Digital Signal .... .... 25
1.4 Aliasing (Spectral Overlapping) .. .... .... .... .... ..... .... 30
1.4.1 The Meaning of the Aliasing (Overlapping) ... ..... .... 33
1.4.2 Drawing the Frequency Response of Digital Signal
in Case of Aliasing (Practical Method)... .... ..... .... 39
1.5 Reconstruction of an Analog Signal from Its Samples . ..... .... 45
1.5.1 Approximation of the Reconstruction Filter ... ..... .... 51
1.6 Discrete Time Processing of Continuous Time Signals. ..... .... 55
1.7 Continuous Time Processing of Digital Signals .. .... ..... .... 61
1.8 Problems... .... .... ..... .... .... .... .... .... ..... .... 68
2 Multirate Signal Processing .... .... .... .... .... .... ..... .... 71
2.1 Sampling Rate Reduction by an Integer Factor (Downsampling,
Compression) ... .... ..... .... .... .... .... .... ..... .... 72
2.1.1 Fourier Transform of the Downsampled Signal. ..... .... 75
2.1.2 How to Draw the Frequency Response of Downsampled
Signal ... .... ..... .... .... .... .... .... ..... .... 78
2.1.3 Aliasing in Downsampling .... .... .... .... ..... .... 80
2.1.4 Interpretation of the Downsampling in Terms
of the Sampling Period... .... .... .... .... ..... .... 83
vii
viii Contents
2.1.5 Drawing the Fourier Transform of Downsampled Signal
in Case of Aliasing (Practical Method)... .... ..... .... 92
2.2 Upsampling: Increasing the Sampling Rate by an Integer
Factor. .... .... .... ..... .... .... .... .... .... ..... .... 97
2.2.1 Upsampling (Expansion).. .... .... .... .... ..... .... 97
2.2.2 Mathematical Formulization of Upsampling ... ..... .... 98
2.2.3 Frequency Domain Analysis of Upsampling... ..... .... 99
2.2.4 Interpolation .. ..... .... .... .... .... .... ..... .... 103
2.2.5 Mathematical Analysis of Interpolation... .... ..... .... 107
2.2.6 Approximation of the Ideal Interpolation Filter. ..... .... 111
2.2.7 Anti-aliasing Filter .. .... .... .... .... .... ..... .... 126
2.3 Practical Implementations of C/D and D/C Converters. ..... .... 128
2.3.1 C/D Conversion .... .... .... .... .... .... ..... .... 129
2.3.2 Sample and Hold ... .... .... .... .... .... ..... .... 130
2.3.3 Quantization and Coding.. .... .... .... .... ..... .... 134
2.3.4 D/C Converter. ..... .... .... .... .... .... ..... .... 136
2.4 Problems... .... .... ..... .... .... .... .... .... ..... .... 139
3 Discrete Fourier Transform .... .... .... .... .... .... ..... .... 145
3.1 Manipulation of Digital Signals .. .... .... .... .... ..... .... 146
3.1.1 Manipulation of Periodic Digital Signals.. .... ..... .... 149
3.1.2 Shifting of Periodic Digital Signals.. .... .... ..... .... 149
3.1.3 Some Well Known Digital Signals.. .... .... ..... .... 156
3.2 Review of Signal Types.... .... .... .... .... .... ..... .... 158
3.3 Convolution of Periodic Digital Signals.... .... .... ..... .... 165
3.3.1 Alternative Method to Compute the Periodic
Convolution .. ..... .... .... .... .... .... ..... .... 166
3.4 Sampling of Fourier Transform .. .... .... .... .... ..... .... 170
3.5 Discrete Fourier Transform.. .... .... .... .... .... ..... .... 172
3.5.1 Aliasing in Time Domain . .... .... .... .... ..... .... 182
3.5.2 Matrix Representation of DFT and Inverse DFT..... .... 184
3.5.3 Properties of the Discrete Fourier Transform... ..... .... 185
3.5.4 Circular Convolution. .... .... .... .... .... ..... .... 188
3.6 Practical Calculation of the Linear Convolution.. .... ..... .... 198
3.6.1 Evaluation of Convolution Using Overlap-Add Method ... 199
3.6.2 Overlap-Save Method.... .... .... .... .... ..... .... 204
3.7 Computation of the Discrete Fourier Transform.. .... ..... .... 207
3.7.1 Fast Fourier Transform (FFT) Algorithms. .... ..... .... 207
3.7.2 Decimation in Time FFT Algorithm. .... .... ..... .... 207
3.7.3 Decimation in Frequency FFT Algorithm. .... ..... .... 217
