LATHAMATHAVAN ENGINEERING COLLEGE
MADURAI
DEPARTMENTOFECE
2MARKSQUESTION-ANSWERS
SignalsandSystems
UNITI
2MarkQuestionsandAnswers
1.DefineSignal.
Signal is a physical quantitythat varieswith respect to time , space oranyother independent variable.
Or
It is a mathematical representation of thesystem
Egy(t) =t. andx(t)=sin t.
2.Definesystem.
Aset of components that are connected together to performthe particular task.
3.Whatarethemajorclassificationsofthesignal?(i)Discrete time signal
(ii)Continuoustime signal
4.Definediscretetimesignalsandclassifythem.
Discrete time signals aredefined onlyat discrete times, and forthese signals, the independentvariabletakes on onlya discreteset ofvalues. Classification of discretetime signal:
1.Periodic and Aperiodicsignal 2.Even and Odd signal
5.Definecontinuoustimesignalsandclassifythem.
Continuoustime signals aredefined for a continuous of values of the independent variable.In the caseof continuous time signals the independent variable is continuous.
For example:
(i)Aspeech signalas afunction of time
(ii)Atmosphericpressure asa function ofaltitude Classification ofcontinuous time signal:
(i)Periodic and Aperiodic signal (ii)Even and Odd signal
6.Definediscretetimeunitstepunitimpulse.Discrete time Unit impulse is defined as
δ[n]={0,n≠0 {1, n=0
Unit impulse isalso knownas unitsample.
Discrete time unit step signal is defined by U[n]={0,n=0
{1,n>=0
7.Definecontinuoustimeunitstepandunitimpulse. Continuoustime unitimpulse is defined as
δ(t)={1, t=0 {0, t≠0
Continuoustime Unit step signal is definedasU(t)={0, t<0
{1, t≥0
8.Defineunitrampsignal.
Continuoustime unit ramp function is defined by r(t)={0,t0
{t, t≥0
Aramp signal starts at t=0 and increases linearlywith time ‘t’.
9.Defineperiodicsignal.andnonperiodicsignal.
Asignal issaid to be periodic ,if it exhibits periodicity.i.e., X(t +T)=x(t), for all values of t.
Periodic signal has the propertythat it is unchanged byatime shift ofT.
Asignal that doesnot satisfytheaboveperiodicitypropertyis calledan aperiodic signal.
10.Defineevenandoddsignal?
A discrete time signal issaid to be even when, x[-n]=x[n].
The continuoustime signalissaid to be even when, x(-t)=x(t)
For example,Cosωn is an even signal.
Thediscretetime signalissaid to be odd when x[-n]= -x[n]
The continuoustime signalissaid to be odd when x(-t)=-x(t)
Odd signalsare also known as nonsymmetrical signal. Sine wavesignal isan odd signal.
11.DefineEnergyandpowersignal.
Asignal issaid to be energysignal if it havefinite energyand zeropower.Asignal issaid to be powersignal if it have infinite energy and finite power.
If the above two conditions arenot satisfied then the signal issaid tobe neigtherenergynorpowersignal
12.Defineunitpulsefunction.
Unit pulse function ∏(t)is obtained from unit step signals∏(t)=u(t+1/2)-u(t-1/2)
Thesignals u(t+1/2)andu(t-1/2)are the unit step signalsshifted by1/2units in the time axistowards the left and right ,respectively.
13.Definecontinuoustimecomplexexponentialsignal.
The continuoustime complexexponential signal isof theform x(t)=Ceat
wherec and a are complexnumbers.
14.Whatiscontinuoustimerealexponentialsignal. Continuoustime real exponential signal is definedby
x(t)=Ceat
wherec and a are complexnumbers.Ifcand a arereal ,then it is called asreal exponential.
15.Whatiscontinuoustimegrowingexponentialsignal? Continuoustime growing exponential signal is defined as
x(t)=Ceat
wherec and a are complexnumbers.
Ifa is positive,as t increases , then x(t) is a growing exponential.
