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%
the function squared, divide the
%
period and display the result
disp([‘(b) Px = ‘,num2str(Px)]) ;
The output of this program is (a) Ex = 21.5177 (b) Px = 75.015
Analytical computations: ∞ 5.5 2 4 e− t / 10 (a) E x = x(t ) dt = −∞ 2.5
∫
∫
2
5.5
dt
5.5
16 e − t / 5 = 16 ∫ e− t / 5 d = − 5 × 16 = 21.888 2.5 2.5
(The small difference difference in results is probably probably due to the the error error inherent in trapezoidal-rule integration. It could be reduced by using time points spaced more closely together. together. (b) P x
=
1
5
(−3t ) 10 ∫
2
dt
=
−5
1
5
9t 5 ∫
2
dt =
0
1 5
(3t 3 )50
=
375 5
= 75
Check.
2.10 SUMMARY OF IMPORT IMPORTANT POINTS 1. 2. 3.
4. 5.
The term continuous and the term continuous-time mean different things. continuous and continuous-time mean A continuous-time impulse, although very useful in signal and system analysis, is not a function in the ordinary sense. Many practical signals can be described by combinations of shifted and/or scaled standard functions, and the order in which scaling and shifting are done is significant. Signal energy is, in general, not the same thing as the actual physical energy delivered by a signal. A signal with finite signal energy is called an energy signal and and a signal with infinite signal energy and finite average power is called a power a power signal .
EXERCISES WITH ANSWERS (On each exercise, the answers listed are in random order.) Signal Functions
1. If g(t ) = 7e−2t − 3 write out and simplify (a) g(3)
(b) g(2 − t )
(c) g((t /10) + 4 )
(d) g( jt ) jt
(e) g( jt ) + g(− jt ) 2
(f ) g(( jt − 3) / 2) + g((− jt − 3)/ 2) 2 7coss(t ), Answers: 7co
7e −7 + 2t ,
7e − j 2t − 3,
7e − ( t / 5) −11,
7e −3 cos(2t ),
7e −9
Exercises with Answers
2. If g( x ) = x 2 (a) g( z ) (d) g(g(t ))
− 4 x + 4 write out and simplify (b) g(u + v ) (c) g(e jt ) (e) g(2)
− 4 z + 4, t 4 − 8t 3 + 20t 2 − 16t + 4
Answers: (e jt − 2)2, z 2
u2 + v2
0,
+ 2 u v − 4 u − 4v + 4,
3. What would be the value of g in each of the following MATLAB instructions? t = 3 ; g = sin(t) ; x = 1:5 ; g = cos(pi*x) ; f = -1:0.5:1 ; w = 2*pi*f ; g = 1./(1+j*w’) ;
Answers: 0.1411, [−1,1, −1, 1, −1],
0.0247 + j 0.155 0.0920 + j 0. 289 1 0.0920 − j 0. 289 j .155 0.0247 − j0
4. Let two functions be defined by
1, x1 (t ) = −1,
sin ( 20t ) ≥ 0 sin ( 20t ) < 0
sin(2t ) ≥ 0
t , −t ,
x 2 (t ) =
and
sin(2t ) < 0
.
