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Chapter 2 Problems

The first section of problems belongs to the basic category, and the answers are provided in the back of the book. The remaining three sections contain problems belonging to the basic, advanced, and extension categories, respectively. Extension problems introduce applications, concepts, or methods beyond those presented in the text.

BASIC PROBLEMS WITH ANSWERS

2.1

Let

$$x[n]=\delta [n]+2\delta [n-1]-\delta [n-3],h[n]=2\delta [n+1]+2\delta [n-1].$$

Compute and plot each of the following convolutions: (a) $y_1[n]=x[n]\ast h[n]$
(b) $y_2[n]=x[n+2]\ast h[n]$
(c) $y_3[n]=x[n]\ast h[n+2]$

2.2

Consider the signal

$$h[n]={\left(\frac{1}{2}\right)}^{n-1}(u[n+3]-u[n-10]).$$

Express $A$ and $B$ in terms of $n$ so that the following equation holds: [Equation missing – original had: equation441, 216, 785, 261]

2.3

Consider an input $x[n]$ and a unit impulse response $h[n]$ given by

$$x[n]={\left(\frac{1}{2}\right)}^{n-2}u[n-2],h[n]=u[n+2].$$

Determine and plot the output $y[n]=x[n]\ast h[n]$.

2.4

Compute and plot $y[n]=x[n]\ast h[n]$, where [Equation missing – original had: equation489, 441, 736, 533]

2.5

Let [Equation missing – original had: equation352, 578, 874, 625] where $N\le 9$ is an integer. Determine the value of $N$, given that $y[n]=x[n]\ast h[n]$ and

$$y[4]=5,y[14]=0.$$

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2.6

Compute and plot the convolution $y[n]=x[n]\ast h[n]$, where

$$x[n]={\left(\frac{1}{3}\right)}^{-n}u[-n-1],h[n]=u[n-1].$$

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2.7

A linear system $S$ has the relationship

$$y[n]=\sum _{k=-\infty }^{\infty }x[k]g[n-2k]$$

between its input $x[n]$ and its output $y[n]$, where $g[n]=u[n]-u[n-4]$.

2.8

Determine and sketch the convolution of the following two signals: [Equation missing – original had: equation455, 210, 713, 293]

$$h(t)=\delta (t+2)+2\delta (t+1).$$

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2.9

Let

$$h(t)=e^{2t}u(-t+4)+e^{-2t}u(t-5).$$

Determine $A$ and $B$ such that [Equation missing – original had: equation427, 427, 744, 492]

2.10

Suppose that [Equation missing – original had: equation471, 545, 689, 591] and $h(t)=x(t/\alpha )$, where $0<\alpha \le 1$. (a) Determine and sketch $y(t)=x(t)\ast h(t)$.
(b) If $dy(t)/dt$ contains only three discontinuities, what is the value of $\alpha$?

2.11

Let

$$x(t)=u(t-3)-u(t-5),h(t)=e^{-3t}u(t).$$

(a) Compute $y(t)=x(t)\ast h(t)$.
(b) Compute $g(t)=(dx(t)/dt)\ast h(t)$.
(c) How is $g(t)$ related to $y(t)$?

2.12

Let

$$y(t)=e^{-t}u(t)\ast \sum _{k=-\infty }^{\infty }\delta (t-3k).$$

Show that $y(t)=A{e}^{-t}$ for $0\le t<3$, and determine the value of $A$.

2.13

Consider a discrete-time system $S_1$ with impulse response

$$h[n]={\left(\frac{1}{5}\right)}^{n}u[n].$$

(a) Find the integer $A$ such that $h[n]-Ah[n-1]=\delta [n]$.
(b) Using the result from part (a), determine the impulse response $g[n]$ of an LTI system $S_2$ which is the inverse system of $S_1$.

2.14

Which of the following impulse responses correspond(s) to stable LTI systems? (a) $h_1(t)=e^{-(1-2j)t}u(t)$
(b) $h_2(t)=e^{-t}\cos (2t)u(t)$

2.15

Which of the following impulse responses correspond(s) to stable LTI systems? (a) $h_1[n]=n\cos \left(\frac{\pi }{4}n\right)u[n]$
(b) $h_2[n]=3^{n}u[-n+10]$

2.16

For each of the following statements, determine whether it is true or false: (a) If $x[n]=0$ for $n<N_1$ and $h[n]=0$ for $n<N_2$, then $x[n]\ast h[n]=0$ for $n<N_1+N_2$.
(b) If $y[n]=x[n]\ast h[n]$, then $y[n-1]=x[n-1]\ast h[n-1]$.
(c) If $y(t)=x(t)\ast h(t)$, then $y(-t)=x(-t)\ast h(-t)$.
(d) If $x(t)=0$ for $t>T_1$ and $h(t)=0$ for $t>T_2$, then $x(t)\ast h(t)=0$ for $t>T_1+T_2$.

