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8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
http://slidepdf.com/reader/full/pages-from-monson-hayes-schaum-s-outline-of-digital-signal-bookseeorg 1/7
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
http://slidepdf.com/reader/full/pages-from-monson-hayes-schaum-s-outline-of-digital-signal-bookseeorg 2/7
CHAP
11 SIGNALSANDSYSTEMS
1 42
The inputtoalinearshift-invariantsystemisperiodicwithperiod N .
(a) ShowthattheoutputofthesystemisalsoperiodicwithperiodN .
(b) Ifthesystemislinearbutshift-varying,istheoutputguaranteedtobeperiodic?
(c) If thesystemisnonlinearbutshift-invariant,istheoutputguaranteedto
be
periodic?
1 43 Ifx(n)
0 forn 0, andtheoddpartisx,,(n) n(0.5)1n1,indx(n)giventhatx(0)
1 .
1 44 Findtheconjugatesymmetricpartofthesequence
1 45
If x(n)isodd,whatisy(n)
x2(n)?
1 46
If x(n) 0 forn 0, e isthepowerintheevenpartofx(n),and oisthepowerintheoddpart,whichofthe
followingstatements re true?
(a) c? o
(b)
o e
(c) e o
d) Noneoftheabovearetrue.
1 47 Expressthesequence
( - 1
x(n)
(0
2
else
asasumofscaledandshiftedunitsteps.
1 48 Synthesizethetriangularpulse
as asumofscaledandshiftedpulses,
Discrete Time Systems
1 49 Listedbelow areseveralsystemsthatrelate theinputx(n )tothe output y(n ). Foreach,determine whether the
systemislinearornonlinear,shift-invariantor shift-varying,stableorunstable,causalornoncausal,andinvertible
ornoninvertihle.
(e) y(n) median(x(n l) ,x(n ),x(n I))
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
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SIGNALSANDSYSTEMS [CHAP. 1
Givenbelowaretheunitsampleresponsesofseverallinearshift-invariantsystems.F oreachsystem,determinethe
conditionsontheparameter
a
inorderforthesystemtobestable.
(a) h ( n ) a n u ( - n )
h)
h ( n ) a [ u ( n )
u (n 100 ) )
( c )
h ( n )
a nl
Isittruethatallmemorylesssystemsareshift-invariant?
Co nsid erthelinearshift-invarian tsystemdesc ribed by thefirst-orderlinearconstantc oefficientdifferenceequ ation
y ( n )
u y(n I x ( n )
Determinetheconditions(ifany)forwhichthissystemisstable.
Supposethattwosystems,
SI
and
SZ,
areconnectedin parallel.
( a )
f both S , and Szarelinear,shift-invariant, stable. andca usal,will theparallelconnection alwaysbe linear,
shift-invariant,stable.andcausal?
( h ) Ifboth
S,
andSzarenonlinear.willtheparallelconnectionnecessarilybenonlinear?
c)
If
bothS I and
S
areshift-varying,willtheparallelconnectionnecessarilybeshift-varying?
onvolution
Findtheconvolutionofthetwosequences
s ( n ) 6 ( n 2 )
2 6 (n 4 ) 3 6 ( n
6
h (n ) 2S (n
3 )
S ( n )
26(n
2 )
6(n
Theunitsampleresponseof alinearshift-invariantsystemis
h ( n )
36(n 3)
0 .5S (n 4 ) 0 .26 (n
5 )
0 . 7 6 ( n .
Findtheresponseofthissystemtotheinput
x ( n ) u ( n
I).
Alinearshift-invariantsystemhasaunitsampleresponse
h ( n )
u ( - n )
Findtheoutputiftheinputis
A- (n )
( i ) r c ( n )
Thestep responseofasystem isdefinedastheresponseofthesystemtoaunitstep
u ( n ) .
( a ) Lets ( n ) bethestepresponseofalinearshift-invariantsystem.Expresss ( n ) intermsoftheunitsampleresponse
h ( n ) ,
andfind
s ( n )
when
h ( n )
u n
u ( n
6 ) .
b)
Derivean expression forh ( n ) in terms of s ( n ) and find the unit sample response For a systemwhosestep
responseis
The unitsampleresponseofalinearshift-invariantsystemisshownbelow.
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
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CHAP. SIGNALSANDSYSTEMS
( a ) Findtheresponseofthesystemtotheinputu(n 4 ) .
( 6 ) Repeatforx ( n ) = ( - l ) u ( n ) .
Ifx ( n )
=
( i ) u ( n 2 ) andh ( n )
=
2 u( -n
5 ,
findtheconvolutiony ( n )
=
x ( n ) * h(n) .
Giventhreesequences,
h ( n ) , g ( n ) ,
and
r ( n ) ,
express
g(n)
intermsof
r ( n )
if
Let
h ( n ) = a n u ( n )
and
x ( n )
=
bnu(n) .
Findtheconvolution
y ( n ) = x ( n ) * h ( n )
assumingthat
a 6 .
If
x ( n ) = a n u ( n ) ,
findtheconvolution
y ( n ) = x ( n )
*
x ( n ) .
The inputto alinearshift-invariantsystemistheunitstep,
x ( n )
=
u ( n ) ,
andtheresponseis
y ( n )
=
S(n).
Findthe
unitsampleresponseofthissystem.
Ifh ( n ) = A 6 ( n )+ ) u (n ) istheunitsampleresponseofalinearshift-invariantsystem,ands ( n ) isthestepresponse
(theresponseofthesystemtoaunitstep),findthevalueoftheconstant A sothatlim,,, s ( n ) = 0.
Theunitsampleresponseofalinearshift-invariantsystemis
Findtheresponseofthesystemtothecomplexexponential
x ( n ) = e x p ( j n r r / 4 ) .
