Signals and Linear Systems

Chapter 2

X(f)

-fW-f, 갸 + w f' - Wf.f, + w f

X(f)

J

쟈

X(f)

-fc f .1

.51 Spectra of three bandpass signa's.

quation (2.7.11) is a 〕 an be represented in ure components.

very useful relation; it basically says that a bandpass terms of two lowpass signals, namely, its in-phase and

i this case, the complex lowpass signal

we define the envelope and the phase of the bandpass signal as

-

lxi (t)l = A(,) = 7x 공 (t) + x옭 (t)

/x1(t) = 9(t) = arctan 쁩， (2.7.13)

xj(t) = xC(t) + jxs(t)

equivalent of the bandpass si

(2.7.12)

x(t). If we represent x1(t) in

Xs(t)x(t)

a1

an

n

ct

g

ar

have

x1(t) = 7x공 (t) +x옭 (t)ej

d the lowpass oordinates, we

，쎌‘,7‘… 계

―l-―

■

―--

--

Problems

Equations

105

xpressing a band- signal in terms of

t express xi(I) as

xj(t) = A(t)e18(t).

Equations (2.7.14) and (2.7.11), we have

x (t) = Re[xj (t)eI22Tit]

= Re[A (t)e' 2rft+9 (t)

= A(t) cos(27rft + 9(t)).

(2.7. - 7) and (2.7.11) represent two methods for

pass signal in terms of two the in-phase and 니 uadrature the bandpass signal.

2.8 FURTHER READING

Numerous references cover the analysis of LTI systems Willsky, and Young (1983)

in both the time and fre- contains wide coverage of

time- and frequency-domain analysis of both discrete-time and contin tems. Papoulis (1962) and Bracewell (1 ％「） provide in-Ueptn a깻lysis series and transform techniques. A more advanced treatment 01 uneai on linear operator theory in

using MATLAB.

PROBLEMS

2.1 Plot the f이 lowing signals:

1. x1(t) = Fl(2t + 5)

2. x2(t) = fl(-2t + 8)

3. x3(t) = E곁0A (t - '/)

4. X4(t) = 2 Ei 」④A （쑤） ; E n=-c'c A(t , 4n)

5. x5(t) = sgn(2t) - sgn(t)

6. x6(t) E 곁」閃（ 'I)IzA(t - 2n)

7. X7(t) = sinc( lot)

8. x8(t) = sine 솖 9. x9(t) = 41J （汕 cos(4rt)

2.2 Plot the discrete version of the given signals. Assume they are sampled at

multiples of T0, i.e., x[n] = x (n T0)

1. x(t) = sinc(3t) and T0 = 승 2. x(t) = I-I （붕

3. : （ㅇ = tu,i(t) ' (t 』 1)u, (t 」 1) and T0 = 糞

= 0 士 l , 士2, denoted by x1 『”」 and x2[n]. Verify that from this observation?

lowpass signals. We can express components or in terms of the envelope and phase of

the

of the Fourier-

systems based can be found in Franks (1969). Digital filters are studied

detail by Proakis and Manolakis (1996). The book by ingle and Proakis (1997)

emphasizes design techniques for digital filters

and T0 = 윳

2.3 Two signals, xl (t) = 1 and X2(t) = cot The resulting discrete-time signals are xl 『끼 = X2 I 끼． What can you conclude

(2.7.14)

(2.7. ] 5)

(2.7.16)

(2.7.17)

2.4 Show that the sum of two discrete

signals is signals is periodic, whereas the

not necessarily periodic. Under what riodic signals periodic?

Sum of two condition is

Continuous periodic the snm nf tx,,-, p^t

Signals and Linear Systems Chapter 2

Classify these signals into energy-type signals, power-type signals, and that are neither energy type nor power type signals. For energy-type and type signals, find the energy or the power content of the signal.

signals power-

() I > 0

I < 0

(e 【 I cost) u 【 1 (t) e,t cost

sgn(t)

A cos 2rJj t + B cos 2rtt

1. x1(t) ==

2. x2(t) =

3. x3(t) =

4. x4(t) =

0 t < 0

내ㅐ que. whereas odd signals is even, even or ㅣ

odd sign al 」｀ odd.

1. sin(4OOOㅍ t) + cos(l I .000Jrt)

2. sin(4OOOj-t) + cos( " , 000t)

3. sin[4OOOㅠ ,1j + cos[l I ,OOOrii]

4. sin[4OQOj,1j + cos[1 l,000nj

2.6 Classify the sign are neither even

signals.