2.17

Consider an LTI system whose input $x(t)$ and output $y(t)$ are related by the differential equation

$$\frac{d}{dt}y(t)+4y(t)=x(t).\text{(P2.17-1)}$$

The system also satisfies the condition of initial rest. (a) If $x(t)=e^{(-1+3j)t}u(t)$, what is $y(t)$?
(b) Note that $\Re \{x(t)\}$ will satisfy eq. (P2.17-1) with $\Re \{y(t)\}$. Determine the output $y(t)$ of the LTI system if

$$x(t)=e^{-t}\cos (3t)u(t).$$

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2.18

Consider a causal LTI system whose input $x[n]$ and output $y[n]$ are related by the difference equation

$$y[n]=\frac{1}{4}y[n-1]+x[n].$$

Determine $y[n]$ if $x[n]=\delta [n-1]$.

2.19

Consider the cascade of the following two systems $S_1$ and $S_2$, as depicted in Figure P2.19:

Figure P2.19 $S_1$: causal LTI, $$w[n]=\frac{1}{2}w[n-1]+x[n];$$

$S_2$: causal LTI,

$$y[n]=\alpha y[n-1]+\beta w[n].$$

The difference equation relating $x[n]$ and $y[n]$ is:

$$y[n]=-\frac{1}{8}y[n-2]+\frac{3}{4}y[n-1]+x[n].$$

(a) Determine $\alpha$ and $\beta$.
(b) Show the impulse response of the cascade connection of $S_1$ and $S_2$.

2.20

Evaluate the following integrals:

$${\int }_{-\infty }^{\infty }{u}_0(t)\cos (t)dt,{\int }_0^5\sin (2\pi t)\delta (t+3)dt,{\int }_{-5}^5{u}_1(1-\tau )\cos (2\pi \tau )d\tau .$$

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BASIC PROBLEMS

2.21

Compute the convolution $y[n]=x[n]\ast h[n]$ of the following pairs of signals: (a) [Equation missing – original had: equation234, 532, 499, 577]
(b) $x[n]=h[n]={\alpha }^{n}u[n]$
(c) $x[n]={\left(-\frac{1}{2}\right)}^{n}u[n-4]$, $h[n]=4^{n}u[2-n]$
(d) $x[n]$ and $h[n]$ are as in Figure P2.21.

Figure P2.21 *[Image: figure P2.21 – original had: image[[477, 691, 924, 766]]]* --- ### 2.22 For each of the following pairs of waveforms, use the convolution integral to find the response $y(t)$ of the LTI system with impulse response $h(t)$ to the input $x(t)$. Sketch your results. (a) $x(t) = e^{-\alpha t}u(t)$, $h(t) = e^{-\beta t}u(t)$ (b) $x(t) = u(t) - 2u(t-2) + u(t-5)$, $h(t) = e^{2t}u(1-t)$ (c) $x(t)$ and $h(t)$ are as in Figure P2.22(a). (d) $x(t)$ and $h(t)$ are as in Figure P2.22(b). (e) $x(t)$ and $h(t)$ are as in Figure P2.22(c). *[Images: figures P2.22(a), P2.22(b), P2.22(c) – original had: image[[348,210,805,358]] etc.]* --- ### 2.23 Let $h(t)$ be the triangular pulse shown in Figure P2.23(a), and let $x(t)$ be the impulse train depicted in Figure P2.23(b). That is, $$x(t)=\sum _{k=-\infty }^{+\infty }\delta (t-kT).$$

Determine and sketch $y(t)=x(t)\ast h(t)$ for the following values of $T$: (a) $T=4$ (b) $T=2$ (c) $T=3/2$ (d) $T=1$

2.24

Consider the cascade interconnection of three causal LTI systems, illustrated in Figure P2.24(a). The impulse response $h_2[n]$ is

$$h_2[n]=u[n]-u[n-2],$$

and the overall impulse response is as shown in Figure P2.24(b).

Figure P2.24 (a) Find the impulse response $h_1[n]$. (b) Find the response of the overall system to the input $$x[n]=\delta [n]-\delta [n-1].$$

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2.25

Let the signal

$$y[n]=x[n]\ast h[n],$$

where

$$x[n]=3^{n}u[-n-1]+{\left(\frac{1}{3}\right)}^{n}u[n],h[n]={\left(\frac{1}{4}\right)}^{n}u[n+3].$$

(a) Determine $y[n]$ without utilizing the distributive property of convolution.
(b) Determine $y[n]$ utilizing the distributive property of convolution.

2.26

Consider the evaluation of

$$y[n]=x_1[n]\ast x_2[n]\ast x_3[n],$$

where $x_1[n]=(0.5{)}^{n}u[n]$, $x_2[n]=u[n+3]$, and $x_3[n]=\delta [n]-\delta [n-1]$. (a) Evaluate the convolution $x_1[n]\ast x_2[n]$.
(b) Convolve the result of part (a) with $x_3[n]$ in order to evaluate $y[n]$.
(c) Evaluate the convolution $x_2[n]\ast x_3[n]$.
(d) Convolve the result of part (c) with $x_1[n]$ in order to evaluate $y[n]$.

