[PROBLEMS](#page-9-0)

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PROBLEMS

• 2.2-1 Determine the constants c1, c2, λ1, and λ² for each of the following second-order systems, which have zero-input responses of the form ^yzir(t) ^{= c}1eλ1^{t +}c2eλ2*^t* .
  • (a) y¨(t) + 2y˙(t) + 5y(t) = ¨x(t) − 5x(t) with yzir(0) = 2 and y˙zir(0) = 0.
  • (b) y¨(t) + 2y˙(t) + 5y(t) = ¨x(t) − 5x(t) with yzir(0) = 4 and y˙zir(0) = −1.
  • (c) ^d2 dt^{2 y}(t) ^{+ 2 d} dt y(t) = x(t) with yzir(0) = 1 and y˙zir(0) = 2.
  • (d) (D² +2D+10){y(t)} = (D⁵ −D){x(t)} with yzir(0) = ˙yzir(0) = 1.
  • (e) (D^{2 + 7 2D + 3 2} ){y(t)} = (D + 2){x(t)} with yzir(0) = 3 and y¨zir(0) = −8. [Caution: The

second IC is given in terms of the second derivative, not the first derivative].

• (f) 13y(t) ^{+ 4 d} dt ^y(t) ^{+ d2} dt^{2 y}(t) ^{= 2}x(t) ^{− 4 d} dt x(t) with yzir(0) = 3 and y¨zir(0) = −15. [Caution: The second IC is given in terms of the second derivative, not the first derivative].
• 2.2-2 Consider a linear time-invariant system with input x(t) and output y(t) that is described by the differential equation

$$(D+1)(D2-1) \{y(t)\} = (D5-1) \{x(t)\}$$

Furthermore, assume y(0) = ˙y(0) = ¨y(0) = 1.

224 CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS

• (a) What is the order of this system?
• (b) What are the characteristic roots of this system?
• (c) Determine the zero-input response yzir(t). Simplify your answer.
• 2.2-3 A real LTIC system with input x(t) and output y(t) is described by the following constant-coefficient linear differential equation:

$$(D3 + 9D) \{y(t)\} = (2D3 + 1) \{x(t)\}.$$

• (a) What is the characteristic equation of this system?
• (b) What are the characteristic modes of this system?
• (c) Assuming yzir(0) = 4, y˙zir(0) = −18, and y¨zir(0) = 0, determine this system’s zero-input response yzir(t). Simplify yzir(t) to include only real terms (i.e., no j’s should appear in your answer).
• 2.2-4 An LTIC system is specified by the equation

$$(D2 + 5D + 6)y(t) = (D + 1)x(t)$$

• (a) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of this system.
• (b) Find y0(t), the zero-input component of the response y(t) for t ≥ 0, if the initial conditions are y0(0−) = 2 and y˙0(0−) = −1.
• 2.2-5 Repeat Prob. 2.2-4 for

$$(D^2 + 4D + 4)y(t) = Dx(t)$$

and y0(0−) = 3, y˙0(0−) = −4.

2.2-6 Repeat Prob. 2.2-4 for

$$D(D+1)y(t) = (D+2)x(t)$$

and y0(0−) = ˙y0(0−) = 1.

2.2-7 Repeat Prob. 2.2-4 for

$$(D^2 + 9)y(t) = (3D + 2)x(t)$$

and y0(0−) = 0, y˙0(0−) = 6.

2.2-8 Repeat Prob. 2.2-4 for

$$(D2 + 4D + 13)y(t) = 4(D+2)x(t)$$

with

$$y_0(0^-) = 5$$

, $\dot{y}_0(0^-) = 15.98$ .

2.2-9 Repeat Prob. 2.2-4 for

$$D^2(D+1)y(t) = (D^2+2)x(t)$$

with y0(0−) = 4, y˙0(0−) = 3, and y¨0(0−) = −1.

2.2-10 Repeat Prob. 2.2-4 for

$$(D+1)(D^2 + 5D + 6)y(t) = Dx(t)$$

with y0(0−) = 2, y˙0(0−) = −1, and y¨0(0−) = 5.