16.Whatiscontinuoustimedecayingexponential? Continuoustime growing exponential signal is defined as
x(t)=Ceat
wherec and a are complexnumbers.
Ifa is negative,as t increases, then x(t) is a decayingexponential.
17.WhatarethetypesofFourierseries?
1.Exponential Fourierseries2.Trigonometric Fourierseries
18.Writedowntheexponentialformofthefourierseriesrepresentationofa
periodicsignal?x(t)=∑akejkωot
Herethe summation is taken from-∞to ∞.
ak=1/T ∫x(t)e-jkωot
Herethe integration istakenfrom 0 to T.
Thesetofcoefficients { ak} areoftencalled thefourierseriescoefficientsorspectralcoefficients.
The coefficient aois thedc or constant component ofx(t).
19.Writedownthetrigonometricformofthefourierseriesrepresentationofaperiodicsignal?
x(t)= ao+∑[ancos nωot + bnsin nωot ]
where
ao=1/T ∫x(t) dt
an=1/T ∫x(t)cos nωotdt
bn=1/T ∫x(t)cos nωot dt
20.Writeshortnotesondirichletsconditionsforfourierseries. a.x(t) mustbe absolutelyintegrable
b.The function x(t)should be single valued within the interval T.
c.The function x(t)should havefinite numberof discontinuities in any finite interval oftime T.
d.The function x(t)should havefinite numberof maxima minima in the interval T.
21.StateTimeShiftingpropertyinrelationtofourierseries.x(t-to)FSake-jkωot
Time shiftingpropertystates that; when aperiodicsignal isshifted in time, themagnitudes ofitsfourierseriescoefficients, remain unaltered.
22.Stateparseval’stheoremforcontinuoustimeperiodicsignals.Parseval’srelation forcontinuoustime periodic signals is
1/T ∫x(t)2 dt =∑ak2
Parseval’srelation states that the total average powerin a periodicsignal equals the sum of the averagepower in all of its harmonic components.
23.Definecontinuoustimesystem.
Acontinuoustime system is a system in which continuoustime input signals are applied and result in continuoustime output signals.
24.Definefouriertransformpair.
Considerthe aperiodic signal x(t)Fourier transform ofx(t) is defined as
X(jω) = ∫x(t)e-jωtdt------(1)
Inversefouriertransformofx(t) isgiven by
x(t)=1/2π∫X(jω)ejωtdω------(2)
Equations(1)(2)arereferred to as the fouriertransform pair.
25.Writeshortnotesondirichletsconditionsforfouriertransform. a.x(t) be absolutelyintegrable
b.x(t) haveafinite numberof maxima and minimawithin anyfinite interval. c.x(t) haveafinite number of discontinuitieswithin anyfinite interval.
Furthermoreeach ofthesediscontinuities mustbe finite.
26.Explainhowaperiodicsignalscanberepresentedbyfouriertransform.
Considerthe aperiodic signal x(t)Fourier transform ofx(t) is defined as
X(jω) = ∫x(t)e-jωtdt------(1)
Inversefouriertransformofx(t) isgiven by
x(t)=1/2Π∫X(jω)ejωtdω------(2)
27.Stateconvolutionpropertyinrelationtofouriertransform.Y(t)=x(t)*h(t)
Y(jω)= H(jω)X(jω)
Convolution propertystates thatconvolution in timedomain corresponds to multiplication inthefrequencydomain.
28.Stateparseval’srelationforcontinuoustimefouriertransform.Ifx(t) and X(jω)areafourier transform pair then
∫x(t)2 dt = 1/2 πi ∫|X(j ω)|2 dω
29.whatistheuseoflaplacetransform?
Laplacetransform is ananother mathematical toolused for analysis ofsignals and systems.Laplacetransform is used foranalysisof unstable systems.
30.Whatarethetypesoflaplacetransform? 1.Bilateralor twosided laplacetransform. 2.Unilateral orsingle sided laplace transform.
.
31.DefineBilateralandunilaterallaplacetransform.
