Digital Signal Processing. Principles, Algorithms and System Design

615 Pages·2016·5.57 MB·English

by  Winser Alexander and Cranos M. Williams (Auth.)

Digital Signal
Processing: Principles,
Algorithms and System
Design

Digital Signal
Processing: Principles,
Algorithms and System
Design
Winser E. Alexander
Cranos M. Williams
North Carolina State University
NC, USA
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List of Tables
Table2.1 Atabularrepresentationofadiscretetimesignal 20
Table2.2 ImpulseresponseofdiscretetimesystemforExample2.12 61
Table2.3 ImpulseresponseofFIRsysteminExample2.23 84
Table2.4 TabulationofasamplesequenceforExample2.35 118
Table2.5 TabulationofaoutputsamplesequenceforExample2.35after
interpolationusingasample-and-holdoperation 120
Table2.6 TabulationofasamplesequenceforExample2.41 136
Table2.7 TabulationofaoutputsamplesequenceforExample2.41after
decimationusingasample-and-holdoperation 137
Table3.1 TableofvaluesforX (k)forExample3.6 184
3
Table3.2 TableofvaluesforX (k)forExample3.6 184
4
Table3.3 FrequencyresponsevaluesforProblem3.1 197
Table3.4 ValuesoftheDFTforProblem3.3 198
Table3.5 TabulationofthesequenceforProblem3.9 200
Table3.6 TabulationofthesequenceforProblem3.12 201
Table3.7 FFTforsequenceforProblem3.13 202
Table4.1 FiltercoefficientsforahighpassFIRfilter 209
Table4.2 FiltercoefficientsforalowpassFIRfilter 214
Table4.3 Theimpulseresponsesforlowpassandhighpassidealfiltersfor
2M+1coefficientsshiftedbyM samples 222
Table4.4 Theimpulseresponsesforbandpassandbandstopidealfiltersfor
2M+1coefficientsshiftedbyM samples 222
Table4.5 Frequencydomaincharacteristicsforwindowfunctions[8] 223
Table4.6 Summary ofthe requiredorder forthe FIRfilters designedfor
Example4.6 238
Table4.7 Frequencytransformationsforanaloglowpassandhighpassfilters
[4] 246
Table4.8 Frequencytransformationsforanalogbandpassandbandstop
filters[4] 247
Table4.9 Digitalfrequencytransformationsforlowpassandhighpassdigital
filters[4] 249
Table4.10 Digitalfrequencytransformationsforbandpassandbandstopdigital
filters[4] 249
Table6.1 Examplesoffloatingpointrepresentationofnumbers 352
Table7.1 Comparisonofthenumberofadditionsandmultiplicationsforthe
twointerpolationmethods 411
Table7.2 Comparisonofthenumberofadditionsandmultiplicationsforthe
twodecimationmethods 417
Table7.3 Tableofcoefficientsforb6 446
Table10.1 Comparisonofcomputationalcomplexityforimplementationsofthe
2-DDFT 559
Table10.2 A2-Dsequenceasinputforthe2-DDCT 560
xiii
xiv List of Tables
Table10.3 Theextendedsequenceasinputforthe2-DDCT 560
Table10.4 The2-DDCTforthe2-Dsequence 561
List of Figures
Fig.1.1 Conceptualblockdiagramofatypicalsystem 5
Fig.1.2 StemplotforsequenceinExample1.1 10
Fig.1.3 StemplotofcosinesequenceforExample1.2 11
Fig.1.4 StemplotofadelayedimpulseforExample1.3 12
