TIME-DOMAIN ANALYSIS OF [CONTINUOUS-TIME](#page-8-0) SYSTEMS

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• (b) Prove E[x1(t)] = E[x1(t − T)]. That is, shifting a signal does not affect its energy.
• (c) If (x1(t) = 0) ⇒ (x2(t) = 0) and (x2(t) = 0) ⇒ (x1(t) = 0), then prove E[x1(t) + x2(t)] = E[x1(t)] + E[x2(t)]. That is, the energy of the sum of two nonoverlapping signals is the sum of the two individual energies.
• (d) Prove E[x1(Tt)] = (1/|T|)E[x1(t)]. That is, time-scaling a signal by T reciprocally scales the signal energy by 1/|T|.
• 1.1-10 Consider the signal x(t) shown in Fig. P1.1-10. Outside the interval shown, x(t) is zero. Determine the signal energy E[x(t)]. [Hint: Use the results of Prob. 1.1-9.]

Figure P1.1-10

1.1-11 (a) Show that the power of a signal

$$x(t) = \sum_{k=m}^{n} D_k e^{j\omega_k t}$$ $$P_x = \sum_{k=m}^{n} |D_k|^2$$

assuming all frequencies to be distinct, that is, ω*ⁱ* = ω*^k* for all i = k.

• (b) Use the result in part (a) to determine the power of each of the signals in Prob. 1.1-6.
• 1.1-12 A binary signal x(t) = 0 for t < 0. For positive time, x(t) toggles between one and zero as follows: one for 1 second, zero for 1 second, one for 1 second, zero for 2 seconds, one for 1 second, zero for 3 seconds, and so forth. That is, the “on” time is always 1 second, but the “off” time successively increases by 1 second between each toggle. A portion of x(t) is shown in Fig. P1.1-12. Determine the energy and power of x(t).
• 1.2-1 For the signal x(t) depicted in Fig. P1.2-1, sketch the signals
  • (a) x(−t)

is

• (b) x(t +6)
• (c) x(3t)
• (d) x(t/2)
• 1.2-2 For the signal x(t) illustrated in Fig. P1.2-2, sketch
  • (a) x(t −4)
  • (b) x(t/1.5)
  • (c) x(−t)
  • (d) x(2t −4)
  • (e) x(2−t)
• 1.2-3 In Fig. P1.2-3, express signals x1(t), x2(t), x3(t), x4(t), and x5(t) in terms of signal x(t) and its time-shifted, time-scaled, or time-reversed versions.
• 1.2-4 For an energy signal x(t) with energy Ex, show that the energy of any one of the signals −x(t), x(−t), and x(t − T) is Ex. Show also that the energy of x(at) as well as x(at − b) is Ex/a, but the energy of ax(t) is a²Ex. This shows that time

Figure P1.2-3

inversion and time shifting do not affect signal energy. On the other hand, time compression of a signal (a > 1) reduces the energy, and time expansion of a signal (a < 1) increases the energy. What is the effect on signal energy if the signal is multiplied by a constant a?

1.2-5 Define 2x(−3t + 1) = t[u(−t − 1) − u(−t + 1)], where u(t) is the unit step function. (a) Plot 2x(−3t +1) over a suitable range of t.

• (b) Plot x(t) over a suitable range of t.

• 1.2-6 Consider the signal ^x(t) ^{= 2}−tu(t) , where u(t) is the unit step function.

  • (a) Accurately sketch x(t) over (−1 ≤ t ≤ 1).
  • (b) Accurately sketch y(t) = 0.5x(1 − 2t) over (−1 ≤ t ≤ 1).

• 1.2-7 Define signals y(t) and z(t) as in Fig. P1.2-7.

  • (a) Determine constants a, b, and c to produce z(t) = ax(bt +c) in Fig. P1.2-7.

• Figure P1.2-7

• (b) Determine and sketch a signal v(t) such that ^z(t) ⁼ $ ^t −∞ ^v(τ )d^τ .

• 1.3-1 Think of a real-world signal that is a personally relevant and interesting. Describe the signal and then classify it according to the six following characteristics:

  • (a) continuous-time or discrete-time
  • (b) analog or digital
  • (c) periodic or aperiodic
  • (d) energy or power
  • (e) causal or noncausal
  • (f) deterministic or random

If possible, think of a second real-world signal that has the opposite six characteristics of your first signal. If such a second signal is not possible, carefully explain why that is the case.

1.3-2 Define signal ^y(t) ⁼ %^{∞ k}=−∞ x(0.5t − 10k), where

$$x(t) = \begin{cases} e^{-2t} & t \ge 1 \\ 0 & t < 1 \end{cases}$$

• (a) Determine the constant a such that the signal x(−2t +a) is borderline anticausal.
• (b) Is the signal y(t) periodic? If so, determine the period Ty. If not, explain why y(t) is not periodic.
• 1.3-3 Determine whether each of the following statements is true or false. If the statement is false, demonstrate this by proof or example.
  • (a) Every continuous-time signal is an analog signal.
  • (b) Every discrete-time signal is a digital signal.
  • (c) If a signal is not an energy signal, then it must be a power signal and vice versa.
  • (d) An energy signal must be of finite duration.
  • (e) A power signal cannot be causal.
  • (f) A periodic signal cannot be anticausal.
• 1.3-4 Determine whether each of the following statements is true or false. If the statement is

false, demonstrate by proof or example why the statement is false.