Thebilateral laplacetransform is defined as
X(s)=∫x(t)e-stdt
Herethe integration istaken from -∞to ∞.Henceitis called bilateral laplace transform
Theunilateral laplacetransform is defined as
X(s)=∫x(t)e-stdt
Herethe integration istakenfrom 0 to ∞.Henceitis called unilateral laplace transform.
32.Defineinverselaplacetransform.
Theinverselaplacetransform isgivenas
x(t)=1/2πj ∫X(s)estds
Herethe integration istakenfrom σ-j∞to σ+j∞.
33.Statethelinearitypropertyforlaplacetransform.
Let x1(t)X1(S) andx2(t)X2(s)bethetwo laplacetransform pairs.Then linearitypropertystates that
L[a1x1(t)+a2x2(t)]=a1X1(s)+a2X2(s)Herea1 and a2areconstants.
34.Statethetimeshiftingpropertyforlaplacetransform.
Let x(t)X(S)be alaplacetransform pair.Ifx(t) is delayed by time t0 ,then itslaplacetransform ismultiplied by e-st0.
L[x(t-t0)=e-st0X(s)
35.Regionofconvergenceofthelaplacetransform.
Therange ofvaluesofsforwhich the integral i.e.,∫x(t)e-stdt converges isreferred toasthe region ofconvergenceof thelaplacetransform.
36.Whatispolezeroplot.
Therepresentation ofX(s) through its polesand zeros in the splane isreferred to as pole zeroplot.
37.Stateinitialvaluetheoremandfinalvaluetheoremforlaplacetransform.
IfL[x(t)]=X(s), then initial valuetheorem states that x(0)=lim(s---∞)SX(S)
IfL[x(t)]=X(s), then final value theorem states that lim(t---∞)x(t)=lim(s---> 0)SX(S)
38.StateConvolutionproperty.
Thelaplace transform ofconvolution of two functionsis equivalent to multiplication oftheir laplace transforms.
L[x1(t)*x2(t)]=X1(s)X2(s)
39.Defineacausalsystem.
The causal systemgenerates the output dependingupon present&past inputsonly.Acausal system isnon anticipatory.
40.Whatismeantbylinearsystem?
A linearsystem should satisfysuperposition principle. Alinearsystem should satisfyF[ax1(t)+bx2(t)]→ay1(t)+by2(t)
y1(t)=F[x1(t)]y2(t)=F[x2(t)]
41.Definetimeinvariantsystem.
Asystem is time invariant if the behaviorandcharacteristics of thesystem arefixed overtime.
Asystem is time invariant if atime shift in the input signal results in an identical time shift in the output signal.
For example ,atime invariant system should producey(t-t0)as the output when x(t-to) is the input.
42.Definestablesystem?
When thesystem produces bounded output forbounded input, then the system is called bounded input&bounded output stable.
If the signal isbounded, then itsmagnitude willalways be finite.
43.Definememoryandmemorylesssystem.
Theoutput of a memorysystem at anyspecified time depends on the inputsat that specified time and at other times.Such systemshave memoryor energystorage elements.
Thesystem issaid to be static or memoryless if its output depends upon the present input only.
44.Defineinvertiblesystem.
Asystem issaid to be invertible if theinput isgetfrom the output input. Otherwise the system isnoninvertiblesystem.
45.Whatissuperpositionproperty?
Ifan input consists of theweighted sum ofseveralsignals, then the output isthe superposition that is, the weighted sum of theresponsesof the system to each ofthose signals
46.WhyCTsignalsarerepresentedbysamples.
A CT signal can not beprocessed in the digital processor orcomputer. To enable thedigital transmission of CT signals.
47.Whatismeantbysampling.
Asamplingis a processbywhich aCTsignal is converted into a sequence of discretesampleswith each samplerepresentingthe amplitudeof the signal
at the particularinstant of time. 48.StateSamplingtheorem.
A band limited signal offinite energy,which has no frequency components higher than theWhertz, is completelydescribed byspecifyingthe values of thesignal at theinstant of time separated by1/2Wsecondsand
A band limited signal offinite energy,which has no frequency components higher than theWhertz, is completelyrecovered fromthe knowledgeofitssamplestaken at the rate of2Wsamples persecond.
