# Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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• 8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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• 8/10/2019 Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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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

followingstatementsre 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)

3/7

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)

4/7

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)

5/7

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

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)

6/7

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)

7/7

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.

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