A Causal Continuous time Lti System is Described by the Following Differential Equation
Problem 1
Let
$$x[n]=\delta[n]+2 \delta[n-1]-\delta[n-3] \text { and } h[n]=2 \delta[n+1]-2 \delta[n-1]$$
Compute and plot each of the following convolutions:
(a) $y_{1}[n]=x[n] * h[n]$
(b) $y_{2}[n]=x[n+2] * h[n]$
(c) $y_{3}[n]=x[n] * h[n+2]$
Problem 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 bolds
$$h[n-k]=\left\{\begin{array}{ll}
\left(\frac{1}{2}\right)^{n-k-1}, & A \leq k \leq B \\
0, & \text { elsewhere }
\end{array}\right.$$
Problem 3
Consider an input $x[n]$ and a unit impulse response $h\{n]$ given by
$$\begin{aligned}
&x[n]=\left(\frac{1}{2}\right)^{n-2} u[n-2]\\
&h[n]=u[n+2]
\end{aligned}$$
Determine and plot the output $y[n]=x[n] * h[n]$
Problem 4
Compute and plot $y[n]=x[n] * h[n],$ where
$$\begin{array}{l}
x[n]=\left\{\begin{array}{ll}
1, & 3 \leq n \leq 8 \\
0, & \text { otherwise }
\end{array}\right. \\
h[n]=\left\{\begin{array}{ll}
1, & 4 \leq n \leq 15 \\
0, & \text { otherwise }
\end{array}\right.
\end{array}$$
Problem 5
Let
$$x[n]=\left\{\begin{array}{ll}
1, & 0 \leq n \leq 9 \\
0, & \text { etsewhere }
\end{array} \text { and } h[n]=\left\{\begin{array}{ll}
1, & 0 \leq n \leq N \\
0, & \text { elsewhere }
\end{array}\right.\right.$$
where $N \leq 9$ is an integer. Determine the value of $N$, giventhat $y[n]=x[n] * h[n]$ and
$$y[4]=5, \quad y[14]=0$$
Problem 6
Compute and plot the convolution $y[n]=x[n] * h[n],$ where
$$x[n]=\left(\frac{1}{3}\right)^{-n} u[-n-1] \text { and } h[n]=u[n-1]$$
Problem 7
A linear system $S$ has the relationship
$$y[n]=\sum_{k=-\infty}^{\infty} x[k] g[n-2 k]$$
between its input $x[n]$ and its output $y[n],$ where $g[n]=u[n]-u[n-4]$
(a) Determine $y[n]$ when $x[n]=\delta[n-1]$
(b) Determine $y[n]$ when $x[n]=\delta\{n-2]$
(c) Is $S$ LTI?
(d) Determine $y\{n]$ when $x[n]=u[n]$
Problem 8
Determine and sketch the convolution of the following two signals:
$$\begin{aligned}
&x(t)=\left\{\begin{array}{ll}
t+1, & 0 \leq t \leq 1 \\
2-t, & 1<t \leq 2 \\
0, & \text { elsewhere }
\end{array}\right.\\
&h(t)=\delta(t+2)+28(t+1)
\end{aligned}$$
Problem 9
Let
$$h(t)=e^{2 t} u(-t+4)+e^{-2 t} u(t-5)$$
Determine $A$ and $B$ such that
$$h(t-\tau)=\left\{\begin{array}{ll}
e^{-2(t-\tau)}, & \tau<A \\
0, & A<\tau<B \\
e^{2(t-r)}, & B<\tau
\end{array}\right.$$
Problem 10
Suppose that
$$x(r)=\left\{\begin{array}{ll}
1, & 0 \leq t \leq 1 \\
0, & \text { elsewhere }
\end{array}\right.$$
and $h(t)=x(t / \alpha),$ where $0<\alpha \leq 1$
(a) Determine and sketch $y(t)=x(t) * h(t)$
(b) If $d y(t) / d t$ contains only three discontinuities, what is the value of $\alpha ?$
Problem 11
Let
$$x(t)=u(t-3)-u(t-5) \quad \text { and } \quad h t t)=e^{-3 t} u[t)$$
(a) Compute $y(t)=x(t) * h(t)$
(b) Compute $g(t)=(d x(t) / d t) * h(t)$
(c) How is $g(t)$ related to $y(t) ?$
Problem 12
Let
$$y(t)=e^{-i} u(t) * \sum_{k=-\infty}^{\infty} \delta(t-3 k)$$
Show that $y(t)=A e^{t}$ for $0 \leq t<3$, and determine the value of $A$
Problem 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 $h[n]-A h[n-1]=\delta[n]$
(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 $S_{1}$
Problem 14
Which of the following impulse responses correspond(s) to stable LTS systems?
(a) $h_{1}(t)=e^{-(1-2 y) t} u(t)$
(b) $h_{2}(t)=e^{-t} \cos (2 t) u(t)$
Problem 15
Which of the following impulse responses correspond(s) to stable $\&$ T stems?
(a) $\left.h_{1} | n\right]=n \cos \left(\frac{\pi}{4} n\right) u[n]$
(b) $h_{2}[n]=3^{n} u[-n+10]$
Problem 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 $x[n] * h[n]=0$ for $n<N_{t}+N_{2}$
(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 $t>T_{2},$ then $x(t) * h(t)=0$ for $t>$ $T_{1}+T_{2}$
Problem 17
Consider an LTI system whose input $x(t)$ and output $y(t)$ are related by the differential equation
$$\frac{d}{d t} y(t)+4 y(t)=x(t)$$
The system also satisfies the condition of initial rest.
(a) If $x(t)=e^{(-1+3) t^{t}} u(t),$ what is $y(t) ?$ put $y(t)$ of the LTI system if
$$x(t)=e^{-t} \cos (3 t) u(t)$$
Problem 18
Consider a causal LT system whose input $x[n]$ and output $y(n]$ are related by the difference equation
$$\left.y[n]=\frac{1}{4} y[n-1]+x | n\right\rceil$$
Determine $y(n]$ if $x[n]=\delta[n-1]$
Problem 19
Consider the cascade of the following two systems $S_{1}$ and $S_{2}$, as depicted in Figure P2. 19
$$\begin{array}{l}
S_{1}: \text { causal LTI, } \\
\qquad w[n]=\frac{1}{2} w[n-1]+x[n]
\end{array}$$
$$\begin{array}{l}
S_{2}: \text { causal LTI, } \\
\qquad y[n]=\alpha y[n-1]+\beta w[n]
\end{array}$$
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) Deterinine $\alpha$ and $\beta$
(b) Show the impulse response of the cascade conbection of $S_{1}$ and $S_{2}$.
Problem 20
Evaluate the following integrals:
(a) $\int_{-\infty}^{\infty} u_{0}(t) \cos (t) d t$
(b) $\int_{0}^{5} \sin (2 \pi t) \delta(t+3) d t$
(c) $\int_{-5}^{5} u_{1}(1-\tau) \cos (2 \pi \tau) d \tau$
Problem 21
Compute the convolution $y[n]=x[n] * h[n]$ of the following pairs of signals:
(a) $\left.\begin{array}{l}x[n]=\alpha^{n} u[n], \\ h[n]=\beta^{n} u[n]\end{array}\right\} \alpha \neq \beta$
(b) $x[n]=h[n]=\alpha^{n} u[n]$
(c) $x[n]=\left(-\frac{1}{2}\right)^{n} u[n-4]$
$h[n]=4^{n}[2-n]$
(d) $x[n]$ and $h[n]$ are as in Figure $P 2.21$
Problem 22
For each of the following pairs of waveforms, use the convolution integral to find the response $y(t)$ of the LT1 system with impulse response $h(t)$ to the input $x(t)$. Sketch your results.
(a) $\left.\begin{array}{l}x(t)=e^{-\alpha t} u(t) \\ h(t)=e^{-\beta t} u(t)\end{array}\right\}(\text { Do this both when } \alpha \neq \beta \text { and when } \alpha=\beta .)$
(b) $x(t)=u(t)-2 u(t-2)+u(t-5)$
$h(t)=e^{2 x} u(1-t)$
(c) $x(t)$ and $h(t)$ are as in Figure $P 2.22(a)$
(d) $x(t)$ and $h(t)$ are as in Figure $P 2.22(b)$
(e) $x(t)$ and $h(t)$ are as in Figure $P 2.22(c)$
Problem 23
Let $h(t)$ be the triangular pulse shown in Figure $P 2.23(a),$ and let $x(t)$ be the impuise train depicted in Figure $P 223(b) .$ That is,
$$x(t)=\sum_{k=-\infty}^{+\infty} \delta(t-k T)$$
Determine and sketch $y(t)=x(t) * h(t)$ for the following values of $T$
(a) $T=4$
(b) $T=2$
(c) $T=3 / 2$
(d) $T=1$
Problem 24
Consider the cascade interconnection of three causal LTT systems, illustrated in Figure $\mathrm{P} 2.24(\mathrm{a})$. The impulse response $\hbar_{2}[n]$ is
$$h_{2}[n]=u[n]-u[n-2]$$
and the overall impulse response is as shown in Figure $\mathrm{P} 2.24(\mathrm{b})$
(a) Find the impulse response $h_{1}[n]$
(b) Find the response of the overall system to the input
$$x[n]=\delta[n]-\delta[n-1]$$
Problem 25
Let the signal
$$y[n]=x[n] * h[n]$$
where
$$x[n]=3^{n} u[-n-2]+\left(\frac{1}{3}\right)^{n} u[n]$$
and
$$h | n]=\left(\frac{1}{4}\right)^{n} u[n+3]$$
(a) Determine $y[n]$ without utilizing the distributive property of convolution.
