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ThisisthetrigonometricformoftheDTFT.∗∗9.2−2∗∗Asignal∗x∗[∗n∗]canbeexpressedasthesumofevenandoddcomponents(Sec.1.5−1):
x[n] = x_e[n] + x_o[n]
(a)If∗x∗[∗n∗]⇐⇒∗X∗(),showthatforreal∗x∗[∗n∗],
x_e[n] \Longleftrightarrow \operatorname{Re}[X(\Omega)]
and
x_o[n] \Longleftrightarrow j \operatorname{Im}[X(\Omega)]
−(b)VerifytheseresultsbyfindingtheDTFToftheevenandoddcomponentsofthesignal(0.8)∗nu∗[∗n∗].−∗∗9.2−3∗∗Forthefollowingsignals,findtheDTFTdirectly,usingthedefinitioninEq.(9.19).Assume∣γ∣<1.−(a)δ[∗n∗](b)
\delta[n-k]
−(c)<sup>γ</sup>∗nu∗[∗n∗−1]−(d)<sup>γ</sup>∗nu∗[∗n∗+1]−(e)(−γ)∗nu∗[∗n∗]−(f)γ<sup>∣</sup>∗n∗<sup>∣</sup>−∗∗9.2−4∗∗UseEq.(9.18)tofindtheinverseDTFTforthefollowingspectra,givenonlyovertheinterval∣∣≤π.Assume∗<sup>c</sup>∗and<sup>0</sup><π.(a)∗ejk∗integer∗k∗−(b)cos∗k∗integer∗k∗−(c)cos2(/2)(d)
\Delta \left( \frac{1}{2\Omega_c} \right)
−(e)2πδ(−0)−(f)π[δ(−0)+δ(+0)]−∗∗9.2−5∗∗(a)DetermineandplottheDTFT∗X∗()ofthetriangularsignal∗x∗[∗n∗]showninFig.P9.2−5.−(b)UsingEx.9.5asaguide,useMATLABandtheFFTtovalidatetheDTFTcalculationsandplotofpart(a).∗∗FigureP9.2−5∗∗−∗∗9.2−6∗∗UsingEq.(9.18),showthattheinverseDTFTofrect((−π/4)/π)is0.5sinc(π∗n∗/2)∗ej∗π∗n∗/4.−∗∗9.2−7∗∗UsingEq.(9.19),findtheDTFTofthesignals∗x∗[∗n∗]inFig.P9.2−7.−∗∗9.2−8∗∗UsingEq.(9.19),findtheDTFTofthesignalsdepictedinFig.P9.2−8.−∗∗9.2−9∗∗UseEq.(9.18)tofindtheinverseDTFTofthespectra(shownonlyfor∣∣≤π)inFig.P9.2−9.∗∗FigureP9.2−9∗∗∗∗9.2−10∗∗UseEq.(9.18)tofindtheinverseDTFTofthespectra(shownonlyfor∣∣≤π)inFig.P9.2−10.∗∗9.2−11∗∗FindtheDTFTforthesignalsshowninFig.P9.2−11.∗∗FigureP9.2−10∗∗(c)(d)∗∗FigureP9.2−11∗∗−∗∗9.2−12∗∗FindtheinverseDTFTof∗X∗()(shownonlyfor∣∣≤π)forthespectraillustratedinFig.P9.2−12.[∗Hint:X∗()=∣∗X∗()∣∗ej<sup>X</sup>∗().Thisproblemillustrateshowdifferentphasespectra(bothwiththesameamplitudespectrum)represententirelydifferentsignals.]−∗∗9.2−13∗∗(a)Showthattime−expandedsignal∗xe∗[∗n∗]inEq.(3.2)canalsobeexpressedas
x_e[n] = \sum_{k=-\infty}^{\infty} x[k]\delta[n - Lk]