• (a) Every bounded periodic signal is a power signal.
• (b) Every bounded power signal is a periodic signal.
• (c) If an energy signal x(t) has energy E, then the energy of x(at) is E/a. Assume a is real and positive.
• (d) If a power signal x(t) has power P, then the power of x(at) is P/a. Assume a is real and positive.
• 1.3-5 Given x1(t) = cos(t), x2(t) = sin(πt), and x3(t) = x1(t)+x2(t).
  • (a) Determine the fundamental periods T¹ and T² of signals x1(t) and x2(t).
  • (b) Show that x3(t) is not periodic, which requires T³ = k1T¹ = k2T² for some integers k¹ and k2.
  • (c) Determine the powers Px¹ , Px² , and Px³ of signals x1(t), x2(t), and x3(t).
• 1.3-6 For any constant ω, is the function f(t) = sin(ωt) a periodic function of the independent variable t? Justify your answer.
• 1.3-7 The signal shown in Fig. P1.3-7 is defined as

$$x(t) = \begin{cases} t & 0 \le t < 1 \\ 0.5 + 0.5 \cos(2\pi t) & 1 \le t < 2 \\ 3 - t & 2 \le t < 3 \\ 0 & \text{otherwise} \end{cases}$$

The energy of x(t) is E ≈ 1.0417.

• (a) What is the energy of y1(t) = (1/3)x(2t)?
• (b) A periodic signal y2(t) is defined as

$$y_2(t) = \begin{cases} x(t) & 0 \le t < 4\\ y_2(t+4) & \forall t \end{cases}$$

What is the power of y2(t)?

(c) What is the power of y3(t) = (1/3)y2(2t)?

Figure P1.3-7

• 1.3-8 Let y1(t) = y2(t) = t ² over 0 ≤ t ≤ 1. Notice, this statement does not require y1(t) = y2(t) for all t.
  • (a) Define y1(t) as an even, periodic signal with period T¹ = 2. Sketch y1(t) and determine its power.
  • (b) Design an odd, periodic signal y2(t) with period T² = 3 and power equal to unity. Fully describe y2(t) and sketch the signal over at least one full period. [Hint: There are an infinite number of possible solutions to this problem—you need to find only one of them!]
  • (c) We can create a complex-valued function y3(t) = y1(t) + jy2(t). Determine whether this signal is periodic. If yes, determine the period T3. If no, justify why the signal is not periodic.
  • (d) Determine the power of y3(t) defined in part (c). The power of a complex-valued function z(t) is

$$P = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} z(\tau) z^*(\tau) d\tau$$

• 1.4-1 Sketch the following signals:
  • (a) u(t −5)−u(t −7)
  • (b) u(t −5) +u(t −7)
  • (c) t ²[u(t −1)−u(t −2)]
  • (d) (t −4)[u(t −2) −u(t −4)]
• 1.4-2 Express each of the signals in Fig. P1.4-2 by a single expression valid for all t.

1.4-3 Letting w(t) = t[u(t)−u(t −1)], define the periodic signal x(t) as

$$x(t) = \sum_{k=-\infty}^{\infty} w(2t + 2k) - 0.5w(2t + 2k - 1)$$

• (a) Sketch w(t) and x(t). What is the fundamental period T⁰ of signal x(t)?
• (b) Sketch ^y(t) ^{= d} dt x(1−0.5t).
• (c) Determine the energy Ez and power Pz of the signal z(t) = x(0.5 − 1.5t)[u(t)−u(t −1)]. Sketching z(t) should help.
• 1.4-4 Define signal x(t) = u(t−1)−u(t−2.5)−2δ(t− 4)+δ(t −6).
  • (a) Sketch ^y(t) ⁼ $ ^t −∞ ^x(τ )d^τ .
  • (b) Describe a simple change that can be made to the right-most delta function in x(t) so that ^y(t) ⁼ $ ^t −∞ ^x(τ )d^τ has finite energy.
  • (c) Sketch ^z(t) ⁼ $ ^{∞ t} x(τ )dτ .
  • (d) Determine real constants A and B so that w(t) = x t−A B has a region of support [−2, 2].
• 1.4-5 Simplify the following expressions:

(a)

$$\left(\frac{\sin t}{t^2 + 2}\right) \delta(t)$$

\n(b) $\left(\frac{j\omega + 2}{\omega^2 + 9}\right) \delta(\omega)$

(c) [e−^t cos(3*^t* ⁻60◦)]δ(t)

(d)

$$\left(\frac{\sin\left[\frac{\pi}{2}(t-2)\right]}{t^2+4}\right)\delta(1-t)$$

(e)

$$\left(\frac{1}{j\omega+2}\right)\delta(\omega+3)$$

(f)

$$\left(\frac{\sin k\omega}{\omega}\right)\delta(\omega)$$

[Hint: Use Eq. (1.10). For part (f) use L’Hôpital’s rule.]