(b) Determine $y[n]$ utilizing the distributive property of convolution.
Problem 26
Consider the evaluation of
$$y[n]=x_{1}[n] * x_{2}[n] * x_{3}[n]$$
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 $x_{1}[n] * x_{2}[n]$
(b) Convolve the result of part (a) with $x_{3}[n]$ in order to evaluate $y[n]$
(c) Evaluate the convolution $x_{2}[n] * x_{3}[n]$
(d) Convolve the result of part (c) with $x_{1}[n]$ in order to evaluate $y[n]$
Problem 27
We define the area under a continuous-time signal $v(t)$ as
$$\mathbf{4}_{\mathbf{v}}=\int_{-\infty}^{+\infty} v(t) d t$$
Show that if $y(t)=x(t) * h(t),$ then
$$A_{y}=A_{x} A_{h}$$
Problem 28
The following are the impulse responses of discrete-time LTI systerns. Deternine 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) $h[n]=n\left(\frac{1}{3}\right)^{n} u[n-1]$
Problem 29
The following are dee impulse responses of continuous-time LTI systems. Determine whether each system is causal and/or stable. Justify your answers.
(a) $h(t)=e^{-4 t} u(t-2)$
(b) $h(t)=e^{-6 r} u(3-t)$
(c) $h(t)=e^{-2 t} u(t+50)$
(d) $h(t)=e^{2 t} u(-1-t)$
(e) $h(t)=e^{||6 |}$
(f) $h(t)=i e^{-i} u(t)$
(g) $h(t)=\left(2 e^{-t}-e^{(t-100) t 100}\right) u(t)$
Problem 30
Consider the first-order difference equation
$$y[n]+2 y[n-1]=x[n]$$
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 $y[0], y[+1], y[+2], \ldots$ in that order.
Problem 31
Consider the LTI system initially at rest and described by the difference equation
$$y[n]+2 y[n-1]=x[n]+2 x\{n-2]$$
Find the response of this system to the input depicted in Figure $P 2.31$ by solving the difference equation recursively.
Problem 32
Consider the difference equation
$$y[n]-\frac{1}{2} y[n-1]=x[n]$$
and suppose that
$$x[n]=\left(\frac{1}{3}\right)^{n} u[n]$$
Assume that the solution $y[n]$ consists of the sum of a particular solution $y_{p}[n]$ to eq. $(\mathrm{P} 2.32-1)$ 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 bomogeneous 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 $y_{p}[n\rangle$ is of the form $B\left(\frac{1}{3}\right)^{n}$ for $n \geq 0,$ 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 $x[x]=0$ 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 $n \geq 0 .$ In order to solve for the unknown constant $A$, we must specify a value for $y[n]$ for some $n \geq 0 .$ Use the condition of initial rest and eqs. $(\mathbf{P} 2.32-1)$ and $(\mathrm{P} 2.32-2)$ to determine $y[0] .$ From this value determine the constant $A$. The result of this calculation yields the solution to the difference equation $(P 2.32-1)$ under the condition of initial rest, when the input is given by eq. $(\mathrm{P} 2.32-2)$
Problem 33
Consider a system whose input $x(t)$ and output $y(t)$ satisfy the first-order differential equation
$$\frac{d y(t)}{d t}+2 y(t)=x(t)$$
The system also satisfies the condition of initial rest.
(a) (i) Determine the system output $y_{1}(t)$ when the input is $x_{1}(t)=e^{3 t} u(t)$
(ii) Determine the system output $y_{2}(t)$ when the input is $x_{2}(t)=e^{2 t} u(t)$
(iii) Determine the system output $y_{3}(t)$ when the input is $x_{3}(t)=\alpha e^{3 t} u(t)+$ $\beta e^{2 t} u(t),$ where $\alpha$ and $\beta$ are real numbers. Show that $y_{3}(t)=\alpha y_{1}(t)+$ $\beta y_{2}(t)$
(iv) Now let $x_{1}(f)$ and $x_{2}(f)$ be arbitrary signals such that
$$\begin{aligned}
&x_{1}(t)=0, \text { for } t<t_{1}\\
&x_{2}(t)=0, \text { for } \iota<t_{2}
\end{aligned}$$
Letting $y_{1}(t)$ he the system output for input $x_{1}(t), y_{2}(t)$ be the system output for input $x_{2}(t),$ and $y_{3}(t)$ be the system output 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) $(i) \quad$ Determine the system output $y_{1}(t)$ when the input is $x_{1}(t)=K e^{2 t} u(t)$
(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 arbitary 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 $y_{2}(t)$ bo the system output 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 171 . Since this system satisfies the condition of initial rest, it is causal as well.
Problem 34
The initial rest assumption corresponds to a zero-valued auxiliary condition being imposed al 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 condstion $y(1)=1,$ 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 $y(\theta)+y(4)=0,$ show that the system is linear but not time invariant.
Problem 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-invanant. 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 $(\mathrm{P} 2.33-1) .$ 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) $x_{2}(t)=\left\{\begin{array}{ll}0 & t<-1 \\ 1, & t>-1\end{array}\right.$
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.
Problem 36
Consider a discrete-time system whose input $x[n]$ and output $y[n]$ are related by
$$\left.y[n]=\left(\frac{1}{2}\right) y | n-1\right]+x[n]$$
(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 $n<n_{4}$ ), 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 $2.35 .$
Problem 37
Consider a system whose input and output are related by the first-order differential equation $(\mathrm{P} 2.33-1) .$ Assume that the system satisfies the condition of final rest [i. e., if $\left.x(t)=0 \text { for } t>t_{0}, \text { then } y(t)=0 \text { for } t>t_{0}\right] .$ Show that this system is $n o t$ 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 $y_{1}(t) \neq y_{2}(t)$ for $t<0 .$
Problem 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) $y[n]=\frac{1}{3} y[n-1]+x[n-1]$
Problem 39
Draw block diagram representations for causal LTI systems described by the follwing differential equations:
(a) $y(t)=-\left(\frac{1}{2}\right) d y(t) / d t+4 x(t)$
(b) $d y(t) / d t+3 y(t)=x(t)$
Problem 40
(a) Consider an LT system with input and output related through the equation
$$y(t)=\int_{-\infty}^{t} e^{-(t-\tau)} x(\tau-2) d \tau$$
What is the irapulse response $h(t)$ for this system?
(b) Determine the response of the system when the input $x(t)$ is as shown in Figure $\mathbf{P} 2,4 \mathbf{O}$
Problem 41
Consider the signal
$$x[n]=\cdot \alpha^{n} u[n]$$
(a) Sketch the signal $g[n]=x[n]-\alpha x[n-1]$
(b) Use the result of part (a) in conjunction with properties of convolution in order to determine a sequence $h[n]$ such diat
$$x[n] * h[n]=\left(\frac{1}{2}\right)^{n}\{u[n+2]-u[n-2]\}$$
Problem 42
Suppose that the signal
$$x(t)=u(t+0.5)-u(t-05)$$
and the signal
$$h(t)=e^{j \omega_{0}^{\prime}}$$
(a) Determine a value of $\omega_{0}$ which ensures that
$$y(0)=0$$
where $y(t)=x(t) * h(t)$
(b) Is your answer to the previous part unique?
Problem 43
One of the important properties of convolution, in both continuous and discrete time, is the associativity property, In this problem, we will check and illustrate this property.
(a) Prove the equality
$$[x(t) * h(t)] * g(t)=x(t) * | h(t) * g(t)\}$$
by showing that both sides of eq. (P2.43-1) equal
$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\pi} x(\tau) h(\sigma) g(t-\tau-\sigma) d \tau d \sigma$$
(b) Consider two LTI systems with the unit sample responses $h_{1}[n]$ and $h_{2}[n]$ shown in Figure $\mathrm{P} 2.43$ (a). These two systems are cascaded as shown in Figure P2.43(b). Let $x$ (n) $=u[n]$
(ij) Compute $y[n]$ by first computing $w[n]=x[n] * h_{1}[n]$ and then computing $y(n]=w[n] * h_{2}[n] ;$ that is, $y[n]=\left[x[n] * h_{1}[n]\right] * h_{2}[n]$
(ii) Now find $y[n]$ by first convolving $h_{\mathrm{j}}[n]$ and $h_{2}[n]$ to obtan $g[n]=$ $h_{1}[n] * h_{2}[n]$ and then convolving $x[n]$ with $g[n]$ to obtain $y[n]=$ $x[n] *\left[h_{1}[n] * h_{2}[n]\right]$
The answers to (i) and (ii) should be identical, illustrating the associativity property of discrete-time convolution
(c) Consider the cascade of two LTI systems as in Figure $P 2.43(b) .$ where in this case
$$h_{1}[n]=\sin 8 n$$
and
$$h_{2}[n]=a^{n} u[n], \quad|a|<1$$
and where the input is
$$x[n]=\delta[n]-a \delta[n-1]$$
Determine the output $y[n] .$ (Hint: The use of the associative and commutative properties of convolution should greatly facilitate the solution )
Problem 44
(a) If
$$x(t)=0,|t|>T_{1}$$
and
$$h(t)=0,|t|>T_{2}$$
then
$$x(t) * h(t)=0,|t|>T_{3}$$
for some positive number $T_{3}$. Express $T_{3}$ in terms of $T_{1}$ and $T_{2}$
(b) A discrete-time LTI system has input $x[n],$ impulse response $h[n],$ and output $y[n] .$ If $h[n]$ is known to be zero everywhere outside the interval $N_{0} \leq n \leq$ $N_{1}$ and $x[n]$ is known to be zero everywhere outside the interval $N_{2} \leq n \leq$ $\left.N_{3}, \text { then the output } y | n\right]$ is constrained to be zero everywhere, except on some interval $N_{4} \leq n \leq N_{5}$
(i) Determine $N_{4}$ and $N_{5}$ in terms of $N_{0}, N_{1}, N_{2},$ and $N_{3}$
(ii) If the interval $N_{0} \leq n \leq N_{1}$ is of length $M_{h}, N_{2} \leq n \leq N_{3}$ is of length $M_{x},$ and $N_{4} \leq n \leq N_{5}$ is of length $M_{y},$ express $M_{y}$ in terms of $M_{h}$ and $M_{x}$
(c) Consider a thiscrete-time LTI system with the property that if the input $x[n]=0$ for all $n \geq 10$, then the output $y[n]=0$ for all $n \geq 15$. What condition must $h[n],$ the impulse response of the system, satisfy for this to be true?