2.27

We define the area under a continuous-time signal $v(t)$ as

$$A_v={\int }_{-\infty }^{+\infty }v(t)dt.$$

Show that if $y(t)=x(t)\ast h(t)$, then

$$A_y=A_x{A}_h.$$

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2.28

The following are the impulse responses of discrete-time LTI systems. Determine whether each system is causal and/or stable. Justify your answers. (a) $h[n]={\left(\frac{1}{5}\right)}^{n}u[n]$
(b) $h[n]=(0.8{)}^{n}u[n+2]$
(c) $h[n]={\left(\frac{1}{2}\right)}^{n}u[-n]$
(d) $h[n]=5^{n}u[3-n]$
(e) $h[n]={\left(-\frac{1}{2}\right)}^{n}u[n]+(1.01{)}^{n}u[n-1]$
(f) $h[n]={\left(-\frac{1}{2}\right)}^{n}u[n]-(1.01{)}^{n}u[1-n]$
(g) $h[n]=n{\left(\frac{1}{3}\right)}^{n}u[n-1]$

2.29

The following are the impulse responses of continuous-time LTI systems. Determine whether each system is causal and/or stable. Justify your answers. (a) $h(t)=e^{-4t}u(t-2)$
(b) $h(t)=e^{-6t}u(3-t)$
(c) $h(t)=e^{-2t}u(t+50)$
(d) $h(t)=e^{2t}u(-1-t)$

2.30

Consider the first-order difference equation

$$y[n]+2y[n-1]=x[n].$$

Assuming the condition of initial rest (i.e., if $x[n]=0$ for $n<n_0$, then $y[n]=0$ for $n<n_0$), find the impulse response of a system whose input and output are related by this difference equation. You may solve the problem by rearranging the difference equation so as to express $y[n]$ in terms of $y[n-1]$ and $x[n]$ and generating the values of $y[0],y[+1],y[+2],\dots$ in that order.

2.31

Consider the LTI system initially at rest and described by the difference equation

$$y[n]+2y[n-1]=x[n]+2x[n-2].$$

Find the response of this system to the input depicted in Figure P2.31 by solving the difference equation recursively.

Figure P2.31 --- ### 2.32 Consider the difference equation $$y[n]-\frac{1}{2}y[n-1]=x[n],\text{(P2.32-1)}$$

and suppose that

$$x[n]={\left(\frac{1}{3}\right)}^{n}u[n].\text{(P2.32-2)}$$

Assume that the solution $y[n]$ consists of the sum of a particular solution $y_p[n]$ to eq. (P2.32-1) and a homogeneous solution $y_h[n]$ satisfying the equation

$$y_h[n]-\frac{1}{2}{y}_h[n-1]=0.$$

(a) Verify that the homogeneous solution is given by $y_h[n]=A{\left(\frac{1}{2}\right)}^n$.
(b) Let us consider obtaining a particular solution $y_p[n]$ such that

$$y_p[n]-\frac{1}{2}{y}_p[n-1]={\left(\frac{1}{3}\right)}^{n}u[n].$$

By assuming that $y_p[n]$ is of the form $B{\left(\frac{1}{3}\right)}^n$ for $n\ge 0$, and substituting this in the above difference equation, determine the value of $B$.
(c) Suppose that the LTI system described by eq. (P2.32-1) and initially at rest has as its input the signal specified by eq. (P2.32-2). Since $x[n]=0$ for $n<0$, we have that $y[n]=0$ for $n<0$. Also, from parts (a) and (b) we have that $y[n]$ has the form

$$y[n]=A{\left(\frac{1}{2}\right)}^n+B{\left(\frac{1}{3}\right)}^n$$

for $n\ge 0$. In order to solve for the unknown constant $A$, we must specify a value for $y[n]$ for some $n\ge 0$. Use the condition of initial rest and eqs. (P2.32-1) and (P2.32-2) to determine $y[0]$. From this value determine the constant $A$. The result of this calculation yields the solution to the difference equation (P2.32-1) under the condition of initial rest, when the input is given by eq. (P2.32-2).

2.33

Consider a system whose input $x(t)$ and output $y(t)$ satisfy the first-order differential equation

$$\frac{dy(t)}{dt}+2y(t)=x(t).\text{(P2.33-1)}$$

The system also satisfies the condition of initial rest. (a) (i) Determine the system output $y_1(t)$ when the input is $x_1(t)=e^{3t}u(t)$.
 (ii) Determine the system output $y_2(t)$ when the input is $x_2(t)=e^{2t}u(t)$.
 (iii) Determine the system output $y_3(t)$ when the input is $x_3(t)=\alpha e^{3t}u(t)+\beta e^{2t}u(t)$, where $\alpha$ and $\beta$ are real numbers. Show that $y_3(t)=\alpha y_1(t)+\beta y_2(t)$.
 (iv) Now let $x_1(t)$ and $x_2(t)$ be arbitrary signals such that