1.4-6 Evaluate the following integrals:

(a)

$$\int_{-\infty}^{\infty} \delta(\tau) x(t-\tau) d\tau$$

\n(b)

$$\int_{-\infty}^{\infty} x(\tau) \delta(t-\tau) d\tau$$

\n(c)

$$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt$$

\n(d)

$$\int_{-\infty}^{\infty} \delta(2t-3) \sin \pi t dt$$

\n(e)

$$\int_{-\infty}^{\infty} \delta(t+3) e^{-t} dt$$

\n(f)

$$\int_{-\infty}^{\infty} (t^3+4) \delta(1-t) dt$$

\n(g)

$$\int_{-\infty}^{\infty} x(2-t) \delta(3-t) dt$$

\n(h)

$$\int_{-\infty}^{\infty} e^{(x-1)} \cos \left[ \frac{\pi}{2} (x-5) \right] \delta(x-3) dx$$

1.4-7 For real and positive constant a, evaluate the following integral:

$$\int_{-\infty}^{\infty} \delta(at) \, dt$$

• 1.4-8 (a) Find and sketch dx/dt for the signal x(t) shown in Fig. P1.2-2.
  • (b) Find and sketch d2x/dt² for the signal x1(t) depicted in Fig. P1.4-2a.
• 1.4-9 Find and sketch $ ^t −∞ ^x(t)dt for the signals ^x(t) illustrated in Fig. P1.4-9.
• 1.4-10 Using the generalized function definition of impulse [Eq. (1.11) with T = 0], show that δ(t) is an even function of t.
• 1.4-11 Using the generalized function definition of impulse [Eq. (1.11) with T = 0], show that

$$\delta(at) = \frac{1}{|a|} \delta(t)$$

1.4-12 Show that

Figure P1.4-9

$$\int_{-\infty}^{\infty} \dot{\delta}(t)\phi(t) dt = -\dot{\phi}(0)$$

where φ(t) and φ(˙ t) are continuous at t = 0, and φ(t) → 0 as t → ±∞. This integral defines δ(˙ t) as a generalized function. [Hint: Use integration by parts.]

• 1.4-13 A sinusoid eσ*^t* cos ωt can be expressed as a sum of exponentials est and e−st [Eq. (1.14)] with complex frequencies s = σ +jω and s = σ −jω. Locate in the complex plane the frequencies of the following sinusoids:

  • (a) cos 3t
  • (b) e−3*^t* cos 3t
  • (c) e2*^t* cos 3t
  • (d) e−2*^t*
  • (e) e2*^t*
  • (f) 5

• 1.5-1 Find and sketch the odd and the even components of the following:

  • (a) u(t)
  • (b) tu(t)
  • (c) sinω0t
  • (d) cosω0t
  • (e) cos(ω0t +θ )
  • (f) sinω0tu(t)
  • (g) cosω0tu(t)

• 1.5-2 Define x(t) = 2u(t+1)−u(t−2)−u(t−3).

  • (a) Letting xo(t) designate the odd portion of x(t), accurately sketch xo(1−2t).

• (b) Letting xe(t) designate the even portion of x(t), accurately sketch xe(2+t/3).

• 1.5-3 (a) Determine even and odd components of the signal ^x(t) ^{= e}−2*^t u*(t).

  • (b) Show that the energy of x(t) is the sum of energies of its odd and even components found in part (a).
  • (c) Generalize the result in part (b) for any finite energy signal.

• 1.5-4 (a) If xe(t) and xo(t) are even and the odd components of a real signal x(t), then show that

$$\int_{-\infty}^{\infty} x_e(t)x_o(t) dt = 0$$

(b) Show that

$$\int_{-\infty}^{\infty} x(t) dt = \int_{-\infty}^{\infty} x_e(t) dt$$

• 1.5-5 An aperiodic signal is defined as x(t) = sin(πt)u(t), where u(t) is the continuous-time step function. Is the odd portion of this signal, xo(t), periodic? Justify your answer.

• 1.5-6 An aperiodic signal is defined as x(t) = cos(πt)u(t), where u(t) is the continuous-time step function. Is the even portion of this signal, xe(t), periodic? Justify your answer.

• 1.5-7 Consider the signal x(t) shown in Fig. P1.5-7.

• (a) Determine and carefully sketch v(t) = 3x(−(1/2)(t +1)).

• (b) Determine the energy and power of v(t).

• (c) Determine and carefully sketch the even portion of v(t), ve(t).

• (d) Let a = 2 and b = 3; sketch v(at + b), v(at) +b, av(t +b), and av(t)+b.

• (e) Let a = −3 and b = −2; sketch v(at + b), v(at)+b, av(t +b), and av(t) +b.