Fig.1.5 PlotofsectionofasinusoidalsignalforExample1.4 14
Fig.1.6 PlotoftherealpartofacomplexexponentialinExample1.5 15
Fig.2.1 ExampleofadiscretetimesignalasrepresentedbyTable2.1 21
Fig.2.2 Example of a discrete time signal represented by a functional
representation 21
Fig.2.3 Example of a discrete time signal represented by a sequence
representation 22
Fig.2.4 Stemplotofasequenceusingunitstepsequences 24
Fig.2.5 Stemplotofsequenceusingtheunitramp 25
Fig.2.6 Stemplotofsequenceusingexponentialsignals 26
Fig.2.7 Stemplotofsequenceusingcomplexexponentialsignals 28
Fig.2.8 Stemplotofanevendiscretetimesignal 29
Fig.2.9 Stemplotofanodddiscretetimesignal 29
Fig.2.10 Stemplotofadiscretetimesignalthatisneitherevennorodd 30
Fig.2.11 StemplotofthediscretetimesignalinFig.2.10afterflippingitleftto
righttoformx(−n) 31
Fig.2.12 Stemplotofthecomparisonoftheoriginalevensignalwiththe
extractedevensignal 31
Fig.2.13 Stemplotofthecomparisonoftheoriginaloddsignalwiththe
extractedoddsignal 32
Fig.2.14 Stemplotofasinusoidalsequence 36
Fig.2.15 Stemplotofasinusoidalsequencewithindependentvariablescaled
byafactorofa=0.5 37
Fig.2.16 Stemplotofasinusoidalsequencewithindependentvariablescaled
byafactorofa=1.5 38
Fig.2.17 Stemplotofasinusoidalsequence 39
Fig.2.18 Stemplotofasinusoidalsequencewithindependentvariableshifted
byafactorofm=6 39
Fig.2.19 Stemplotofasinusoidalsequencewithindependentvariableshifted
byafactorofm=−6 40
Fig.2.20 Stemplotofasamplesequence 41
Fig.2.21 Stemplotofthereversedsamplesequence 41
Fig.2.22 OriginaldiscretetimesignalforExample2.7 42
Fig.2.23 Transformationy1(n)=x(n−4)forExample2.7 43
Fig.2.24 Transformationy2(n)=x(3−n)forExample2.7 45
Fig.2.25 Transformationy3(n)=x(3n)forExample2.7 45
Fig.2.26 Transformationy4(n)=x(3n+1)forExample2.7 46
xv
xvi List of Figures
Fig.2.27 Transformationy5(n)=x(n)u(2−n)forExample2.7 46
Fig.2.28 Transformationy6(n)=x(n−2)δ(n−2)forExample2.7 47
Fig.2.29 Transformationy7(n)= 12x(cid:2)(n(cid:3))+ 12(−1)nx(n)forExample2.7 47
Fig.2.30 Transformationy (n)=x n forExample2.7 48
8 2
Fig.2.31 Modulation 48
Fig.2.32 Addition 48
Fig.2.33 Multiplication 49
Fig.2.34 Delay 49
Fig.2.35 Signalsx (t),x (t),andtheirsamplesonthesameplot 51
1 2
Fig.2.36 Input1forcross-correlationexample 70
Fig.2.37 Input2forcross-correlationexample 71
Fig.2.38 Blockdiagramoftheoverallsystem 71
Fig.2.39 Pole–zeroplotforExample2.25 88
Fig.2.40 Blockdiagramforadiscretetimesystem 89
Fig.2.41 RegionofconvergenceforS1(z)withb=3 95
Fig.2.42 RegionofconvergenceforS2(z)witha=0.5 95
Fig.2.43 RegionofconvergenceforX(z)witha=0.5,b=3.0 96
Fig.2.44 Stemplotofx(n)inExample2.28forn=−25ton=25 100
Fig.2.45 Comparisonofthesample-and-holdandtheideallowpassfilter 117
Fig.2.46 Reconstructionusingasample-and-holdandalowpassfilter 118
Fig.2.47 StemplotofthesamplesequenceforExample2.35 119