−(b)FindtheDTFTof∗xe∗[∗n∗]byfindingtheDTFToftheright−handsideoftheequationinpart(a).−(c)Usetheresultinpart(b)andTable9.1tofindtheDTFTof∗z∗[∗n∗],showninFig.P9.2−13.−∗∗9.2−14∗∗(a)AglanceatEq.(9.18)showsthattheinverseDTFTequationisidenticaltotheinverse(continuous−time)FouriertransformEq.(7.10)forasignal∗x∗(∗t∗)bandlimitedtoπrad/s.Hence,weshouldbeabletousethecontinuous−timeFouriertransformTable7.1tofindDTFTpairsthatcorrespondtocontinuous−timetransformpairsforbandlimitedsignals.UsethisfacttoderiveDTFTpairs8,9,11,12,13,and14inTable9.1bymeansoftheappropriatepairsinTable7.1.−(b)Canthismethodbeusedtoderivepairs2,3,4,5,6,7,10,15,and16inTable9.1?Justifyyouranswerwithspecificreason(s).−∗∗9.2−15∗∗Arethefollowingfrequency−domainsignalsvalidDTFT′s?Answeryesorno,andjustifyyouranswers.−(a)∗X∗()=+π−(b)∗X∗()=∗j∗+π−(c)∗X∗()=sin(10)−(d)∗X∗()=sin(/10)−(e)∗X∗()=δ()−∗∗9.3−1∗∗Usingonlypairs2and5(Table9.1)andthetime−shiftingpropertyofEq.(9.31),findtheDTFTofthefollowingsignals,assuming∣∗a∗∣<1.−(a)∗u∗[∗n∗]−∗u∗[∗n∗−9]−(b)∗an∗−∗mu∗[∗n∗−∗m∗]−(c)∗an∗−3(∗u∗[∗n∗]−∗u∗[∗n∗−10])1(b)∗∗FigureP9.2−13∗∗−(d)∗<sup>a</sup>n∗−∗mu∗[∗n∗]−(e)∗anu∗[∗n∗−∗m∗]−(f)(∗n∗−∗m∗)∗an∗−∗mu∗[∗n∗−∗m∗]−(g)(∗n∗−∗m∗)∗anu∗[∗n∗]−(h)∗nan∗−∗mu∗[∗n∗−∗m∗]−∗∗9.3−2∗∗Thetriangularpulse∗x∗[∗n∗]showninFig.P9.3−2aisgivenby
X(\Omega) = \frac{4e^{j6\Omega} - 5e^{j5\Omega} + e^{j\Omega}}{(e^{j\Omega} - 1)^2}
UsethisinformationandtheDTFTpropertiestofindtheDTFTofthesignals∗x∗1[∗n∗],∗x∗2[∗n∗],∗x∗3[∗n∗],and∗x∗4[∗n∗]showninFigs.P9.3−2b,P9.3−2c,P9.3−2d,andP9.3−2e,respectively.∗∗9.3−3∗∗Supposesignal∗<sup>x</sup>∗[∗n∗]=sinc2(π∗n∗/2)modulatesacarriercos(c∗n∗)toproducesignal∗y∗[∗n∗]=∗x∗[∗n∗]cos(c∗n∗).FindandsketchtheDTFTof:(a)∗x∗[∗n∗]−(b)∗y∗[∗n∗]for<sup>c</sup>=π/2−(c)∗y∗[∗n∗]for<sup>c</sup>=3π/4−(d)∗y∗[∗n∗]for<sup>c</sup>=π∗∗9.3−4∗∗Showthatperiodicconvolution∗X∗()−∗∗Y∗()=2π∗X∗()if
X(\Omega) = \sum_{k=0}^{4} a_k e^{-jk\Omega}
and
Y(\Omega) = \frac{\sin(5\Omega/2)}{\sin(\Omega/2)} e^{-j2\Omega}
where∗ak∗isasetofarbitraryconstants.−∗∗9.3−5∗∗Usingonlypair2(Table9.1)andpropertiesofDTFT,findtheDTFTofthefollowingsignals,assuming∣∗a∗∣<1and<sup>0</sup><π.−(a)∗an∗cos0∗nu∗[∗n∗]−(b)∗<sup>n</sup>∗<sup>2</sup>∗anu∗[∗n∗]−(c)(∗n∗−∗k∗)∗a∗<sup>2</sup>∗nu∗[∗n∗−∗m∗]−∗∗9.3−6∗∗Usepair10inTable9.1,andsuitablepropertiesoftheDTFT,toderivepairs11,12,13,14,15,and16.−∗∗9.3−7∗∗Usethetime−shiftingpropertytoshowthat
x[n+k]+x[n-k] \Longleftrightarrow 2X(\Omega)\cos k\Omega
(e)∗∗FigureP9.3−2∗∗∗∗FigureP9.3−7∗∗UsethisresulttofindtheDTFTofthesignalsshowninFig.P9.3−7.∗∗9.3−8∗∗Usethetime−shiftingpropertytoshowthat
x[n+k] - x[n-k] \Longleftrightarrow 2jX(\Omega) \sin k\Omega
UsethisresulttofindtheDTFTofthesignalshowninFig.P9.3−8.∗∗9.3−9∗∗Supposesignal∗x∗[∗n∗]hasspectrum∗X∗()thatisbandlimitedtoπ/2rad/sample.Next,definesignal∗y∗[∗n∗]as