(d) Consider an LT1 system with impulse response in Figure $P 2.44 .$ Over what in terval must we know $x(t)$ in order to determine $y(0) ?$
Problem 45
(a) Show that if the response of an LT1 system to $x(t)$ is the output $y(t),$ then the response of the system to
$$x^{\prime}(t)=\frac{d x(t)}{d t}$$
is $y^{\prime}(t) .$ Do this problem in three different ways:
(i) Directly from the properties of linearity and time invariance and the fact that
$$x^{\prime}(t)=\lim _{h \rightarrow 0} \frac{x(t)-x(t-h)}{h}$$
(ii) By differentiating the convolution integral.
(iii) By examining the system in Figure P2.45.
(b) Demonstrate the validity of the following relationships:
(i) $y^{\prime}(t)=x(t) * h^{\prime}(t)$
(ii) $y(t)=\left(\int_{-\infty}^{t} x(\tau) d \tau\right) * h^{\prime}(t)=\int_{-x}^{r}\left[x^{\prime}(\tau) * h(\tau)\right] d \tau=x^{\prime}(t) *\left(\int_{-x}^{f} h(\tau) d \tau\right)$
[Hint: These are easily done using block diagrams as in (iii) of part (a) and the fact that $\left.u_{1}(f) * u_{-1}(t)=\delta(t) .\right]$
(c) An LTT system has the response $y(t)=\sin \omega_{0} t$ to input $x(t)=e^{-5 t} u\{t\rangle$. Use the result of part (a) to aid in determining the impulse response of this system.
(d) Let $s(t)$ be the unit step response of a continuous-time LTI system. Use part (b) to deduce that the response $y(t)$ to the input $x(t)$ is
$$y(t)=\int_{-\infty}^{+x} x^{\prime}(\tau) * s(t-\tau) d \tau$$
Show also that
$$x(t)=\int_{-x}^{+\infty} x^{\prime}(\tau) u(t-\tau) d \tau$$
(e) Use eq. $(\mathrm{P} 2.45-1)$ to determine the response of an LT system with step response
$$s(t)=\left(e^{-3 t}-2 e^{-2 t}+1\right) u(t)$$
to the input $x(t)=e^{t} u(t)$
(f) Let $s[n]$ be the unit step response of a discrete-time LTI system. What are the discrete-time counterparts of eqs. (P2.45-1) and (P2.45-2)?
Problem 46
Consider an LTI system $S$ and a signal $x(t)=2 e^{-3 r} u(t-1)$. If
$$x(t) \rightarrow y(t)$$
and
$$\frac{d x(t)}{d t} \longrightarrow-3 y(t)+e^{-2 t} u(t)$$
determine the impulse response $h(t)$ of $S$.
Problem 47
We are given a certain linear time-invariant system with impulse response $h_{0}(t)$. We are told that when the input is $x_{0}(t)$ the output is $y_{0}(t) .$ which is sketched in Figure P2.47. We are then given the following set of inputs to linear time-invariant systems with the indicated impulse responses:
[Here $x_{0}^{\prime}(t)$ and $h_{0}^{\prime}(t)$ denote the first derivatives of $x_{0}(t)$ and $h_{0}(t),$ respectively.]
In each of these cases, determine whether or not we have enough information to determine the output $y(t)$ when the input is $x(t)$ and the system has impulse response $h(t)$. If it is possible to determine $y(t),$ provide an accurate sketch of it with numerical values clearly indicated on the graph.
Problem 48
Determine whether each of the following statements concerning LTI systems is me or false. Justify your answers.
(a) If $h(r)$ is the impulse response of an LTI system and $h(t)$ is periodic and nonzero. the system is unstable.
(b) The inverse of a causal LTI system is always causal.
(c) $\mathbf{f}|h[n]| \leq K$ for each $n$, where $K$ is a given number, then the $L T$ is system with $h[n]$ as its impulse response is stable.
(d) If a discrete-time LTI system has an impulse response $h[n]$ of finite duration. the system is stable
(e) If an LTI system is causal, it is stable.
(f) The cascade of a noncausal LTI system with a causal one is necessarily noncausal.
(g) A continuous-time LTI system is stable if and only if its step response $s(t)$ is absolutely integrable - that is, if and only if
$$\int_{-\infty}^{+\infty}|s(t)| d t<\infty$$
(h) A discrete-time LTI systew is causal if and only if its slep response $s[n]$ is zero for $n<0$
Problem 49
In the text, we showed that if $h[n]$ is absolutely summable, i.e., if
$$\sum_{k=-x}^{+\infty}|h[k]|<\infty$$
then the LTI system with impulse response $h[n]$ is stable. This means that absolute summability is a sufficient condition for stability. In this problem, we shall show that it is also a necessary condition. Consider an LTI system with impulse response $h[n]$ that is not absolutely summable; that is,
$$\sum_{k=-x}^{+\infty}|h[k]|=\infty$$
(a) Suppose that the input to this system is
$$x[n]=\left\{\begin{array}{ll}
0, & \text { if } h[-n]=0 \\
\frac{\left.h_{1} \cdot n\right]}{|h|-n|)}, & \text { if } h[-n] \neq 0
\end{array}\right.$$
Does this input signal represent a bounded input? If so, what is the smallest number $B$ such that
$$|x| n|| \leq B \text { for all } n ?$$
(b) Calculate the output at $n=0$ for this particular choice of input. Does the result prove the contention that absolute summability is a necessary condition for stability?
(c) In a similar fashion, show that a continuous-time LTI system is stable if and only if its impulse response is absolutely integrable.
Problem 50
Consider the cascade of two systems shown in Figure P2.50. The first system. A, is known to be LTI. The second system, $B$, is known to be the inverse of system A. Let $y_{1}(t)$ denote the response of system $A$ to $x_{1}(t),$ and let $y_{2}(t)$ denote the response of system $A$ to $x_{2}(t)$
(a) What is the response of system $B$ to the $\ln$ put $a y_{1}(t)+b y_{2}(t),$ where $a$ and $b$ are constants?
(b) What is the response of system $B$ to the input $y_{1}(t-\tau)^{\prime}$
Problem 51
In the text, we saw that the overall input-output relationship of the cascade of two LTI systems does not depend on the order in which they are cascaded. This fact, known as the commutativity property, depends on both the linearity and the time invariance of both systems. In this problem, we illustrate the point.
(a) Consider two discrete-time systems $A$ and $B$, where system $A$ is an LTI system with 'unit sample response $h[n]=(1 / 2)^{n} u[n] .$ System $B,$ on the other hand, is linear but time varying. Specifically, if the input to system $B$ is $w[n],$ its output is
$$z[n]=n w[n]$$
Show that the commutativity property does not hold for these two systerns by computing the impulse responses of the cascade combinations in Figures P2.5 $\mathrm{J}(\mathrm{a})$ and $\mathrm{P} 2.5 \mathrm{l}(\mathrm{b}),$ respectively.
(b) Suppose that we replace system $B$ in each of the interconnected systems of Figure $\mathbf{P} 2.51$ by the system with the following relationship between its input $w[n]$ and output $z[n]$
$$z|n|=w[n\}+2$$
Repeat the calculations of part (a) in this case.
Problem 52
Consider a discrete-time $L T I$ system with unit sample response
$$h[n]=(n+1) \alpha^{n} u[n]$$
where $|\alpha|<1 .$ Show that the step response of this system is
$$s[n]=\left[\frac{1}{(\alpha-1)^{2}}-\frac{\alpha}{(\alpha-1)^{2}} \alpha^{n}+\frac{\alpha}{(\alpha-1)}(n+1) \alpha^{n}\right] u[n]$$
(Hint: Note that
$$\sum_{k=0}^{N}(k+1) \alpha^{k}=\frac{d}{d \alpha} \sum_{k=0}^{N+1} \alpha^{k}$$)
Problem 53
(a) Consider the homogeneous differential equation
$$\sum_{k=0}^{N} a_{k} \frac{d^{k} y(t)}{d t^{k}}=0$$
Show that if $s_{0}$ is a solution of the equation
$$p(s)=\sum_{k=0}^{N} a_{k} s^{k}=0$$
then $A e^{s_{n} f}$ is a solution of eq. $\{\mathbf{P} 2.53-1 \text { ), where } A$ is an arbitrary complex constant.