• 2.2-11 A system is described by a constant-coefficient linear differential equation and has zero-input response given by ^y0(t) ^{= 2}e−^{t +}3.
  • (a) Is it possible for the system’s characteristic equation to be λ + 1 = 0? Justify your answer.
  • (b) Is it possible for the system’s characteristic equation to be ^√3(λ^{2 +}λ) ⁼ 0? Justify your answer.
  • (c) Is it possible for the system’s characteristic equation to be λ(λ + 1)² = 0? Justify your answer.
• 2.2-12 Consider the circuit of Fig. P2.2-12. Using operator notation, this system can be described as (D+a1){y(t)} = (b0D+b1){x(t)}.
  • (a) Determine the constants a1, b0, and b¹ in terms of the system components R, Rf , and C.
  • (b) Assume that R = 300 k, Rf = 1.2 M, and C = 5 µF. What is the zero-input response y0(t) of this system, assuming vC(0) = 1 V?

2.3-1 Determine the characteristic equation, characteristic modes, and impulse response h(t) for each of the following real LTIC systems. Since the systems are real, express each h(t) using only real terms (i.e., no j’s should appear in your answers).

Problems 225

• (a) (D² +1){y(t)} = 2D{x(t)}
• (b) (D³ +D){y(t)} = (2D³ +1){x(t)}

(c)

$$\frac{d^2}{dt^2}y(t) + 2\frac{d}{dt}y(t) + 5y(t) = 8x(t)$$

2.3-2 Find the unit impulse response of a system specified by the equation

$$(D2 + 4D + 3)y(t) = (D + 5)x(t)$$

2.3-3 Repeat Prob. 2.3-2 for

$$(D2 + 5D + 6)y(t) = (D2 + 7D + 11)x(t)$$

2.3-4 Repeat Prob. 2.3-2 for the first-order allpass filter specified by the equation

$$(D+1)y(t) = -(D-1)x(t)$$

2.3-5 Find the unit impulse response of an LTIC system specified by the equation

(D^{2 +}6D+9)y(t) ⁼ (2D+9)x(t)

• 2.3-6 Determine and plot the unit impulse response h(t) of the op-amp circuit of Fig. P2.2-12, assuming that R = 300 k, Rf = 1.2 M, and C = 5 µF.
• 2.3-7 A causal LTIC system with input x(t) and output y(t) is described by the constant coefficient integral equation

$$y(t) + \int 3y(t) dt + \int \int 2y(t) dt =$$ $$\int \int x(t) dt - \int \int \int x(t) dt.$$

• (a) Express this system as a constant coefficient linear differential equation in standard operator form.
• (b) Determine the characteristic modes of this system.

Figure P2.4-1

• (c) Determine the impulse response h(t) of this system.
• 2.4-1 Let f(t) = h1(t)∗h2(t), where h1(t) and h2(t) are shown in Fig. P2.4-1. In the following, use the graphical convolution procedure where you flip and shift h2(t).
  • (a) Plot h1(τ ) and h2(t − τ ) as functions of τ . Clearly label the plots, including necessary function parameterizations.
  • (b) Determine the (piecewise) regions of f(t) and set up the corresponding integrals that describe f(t) in those regions. Do not evaluate the integrals, only set them up!
  • (c) Determine f(1), which is f(t) evaluated at t = 1. Provide a number, not a formula.
• 2.4-2 Consider signals h(t) = u(t + 3) − 2u(t + 1) + u(t − 1) and x(t) = cos(t) u(t − π/2)− u(t −3π/2) . Let y(t) = x(t) ∗ h(t).
  • (a) Determine the last time tlast that y(t) is nonzero. That is, find the smallest value tlast such that y(t) = 0 for all t > tlast.
  • (b) Determine the approximate time tmax where y(t) is a maximum.
• 2.4-3 Consider signals h(t) = −u(t + 2) + ³u(^{t − 1}) ^{− 2}u(^{t − 5 2} ) and x(t) = sin(t)[u(t + 2π ) −u(t +π )]. Determine the approximate time tmin where y(t)=x(t)∗h(t) is a minimum. Note, the minimum value of y(t) = 0!
• 2.4-4 An LTIC system has impulse response h(t) = 3u(t − 2). For input x(t) shown in Fig. P2.4-4, use the graphical convolution procedure to determine yzsr(t) = h(t) ∗ x(t). Accurately sketch yzsr(t). When solving for yzsr(t), flip and shift x(t) and explicitly show all integration steps—even if apparently trivial!

