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)

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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)

    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)

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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)

    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.

    $165.02.

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