Fig.2.48 StemplotoftheoutputsamplesequenceforExample2.35after
usingthesample-and-holdoperation 120
Fig.2.49 Stem plot of the input and output sample sequences for
Example2.35onthesameplot 121
Fig.2.50 StemplotoftheinputandoutputsamplesequencesforExample2.35
aftercompensatingforthedelayinthesample-and-holdoperator 121
Fig.2.51 Stem plot of the input and filtered output sample sequences
for Example 2.35 after compensating for the delays in the
sample-and-holdoperatorandthefilter 122
Fig.2.52 Impulseresponseforthelinearpointconnector 122
Fig.2.53 Comparisonofthelinearpointconnectorandtheideallowpassfilter 123
Fig.2.54 Reconstructionusingalinearpointconnectorandalowpassfilter 124
Fig.2.55 SamplesequenceforExample2.36 124
Fig.2.56 Magnitude and phase responses for the sample sequence in
Example2.36 125
Fig.2.57 Magnitudeandphaseresponsesforthemodifiedsamplesequence
inExample2.36 126
Fig.2.58 SamplesequenceforExample2.36afterinterpolation 127
Fig.2.59 StemplotofsamplesequenceforExample2.37 128
Fig.2.60 Samplesequenceafterpaddingwithzeros 129
Fig.2.61 InterpolatedoutputsequenceforExample2.37 130
Fig.2.62 Abandlimitedsignalwith256samples 130
Fig.2.63 Interpolatedsignalupsampledbyafactorof3 131
Fig.2.64 Samplesequencedecimatedbyafactorof3 133
List of Figures xvii
Fig.2.65 Samplesequencedecimatedbyafactorof3usingthedecimate
Matlabfunction 133
Fig.2.66 StemplotforinterpolatedsequenceforExample2.40 136
Fig.2.67 StemplotofthesamplesequenceforExample2.41afterfiltering
usingtheantialiasinglowpassfilter 137
Fig.2.68 StemplotoftheoutputsamplesequenceforExample2.41after
usingthesample-and-holdoperation 138
Fig.2.69 Stem plot of the input and output sample sequences for
Example2.41onthesameplot 138
Fig.2.70 StemplotoftheinputandoutputsamplesequencesforExample2.41
aftercompensatingforthedelayinthesample-and-holdoperator 139
Fig.2.71 PlotofSpectrumforProblem2.28 148
Fig.2.72 PlotofspectrumforProblem2.30 148
Fig.3.1 Generaldiscretetimesystemwithinputejωn 161
Fig.3.2 MagnitudespectrumforFIRfilter 163
Fig.3.3 PhasespectrumforFIRfilter 164
Fig.3.4 MagnitudeplotofH(ω)forExample3.2 166
Fig.3.5 PhaseplotofH(ω)forExample3.2 166
Fig.3.6 Pole–zeroplotof H(z)forlowpassfilterwithcutofffrequency
ωc =0.6π 167
Fig.3.7 MagnitudeandphaseplotsofH(z)forlowpassfilterwithcutoff
frequencyωc =0.6π 168
Fig.3.8 Pole–zeroplotofH(z)forhighpassfilterwithcutofffrequency
ωc =0.6π 168
Fig.3.9 MagnitudeandphaseplotsofH(z)forhighpassfilterwithcutoff
frequencyωc =0.6π 169
Fig.3.10 Pole–zeroplotofH(z)forbandstopfilterwithcutofffrequenciesat
ω1=0.6π andω2=0.75π 170
Fig.3.11 MagnitudeandphaseplotsofH(z)forbandstopfilterwithcutoff
frequencyω1=0.6π andω2=0.75π 170
Fig.3.12 Pole–zeroplotofH(z)forbandpassfilterwithcutofffrequenciesat
ω1=0.6π andω2=0.75π 171
Fig.3.13 MagnitudeandphaseplotsofH(z)forbandpassfilterwithcutoff
