4 22 Determine the Continuous time Signal Corresponding to Each of the Following Transforms
Problem 1
Use the Fourier transform analysis equation (4.9) to calculate the Fourier transforms of:
(a) $e^{-2(t-1)} u(t-1)$
(b) $e^{-\mathbf{2}|t-1|}$
Sketch and label the magnitude of each Fourier transform.
Problem 2
Use the Fourier transform analysis equation (4.9) to calculate the Fourier transforms of:
(a) $\delta(t+1)+\delta(t-1)$
(b) $\frac{d}{d t}\{u(-2-t)+u(t-2)\}$
Sketch and label the magnitude of each Fourier transform.
Problem 3
Determine the Fourier transform of each of the following periodic signals:
(a) $\sin \left(2 \pi t+\frac{\pi}{4}\right)$
(b) $1+\cos \left(6 \pi t+\frac{\pi}{8}\right)$
Problem 4
Use the Fourier transform synthesis equation (4.8) to determine the inverse Fourier transforms of:
(a) $X_{1}(j \omega)=2 \pi \delta(\omega)+\pi \delta(\omega-4 \pi)+\pi \delta(\omega+4 \pi)$
(b) $\mathbf{X}_{2}(j \omega)=\left\{\begin{aligned} 2, & 0 \leq \omega \leq 2 \\-2, &-2 \leq \omega<0 \\ 0, &|\omega|>2 \end{aligned}\right.$
Problem 5
Use the Fourier transform synthesis equation (4.8) to determine the inverse Fourier transform of $X(j \omega)=|X(j \omega)| e^{j \not < X(j, w)}$, where $$| X(j \omega)|=2\{u(\omega+3)-u(\omega-3)\}$$ $$\not < X(j \omega)=-\frac{3}{2} \omega+\pi$$ Use your answer to determine the values of $t$ for which $x(t)=0$.
Problem 6
Given that $x(t)$ has the Fourier transform $X(j \omega$ ), express the Fourier transforms of the signals listed below in terms of $X(j \omega) .$ You may find useful the Fourier transform properties listed in Table 4.1.
(a) $x_{1}(t)=x(1-t)+x(-1-t)$
(b) $x_{2}(t)=x(3 t-6)$
(c) $x_{3}(t)=\frac{d^{2}}{d t^{2}} x(t-1)$
Problem 7
For each of the following Fourier transforms, use Fourier transform properties (Table 4.1) to determine whether the corresponding time-domain signal is (i) real, imaginary, or either and (ii) even, odd, or neither. Do this without evaluating the inverse of any of the given transforms.
(a) $X_{1}(j \omega)=u(\omega)-u(\omega-2)$
(b) $X_{2}(j \omega)=\cos (2 \omega) \sin \left(\frac{\omega}{2}\right)$
(c) $X_{3}(j \omega)=A(\omega) e^{j B(\omega)},$ where $A(\omega)=(\sin 2 \omega) / \omega$ and $B(\omega)=2 \omega+\frac{\pi}{3}$
(d) $X(j \omega)=\sum_{k=-\infty}^{\infty}\left(\frac{1}{2}\right)^{|k|} \delta\left(\omega-\frac{k \pi}{4}\right)$
Problem 8
Consider the signal $$x(t)=\left\{\begin{array}{ll}0, & t<-\frac{1}{2} \\t+\frac{1}{2}, & -\frac{1}{2} \leq t \leq \frac{1}{2} \\1, & t>\frac{1}{2}\end{array}\right.$$
(a) Use the differentiation and integration properties in Table 4.1 and the Fourier transform pair for the rectangular pulse in Table 4.2 to find a closed-form expression for $X(j \omega)$.
(b) What is the Fourier transform of $g(t)=x(t)-\frac{1}{2} ?$
Problem 9
Consider the signal $$x(t)=\left\{\begin{array}{ll}0, & |t|>1 \\(t+1) / 2, & -1 \leq t \leq 1\end{array}\right.$$
(a) With the help of Tables 4.1 and $4.2,$ determine the closed-form expression for $X(j \omega)$.
(b) Take the real part of your answer to part (a), and verify that it is the Fourier transform of the even part of $x(t)$
(c) What is the Fourier transform of the odd part of $x(t) ?$
Problem 10
(a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal:
$$x(t)=t\left(\frac{\sin t}{\pi t}\right)^{2}$$
(b) Use Parseval's relation and the result of the previous part to determine the numerical value of
$$A=\int_{-\infty}^{+\infty} t^{2}\left(\frac{\sin t}{\pi t}\right)^{4} d t$$
Problem 11
Given the relationships $$y(t)=x(t) * h(t)$$ and $$g(t)=x(3 t) * h(3 t)$$ and given that $x(t)$ has Fourier transform $X(j \omega)$ and $h(t)$ has Fourier transform $H(j \omega),$ use Fourier transform properties to show that $g(t)$ has the form $$g(t)=A y(B t)$$ Determine the values of $A$ and $B$.
Problem 12
Consider the Fourier transform pair $$e^{-|t|} \stackrel{\mathfrak{F}}{\longleftrightarrow} \frac{2}{1+\omega^{2}}$$
(a) Use the appropriate Fourier transform properties to find the Fourier transform of $t e^{-|t|}$.
