Digital Filters Using MATLAB

807 Pages·2020·36.938 MB·English

by  Lars Wanhammar, Tapio Saramäki

Lars Wanhammar
Tapio Saramäki
Digital Filters Using
MATLAB
Digital Filters Using MATLAB
Lars Wanhammar Tapio Saramäki
(cid:129)
Digital Filters Using
MATLAB
123
LarsWanhammar TapioSaramäki
LinköpingUniversity Tampere University of Technology
Linköping,Sweden Tampere,Finland
The additional materials will be available on https://www.springer.com/in/book/
9783030240622.
ISBN978-3-030-24062-2 ISBN978-3-030-24063-9 (eBook)
https://doi.org/10.1007/978-3-030-24063-9
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Preface
This book is the end result of a course in Digital Filters given at Linköping
University for the last 45 years and some material from the Tampere
University of Technology. The aim is to aid the electrical engineer to
understand and design digital filters.
Most of the advanced mathematics required for the synthesis of digital
filters have been avoided by providing a set of MATLAB functions that
allows sophisticated filters to be designed. The toolbox deviates from tra-
ditional design software, that is, from a large and complex program that can
onlysolveagivensetofdesignproblems.Instead,thetoolboxcontainsaset
of simple functions that only solve a small elementary task. These code
snippetscaneasilybecombinedintoaworkingprogramforagivenproblem.
Thereaderisurgedtodevelophis/herdesignsbymodifyingthesefunctions.
Most of these functions can easily be converted to run under Octave and
Scilab as well.
Chapter 1 reviews concepts like analog and digital signals, their Fourier
and z-transform as well as sampling of analog signals.
InChap.2wediscusstechnologiesfortheimplementationofdigitalfilters
and their properties. Also, behavioral properties such as poles and zeros,
magnitude,attenuation,phase,andgroupdelayresponsesaswellasimpulse
and step responses. Finally, structural descriptions of digital filter structures
like difference equations, signal-flow graphs, direct form structures, cascade
and parallel form, and state-space representations, are discussed.
In Chap. 3 we discuss the fundamental properties of DSP algorithms and
their representations, i.e., precedence graphs, parallelism, sequentially com-
putable algorithms, and computation graphs. Also, we discuss an upper
bound on maximal sample frequency, pipelining, and interleaving. We also
discuss the concepts of multiple constant multiplication techniques and the
important issue of power consumption.
In Chap. 4 we discuss finite wordlength effects such as number repre-
sentation, quantization, overflow, roundoff and quantization noise, and par-
asitic oscillations. Techniques for scaling of signals, the effect on roundoff
and quantization noise. Finally, we discuss the effect of finite wordlength
of the filter coefficients.
In Chap. 5 we discuss basic properties and several computer-based
methods for the design of FIR filters while in Chap. 6 we discuss several
structures for the realization of these FIR filters.
v
vi Preface
The design of an analog filter is often the first step in the design of IIR
digital filters. The aim is to obtain a frequency-selective digital filter. In
Chap.7we,therefore,discussthesynthesisofclassicaltransferfunctionslike
Butterworth, Chebyshev I, Chebyshev II, Cauer filters, and filters with
constant pole radius as well as techniques to transform, for example, a low
pass filter into bandpass filter, etc. The group delay response of the
approximations is an essential property for low-sensitivity filter realization.
Chapter8dealswithpassiveLCfilterswithlumpedanddistributedcircuit
elements. The reader may believe that this is an outdated technology.
However, it is still being used, and more importantly, the theory behind all
advancedfilterstructureswithlowelementsensitivityisbasedonpassiveLC
filters. This is also the case for digital and switched-capacitor filters. The
readerisstronglyrecommendedtocarefullystudytheprincipleofmaximum
power transfer, sensitivity to element errors, and the implications of Equa-
tion†. In Chaps. 10, 11, and 12, these structures are used for the design of
low-sensitive digital filters.
InChap.9wediscussseveraltechniquestotransformthetransferfunction
ofananalogfilterintoacorrespondingdigitalfilter,butalsocomputer-based
techniques to directly design the digital filter in the z-domain.
The principle of maximum power transfer of doubly resistive terminated
reactance networks that were discussed in Chap. 8 resulted in optimal
passband sensitivity. Hence, a corresponding digital filter structure must
mimic the power concept in the digital domain. That is, the currents and
voltagesintheanalogstructuresmustberepresentedbyalinearcombination
oftwosignalvariablesinthedigitaldomain.InChaps.10,11,and12weuse
so-called voltage waves.
InChap.10,thebasicbuildingblocksofwavedigitalfiltersarediscussed.
In addition, the power representation allows the nonlinearities in the digital
filter, due to quantization and overflow, to be suppressed by introducing
energy losses.
InChap.11,wediscusswavedigitalfiltersthatarebasedonanalogladder
structures.
In Chap. 12, we discuss a particular class of wave digital filter structures
thatarebasedonsymmetricalreactancetwo-ports.Latticeandcirculator-tree
structures belong to this class of structures. We also show that filters with
constant pole radius are of particular interest. Also, we show that all wave
digital filters discussed can be efficiently realized using numerically equiv-
alentstate-spacerepresentation. Finally,we comparethedifferent structures.
