Chapter1Problems

1.1Sketchthewaveformoffollowingsignals.

1.f(t)=(2-e~l)u(t)

[于⑷

一;

------------------------------十

0

1-2Writeouttheexpressionforallwareformasshownas

below.

1.

zK.

-2°2上

f(t尸+1[u(t+2)-u(t-2)]

2.

3-----------

1-------

/--------------

______________________________i

T005L5

f(t)=u(t+1)+u(t-0.5)+w(z-1.5)

3.

fctj

0丁七

Letf}(f)=Esxktthenf}(T)=ESxkT=0

7T

=S-t[u(t)-u(t-T)]

1-3Determingwhetherthe1,cost,...,cosntareorthogonal

functiononinterval(O,2乃).

「1Rn

Proof:I°”cosmtcosntdt=—I[cos(m+/?)，+cos(m-n)t]dt

m^n

m=n

1.4Determingwhetherl,x,x2,x3areorthogonalfunctionon

theinterval[0.1].

1

Proof:(l-x"'dx=(x"'dx=-x",+'=—^0

X)J)mm

0

Sothatontheinterval[0.1]theyarenotorthogonal.

1-5Findthevalueoftheintegral.

i.]7?-0）即辿=]7（-。）即）出=/（工）

f0>0

2.ju(t-t0)dt=-4)力="

。<0

2,1。

3￡e-8(t-tQ)dt^J產即70)沒=

0

f+4)出^^8(t+4)力=J

[(3戶+/—5)3(2)—3)力

193

=-[3-+——5]

242

_13

~~8

m-J"''tt

6.je[5。+m)+/’一寸)]力

1-6Thewareformoff(t)isdepictedasbelow，pleasesketchthewaveformof

f(t?3),f(?2t?1).

1-7f(t)asshownasFig.Tryingtosketchthewaveformoffollowingsignals.

的）

2.f(1)u(3-t)

df

3.。)

dt

一七

0

Ml)

4.

3

1.8Plotthewareformofgivendiscretesignals.

1.(仙⑺

I<?

A：_I__L

°12

2.(-2)n?(n)

3.〃?〃(〃)

/

0.-4I

八

Tr、

/卜、,處

0Izi4'JI\/

1-9Calculatetheconvolutionof/((f)andf2(t)

L/i(O=u(t),72(O=u(t)

=j*u(T)u(t-r)dr“(f)

=w(r)

=tu(t)

al

2.7,(O=u(t),f2(t)=e-u(t)

/W)*力(f)=xfx(t-r)f2(r)dr

e~aTu(r)u(t-T)C!T

-00

?(r)

0

-e-arM(f)

a

=—Fi-e』(f)

tzL

3./,(0=cos0°f,%。)=50+1)-b(f-1)

工(。*似。=<C)*W(f+l-]

r(f+i)一I)

=cos0q(f+1)-cosg(t-1)

4.

=￡2e~r[w(r)-w(r-3)]w(Z-T)t/r

=2^e~rdni(t)-2^e~rdru(t-3)

=2[1_"1〃(￡)-2[1_/]〃(,_3)

2-2/0<r<3

=?

2(1-e-3)3<t

1-10Determingtheperiodicforthefollowingsignals.

jr

(1)X(n)=cos—n

71,

LetX(n+N)=cos—(n+N)

/乃TC、

=cos(—〃+—N)

2020

71

=cos——n

20N=40

=X(n)

SotheperrodisN=40

(2)X(n)=5a(—+y)

LetX(n+N)=sin(——+—+——)

40240

./〃)乃、

=sin(——+一)

402

=X(〃)

N=80

SotheperiodisN=80

1-11FindthewaveformoftheRr(t)*fi(t)

br,

(a)Solution:fj(t)=u(t)-u(t-1),f2(t)=—t[u⑴-u(t-2)]

2

f1(t)*f2(t)=￡f2(r)f1(t-r)dr

—T[u(z)-u(r-2)][u(t-2)-u(t-r-l)]dT

b

()rdru(t)-1rdru(t-l)-TATu(t-2)+

2

((((

一

bJ3一2I

c(—t))((

))4一

Twl)2

eeff

hHbt(-)

eey

z(。

nr(z

)公

fef一21)W

i—fO

△八

r--32八rGer

shawAz

一

一

-)八

wenh1))(sc

f(八dT

ededea

-ro32z

e=a:)el

fx3ce;c

pofndu

lru￡nod4T

ffs—Nl

oevf(

f『

sToe(t2Ha

*-o2t

st(b+t)

fo*c八e

tD2r

2)twnH))02

h=z=

(t*B(oHbM

ffhe_〔n=+-)52-y

tefTJ22Wb

((6+oM—t

6一

一-

)t(4Gt2m

f〔

uHuuHTT)dt,e

Nf(-b++->(!S

uCeje-tr2e(2a2

一u

,±ju3—omff-)H&n

ot)o)i+bU

—

(±ou2『s5

+_(()u-3+n±3

)H二

一+fi)d

8±(+fa4?d

)Hfon+o

))uu8aTg-:f—5

a(Mnfto(

+(()(t

dt6『

_5t(o_g-(U

+—4

(±1工)-

)》「

+)2wr^7T

1u+lu+〈a

—(fO)err2^13

(8d)Jf1lg

e〕2))

-t2o)h-}

u+))*)r2

e—(2

lu

c8z7l0_hz

)du)1baV)(

(5(-4v)2

t+b)er

)、((

(uyv

e'-

-a4(

，)+h

+u-

l”

ls22

e)i7

)+n)

e-—

u8,g+22

T)

()5

tzu

+±rr(

))h

)j2

Ue)7-2

(J)

Tu+〕

e5

z3〔

+^(

dl

57

)2^1

ud)

(5

t