(b) Use the result from part (a), along with the duality property, to determine the Fourier transform of
$$\frac{4 t}{\left(1+t^{2}\right)^{2}}$$
Hint: See Example 4.13
Problem 13
Let $x(t)$ be a signal whose Fourier transform is $$X(j \omega)=\delta(\omega)+\delta(\omega-\pi)+\delta(\omega-5)$$ and let $$h(t)=u(t)-u(t-2)$$.
(a) Is $x(t)$ periodic?
(b) Is $x(t) * h(t)$ periodic?
(c) Can the convolution of two aperiodic signals be periodic?
Problem 14
Consider a signal $x(t)$ with Fourier transform $X(j \omega) .$ Suppose we are given the following facts:
1. $x(t)$ is real and nonnegative
2. $\mathfrak{F}^{-1}\{(1+j \omega) X(j \omega)\}=A e^{-2 t} u(t) .$ where $A$ is independent of $t$.
3. $\int_{-\infty}^{\infty}|X(j \omega)|^{2} d \omega=2 \pi$.
Determine a closed-form expression for $x(t)$.
Problem 15
Let $x(t)$ be a signal with Fourier transform $X(j \omega) .$ Suppose we are given the following facts:
1. $x(t)$ ts real
2. $x(t)=0$ for $t \leq 0$
3. $\frac{1}{2 \pi} \int_{-\infty}^{\infty} \operatorname{Re}\{X(j \omega)\} e^{f \omega t} d \omega=|t| e^{-|t|}$
Determine a closed-form expression for $x(t)$.
Problem 16
Consider the signal $$x(t)=\sum_{k=-\infty}^{\infty} \frac{\sin \left(k \frac{\pi}{4}\right)}{\left(k \frac{\pi}{4}\right)} \delta\left(t-k \frac{\pi}{4}\right)$$
(a) Determine $g(t)$ such that $$x(t)=\left(\frac{\sin t}{\pi t}\right) g(t)$$
(b) Use the multiplication property of the Fourier transform to argue that $X(j \omega)$ is periodic. Specify $X( j \omega)$ over one period.
Problem 17
Determine whether each of the following statements is true or false. Justify your answers.
(a) An odd and imaginary signal always has an odd and imaginary Fourier transform.
(b) The convolution of an odd Fourier transform with an even Fourier transform is always odd.
Problem 18
Find the impulse response of a system with the frequency response
$$H(j \omega)=\frac{\left(\sin ^{2}(3 \omega)\right) \cos \omega}{\omega^{2}}$$
Problem 19
Consider a causal LTI system with frequency response
$$H(j \omega)=\frac{1}{j(t)+3}$$
For a particular input $x(t)$ this system 1 s observed to produce the output
$$y(t)=e^{-3 t} u(t)-e^{-4 t} u(t)$$
Determine $x(t)$.
Problem 20
Find the impulse response of the causal LTI system represented by the $R L C$ circuit considered in Problem $3.20 .$ Do this by taking the inverse Fourier transform of the circuit's frequency response. You may use Tables 4.1 and 4.2 to help evaluate the inverse Fourier transform.
Problem 21
Compute the Fourier transform of each of the following signals:
(a) $\left[e^{-a t} \cos \omega_{0} t\right] u(t), \alpha>0$
(b) $e^{-3 |t|} \sin 2 t$
(c) $x(t)=\left\{\begin{array}{ll}1+\cos \pi t, & |t| \leq 1 \\ 0, & |t|>1\end{array}\right.$
(d) $\sum_{k=0}^{x} \alpha^{k} \delta(t-k T)|\alpha|<1$
(e) $\left[ t e^{-2 t} \sin 4 t\right] u(t)$
(f) $\left|\frac{\sin \pi t}{\pi t}\right|\left[\frac{\sin 2 \pi(t-1)}{\pi(t-1)}\right]$
(g) $x(t)$ as shown in Figure $P 4.21(a)$
(h) $x(t)$ as shown in Figure $\mathbf{P 4 . 2 1 ( b )}$
(i) $x(t)=\left\{\begin{array}{ll}1-t^{2}, & 0<t<1 \\ 0, & \text { otherwise }\end{array}\right.$
(j) $\sum_{n=-\infty}^{+\infty} e^{-|t-2 n|}$
Problem 22
Determine the continuous-time signal corresponding to each of the following transforms.
(a) $X(j \omega)=\frac{2 \sin [3(\omega-2 \pi)]}{(\omega-2 \pi)}$
(b) $X(y \omega)=\cos (4 \omega+\pi / 3)$
(c) $X(j \omega)$ as given by the magnitude and phase plots of Figure $P 4.22(a)$
(d) $X(j(\nu)=2[\delta(\omega-1)-\delta(\omega+1)]+3[\delta(\omega-2 \pi)+\delta(\omega+2 \pi)]$
(e) $X(j \omega)$ as in Figure $P 4.22(b)$
Problem 23
Consider the signal $$x_{0}(t)=\left\{\begin{array}{ll}e^{t}, & 0 \leq t \leq 1 \\0, & \text { elsewhere }
\end{array}\right.$$
Determine the Fourier transform of each of the signals shown in Figure $\mathrm{P} 4.23 .$ Yow should be able to do this by explicitly evaluating only the transform of $x_{0}(t)$ and then using properties of the Fourier transform.