$$x_1(t)=0\text{ for }t<t_1,x_2(t)=0\text{ for }t<t_2.$$

Letting $y_1(t)$ be the system output for input $x_1(t)$, $y_2(t)$ for input $x_2(t)$, and $y_3(t)$ for $x_3(t)=\alpha x_1(t)+\beta x_2(t)$, show that

$$y_3(t)=\alpha y_1(t)+\beta y_2(t).$$

We may therefore conclude that the system under consideration is linear. (b) (i) Determine the system output $y_1(t)$ when the input is $x_1(t)=K{e}^{2t}u(t)$.
 (ii) Determine the system output $y_2(t)$ when the input is $x_2(t)=K{e}^{2(t-T)}u(t-T)$. Show that $y_2(t)=y_1(t-T)$.
 (iii) Now let $x_1(t)$ be an arbitrary signal such that $x_1(t)=0$ for $t<t_0$. Letting $y_1(t)$ be the system output for input $x_1(t)$ and $y_2(t)$ for $x_2(t)=x_1(t-T)$, show that

$$y_2(t)=y_1(t-T).$$

We may therefore conclude that the system under consideration is time invariant. In conjunction with the result derived in part (a), we conclude that the given system is LTI. Since this system satisfies the condition of initial rest, it is causal as well.

2.34

The initial rest assumption corresponds to a zero-valued auxiliary condition being imposed at a time determined in accordance with the input signal. In this problem we show that if the auxiliary condition used is nonzero or if it is always applied at a fixed time (regardless of the input signal) the corresponding system cannot be LTI. Consider a system whose input $x(t)$ and output $y(t)$ satisfy the first-order differential equation (P2.33-1). (a) Given the auxiliary condition $y(1)=1$, use a counterexample to show that the system is not linear.
(b) Given the auxiliary condition $y(1)=1$, use a counterexample to show that the system is not time invariant.
(c) Given the auxiliary condition $y(1)=1$, show that the system is incrementally linear.
(d) Given the auxiliary condition $y(1)=0$, show that the system is linear but not time invariant.
(e) Given the auxiliary condition $y(0)+y(4)=0$, show that the system is linear but not time invariant.

2.35

In the previous problem we saw that application of an auxiliary condition at a fixed time (regardless of the input signal) leads to the corresponding system being not time-invariant. In this problem, we explore the effect of fixed auxiliary conditions on the causality of a system. Consider a system whose input $x(t)$ and output $y(t)$ satisfy the first-order differential equation (P2.33-1). Assume that the auxiliary condition associated with the differential equation is $y(0)=0$. Determine the output of the system for each of the following two inputs: (a) $x_1(t)=0$, for all $t$
(b) [Equation missing – original had: equation233, 620, 460, 662]
Observe that if $y_1(t)$ is the output for input $x_1(t)$ and $y_2(t)$ is the output for input $x_2(t)$, then $y_1(t)$ and $y_2(t)$ are not identical for $t<-1$, even though $x_1(t)$ and $x_2(t)$ are identical for $t<-1$. Use this observation as the basis of an argument to conclude that the given system is not causal.

2.36

Consider a discrete-time system whose input $x[n]$ and output $y[n]$ are related by

$$y[n]=\frac{1}{2}y[n-1]+x[n].$$

(a) Show that if this system satisfies the condition of initial rest (i.e., if $x[n]=0$ for $n<n_0$, then $y[n]=0$ for $n<n_0$), then it is linear and time invariant.
(b) Show that if this system does not satisfy the condition of initial rest, but instead uses the auxiliary condition $y[0]=0$, it is not causal. [Hint: Use an approach similar to that used in Problem 2.35.]

2.37

Consider a system whose input and output are related by the first-order differential equation (P2.33-1). Assume that the system satisfies the condition of final rest [i.e., if $x(t)=0$ for $t>t_0$, then $y(t)=0$ for $t>t_0$]. Show that this system is not causal. [Hint: Consider two inputs to the system, $x_1(t)=0$ and $x_2(t)=e^t(u(t)-u(t-1))$, which result in outputs $y_1(t)$ and $y_2(t)$, respectively. Then show that $y_1(t)\ne y_2(t)$ for $t<0$.]