• 1.5-8 Consider the signal y(t) = (1/5)x(−2t − 3) shown in Fig. P1.5-8.

Figure P1.5-8

• (a) Does y(t) have an odd portion, yo(t)? If so, determine and carefully sketch yo(t). Otherwise, explain why no odd portion exists.
• (b) Determine and carefully sketch the original signal x(t).
• 1.5-9 Consider the signal −(1/2)x(−3t + 2) shown in Fig. P1.5-9.

Figure P1.5-9

• (a) Determine and carefully sketch the original signal x(t).
• (b) Determine and carefully sketch the even portion of the original signal x(t).
• (c) Determine and carefully sketch the odd portion of the original signal x(t).
• 1.5-10 The conjugate symmetric (or Hermitian) portion of a signal is defined as wcs(t) = (w(t) + w∗(−t))/2. Show that the real portion of wcs(t) is even and that the imaginary portion of wcs(t) is odd.
• 1.5-11 The conjugate antisymmetric (or skew-Hermitian) portion of a signal is defined as wca(t) = (w(t) − w∗(−t))/2. Show that the real portion of wca(t) is odd and that the imaginary portion of wca(t) is even.

1.5-12 Define ^w(t) ⁼ ej(t+π/4) .

• (a) Referring to the definition in Prob. 1.5-10, determine wcs(t). Express your simplified answer in standard rectangular form.
• (b) Referring to the definition in Prob. 1.5-11, determine wca(t). Express your simplified answer in standard polar form.

144 CHAPTER 1 SIGNALS AND SYSTEMS

1.5-13 Figure P1.5-13 plots a complex signal w(t) in the complex plane over the time range (0 ≤ t ≤ 1). The time t =0 corresponds with the origin, while the time t = 1 corresponds with the point (2, 1).

Figure P1.5-13

• (a) In the complex plane, plot w(t) over (−1 ≤ t ≤ 1) if w(t) is an even signal.
• (b) In the complex plane, plot w(t) over (−1 ≤ t ≤ 1) if w(t) is an odd signal.
• (c) In the complex plane, plot w(t) over (−1 ≤ t ≤ 1) if w(t) is a conjugate symmetric signal. [Hint: See Prob. 1.5-10.]
• (d) In the complex plane, plot w(t) over (−1 ≤ t ≤ 1) if w(t) is a conjugate antisymmetric signal. [Hint: See Prob. 1.5-11.]
• (e) In the complex plane, plot as much of w(3t) as possible.
• 1.5-14 Define complex signal x(t) = t ²(1 + j) over interval (1 ≤ t ≤ 2). The remaining portion is defined such that x(t) is a minimum-energy, skew-Hermitian signal.
  • (a) Fully describe x(t) for all t.
  • (b) Sketch y(t) = Re{x(t)} versus the independent variable t.
  • (c) Sketch z(t) = Re{jx(−2t + 1)} versus the independent variable t.
  • (d) Determine the energy and power of x(t).

[Hint: See Prob. 1.5-11 for a definition of skew-Hermitian signals.]

• 1.6-1 Write the input–output relationship for an ideal integrator. Determine the zero-input and zero-state components of the response.
• 1.6-2 A force x(t) acts on a ball of mass M (Fig. P1.6-2). Show that the velocity v(t) of the ball at any instant t > 0 can be determined if we know the force x(t) over the interval from 0 to t and the ball’s initial velocity v(0).

Figure P1.6-2

• 1.6-3 From your personal experience, provide an example of:
  • (a) a single-input, single-output (SISO) system
  • (b) a multiple-input, single-output (MISO) system
  • (c) a single-input, multiple-output (SIMO) system
  • (d) a multiple-input, multiple-output (MIMO) system
• 1.7-1 For the systems described by the following equations, with the input x(t) and output y(t), determine which of the systems are linear and which are nonlinear.

(a)

$$\frac{dy(t)}{dt} + 2y(t) = x^2(t)$$

\n(b)

$$\frac{dy(t)}{dt} + 3ty(t) = t^2x(t)$$

\n(c)

$$3y(t) + 2 = x(t)$$

\n(d)

$$\frac{dy(t)}{dt} + y^2(t) = x(t)$$

\n(e)

$$\left(\frac{dy(t)}{dt}\right)^2 + 2y(t) = x(t)$$

\n(f)

$$\frac{dy(t)}{dt} + (\sin t)y(t) = \frac{dx(t)}{dt} + 2x(t)$$

(g)

$$\frac{dy(t)}{dt} + 2y(t) = x(t) \frac{dx(t)}{dt}$$

(h)

$$y(t) = \int_0^t x(\tau) d\tau$$

−∞ 1.7-2 For the systems described by the following equations, with the input x(t) and output y(t), explain with reasons which of the systems are time-invariant parameter systems and which are time-varying-parameter systems.