Problem 24
(a) Determine which, if any, of the real signals depicted in Figure $\mathrm{P} 4.24$ have Fourier transforms that satisfy each of the following conditions:
(1) $\operatorname{Re}\{X(j \omega)\}=0$
(2) $\operatorname{gn}\{X(j \omega)\}=0$
(3) There exists a real $\alpha$ such that $e^{j \alpha \omega} X(j \omega)$ is real
(4) $\int_{-x}^{\infty} X(j \omega) d \omega=0$
(5) $\operatorname{l}_{-x}^{x} \omega X(j \omega) d \omega=0$
(6) $X(j \omega)$ is periodic
(b) Construct a signal that has properties $(1),(4),$ and (5) and does not have the others.
Problem 25
Let $X(j \omega)$ denote the Fourier transform of the signal $x(t)$ depicted in Figure $\mathrm{P} 4.25$
(a) Find $\Varangle X(j \omega)$
(b) Find $X\left(j^{()}\right)$
(c) Find $\int_{-\infty}^{x} X(j \omega) d \omega$
(d) Evaluate $\int_{-x}^{\infty} X(j \omega) \frac{2 \sin \omega}{\omega} e^{J 2 \omega} d \omega$
(e) Evaluate $\int_{-\infty}^{\infty}|X(j \omega)|^{2} d \omega$
(f) Sketch the inverse Fourier transform of $\operatorname{Re}\{X(j \omega)\}$.
Note: You should perform all these calculations without explicitly evaluating $X ( j \omega$ ).
Problem 26
(a) Compute the convolution of each of the following pairs of signals $x(f)$ and $h(t)$ by calculating $X(j \omega)$ and $H(j \omega),$ using the convolution property, and inverse transforming.
(i) $x(t)=t e^{-2 t} u(t), h(t)=e^{-4 t} u(t)$
(ii) $x(t)=t e^{-2 t} u(t), h(t)=t e^{-4 t} u(t)$
(iii) $x(t)=e^{-t} u(t), h(t)=e^{t} u(-t)$
(b) Suppose that $x(t)=e^{-(t-2)} u(t-2)$ and $h(t)$ is as depicted in Figure $P 426$ Verify the convolution property for this pair of signals by showing that the Fourier transform of $y(t)=x(t)$; $h(t)$ equals $H(j \omega) X(j \omega)$.
Problem 27
Consider the signals $$x(t)=u(t-1)-2 u(t-2)+u (t-3)$$ and $$x(t)=\sum_{k=-\infty}^{\infty} x(t-k T)$$ where $T>0 .$ Let $a_{k}$ denote the Fourier series coefficients of $\tilde{x}(t),$ and let $X(j \omega)$ denote the Fourier transform of $x(t)$.
(a) Determine a closed-form expression for $X(j \omega)$
(b) Determine an expression for the Fourier coefficients $a_{k}$ and verify that $a_{k}=$$\frac{1}{T} X\left(j \frac{2 \pi k}{T}\right)$.
Problem 28
(a) Let $x(t)$ have the Fourier transform $X(j \omega)$, and let $p(t)$ be periodic with fundamental frequency $\omega_{0}$ and Fourier series representation $$p(t)=\sum_{n=-\infty}^{+\infty} a_{n} e^{j n \omega_{0}}$$ Determine an expression for the Fourier transform of $$y(t)=x(t) p(t)$$.
(b) Suppose that $X(j \omega)$ is as depicted in Figure $P 4.28(a) .$ Sketch the spectrum of $y(t)$ in eq. $(P 4.28-1)$ for each of the following choices of $p(t)$.
(i) $p(t)=\cos (t / 2)$
(ii) $p(t)=\cos t$
(iii) $p(t)=\cos 2 \mathrm{r}$
(iv) $p(t)=(\sin t)(\sin 2 t)$
(v) $p(t)=\cos 2 t-\cos t$
(vi) $p(t)=\sum_{n=-\infty}^{+\infty} \delta(t-\pi n)$
(vii) $p(t)=\sum_{n=-\infty}^{+\infty} \delta(t-2 \pi n)$
(viii) $p(t)=\sum_{n=-\infty}^{+\infty} \delta(t-4 \pi n)$
(ix) $p(t)=\sum_{n=-\infty}^{+\infty} \delta(t-2 \pi n)-\frac{1}{2} \sum_{n=-\infty}^{+\infty} \delta(t-\pi n)$
(x) $p(t)=$ the periodic square wave shown in Figure $P 4.28(b)$
Problem 29
A real-valued continuous-time function $x(t)$ has a Fourier transform $X(j \omega)$ whose magnitude and phase are as illustrated in Figure $\mathrm{P} 4.29(\mathrm{a})$.
Problem 30
Suppose $g(t)=x(t) \cos t$ and the Fourier transform of the $g(t)$ is
$$G(j \omega)=\left\{\begin{array}{ll}1, & |\omega| \leq 2 \\0, & \text { otherwise }\end{array}\right.$$
(a) Deternnine $x(t)$
(b) Specify the Fourier transform $X_{1}(j \omega)$ of a signal $x_{1}(t)$ such that $$g(t)=x_{1}(t) \cos \left(\frac{2}{3} t\right)$$
Problem 31
(a) Show that the three LTI systems with impulse responses $$\begin{aligned}&h_{1}(t)=u(t)\\&h_{2}(t)=-2 \delta(t)+5 e^{-2 t} u(t)\end{aligned}$$ and $$h_{3}(t)=2 t e^{-t} u(t)$$ all have the same response to $x(t)=\cos t$.
(b) Find the impulse response of another LTI system with the same response to $\cos t$. This problem illustrates the fact that the response to cos $t$ cannot be used to specify an LTI system uniquely.