2.38

Draw block diagram representations for causal LTI systems described by the following difference equations: (a) $y[n]=\frac{1}{3}y[n-1]+\frac{1}{2}x[n]$
(b) $y[n]=\frac{1}{3}y[n-1]+x[n-1]$

2.39

Draw block diagram representations for causal LTI systems described by the following differential equations: (a) $y(t)=-\frac{1}{2}\frac{dy(t)}{dt}+4x(t)$
(b) $\frac{dy(t)}{dt}+3y(t)=x(t)$

Chap.2 Problems 137 2.6 SUMMARY In this chapter, we have developed important representations for LTI systems, both in dis crete time and in continuous time. In discrete time we derived a representation of signals as weighted sums of shifted unit impulses·, and we then used this to derive the convolution sum representation for the response of a discrete-time LTI system. In continuous time we .derived an analogous representation of continuous-time signals as weighted integrals of shifted unit impulses, and we used this to derive the convolution integral representation for continuous-time LTI systems. These representations are extremely important, as they allow us to compute the response of an LTI system to an arbitrary input in terms of the sys tem’s response to a unit impulse. Moreover, in Section 2.3 the convolution sum and integral provided us with a means of analyzing the properties of LTI systems and, in particular, of relating LTI system properties, including causality and stability, to corresponding proper ties of the unit impulse response. Also, in Section 2.5 we developed an interpretation of the continuous-time unit impulse and other related singularity functions in terms of their behavior under convolution. This interpretation is particularly useful in the analysis of LTI systems. An important class of continuous-time systems consists of those described by linear constant-coefficient differential equations. Similarly, in discrete time, linear constant coefficient difference equations play an equally important role. In Section 2.4, we exam ined simple examples of differential and difference equations and discussed some of the properties of systems described by these types of equations. In particular, systems de scribed by linear constant-coefficient differential and difference equations together with the condition of initial rest are causal and LTI. In subsequent chapters, we will develop additional tools that greatly facilitate our ability to analyze such systems. Chapter 2 Problems The first section of problems belongs to the basic category, and the answers are pro vided in the back of the book. The remaining three sections contain problems belonging to the basic, advanced, and extension categories, respectively. Extension problems introduce applications, concepts, or methods beyond those pre sented in the text. BASIC PROBLEMS WITH ANSWERS 2.1. Let x[n] = S[n] + 2S[n- 1] - S[n - 3] and h[n] = 2S[n + 1] + 2S[n - 1]. Compute and plot each of the following convolutions: (a) Yt [n] = x[n] * h[n] (b) Y2[n] = x[n + 2] * h[n] (c) y3[n] = x[n] * h[n + 2] Linear Time-Invariant Systems Chap. 2 138 2.2. Consider the signal n-1 ~ h[n] = {u[n + 3] - u[n - 10]}.

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	(	)			

Express A and B in terms of n so that the following equation holds: k] = (! ) n-k-1 A < k < B

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h[n -		2   '	-	- .	
		{ 0,	elsewhere		

2.3. Consider an input x[n] and a unit impulse response h[n] given by J- G 2 x[n] = u[n - 2],

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h[n] = u[n + 2].

Determine and plot the output y[n] = x[n] h[n]. * 2.4. Compute and plot y[n] = x[n] h[n], where 6: 3:sn:s8 x[n] = { otherwise ’

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		1 4 :s n :s 15			

h[n] = { • 0, otherwise

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2.5. Let 0 :s n :s 9 d 0 :s n :s N

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x[ n] = 1, an h[ ] { 1, { 0, n = elsewhere 0, elsewhere ’ :s 9 is an integer. Determine the value of N, given that y[n] = x[n] * where N h[n] and y[4] = 5, y[14] = 0.

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2.6. Compute and plot the convolution y[n] = x[n] * h[n], where )-n 1 1] and 1].

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x[n] = (3 u[ -n - h[n] = u[n- 2.7. A linear systemS has the relationship 00 y[n] = ~ x[k]g[n - 2k]

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k= -oo between its input x[n] and its output y[n], where g[n] = u[n] - u[n - 4]. Chap. 2 Problems 139 (a) Determine y[n] when x[n] = 8[n - 1]. (b) Determine y[n] when x[n] = 8[n- 2]. (c) Is SLTI? (d) Determine y[n] when x[n] = u[n]. 2.8. Determine and sketch the convolution of the following two signals: t + 1, 0 ~ t ~ 1 x(t) = 1 < t ~ 2 , 2 - t, { 0, elsewhere

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h(t) = 8(t + 2) + 28(t + 1). 2.9. Let h(t) = e21u(-t + 4) + e-21u(t- 5). Determine A and B such that e - 2(t-r) , T<A

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h(t- T) = 0, A<T<B. { e2(t-r) , B<T

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2.10. Suppose that 6: 0 ~ t ~ 1

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x(t) = { elsewhere and h(t) = x(tla), where 0 <a ~ 1. (a) Determine and sketch y(t) = x(t) * h(t).

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(b) If dy(t)/dt contains only three discontinuities, what is the value of a? 2.11. Let x(t) = u(t - 3) - u(t - 5) and h(t) = e-31 u(t).

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(a) Compute y(t) = x(t) * h(t). (b) Compute g(t) = (dx(t)/dt) * h(t). (c) How is g(t) related to y(t)? 2.12. Let L00 y(t) = e-1u(t) * 8(t- 3k).