(a)

$$y(t) = x(t-2)$$

\n(b) $y(t) = x(-t)$
\n(c) $y(t) = x(at)$
\n(d) $y(t) = tx(t-2)$
\n(e) $y(t) = \int_{-5}^{5} x(\tau) d\tau$
\n(f) $y(t) = \left(\frac{dx(t)}{dt}\right)^2$

• 1.7-3 Two inputs, temperature T(t) and wind speed V(t), produce an output, wind chill W(t), according to W(t) = 35.74 + 0.6215T(t) − 35.75{V(t)} 0.16 ⁺ 0.4275T(t){V(t)} 0.16. The independent variable here is time, t. Answer the following questions yes or no, and provide mathematical justification for each answer.
  • (a) Is this system BIBO-stable?
  • (b) Is the system memoryless?
  • (c) Is the system causal?
  • (d) For simplicity, let the wind speed be constant, V(t) = kV . Thus, W(t) = k¹ + k2T(t) for some constants k¹ and k2. Is this simplified system linear?
  • (e) For simplicity, let the temperature be constant, T(t) = kT . Thus, W(t) = k³ + k⁴ {V(t)} 0.16 for some constants k³ and k4. Is this simplified system linear?
• 1.7-4 Input voltage x(t) applied to an inverting op-amp follower circuit produces output y(t) according to

$$y(t + t_{p}) = \begin{cases} -V_{ref} & x(t) > V_{ref} \\ V_{ref} & x(t) < -V_{ref} \\ -x(t) & \text{otherwise} \end{cases}$$

where op-amp reference voltage Vref and propagation delay t^p are both positive constants. Answer the following questions yes or no, and provide mathematical justification for each answer.

• (a) Is this system BIBO-stable?
• (b) Is the system causal?
• (c) Is the system invertible?
• (d) Is the system linear?
• (e) Is the system memoryless?
• (f) Is the system time invariant?
• 1.7-5 Repeat Prob. 1.7-4 for a system with input x(t) that produces output y(t) according to

$$y(t+1) = \begin{cases} -2x(t) & \text{when } x(t) \ge 0\\ 0 & \text{otherwise} \end{cases}$$

1.7-6 Repeat Prob. 1.7-4 for a system with input x(t) that produces output y(t) according to

$$y(t-1) = \begin{cases} x(t-1) & \text{when } \frac{d}{dt}x(t) \ge 0\\ x(t-2) & \text{otherwise} \end{cases}$$

• 1.7-7 Repeat Prob. 1.7-4 for a system that multiplies a given input by a ramp function, r(t) = tu(t). That is, y(t) = x(t)r(t).
• 1.7-8 Repeat Prob. 1.7-4 for a system with input x(t) that produces output y(t) according to

$$y(t) = \frac{d}{dt}x(t-1)$$

1.7-9 Repeat Prob. 1.7-4 for a system with input x(t) that produces output y(t) according to

$$y(t) = \begin{cases} x(t) & \text{if } x(t) > 0\\ 0 & \text{if } x(t) \le 0 \end{cases}$$

1.7-10 A continuous-time system is given by

$$y(t) = 0.5 \int_{-\infty}^{\infty} x(\tau) [\delta(t-\tau) - \delta(t+\tau)] d\tau$$

Recall that δ(t) designates the Dirac delta function.

• (a) Explain what this system does.
• (b) Is the system BIBO-stable? Justify your answer.
• (c) Is the system linear? Justify your answer.
• (d) Is the system memoryless? Justify your answer.
• (e) Is the system causal? Justify your answer.
• (f) Is the system time invariant? Justify your answer.
• 1.7-11 For a certain LTI system with the input x(t), the output y(t) and the two initial conditions q1(0) and q2(0), the following observations were made:

x(t)	q1(0)	q2(0)	y (t)
0	1	−1	e−t u(t)
0	2	1	e−t (3t +2)u(t)
u(t)	−1	−1	2u(t)

Determine y(t) when both the initial conditions are zero and the input x(t) is as shown in Fig. P1.7-11. [Hint: There are three causes: the input and each of the two initial conditions. Because of the linearity property, if a cause is increased by a factor k, the response to that cause also increases by the same factor k. Moreover, if causes are added, the corresponding responses add.]

Figure P1.7-11

1.7-12 A system is specified by its input–output relationship as

$$y(t) = \frac{x^2(t)}{dx(t)/dt}$$

Show that the system satisfies the homogeneity property but not the additivity property.

1.7-13 Show that the circuit in Fig. P1.7-13 is zero-state linear but not zero-input linear. Assume all diodes to have identical (matched) characteristics. The output is the current y(t).

Figure P1.7-13

1.7-14 The inductor L and the capacitor C in Fig. P1.7-14 are nonlinear, which makes the circuit nonlinear. The remaining three elements are linear. Show that the output y(t) of this nonlinear circuit satisfies the linearity conditions with respect to the input x(t) and the initial conditions (all the initial inductor currents and capacitor voltages).

1.7-15 For the systems described by the following equations, with the input x(t) and output y(t), determine which are causal and which are noncausal.