In Chap. 13, we discuss filter structures that are particularly suitable for
filters with very small transition bands, so-called frequency-response mask-
ing filters.
InChap.14,wediscussthedesignoffilteringstructuresforsamplingrate
conversion and efficient methods for their realization.
InChap.15,wediscussthedesignandrealizationofmultiratefilters that
reduce the arithmetic workload by using different internal sampling rates.
Finally, in Chap. 16, we discuss several architectures for the implemen-
tation of digital filters. In particular, we discuss the implementation of
arithmetic operations with bit-serial arithmetic and the implementation of
Preface vii
sum-of-products using distributed arithmetic. We also show how maximally
fast implementations can be obtained.
The authors are indebted to a large number of students and faculty col-
leagueswhoprovidedvaluablesuggestionsandassistanceinthepreparation
of the final manuscript and to allow us to use extracts of some of their
publications and MATLAB programs. We wish to particularly thank Pro-
fessors:SvenEriksson,AlfredFettweis,OscarGustafsson,HåkanJohansson,
Kent Palmkvist, Mark Vesterbacka, and Jacob Wikner.
Linköping, Sweden Lars Wanhammar
Tampere, Finland Tapio Saramäki
Contents
1 Signals and Transforms . .... .... .... .... .... ..... .... 1
1.1 Introduction. ..... .... .... .... .... .... ..... .... 1
1.2 Communication Systems.... .... .... .... ..... .... 2
1.2.1 Multimedia Communication Systems.... .... 3
1.2.2 Satellite-Based Communication Networks .... 3
1.3 Signals and Signal Carriers.. .... .... .... ..... .... 4
1.4 Fourier Transform. .... .... .... .... .... ..... .... 7
1.4.1 Fourier Transform of Continuous-Time
Signals.. .... .... .... .... .... ..... .... 7
1.4.2 Fourier Transform of Discrete-Time
Signals.. .... .... .... .... .... ..... .... 8
1.5 z-Transform. ..... .... .... .... .... .... ..... .... 10
1.5.1 Inverse z-Transform.... .... .... ..... .... 12
1.5.2 Inversion Using Residue Calculus. ..... .... 13
1.6 Sampling of Analog Signals. .... .... .... ..... .... 14
1.6.1 Poisson’s Summation Formula.... ..... .... 14
1.6.2 Sampling of Bandpass Signals.... ..... .... 17
1.6.3 Selection of the Sampling Frequency.... .... 19
1.6.4 Reconstruction of the Analog Signal .... .... 22
1.7 Analog-to-Digital Conversion.... .... .... ..... .... 22
1.7.1 ADC Resolution .. .... .... .... ..... .... 23
1.7.2 High-Speed Analog-to-Digital Converters .... 26
1.7.3 Oversampling Analog-to-Digital
Converters ... .... .... .... .... ..... .... 27
1.8 Digital-to-Analog Conversion.... .... .... ..... .... 28
1.9 Problems... ..... .... .... .... .... .... ..... .... 31
References.... .... ..... .... .... .... .... .... ..... .... 33
2 Discrete-Time and Digital Filters .. .... .... .... ..... .... 35
2.1 Introduction. ..... .... .... .... .... .... ..... .... 35
2.1.1 In the Beginning .. .... .... .... ..... .... 35
2.2 Discrete-Time Filters... .... .... .... .... ..... .... 36
2.2.1 CTD Filters .. .... .... .... .... ..... .... 36
2.2.2 Switched-Capacitor Filters... .... ..... .... 36
2.2.3 Switched-Current Filters. .... .... ..... .... 37
ix
x Contents
2.3 Digital Filters .... .... .... .... .... .... ..... .... 37
2.3.1 Coefficient Sensitivity .. .... .... ..... .... 39
2.3.2 Robustness... .... .... .... .... ..... .... 40
2.4 Frequency-Selective Filters.. .... .... .... ..... .... 40
2.4.1 Difference Equations ... .... .... ..... .... 40
2.4.2 Signal-Flow Graphs.... .... .... ..... .... 40
2.4.3 FIR and IIR Filters .... .... .... ..... .... 41
2.4.4 Transfer Function.. .... .... .... ..... .... 45
2.4.5 Poles and Zeros... .... .... .... ..... .... 46
2.4.6 Frequency Response ... .... .... ..... .... 47
2.4.7 Magnitude Response ... .... .... ..... .... 48
2.4.8 Attenuation .. .... .... .... .... ..... .... 49
2.4.9 Phase Response... .... .... .... ..... .... 49
2.4.10 Phase Delay.. .... .... .... .... ..... .... 51
2.4.11 Group Delay . .... .... .... .... ..... .... 52
2.4.12 Impulse and Step Response.. .... ..... .... 53
2.5 Filter Realizations. .... .... .... .... .... ..... .... 55
2.6 Direct Form Structures . .... .... .... .... ..... .... 56
2.6.1 Direct Form I. .... .... .... .... ..... .... 56