Figure P2.4-4

2.4-5 Suppose an LTIC system has impulse response h(t) and input x(t) = u(t). Figure P2.4-5 shows x(t) and h(t + 1), respectively. Be careful! Figure P2.4-5 shows h(t +1), not h(t).

Figure P2.4-5

• (a) Is system h(t) causal? Mathematically justify your answer.
• (b) Use the graphical convolution procedure to determine yzsr(t) = x(t) ∗ h(t). Accurately sketch yzsr(t). When solving for yzsr(t), flip and shift x(t) and explicitly show all integration steps.

2.4-6 Repeat Prob. 2.4-5 using the signals of Fig. P2.4-6, rather than those of Fig. P2.4-5. Be careful! Figure P2.4-6 shows h(t −1), not h(t).

Figure P2.4-6

• 2.4-7 Suppose an LTIC system has impulse response h(t) and input x(t), both shown in Fig. P2.4-7. Use the graphical convolution procedure to determine yzsr(t) = x(t) ∗ h(t). Accurately sketch yzsr(t). When solving for yzsr(t), flip and shift x(t) and explicitly show all integration steps.
• 2.4-8 An LTIC system has impulse response h(t), as shown in Fig. P2.4-8. Let t have units of seconds. Let the input be x(t) = u(−t − 2) and designate the output as yzsr(t) = x(t) ∗ h(t).
  • (a) Use the graphical convolution procedure where h(t) is flipped and shifted to determine yzsr(t). Accurately plot your result.
  • (b) Use the graphical convolution procedure where x(t) is flipped and shifted to determine yzsr(t). Accurately plot your result.
• 2.4-9 If c(t) = x(t) ∗ g(t), then show that Ac = AxAg, where Ax,Ag, and Ac are the areas under x(t), g(t), and c(t), respectively. Verify this area property of convolution in Exs. 2.10 and 2.12.

Figure P2.4-7

2.4-10 If x(t) ∗ g(t) = c(t), then show that x(at) ∗ g(at) = |1/a|c(at). This time-scaling property of convolution states that if both x(t) and g(t) are time-scaled by a, their convolution is also time-scaled by a (and multiplied by |1/a|).

Figure P2.4-8

• 2.4-11 Show that the convolution of an odd and an even function is an odd function and the convolution of two odd or two even functions is an even function. [Hint: Use the time-scaling property of convolution in Prob. 2.4-10.]
• 2.4-12 Suppose an LTIC system has impulse response h(t) = (1 − t)[u(t) − u(t − 1)] and input x(t) = u(−t −1)+u(t −1). Use the graphical convolution procedure to determine yzsr(t) = x(t) ∗ h(t). Accurately sketch yzsr(t). When solving for yzsr(t), flip and shift h(t), explicitly show all integration steps, and simplify your answer.
• 2.4-13 Using direct integration, find ^e−atu(t) ^{∗ e}−btu(t).
• 2.4-14 Using direct integration, find u(t) ∗ u(t), ^e−atu(t) ^{∗ e}−atu(t), and tu(t) ^{∗ u}(t).
• 2.4-15 Using direct integration, find sin t u(t) ∗ u(t) and cos t u(t) ∗ u(t).
• 2.4-16 The unit impulse response of an LTIC system is

$$h(t) = e^{-t}u(t)$$

Find this system’s (zero-state) response y(t) if the input x(t) is:

$$(a) u(t)$$ $$(b) e^{-t}u(t)$$ $$(c) e^{-2t}u(t)$$

(d) sin 3t u(t)

Use the convolution table (Table 2.1) to find your answers.

2.4-17 Repeat Prob. 2.4-16 for

$$h(t) = [2e^{-3t} - e^{-2t}]u(t)$$

and if the input

$$x(t)$$

is:
\n(a) $u(t)$
\n(b) $e^{-t}u(t)$
\n(c) $e^{-2t}u(t)$

2.4-18 Repeat Prob. 2.4-16 for

$$h(t) = (1 - 2t)e^{-2t}u(t)$$

and input x(t) = u(t).