(b) The polynomial $p(s)$ in eq. $(P 2.5 .3-2)$ can be factored in terms of its roots $s_{1}, \ldots, s_{r}$ as
$$p(s)=a_{N}\left(s-s_{1}\right)^{r_{1}}\left(s-s_{2}\right)^{\sigma_{2}} \ldots\left(s-s_{r}\right)^{\sigma_{r}}$$
where the $s_{1}$ are the distinct solutions of eq. $(P 2.53-2)$ and the $\sigma$, are their multiplicities- -that $1 \mathrm{s}$, the number of times each root appears as a solution of the equation. Note that
$$\boldsymbol{\sigma}_{\mathrm{l}}+\sigma_{2}+\ldots+\boldsymbol{\sigma}_{r}=N$$
In general, if $\sigma_{i}>1,$ then not only is $A e^{s, t}$ a solution of eq. (P2.53-1). but so is $A t^{\prime} e^{x_{1} f},$ as long as $j$ is an integer greater than or equal to zero and less than or equal to $\sigma_{i}-1 .$ To illustrate this, show that if $\sigma_{i}=2,$ then Ale'. Is a solution of eq. $(\mathrm{P} 2.53-1) .[\text { Hint: } \text { Sow that if } s$ is an arbitrary complex number, then
$$\left.\sum_{k=0}^{N} \frac{d^{k}\left(A t e^{v}\right)}{d t^{k}}=A p(s) t e^{s t}+A \frac{d p(s)}{d s} e^{s t} \cdot\right]$$
Thus, the most general solution of eq. (P2.53-1) is
$$\sum_{i=1}^{r} \sum_{j+0}^{\sigma_{1}-1} A_{i j} r^{\prime} e^{k_{1} t}$$
where the $A_{t}$, are arbitrary complex constants.
(c) Solve the following homogeneous differential equations with the specified auxiliary conditions:
(i) $\frac{d^{2} y^{\prime}(1)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=0, y\langle 0\rangle=0, \quad y^{\prime}(0)=2$
(ii) $\frac{d^{2} y^{2}(1)}{d t^{2}}+3 \frac{d y(1)}{d t}+2 y(t)=0, y(0)=1, \quad y^{\prime}(0)=-1$
(iii) $\frac{d^{2} y(t)}{d f^{2}}+3 \frac{d y(f)}{d t}+2 y(t)=0, y(0)=0, \quad y^{\prime}(0)=0$
$(iv) \frac{d^{2} y(1)}{d t^{2}}+2 \frac{d y(t)}{d t}+y(t)=0, y(0)=1, y^{\prime}(0)=1$
$(v) \frac{d^{1} y(t)}{d t}+\frac{d^{2} y(t)}{d t^{2}}-\frac{d y(t)}{d t}-y(t)=0, y(0)=1, y^{\prime}(0)=1, y^{\prime \prime}(0)=-2$
(vi) $\frac{d^{2} y(t)}{d t^{2}}+2 \frac{d y(t)}{d t}+5 y(t)=0, y(0)=1, \quad y^{\prime}(0)=1$
Problem 54
(a) Consider the homogeneous difference equation
$$\sum_{k=0}^{N} a_{k} y[n-k]=0$$
Show that if $z_{0}$ is a solution of the equation
$$\sum_{k=0}^{N} a_{k} z^{-i}=0$$
then $A_{\bar{\ell}}^{n}$ is a solution of eq. $(P 2.54-1),$ where $A$ is an arbitrary constant.
(b) As it is more convenient for the moment to work with polynomials that have only nonnegative powers of $z,$ consider the equation obtained by multiplying both sides of eq. $(\mathbf{P} 2.54-2)$ by $z^{N}$
$$p(z)=\sum_{k=0}^{N} a_{k} z^{N-k}=0$$
The polynomial p(z) can be factored as
$$p(z)=a_{0}\left(z-z_{1}\right)^{\sigma_{1}} \ldots\left(z-z_{r}\right)^{p_{r}}$$
where the $z_{1}, \ldots, z_{r}$ are the distinct roots of $p(z)$ Show that if $y[n]=n z^{n-1},$ then
$$\sum_{i=0}^{N} a_{k} y[n-k]=\frac{d p(z)}{d z} z^{n-n}+(n-N) p(z) z^{n-N \cdot 1}$$
Use this fact to show that if $\sigma_{i}=2,$ then both $A z_{i}^{n}$ and $B n z_{i}^{n-1}$ are solutions of eq. $(\mathbf{P} 2.54-1),$ where $A$ and $B$ are arbitrary complex constants. More generally, one can use this same procedure to show that if $\sigma_{1}>1,$ then
$$A \frac{n !}{r !(n-r) !} z^{n-r}$$
is a solution of eq. $(P 2.54-1)$ for $r=0,1, \ldots, \sigma_{i}-1.7$
(c) Solve the following homogeneous difference equations with the specified auxiliary conditions:
(i) $y[n]+\frac{3}{4} y[n-1]+\frac{1}{8} y[n-2]=0 ; y[0]=1, y(-1]=-6$
(ii) $y[n]-2 y[n-1]+y[n-2]=0 ; y[0]=1, y[1]=0$
(iii) $y[n]-2 y[n-1]+y[n-2]=0 ; y[0]=1, y[10]=21$
$(\text { iv }) \quad y[n]-\frac{\sqrt{2}}{2} y[n-1]+\frac{1}{4} y[n-2]=0 ; y[0]=0, y[-1]=1$
Problem 55
In the text we described one method for solving linear constant-coefficient difference equations, and another method for doing this was illustrated in Problem $2.30 .$ If the assumption of initial rest is made so that the system described by the difference equation is LTI and causal, then, in principle, we can determine the unit impulse response $h[n]$ using either of these procedures. In Chapter $5,$ we describe another method that allows us to determine $h[n]$ in a mare elegant way. In this problem we describe yet another approach, which basically shows that $h[n]$ can be determined by solving the homogeneous equation with appropriate initial conditions.
(a) Consider the system initially at rest and described by the equation
$$y[n]-\frac{1}{2} y[n-1]=x[n]$$
Assuming that $x[n]=\delta[n],$ what is $y\{0] ?$ What equation does $h[n]$ satisfy for $n \geq 1,$ and with what auxiliary condition? Solve this equation to obtain a closed-form expression for $h[n]$
(b) Consider next the LTI system initially at rest and described by the difference equation
$$y[n]-\frac{1}{2} y[n-1]=x[n]$$
This system is depicted in Figure $\mathrm{P} 2.55$ (a) as a cascade of two LTS systems that are initially at rest: Because of the properties of LTI systems, we can reverse the order of the systems in the cascade to obtain an alternative representation of the same overall system, as illustrated in Figure $\mathrm{P} 2.55$ ( b). From this fact. use the result of part (a) to determine the impulse response for the system described by eq. (P2.55-2).
(c) Consider again the system of part (a), with $h[n]$ denoting its impulse response. Show, by verifying that eq. $(\mathrm{P} 2.55-3)$ satisfies the difference equation (P2.55-
1), that the response $y[n]$ to an arbitrary input $x[n]$ is in fact given by the convotution sum
$$y[n]=\sum_{n=-\infty}^{+\infty} h[n-m] x[m]$$
'Here, we are using factorial rotation- - that $is, k !=k(k-1)(k-2) .(2)(1),$ where $u !$ is defined to be 1
(d) Consider the LTI system initially at rest and described by the difference equation
$$\sum_{k=0}^{N} a_{k} y[n-k]=x[n]$$
Assuming that $a_{0} \neq 0,$ what is $y[0]$ if $x[n]=\delta[n] ?$ Using this result, specify the homogeneous equation and initial conditions that the impulse response of the system must satisfy.
Consider next the causal LTT system described by the difference equation
$$\sum_{k=0}^{N} a_{k} y[n-k]=\sum_{k=0}^{M} b_{k} x[n-k]$$
Express the impulse response of this system in terms of that for the LIT system described by eq. (P2.55-4).
(e) There is an alternative method for determining the impulse response of the LTT system described by eq. (P2.55-5), Specifically, given the condition of initial rest, i.e., in this case, $y[-N]=y[-N+1]=\ldots=y[-1]=0,$ solve eq. $(P 2.55-5)$ recursively when $x[n]=\delta[n]$ in order to determine $y[0], \therefore \ldots y[M]$ What equation does $h[n]$ satisfy for $n \geq M ?$ What are the appropriate initial conditions for this equation?
(0) Using either of the methods outlined in parts (d) and (e), find the impulse responses of the causal LTT systems described by the following equations:
(i) $y[n]-y[n-2]=x[n]$
(ii) $y[n]-y[n-2]=x[n]+2 x[n-1]$
(iii) $y[n]-y[n-2]=2 x[n]-3 x[n-4]$
(iv) $y[n]-(\sqrt{3} / 2) y\{n-1\}+\frac{1}{4} y[n-2]=x[n]$
Problem 56
In this problem, we consider a procedure that is the continuous-time counterpart of the technique developed in Problem 2.55. Again, we will see that the problem of determining the impulse response $h(t)$ for $t>0$ for an LTI system initially at rest and described by a linear constant-coefficient differential equation reduces to the problem of solving the homogeneous equation with appropriate initial conditions.
