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Unformatted text preview: 78 ContinuousTime Signals and Systems Chap_ Procedures are developed for expressing certain types of signals as mathema _
ical functions. Nonsinusoidal periodic functions are introduced, and a technique for
writing equations for these periodic signals is presented. A general technique is then given for expressing the output of a continuous.
time system that is an interconnection of systems. An example is given of a feedback
control system. Several important properties of systems are defined, and procedures are given to determine whether a system possesses these properties. This chapter is devoted to continuous—time signals and systems. In Chapter 9,
the same topics are developed as they relate to discretetime signals and systems.
Many of the topics are identical; however, in some cases, there are significant differ— ences. (See Table 2.4.) 2.1. The signals in Figure P2.1 are zero except as shown. (a) For the signal x(t) of Figure P2.1(a), plot
(i) x(t/3) (ii) x(—t)
(iii) x(3 + t) (iv) x(2 — t) Verify your results by checking at least two points. x (t) (C) Figure P2.1 Chap. 2 Problems 79 (b) Repeat (a) for the signal x(t) of Figure P2.1(b).
(c) Repeat (a) for the signal x(t) of Figure P2.1(c). 2.2. The signals in Figure P2.1 are Zero except as shown.
(a) For the signal x(t) of Figure P2.1(a), plot (i) 4x(t) — 2 (ii) 2x(:) + 2
(iii) 2x(2t) + 2 (iv) 4x(t) + 2 Verify your results by checking at least two points.
(b) Repeat part (a) for the signal x(t) of Figure P2.1(b).
(c) Repeat part (a) for the signal x(t) of Figure P2.1(c). 2.3. You are given the two signals in Figure P23. (a) Express y(t) as a function of x(t).
(b) Verify your result by checking at least three points in time. 2.4. You are given the signals x(t) and y(t) in Figure P24. (3) Express y(t) as a function of x(t).
(b) Verify your results by checking at least three points in time. (c) Express x(t) as a function of y(t).
((1) Verify the results of part (C) by checking at least three points in time. 2.5. Given x(t) = 4(t + 2)u(t + 2) 4tu(t) 4u(t 2) 4(t 4)u(t 4) + 4(t ~ 5)u(t — 5),
find and sketch x(2t ~ 4). (b) Figure P23 80 ContinuousTime Signals and Systems Chap, 2 (b) Figure P24 2.6. Given
x(t) = 5u(t + 2)  u(t) + 3u(t  2) — 7u(t  4),
find and sketch x(—2t — 4).
2.7. Plot the even and odd parts of the signal of (a) Figure P2.1(a)
(b) Figure P2.1(b)
(c) Figure P2.1(c)
((1) Figure P2.4(a)
(e) Verify your results, using (2.11). 2.8. For each of the signals given, determine mathematically whether the signal is even,
odd, or neither. Sketch the waveforms to verify your results. (a) x(t) = 4t (b) x(t) = all (e) x(t) = 5005 3t
((1) x(t) = sin(3t  g
(e) x(t) = u(t) is Chap. 2 Chap. 2 Problems 81 2.9. The average value Ax of a signal x(t) is given by 1 T
Ax = Tiling? _Tx(t)dt, Let xe(t) be the even part and x0(t) be the odd part of x(t).
(a) Show that 1 T Tx0(t)dt = 0.
(b) Show that
1 T _Txe(t)dt = Ax. (c) Show that x0(0) = O and xe(0) = x(0). 2.10. Give proofs of the following statements: (a) The sum of two even functions is even. (b) The sum of two odd functions is odd. (c) The sum of an even function and an odd function is neither even nor odd.