Graph the product of these two functions versus time over the time range, −2 < t < 2. Answer: x(t ) 2
-2
t
2 -2
Scaling and Shifting
g(−t ), − g(t ), g(t − 1), and g(2t ). 5. For each function g(t graph ) (a)
(b)
g(t )
g(t )
4
3 2
t
-1 1 -3
t
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Answers: g(-t )
-g(t )
g(-t ) 3
4
3 1
t
-2
2
t
-1 -3
,
4
1 2
t
-3
t
-3
,
,
g(2t )
4
t
,
-1
g(2t )
3 3
1
t
-4
,
g(t -1)
g(t -1)
1
-g(t )
3 -12
t
1
,
1 2
t
-3
,
6. Find the values of the following signals at the indicated times. (a) x(t ) = 2 rect(t / 4), x(−1) (b) x(t ) = 5rect (t / 2 ) sgn(2t ), x(0 .5) (c) x(t ) = 9 rect(t /10) sgn(3( t − 2)), x(1) Answers: −9, 2, 5 7. For each pair of functions in Figure E.7 provide the values of the constants A, t 0 and w in the shifting and/or scaling to g 2 (t ) = A g1 ((t − t0 ) / w ). (a) 2 ) 1 ( 0 g -1 -2 -4
(a)
t
2 ) 1 ( 0 g -1 -2 -4 t
1
2
-2
0
2
4
-2
t
0
(b) 2 ) 1 ( 0 g -1 -2 -4
2
4
2
4
2
4
t
(b)
t
2 ) 1 ( 0 g -1 -2 -4 t
1
2
-2
0
2
4
-2
t
0 t
(c) 2 ) 1 ( 0 g -1 -2 -4
(c)
t
2 ) 1 ( 0 g -1 -2 -4 t
1
2
-2
0
2
4
t
-2
0 t
Figure E.7
Answers: A = 2 , t0
= 1, w = 1 ;
A = −1 / 2 , t0
= −1, w = 2 ;
A = −2 , t0
= 0 , w = 1/ 2
Exercises with Answers
8. For each pair of functions in Figure E.8 provide the values of the constants A, t 0 and a in the functional shifting and/or scaling to g 2 (t ) = A g1 (w (t − t0 )). 8
8
4 ) t (
(a)
1
g
4 ) t (
0 -4 -8 -10
(b)
) t (
1
g
0
2
g
-4 -5
0 t
5
-8 -10
10
8
8
4
4 ) t (
0
1
g
-5
0 t
5
-8 -10
10
1
-5
0 t
5
-8 -10
10
(e)
1
g
2
g
10
-5
0 t
5
10
-5
0 t
5
10
-4 -5
0 t
5
-8 -10
10
8
4
4 ) t (
0
2
g
-4
Figure E.8
5
0
8
-8 -10
0 t
4 ) t (
-4
) t (
-5
8
0
-8 -10
10
-4
4 g
5
0
2
g
8
(d)
0 t
4 ) t (
-4
) t (
-5
8
0
-8 -10
10
-4
4
(c)
5
0
2
8
) t (
0 t
g
-4 -8 -10
-5
0 -4
-5
0 t
5
10
-8 -10
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Answers: A = 3, t0 = 2, w = 2 A = −3, t 0 = −6, w = 1 / 3 or A = −3, t0 A = −2, t0 = −2, w = 1/ 3, A = 3, t0 = −2, w = 1/ 2, A = 2, t0 = 2, w = −2
= 3, w = −1/ 3,
9. Figure E.9 shows a graphed function g1 (t ), which is zero for all time outside the range graphed. Let some other functions be defined by g2 (t ) = 3 g1 (2 − t ), g3 (t ) = − 2 g1 (t / 4 ),
g4 (t ) = g1
t − 3 2
Find these values. (a) g 2 (1)
(b) g3 ( −1)
(c) [g4 (t ) g3 (t )]t = 2
−1
(d)
∫ − g (t ) dt 4
3
g1(t ) 4 3 2 1 -4 -3 -2 -1
1
-1 -2 -3 -4
2
3
4
t
Figure E.9
Answers: −7/2, −3/2, −2, −3 10. A function G( f ) is defined by G( f ) = e − j 2 f rect( f / 2). Graph the magnitude and phase of G( f −20 < f < 20.
