pages from [monson hayes] schaum s outline of digital signal (booksee.org)

8
SIGNALS AND SYSTEMS [CHAP.  Given that there is a single  particle in the reactor at time n 0, find an expression for the total number of particles within the reactor at time n Let a(n) and B(n) be the number of a particles and B particles within the reactor at time n . The behavior within the reactor may be described by the following pair of coupled difference equations: Before we can solve these difference equations, we must uncouple them. Th erefore, let us derive a single difference equation for B(n). From the first equation we see that a(n) #?(n  I). Substituting this relation into the second difference equation, we have B(n+ 1)=8B(n- 1)+2B(n) or, equivalently, B(n)  2B(n 8B(n 2) The characteristic equation for this difference equation is which gives the following homogeneous solulion Similarly, because a(n) B(n I), the solution for a(n) is With the initial conditions a(0) and B(0) 0, we may solve for A1 and A2 as follows: and the solutions for a(n) and B(n) are a(n) i(4)" + (-2)" n 0 B(n)  :(4)" ?(-2) n 0  Because we are interested in the total number of particles within the reactor at time n , with Supplementary Problems Discrete Time Signals 1 41 Find the period N of the sequence

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Page 1: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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

Page 2: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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))

Page 3: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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.

Page 4: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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)

Page 5: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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.

Page 6: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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).

Page 7: Pages From [Monson Hayes] Schaum s Outline of Digital Signal (BookSee.org)

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.