(a) Consider the LTT system initially at rest and described by the differential equation
$$\frac{d y(t)}{d t}+2 y(t)=x(t)$$
Suppose that $x(t)=\delta(t) .$ In order to determine the value of $y(t)$ immediately after the application of the unit impulse, consider integrating eq. $\{\mathrm{P} 2.56-1$ ) from
$t=0^{-}$ to $t=0^{+}$ (i.e., from "just before" to "just after" the application of the impulse). This yields
$$y\left(0^{\circ}\right)-y\left(0^{-}\right)+2 \int_{0}^{0^{+}} y(\tau) d \tau=\int_{0}^{0^{+}} \delta(\tau) d \tau=1$$
since die system is initially at rest and $x(f)=0$ for $t<0, y\left(0^{-}\right)=0 .$ To satisfy eq. $(\mathbf{P} 2.56-2)$ we must have $y\left(0^{+}\right)=1 .$ Thus, since $x(t)=0$ for $f>0,$ the impulse response of our system is the solution of the homogeneous differential equation
$$\frac{d y(t)}{d t}+2 y(t)=0$$
with initial condition
$$y\left(0^{+}\right)=1$$
Solve this differential equation to obtain the impulse response $h(t)$ for the system. Check your result by showing that
$$y(t)=\int_{-\infty}^{+\infty} h(t-\tau) x(\tau) d \tau$$
satisfies eq. $(\mathrm{P} 2.56-1)$ for any input $x(t)$
(b) To generalize the preceding argument, consider an LTI system initially at rest and described by the differential equation
$$\sum_{k=0}^{N} a_{k} \frac{d^{k} y(t)}{d t^{k}}=x(t)$$
with $x(t)=\delta(t)$. Assume the condition of initial rest, which, since $x(t)=0$ for $r<0,$ implies that
$$y\left(0^{-}\right)=\frac{d y}{d t}\left(0^{-}\right)=\ldots=\frac{d^{N-1} y}{d t^{N-1}}\left(0^{-}\right)=0$$
$$y\left(0^{-}\right)=\frac{d y}{d t}\left(0^{-}\right)=\ldots=\frac{d^{\mathrm{N}-1} y}{d t^{N-1}}\left(0^{-}\right)=0$$
Integrate both sides of eq. (P2.56-3) once from $t=0^{-}$ to $t=0^{+},$ and use eq. $(\mathbf{P} 2.56-4)$ and an argument similar to that used in part (a) to show that the
resulting equation is satisfied with
$$y\left(0^{+}\right)=\frac{d y}{d t}\left(0^{+}\right)=\ldots=\frac{d^{N-2} y}{d t^{N-2}}\left(0^{+}\right)=0$$
and
$$\frac{d^{N \cdots 2} y}{d f^{N-2}}\left(0^{+}\right)=\frac{1}{a^{N}}$$
Consequently, the system's impulse response for $t>0$ can be obtained by solving the homogeneous equation
$$\sum_{k=0}^{N} a_{k} \frac{d^{k} y(t)}{d t^{k}}=0$$
with initial conditions given by eqs. (P2.56-5).
(c) Consider now the causal LTI system described by the differential equation
$$\sum_{k=0}^{N} a_{k} \frac{d^{k} y(t)}{d t^{k}}=\sum_{k=0}^{M} b_{k} \frac{d^{k} x(t)}{d t^{k}}$$
Express the impulse response of this system in terms of that for the system of part (b). (Hint: Examine Figure P2.56.)
(d) Apply the procedures outlined in parts (b) and (c) to find the impulse responses for the LTI systems initially at rest and described by the following differential equations:
(i) $\frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=x(t)$
(ii) $\frac{d^{2} y(1)}{d t^{2}}+2 \frac{d y(t)}{d t}+2 y(t)=x(t)$
(e) Use the results of parts (b) and (c) to deduce that if $M \geq N$ in eq. $(P 2.56-6)$, then the impulse response $h(t)$ will contain singularity terms concentrated at $t=0 .$ In particular, $h(t)$ will contain a term of the form
$$\sum_{r=0}^{M-N} \alpha_{r} u_{r}(t)$$
where the $\alpha_{r}$ are constants and the $u_{r}(t)$ are the singularity functions defined in Section 2.5.
(f) Find the impulse responses of the causal LT1 systerss described by the following differential equations:
(i) $\frac{d y(1)}{d t}+2 y(t)=3 \frac{d x(t)}{d t}+x(t)$
(ii) $\frac{d^{2} y(x)}{d t^{2}}+5 \frac{d y(t)}{d t}+6 y(t)=\frac{d^{3} x(t)}{d t^{4}}+2 \frac{d^{2} x(t)}{d t^{2}}+4 \frac{d x(t)}{d t}+3 x(t)$
Problem 57
Consider a causal LTI system $S$ whose input $x[n]$ and output $y\{n\}$ are related by the difference equation
$$y[n]=-a y[n-1]+b_{0} x[n]+b_{1} x[n-1]$$
(a) Verify that $S$ may be considered a cascade connection of two causal LTT systems $S_{1}$ and $S_{2}$ with the following input-output relationship:
$$\begin{aligned}
&\left.S_{1}: y_{1}[n]=b_{0} x_{1}[n]+b_{1} x_{1} | n-1\right]\\
&S_{2}: y_{2}[n]=-a y_{2}[n-1]+x_{2}[n]
\end{aligned}$$
(b) Draw a block diagram representation of $S_{1}$
(c) Draw a block diagram representation of $S_{2}$
(d) Draw a block diagram representation of $S$ as a cascade connection of the black diagram representation of $S_{1}$ followed by the block diagram representation of $S_{2}$
(e) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{2}$ followed by the block diagram representation of $S_{1}$
(f) Show that the two unit-delay elements in the block diagram representation of $S$ obtained in part (e) may be collapsed into one unit-delay element. The resulting block diagram is referred to as a Direct Form $l /$ realization of $S$, while the block diagrams obtained in parts (d) and (e) are referred to as Direct Form $I$ realizations of $S$
Problem 58
Consider a causal LTI system $S$ whose input $x[n]$ and output $y[n]$ are related by the difference equation
$$2 y[n]-y[n-1]+y[n-3]=x[n]-5 x[n-4]$$
(a) Verify that $S$ may be considered a cascade connection of two causal LII systems $S_{1}$ and $S_{2}$ with the following input-output relationship:
$$\begin{aligned}
&S_{1}: 2 y_{1}[n]=x_{1}[n]-5 x_{1}\{n-4\}\\
&S_{2}: y_{2}[n]=\frac{1}{2} y_{2}[n-1]-\frac{1}{2} y_{2}[n-3]+x_{2}[n]
\end{aligned}$$
(b) Draw a block diagram representation of $S_{1}$
(c) Draw a block diagram representation of $S_{2}$
(d) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{1}$ followed by the block diagram representation of $S_{2}$
(e) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{2}$ followed by the block diagram representation of $S_{1}$
(f) Show that the four delay elements in the block diagram representation of $S$ obtained in part (e) may be collapsed to three. The resulting block diagram is referred to as a Direct Form $I I$ realization of $S,$ while the block diagrams obtained in parts $(\mathrm{d})$ and (e) are referred to as Direct Form $I$ realizations of $S$.
Problem 59
Consider a causal LTI system $S$ whose input $x(t)$ and output $y(t)$ are related by the differential equation
$$a_{1} \frac{d y(t)}{d t}+a_{n} y(t)=b_{0} x(t)+b_{1} \frac{d x(t)}{d t}$$
(a) Show that
$$y(t)=A \int_{-x}^{t} y(\tau) d \tau+B x(t)+C \int_{x}^{t} x(\tau) d \tau$$
and express the constants $A, B,$ and $C$ in terms of the constants $a_{0}, a_{1}, b_{0}$ and $b_{1}$
(b) Show that $S$ may be considered a caseade connection of the following two causal LTI systems:
$$\begin{array}{l}
S_{1}: y_{1}(t)=B x_{1}(t)+C \int_{-x}^{t} x(\tau) d \tau \\
S_{2}: y_{2}(t)=A \int_{x}^{\prime} y_{2}(\tau) d \tau+x_{2}(t)
\end{array}$$
(c) Draw a block diagram representation of $S_{1}$
(d) Draw a block diagram representation of $S_{2}$
(e) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{1}$ followed by the block diagram representation of $S_{2}$
(f) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{2}$ followed by the block diagram of representation $S_{1}$
(g) Show that the two integrators in your answer to part (f) may be collapsed into one. The resulting block diagram is referred to as a Direct Form $\mathrm{H}$ realization of $S,$ while the block diagrams obtained in parts (e) and (f) are referred to as Direct Form $I$ realizations of $S$
Problem 60
Consider a causal LTI system $S$ whose input $x\{t \text { ) and output } y(t)$ are related by the differential equation
$$a_{2} \frac{d^{2} y(t)}{d t^{2}}+a_{1} \frac{d y(t)}{d t}+a_{0} y(t)=b_{0} x(t)+b_{1} \frac{d x(t)}{d t}+b_{2} \frac{d^{2} x(t)}{d t^{2}}$$
(a) Show that
$$\begin{aligned}
y(t)=& A \int_{-x}^{l} y(\tau) d \tau+B \int_{-x}^{t}\left(\int_{-x}^{t} y(\sigma) d \sigma\right) d \tau \\
&+C x(t)+D \int_{-x}^{t} x(\tau) d \tau+E \int_{-\infty}^{t}\left(\int_{-x}^{\tau} x(\sigma) d \sigma\right) d \tau
\end{aligned}$$
and express the constants $A, B, C, D,$ and $E$ in terms of the constants $a_{0}, a_{1}, a_{2}$ $b_{0}, b_{1},$ and $b_{2}$
(b) Show that $\$$ may be considered a cascade connection of the following two causal LTI systems:
$$\begin{array}{l}
S_{1}: y_{1}(t)=C x_{1}(t)+D \int_{-x} x_{1}(\tau) d \tau+E \int_{-x}^{1}\left(\int_{-x}^{\tau} x_{1}(\sigma) d \sigma\right) d \tau \\
S_{2}: y_{2}(t)=A \int_{-\infty}^{t} y_{2}(\tau) d \tau+B \int_{-\infty}^{t}\left(\int_{-x}^{\tau} y_{2}\{\sigma) d \sigma\right) d \tau+x_{2}(t)
\end{array}$$
(c) Draw a block diagram representation of $S_{1}$
(d) Draw a block diagram representation of $S_{2}$
(e) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{1}$ followed by the block diagram representation of $S_{2}$
(f) Draw a block diagram representation of $S$ as a cascade connection of the block diagram representation of $S_{2}$ followed by the block diagram representation of $\mathcal{S}_{1}$
(g) Show that the four integrators in your answer to part (f) may be collapsed into two. The resulting block diagram is referred to as a Direct Form 11 realization of $S,$ while the block diagrams obtained in parts (e) and (f) are referred to as Direct Form $I$ realizations of $S$
Problem 61
(a) In the circuit shown in Figure $P 2.61(a), x(t)$ is the input voltage. The voltage $y(t)$ across the capacitor is considered to be the system output.