− 10) + G( f + 10)
over the range,
|G( f )| 1
-20
20
f
G( f )
π
-20
20 -π
f
Exercises with Answers
Answer: G( f
f − 10 − j 2 ( f f +10) f + 10 e − 10) + G( f + 10) = e− j 2( f −10) rect + rect 2
2
11. Write an expression consisting of a summation of unit-step functions to represent a signal that consists of rectangular pulses of width 6 ms and height 3, which occur at a uniform rate of 100 pulses per second with the leading edge of the first pulse occurring at time t = 0. Answer: x(t ) = 3
∞
∑ [u(t − 0.01n) − u(t − 0.01n − 0.006)]
n=0
Derivatives and Integrals
12. Graph the derivative of x(t ) = (1 − e − t ) u(t ). Answer: x(t ) 1 -1
t
4 -1 dx/dt 1
4 t
-1 -1
13. Find the numerical value of each integral. 5 / 2
8
(a)
∫ − [ (t + 3) − 2 (4t )] dt
∫
(b)
2 (3t ) dt
1 / 2
1
Answers: −1/2, 1 14. Graph the integral from negative infinity to time t of the functions in Figure E.14, which are zero for all time t < 0. g(t )
g(t )
1
1 1
2
3
t
1 2
1
2
3
t
Figure E.14
Answers: ∫ g(t ) dt
∫ g(t ) dt
1
1
1 2
1
2
3
t
,
1
2
3
t
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Mathematical Description of Continuous-Time Signals
Even and Odd Signals
) described over the time range 0 < t < 10 by 15. An even function g(t is
2t , g(t ) = 15 − 3t , −2,
0 < t < 3 3 < t < 7 . 7 < t < 10
(a) What is the value of g(t ) at time t = −5? (b) What is the value of the first derivative of g(t at ) time t = −6? Answers: 3, 0 16. Find the even and odd parts of these functions. (a) g(t ) = 2t 2
(b) g(t ) = 20 cos(40t − / 4)
(c)
(d) g(t ) = t (2 − t 2 )(1 + 4 t 2 )
(e)
− 3t + 6 2t 2 − 3t + 6 g(t ) = 1 + t g(t ) = t (2 − t )(1 + 4t )
Answers: t (2 − 4t ), (20 / 2 ) cos(40t ), 0, 2
(20 / 2 ) sin(40t ), 2t 2 + 6
−t
t (2 − t 2 )(1 + 4t 2 ),
2t 2 + 9 1 − t 6 + 5t 2
, 7t 2,
2
1 − t 2
, −3t
17. Graph the even and odd parts of the functions in Figure E.17. (a)
(b)
g(t )
g(t )
1
1 t
1
1 2 -1
Figure E.17
Answers: ge(t )
ge(t )
1
1 1
t
1 2 -1
t
go(t )
go(t ) 1
1
1
t
,
1 2 -1
t
t
Exercises with Answers
67
18. Graph the indicated product or quotient g(t ) of the functions in Figure E.18. (b) (a) 1 -1
1 -1
t
1 -1
g(t ) g(t )
1 -1
t
1 -1
-1
Multiplication
t
1
1 1 -1
(d)
(c)
1
1
t
-1
Multiplication
t
t
1
g(t )
g(t ) 1
Multiplication
t
1
1
Multiplication
t
1
(f) (e) 1 1 ...
...
-1
1
1
t
t
-1 g(t )
-1 g(t )
1 -1
t
1
1
Multiplication
1 -1
(h)
(g)
1
1
t
-1
-1 g(t )
1 1 Figure E.18
Multiplication
t
t
Division
-1 1
t g(t )
π
1
t
Division
68
Mathematical Description of Continuous-Time Signals
C h a p t e r 2
Answers:
g(t )
g(t )
g(t ) 1
1
1
-1
t
1
...
-1
-1
t
1 -1
,
1
g(t ) g(t )
1
t
1
t
1
-1
,
g(t )
1 -1
,
g(t )
1
-1
-1
1
t
1 -1
t
-1
,
19. Use the properties of integrals of even and odd functions to evaluate these integrals in the quickest way. 1
(a)
1 / 20
∫ − (2 + t) dt
(b)
1
4 t cos(10 ∫ −
t ) dt
(d)
+ 8 sin(5t )] dt
t sin(10 ∫ −
t ) dt
1 / 10
1
1
∫ −
e − t dt
(f )
1
Answers: 0,
t )
1 / 10
1 / 20
(e)
[4 cos(10 ∫ − 1 / 20
1 / 20
(c)
t
-1
,
g(t )
-1
...
∫ − te−
t
dt
1
8
,
1
10 50
, 0, 1.264, 4
Periodic Signals
20. Find the fundamental period and fundamental frequency of each of these functions. (a) g(t ) = 10 cos(50t )
(b) g(t ) = 10 cos(50t + / 4)
(c) g(t ) = cos(50t ) + sin(15t ) (d) g(t ) = cos(2t ) + sin(3t ) + cos(5t − 3 / 4 ) Answers: 2 s, 1 / 25 s, 2.5 Hz, 1 / 25 s, 1 / 2 Hz, 0.4 s, 25 Hz, 25 Hz
t
Exercises with Answers
21. One period of a periodic signal x(t ) with fundamental period T 0 is graphed in Figure E.21. What is the value of x(t ) at time t = 220ms?
x(t ) 4 3 2 1 5ms 10ms15ms20ms
-1 -2 -3 -4
t
T 0
Figure E.21
Answer: 2 22. In Figure E.22 find the fundamental period and fundamental frequency ofg(t ).
g(t )
...