(a)

$$y(t) = x(t-2)$$

(b) $y(t) = x(-t)$

(c) y(t) = x(at) a > 1

(d) y(t) = x(at) a < 1

1.7-16 For the systems described by the following equations, with the input x(t) and output y(t), determine which are invertible and which are noninvertible. For the invertible systems, find the input–output relationship of the inverse system.

(a)

$$y(t) = \int_{-\infty}^{t} x(\tau) d\tau$$

\n(b) $y(t) = x^{n}(t), x(t)$ real, n integer
\n(c) $y(t) = \frac{dx(t)}{dt}$

$$(c) \ y(t) = \frac{t}{dt}$$

(d)

$$y(t) = x(3t - 6)$$

(e)

$$y(t) = \cos [x(t)]$$

(f)

$$y(t) = e^{x(t)}$$

, $x(t)$ real

• 1.7-17 Figure P1.7-17 displays an input x1(t) to a linear time-invariant (LTI) system H, the corresponding output y1(t), and a second input x2(t).

  • (a) Bill suggests that x2(t) = 2x1(3t)−x1(t−1). Is Bill correct? If yes, prove it. If not, correct his error.
  • (b) Bill wants to know the output y2(t) in response to the input x2(t). Provide him with an expression for y2(t) in terms of y1(t). Use MATLAB to plot y2(t).

• 1.7-18 A linear time-invariant system H acts on input x(t) = u(t − 0.5) − u(t − 1.5) to produce output y(t) = H {x(t)} = 0.5u(t) + 0.5u(t − 1) −u(t −2).

  • (a) Is it possible that the system is causal? Explain your answer. If not causal, determine the shift necessary to make the system causal.
  • (b) Is it possible that the system is memoryless? Explain your answer.
  • (c) Suppose the output y(t) is applied to an identical system H to produce output

$$z(t) = H\{y(t)\} = H\{H\{x(t)\}\}$$

If possible, determine and sketch z(t). If not possible, explain why z(t) cannot be determined using the information given.

1.8-1 For the circuit depicted in Fig. P1.8-1, find the differential equations relating outputs y1(t) and y2(t) to the input x(t).

1.8-2 For the circuit depicted in Fig. P1.8-2, find the differential equations relating outputs y1(t) and y2(t) to the input x(t).

Figure P1.8-2

• 1.8-3 A simplified (one-dimensional) model of an automobile suspension system is shown in Fig. P1.8-3. In this case, the input is not a force but a displacement x(t) (the road contour). Find the differential equation relating the output y(t) (auto body displacement) to the input x(t) (the road contour).
• 1.8-4 A field-controlled dc motor is shown in Fig. P1.8-4. Its armature current ia is maintained constant. The torque generated by this motor is proportional to the field current if (torque= Kf if). Find the differential equation relating the output position θ to the input voltage x(t). The motor and load together have a moment of inertia J.
• 1.8-5 Water flows into a tank at a rate of qi units/s and flows out through the outflow valve at a rate of q⁰ units/s (Fig. P1.8-5). Determine the equation relating the outflow q⁰ to the input qi. The outflow rate is proportional to the head h. Thus q⁰ = Rh, where R is the valve resistance. Determine also the differential equation relating the head h to the input qi. [Hint: The net inflow of water in time Δt is (qi − q0)Δt. This inflow is also AΔh, where A is the cross section of the tank.]
• 1.8-6 Consider the circuit shown in Fig. P1.8-6, with input voltage x(t) and output currents y1(t), y2(t), and y3(t).
  • (a) What is the order of this system? Explain your answer.
  • (b) Determine the matrix representation for this system.
  • (c) Use Cramer’s rule to determine the output current y3(t) for the input voltage x(t) = [2− | cos(t)|]u(t −1).
• 1.10-1 Write state equations for the parallel RLC circuit in Fig. P1.8-2. Use the capacitor voltage q¹ and the inductor current q² as your state variables.

Show that every possible current or voltage in the circuit can be expressed in terms of q1, q² and the input x(t).

1.10-2 Write state equations for the third-order circuit shown in Fig. P1.10-2, using the inductor currents q1, q² and the capacitor voltage q³ as state variables. Show that every possible voltage or current in this circuit can be expressed as a linear combination of q1, q2, q3, and the input x(t). Also, at some instant t, it was found that

q¹ = 5, q² = 1, q³ = 2, and x = 10. Determine the voltage across and the current through every element in this circuit.

• 1.11-1 Provide MATLAB code and output that plots the odd portion xo(t) of the function x(t) = ²−^t cos(2πt)u(t−π ) over a suitable-length interval using a suitable number of points.
• 1.11-2 Provide MATLAB code and output that plots the even portion xe(t) of the function x(t) = ²−t/² cos(4πt)u(^{t −} 0.5) over a suitable ^t using t = 0.002 second between points.
• 1.11-3 Define ^x(t) ⁼ et(1+j2π )u(−t) and ^y(t) ⁼ Re* 2x −5−t 2 +.
  • (a) Use MATLAB to plot Re{x(t)} versus Im{x(at)} for a = 0.5, 1, and 2 and −10 ≤

Figure P1.10-2

t ≤ 10. How important is the scale factor a on the shape of the resulting figure?