49.Whatismeantbyaliasing.
When thehighfrequencyinterfereswith lowfrequencyandappearsas lowthen the phenomenon is called aliasing.
50.Whataretheeffectsaliasing.
Sincethe highfrequencyinterfereswithlowfrequencythen the distortion isgenerated.
Thedata is lostand it can not be recovered. 51.Howthealiasingprocessiseliminated.
i). Sampling ratefs≥2W.
ii). Strictlyband limitthe signal to ‘W’.
This canbe obtained byusingtheLowpassfilerbefore the samplingprocess.It is also called as antialiasing filter.
52..DefineNyquistrate.andNyquistinterval.
When thesamplingrate becomesexactlyequal to ‘2W’samples/sec, foragivenbandwidth of Whertz, then it is called Nyquist rate.
Nyquistinterval is the time interval betweenanytwo adjacent samples. Nyquistrate=2WHz
Nyquistinterval =1/2Wseconds. 53.Definesamplingofbandpasssignals.
A bandpasssignal x(t)whosemaximum bandwidth is‘2W’can be completelyrepresented into and recoveredfrom itssamples, if it issampled at the minimumrate oftwicetheband width.
54.DefineZtransform.
TheZtransform of adiscrete time signalx[n]isdenoted byX(z)and it isgivenasX(z)=∑x[n]z-n.and the valuen range from-∞to +∞. Here ‘z’ isthe complexvariable.This Z transform is also calledas bilateral or two sidedZ transform.
55.WhatarethetwotypesofZtransform?(i)UnilateralZtransform
(ii)BilateralZtransform
56.DefineunilateralZtransform.
TheunilateralZtransform ofsignal x[n] isgiven asX(z)=∑x[n]z-n
Theunilateral andbilateral Ztransformsare sameforcausal signals.
57.WhatisregionofConvergence.
Theregion ofconvergenceorROC isspecifiedfor Ztransform ,where it converges.
58.WhatarethePropertiesofROC.
i.The ROC of afinite duration sequenceincludes the entirez- plane, exceptz= 0 and|z|=∞.
ii.ROC does not contain anypoles.
iii.ROC isthe ringin the z-plane centeredabout origin.
iv.ROC of causal sequence(right handed sequence)is of the form|z|r. v.ROC of left handed sequenceis of theform|z|r.
vi.ROC of two sided sequenceis the concentricring in the zplane.
59.WhatisthetimeshiftingpropertyofZtransform.
x[n]X(Z)then
x[n-k]Z-kX[Z].
60.WhatisthedifferentiationpropertyinZdomain. x[n]X(Z)then
nx[n]-z d/dz{X[Z].}.
61.StateconvolutionpropertyofZtransform. The convolution propertystates that if
x1[n]X1(Z)and
x2[n]X2(Z)then
x1[n]*x2[n]X1(Z)X2(Z)
That isconvolution of two sequences in time domain is equivalent to multiplication of their Ztransforms.
62.StatethemethodstofindinverseZtransform. a.Partial fraction expansion
b.Contour integration
c.Powerseriesexpansion d.Convolution method.
63.StatemultiplicationpropertyinrelationtoZtransform. This propertystates that if,
x1[n]X1(Z)and
x2[n]
x1[n]x2[n]
X2(Z)then
1/2Πj ∫X1(v). X2(Z/v)v-1dv
Herec is a closedcontour.Itencloses the originand lies in the Roc which is common to both X1(v). X2(1/v)
64.Stateparseval’srelationforZtransform.
Ifx1[n]andx2[n] are complexvalued sequences,then the parseval’srelation states that
∑x1[n]x2*[n]= 1/2Πj ∫X1(v). X2*(1/v*)v-1dv.
65.WhatistherelationshipbetweenZtransformandfouriertransform.X(z)=∑x[n] z-n.------1.
X(w)=∑x[n]e-jωn. ------2
X(z)at z= ejωis = X(w).