(a) ...
... 1
t
... 1
(b)
g(t ) ...
... 1
...
... 1
(c)
t
t
t g(t )
...
... 1
t
Figure E.22
Answers: 1 Hz, 2 Hz, 1 / 2 s, 1 s,1 / 3 s, 3 Hz Signal Energy and Power
23. Find the signal energy of these signals. (a) x(t) = 2 rect(t ) (c) x(t ) = u(t ) − u(10 − t ) (e) x(t ) = rect(t ) cos(4 t ) Answers: 1 / 2,
(b) x(t ) = A(u(t ) − u(t − 10 )) (d) x(t ) = rect(t ) cos(2t ) (f) x(t ) = rect(t ) sin(2t )
∞ , 10 A2, 1 / 2, 4, 1 / 2
24. A signal is described by x(t ) = A rect(t ) + B rect(t − 0 .5 ). What is its signal energy? Answer: A2
+ B 2 + AB
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25. Find the average signal power of the periodic signal x(t ) in Figure E.25.
x(t ) 3 2 1 -4 -3 -2 -1 -1 -2 -3
1 2 3 4
t
Figure E.25
Answer: 8/9 26. Find the average signal power of these signals. (a) x(t ) = A (c) x(t ) = A cos(2f0 t + ) Answers: A2, A2 / 2, 1 / 2
(b) x(t ) = u(t )
EXERCISES WITHOUT ANSWERS Signal Functions
27. Given the function definitions on the left, find the function values on the right. (a) g(t ) = 100 sin(200t + / 4) (b) g(t ) = 13 − 4t + 6t 2 (c) g(t ) = −5e −2t e− j 2 t
g(0.001) g(2) g(1 / 4)
28. Let the continuous-time unit impulse function be represented by the limit (x)
= lim (1 /a) rect(x / a), a > 0. a→0
The function (1/a) rect( x / a) has an area of one regardless of the value of a. (a) What is the area of the function (4 x ) = lim (1 /a) rect( 4 x / a)? a→0
(b) What is the area of the function ( −6 x ) = lim (1 /a) rect( −6 x / a)? a→0
(c) What is the area of the function (bx ) = lim (1/a) rect(bx /a) for b positive and a→0 for b negative? 29. Using a change of variable and the definition of the unit impulse, prove that (a (t
− t0 )) = (1/ a )(t − t0 ) .
30. Using the results of Exercise 29, show that ∞
∑
( x − n / a ) a n =−∞ (b) The average value of 1 (ax ) is one, independent of the value of a. (c) Even though (at ) = (1/ a )(t ), 1 (ax ) ≠ (1 / a )1 ( x )
(a)
1 (ax )
=
1
Exercises without Answers
Scaling and Shifting
31. Graph these singularity and related functions. (a) g(t ) = 2 u(4 − t )
(b) g(t ) = u(2t )
(c) g(t ) = 5 sgn(t − 4 )
(d) g(t ) = 1 + sgn(4 − t )
(e) g(t ) = 5 ramp(t + 1)
(f ) g(t ) = −3 ramp(2t )
(g) g(t ) = 2(t + 3)
(h) g(t ) = 6(3t + 9)
(i) g(t ) = −4(2(t − 1))
( j) g(t ) = 21 (t − 1 / 2)
(k) g(t ) = 81 (4t )
(l) g(t ) = −62 (t + 1)
(m) g(t)
= 2 rect(t / 3)
(o) g(t ) = −3 rect(t − 2)
(n) g(t ) = 4 rect((t + 1) / 2) ( p) g(t ) = 0 .1 rect((t − 3) / 4 )
32. Graph these functions. (a) g(t ) = u(t ) − u(t − 1)
(b) g(t ) = rect(t − 1 / 2)
(c) g(t ) = −4 ramp(t ) u(t − 2)
(d) g(t) = sgn(t) sin(2t )
(e) g(t ) = 5e − t / 4 u(t )
(f ) g(t ) = rect(t ) cos(2t )
(g) g(t ) = −6 rect(t ) cos(3t ) (h) g(t ) = u(t + 1 / 2) ramp(1/ 2 − t ) (i) g(t ) = rect(t + 1 / 2) − rect(t − 1/ 2) t
( j) g(t ) =
∫ −∞[ (
+ 1) − 2() + ( − 1)] d
(k) g(t) = 2 ramp(t) rect((t − 1) / 2) (l) g(t ) = 3 rect(t / 4) − 6 rect(t / 2) 33. Graph these functions. (a) g(t ) = 3(3t ) + 6(4(t − 2)) (b) g(t ) = 21 (− t / 5) t
(c) g(t ) = 1 (t ) rect(t /11)
(d) g(t ) =
∫ −∞[
2 ( )
− 2 ( − 1)]d
34. A function g(t ) has the following description. It is zero for t < −5. It has a slope of –2 in the range −5 < t < −2. It has the shape of a sine wave of unit amplitude and with a frequency of 1 / 4 Hz plus a constant in the range −2 < t < 2. For t > 2 it decays exponentially toward zero with a time constant of 2 seconds. It is continuous everywhere. (a) Write an exact mathematical description of this function. (b) Graph g(t ) in the range −10 < t < 10.