frequencyω1=0.6π andω2=0.75π 172
Fig.3.14 Pole–zeroplotofH(z)fordigitalresonatorwithzerosatz = ±1,
ω=0.7π 173
Fig.3.15 Frequency response ofH(z)fordigitalresonatorwithzeros at
z=±1,ω=0.7π 174
Fig.3.16 Pole–zeroplotofH(z)fordigitalresonatorwithzerosatz = 0,
ω=0.7π 174
Fig.3.17 FrequencyresponseofH(z)fordigitalresonatorwithzerosatz=0,
ω=0.7π 175
Fig.3.18 Blockdiagramoftheoverallsystem 175
Fig.3.19 Plotof50samplesoftheinputsequenceforExample3.4 177
Fig.3.20 Plotof50samplesoftheoutputsequenceforExample3.4 178
xviii List of Figures
Fig.3.21 MagnitudeandphaseplotsforExample3.5 181
Fig.3.22 Circularconvolutionofx (n)andx (n) 183
1 2
Fig.3.23 Linearconvolutionusingconvolutioninthetimedomain 183
Fig.3.24 LinearconvolutionusingtheDFT 184
Fig.3.25 StemplotofatestsequenceforExample3.11 195
Fig.3.26 StemplotsofthemagnitudesoftheDFTandtheDCTforthetest
sequenceforExample3.11 196
Fig.4.1 Specificationsforalowpassfilter 206
Fig.4.2 StemplotforthecoefficientsinH(z)forExample4.1 210
Fig.4.3 MagnitudeandphaseplotsforH(z)forExample4.1 212
Fig.4.4 StemplotforthecoefficientsinH(z)forExample4.2 212
Fig.4.5 MagnitudeandphaseplotsforH(z)forExample4.2 214
Fig.4.6 Exampleofanideallowpassfilter(ωc =0.67π) 216
Fig.4.7 ImpulseresponseforthelowpassfilterinExample4.3 218
Fig.4.8 MagnitudeandphaseresponsesforthelowpassfilterinExample4.3 218
Fig.4.9 Exampleofanidealhighpassfilter(ωc =0.391π) 219
Fig.4.10 ImpulseresponseforthehighpassfilterinExample4.4 221
Fig.4.11 Magnitude and phase responses for the high pass filter in
Example4.4 222
Fig.4.12 ImpulseresponsesforthetwoFIRfilterdesignsusingtherectangular
windowforcomparison 226
Fig.4.13 Magnitudeandphaseresponseofbandstopfilterusingarectangular
window 227
Fig.4.14 ImpulseresponsesfortheFIRfilterdesignusingtheHamming
window 227
Fig.4.15 MagnitudeandphaseresponseofbandstopfilterusingMatlab’sfir1
functionandaHammingwindow 228
Fig.4.16 MagnituderesponsefortheFIRfilterusingtheHammingwindow 235
Fig.4.17 MagnituderesponsefortheFIRfilterusingtheKaiserwindow 236
Fig.4.18 Magnituderesponseforthedesignusingfirpm 236
Fig.4.19 DesignusingfirlsforExample4.6 237
Fig.4.20 DesignusingfirclsforExample4.6 237
Fig.4.21 Designusingfir2forExample4.6 238
Fig.4.22 Aplotofthemappingofcontinuoustimefrequenciestodiscretetime
frequenciesforthebilineartransformation 243
Fig.4.23 MagnitudeplotforExample4.13 255
Fig.4.24 MagnitudeplotforExample4.14 256
Fig.5.1 BlockdiagramforthedirectrealizationofanFIRdiscretetime
system 278
Fig.5.2 BlockdiagramforthedirectimplementationofanFIRdigitalfilter 279
Fig.5.3 BlockdiagramforalinearphaseFIRdigitalfilter 281
Fig.5.4 BlockdiagramforalinearphaseFIRdigitalfilter 282
Fig.5.5 PolyphaseFIRimplementationusing2filters 285
Fig.5.6 PolyphaseFIRimplementationusing3filters 287
Fig.5.7 FIRlatticefiltersection 288