When z- transform is evaluated on unitcircle(ie.|z|= 1) then it becomesfouriertransform.
66.WhatismeantbystepresponseoftheDTsystem.
Theoutput of the systemy(n) isobtained fortheunitstep input u(n) then it issaid to be step response ofthe system.
67.DefineTransferfunctionoftheDTsystem.
TheTransferfunction ofDT system is defined asthe ratio ofZtransform of thesystem outputto theinput.
That is,H(z)=Y(z)/X(z),
68.DefineimpulseresponseofaDTsystem.
Theimpulse response isthe output produced byDT system when unit impulse is applied at the input.The impulse responseis denoted byh(n).
Theimpulse response h(n) is obtained bytakinginverseZtransform from thetransferfunction H(z).
69..Statethesignificanceofdifferenceequations.
Theinput and output behaviour ofthe DT systemcanbecharacterized with the help oflinear constant coefficient differenceequations.
70.WritethediffereceequationforDiscretetimesystem.
Thegeneral form ofconstant coefficient difference equation isY(n) =-Σaky(n-k)+Σbkx(n-k)
Heren isthe orderof difference equation.x(n) is theinput andy(n) isthe output.
71.DefinefrequencyresponseoftheDTsystem.
Thefrequencyresponseof the system is obtained from the Transferfunction byreplacingz=ejω
Ie,H(z)=Y(z)/X(z), Wherez= ejω72.Whatistheconditionforstablesystem.
ALTIsystem isstableifΣ│h(n)│∞.
Herethe summation is absolutelysummable
73.Whataretheblocksusedforblockdiagramrepresentation.
Theblock diagrams areimplemented with the help ofscalarmultipliers,adders and multipliers
74.Statethesignificanceofblockdiagramrepresentation.
TheLTIsystems are represented with the help ofblock diagrams. The block diagrams are moreeffective wayofsystem description. Block Diagrams indicatehow individual calculations areperformed. Various blocks are usedforblock diagram representation.
75.Whatarethepropertiesofconvolution? i.Commutative
ii.Assosiative. iii.Distributive
76.StatetheCommutativepropertiesofconvolution?
Commutative propertyof Convolution isx(t)*h(t)=h(t)*x(t)
77.StatetheAssociativepropertiesofconvolutionAssociative Propertyofconvolution is[x(t)*h1(t)]*h2(t)=x(t)*[h1(t)*h2(t)]
78..StateDistributivepropertiesofconvolution TheDistributive Propertyofconvolution is
{x(t)*[h1(t)+h2(t)]}=x(t)*h1(t) +x(t)*h2(t)]79.Definecausalsystem.
ForaLTIsystem to be causal if h(n)=0,forn<0. 80.Whatistheimpulseresponseofthesystemy(t)=x(t-t0).
Answer: h(t)=δ(t-t0)
81.WhatistheconditionforcausalityifH(z)isgiven.
AdiscreteLTIsystem withrational system function H(z) is causalifand onlyif i.TheROC is the exteriorof the circle outside the outermost pole.
ii. When H(z) is expressed as aratio of polynomialsin z, the order of the numeratorcan not begreater than the order ofthe denominator.
82.WhatistheconditionforstabilityifH(z)isgiven.
AdiscreteLTIsystem with rational system function H(z) isstableifand only ifall of the polesH(z) lies inside the unit circle. That istheymust all have magnitude smaller than 1.
83.Checkwhetherthesystemiscausalornot,theH(z)isgivenby(z3+z)/(z+1). Thesystem is notcausal because theorder ofthenumeratorisgreaterthan
denominator.
84.Checkwhetherthesystemisstableornot,theH(z)isgivenby(z/z-a).,|a|<1. Thesystem isstable because thepolesat z= alies inside theunit circle.
85.Determinethetransferfunctionforthesystemdescribedbythedifferenceequationy(n)-y(n-1)=x(n)-x(n-2).
Bytakingz transformon both sides the transferfunction H(z)=(z2- 1)/(z2 -z).