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k= - oo Show that y(t) = Ae-1 for 0 ~ t < 3, and determine the value of A. 140 Linear Time-Invariant Systems Chap.2 2.13. Consider a discrete-time system S with impulse response 1 J G h[n] = u[n]. (a) Find the integer A such that h[n] - Ah[n- 1] = 5[n]. (b) Using the result from part (a), determine the impulse response g[n] of an LTI system S2 which is the inverse system of S I· 2.14. Which of the following impulse responses correspond(s) to stable LTI systems? (a) h1(t) = e-<l-2j)ru(t) (b) h2(t) = e-r cos(2t)u(t) 2.15. Which of the following impulse responses correspond(s) to stable LTI systems? (a) h1[n] = ncos(-n)u[n] (b) h2[n] = 3nu[-n + 10] 2.16. For each of the following statements, determine whether it is true or false: (a) If x[n] = 0 for n < N 1 and h[n] = 0 for n < N2, then x[n] * h[n] = 0 for n <N1 +N2. (b) If y[n] = x[n] * h[n], then y[n - 1] = x[n- 1] * h[n- 1]. (c) If y(t) = x(t) * h(t), then y( -t) = x( -t) * h( -t). (d) If x(t) = 0 for t > T 1 and h(t) = 0 for t > T2, then x(t) * h(t) = 0 for t > T1 + T2. 2.17. Consider an LTI system whose input x(t) and output y(t) are related by the differ ential equation d dty(t) + 4y(t) = x(t). (P2.17-1) The system also satisfies the condition of initial rest. (a) If x(t) = e<-l+3j)tu(t), what is y(t)? (b) Note that <Re{x(t)} will satisfy eq. (P2.17-1) with (}le{y(t)}. Determine the out put y(t) of the LTI system if · x(t) e-r cos(3t)u(t). 2.18. Consider a causal LTI system whose input x[n] and output y[n] are related by the difference equation 1 y[n] = y[n- 1] + x[n]. 4 Determine y[n] if x[n] = 5[n - 1]. 2.19. Consider the cascade of the following two systems S 1 and S 2, as depicted in Figure P2.19: w[nJ.,I s2 1----t• y[n) Figure P2. 1 9 Chap.2 Problems 141 S 1 : causal LTI, 1 w[n] = 2w[n - 1] + x[n]; S2 : causal LTI, y[n] = ay[n- 1] + {3w[n]. The difference equation relating x[n] and y[n] is: 1 3 y[n] = - 8 y[n- 2] + 4y[n- 1] + x[n]. (a) Determine a and {3. (b) Show the impulse response ofthe cascade connection of S 1 and S 2. 2.20. Evaluate the following integrals: f (a) .:0"" uo(t) cos(t) dt (b) Jr? sin(27rt)8(t + 3)dt (C) e5 UJ (1 - T) COS(27TT) dT BASIC PROBLEMS 2.21. Compute the convolution y[n] = x[n] * h[n] of the following pairs of signals: (a) x[n] = a 11 u[n], ) a ""’ {3 h[n] = {311u[n], (b) x[n] = h[n] = a11u[n] (c) x[n] = (-! )11u[n- 4] h[n] = 4”u[2 - n] (d) x[n] and h[n] are as in Figure P2.21. x[n] h[n) 1I I I I I ••••• 1I I I I I I ••• I I I I I I •••• •••• • ••• -1 0 1 2 3 4 5 n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 n Figure P2.21 2.22. For each of the following pairs of waveforms, use the convolution integral to find the response y(t) of the LTI system with impulse response h(t) to the input x(t). Sketch your results. (a) x(t) = e-ar u(t)} (Do this both when a ""’ {3 and when a = {3.) h(t) = e-f3ru(t) 142 Linear Time-Invariant Systems Chap. 2 x(t) = u(t) - 2) + u(t (b) 2u(t - - 5) h(t) = e21u (l - t) (c) x(t) and h(t) are as in Figure P2.22(a). (d) x(t) and h(t) are as in Figure P2.22(b). x(t) and h(t) are as in Figure P2.22(c). (e) x(t) h(t)

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2 One period of sin ‘ITt 2 (a) x(t) h(t)

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4 3 2 - 1

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3 (b) x(t) h(t)

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1 (c) Figure P2.22 2.23. Let h(t) be the triangular pulse shown in Figure P2.23(a), and let x(t) be the impulse train depicted in Figure P2.23(b). That is, L+o o

			x(t) =	B(t - kT).
			k= - oo
Determine and sketch y(t) = .x(t) * h(t) for the following values ofT:
	= 4	= 2	= 3/2		= 1
----	----	-------	-------	----	----	---
(a)	T	(b) T	(c) T	(d)	T
Chap.2 Problems 143
h(t)
(a)
···t t t t :r t t t t t t···

- 2T  -T	0  T  2T  3T		t
	(b)		Figure P2.23

2.24. Consider the cascade interconnection of three causal LTI systems, illustrated in Fig ure P2.24(a). The impulse response h [n] is 2 hz[n] = u[n] - u[n - 2],

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and the overall impulse response is as shown in Figure P2.24(b). x[n] h [n] y[n]

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1 (a) -1 0 1 2 3 4 5 6 7

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n (b) Figure P2.24

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(a) Find the impulse response h [n]. 1 (b) Find the response of the overall system to the input x[n] = 5[n] - 5[n- 1].