2.4-19 Repeat Prob. 2.4-16 for

$$h(t) = 4e^{-2t}\cos 3t u(t)$$

and each of the following inputs x(t):

• (a) u(t)
• (b) e−^t u(t)
• 2.4-20 Repeat Prob. 2.4-16 for

$$h(t) = e^{-t}u(t)$$

and each of the following inputs x(t):

• (a) e−2*^t u*(t)
• (b) e−2(t−3) u(t)
• (c) e−2*^t u*(t −3)
• (d) The gate pulse depicted in Fig. P2.4-20—and provide a sketch of y(t).

Figure P2.4-20

2.4-21 A first-order allpass filter impulse response is given by

$$h(t) = -\delta(t) + 2e^{-t}u(t)$$

• (a) Find the zero-state response of this filter for the input et u(−t).
• (b) Sketch the input and the corresponding zero-state response.
• 2.4-22 Figure P2.4-22 shows the input x(t) and the impulse response h(t) for an LTIC system. Let the output be y(t).
  • (a) By inspection of x(t) and h(t), find y(−1), y(0), y(1), y(2), y(3), y(4), y(5), and

Figure P2.4-22

y(6). Thus, by merely examining x(t) and h(t), you are required to see what the result of convolution yields at t = −1, 0, 1, 2, 3, 4, 5, and 6.

• (b) Find the system response to the input x(t).

• 2.4-23 The zero-state response of an LTIC system to an input ^x(t) ^{= 2}e−2*^{t u}(t) is ^y(t) = [4e*−2*^t* ⁺ 6e−3*^t* ]u(t). Find the impulse response of the system. [Hint: We have not yet developed a method of finding h(t) from the knowledge of the input and the corresponding output. Knowing the form of x(t) and y(t), you will have to make the best guess of the general form of h(t).]

• 2.4-24 Sketch the functions x(t) = 1/(t ² +1) and u(t). Now find x(t) ∗ u(t) and sketch the result.

• 2.4-25 Figure P2.4-25 shows x(t) and g(t). Find and sketch c(t) = x(t) ∗ g(t).

• 2.4-26 Find and sketch c(t) = x(t) ∗ g(t) for the functions depicted in Fig. P2.4-26.

• 2.4-27 Find and sketch c(t) = x1(t) ∗ x2(t) for the pairs of functions illustrated in Fig. P2.4-27.

• 2.4-28 Use Eq. (2.37) to find the convolution of x(t) and w(t), shown in Fig. P2.4-28.

• 2.4-29 Determine H(s), the transfer function of an ideal time delay of T seconds. Find your answer by two methods: using Eq. (2.39) and using Eq. (2.40).

• 2.4-30 Determine y(t) = x(t) ∗ h(t) for the signals depicted in Fig. P2.4-30.

• 2.4-31 Two linear time-invariant systems, each with impulse response h(t), are connected in cascade. Refer to Fig. P2.4-31. Given input x(t) = u(t), determine y(1). That is, determine the step response at time t = 1 for the cascaded system shown.

• 2.4-32 Consider the electric circuit shown in Fig. P2.4-32.

  • (a) Determine the differential equation that relates the input x(t) to output y(t). Recall that iC(t) ^{= C}dvC(t) dt and vL(t) ^{= L}diL(t) dt .
  • (b) Find the characteristic equation for this circuit, and express the root(s) of the characteristic equation in terms of L and C.
  • (c) Determine the zero-input response given an initial capacitor voltage of one volt and an initial inductor current of zero amps. That is, find y0(t) given vC(0) = 1 V and

Figure P2.4-26

Problems 229

Figure P2.4-27

iL(0) = 0 A. [Hint: The coefficient(s) in y0(t) are independent of L and C.]

• (d) Plot y0(t) for t ≥ 0. Does the zero-input response, which is caused solely by initial conditions, ever “die out”?
• (e) Determine the total response y(t) to the input ^x(t) ^{= e}−^t u(t). Assume an initial inductor current of iL(0−) = 0 A, an initial capacitor voltage of vC(0−) = 1 V, L = 1 H, and C = 1 F.