(i) Determine the differential equation relating $x(t)$ and $y(t)$
(ii) Show that the homogeneous solution of the differential equation from part
(i) has the form $K_{1} e^{j \alpha_{1} t}+K_{2} e^{j \omega_{2} t}$. Specify the values of $\omega_{1}$ and $\omega_{2}$
(iii) Show that, since the voltage and current are restricted to be real, the natural response of the system is sinusoidal.
(b) In the circuit shown in Figure $\mathrm{P} 2.61(\mathrm{b}), x(t)$ is the input voltage. The voltage $y(t)$ across the capacitor is considered to be the system output.
(i) Determine the differential equation relasing $x(t)$ and $y(t)$
(ii) Show that the natural response of this system has the form $K e^{-a t}$, and specify the value of $a$
(c) In the circuit shown in Figure $\mathrm{P} 2.6 \mathrm{f}(\mathrm{c}), x(t)$ is the inpot voltage. The voltage $y(t)$ across the capactor is considered to be the system output.
(i) Determine the differential equation relating $x(t)$ and $y(t)$
(ii) Show that the homogeneous solution of the differential equation from part
(i) has the form $e^{-\alpha f}\left\{K_{1} e^{j 2 t}+K_{2} e^{-j 2 r}\right\},$ and specify the value of $a$
(iii) Show that, since the voltage and current are restricted to be real, the natural response of the system is a decaying sinusoid.
Problem 62
(a) In the mechanical system shown in Figure $P 2.62(a),$ the force $x(t)$ applied to the mass represents the input, while the displacement $y(t)$ of the mass represents the output. Determine the differential equation relating $x(t)$ and $y(t)$. Show that the natural response of this system is periodic
(b) Consider Figure $\mathrm{P} 2,62(\mathrm{b}),$ in which the force $x(t)$ is the input and the velocity $y(t)$ is the output. The mass of the car is $m$, while the coefficient of kinetic friction is $\rho .$ Show that the natural response of this system decays with increasing time
(c) In the mechanical system shown figure $P 2.62(c),$ the force $x(t)$ applied to the mass represents the input, while the displacement $y(t)$ of the mass represents the output.
(i) Determine the differential equation relating $x(f)$ and $y(t)$
(ii) Show that the homogeneous solution of the differential equation from part
(i) has the form $\left.e^{-\operatorname{ai}\{} K_{1} e^{j \mathrm{i}}+K_{2} e^{-j r}\right\},$ and specify the value of $a$
(iii) Show that, since the force and displacement are restricted to be real, the natural response of the system is a decaying sinusoid.
Problem 63
A $\$ 100,000$ mortgage is to be retired by equal monthly payments of $D$ dollars. In terest, compounded monthly, is charged at the rate of $12 \%$ per annum on the unpaid balance; for example, after the first month, the total debt equals
$$\$ 100,000+\left(\frac{0.12}{12}\right) \$ 100.000=\$ 101,000$$
The problem is to determine $D$ such that after a specified time the mortgage is paid in full, leaving a net balance of zero.
(a) To set up the problem, let $y[n]$ denote the unpaid balance after the $n$ th monthly payment. Assume that the principal is borrowed in month 0 and monthly payments begin in month 1. Show that $y[n]$ satisfies the difference equation
$$y[n]-\gamma y[n-1]=-D \quad n \geq 1$$
with initial condition
$$y[0]=\$ 100,000$$
where $\gamma$ is a constant. Determine $\gamma$
(b) Solve the difference equation of part (a) to determine
$$y[n] \quad \text { for } n \geqslant 0$$
(Hink: The particular solution of eq. $(\mathrm{P} 2.63-1)$ is a constant $Y$. Find the value of $Y,$ and express $y(n]$ for $n \geq 1$ as the sum of particular and homogeneous solutions. Deternine the unknown constant in the homogeneous solution by directly calculating $y[1]$ from eq. $(P 2.63-1)$ and comparing it to your solution.)
(c) If the mortgage is to be retired in 30 years after 360 monthly payments of $D$ dollars, determine the appropriate value of $D$
(d) What is the total payment to the bank over the 30 -year period?
(e) Why do banks make loans?
Problem 64
One important use of inverse systems is in situations in which one wishes to remove distortions of some type. A good example of this is the problem of removing echoes from acoustic signals. For example. if an auditorium has a perceptible echo, then an initial acoustic impulse will be followed by attenuated versions of the sound at regularly spaced intervals. Consequently, an often-used model for this phenomenon is an LTI system with an impulse response consisting of a train of impulses, is...
$$h(t)=\sum_{k=0}^{x} h_{k} \delta(t-k T)$$
Here the echoes occur $T$ seconds apart, and $h_{k}$ represents the gain factor on the $k$ th echo resulting from an initial acoustic impulse.
(a) Suppose that $x(t)$ represents the original acoustic signal (the music produced by an orchestra, for example) and that $y(t)=x(t) * h(t)$ is the actual signal that is heard if no processing is done to remove the echoes. In order to remove the distortion introduced by the echoes, assume that a microphone is used to sense $y(t)$ and that the resulting signal is transduced into an electrical signal. We will also use $y(t)$ to denote this signal, as it represents the electrical equivalent of the acoustic signal, and we can go from one to the other via acoustic-electrical conversion systems.
The important point to note is that the system with impulse response given by eq. (P2.64-1) is invertible. Therefore, we can find an LTI system with impulse response $g(r)$ such that
$$y(t) * g(t)=x(t)$$
and thus, by processing the electrical signal $y(t)$ in this fashicn and then converting back to an acoustic signal, we can remove the troublesome echoes. The required impulse response $g(t)$ is also an impulse train:
$$g(t)=\sum_{k=0}^{x} g_{k} \delta(t-k T)$$
Determine the algebraic equations that the successive $g_{k}$ must satisfy, and solve these equations for $g_{0}, g_{1},$ and $g_{2}$ in terms of $h_{k}$
(b) Suppose that $h_{0}=1, h_{1}=1 / 2,$ and $h_{t}=0$ for all $i \geq 2$ What is $g(t)$ in this case?
(c) A good model for the generation of echocs is illustrated in Figure P2.64. Hence, each successive echo represents a fed-back version of $y(t),$ delayed by $T$ seconds and scaled by $\alpha .$ Typically, $0<\alpha<1,$ as successive echoes are attendated.
(i) What is the impulse response of this systern's (Assume initial rest, i.e., $y(t)=0$ for $t<0$ if $x(t)=0$ for $t<0 .$
(ii) Show that the system is stable if $0<\alpha<1$ and unstable if $\alpha>1$
(iii) What is $g(t)$ in this case? Construct a realization of the inverse system using adders, coefficient multipliers, and $T$ -second delay elements.
(d) Although we have phrased the preceding discussion in terms of continuous-time systems because of the application we have been considering, the same general ideas hold in diserete tome. That is, the CTI systern with impulse response
$$h[n]=\sum_{k=0}^{\infty} \delta | n-k N$$
is invertible and has as its inverse an LTI sysiern with impulse response
$$g[n]=\sum_{k=0}^{\infty} g_{k} \delta | n-k N$$
It is not difficult to check that the $g_{r}$ satisfy the same algebrace equations as in part (a) Consider bow the discrete-time LTI system with impulse response
$$h[n]=\sum_{k=-\infty}^{\infty} \delta[n-k N]$$
This system is nor invertible. Find two inputs that produce the same output.
Problem 65
In Problem $1.45,$ we introduced and examined some of the basic properties of $\cos$ relation functions for continuous-time signals. The discrete-time counterpart of the correladon function has esseatially the same properties as those in continuous time, and both are extremely important in numerous applications (as is discussed in Problems 2.66 and 2.67 ). In this problem, we introduce the discrete-time correlation function and examine several more of its properties.