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144 Linear Time-Invariant Systems Chap.2 2.25. Let the signal

y[n] = x[n] h[n], r where G x[n] = 311u[ -n- 1] + u[n] and r h[n] = (~ u[n + 3]. (a) Determine y[n] without utilizing the distributive property of convolution. (b) Determine y[n] utilizing the distributive property of convolution. 2.26. Consider the evaluation of y[n] = x, [n] * *

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xz[n] x3[n], where x, [n] = (0.5)11u[n], Xz[n] = u[n + 3], and x3[n] = 5[n] - 5[n - 1].

(a) Evaluate the convolution x 1 [n] xz[n]. (b) Convolve the result of part (a) with x3[n] in order to evaluate y[n]. (c) Evaluate the convolution x [n] * x3[n].

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2 (d) Convolve the result of part (c) with x [h] in order to evaluate y[n]. 1 2.27. We define the area under a continuous-time signal v(t) as L+: Av = v(t) dt. Show that if y(t) = x(t) *

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h(t), then Ay = AxAh. 2.28. The following are the impulse responses of discrete-time LTI systems. Determine whether each system is causal and/or stable. Justify your answers. h[n] = (~)11u[n] (a) (b) h[n] = (0.8)“u[n + 2] = <!) (c) h[n] 11u[ - n]

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(d) = (5)11u[3 - n] h[n] !) (e) h[n] = (- 11u[n] + (1.01)11u[n- 1]

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!) (f) h[n] = (- 11u[n] + (l.Ol)11u[l - n]

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(g) h[n] = n(~)11u[n- 1] 2.29. The following are the impulse responses of continuous-time LTI systems. Determine whether each system is causal and/or stable. Justify your answers. (a) h(t) = e-41 u(t - 2)

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(b) h(t) = e-61u(3- t) h(t) = e-21u(t +50) (c) (d) h(t) = e21u( -1- t)

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Chap.2 Problems 145 (e) h(t) = e-6ltl (f) h(t) = te-1 u(t) h(t) = (2e-t - (g) e<r-IOO)/JOO)u(t) 2.30. Consider the first-order difference equation + 1] = y[n] 2y[n - x[n]. Assuming the condition of initial rest (i.e., if x[n] = 0 for n < no, then y[n] = 0 for n < n 0), find the impulse response of a system whose input and output are related by this difference equation. You may solve the problem by rearranging the difference equation so as to express y[ n] in terms of y[ n-1] and x[ n] and generating the values of y[O], y[ + 1], y[ +2], … in that order. 2.31. Consider the LTI system initially at rest and described by the difference equation

•   1] = x[n]	+	
  

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y[n] 2y[n - 2x[n - 2]. Find the response of this system to the input depicted in Figure P2.31 by solving the difference equation recursively. x[n] … ‘r’fl I J’,,_ •••• -2-1 0 1 2 3 4 n Figure P2.31

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2.32. Consider the difference equation 1 1] = y[n]- zy[n - x[n], (P2.32-1)

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and suppose that J (~ x[n] = u[n]. (P2.32-2)

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Assume that the solution y[n] consists of the sum of a particular solution Yp[n] to eq. (P2.32-1) and a homogeneous solution Yh[n] satisfying the equation 1 Yh[n] - zYh[n - 1] = 0.

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(a) Verify that the homogeneous solution is given by AGJ Yh[n] =

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(b) Let us consider obtaining a particular solution Yp[n] such that 1 (1

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)n = 3’

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Yp[n]- zYp[n - 1] u[n].

’ 146 Linear Time-Invariant Systems Chap.2 By assuming that Yp[n] is of the form B( ~ )n for n 2: 0, and substituting this in the above difference equation, determine the value of B. (c) Suppose that the LTI system described by eq. (P2.32-1) and initially at rest has = 0 for n < 0, we as its input the signal specified by eq. (P2.32-2). Since x[n] have that y[n] = 0 for n < 0. Also, from parts (a) and (b) we have that y[n] has the form for n :::: 0. Iri order to solve for the unknown constant A, we must specify a value for y[n] for some n :::: 0. Use the condition of initial rest and eqs. (P2.32-1) and (P2.32-2) to determine y[O]. From this value determine the constant A. The result of this calculation yields the solution to the difference equation (P2.32- 1) under the condition of initial rest, when the input is given by eq. (P2.32-2). 2.33. Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation dy(t) dt + =

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	2y(t)	x(t).	(P2.33-1)

The system also satisfies the condition of initial rest. (a) (i) Determine the system output y1( t) when the input is x1( t) = 31 e u(t). = e21 (ii) Determine the system output y2(t) when the input is x2(t) u(t). (iii) Determine the system output y (t) when the input is x (t) = ae31u(t) + 3 3