−10 < t < 10 . Graph 2 g(3 − t ) in the range −10 < t < 10. Graph −2 g((t + 1) / 2) in the range −10 < t < 10.
(c) Graph g(2t ) in the range (d) (e)
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35. Using MATLAB, for each function below graph the original function and the shifted and/or scaled function. t < −1 −2, (a)
2t , − 1 < t < 1 g(t ) = 2 3 − t , 1 < t < 3 −6, t > 3
(b) g(t ) = Re(e jt (c) G( f ) =
+ e j1.1t )
5 f 2
−3 g(4 − t ) vs. t
g(t /4) vs. t G(10( f
− j2 + 3
− 10)) + G(10( f + 10))
vs. f
36. A signal occurring in a television set is illustrated in Figure E.36. Write a mathematical description of it.
Signal in Television x(t ) 5 -10
60
t (µs)
-10 Figure E.36
Signal occurring in a television set
37. The signal illustrated in Figure E.37 is part of a binary-phase-shift-keyed (BPSK) binary data transmission. Write a mathematical description of it.
BPSK Signal x(t ) 1
4
t (ms)
-1 Figure E.37
BPSK signal
38. The signal illustrated in Figure E.38 is the response of an RC lowpass filter to a sudden change in its input signal. Write a mathematical description of it.
RC Filter Signal x(t )
4 20
-1.3333 -4 -6 Figure E.38
Transient response of an RC filter
t (ns)
Exercises without Answers
39. Describe the signal in Figure E.39 as a ramp function minus a summation of step functions.
x(t ) 15
... t
4 Figure E.39
40. Mathematically describe the signal in Figure E.40. x(t ) Semicircle ...
9
... t
9 Figure E.40
41. Let two signals be defined by
1 , cos(2t ) ≥ 0 and x1 (t ) = 0, cos(2t ) < 0
x 2 (t ) = sin(2t / 10).
Graph these products over the time range −5 < t < 5. (a) x1 (2t ) x2 (− t )
(b) x1 (t / 5) x2 (20t )
(c) x1 (t / 5) x2 (20(t + 1))
(d) x1 ((t − 2) / 5) x2 (20t )
42. Given the graphical definition of a function in Figure E.42, graph the indicated shifted and/or scaled versions.
g(t ) 2
(a)
1
t → 2t 1
-2
2
3
4
5
6
t
g(t) → −3 g(− t )
-2
g(t ) = 0 , t < −2 or t > 6
g(t ) 2
(b)
t → t + 4
1 t
-2
1
2
3
4
5
6
-2
g(t ) is periodic with fundamental period, 4 Figure E.42
g(t ) → −2 g((t − 1) / 2)
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74
Mathematical Description of Continuous-Time Signals
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43. For each pair of functions graphed in Figure E.43 determine what shifting and/or scaling has been done and write a correct functional expression for the shifted and/or scaled function.