y[n] = \begin{cases} x[n] & n \text{ even} \ 0 & n \text{ odd} \end{cases}
Determinethespectrumof∗Y∗()intermsof∗X∗().Sketch∗Y∗()if,over−π≤≤π,
Y(\Omega) = \begin{cases} |2\Omega/\pi| & -\pi/2 \le \Omega \le \pi/2\ 0 & \text{otherwise} \end{cases}
∗∗9.3−10∗∗RepeatProb.9.3−9if∗y∗[∗n∗]isinsteaddefinedas
y[n] = \begin{cases} x[n] & n \text{ odd} \ 0 & n \text{ even} \end{cases}
−∗∗9.3−11∗∗Usingonlypair2inTable9.1andtheconvolutionproperty,findtheinverseDTFTof∗X∗()=∗<sup>e</sup>∗2∗<sup>j</sup>∗/(∗ej∗<sup>−</sup>γ)2.−∗∗9.3−12∗∗InTable9.1,youaregivenpair1.FromthisinformationandusingsuitablepropertiesoftheDTFT,derivepairs2,3,4,5,6,and7ofTable9.1.Forexample,startingwithpair1,derivepair2.Frompair2,usesuitablepropertiesoftheDTFTtoderivepair3.Frompairs2and3,derivepair4,andsoon.−∗∗9.3−13∗∗Fromthepair∗ej∗(0/2)∗<sup>n</sup>∗⇐⇒<sup>2</sup>πδ(<sup>−</sup>(0/2))overthefundamentalband,andthefrequency−convolutionproperty,findtheDTFTof∗ej∗0∗n∗.Assume<sup>0</sup><π/2.−∗∗9.3−14∗∗FromthedefinitionandpropertiesoftheDTFT,showthat(a)
\sum_{n=-\infty}^{\infty} \operatorname{sinc}(\Omega_c n) = \frac{\pi}{\Omega_c} \quad \Omega_c < \pi
\n(b)
\sum_{n=-\infty}^{\infty} (-1)^n \operatorname{sinc}(\Omega_c n) = 0 \quad \Omega_c < \pi
\n(c)
\sum_{n=-\infty}^{\infty} \operatorname{sinc}^2(\Omega_c n) = \frac{\pi}{\Omega_c} \quad \Omega_c < \pi/2
\n(d)
\sum_{n=-\infty}^{\infty} (-1)^n \operatorname{sinc}^2(\Omega_c n) = 0 \quad \Omega_c < \pi/2
\n(e)
\int_{-\pi}^{\pi} \frac{\sin(M\Omega/2)}{\sin(\Omega/2)} = 2\pi \quad \text{odd } M
\int_{-\pi}^{\infty} \frac{\sin(\alpha z/2)}{\sin(\alpha z/2)} \sin(\alpha z/2)
(f)
\sum_{n=-\infty}^{\infty} |\sin(\alpha z/2)|^4 = 2\pi/3\Omega_c \quad \Omega_c < \pi/2
∗∗9.3−15∗∗Showthattheenergyofsignal∗xc∗(∗t∗)specifiedinEq.(9.41)isidenticalto∗T∗timestheenergyofthediscrete−timesignal∗x∗[∗n∗],assuming∗xc∗(∗t∗)isbandlimitedto∗B∗≤1/2∗T∗Hz.[∗Hint:∗Recallthat
\int_{-\infty}^{\infty} \operatorname{sinc} [\pi(t-m)] \operatorname{sinc} [\pi(t-n)] dt
= \begin{cases} 0 & m \neq n \ 1 & m = n \end{cases}
Thatis,sincfunctionsareorthogonal.]∗∗9.4−1∗∗UsetheDTFTmethodtofindthezero−stateresponse∗y∗[∗n∗]ofacausalsystemwithfrequencyresponse
H(\Omega) = \frac{e^{i\Omega} + 0.32}{e^{i2\Omega} + e^{i\Omega} + 0.16}
andtheinput∗<sup>x</sup>∗[∗n∗]=(−0.5)∗nu∗[∗n∗].∗∗9.4−2∗∗RepeatProb.9.4−1for