Problem 32
Consider an LTI system $S$ with impulse response $$h(t)=\frac{\sin (4(t-1))}{\pi(t-1)}$$ Determine the output of $S$ for each of the following inputs:
(a) $x_{1}(t)=\cos \left(6 t+\frac{\pi}{2}\right)$
(b) $x_{2}(t)=\sum_{k=0}^{\infty}\left(\frac{1}{2}\right)^{k} \sin (3 k t)$
(c) $x_{3}(t)=\frac{\sin (4(t+1))}{\pi(t+1)}$
(d) $x_{4}(t)=\left(\frac{\sin 2 t}{\pi}\right)^{2}$
Problem 33
The input and the output of a causal LTI system are related by the differential equation $$\frac{d^{2} y(t)}{d t^{2}}+6 \frac{d y(t)}{d t}+8 y(t)=2 x(t)$$
(a) Find the impulse response of this system.
(b) What is the response of this system if $x(t)=t e^{-2 t} u(t) ?$
(c) Repeat part (a) for the causal LTI system described by the equation
$$\frac{d^{2} y(t)}{d t^{2}}+\sqrt{2} \frac{d y (t)}{d t}+y(t)=2 \frac{d^{2} x(t)}{d t^{2}}-2 x(t)$$
Problem 34
A causal and stable LTI system $S$ has the frequency response $$H(j \omega)=\frac{j \omega+4}{6-\omega^{2}+5 j \omega}$$.
(a) Determine a differential equation relating the input $x(f)$ and output $y(t)$ of $S$
(b) Determine the impulse response $h(t)$ of $S$
(c) What is the output of $S$ when the input is $$x(t)=e^{-4 t} u(t)-t e^{4 t} u(t) ?$$
Problem 35
In this problem, we provide examples of the effects of nonlinear changes in phase.
(a) Consider the continuous-time LTI system with frequency response $$H(j \omega)=\frac{a-j \omega}{a+j \omega^{\prime}}$$ where $a>0 .$ What is the magnitude of $H(j \omega) ?$ What is $\not < H(j \omega) ?$ What is the impulse response of this system?
(b) Determine the output of the system of part (a) with $a=1$ when the input is $$\cos (t / \sqrt{3})+\cos t+\cos \sqrt{3} t.$$ Roughly sketch both the input and the output.
Problem 36
Consider an LTI system whose response to the input $$x(t)=\left[e^{-t}+e^{-3 t}\right] u(t)$$ is $$y(t)=\left[2 e^{-t}-2 e^{-4 t}\right] u(t)$$.
(a) Find the frequency response of this system.
(b) Determine the system's impulse response.
(c) Find the differential equation relating the input and the output of this system.
Problem 37
Consider the signal $x(t)$ in Figure $\mathrm{P} 4.37$
(a) Find the Fourier transform $X(j \omega)$ of $x(t)$
(b) Sketch the signal $$\bar{x}(t)=x(t) * \sum_{k=-{\infty}}^{\infty} \delta(t-4 k)$$.
(c) Find another signal $g(t)$ such that $g(t)$ is not the same as $x(t)$ and $$\bar{x}(t)=g(t) * \sum_{k=-\infty}^{\infty} \delta(t-4 k)$$.
(d) Argue that, although $G(j \omega)$ is different from $X(j w), G\left(j \frac{m k}{2}\right)=X\left(j \frac{\pi k}{2}\right)$ for all integers $k .$ You should not explicitly evaluate $G(j \omega)$ to answer this question.
Problem 38
Let $x(t)$ be any signal with Fourier transform $X(j \omega)$. The frequency-shift property of the Fourier transform may be stated as
(a) Prove the frequency-shift property by applying the frequency shift to the analysis equation $$X(j \omega)=\int_{-\infty}^{\infty} x(t) e^{-j \omega t} d t$$
(b) Prove the frequency-shift property by utilizing the Fourier transform of $e^{j \omega_{0} f}$ in conjunction with the multiplication property of the Fourier transform.
Problem 39
Suppose that a signal $x(t)$ has Fourier transform $X(j \omega) .$ Now consider another signal $g(t)$ whose shape is the same as the shape of $X(j \omega) ;$ that is, $$g(t)=X(j t)$$
(a) Show that the Fourier transform $G(j \omega \text { ) of } g(t) \text { has the same shape as } 2 \pi x(-t)$ that is, show that $$G(j \omega)=2 \pi x(-\omega)$$
(b) Using the fact that $$\mathfrak{F}\{\delta(t+B)\}=e^{j \boldsymbol{B} \omega}$$ in conjunction with the result from part (a), show that $$\mathfrak{F}\left\{e^{j B r}\right\}=2 \pi \delta(\omega-B)$$
Problem 40
Use properties of the Fourier transform to show by induction that the Fourier transform of $$x(t)=\frac{t^{n-1}}{(n-1) !} e^{-u t} u(t), a>0$$ is $$\frac{1}{(a+j \omega)^{n}}$$
Problem 41
In this problem, we derive the multiplication property of the continuous-time Fourier transform. Let $x(t)$ and $y(t)$ be two continuous-tine signals with Fourier transforms $X(j \omega)$ and $Y(j \omega),$ respectively. Also, let $g(t)$ denote the inverse Fourier transform of $\frac{1}{2 \pi}\{X(j \omega) * Y(j \omega)\}$
(a) Show that $$g(t)=\frac{1}{2 \pi} \int_{-x}^{+x} X(j \theta)\left[\frac{1}{2 \pi} \int_{-x}^{+\infty} Y(j(\omega-\theta)) e^{j \omega t} d \omega\right] d \theta$$
(b) Show that $$\frac{1}{2 \pi} \int_{\infty}^{-\infty} Y(j(\omega-\theta)) e^{j \omega t} d \omega=e^{j \theta f} y(t)$$
(c) Combine the results of parts (a) and (b) to conclude that $$g(t)=x(t) y(t)$$
Problem 42
Let $$g_{1}(t)=\left\{\left[\cos \left(\omega_{0} t\right)\right] x(t)\right\} * h(t) \quad \text { and } \quad g_{2}(t)=\left\{\left[\sin \left(\omega_{0} t^{\prime}\right)\right] x(t)\right\} * h(t)$$ where $$x(t)=\sum_{k=\infty}^{x} a_{k} e^{j k 100 t}$$ is a real-valued periodic signal and $h(t)$ is the impulse response of a stable LTI system.