g(t ) 2
(a)
2
-2 -1
t
1 2 3 4 5 6
-4 -3 -2 -1-1
t
1 2 3 4
g(t ) 2
(b) -2
1 2 3 4 5 6
t
-2 -1
t
1 2 3 4 5 6
Figure E.43
In (b), assuming g(t ) is periodic with fundamental period 2, find two different shifting and/or scaling changes that yield the same result. 44. Write a function of continuous time t for which the two successive changes t → −t and t → t − 1 leave the function unchanged. 45. Graph the magnitude and phase of each function versus f . (a) G( f ) =
j f 1 + j f / 10
(b) G( f ) = rect (c) G( f ) =
f − 1000 + rect f + 1000 e − jf /500 100 100 1
250 − f 2
+ j 3 f
46. Graph versus f , in the range −4 < f < 4 the magnitudes and phases of (a) X( f ) = 5 rect(2 f )e+ j 2f
(b) X( f ) = j5( f
+ 2) −
(c) X( f ) = (1 / 2)1 / 4 ( f )e − jf Generalized Derivative
47. Graph the generalized derivative of g(t ) = 3 sin(t / 2 )rect(t ). Derivatives and Integrals
48. What is the numerical value of each of the following integrals? ∞
(a)
∫ −∞ (t ) cos(48
t ) dt
∞
(b)
20
(c)
∫ (t − 8) rect(t / 16) dt
0
∫ −∞ (t − 5) cos(
t ) dt
j5( f − 2)
Exercises without Answers
49. What is the numerical value of each of the following integrals? ∞
(a)
∞
∫ −∞
1 (t ) cos(48t ) dt
(b)
∫ −∞
1 (t ) sin(2t ) dt
20
(c) 4
∫
4 (t
− 2) rect(t) dt
0
50. Graph the time derivatives of these functions. (a) g(t ) = sin(2t ) sgn(t )
(b) g(t ) = cos(2t )
Even and Odd Signals
51. Graph the even and odd parts of these signals. (a) x(t ) = rect(t − 1)
(b) x(t ) = 2 sin(4t − / 4) rect(t )
52. Find the even and odd parts of each of these functions. (a) g(t ) = 10 sin(20t )
(b) g(t ) = 20t 3
(c) g(t ) = 8 + 7t 2
(d) g(t ) = 1 + t
(e) g(t ) = 6t
(f ) g(t ) = 4t cos(10 t )
(g) g( t) = cos(t ) / t
(h) g(t ) = 12 + sin(4t ) / 4t
(i) g(t ) = (8 + 7t ) cos(32t )
( j) g(t ) = (8 + 7t 2 ) sin(32t )
53. Is there a function that is both even and odd simultaneously? Discuss. 54. Find and graph the even and odd parts of the function x(t ) in Figure E.54.
x(t ) 2 1 -5 -4 -3 -2 -1 -1
1 2 3 4 5
t
Figure E.54
Periodic Signals
55. For each of the following signals, decide whether it is periodic and, if it is, find the period. (a) g(t ) = 28 sin(400t )
(b) g(t ) = 14 + 40 cos(60t )
(c) g(t ) = 5t − 2 cos(5000t )
(d) g(t ) = 28 sin(400t ) + 12 cos(500t )
(e) g(t ) = 10 sin(5t ) − 4 cos(7t )
(f ) g(t ) = 4 sin(3t ) + 3 sin( 3t )
56. Is a constant a periodic signal? Explain why it is or is not periodic and, if it is periodic, what is its fundamental period?
75
76
C h a p t e r 2
Mathematical Description of Continuous-Time Signals
Signal Energy and Power
57. Find the signal energy of each of these signals. (a) x(t) = 2 rect(− t ) (c) x(t ) = 3 rect(t / 4) (e) x(t ) = (t )
(b) x(t) = rect(8t ) (d) x(t ) = 2 sin(200t )
(Hint: First find the signal energy of a signal that approaches an impulse in some limit, then take the limit.) (f ) x(t ) =
d dt
t
(rect(t ))
(g) x(t ) =
∫ −∞rect(
) d
(h) x(t ) = e( −1− j 8 )t u(t ) 58. Find the average signal power of each of these signals. (a) x( t) = 2 sin(200t ) (c) x(t ) = e j100 t
(b) x( t) = 1 (t )
59. A signal x is periodic with fundamental period T 0 over the time period 0 < t < 6 by
= 6. This signal is described
rect((t − 2) / 3) − 4 rect(( t − 4) / 2). What is the average signal power of this signal?