• (b) Use MATLAB to plot y(t) over −10 ≤ t ≤ 10. Analytically determine the time t⁰ where y(t) has a jump discontinuity. Verify your calculation of t⁰ using the plot of y(t).
• (c) Use MATLAB and numerical integration to compute the energy Ex of signal x(t).
• (d) Use MATLAB and numerical integration to compute the energy Ey of signal y(t).
• 1.11-4 Consider the signal ^x(t) ^{= u}( ^{t 2} + 1) − u(t − ¹) ⁻ δ( ^{t 2} ). Define ^y(t) ⁼ $ ^t−³ −∞ ^x(τ )d^τ and ^z(t) ⁼ $ ^{∞ t} x(τ )dτ .
  • (a) Using MATLAB, accurately plot y(t).
  • (b) Using MATLAB, accurately plot z(t).
  • (c) Using MATLAB, accurately plot ^w(t) ^{= d} dt y(t) +z(t) .

TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS

In this book we consider two methods of analysis of linear time-invariant (LTI) systems: the time-domain method and the frequency-domain method. In this chapter we discuss the time-domain analysis of linear, time-invariant, continuous-time (LTIC) systems.

2.1 INTRODUCTION

For the purpose of analysis, we shall consider linear differential systems. This is the class of LTIC systems introduced in Ch. 1, for which the input x(t) and the output y(t) are related by linear differential equations of the form

$$\frac{d^{N}y(t)}{dt^{N}} + a_{1} \frac{d^{N-1}y(t)}{dt^{N-1}} + \dots + a_{N-1} \frac{dy(t)}{dt} + a_{N}y(t)$$

\n

$$= b_{N-M} \frac{d^{M}x(t)}{dt^{M}} + b_{N-M+1} \frac{d^{M-1}x(t)}{dt^{M-1}} + \dots + b_{N-1} \frac{dx(t)}{dt} + b_{N}x(t) \tag{2.1}$$

where all the coefficients ai and bi are constants. Using operator notation D to represent d/dt, we can express this equation as

$$(DN + a1DN-1 + ··· + aN-1D + aN)y(t)$$

= (b_N-MD^M + b_N-M+1D^M-1 + ··· + b_N-1D + b_N)x(t)

or

$$Q(D)y(t) = P(D)x(t)$$

\n(2.2)

where the polynomials Q(D) and P(D) are

$$Q(D) = DN + a1DN-1 + \dots + aN-1D + aN$$ $$P(D) = bN-MDM + bN-M+1DM-1 + \dots + bN-1D + bN$$

Theoretically the powers M and N in the foregoing equations can take on any value. However, practical considerations make M > N undesirable for two reasons. In Sec. 4.3-3, we shall show that an LTIC system specified by Eq. (2.1) acts as an (M − N)th-order differentiator. A differentiator represents an unstable system because a bounded input like the step input results in an unbounded output, δ(t). Second, noise is enhanced by a differentiator. Noise is a wideband signal containing components of all frequencies from 0 to a very high frequency approaching ∞. † Hence, noise contains a significant amount of rapidly varying components. We know that the derivative of any rapidly varying signal is high. Therefore, any system specified by Eq. (2.1) in which M > N will magnify the high-frequency components of noise through differentiation. It is entirely possible for noise to be magnified so much that it swamps the desired system output even if the noise signal at the system’s input is tolerably small. Hence, practical systems generally use M ≤ N. For the rest of this text we assume implicitly that M ≤ N. For the sake of generality, we shall assume M = N in Eq. (2.1).

In Ch. 1, we demonstrated that a system described by Eq. (2.2) is linear. Therefore, its response can be expressed as the sum of two components: the zero-input response and the zero-state response (decomposition property).‡ Therefore,

total response = zero-input response + zero-state response

The zero-input response is the system output when the input x(t) = 0, and thus it is the result of internal system conditions (such as energy storages, initial conditions) alone. It is independent of the external input x(t). In contrast, the zero-state response is the system output to the external input x(t) when the system is in zero state, meaning the absence of all internal energy storages: that is, all initial conditions are zero.

2.2 SYSTEM RESPONSE TO INTERNAL CONDITIONS: THE ZERO-INPUT RESPONSE

The zero-input response y0(t) is the solution of Eq. (2.2) when the input x(t) = 0 so that

$$Q(D)y_0(t) = 0$$ $$Q(D)y_0(t) = 0$$

If y(t) is the zero-state response, then y(t) is the solution of

$$Q(D)y(t) = P(D)x(t)$$

subject to zero initial conditions (zero-state). Adding these two equations, we have

$$Q(D)[y_0(t) + y(t)] = P(D)x(t)$$

Clearly, y0(t)+y(t) is the general solution of Eq. (2.2).

^† Noise is any undesirable signal, natural or manufactured, that interferes with the desired signals in the system. Some of the sources of noise are the electromagnetic radiation from stars, the random motion of electrons in system components, interference from nearby radio and television stations, transients produced by automobile ignition systems, and fluorescent lighting.