2.6.2 Direct Form II.... .... .... .... ..... .... 57
2.7 Transposed Structures.. .... .... .... .... ..... .... 59
2.7.1 Direct Form It .... .... .... .... ..... .... 59
2.7.2 Direct Form IIt.... .... .... .... ..... .... 60
2.8 Cascade Form Structures.... .... .... .... ..... .... 60
2.9 Parallel Form Structures .... .... .... .... ..... .... 62
2.10 State-Space Structures.. .... .... .... .... ..... .... 64
2.10.1 State-Space Realization of Direct Form II .... 64
2.10.2 State-Space Realization of the Parallel
Form ... .... .... .... .... .... ..... .... 66
2.11 Problems... ..... .... .... .... .... .... ..... .... 67
References.... .... ..... .... .... .... .... .... ..... .... 71
3 Filter Algorithms.. ..... .... .... .... .... .... ..... .... 73
3.1 Introduction. ..... .... .... .... .... .... ..... .... 73
3.2 Filter Algorithms.. .... .... .... .... .... ..... .... 73
3.2.1 Arithmetic Operations .. .... .... ..... .... 75
3.2.2 Precedence Graphs. .... .... .... ..... .... 75
3.2.3 Parallelism in Algorithms.... .... ..... .... 76
3.2.4 Sequentially Computable Algorithms.... .... 77
3.3 Precedence Form.. .... .... .... .... .... ..... .... 78
3.4 Difference Equations... .... .... .... .... ..... .... 81
3.5 Computation Graphs... .... .... .... .... ..... .... 84
3.5.1 Fully Specified Signal-Flow Graphs..... .... 84
3.5.2 Critical Paths. .... .... .... .... ..... .... 85
3.5.3 Equalizing Delay.. .... .... .... ..... .... 85
3.5.4 Shimming Delay .. .... .... .... ..... .... 85
Contents xi
3.6 Maximum Sampling Frequency... .... .... ..... .... 86
3.6.1 Cyclic Scheduling . .... .... .... ..... .... 88
3.7 Equivalence Transformations .... .... .... ..... .... 88
3.8 Pipelining and Interleaving.. .... .... .... ..... .... 90
3.8.1 Pipelining.... .... .... .... .... ..... .... 90
3.8.2 Interleaving .. .... .... .... .... ..... .... 93
3.9 Arithmetic Operations.. .... .... .... .... ..... .... 94
3.9.1 Adders and Subtractors . .... .... ..... .... 94
3.9.2 Multiplication. .... .... .... .... ..... .... 94
3.9.3 Constant Multiplication . .... .... ..... .... 95
3.9.4 Composite Arithmetic Operations . ..... .... 96
3.10 Power Consumption in CMOS Circuits. .... ..... .... 98
3.10.1 Switching Power Dissipation. .... ..... .... 99
3.10.2 Short-Circuit Power Dissipation... ..... .... 100
3.10.3 Static Power Dissipation .... .... ..... .... 100
3.10.4 Leakage Power Dissipation .. .... ..... .... 100
3.10.5 Activity Factor.... .... .... .... ..... .... 101
3.10.6 Clock Frequency .. .... .... .... ..... .... 102
3.10.7 Switched Capacitance .. .... .... ..... .... 102
3.10.8 Power Supply Voltage Scaling.... ..... .... 102
3.11 Problems... ..... .... .... .... .... .... ..... .... 103
References.... .... ..... .... .... .... .... .... ..... .... 106
4 Finite Wordlength Effects.... .... .... .... .... ..... .... 109
4.1 Introduction. ..... .... .... .... .... .... ..... .... 109
4.2 Number Representations.... .... .... .... ..... .... 110
4.2.1 Fixed-Point Representations.. .... ..... .... 110
4.2.2 Redundant Representations .. .... ..... .... 112
4.2.3 Quantization.. .... .... .... .... ..... .... 113
4.2.4 Floating-Point Representation .... ..... .... 115
4.3 Parasitic Oscillations... .... .... .... .... ..... .... 116
4.3.1 Zero-Input Oscillations.. .... .... ..... .... 116
4.3.2 Overflow Characteristics .... .... ..... .... 119
4.3.3 Periodic Input Oscillations... .... ..... .... 121
4.3.4 Unobservable Oscillations ... .... ..... .... 123
4.3.5 Parasitic Oscillations in Algorithms Using
Floating-Point Arithmetic.... .... ..... .... 123
4.4 Stability.... ..... .... .... .... .... .... ..... .... 124
4.5 Scaling of Signal Levels.... .... .... .... ..... .... 124
4.5.1 Safe Scaling.. .... .... .... .... ..... .... 125
4.5.2 Norms .. .... .... .... .... .... ..... .... 127
4.5.3 Scaling for Wideband Signals .... ..... .... 129
4.5.4 Scaling for Narrowband Signals... ..... .... 132
4.5.5 Scaling of State-Space Structures.. ..... .... 135
4.6 Quantization Noise .... .... .... .... .... ..... .... 136
4.7 Roundoff Noise... .... .... .... .... .... ..... .... 137
4.7.1 Short Coefficients.. .... .... .... ..... .... 140
4.7.2 Roundoff Noise in State-Space Structures .... 143