H(\Omega) = \frac{e^{i\Omega} + 0.32}{e^{i2\Omega} + e^{i\Omega} + 0.16}
andinput∗x∗[∗n∗]=∗u∗[∗n∗].∗∗9.4−3∗∗RepeatProb.9.4−1for
H(\Omega) = \frac{e^{i\Omega}}{e^{i\Omega} - 0.5}
and
x[n] = 0.8nu[n] + 2(2)nu[-(n+1)]
∗∗9.4−4∗∗DetermineandsketchthemagnitudeandphaseresponseforanLTIDsystemspecifiedbytheequation
y[n] + 0.5y[n-1] = x[n] - 0.9x[n-1]
Determinethesystemoutput∗y∗[∗n∗]fortheinput∗<sup>x</sup>∗[∗n∗]=cos(<sup>π</sup>∗<sup>n</sup>∗<sup>3</sup>+0.5).∗∗9.4−5∗∗RepeatProb.9.4−4iftheLTIDsystemisinsteadspecifiedbytheequation
y[n] - 0.5y[n-1] = x[n] + 0.9x[n-1]
∗∗9.4−6∗∗Anaccumulatorsystemhasthepropertythataninput∗x∗[∗n∗]resultsintheoutput
y[n] = \sum_{k=-\infty}^{n} x[k]
−(a)Findtheunitimpulseresponse∗h∗[∗n∗]andthefrequencyresponse∗H∗()fortheaccumulator.−(b)Usetheresultsofpart(a)tofindtheDTFTof∗u∗[∗n∗].−∗∗9.4−7∗∗Anoncausal7−pointmovingaverageisdescribedbytheequation
y[n] = \frac{1}{7} \sum_{k=-3}^{3} x[n-k]
−(a)Findandsketchthemagnitudeandphaseresponsesofthesystem.−(b)Howcanthissystembemadecausal?Plotthemagnitudeandphaseresponsesofthecausalsystem,andcommentonanydifferencesfrompart(a).−∗∗9.4−8∗∗AnLTIDsystemfrequencyresponseover∣∣≤πis
H(\Omega) = \text{rect}\bigg(\frac{\Omega}{\pi}\bigg)e^{-j2\Omega}
Findtheoutput∗y∗[∗n∗]ofthissystem,iftheinput∗x∗[∗n∗]isgivenby−(a)sinc(π∗n∗/2)−(b)sinc(π∗n∗)−(c)sinc2(π∗n∗/4)−∗∗9.4−9∗∗(a)If∗x∗[∗n∗]⇐⇒∗X∗(),then,showthat(−1)∗nx∗[∗n∗]⇐⇒∗<sup>X</sup>∗(−π).−(b)Sketch<sup>γ</sup>∗nu∗[∗n∗]and(−γ)∗nu∗[∗n∗]for<sup>γ</sup><sup>=</sup>0.8;seethespectrafor<sup>γ</sup>∗nu∗[∗n∗]inFigs.9.5band9.5c.Fromthesespectra,sketchthespectrafor(−γ)∗nu∗[∗n∗].−(c)Anideallowpassfilterofcutofffrequency∗<sup>c</sup>∗isspecifiedbythefrequencyresponse∗H∗()=rect(/2∗c∗).Finditsimpulseresponse∗h∗[∗n∗].Findthefrequencyresponseofafilterwhoseimpulseresponseis(−1)∗nh∗[∗n∗].Sketchthefrequencyresponseofthisfilter.Whatkindoffilteristhis?−∗∗9.4−10∗∗Ananalogdifferentiator∗<sup>y</sup>∗(∗t∗)<sup>=</sup>∗<sup>d</sup>dtx∗(∗t∗)canbeapproximatedusingabackwarddifferencesystemdescribedas
y[n] = \frac{x[n] - x[n-1]}{T}