Figure P2.4-33

• 2.4-33 Two LTIC systems have impulse response functions given by h1(t) = (1−t)[u(t)−u(t−1)] and h2(t) = t[u(t +2)−u(t −2)].

  • (a) Carefully sketch the functions h1(t) and h2(t).
  • (b) Assume that the two systems are connected in parallel, as shown in Fig. P2.4-33a. Carefully plot the equivalent impulse response function, hp(t).
  • (c) Assume that the two systems are connected in cascade, as shown in Fig. P2.4-33b. Carefully plot the equivalent impulse response function, hs(t).

• 2.4-34 Consider the circuit shown in Fig. P2.4-34.

  • (a) Find the output y(t) given an initial capacitor voltage of y(0) = 2 volts and an input x(t) = u(t).
    • (b) Given an input x(t) = u(t − 1), determine the initial capacitor voltage y(0) so that the output y(t) is 0.5 volt at t = 2 seconds.

Figure P2.4-34

2.4-35 An analog signal is given by x(t) = t[u(t) − u(t − 1)], as shown in Fig. P2.4-35. Determine and plot y(t) = x(t) ∗ x(2t).

Figure P2.4-35

• 2.4-36 Consider the electric circuit shown in Fig. P2.4-36.
  • (a) Determine the differential equation that relates the input current x(t) to output current y(t). Recall that

$$v_L(t) = L \frac{di_L(t)}{dt}$$

• (b) Find the characteristic equation for this circuit, and express the root(s) of the characteristic equation in terms of L1, L2, and R.
• (c) Determine the zero-input response given initial inductor currents of one ampere each. That is, find y0(t) given iL¹ (0) = iL² (0) = 1 A.

Figure P2.4-36

Figure P2.4-38

• 2.4-37 An LTI system has step response given by ^g(t) ^{= e}−^{t u}(t) ^{− e}−2*^t u*(t). Determine the output of this system y(t) given an input x(t) = δ(t −π ) −cos( ^√3)u(t).
• 2.4-38 The periodic signal x(t) shown in Fig. P2.4-38 is input to a system with impulse response function h(t) = t[u(t) − u(t − 1.5)], also shown in Fig. P2.4-38. Use convolution to determine the output y(t) of this system. Plot y(t) over (−3 ≤ t ≤ 3).
• 2.4-39 Consider the electric circuit shown in Fig. P2.4-39.
  • (a) Determine the differential equation relating input x(t) to output y(t).
  • (b) Determine the output y(t) in response to the input ^x(t) ^{= 4}te−3t/2u(t). Assume component values of R = 1 , C¹ = 1 F, and C² = 2 F, and initial capacitor voltages of VC¹ = 2 V and VC² = 1 V.

Figure P2.4-39

• 2.4-40 An LTIC system has impulse response h(t) = 3e−|t^| .
  • (a) Is the system causal? Mathematically justify your answer.
  • (b) Determine the zero-state response of this system if the input is x(t) = u(2−t).
• 2.4-41 A cardiovascular researcher is attempting to model the human heart. He has recorded ventricular pressure, which he believes corresponds to the heart’s impulse response

function h(t), as shown in Fig. P2.4-41. Comment on the function h(t) shown in Fig. P2.4-41. Can you establish any system properties, such as causality or stability? Do the data suggest any reason to suspect that the measurement is not a true impulse response?

• 2.4-42 Consider an integrator system, ^y(t) ⁼ $ ^t −∞ ^x(τ )d^τ .
  • (a) What is the unit impulse response hi(t) of this system?
  • (b) If two such integrators are put in parallel, what is the resulting impulse response hp(t)?
  • (c) If two such integrators are put in series, what is the resulting impulse response hs(t)?
• 2.4-43 The autocorrelation of a function x(t) is given by rxx(t) ⁼ $ ^∞ −∞ ^x(τ )x(τ ^{− t})d^τ . This equation is computed in a manner nearly identical to convolution.
  • (a) Show rxx(t) = x(t) ∗ x(−t).
  • (b) Determine and plot rxx(t) for the signal x(t) depicted in Fig. P2.4-43. [Hint: rxx(t) = rxx(−t).]