(a) Specify a value for $\omega_{0}$ and any necessary constraints on $H(j \omega)$ to ensure that $$g_{1}(t)=\mathcal{G}_{\ell}\left\{a_{5}\right\} \quad \text { and } \quad g_{2}(t)=\mathcal{G}_{m}\left\{a_{5}\right\}$$
(b) Give an example of $h(\tau)$ such that $H(j \omega)$ satisfies the constraints you specified in part (a).
Problem 43
Let $$g(t)=x(t) \cos ^{2} t * \frac{\sin t}{\pi t}$$ Assuming that $x(t)$ is real and $X(j \omega\}=0$ for $|\omega| \geq 1$, show that there exists an LTI system $S$ such that $$x(t) \stackrel{s}{\longrightarrow} g(t)$$
Problem 44
The output $y(t)$ of a causal LTI system is related to the input $x(t)$ by the equation $$\frac{d y(t)}{d t}+10 y(t)=\int_{-\infty}^{+x} x(\tau) z(t-\tau) d \tau-x(t)$$ where $z(t)=e^{-t} u(t)+3 \delta(t)$
(a) Find the frequency response $H(j \omega)=Y(j \omega) Y X(j \omega)$ of this system.
(b) Determine the impulse response of the system.
Problem 45
In the discussion in Section 4.3 .7 of Parseval's relation for continuous-time signals, we saw that $$\left.\int_{\infty}^{+\infty} x(t)\right|^{2} d t=\frac{1}{2 \pi} \int_{-x}^{+\infty}|X(j \omega)|^{2} d \omega$$
This says that the total energy of the signal can be obtained by integrating $|X(j \omega)|^{2}$ over all frequencies. Now consider a real-valued signal $x(t)$ processed by the ideal bandpass filter $H(j \omega)$ shown in Figure $P 4.45 .$ Express the energy in the output signal $y(t)$ as an integration over frequency of $|X(j \omega)|^{2} .$ For $\Delta$ sufficiently small so that $|X(j \omega)|$ is approximately constant over a frequency interval of width $\Delta,$ show that the energy in the output $y(f)$ of the bandpass filter is approximately proportional to $\Delta\left|X\left(j \omega_{0}\right)\right|^{2}$.
On the basis of the foregoing result. $\Delta .\left.X\left(j \omega_{0}\right)\right|^{2}$ is proportional to the energy in the signal in a bandwidth $\Delta$ around the frequency $\omega_{k} .$ For this reason, $|X(f \omega)|^{2}$ is often referred to as the energy-density spectrum of the signal $x(t)$.
Problem 46
In Section $4.5 .1,$ we discussed the use of amplitude modulation with a complex exponential carrier to implement a bandpass filter. The specific system was shown in Fignre $4.26,$ and if only the real part of $f(t)$ is retained, the equivalent bandpass filter is that shown in Figure 4.30.
In Figure $P 4.46,$ we indicate an implementation of a bandpass filter using sinusoidal modulation and lowpass filters. Show that the output $y(t)$ of the system is identical to that which would be obtained by retaining only $\operatorname{Re}\{f(x)\}$ in Figure 4.26.
Problem 47
An important property of the frequency response $H(j \omega)$ of a continuous-time LTT system with a real, causal impulse response $h(f)$ is that $H(j \omega)$ is completely specified by its real part, $\mathfrak{F} e\{H(j \omega)\}$. The current problem is concerned with deriving and examining some of the implications of this property, which is generally referred to as real-part sufficiency.
(a) Prove the property of real-part sufficiency by examining the signal $h_{e}(t),$ which is the even part of $h(t) .$ What is the Fourier transform of $h_{c}\{t$ )? Indicate bow $h(t)$ can be recovered from $h_{e}(t)$.
(b) If the real part of the frequency response of a causal system is $$\mathfrak{Fe}\{H(j \omega)\}=\cos \omega$$ what is $h(t) ?$
(c) Show that $h(t)$ can be recovered from $h_{0}(t),$ the odd part of $h(t),$ for every value of $t$ except $t=0 .$ Note that if $h(t)$ does not contain any singularities $\left[\delta(t), u_{1}(t), u_{2}(t), \text { etc. }\right]$ at $t=0,$ then the frequency response $$H\{j \omega)=\int_{-\infty}^{+\infty} h(t) e^{-j \omega t} d t$$ will not change it $f(t)$ is set to some arbitrary finite value at the single point $\mathrm{t}=0 .$ Thus, in this case, show that $H(j \omega)$ is also completely specified by its imaginary part.