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{3 e21			(t) =  ay1( t) +
u(t), where a	and {3  are real numbers. Show that y		3

{3 Y2(t). (iv) Now let x (t) and x (t) be arbitrary signals such that

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1	2		

= 0, xi(t) fort< t1,

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= 0, fort < t2. x2(t) Letting y (t) be the system output for input x (t), Y2(t) be the system output 1 1

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= ax1( t) + f3x2(t), for input x2(t), and y3(t) be the system output for x3(t) show that We may therefore conclude that the system under consideration is linear. = K e 21 (b) (i) ’ Determine the system output y1( t) when the input is x1( t) u(t). = Ke2 (ii) Determine the system output y2(t) when the input is x2(t) <t-T) u(t ..:.. T). Show that Y2(t) = Yt (t - T).

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= 0 fort < t0. Letting (iii) Now let x1( t) be an arbitrary signal such that Xi (t) Yi (t) be the system output for input Xi (t) and Y2(t) be the system output

for x2(t)  Xi (t -	T), show that		

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= Yz(t) Y1 (t - T).

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Chap.2 Problems 147 We may therefore conclude that the system under consideration is time invariant. In conjunction with the result derived in part (a), we conclude that the given system is LTI. Since this system satisfies the condition of initial rest, it is causal as well. 2.34. The initial rest assumption corresponds to a zero-valued auxiliary condition being imposed at a time determined in accordance with the input signal. In this problem we show that if the auxiliary condition used is nonzero or if it is always applied at a fixed time (regardless of the input signal) the corresponding system cannot be LTI. Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation (P2.33-1). (a) Given the auxiliary condition y(l) = 1, use a counterexample to show that the system is not linear. (b) Given the auxiliary condition y(l) = 1, use a counterexample to show that the system is not time invariant. (c) Given the auxiliary condition y(l) = 1, show that the system is incrementally linear. (d) Given the auxiliary condition y(1) = 0, show that the system is linear but not time invariant. (e) Given the auxiliary condition y(O) + y(4) = 0, show that the system is linear but not time invariant. 2.35. In the previous problem we saw that application of an auxiliary condition at a fixed time (regardless of the input signal) leads to the corresponding system being not time-invariant. In this problem, we explore the effect of fixed auxiliary conditions on the causality of a system. Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation (P2.33-1). Assume that the auxiliary condition associated with the differential equation is y(O) = 0. Determine the output of the system for each of the following two inputs: (a) x (t) = 0, for all t 1 (b) X2(t) = { 01,, t < - 1 t > - 1 Observe that if y (t) is the output for input x (t) and y (t) is the output for input 1 1 2 x2(t), then Yt (t) and Y2(t) are not identical fort< -1, even though x 1 ( t) and x2(t) are identical fort < - 1. Use this observation as the basis of an argument to conclude that the given system is not causal. 2.36. Consider a discrete-time system whose input x[n] and output y[n] are related by G y[n] = )y[n - 1] + x[n]. (a) Show that if this system satisfies the condition of initial rest (i.e., if x[n] = 0 for n < n0, then y[n] = 0 for n < n 0), then it is linear and time invariant. (b) Show that if this system does not satisfy the condition of initial rest, but instead uses the auxiliary condition y[O] = 0, it is not causal. [Hint: Use an approach similar to that used in Problem 2.35.] 148 Linear Time-Invariant Systems Chap. 2 2.37. Consider a system whose input and output are related by the first-order differential equation (P2.33-1). Assume that the system satisfies the condition of final rest [i.e., if x(t) = 0 fort > to, then y(t) = 0 fort > to]. Show that this system is not causal. [Hint: Consider two inputs to the system, x 1 (t) = 0 and x2(t) = e1(it(t)- u(t- 1)), which result in outputs YI (t) and Y2(t), respectively. Then show that y 1 ( t) .:P y2(t) fort< 0.] 2.38. Draw block diagram representations for causal LTI systems described by the fol lowing difference equations: (a) y[n] = 1y[n- 1] + !x[n] (b) y[n] = 1y[n- 1] + x[n- 1] 2.39. Draw block diagram representations for causal LTI systems described by the fol lowing differential equations: (a) y(t) = -(!)dy(t)ldt + 4x(t) (b) dy(t)ldt + 3y(t) = x(t) ADVANCED PROBLEMS 2.40. (a) Consider an LTI system with input and output related through the equation L”’ y(t) = e-<r—r)x(T- 2)dr. What is the impulse response h(t) for this system? (b) Determine the response of the system when the input x(t) is as shown in Figure P2.40. x(t) ,I 1 2 Figure P2.40 2.41. Consider the signal x[n] = anu[n]. (a) Sketch the signal g[n] = x[n] - ax[n - 1]. (b) Use the result of part (a) in conjunction with properties of convolution in order to determine a sequence h[n] such that J G x[n] * h[n] = {u[n + 2] - u[n - 2]}. 2.42. Suppose that the signal x(t) = u(t + 0.5) - u(t - 0.5)