Figure P2.4-43

2.4-44 Consider the circuit shown in Fig. P2.4-44. This circuit functions as an integrator. Assume ideal op-amp behavior and recall that

dVC(t)

• (a) Determine the differential equation that relates the input x(t) to the output y(t).
• (b) This circuit does not behave well at dc. Demonstrate this by computing the zero-state response y(t) for a unit step input x(t) = u(t).
• 2.4-45 Derive the result in Eq. (2.37) in another way. As mentioned in Ch. 1 (Fig. 1.27b), it is possible to express an input in terms of its step components, as shown in Fig. P2.4-45. Find the system response as a sum of the responses to the step components of the input.

2.4-46 Show that an LTIC system response to an everlasting sinusoid cosω0t is given by

$$y(t) = |H(j\omega_0)| \cos [\omega_0 t + \angle H(j\omega_0)]$$

where

$$H(j\omega) = \int_{-\infty}^{\infty} h(t)e^{-j\omega t} dt$$

assuming the integral on the right-hand side exists.

2.4-47 A line charge is located along the x axis with a charge density Q(x) coulombs per meter. Show that the electric field E(x) produced by this line charge at a point x is given by

$$E(x) = Q(x) * h(x)$$

where h(x) = 1/4π x2. [Hint: The charge over an interval τ located at τ = nτ is Q(nτ )τ . Also by Coulomb’s law, the electric field E(r) at a distance r from a charge q coulombs is given by E(r) = q/4π r2.]

• 2.4-48 A system is called complex if a real-valued input can produce a complex-valued output. Suppose a linear time-invariant complex system has impulse response h(t) = j[u(−t + 2) − u(−t)].

  • (a) Is this system causal? Explain.
  • (b) Use convolution to determine the zero-state response y1(t) of this system in response to the unit-duration pulse x1(t) = u(t) − u(t − 1).
  • (c) Using the result from part (a), determine the zero-state response y2(t) in response to x2(t) = 2u(t −1)−u(t −2)−u(t −3).

• 2.5-1 Explain, with reasons, whether the LTIC systems described by the following equations are (i) stable or unstable in the BIBO sense; (ii) asymptotically stable, unstable, or marginally stable. Assume that the systems are controllable and observable.

  • (a) (D² +8D+12)y(t) = (D−1)x(t)
  • (b) D(D² +3D+2)y(t) = (D+5)x(t)
  • (c) D2(D² +2)y(t) = x(t)
  • (d) (D+1)(D² −6D+5)y(t) = (3D+1)x(t)

• 2.5-2 Repeat Prob. 2.5-1 for the following:

  • (a) (D+1)(D² +2D+5)2y(t) = x(t)
  • (b) (D+1)(D² +9)y(t) = (2D+9)x(t)
  • (c) (D+1)(D² +9)2y(t) = (2D+9)x(t)
  • (d) (D² +1)(D² +4)(D² +9)y(t) = 3Dx(t)

• 2.5-3 Consider an LTIC system with unit impulse response ^h(t) ⁼ et 2 ³ cos( ^{3 2 t}) ^{+ 1 3} sin(πt) u(123 − t). Is this system BIBO-stable? Mathematically justify your answer.

• 2.5-4 Consider an LTIC system with unit impulse response ^h(t) ^{= 1 t} u(t −T).

  • (a) Determine, if possible, the value(s) of T for which this system is causal.
  • (b) Determine, if possible, the value(s) of T for which this system is BIBO-stable. Justify all answers mathematically.

• 2.5-5 You are given the choice of a system that is guaranteed internally stable or a system that is guaranteed externally stable. Which do you choose? Why?

• 2.5-6 For a certain LTIC system, the impulse response h(t) = u(t).

  • (a) Determine the characteristic root(s) of this system.
  • (b) Is this system asymptotically or marginally stable, or is it unstable?
  • (c) Is this system BIBO-stable?
  • (d) What can this system be used for?