Problem 48
Let us consider a system with a real and causal impulse response $h(t)$ that does not have any singularities at $t=0 .$ En Problem $4.47,$ we saw that either the real or the imaginary part of $H(j \omega)$ completely determines $H(j \omega) .$ In this problem we derive an explicit relationship between $H_{R}(j \omega)$ and $H_{r}(j \omega),$ the real and imaginary parts $o[H(j \omega)$.
(a) To begin, note that since $h(t)$ is causal, $$h(t)=h(t) u(t)$$ except perhaps at $t=0 .$ Now, since $h(t)$ contains no singularities at $t=0,$ the Fourier transforms of both sides of eq. $(P 4.48-1)$ must be identical. Use this fact, together with the multiplication property, to show that
Use eq. $(\mathrm{P} 4.48-2)$ to determine an expression for $H_{R}(j \omega)$ in terms of $H,(j \omega)$ and one for $H,(j \omega)$ in terms of $H_{p}(j \omega)$
(b) The operation $$y(t)=\frac{1}{\pi} \int_{-x}^{+x} \frac{x(t)}{t - \tau} d \tau$$
is called the Hilbert transform. We have just seen that the real and imaginary parts of the transform of a real, causal impulse response $h(t)$ can be determined from one another using the Hilbert transform.
Now consider eq. $(\mathrm{P} 4.48-3),$ and regard $y(t)$ as the output of an LTI system with input $x(t) .$ Show that the frequency response of this system is $$H(j \omega)=\left\{\begin{array}{ll}-j, & \omega>0 \\
j, & \omega<0\end{array}\right.$$
(c) What is the Hilbert transform of the signal $x(t)=\cos 3 t^{\prime}$
Problem 49
Let $H(j \omega)$ be the frequency response of a continuous-time LTI system, and suppose that $H(j \omega)$ is real, even, and positive. Also, assume that $$\max _{\omega}\{H(j \omega)\}=H(0)$$
(a) Show that:
(i) The impulse response, $h(t),$ is real.
(ii) $\max \{|h(t)|\}=h(0)$.
Hint: If $f(t, \omega)$ is a complex function of two variables, then
$$\left|\int_{-\infty}^{+\infty} f(t, \omega) d \omega \leq \int_{-\infty}^{+\infty}\right| f(t, \omega) | d \omega$$
(b) One important concept in system analysis is the bandwidth of an LTI system. There are many different mathematical ways in which to define bandwidth, but they are related to the qualitative and intuitive idea that a system with frequency response $G(j \omega)$ essentially "stops" signals of the form $e^{j \omega t}$ for values of $\omega$ where $G(j \omega)$ varnishes or is small and "passes" those complex exponentials in the band of frequency where $G(j \omega)$ is not small. The width of this band is the bandwidth. These ideas will be made much clearer in Chapter $6,$ but for now we will consider a special definition of bandwidth for those systems with frequency responses that have the properties specified previously for $H(j \omega) .$ Specifically, one definition of the bandwidth $B_{w}$ of such a system is the width of the rectangle of height $H(j 0)$ that has an area equal to the area under $H(j \omega) .$ This is illustrated in Figure $P 4.49(a) .$ Note that since $H(j 0)=\max _{\omega} H(j \omega),$ the frequencies within the band indicated in the figure are those for which $H(j \omega\}$ is largest. The exact choice of the width in the figure is, of course, a bit arbitrary, but we have chosen one definition that allows us to compare different systems and to make precise a very important relationship between time and frequency. What is the bandwidth of the system with frequency response $$H(j \omega)=\left\{\begin{array}{ll}\mathbf{1}, & |\omega|<W \\0, & |\omega|>W\end{array}\right.$$
(c) Find an expression for the bandwidth $B_{w}$ in terms of $H(j \omega)$
(d) Let $s(t)$ denote the step response of the system set out in part (a). An important measure of the speed of response of a system is the rise time, which, like the bandwidth, has a qualitative definition, leading to many possible mathematical definitions, one of which we will use. Intuitively, the rise time of a system is a measure of how fast the step response rises from zero to its final value, $$s(\alpha)=\lim _{t \rightarrow \infty} s(t)$$
Thus, the smaller the rise time, the faster is the response of the system. For the system under consideration in this problem, we will define the rise time as $$t_{r}=\frac{s\left(\infty\right)}{h(0)}$$ since $$s^{\prime}(t)=h(t)$$ and also because of the property that $h(0)=\max _{t} h(t), t_{r}$ is the time it would take to go from zero to $s(\infty)$ while maintaining the maximum rate of change of $s(t) .$ This is illustrated in Figure $\mathbf{P 4} . \mathbf{4 9}(\mathbf{b})$ Find an expression for $t_{r}$ in terms of $H(j \omega)$.
(e) Combine the results of parts (c) and (d) to show that $$B_{w} t_{r}=2 \pi$$ Thus, we cannot independently specify both tha rise time and the bandwidth of our system. For example, eq. (P4.49-1) implies that, if we want a fast system ( $t_{r}$ small), the system must have a large bandwidth. This is a fundamental trade-off that is of central importance in many problems of system design.