^‡ We can verify readily that the system described by Eq. (2.2) has the decomposition property. If y0(t) is the zero-input response, then, by definition,

or

$$(DN + a1DN-1 + \dots + aN-1D + aN)y0(t) = 0$$

\n(2.3)

A solution to this equation can be obtained systematically [1]. However, we will take a shortcut by using heuristic reasoning. Equation (2.3) shows that a linear combination of y0(t) and its N successive derivatives is zero, not at some values of t, but for all t. Such a result is possible if and only if y0(t) and all its N successive derivatives are of the same form. Otherwise their sum can never add to zero for all values of t. We know that only an exponential function eλ*^t* has this property. So let us assume that

$$y_0(t) = ce^{\lambda t}$$

is a solution to Eq. (2.3). Then

$$Dy_0(t) = \frac{dy_0(t)}{dt} = c\lambda e^{\lambda t}$$ $$D^2y_0(t) = \frac{d^2y_0(t)}{dt^2} = c\lambda^2 e^{\lambda t}$$ $$\vdots$$ $$D^Ny_0(t) = \frac{d^Ny_0(t)}{dt^N} = c\lambda^N e^{\lambda t}$$

Substituting these results in Eq. (2.3), we obtain

$$c(\lambda^N + a_1 \lambda^{N-1} + \dots + a_{N-1} \lambda + a_N)e^{\lambda t} = 0$$

For a nontrivial solution of this equation,

$$\lambda^{N} + a_{1}\lambda^{N-1} + \dots + a_{N-1}\lambda + a_{N} = 0$$

\n(2.4)

This result means that ce^λt is indeed a solution of Eq. (2.3), provided λ satisfies Eq. (2.4). Note that the polynomial in Eq. (2.4) is identical to the polynomial Q(D) in Eq. (2.3), with λ replacing D. Therefore, Eq. (2.4) can be expressed as

$$Q(\lambda) = 0$$

Expressing Q(λ) in factorized form, we obtain

$$Q(\lambda) = (\lambda - \lambda_1)(\lambda - \lambda_2) \cdots (\lambda - \lambda_N) = 0$$

\n(2.5)

Clearly, λ has N solutions: λ1, λ2, …, λN, assuming that all λ*ⁱ* are distinct. Consequently, Eq. (2.3) has N possible solutions: c1e^λ1*^t* , c2e^λ2*^t* , …, cNe^λNt , with c1, c2,…, cN as arbitrary constants. We

can readily show that a general solution is given by the sum of these N solutions† so that

$$y_0(t) = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t} + \dots + c_N e^{\lambda_N t}$$

(2.6)

where c1, c2, …, cN are arbitrary constants determined by N constraints (the auxiliary conditions) on the solution.

Observe that the polynomial Q(λ), which is characteristic of the system, has nothing to do with the input. For this reason the polynomial Q(λ) is called the characteristic polynomial of the system. The equation

$$Q(\lambda) = 0$$

is called the characteristic equation of the system. Equation (2.5) clearly indicates that λ1, λ2, …, λ*^N* are the roots of the characteristic equation; consequently, they are called the characteristic roots of the system. The terms characteristic values, eigenvalues, and natural frequencies are also used for characteristic roots.‡ The exponentials e^λit (i = 1, 2,…,n) in the zero-input response are the characteristic modes (also known as natural modes or simply as modes) of the system. There is a characteristic mode for each characteristic root of the system, and the zero-input response is a linear combination of the characteristic modes of the system.

An LTIC system’s characteristic modes comprise its single most important attribute. Characteristic modes not only determine the zero-input response but also play an important role in determining the zero-state response. In other words, the entire behavior of a system is dictated primarily by its characteristic modes. In the rest of this chapter we shall see the pervasive presence of characteristic modes in every aspect of system behavior.

REPEATED ROOTS

The solution of Eq. (2.3) as given in Eq. (2.6) assumes that the N characteristic roots λ1, λ2, …, λ*^N* are distinct. If there are repeated roots (same root occurring more than once), the form of the solution is modified slightly. By direct substitution we can show that the solution of the equation

$$(D - \lambda)^2 y_0(t) = 0$$

is given by

$$y_0(t) = (c_1 + c_2 t)e^{\lambda t}$$

† To prove this assertion, assume that y1(t), y2(t), …, yN(t) are all solutions of Eq. (2.3). Then

$$Q(D)y_1(t) = 0$$

\n

$$Q(D)y_2(t) = 0$$

\n

$$\vdots$$

\n

$$Q(D)y_N(t) = 0$$

Multiplying these equations by c1, c2, …, cN, respectively, and adding them together yield

$$Q(D)[c_1y_1(t) + c_2y_2(t) + \cdots + c_Ny_n(t)] = 0$$

This result shows that c1y1(t) + c2y2(t) +···+ cNyn(t) is also a solution of the homogeneous equation [Eq. (2.3)].

‡ Eigenvalue is German for “characteristic value.”