FindandsketchthemagnitudeandphaseresponsesofthisDTsystem.Forwhatfrequenciesdoesthesystemmostbehaveasadifferentiator?Forwhatfrequenciesdoesthesystemleastbehaveasadifferentiator?∗∗9.4−11∗∗Afilterwithimpulseresponse∗h∗[∗n∗]ismodifiedasshowninFig.P9.4−11.Determinetheresultingfilterimpulseresponse∗h∗1[∗n∗].Findalsotheresultingfilterfrequencyresponse∗H∗1()intermsofthefrequencyresponse∗H∗().Howare∗H∗()and∗H∗1()related?−∗∗9.4−12∗∗(a)ConsideranLTIDsystem∗S∗1,specifiedbyadifferenceequationoftheformofEqs.(3.15)or(3.16)or(3.20)inCh.3.Weconstructanothersystem∗S∗<sup>2</sup>byreplacingcoefficients∗ai∗(∗i∗=0,1,2,...,∗N∗)bycoefficients(−1)∗<sup>i</sup>ai∗andreplacingallcoefficients∗bi∗(∗i∗=0,1,2,...,∗N∗)withcoefficients(−1)∗<sup>i</sup>bi∗.Howarethefrequencyresponsesofthetwosystemsrelated?−(b)If∗S∗<sup>1</sup>representsalowpassfilter,whatkindoffilterisspecifiedby∗S∗2?−(c)Whattypeoffilter(lowpass,highpass,etc.)isspecifiedbythedifferenceequation
y[n] - 0.8y[n-1] = x[n]
Whatkindoffilterisspecifiedbythefollowingdifferenceequation?
y[n] + 0.8y[n-1] = x[n]
∗∗9.4−13∗∗(a)ThesystemshowninFig.P9.4−13containstwoidenticalLTIDfilterswithfrequencyresponse∗H∗0()andcorrespondingimpulseresponse∗h∗0[∗n∗].Itiseasytoseethatthesystemislinear.Showthatthissystemisalsotime−invariant.Dothisbyfindingthe∗∗FigureP9.4−11∗∗responseofthesystemtoinputδ[∗n∗−∗k∗]intermsof∗h∗0[∗n∗].−(b)If∗H∗0()=rect(/2∗W∗)overthefundamentalband,and∗<sup>c</sup>∗+∗W∗≤π,find∗H∗(),thefrequencyresponseofthissystem.Whatkindoffilteristhis?−∗∗9.5−1∗∗DeterminetheDTFTof∗x∗[∗n∗]=sin(0∗n∗)fromtheCTFTof∗x∗c(∗t∗)=sin(ω0∗t∗).−∗∗9.5−2∗∗ACTsignal∗x∗(∗t∗),bandlimitedto25kHz,issampledat50kHztoproduce
x[n] = \delta[n+4] - 2\delta[n+2] + \delta[n+1] - 3\delta[n] - \delta[n-1] - 2\delta[n-2] - \delta[n-4]
DeterminetheCTFT∗X∗(ω).−∗∗9.7−1∗∗Thisproblemusesamatrix−basedapproachtoinvestigatethecomputationoftheinverseDTFS.−(a)ImplementEq.(9.3),theinverseDTFS,usingamatrix−basedapproach.−(b)Comparetheexecutionspeedofthematrix−basedapproachtotheIFFT−basedapproachforinputvectorsofsizes10,100,and1000.∗∗FigureP9.4−13∗∗−(c)WhatistheresultofmultiplyingtheDFTmatrix∗∗W∗∗∗N∗<sup>0</sup>bytheinverseDTFSmatrix?Discussyourresult.−∗∗9.7−2∗∗Astable,first−orderhighpassIIRdigitalfilterhastransferfunction
H[z] = \left(\frac{1+\alpha}{2}\right) \left(\frac{1-z^{-1}}{1-\alpha z^{-1}}\right)