Problem 50
In Problems 1.45 and $2.67,$ we defined and examined several of the properties and uses of correlation functions. In the current problem, we examine the properties of such functions in the frequency domain. Let $x(t)$ and $y(t)$ be two real signals. Then the cross-correlation function of $x(t)$ and $y(t)$ is defined as $$\phi_{x y}(t)=\int_{-\infty}^{+\infty} x(t+\tau) y(\tau) d \tau$$
Similarly, we can define $\phi_{y x}(t), \phi_{z x}(t),$ and $\phi_{y y}(r) .$ [The last two of these are called the autocorrelation functions of the signals $x(t) \text { and } y(t) \text { , respectively }]$ Let $\Phi_{x v}(j \omega)$ $\Phi_{y x}(j \omega), \Phi_{x x}\{j \omega),$ and $\Phi_{y y}(j \omega)$ denote the Fourier transforms of $\phi_{x y}(t), \phi_{v x}(t)$ $\phi_{x x}(t),$ and $\phi_{y y}(t),$ respectively.
(a) What is the relationship between $\Phi_{x y}(j \omega)$ and $\Phi_{y x}(j \omega) ?$
(b) Find an expression for $\Phi_{x y}(j \omega)$ in terms of $X(j \omega)$ and $Y(j \omega)$
(c) Show that $\Phi_{x x}(j \omega)$ is real and nonnegative for every $\omega$
(d) Suppose now that $x(t)$ is the input to an LTI system with a real-valued impulse response and with frequency response $H(j \omega)$ and that $y(t)$ is the output. Find expressions for $\Phi_{\mathrm{ry}}(j \omega)$ and $\Phi_{y y}(j \omega)$ in terms of $\Phi_{x x}(j \omega)$ and $H(j \omega)$.
(e) Let $x(t)$ be as is illustrated in Figure $P 4.50,$ and let the LTS system impulse using the results of parts $(a)-(d)$.
(f) Suppose that we are given the following Fourier transform of a function $\phi(t)$:
$$\Phi(j \omega)=\frac{\omega^{2}+100}{\omega^{2}+25}$$
Find the impulse responses of $t w o$ causal, stable LTI systems that have autocorrelation functions equal to $\phi(t) .$ Which one of these has a causal, stable inverse?
Problem 51
(a) Consider two 1.7 s systems with impulse responses $h(t)$ and $g(t),$ respectively. and suppose that these systems are inverses of one another. Suppose also that the systems have frequency responses denoted by $H(j \omega)$ and $G(j \omega),$ respectively. What is the relationship between $H(j \omega)$ and $G(j \omega) ?$
(b) Consider the continuous-time $L$ TI system with frequency response $$H(j \omega)=\left\{\begin{array}{ll}
1, & 2<|\omega|<3 \\0, & \text { otherwise }\end{array}\right.$$
(i) $\mathrm{fs}$ it possible to find an input $x(t)$ to this system such that the output is as depicted in Figure $\mathrm{P} 4.50 ?$ If $\mathrm{s} 0,$ find $x(t) .$ If not, explain why not.
(ii) Is this system invertible? Explain your answer.
(c) Consider an auditorium with an echo problem. As discussed m Problem 2.64 we can model the acoustics of the auditorium as an LTI system with an impulse response consisting of an impulse train, with the $k$ th impulse in the train corresponding to the $k$ th echo. Suppose that in this particular case the impulse response is $$h(r)=\sum_{k=0}^{\infty} e^{-k T} \delta(t-k T)$$ where the factor $e^{-k t}$ represents the attenuation of the $k$ th echo.
In order to make a high-quality recording from the stage, the effect of the echoes must be removed by performing some processing of the sounds sensed by the recording equipment. In Problem 2.64 . we used constitutional techniques to consider one example of the design of such a processors (for a different acoustic model, In the current problem, we will use frequency-domain techniques. Specifically, let $G(j \omega)$ denote the frequency response of the $L T 1$ system to be used to process the sensed acoustic signal. Choose $G(j \omega)$ so that the echoes are completely removed and the resulting signal is a faithful reproduction of the original stage sounds.
(d) Find the differential equation for the inverse of the system with impulse response $$h(t)=2 \delta(t)+u_{1}(t)$$.
(e) Consider the LTI system initially at rest and described by the differential equation $$\frac{d^{2} y(t)}{d t^{2}}+6 \frac{d y(t)}{d t}+9 y(t)=\frac{d^{2} x(t)}{d t^{2}}+3 \frac{d x(t)}{d t}+2 x(t)$$ The inverse of this system is also initially at rest and described by a differential equation. Find the differential equation describing the inverse, and find the impulse responses $h(t)$ and $g(t)$ of the original system and its inverse.
Problem 52
Inverse systems frequently find application in problems involving imperfect measuring devices. For example, consider a device for measuring the temperature of a liquid. It is often reasonable to model such a device as an LTI system that, because of the response characteristics of the measuring element (e.g., the mercury in a thermometer $),$ does not respond instantaneonsly to temperature changes. In particular, assurne that the response of this device to a unit step in temperature is $$s(t)=\left(1-e^{-t / 2}\right) u(t)$$.
(a) Design a compensatory system that, when provided with the output of the measuring device, produces an output equal to the instantarieous temperature of the liquid.