86.Howthediscretetimesystemisrepresented.
TheDT system isrepresented eitherBlock diagramrepresentation ordifferenceequation representation.
87.Whatartetheclassificationofthesystembasedonunitsampleresponse. a.FIR (Finite impulse Response)system.
b.III(InfiniteImpulse Response)system.
88.WhatismeantbyFIRsystem.
If the system havefinite durationimpulse response then the system issaid to be FIR system.
89.WhatismeantbyIIRsystem.
If the system haveinfinite durationimpulse response then thesystem issaid to be FIR system.
90.Whatisrecurssivesystem.
If the present output isdependent upon the presentand past valueof input then thesystem issaid to be recursive system
91.WhatisNonrecursivesystem.
If the present output isdependent upon the presentand past valueofinput and past value ofoutput then the system issaid to be nonrecursssive system.
92.Whatistheblockdiagramrepresentationofrecursivesystem.
X(n)
Non recursivesystem. F[x(n),x(n-1)------x(n-m)]93.Whatistheblockdiagramrepresentationofnonrecurssivesystem.
x(n)
\
94.Whatisthedifferencebetweenrecursiveandnonrecursivesystem
Arecursive system havethe feed backand thenon recursive system have no feed back .And also the need of memoryrequirement forthe recursive system isless than non recursive system.
95.Definerealizationstructure.
Theblock diagram representation of adifferenceequation is called realization structure.These diagram indicate themanner in which the computations areperformed.
96.Whatarethedifferenttypesofstructurerealization.
i.Direct formI
ii. Direcct formII iii. Cascadeform iv. Parallel Form.
97.Whatisnaturalresponse?
This isoutputproduced bythesystem onlyduetoinitial conditions .Input iszero for natural response.Hence it is also calledzero input Response.
98.WhatiszeroinputResponse?
This isoutputproduced bythesystem onlyduetoinitial conditions .Input iszero forzero input response.
99.Whatisforcedresponse.
This isthe output produced bythe system onlydue to input .Initial conditions are consideredzero forforced response.It isdenoted by y(f)(t).
100.Whatiscompleteresponse?
The complete responseof the system is equal to the sum of natural response andforced response .Thus initial conditionsaswell as input both are considered for completeresponse.
101.GivethedirectformIstructure.
X(n)
z-1102.GivethedirectformIIstructure..
x(n) y(n) + +
++
103.HowtheCascaderealizationstructureobtained..
Thegiven transferfunction H(z) isspilited into two or moresub systems. That isforeg.
H(z) =H1(z)H2(z)
X(n)
H1(z) / H2(z)y(n)
104.GivetheparallelforRealizationstructure.
H(z) =c+H1(z) +H2(z)+------+Hk(z)
+
+
X(n)
H1(z)Y(n) +
105.Whatistransformedstructurerepresentation.
Theflow graph reversal theoremstates that if the directions of all branchesare reversedand positions of inputand output isinterchanged, thesystem functionremain unchanged. Such structureiscalled transposed structure
16MARKQUESTIONS
1.Findthetrigonometricfourierseriesrepresentationofaperiodicsignalx(t)=t,fortheintervaloft=-1tot=1?
Thetrigonometricfourierseriesrepresentation of aperiodic signal isx(t)= ao+∑[ancos nωot + bnsin nωot ]
where
ao=1/T ∫x(t) dt =0
an=1/T ∫x(t)cos nωotdt =0
bn=1/T ∫x(t)cos nωot dt ==2/π[-(-1)n/n]
x(t)=∑[2/π[-(-1)n/ n]]sin nπt. Summation variesn =1 to∞..
2.Findtheexponentialfourierseriesforhalfwaverectifiedsinewave.Answer:
ak=A/л(n2-1)
3.Findtheenergyandpowerofthesignal.
i.X(t) =r(t)-r(t-2).
Answer:E =Lt∫x(t)2 dt= ∞.. T →∞.
P=Lt1/2T∫x(t)2 dt= 2W.. T →∞.
ii.x(n)=(1/3)nu(n).