I ’』， ,f,, - 〔 · !' *'l`'' , -, ；【

cos (l20t + 쯤） X2 (t) = e

t 굳 0

t = 0

X5(t) =

X6 (I) =

2.7

Problems

2. Show that x(t)

3. Show that the

power content.

4. Show that the signal

2. Show that the product of two the product of an even and an

3. Try to find an example of two have an even product.

2.12 Plot the f이 lowing signals:

1. x1(t) = LI (t) + fT(-t)

2. x2(t) = LI (t) , U (t - I)

3. x3(t) = A(t)IJ(t)

4. x4(t) = E곁－_ A(t 」 2n)

5. x5(t) = E 곁』閃（-l）〃 A(t , 〃）

6. x(t) = sgn (t) + sgn(l - t)

7. x7(t) = 1 + sgn(t)

8. x8(t) = sgn2 (t)

9. x(t) = sinc(t) sgn(t)

107

Kt聳 t > 0

-’

·

！

……e

-

I

ee■．1

■11

sin t + cos t

x] (t) . x2(t), where x1 (t) is even and x2(t) is odd

，욜

3 1

1

ㅡ미0

1

6. 7.

-I

ㅡㅡ

j 구

￡

L X

signals that are neither even nor odd, but

0 0 0

t >

< ！ㅡ

l'l`

－ㅡ

5. x(t)

I-' ㅡ

－

4. x3(t)

--

given signals are periodic. For periodic signals, deter-

periodic

als that follow

nor odd. In the latter case, find the signals, and signals that

even and odd parts of the

!nto even signals, odd

.8 Classify these signals into periodic and nonperiodic:

1. x1(t) = 2 E곁．④ A （부L) , E곁【閃 A(t . 4n)

2. x2(t) = E곁，凶 A(t ' n)

3. xi(t) = sin t + sin 27rt

4. X4[fl] = sin n

5. xs(t) = E 큐헬-鬧 y(t ; nT), where y(t) is an arbitrary signal and T is an arbitrary constant

2.9 Using the definition of power-type and energy-type signals,

1. Show that x(t) = Ae3(2J0t+9) is a power-type signal and its power content is A 긱

= A cos(27rfht + 。） is power-type and its power is 뽕． unit-step signal U_I (t) is a power-type signal and find its

is neither an energy- nor a power-type signal.

2.10 Find the even and odd parts of the signal x(t) = A(t)u_1 (t).

2.11 Using the definition of even and odd signals,

1. Show that the decomposition of a signal into even and odd parts is

2.5 Determine whether the

mine the period

· Ub Signals and Linear Systems Chapter 2

1. xl(t)

2. x1(t)

3. x1(t)

4. x(t)

5. x3(t)

= sinc(f) 8(t)

= sinc(t) 8(t , 3)

= sinc(t - 2)8(t)

= A(I)* E곁，閃 8(t . 2n) = A (t) ★ 8'(t)

= cost 8(3t)

= cos (2t + 쯤） 8(3t)

더（,)

x6(t)

x7(t)

X8 (t)

= cost 8(3t + 1)

= 8(5t) * 8(4i')

= 8(5t) * 8'(3t)

= cos I 8'(t)

눼긱 fI(t)8(2t - 1) dt

2.15 We have seen that x(t) * 8(t) = x(t). Show that

d" = j-X (t)

dt' x (t) * 紗 (t)

109 Problems

t > 0

t < 0

t = 0 -

옭 x (t) r (t) 0

12. y(t) =

13. y(t) = x (t) + y (t - 1)

14. y(t) = Algebraic sum of jumps in x(t) in the interval (-oc, t]

2.17 Prove that a system is linear if and only if

10.

11.

x10(t) = E곁，閃 (-1 )'n8 (t , n)

xl(t) = :00n=1 去 fl()

2.13 By using the properties expressions:

of the impulse function, find the values of these

0(t)l 【귿 (t) lx(1)

0

= .T [Xi (t)1 + I?T [ x2(t)I .

definitions of linear systems given by

TIixi(t) + x2(t)j

words, show that the two In other

Equations (2.1.39) and (2.1.40) are equivalent.

sinc(t) 8(t) (It

si nc (t + 1)8(t) (It

LEoc, -fI()j8(t)dtZ_n=J 2

8(2't)] dt

2.14 Show that you say about evenness or

and

x (t) ＊ …(t) = 1' · （·。 dr.

