3.3DifferenceEqua/onsforDiscrete-TimeSystems
ADiscrete-/mesystemcanbemodeledwithdifferenceequa3ons
involvingcurrent,past,orfuturesamplesofinputandoutputsignals
Example3.3Moving-averagefilter
Alength-Nmovingaveragefilterisasimplesystemthatproduces
anoutputequaltothearithme=caverageofthemostrecentN
samplesoftheinputsignal.
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.3Moving-averagefilter
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.3Moving-averagefilter
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.4Length-2moving-averagefilter
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.4Length-2moving-averagefilter
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.6Exponen=alsmoother
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.6Exponen=alsmoother
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.7Loanpayments
Expresstheactoftakingabankloananpayingitbackover=measa
discrete-=mesystemproblemwithmonthlypaymentsrepresen=ngthe
Inputsignalandmonthlybalancerepresen=ngtheoutputsignal.
y[n]=(1+c)y[n-1]+x[n]
“c”isthemonthlyinterestrate
3.3DifferenceEqua/onsforDiscrete-TimeSystems
Example3.7Loanpayments
Expresstheactoftakingabankloananpayingitbackover=measa
discrete-=mesystemproblemwithmonthlypaymentsrepresen=ngthe
Inputsignalandmonthlybalancerepresen=ngtheoutputsignal.
y[n]=(1+c)y[n-1]+x[n]
“c”isthemonthlyinterestrate
Wecanfindoutthemonthlypaymentx[n]bysolvingthedifferenceequa=on
3.4Constant-CoefficientLinearDifferenceEqua/ons
Constant-coefficientdifferenceequa3on
N
M
∑ ak y[n − k] = ∑ bk x[n − k]
k=0
k=0
Ini/alcondi/ons:
y[n0-1],y[n0-2],….y[n0-N]
Itistypical,butnotrequired,tohaven0=0.
3.4Constant-CoefficientLinearDifferenceEqua/ons
Itera3velysolvingdifferenceequa3ons.
Considerthedifferenceequa=onfortheexponen=alsmootherofEx.3.6
y[n]=(1-α)y[n-1]+αx[n]
²  Giventheini=alvaluey[-1]oftheoutputsignal,y[0]isfoundby
y[0]=(1-α)y[-1]+αx[0]
²  Knowingy[0],thenextoutputsampley[1]isfoundby
y[1]=(1-α)y[0]+αx[1]
²  Knowingy[1],thenextoutputsampley[2]isfoundby
y[2]=(1-α)y[1]+αx[2]
3.5SolvingDifferenceEqua/ons
N
M
∑ ak y[n − k] = ∑ bk x[n − k]
k=0
k=0
Ini=alcondi=ons:
y[n0 − N ]
y[n0 −1], y[n0 − 2], ,…..,
Generalsolu=on:
y[n] = yh [n]+ y p [n]
²  yn([n]:isthehomogeneoussolu=onofthedifferen=alequa=on
naturalresponse
²  yp[n]:isthepar3cularsolu3onofthedifferen=alequa=on
²  y[n]=yh[n]+yp[n]istheforcedsolu3onofthedifferen=alequa=on
forcedresponse
3.5SolvingDifferenceEqua/ons
Example3.11Naturalresponseofexponen=alsmoother
Determinethenaturalresponseoftheexponen=alsmootherdefined
as:
y[n] = (1− α ) y[n −1]+ α x[n]
Ify[-1]=2
3.5.1Findingthenaturalresponseofadiscrete/mesystem
Generalhomogeneousdifferen=alequa=on:
N
∑ ak y[n − k] = 0
k=0
Characteris/cequa/on:
N
∑ ak z −k = 0
k=0
Toobtainthecharacteris=cequa=on,subs=tute:
y[n − k] → z −k
3.5.1Findingthenaturalresponseofadiscrete/mesystem
Writethecharacteris=cequa=oninopenform:
a0 + a1z −1 + .... + a N −1z −N +1 + a N z −N = 0
Mul=plybothsidesbyzNtoobtain
a0 z N + a1z N −1 + .... + a N −1z1 + a N = 0
Infactoredform:
a0 (a − z1 )(a − z2 )...(a − z N ) = 0
Homogeneoussolu=on(assumingrootsaredis=nct):
N
n
yh [n] = c1z1n + c2 z2n + ... + cN z N
= ∑ ck zkn
k=1
Whereunknowncoefficientsc1,c2,….,cNaredeterminedbythe
ini=alcondi=on,thetermsarecalledthemodesofthesystem
zin
3.5.1Findingthenaturalresponseofadiscrete/mesystem
Example3.12Naturalresponseofsecond-ordersystem
Asecond-ordersystemisdescribedbythedifferenceequa=on:
5
1
y[n]− y[n −1]+ y[n − 2] = 0
6
6
Determinethenaturalresponseofthissystemforn>=0subjectto
ini=alcondi=on:
y[-1]=19,andy[-2]=53
3.5.2Findingtheforcedresponseofadiscrete/mesystem
Choosingapar3cularsolu3onforvariousdiscrete-=meinputsignals
Thecoefficientofthepar3cularsolu3onaredeterminedfromthe
Differenceequa=onbyassumingallini=alcondi=onsareequaltozero.
3.5.2Findingtheforcedresponseofadiscrete/mesystem
Writethehomogeneousdifferenceequa=on
Solvehomogeneousdifference
equa=onwithundeterminedcoefficient
Findtheformofthepar=cularsolu=on
Findthecoefficientsofthepar=cularsolu=on
Addthehomogeneousandpar=cularsolu=on
Togethertoobtainthetotalsolu=on
3.5.2Findingtheforcedresponseofadiscrete/mesystem
Example3.14Forcedresponseofexponen=alsmoother
Determinetheforcedresponseoftheexponen=alsmootherdefined
as:
y[n] = (1− α ) y[n −1]+ α x[n]
Theinputsignalisaunit-stepfunc=on,andy[-1]=2.5