(b) One of the problems that often arises in using inverse systems as cornpensators for measuring devices is that gross inaccuracies in the indicated temperature may occur if the actual output of the measuring device produces errors due to small, erratic phenomena in the device. since there always are such sources of error in real systems, one must take them into account. To illustrate this, consider a measuring device whose overall output can be modeled as the sum of the response of the measuring device characterized by eq. (P4.52-1) and an interfering "noise" signal $n(t) .$ Such a model is depicted in Figure $\mathbf{P 4 . 5 2}(\mathrm{a})$ where we have also included the inverse system of part (a), which now has as its input the overall output of the measuring device. Suppose that $n(t)=\sin \omega t$. What is the contribution of $n(t)$ to the output of the inverse system, and how does this output change as $\omega$ is increased?
(c) The issue raised in part (b) is an important one in many applications of LTI system analysis. Specifically, we are confronted with the fundamental trade-off between the speed of response of the system and the ability of the system to attenuate high-frequency interference. In part (b) we saw that this trade-off implied that, by attempting to speed up the response of a measuring device (by means of an inverse system), we produced a system that would also amplify corrupting sinusoidal signals. To illustrate this concept furthes, consider a measuring device that responds instantaneously to changes in temperature, but that also is corrupted by noise. The response of such a system can be modeled, as depicted in Figure $\mathrm{P} 4.52(\mathrm{b}),$ as the sum of the response of a perfect measuring device and a corrupting signal $n(t) .$ Suppose that we wish to design a compensatory system that will slow down the response to actual temperature variations. but also will attenuate the noise $n(t)$. Let the impulse response of this system be $$h(t)=a e^{u t} u(t)$$ Choose $a$ so that the overall system of Figure $\mathbf{P} 4.52(\mathrm{b})$ responds as quickly as possible to a step change in temperature, subject to the constraint that the amplitude of the portion of the output due to the noise $n(t)=\sin 6 t$ is no larger than $1 / 4$.
Problem 53
As mentioned in the text, the techniques of Fourier analysis can be extended to signals having two independent variables. As their one-dimensional counterparts do in some applications, these techniques play an important role in other applications, such as image processing. In this problem, we introduce some of the elementary ideas of two-dimensional Fourier analysis.
Let $x\left(t_{1}, t_{2}\right)$ be a signal that depends upon two independent variables $'_{1}$ and
$t_{2} .$ The two -dimensional Fourier transform of $x\left(t_{1}, f_{2}\right)$ is defined as
$$X\left(j \omega_{1}, j \omega_{2}\right)=\int_{-\infty}^{+\infty} \int_{-\infty}^{-\infty} x\left(t_{1}, t_{2}\right) e^{-j\left(\omega_{1}^{\prime} 1_{1}+\omega_{2} t_{2}\right)} d t_{1} d t_{2}$$
(a) Show that this double integral can be performed as two successive one-dimensional Fourier transforms, first in $t_{1}$ with $t_{2}$ regarded as fixed and then $\operatorname{in} t_{2}$.
(b) Use the result of part (a) to determine the inverse transform-that is, an expression for $x\left(t_{1}, t_{2}\right)$ in terms of $X\left(j \omega_{1}, j \omega_{2}\right)$.
(c) Determine the two-dimensional Fourier transforms of the following signals:
(i) $x\left(t_{1}, t_{2}\right)=e^{-t_{1}+2 t_{2}} u\left(t_{1}-1\right) u\left(2-t_{2}\right)$
(ii) $x\left(t_{1}, t_{2}\right)=\left\{\begin{array}{ll}e^{-t_{1}^{1}-\left|t_{2}\right|}, & \text { if }-1<t_{1} \leq 1 \text { and }-1 \leq t_{2} \leq 1 \\ 0, & \text { otherwise }\end{array}\right.$
(iii) $x\left(t_{1}, t_{2}\right)=\left\{\begin{array}{ll}e^{-, t_{1}-\left|x_{2}\right|}, & \text { if } 0 \leq t_{1} \leq 1 \text { or } 0 \leq t_{2} \leq 1 \text { (or both) } \\ 0, & \text { otherwise }\end{array}\right.$
(iv) $x\left(t_{1}, t_{2}\right)$ as depicted in Figure P4.53.
(v) $e^{-| t |t_{2}-| t_{1} -t|}$
(d) Determine the signal $x\left(t_{1}, t_{2}\right)$ whose two-dimensional Fourier transform is $$X\left(j \omega_{1}, j \omega_{2}\right)=\frac{2 \pi}{4+j \omega_{1}} \delta\left(\omega_{2}-2 \omega_{1}\right)$$
(e) Let $x\left(t_{1}, t_{2}\right)$ and $h\left(t_{1}, t_{2}\right)$ be two signals with two-dimensional Founer transforms $X\left(j \omega_{1}, j \omega_{2}\right)$ and $H\left(j \omega_{1}, j \omega_{2}\right),$ respectively. Determine the transforms of the following signals in terms of $X\left(j \omega_{1}, j \omega_{2}\right)$ and $H\left(j \omega_{1}, j \omega_{2}\right)$
(i) $x\left(t_{1}-T_{1}, t_{2}-T_{2}\right)$
(ii) $x\left(a t_{1}, b t_{2}\right)$
(iii) $y\left(t_{1}, t_{2}\right)=\int_{-\infty}^{+\infty}\int_{-\infty}^{-\infty} x\left(\eta_{7}, \tau_{2}\right) h\left(t_{1}-\tau_{7}, t_{2}-\tau_{2}\right) d \tau_{1} d \tau_{2}$
Source: https://www.numerade.com/books/chapter/the-continuos-time-fourier-transform/
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