2.16 Classify these systems into linear and nonlinear:

1. y(t) = 2x(t) , 3

2. y(t) == 」x（이

3. y(t)

4. y(t)

5. y(t)

8. y(t) = e-'x(t)

9. y(t) = x(t)u1 (t)

10. y(t) = x(t)6(t)

11. y(t) = x (t) E 곁－鬧 5(t - 〃 T)

2.18 Verify whether any (or both) of the conditions described in Problem 2.17 are satisfied by the systems given in Problem 2.16.

2.19 Prove that if a system satisfies the additivity property described in Problem 2.17. then it is homogeneous for all rational α ．

2.20 Show that the system described by

x'(t) 굳 0 r(t) = 0

is homogeneous but nonlinear. Can you give another example of such a system?

6. x4(t)

7. x4(t)

8. x4(t)

oddness of its th

9 0 1· 2. 3. 읗·

■乂

").

r l L L 겨 15 16

嚥f-L燭燭

cos t [E00fl = I

the impulse derivatjve?

signal is even. What can

t) > 0

t) <0

x(t) 굳 0

x (t) = 0

x(t) 굳 0 otherwise

X X

0 2s( nl()

`

】ee

es一

raㅡ

쓰x'(t)

0

l-

ㅡㅡ

꺾，

6. y(t)=-

7. y(t) ＝ㅣ

1. It 15 nomogeneous, 1.l

we have ''[ax(t)] = 2. It is additive, i.e., for

signals x(t) and all real numbers α ，

a .T [x (I)]. all input signals x1(t) and X2(1), we have

1. It is horn i.e., for all input

x (1)

10. y(t):I

x(t) x(t) 굳 0

x(t) = 0

隧 0

2.25 Prove that if the response

of this system to 붊x(t) is

ofdㅡ嫩

an LTI

y (t). system to x(t) is y(t), then the response

2.26 Prove of this

that if the response of an

system to J즈 ,0x(r)dr is

LTI system to f스鬧y（『）d『·

x(t) is y(t), then the response

Signals and Linear Systems Chapter 2

2.21 Show that the response of a linear system is an output which is identically zero.

to the input which is identically zero

2.22 The system defined by the input-output relation

y (t) = x(t) cos(2jrf0t),

where ．凡 invariant? is a constant, is called a modulator. Is this system linear? Is it time

2.23 Are these statements true or false? Why?

1. A system whose components

2. A system whose components system.

3. The response of a causal system to a causal signal is itself causal.

2.24 Determine whether the following systems are time variant or time invariant:

1. y(t) = 2x(t) + 3

2. y(t) = (t + 2)x(t)

3. y(t) = x (t) + t

4. y(t) = 시 -1)

5. y(t) = x(t)u-1 (t)

6. y(t) = x(t)8(t)

7. y(t) = x (t) E곁－閃 '(l . nT)

8. y(t) = f스oox(r)dr

9. y(t) = x(t) + y (t - 1)

Pro 비 ems

111

2.28 Let a system be defined by

y (t) r+T ··

2T ' _ X(T) at. J t- I

Is this system causal?

2.29 For an LTT system to be causal, it is required that /](t) be zero for t < 0. Give an example of a nonlinear system which is causal, but its impulse response is nonzero for t < α

2.30 Determine whether the impulse response of these LTI systems is causal:

1. h(t) = sinc (t)

2. h(t) = 미븟） 3. h(t) = sine (t)u_i (t)

2.31 Using the convolution integral, show that the response of an LTI system to u_i(t) is given by j·스凶 h(r)dt.

2.32 What is the impulse response of a differentiator? Find the output of this system to an arbitrary input x (t) by finding the convolution of the input and the impulse response. Repeat for the delay system.

2.33 The system defined by

y (t)= [ 『 x(r)dT. Jt- 7

(T is a constant) is a finite-time integrator. Is this system LTI? Find the impulse response of this system.

2.34 Compute the following convolution integrals:

1. e_tu_i(t) ★ etul(t)

2. e-tu_i(t) ★ u_1(t)

3. rI(t) * A (1)

4. (A (t)sgn(t)) *u1(t)

5. A(t) * sgn (t)

6. (A(t)u-1(t)) * H (t)

110 , . '－며．ㅣ卜

are nonlinear is necessarily nonlinear.

are time variant is necessarily a time-variant

2.27 The response of a linear time-invariant system to the input x(t) = e'atu[j (t) is 石 (t). Using time-domain analysis and the result of Problem 2.25, determine the impulse response of this system. What is the response of the system to a general input x(t)?

2.35 Show that in a causal LTI system, the convolution integral reduces to

r+oc ,. ,, , , , r

y tI) = 人 ㅉ『 - t)/1 「『）a『 = / )Ctj)fltt - t) U t

；체