• 2.5-7 In Sec. 2.5 we demonstrated that for an LTIC system, the condition of Eq. (2.45) is sufficient for BIBO stability. Show that this is also a necessary condition for BIBO stability in such systems. In other words, show that if Eq. (2.45) is not satisfied, then there exists a bounded input that produces an unbounded output. [Hint: Assume that a system exists for which h(t) violates Eq. (2.45) and yet produces an output that is bounded for every bounded input. Establish the contradiction in this statement by considering an input x(t) defined by x(t1−τ )=1 when h(τ ) ≥ 0 and x(t¹ − τ ) = −1 when h(τ ) < 0, where t¹ is some fixed instant.]

• 2.5-8 An analog LTIC system with impulse response function h(t) = u(t + 2) − u(t − 2) is presented with an input x(t) = t(u(t) −u(t −2)).

  • (a) Determine and plot the system output y(t) = x(t) ∗ h(t).
  • (b) Is this system stable? Is this system causal? Justify your answers.

• 2.5-9 A system has an impulse response function shaped like a rectangular pulse, h(t) = u(t) − u(t − 1). Is the system stable? Is the system causal?

• 2.5-10 A continuous-time LTI system has impulse response function ^h(t) ⁼%^{∞ i}=0(0.5)ⁱ δ(t−i).

  • (a) Is the system causal? Prove your answer.
  • (b) Is the system stable? Prove your answer.

• 2.6-1 Data at a rate of 1 million pulses per second are to be transmitted over a certain communications channel. The unit step response g(t) for this channel is shown in Fig. P2.6-1.

  • (a) Can this channel transmit data at the required rate? Explain your answer.
  • (b) Can an audio signal consisting of components with frequencies up to 15 kHz be transmitted over this channel with reasonable fidelity?

Figure P2.6-1

• 2.6-2 Determine a frequency ω that will cause the input x(t) = cos(ωt) to produce a strong response when applied to the system described by (D² + 2D + 13/4){y(t)} = x(t). Carefully explain your choice.
• 2.6-3 Figure P2.6-3 shows the impulse response h(t) of a lowpass LTIC system. Determine the peak amplitude A and time constant Th so that rectangular impulse response hˆ(t) is an appropriate approximation of h(t). The two graphs of Fig. P2.6-3 are not necessarily drawn to the same scale.

Figure P2.6-3

• 2.6-4 A certain communication channel has a bandwidth of 10 kHz. A pulse of 0.5 ms duration is transmitted over this channel.

  • (a) Determine the width (duration) of the received pulse.
  • (b) Find the maximum rate at which these pulses can be transmitted over this channel without interference between the successive pulses.

• 2.6-5 A first-order LTIC system has a characteristic root ^λ = −104.

  • (a) Determine Tr, the rise time of its unit step input response.
  • (b) Determine the bandwidth of this system.
  • (c) Determine the rate at which the information pulses can be transmitted through this system.

• 2.6-6 A lowpass system with a 6 MHz cutoff frequency needs to transmit data pulse that are 500 ⁶ ns wide. Determine a suitable transmission rate Frate (pulses/s) for this system.

• 2.6-7 Sketch an impulse response h(t) of a non-causal LP system that has an approximate cutoff frequency of 5 kHz. Since many solutions are possible, be sure to properly justify your answer.

• 2.6-8 Two LTIC transmission channels are available: the first has impulse response h1(t) = u(t) − u(t − 1) and the second has impulse response h2(t) = δ(t) + 0.5δ(t − 1) + 0.25δ(t − 2). Explain which channel is better suited for the transmission of high-speed digital data (pulses).

• 2.6-9 Consider a linear time-invariant system with impulse response h(t) shown in Fig. P2.6-9. Outside the interval shown, h(t) = 0.

Figure P2.6-9

• (a) What is the rise time Tr of this system? Remember, rise time is the time between the application of a unit step and the moment at which the system has “fully” responded.
• (b) Suppose h(t) represents the response of a communication channel. What conditions might cause the channel to have such an impulse response? What is the maximum

average number of pulses per unit time that can be transmitted without causing interference? Justify your answer.