(d) The product of two even functions is even. (e) The product of two odd functions is even. (f) The product of an even function and an odd function is odd. 2.11. Given in Figure P211 are the parts of a signal x(t) and its odd part x0(t), fort 2 0 only;
that is, x(t) and xo(t) for t < 0 are not given. Complete the plots of x(t) and xe(t), and
give a plot of the even part, xe(t), of x(t). Give the equations used for plotting each part
of the signals. 2.12. Prove mathematically that the signals given are periodic. For each signal, find the fun
damental period To and the fundamental frequency too. (a) x(t) = 7sin31
(b) x(t) = sin(8t + 30°) .gnal is even, —2 —1 0 1 2 t Figure P2.11 82 2.13. 2.14. 2.15. 2.16. 2.17. ContinuousTime Signals and Systems Chap. 2 (c) x(t) = 612' (d) x(t) = cost + sinZt
(e) x(t) = elm”) (f) X“) = e—let + elet For each signal, if it is periodic, find the fundamental period To and the fundamental
frequency coo. Otherwise, prove that the signal is not periodic. (a) x(t) = cos 3t + sin Sr. (b) 150) = cos 6t + sin 8t + eiz‘. (c) x(t) = cost + sin 77:. (d) x(t) = x1(t) + x2(3t) where x1(t) = sin(%) and x20) = sin(%’). (a) Consider the signal x(t) = 4cos(12t + 40°) + sin16t. If this signal is periodic, find its fundamental period To and its fundamental
frequency coo. Otherwise, prove that the signal is not periodic. (b) Repeat Part (a) for the signal x(t) = cos 4t + 3e”m’ (c) Repeat Part (a) for the signal x(t) = cos 2m + sin 6t.
((1) Repeat Part (a) for the signal x4(t) = x1(t) + x20) + x3(t), where °° +
x1(t) = cos(7rt), x20) = E rect<t 02”), and x3(t) = 4sin(5%7t + n=—oo Suppose that x10) is periodic with period T1 and that x2(t) is periodic with period T2.
(a) Show that the sum XV) = 3‘10) + X2“) is periodic only if the ratio T1/T2 is equal to a ratio of two integers kz/kl.
(b) Find the fundamental period T0 of x(t), for Tl/Tz = k2/k1. Find / 6(at — b)sin2(t — 4)dt, where a > 0. (Hint: Use a change of variables.) Express the following in terms of x(t): 1 DO y(t) = E/mx('r)[8(7 — 2) + 8(7 + 2)]d7‘. is Chap. Chap. 2 Problems 33
x10)
2
3 fundamental T —1 O 1 2 t
(a)
x(t)
2
—2 —1 O 1 2 3 t
(b) Figure P2.18 fundamenta 2.18. Consider the triangular pulse of Figure P2.18(a). (3) Write a mathematical function for this waveform. (b) Verify the results of Part (a), using the procedure of Example 2.12. (0) Write a mathematical function for the triangular wave of Figure P2.18(b), using the
results of Part (a). 2.19. Consider the trapezoidal pulse of Figure P2.19(a). (3) Write a mathematical function for this waveform.
(b) Verify the results of Part (a), using the procedure of Example 2.12. 1 period T2. Figure P2.19 84 2.20. 2.21. 2.22. 2.23. Continuous—Time Signals and Systems Chap. 2 (c) Write a mathematical function for the waveform of Figure P2.19(b), using the re
sults of Part (a). (a) Prove the timescaling relation in Table 2.3: [:5(at)dt = ﬁ[:5(t)dt. (Hint: Use a change of variable.)