E=Infinite and power isfinite.So it is powersignal
Answer:E =Lt
N→∞.
∑Nx(n)2=9/8.. n = -N
P=Lt1/(2N+1)∫x(t)2 dt= 0.. N→∞.
E=Finite and power iszero. So it is energysignal.
4.FindtheDTFSofx(n)=5+sin(nπ/2)+cos(nπ/4).
x(n)=∑akejkωot
Herethe summation is taken from-∞to ∞.
ak=1/T ∫x(t)e-jkωot
5.FindthefourierseriesrepresentationforFullwaverectifiedsinewave.Answer:
X(t)=2/ л+4/ л∑(1/1-4n2)Cosnωot
6.Consideracontinuoustimesystemwithimpulseresponseh(t)=e-atu(t)totheinputsignalx(t)=e-btu(t).Findthesystemresponse.
Answer:
y(t)=1/(b-a)[e-atu(t)-e-btu(t)]
7.Findthelaplacetransformofx(t)=δ(t)-4/3e-tu(t)+1/3e2tu(t).Answer:
X(s)=(s-1)2/((s+1)(s-2))
8.FindtheinverslaplacetransformofX(s)=1/((s+1)(s+2)).Answer:
x(t)= e-tu(t)-e-2tu(t)
9.Stateandproveparseval’stheoremforFouriertransform.Answer:
Ifx(t) and X(jω)areafourier transform pair then
∫x(t)2 dt = 1/2 Π∫|X(jω)|2 dω
10.DeterminetheFouriertransformofthesignalx(t)=e-atu(t).,a>0,plottheMagnitudeandPhaseSpectrum.
11.DeterminetheSystemreponseofthegivendifferentialequationy’’(t)+3y’(t)=x(t).,Wherex(t)=e-2tu(t).
12.i.DeterminetheZtransformoffollowingfunctions
x[n]=(-1)n2-nu(n)Answer:
X(z)=1/(1+1/2Z-1)
ii.FindtheZtransformofthefollowinganddetermineROCx[n]={8,3,-2,0,4,6}
13.i.DeterminetheZtransformoffollowingfunctionsx[n]=(-1)n2-nu(n)
Answer:
X(z)=1/(1+1/2Z-1)
ii. StateandprovetheTimeshiftingConvolutionpropertiesofZtransform.
x[n]X(Z)then
Proof:
x[n-k]Z-kX[Z].
convolutionpropertyofZtransform.
The convolution propertystates that ifx1[n]X1(Z)and
x2[n]X2(Z)then
Proof:
x1[n]*x2[n]X1(Z)X2(Z)
That isconvolution of two sequences in time domain is equivalent to multiplication of their Ztransforms.
14.(i).DeterminetheZTransformandPlottheROCforthesequencex[n]=anu[n]-bnu[n],.b>a
Answers:X(Z)=Z/(Z-a)–Z/(Z-b).,|z|b.
(ii)ComputetheinversZtransformofX(Z)=(z+0.5)/(z+0.6)(z+0.8).
|z|0.8,usingresiduemethod.Answers.: -0.5(-0.6)n-1u(n-1)+1.5(-0.8)n-1u(n-1).
15.FindtheinversezTransformofthefunctionX(Z)=1/(1-1.5Z-1+0.5Z-2).Usingpowerseriesmethodfor|Z|1and|Z|1.
Answers: 1.For|Z|>1,x(n)={1,1.5,1.75,1.875 …………} 2 For|Z|1,x(n)={………….62,30,14,6,2,0,0}
16.FindtheinversezTransformofthefunctionX(Z)=Z/(Z–1)(Z-2)(Z-3).
UsingpartialfractionmethodforROC|Z|3,3>|Z|>2and|Z|<1.Answers: 1.For|Z|3 ,x(n)=0.5u(n)– 2nu(n)+0.5(3)nu(n).
2 . For3 >|Z|2, x(n)=0.5u(n)– 2nu(n)-0.5(3)nu(-n-1).