Let $x[n]$ and $y[n]$ be two real-valued discrete-time signals. The aulocorrelation functions $\phi_{\mathrm{rx}}[n]$ and $\phi_{y v}[n]$ of $x[n]$ and $y[n],$ respectively, are defined by the expressions
$$\phi_{x x}[n]=\sum_{m=-\infty}^{+\infty} x[m+n] x[m]$$
and
$$\phi_{v v}[n]=\sum_{m=-\infty}^{+\infty} y[m+n] y\{m\}$$
and the cross-correlation functions are given by
$$\phi_{x v}[n]=\sum_{m=-\infty}^{+\infty} x[m+n] y[m]$$
and
$$\phi_{x}[n]=\sum_{m=-x}^{+\infty} y[m+n] x[m]$$
As in continuous time, these functions possess certain symmetry properties. Specifically, $\phi_{\mathrm{rr}}[n]$ and $\phi_{\mathrm{rv}}[n]$ are even functions, while $\phi_{\mathrm{r}},[n]=\phi_{\mathrm{y}},[\cdots n]$
(a) Compute the autocorrelation sequences for the signals $x_{1}[n], x_{2}[n], x_{3}[n,$ and $x_{4}[n]$ depicted in Figure $P 2.65$
(b) Compute the cross-correlation sequences
$$\phi_{x, x_{f}}[n], \quad i \neq j, \quad j, j=1,2,3,4$$
for $x,[n], i=1,2,3,4,$ as shown in Figure $P 2.65$
(c) Let $x[n] \text { be the input to an LTI systern with unit sample respense } h | n]$, and let the corresponding output be $y[n] .$ Find expressions for $\phi_{r s}[n]$ and $\phi_{\curlyvee}[n]$ in terms of $\phi_{x 1}[n]$ and $h[n] .$ Show how $\phi_{1},[n]$ and $\phi_{r r}[n]$ can be viewed as the output of LTI systems with $\phi_{x x}[n]$ as the input. (Do this by explicitly specifying the impulse response of each of the two systems.
(d) Let $h[n]=x_{1}[n]$ in Frgure $P 2.65,$ and let $y[n]$ be the output of the LTI system with impulse response $h[n]$ when the input $x[n]$ also equals $x_{1}[n] .$ Calculate $\phi_{x y}[n]$ and $\phi_{y y}[n]$ using the results of part (c).
Problem 66
Let $h_{1}(t), h_{2}(t),$ and $h_{3}(t),$ as sketched in Figure $P 2.66,$ be the impulse responses of three LTI systems. These three signals are known as Walsh functions and are of considerable practical importance because they can be easily generated by digital logic circuitry and because multiplication by each of them can be implemented in a simple fashion by a polarity-reversing switch.
(a) Determine and sketch a choice for $x_{1}(t)$, a continuous-tine signal with the following properties:
(i) $x_{1}(t)$ is real
(ii) $x_{1}(t)=0$ for $t<0$
(iii) $\left|x_{1}(t)\right| \leq 1$ for $t \geq 0$
(iv) $y_{i}(t)=x_{1}(t) * h(t)$ is as large as possible at $t=4$
(b) Repeat part (a) for $x_{2}(t)$ and $x_{3}(t)$ by making $y_{2}(t)=x_{2}(t) * h_{2}(t)$ and $y_{3}(t)=$ $x_{3}(t) * h_{3}(t)$ each as large as possible at $t=4$
(c) What is the value of
$$y_{i j}(t)=x_{i}(t) * h_{j}(t), i \neq j$$
at time $t=4$ for $i, j=1,2,3 ?$
The system with impulse response $h_{1}(t)$ is known as the marched filter for the signal $x_{f}(t)$ because the impulse respunse is tuned to $x,(t)$ in order to produce the maximum output signal. In the next problem, we relate the concept of a matched filter to that of the correlation function for continuous-time signais.
Problem 67
The cross-correlation function between two continuous-time real signals $A(t)$ and $y(t)$ is
$$\left.\phi_{r,(t)}=\int_{s}^{+\infty} x ! t+\tau\right\rfloor y(\tau) d \tau$$
The ctutocorvelation function of a signal $x(t)$ is obtained by setting $y(t)=r(t)$ in eq. $(P 2.67-1)$
$$\left.\phi_{1 r}(t)=\int_{-\infty}^{r+t} x+t+\tau\right) \lambda(\tau) d \tau$$
(a) Compute the autocorrelation funetion for each of the two signals $x_{1}(t)$ and $x_{2}(t)$ depicted in Figure $\mathrm{P} 2.67(\mathrm{a})$
(b) Let $\lambda(t)$ be a given signal, and assume that $x(t)$ is of finite duration - i.e., that $x(t)=0$ for $t<0$ and $t>T .$ Find the impulse response of an LTI system so that $\phi_{1 r}(1-T)$ is the output if $x(t)$ is the input.
(c) The system determired in part (b) is a matched fuiter for the signal $(t)$ That this definition of a matched filter is identical to the one introduced in Problem $2.66 \mathrm{can} \mathrm{be}$ seen from the following:
Let $x(t)$ be as in part (b), and let $y(t)$ denote the response to $x(t)$ of an LTI system with real impuise response $h(t)$. Assume that $h(t)=0$ for $r<0$ and for $t>T .$ Show that the choice for $h(t)$ that maximizes $y(T),$ subject to the constraint that
$\int_{0}^{T} h^{2}(t) d t=M,$ a fixed positive number
is a scalar multiple of the impulse response determined in part $(\mathbf{b}) .[$ Hnt. Schwartz's inequality states that
$$\int_{b}^{a} u(t) v(t) d t \leq\left[\int_{a}^{b} u^{2}(t) d t\right]^{1 / 2}\left[\int_{t u}^{b} v^{2}(t) d t\right]^{1 / 2}$$
for any two signals $u(t)$ and $v(t)$. Use this to obtain a bound on y $y$ T.
(d) The constraint given by eq. (P2.67 -2 ) simpiy provides a scaling to the impulse response, as increasing $M$ merely changes the scalar maltuplier mentioned in part (c). Thus, we see that the particular choice for $h(t)$ in pars $(b)$ and $(c)$ is matched to the signal $x(t)$ ta produce maximum ourpul. This is an extremety important propeny in a number of applications, as we will now indicate. In communication problems, one often wishes to transmit one of a small number of possible pieces of information. For example, if a complex message is encoded into a sequence of binary digits, we can imagine a system that transmits the information bit by bit. Each bit can bhen be transmitted by sending one signal, say, $x_{0}(t)$, if the bit is a 0 , or a different signal $x_{1}(t)$ if a 1 is to be communicated. In this case, the receiving system for these signals must be capable of recognizing whether $x_{0}(t)$ or $x_{1}(t)$ has been received. Intuitively, what makes sense is to have two systems in the receiver, one tuned to $x_{0}(t)$ and one tuned
the signal to which it is cuned is received. The property of protucing a large outpus when a particular signal is received is exaclly what the matched filter possesses
In practice, there is always distortion and interference in the transmission and reception processes. Consequently, we want to maximize the difference between the response of a matched filter to the input to which it is matched and the response of the filter to one of the other signats that can be transmitted. To illustrate this point, consider the two signals $x_{0}(t)$ and $x_{1}(t)$ depicted in Figure $\mathrm{P} 2.67(\mathrm{b})$. Let $L_{0}$ denote the matched filter for $x_{0}(t),$ and let $L_{1}$ denote the matched filter for $x_{1}(t)$
(i) Sketch the responses of $L_{0}$ to $x_{0}(t)$ and $x_{1}(t) .$ Do the same for $L_{1}$
(ii) Compare the values of these responses at $t=4 .$ How might you modify $x_{3}(t)$ so that the receiver would bave an even easier job of distinguishing between $x_{0}(t)$ and $x_{1}(t)$ in that the response of $L_{n}$ to $x_{1}(t)$ and $L_{1}$ to $x_{0}(t)$ would both be zero at $t=4 ?$
Problem 68
Another application in which matched filters and correlation functions play an important role is radar systems. The underlying principle of radar is that an electromaguetic pulse transmitled at a target will be reflected by the targe and will subsequently return to the sender wath a delay proportional to the distance to the target Ideally, the received sigral wil? simply be a shifted and possibly scaled version of the original transmitted signal Let $p(t)$ he the original pulse that is sent out. Show that
$$\phi_{\mu p}(0)=\max \phi_{, p}(t)$$
if the waveform that comes back to the sender is
$$x(t)=\alpha p\left(t-t_{0}\right)$$
where $\alpha$ is a positive constant, then
$$\phi_{r p}\left(t_{t}\right)=\max _{t} \phi_{r p}(t)$$
(Hint: Use Schwartz's inequality.) Thus, the way in which simple radar ranging systems work is hased on using a matched filter for the transmitted waveform $p(t)$ and noting the tome at which the output of this system reaches its maximum value.