−(a)Deriveanexpressionrelatingαtothe3dBcutofffrequency∗c∗.−(b)Testyourexpressionfrompart(a)inthefollowingmanner.First,computeαtoachievea3dBcutofffrequencyof1kHz,assumingasamplingrateof∗F<sup>s</sup>∗=5kHz.Determineadifferenceequationdescriptionofthesystem,andverifythatthesystemisstable.Next,computeandplotthemagnituderesponseoftheresultingfilter.Verifythatthefilterishighpassandhasthecorrectcutofffrequency.−(c)Holdingαconstant,whathappenstothecutofffrequency∗<sup>c</sup>∗as∗F<sup>s</sup>∗isincreasedto50kHz?Whathappenstothecutofffrequency∗fc∗as∗F<sup>s</sup>∗isincreasedto50kHz?−(d)Isthereawell−behavedinversefilterto∗H∗[∗z∗]?Explain.−(e)Determineαfor∗<sup>c</sup>∗=π/2.Commentontheresultingfilter,particularly∗h∗[∗n∗].−∗∗9.7−3∗∗Usingthefrequencysamplingmethod,designalength−35linearphaseFIRhighstopfilterthathascutofffrequency<sup>c</sup>=2π/3.Plottheresultingfilter′simpulseresponse∗h∗[∗n∗]andmagnituderesponse∣∗H∗()∣.−∗∗9.7−4∗∗Usingthefrequency−samplingmethod,designalength−71linearphaseFIRbandstopfilterthathasstopband(π/3<∣∣<π/2).Plottheresultingfilter′simpulseresponse∗h∗[∗n∗]andmagnituderesponse∣∗H∗()∣.−∗∗9.7−5∗∗FigureP9.7−5providesthedesiredmagnituderesponse∣∗H∗()∣ofarealfilter.Mathematically,
|H(\Omega)| = \begin{cases} 2\frac{4\Omega}{\pi} & 0 \leq \Omega < \frac{\pi}{4} \ 2 - \frac{4\Omega}{\pi} & \frac{\pi}{4} \leq \Omega < \frac{\pi}{2} \ 0 & \frac{\pi}{2} \leq \Omega \leq \pi \end{cases}
Sincethedigitalfilterisreal,∣∗H∗()∣=∣∗H∗(−)∣and∣∗H∗()∣=∣∗H∗(+2π)∣forall.−(a)Canarealizablefilterhavethisexactmagnituderesponse?Explainyouranswer.−(b)Usethefrequency−samplingmethodtodesignanFIRfilterwiththismagnituderesponse(orareasonableapproximation).UseMATLABtoplotthemagnituderesponseofyourfilter.−∗∗9.7−6∗∗ArealFIRcombfilterisneededthathasmagnituderesponse∣∗H∗()∣=[0,3,0,3,0,3,0,3]for=[0,π/4,π/2,3π/4,π,5π/4,3π/2,7π/4],respectively.Providetheimpulseresponse∗h∗[∗n∗]ofafilterthataccomplishesthesespecifications.−∗∗9.7−7∗∗Apermutationmatrix∗∗P∗∗hasasingleoneineachrowandcolumnwiththeremainingelementsallzero.Permutationmatricesareusefulforreorderingtheelementsofavector;theoperation∗∗Px∗∗reorderstheelementsofacolumnvector∗∗x∗∗basedontheformof∗∗P∗∗.−(a)Fullydescribean∗N∗<sup>0</sup>×∗N∗<sup>0</sup>permutationmatrixnamed∗∗R∗∗∗N∗<sup>0</sup>thatreversestheorderoftheelementsofacolumnvector∗∗x∗∗.−(b)GivenDFTmatrix∗∗W∗∗∗N∗<sup>0</sup>,verifythat(∗∗W∗∗∗N∗<sup>0</sup>)(∗∗W∗∗∗N∗<sup>0</sup>)<sup>=</sup>∗∗<sup>W</sup>∗∗<sup>2</sup>∗<sup>N</sup>∗<sup>0</sup>producesascaledpermutationmatrix.Howdoes∗∗W∗∗<sup>2</sup>∗N∗0∗∗x∗∗reordertheelementsof∗∗x∗∗?−(c)Whatistheresultof(∗∗W∗∗<sup>2</sup>∗N∗0)(∗∗W∗∗<sup>2</sup>∗N∗0)∗∗x∗∗=∗∗W∗∗<sup>4</sup>∗N∗0∗∗x∗∗? 