3.8 Total Computation Amount of the FFT Algorithm.... ..... .... 225
3.9 Problems... .... .... ..... .... .... .... .... .... ..... .... 230
Contents ix
4 Analog and Digital Filter Design .... .... .... .... .... ..... .... 233
4.1 Review of Systems... ..... .... .... .... .... .... ..... .... 233
4.1.1 Z-Transform .. ..... .... .... .... .... .... ..... .... 236
4.1.2 Laplace Transform .. .... .... .... .... .... ..... .... 239
4.2 Transformation Between Continuous and Discrete
Time Systems... .... ..... .... .... .... .... .... ..... .... 240
4.2.1 Conversion of Transfer Functions of LTI Systems ... .... 245
4.2.2 Forward Difference Transformation Method ... ..... .... 246
4.2.3 Bilinear Transformation... .... .... .... .... ..... .... 248
4.3 Analogue Filter Design..... .... .... .... .... .... ..... .... 253
4.3.1 Ideal Filters... ..... .... .... .... .... .... ..... .... 254
4.3.2 Practical Analog Filter Design . .... .... .... ..... .... 258
4.3.3 Practical Filter Design Methods .... .... .... ..... .... 260
4.3.4 Analog Frequency Transformations.. .... .... ..... .... 272
4.4 Implementation of Analog Filters. .... .... .... .... ..... .... 273
4.4.1 Low Pass Filter Circuits .. .... .... .... .... ..... .... 273
4.4.2 Analog High-Pass Filter Circuit Design .. .... ..... .... 279
4.4.3 Analog Bandpass Active Filter Circuits .. .... ..... .... 282
4.4.4 Analog Bandstop Active Filter Circuits... .... ..... .... 282
4.5 Infinite Impulse Response (IIR) Digital Filter Design
(Low Pass). .... .... ..... .... .... .... .... .... ..... .... 283
4.5.1 Generalized Linear Phase Systems .. .... .... ..... .... 289
4.6 Finite Impulse Response (FIR) Digital Filter Design .. ..... .... 290
4.6.1 FIR Filter Design Techniques.. .... .... .... ..... .... 291
4.7 Problems... .... .... ..... .... .... .... .... .... ..... .... 297
Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 299
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 301
Chapter 1
Sampling of Continuous Time Signals
Signal is a physical phenomenon that carries information. This physical phe-
nomenon is described by mathematical functions, and usually the signal and its
mathematical function are used for one another, i.e., synonymous. For instance,
when we talk about a sinusoidal signal, we use the sinusoidal function, a mathe-
matical function,tocharacterizethesignal,andthenamesinusoidalisusedforthe
signal.Signalsareusuallydepictedingraphstoobservetheirbehaviorandanalyze
them. Sinusoidal signals are the main signals and all the other signals can be
considered as being made up of sinusoidal signals with different frequencies and
amplitudes. That is to say, any continuous time signal can be written as sum of
sinusoidal signals with different frequencies and amplitudes. Rectangular signal,
squarepulsesignal,impulsetrainsignal,trianglesignalcanbegivenasexamplesof
continuous time signals.
Digital signals are obtained from continuous time signals via sampling opera-
tion.Digitalsignalsarerepresentedasmathematicalsequences,andtheelementsof
these sequences are nothing but the amplitude values taken from continuous time
signalsateverymultipleofthesamplingperiod.Sinceinthelastseveraldecadesa
huge improvement is achieved at the development of the digital devices, it has
becomealmostamustespeciallyforelectricalengineerstohaveagoodknowledge
of digital signals. Digital signals are almost available in every part of our life.
Computers,TVs,speakers,mobilephones,houseequipment,andmostoftheother
electronic devices process digital signals. In this chapter, we discuss the con-
struction of digital signals via sampling operation, their spectral analyses, the case
of aliasing, and reconstruction of a continuous time signal from its samples.
©SpringerNatureSingaporePteLtd.2018 1
O.Gazi,UnderstandingDigitalSignalProcessing,SpringerTopics
inSignalProcessing13,DOI10.1007/978-981-10-4962-0_1