3. For|Z|1,x(n)=-0.5u(-n-1) +2nu(-n-1)–0.5(3)nu(-n-1).
17.Determinethediscretetimefouriertransformofx[n]=anu(n)for-1<a<1
Answer:
X(ejω)=1/(1-ae-jω)
18.Determinetheoutputofthediscretetimelineartimeinvariantsystemwhose
inputandunitsampleresponsearegivenasfollows. x[n]=(1/2)nu[n]
h[n]=(1/4)nu[n]Solution:
Theimpulse response ofthe discrete time linear time invariant system isY[n]=∑x[k]h[n-k];
19.Determinetheoutputofthediscretetimelineartimeinvariantsystemwhose
inputandunitsampleresponsearegivenasfollows. x[n]=bnu[n]
h[n]=anu[n]Solution:
Theimpulse response ofthe discrete time linear time invariant system isY[n]=∑x[k]h[n-k];
20.Thesequencesx[n]andh[n]aregivenasfollows. x[n]={1,1,0,1,1}
h[n]={1,-2,-3,4}
Compute the convolution of thesetwo sequences. Answer:
Y[n]={1,-1,-5,2,3,-5,1,4}
21.Writeshortnotesonpropertiesofconvolution. Explain each prppertywith mathematical definition.
1.Commutative property 2.Distributive property3.Associativeproperty
22.Determinesystemtransferfunctionandimpulseresponseofdiscretetimesystemdescribedbythedifferenceequation
y(n)-5/6y(n-1)+1/6y(n-2)=x(n)-1/2x(n-1).
Answers.
H(z)= 1/(1-1/3 z-1)And h(n)=(1/3)nu(n),|z|1/3.
23.ObtaintheDirectformIandDirectformIIrealizationforthesystem
describedbythedifferenceequationy(n)-5/6y(n-1)+y(n-2)=x(n)+x(n-1).Answers.Y(Z)-5/6Z-1Y(Z)+1/6Z-2Y(Z)=W(Z)
Y(Z) =W(Z)+ 5/6Z-1Y(Z)-1/6Z-2Y(Z)
Draw theblockdiagram representation fordirectIand DirecrIIform.
24.ObtaintheCascadeandparallelform.
realizationforthesystemdescribedbythedifferenceequationy(n)-1/4y(n-1)-1/8y(n-2)=x(n)+3x(n-1)+2x(n-2)
Answers.
H(Z)=Y(Z)/X(Z)= (1+3Z-1+2Z-2) /(1-1/4Z-1-1/8Z-2)H(Z)=H(z)H(z)
WhereH1(z)=(1+z-1)/(1-1/2z-1) andH2(z) = (1+2z-1)/(1+1/4z-1 )Draw theblockdiagram representation forCascadeform.
Forparallel form obtain the transferfunction in the form ofH(Z)=Y(Z)/X(Z)=16+ 22/(1+Z-1) – 37/(1+2Z-1)
Draw theblockdiagram representation forparallel form.
25.ObtaintheDirectformIandDirectformIIrealizationforthesystemdescribedbythedifferentialequationd2y(t)/dt2+5dy(t)/dt+4y(t)=dx(t)/dt.
Soution :The procedure issame as to DT system . Heretake the laplace transform and find the transferfunction H(S) =Y(S)/X(S).
Draw theblockdiagram representation fordirectIand DirecrIIform.Here replaceZ-1by1/S.
26.ObtaintheCascadeandparallelformrealizationforthesystemdescribedbythedifferentialequationd2y(t)/dt2+5dy(t)/dt+4y(t)=dx(t)/dt
Answers.
H(S)=Y(S)/X(S)H(S)=H1(s)H2(s).
Draw theblock diagramrepresentation forCascade form.And Parallel form.
27.ObtainthedirectformI,DirectformII,Cascadeandparallelrealizationofthesystemdescribedbythedifferenceequation
y(n)=0.75y(n-1)-0.125y(n-2)+x(n)+7x(n-1)+x(n-2).
Solution : Draw theFourForm realization structureusingthe aboveprocedure.