• (c) Determine the system output y(t) = x(t) ∗ h(t) for x(t) = [u(t − 2) − u(t)]. Accurately sketch y(t) over (0 ≤ t ≤ 10).
• 2.6-10 A lowpass LTIC system has impulse response ^h(t) = −te−^t u(t).
  • (a) Accurately sketch h(t).
  • (b) Describe a rectangular impulse response hˆ(t) as an appropriate approximation of h(t). What is the approximate cutoff frequency of this system?
• 2.6-11 A lowpass LTIC system has impulse response h(t), as shown in Fig. P2.4-8.
  • (a) As discussed in Sec. 2.6-2, determine a rectangular approximation hˆ(t) to h(t).
  • (b) Using hˆ(t), what is the time constant Th of this lowpass system?
  • (c) Using hˆ(t), what is the approximate radian cutoff frequency ω*^c* of this lowpass system?
  • (d) Assuming a frequency ω⁰ ωc, what is the system response y(t) to the input x(t) = sin(ω0t +π/3)?
• 2.6-12 A first CT lowpass system with time constant T¹ = 4 µs is put in series with a second CT lowpass system with time constant T² = 2 µs. Make an educated sketch of the overall impulse response function hseries(t). What is the time constant Tseries of the overall series-connected system?
• 2.7-1 An LTIC system with input x(t) and output y(t) is described by the following constant coefficient linear differential equation:

$$(D4 - 16) \{y(t)\} = (D - 2) \{x(t)\}.$$

• (a) What are the 4 characteristic roots of this system (λ1, λ2, λ3, and λ4)? Determine the roots by hand and then verify your answers using MATLAB’s roots command.
• (b) From Eq. (2.17), computing h(t) requires a signal ^y˜n(t) ⁼ %^{4 k}=¹ cke^λkt . First, determine a matrix representation of the system of equations needed to solve for the four coefficients ck. Second, write MATLAB code that computes the length-4 column vector of coefficients ck.

2.7-2 Define ^x(t) ^{= 2}u(^{t + 2 3} ) − 2u(t). Further, define the periodic signal h1(t) as

$$h_1(t) = \begin{cases} t & 0 \le t < 1 \\ h_1(t+1) & \forall t \end{cases}$$

Lastly, define the aperiodic signal h2(t) in terms of h1(t) as

$$h_2(t) = h_1(t)[u(t-1) - u(t-2)]$$

• (a) Use MATLAB to plot x(t), h1(t), and h2(t) over the interval −2.5 ≤ t ≤ 3.5.
• (b) Using the graphical convolution procedure, compute y2(t) = x(t) ∗ h2(t).
• (c) Compute by hand and then MATLAB plot y1(t) = x(t) ∗ h1(t). Modify program CH2MP4.m in Sec. 2.7-4 to validate your analytical result.
• 2.7-3 Consider the circuit shown in Fig. P2.7-3. Assume ideal op-amp behavior and recall that

$$i_C(t) = C \frac{dV_C(t)}{dt}$$

Without a feedback resistor Rf , the circuit functions as an integrator and is unstable, particularly at dc. A feedback resistor Rf corrects this problem and results in a stable circuit that functions as a “lossy” integrator.

Figure P2.7-3

• (a) Determine the differential equation that relates the input x(t) to the output y(t). What is the corresponding characteristic equation?
• (b) To demonstrate that this “lossy” integrator is well behaved at dc, determine the zero-state

response y(t) given a unit step input x(t) = u(t).

• (c) Investigate the effect of 10% resistor and 25% capacitor tolerances on the system’s characteristic root(s).

• 2.7-4 Consider the electric circuit shown in Fig. P2.7-4. Let C¹ = C² = 10 µF, R¹ = R² = 100 k, and R³ = 50 k.

  • (a) Determine the corresponding differential equation describing this circuit. Is the circuit BIBO-stable?
  • (b) Determine the zero-input response y0(t) if the output of each op amp initially reads one volt.

• (c) Determine the zero-state response y(t) to a step input x(t) = u(t).

• (d) Investigate the effect of 10% resistor and 25% capacitor tolerances on the system’s characteristic roots.

• 2.7-5 Input x(t)=3[u(t)−u(t−1)]+2[u(t−2)−u(t− 3)] is applied to a lowpass LTIC system with impulse response h(t) = (4 − t)[u(t) − u(t − 2)] to produce output y(t) = x(t) ∗ h(t). Modify program CH2MP4.m in Sec. 2.7-4 to perform the graphical convolution procedure to produce a plot of y(t).