Problem 69
In Section $2.5,$ we characterized the unit doublet through the equation
$$x(t) * u_{1}(t)=\int_{x}^{+x} x(t-\tau) u_{1}(\tau) d \tau=x^{\prime}(t)$$
for any signal $x(t)$. From this equation, we derived the relationship
$$\int_{-x}^{1 x} g(\tau) u_{1}(\tau) d \tau=-g^{\prime}(0)$$
(a) Show that eq. $(P 2.69-2)$ is an equivalent characterization of $u_{1}(t)$ by showing that eq. $(\mathrm{P} 2.69-2)$ implies eq. $(\mathrm{P} 2.69-1)$. $[\mathrm{Hint} \text { ; Fix } t$, and define the signal
\[
g(\tau)=x(t-\tau) \cdot\}
\]
Thus, we have seen that characterizing the unit impulse or unit doublet by how ic behaves under convolution is equivalent to characterizing how it behaves under integration when multiplied by an arbitrary signal $g(t)$. In fact, as indicated in Section $2.5,$ the equivalence of these operatisnal definitions holds for all signals and, in particalar. for all singularity functions.
(b) Let $f(t)$ be a given signal. Show that
$$f(t) u_{1}(t)=f(0) u_{1}(t)-f^{\prime}(0) \delta(t)$$
by showing that both functions have the same operational definitions.
(c) What is the value of
$$\int_{-x}^{x} x(\tau) u_{2}(\tau) d \tau ?$$
Find an expression for $f(t) u_{2}(t)$ analogous to that in part (b) for $f(t) a_{1}(t)$
Problem 70
In analogy with continuous-time singularity functions, we can define a set of discrete-time signals, Specifically, let
$$\begin{aligned}
u_{-1}[n] &=u[n] \\
u_{0}[n] &=\delta[n]
\end{aligned}$$
and
$$u_{1}[n]=\delta[n]-\delta[n-1]$$
$$u_{1}[n]=\delta[n]-\delta[n-1]$$
$$u_{k}[n]=\underbrace{u_{1}[n] * u_{1}[n] * \cdots * u_{1}[n]}_{k \text { times }}, k>0$$
3$$u_{k}[n]=\underbrace{u_{1}[n] * u_{1}[n] * \cdots * u_{1}[n]}_{k \text { times }}, k>0$$
$$u_{k}[n]=\underbrace{\mu_{-1}[n] * u_{-1}[n] * \cdots \cdot u_{-1}[n],}_{|k| \text { times }} k<0$$
Note that
$$\begin{aligned}
&x[n] * \delta[n]=x[n]\\
&x(n] \cdot u[n]=\sum_{m=-x}^{\infty} x[m]
\end{aligned}$$
(a) What is
$$\sum_{m=\infty}^{\infty} x[m] u_{1}[m] ?$$
(b) Show that
$$\begin{aligned}
x[n] u_{1}[n] &=x[0] u_{1}[n]-[x[1]-x[0]] \delta[n-1] \\
&=x[1] u_{1}[n]-[x[1]-x[0]] \delta[n]
\end{aligned}$$
(c) Sketch the signals $u_{2}[n]$ and $u_{3}[n]$
(d) Sketch $u_{-2}[n]$ and $u_{-},[n]$
(e) Show that, in general, for $k>0$
\begin{equation}u_{k}[n]=\frac{(-1)^{n} k !}{n !(k-n) !}[u[n]-u[n-k-1]]\end{equation}
(Hint: Use induction. From part (c), it is evident that $u_{k}[n]$ satisfies eq. $(\mathrm{P} 2.70-1)$ for $k=2$ and $3 .$ Then, assurning that eq. $(\mathrm{P} 2.70-1)$ satisfies $u_{k}[n]$ write $u_{k+1}[n]$ in terms of $u_{k}[n],$ and show that the equation also satisfies $4 k+1[n] .)$
(f) Show that, in general, for $k>0$
$$u_{-\lambda}[n]=\frac{(n+k-1) !}{n !(k-1) !} u[n]$$
(Hint: Again, use induction. Note that
$$u_{(k+1)}[n] \cdots u_{-1 k+1},[n-1]=u_{-2}[n]$$
that eq. $(\mathrm{P} 2.70-2)$ is valid for $u$, $_{\{-1}[n]$ as well.
Problem 71
In this chapter, we have used several properties and ideas that greatly facilitate the analysis of LT1 systems. Among these are two that we wish to examine a bit more closely. As we will see, in cerlain very special cases onc must he careful in using these properties, which otherwise hold without qualification.
(a) Onc of the basic and most important properties of convalution (in both contin uous and discrete time) is associativity. That is, if $x(t) . h(t),$ and $g(t)$ are three signals, Ihen
$$x(t) *[\beta(t) * h(t)]=[x(t) * g(t)] * h(t)=[x(t) * h(t)] *_{W}(t)$$
This relationship holds as long as all three expressions arc well defined and finite. As that is usually the case in practice, we will in general use the associativity property without comments or assumptions. However, there are some cases in which it does $n o t$ hold. For example, consider the system depreted in Figure $P 2.71,$ with $h(t)=u_{1}(t)$ and $g(t)=u(t) .$ Compute the response of thrs system to the input
$x(t)=1$ for all $t$
Do this in the three different ways suggested by eq. $(\mathrm{P} 2.71-1)$ and by the figure.
(i) By first convolving the 1 wo impulse responses and then convolving the result with $x(t)$
(ii) By first convolving $x(t)$ with $u_{1}(t)$ and then convolving the result with $u(t)$
(iii) By first convolving $\lambda(f)$ with $u(t)$ and then convolving the resuit with $i(f)$
(b) Repeat part (a) for
$$x(t)=e^{-t}$$
and
$$\begin{array}{l}
h(t)=e^{t} u(t) \\
g(t)=u_{1}(t)+\delta(t)
\end{array}$$
(c) Do the same for
$$\begin{array}{l}
x[n]=\left(\frac{1}{2}\right)^{n} \\
h[n]=\left(\frac{1}{2}\right)^{n} u[n] \\
g[n]=\delta[n]-\frac{1}{2} \delta[n-1]
\end{array}$$
Thus, in general, the associativity property of convolution holds if and only if the three expressions in eq. $(\mathrm{P} 2.71-1)$ make sense (i.e., if and only if their interpretations in terms of LTT systems are meaningful). For example, in part (a) differentiating a constant and then integrating makes sense, but the process of integrating the constant from $t=-\infty$ and then differentiating does not, and it is only in such cases that associativity breaks down.
Closely related to the foregoing discussion is an issue involving inverse systems. Consider the LTI system with impulse response $h(r)=u(t)$. As we saw in part (a), there are inputs-mpecifically, $x(t)=$ nonzero constant- -for which the output of this system is infinite, and thus, the meaningless to consider the question of inverting such outputs to recover the input. However, if we limit ourselves to inputs that do yield finite outputs, that is, inputs which satisfy
$$\left|\int_{-\infty}^{t} x(\tau) d \tau\right|<\alpha$$
then the system is invertible, and the LTI system with impulse response $u_{1}(t)$ is its inverse.
(d) Show that the LTI system with impulse response $u_{1}(t)$ is not invertible. (Hint:
Find two different inputs that both yield zero output for all time.) However, show that the system is invertible if we limit ourselves to inputs that satisfy eq. (P2.71-2). $[\text {Hint}: \text { In Problem } 1.44,$ we showed that an LTI system is invertible if no input other than $x(t)=0$ yields an ontput that is zero for all time, are there two inputs $x(t)$ that satisfy eq. $(\mathrm{P} 2.71-2)$ and that yield identically zero responses when convolved with $u_{1}(t) ?$ What we have illustrated in this prablem is the following:
(1) If $x(t), h(t),$ and $g(t)$ are three signals, and if $x(t) * g(t), x(t) * h(t),$ and $h(t) * g(t)$ are $a l l$ well defined and finite, then tha associativity property, eq. $(\mathrm{P} 2.7 \mathrm{l}-1),$ bolds.
(2) Let $h(t)$ be the impulse response of an LTl system, and suppose that the impulse response $g(t)$ of a second system has the property
$$h(t) * g(t)=\delta(t)$$
Then, from (I), for all inputs $x(t)$ for which $x(t) * h(t)$ and $x(t) * g(t)$ are both well defined and finite, the two cascades of systems depicted m Frgure $\mathrm{P} 2.71$ act as the identity system, and thus, the two LTI systems can be regarded as inverses of one another For example, if $h(t)=u(t)$ and $g(t)=\mu_{1}(t),$ then, as long as we restrict ourselves to inputs satisfy cq. $(\mathrm{P} 2.71-2),$ we can regard these two systems as inverses.
Therefore, we see that the associativity property of eq. $(\mathrm{P} 2.71-1)$ and the definition of LTI inverses as given in eq. $(\mathrm{P} 2.7 \mathrm{l}-3)$ are valid, as long as all convolutions that are involved are finite. As this is certainiy the case in any realistic probiem, we will in general use these properties without comment or qualification. Note that, although we have phrased most of our discussion in terms of continuous-time signals and systems, the same points can also be made in discrete time [as should be evident from part (c)].
Problem 72
Let $\delta_{\Delta}(t)$ denote the rectangular pulse of height $\frac{1}{d}$ for $0<t \leq \Delta$. Verify that
$$\left.\frac{d}{d t} \delta_{\Delta}(t)=\frac{1}{\Delta} | \delta(t) \cdots \delta(t-\Delta)\right]$$
Problem 73
Show by induction that
$$\left.u_{-k}^{i} t\right)=\frac{t^{k-t}}{(k-1) !} u(t) \text { for } k=1,2,3 \dots$$
Source: https://www.numerade.com/books/chapter/linear-time-invariant-systems/
0 Response to "A Causal Continuous time Lti System is Described by the Following Differential Equation"
Postar um comentário