(b) Prove the following relation from Table 2.3: u(t — to) = /t 5(r — t0)dr. (c) Evaluate the following integrals:
(i) /_:cos(2t)6(t)d1
(ii) [:sin(2t)6(t — 7mm
(iii) 1:cos[2(t — 7’/4)]5(t — mm
(iv) [:sinﬂt — 1)]5(r — 2)dt (v) foosinﬁt — 1)]5(2t — 4)dt Express the following functions in the general form of the unit step function
ll(::t — to): (a) u(21 + 6) (b) u(2t + 6) (c) 14“; + 2) ((1) MG " 2)
In each case, sketch the function derived. Express each given signal in terms of u(t — to). Sketch each expression to verify the
results. (a) u(—t) (b) u(3 — t) (c) tu(—t) (d) (t — 3)u(3 ~ I) (a) Express the output y(t) as a function of the input and the system transformations,
in the form of (2.56), for the system of Figure P2.23(a). (b) Repeat Part (a) for the system of Figure P2.23(b). rawwaswwwnmwxwmmzmmawowWnWWaWWWWWM» l l » Chap. 2 Problems 85 Figure P223 (c) Repeat Part (a) for the case that the summing junction with inputs y3(t) and y5(t) is
replaced with a multiplication junction, such that its output is the product of these two signals. (d) Repeat Part (b) for the case that the summing junction with inputs y3(t), y4(t), and
y5(t) is replaced with a multiplication junction, such that its output is the product of these three signals. 2.24. Consider the feedback system of Figure P224. Express the output signal as a function
of the transformation of the input signal, in the form of (2.58). 2.25. Consider the feedback system of Figure P225. Express the output signal as a function
of the transformation of the input signal, in the form of (2.58). The minus sign at the summing junction indicates that the signal is subtracted. 2.26. (a) Determine whether the system described by :+1
y(t):X x(T—a)d7 Figure P224 ContinuousTime Signals and Systems Chap. 2 x (t) + y (t) Figure P225
(where a is a constant) is
(i) memoryless, (ii) invertible,
(iii) stable, (iv) time invariant, and (v) linear.
(b) For what values of the constant a is the system causal? 2.27. (3) Determine Whether the system described by y(t) = cos[x(t — 1)] is (i) memoryless, (ii) invertible,
(iii) causal, (iv) stable,
(v) time invariant, and (vi) linear. (b) Repeat Part (a) for y(t) = 3x(3t + 3). (c) Repeat Part (a) for y(t) = In [160)]. ((1) Repeat Part (a) for ya) = em. (e) Repeat Part (a) for y(t) = 7x(t) + 6. (f) Repeat Part (a) for y(t) = [00x(57)d7. (g) Repeat Part (a) for y(t) = e"j“’"/ x('r)e_j“"d7.
—oo (h) Repeat Part (a) for y(t) = [ x(7)d7. —1 ems Chap. 2 Problems 87 x20)
2
t
—5 —4 ~3 —2 —1 0 1 2 3 4 5 6 Figure P228
2.28. (a) You are given an LTI system. The response of the system to an input 2.29. 2.30. 2.31. x1(t) = u(t) — u(t  1) is a function y1(t). What is the response of the system to the input x2(t) in
Figure P228 in terms of y1(t)? (b) You are given another LTI system with the input shown in Figure P228. Find the
output y2(t) in terms of the system’s output, y1(t), if y1(t) is in response to the input
x1(t) “ 2u(t 1) u(t 2) u(t 3). Determine whether the ideal time delay W) = x(t " to) is (i) memoryless, (ii) invertible,
(iii) causal, (iv) stable,
(v) time invariant, and (vi) linear. Let h(t) denote the response of a system for which the input signal is the unit impulse func
tion 8(t). Suppose that h(t) for a causal system has the given even part he(t) for t > O: he(t) = t[u(t) — u(t ~— 1)] + u(t — 1),t > 0.
Find h(t) for all time, with your answer expressed as a mathematical function. (3) Sketch the characteristic y versus x for the system y(t) = ix(t)}. Determine '
whether this system is (i) memoryless, (ii) invertible,
(iii) causal, (iv) stable,
(v) time invariant, and (vi) linear. (b) Repeat Part (a) for y(t) = {303’ : : 8. 88 ContinuousTime Signals and Systems Chap. 2 (c) Repeat Part (a) for —10, x < —1
y(t) = 10x(t), {xi E 1.
10, x > 1 (d) Repeat Part (a) for 2, 2<x 1, 1<x§
y(t)= O, 0<x§1. —1, —1<x§ ~2, x5— ...
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 Fall '09
 EARLES

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