Evaluatetheconvolutionofthesequence
x ( n )
=
n( i ) cos ( r r n )
withtheunitstep,
h ( n ) = u ( n ) .
Let
n(0.5)
0 n
n c O
andh ( n ) = e j % u ( - n ) . Ify ( n )
=
x ( n ) * h ( n ) , whatisthenumericalvalueofy( -2)?
Given
andh ( n ) = S(n - 2 ) + S(n 3 )
+
6(n
4 ,
atwhatvalueofn willtheconvolutiony ( n ) = x ( n ) * h ( n ) attainits
maximumvalue,andwhatisthismaximumvalue?
Alinearsystemhasaresponseh k ( n )= S(2n k ) totheunitsample6(n k ) . Findtheresponseofthesystemto
theinput
x ( n ) = u(n) .
Considertheinterconnectionofthreelinearshift-invariantsystemsshowninthefigurebelow.
-
h A n )
I
+
x (n )
h ~ ( n )
+>
= ~ n )
+
-
-
h3(n)
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
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IGN LS ND SYSTEMS [CHAP.
If hl(n)= u(n 2),hn(n)= nu@)andh3(n)= 6(n 2).findtheunitsampleresponseoftheoverallsystem.
ifference Equations
1 71
Considerthelinearshift-invariantsystemdescribedbytheLCCDE
y(n)
=
- i y ( n I ) +2x(n)
Findtheresponseofthissystemtotheinput
x (n) =
2
n = 0 , 2 , 4 , 6 ...
otherwise
Hint: Writex(n)as(1 (-1)")u(n)anduselinearity.
1 72
Considerasystemwithinputx (n )andoutputy(n )thatsatisfiesthedifferenceequation
y(n)
=
ny(n
I)
x ( n )
Ifx(n)= 6(n),determiney(n)foralln.
1 73 linearshift-invariantsystemisdescribedbytheLCCDE
y ( n ) - 5 y ( n I )+ 6 y( n - 2 ) = x ( n 1)
Findthestepresponse ofthesystem(i.e.,theresponsetotheinputx(n )= u(n)).
1 74
Asystemischaracterizedbythedifferenceequation
Iftheinputisx(n)
=
2u(n)
3nu (n),findtheresponseofthesystemassuminginitialconditionsofy(-1) = 2and
y(-2) = 1.
1 75
Considerthesystemdescribedbythedifferenceequation
y(n) y(n
1 )
0.25y(n 2)
=
x(n)
0.25x(n
1)
a) Findtheunitsampleresponseofthesystem.
(b) Findtheresponseofthesystemtox(n )
=
(0.25)"u(n).
1 76 Forasavingsaccountthatpaysinterestattherateof
I
percentpermonth,ifdepositsaremadeonthefirstofeach
monthattherateof$50permonth,howmuchmoneywilltherebe intheaccountattheendof
1
year?
1 77 A savingsaccountpaysinterestattherateof percentpermonth. Withaninitialdepositof$50,howmuchwill
therebeintheaccountafter 10years?
nswers to Supplementary Problems
1 42 a) If x ( n ) = x(n N),byshift-invariance,y(n)= y(n N).Therefore,y(n)isperiodicwithperiodN.
(b) No. (c) Yes.
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
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CHAP.
11
SIGNALSANDSYSTEMS
1 45 Even.
1 46 (a ) istrue.
1 49
(a ) Linear,shift-varying,stable,noncausal,noninvertible.( 6) Linear,shift-varying,unstable,noncausal,
invertible. c ) Linear,shift-invariant,stable.noncausal,invertible. (d) Nonlinear,shift-invariant,stable,causal,
invertible.
0 )
Nonlinear,shift-invariant,stable,noncausal,noninvertible.
1 50 (a) la1 1 . (6) Anyfinite
a .
c ) la[ 1 .
1 51 No. Considerthesystemy(n )
=
x(n)cos (nr /2 ) .
1 53 (a) Yes. (6) No. c ) No.
1 54
Thesequencevalues.beginningatindexn
=
- I , are y(n)= { 2 , 0 , -4, 1 , 6 , 0 , I -1, - 2 , 6 , 3 ) .
n
1 57 ( a) s (n )= h(k).Withh(n)
=
u(n)
u(n
6)thestepresponseis
k=-cc
(6 ) h (n )
=
s (n)
s ( n
1 . Ifs(n)
=
(-0.5)nu(n), thenh(n)
=
S(n)
+
3(-0.5)"u(n
1).
1 58
(a) y(n)= S(n
2) + 2S(n
3)
2S(n
5 )
S(n
6).
b ) y(n)
=
S(n
+
2)
26(n)
+
S(n
2).
bn+l
an l
1 61 y(n) =
u(n).
h - a
1 63
h(n)= S(n) S(n I).
8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)
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5
SIGNALS AND
SYST MS
[CHAP.
y ( n )
[ (4n 3 ) ( - i ) " 3 ] u ( n ) .
y ( -2 )
7 j n s *
3
m a x ( y ( n ) ]= which occurs
at
index n 8.
y ( n )
=
4 n ) .
Y @ ) u ( n 4 ) f (n 2) (n 1) u(n 2) .
y ( n ) [ 4 (- 1 )" ( - f ) " ] u ( n ) .
y (n ) n u (n ) .
~ ( n )
[i (;)(3)
2 ( 2 " ) ] u ( n ) .
( n ) lj?.[4"
4n 12(2 )" ]u (n ) .
0 ) h(n)
[ i n ~ ] ( i ) ~ u ( n ) .b) y ( n ) ( n ) ( f ) n u ( n ) .
$690.46.
$165.02.
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