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Electric Circuits

James W. Nilsson, Susan A. Riedel

Chapter 17

The Fourier Transform - all with Video Answers

Educators

Chapter Questions

01:19

Problem 1

Use the defining integral to find the Fourier transform of the following functions:
$$\text { a) } f(t)=A \sin \frac{\pi}{2} t, \quad-2 \leq t < 2$$
$$f(t)=0, \quad \text { elsewhere }$$
$$\text { b) } f(t)=\frac{2 A}{\tau} t+A, \quad-\frac{\tau}{2} \leq t \leq 0$$
$$f(t)=-\frac{2 A}{\tau} t+A, \quad 0 \leq t \leq \frac{\tau}{2}$$
$$f(t)=0, \quad \text { elsewhere }$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:26

Problem 2

a) Find the Fourier transform of the function shown in Fig. $P 17.2$.
b) Find $F(\omega)$ when $\omega=0$.
c) Sketch $|F(\omega)|$ versus $\omega$ when $A=2$ and $\tau=1$ Hint: Evaluate $|F(\omega)|$ at $\omega=\pm 1,\pm 2,\pm 3, \ldots$
$\pm 15 .$ Then use the fact that $|F(\omega)|$ is an even function of $\omega$.

Amit Srivastava
Amit Srivastava
Numerade Educator
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Problem 3

The Fourier transform of $f(t)$ is shown in Fig. P17.3.
a) Find $f(t)$.
b) Evaluate $f(0)$.
c) Sketch $f(t)$ for $-10 \leq t \leq 10$ s when $A=2 \pi$ and $\omega_{0}=2 \mathrm{rad} / \mathrm{s}$. Hint: Evaluate $f(t)$ at $t=0$ $1,2,3, \ldots, 10 \mathrm{s}$ and then use the fact that $f(t)$ is even.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:09

Problem 4

Find the Fourier transform of each of the following functions. In all of the functions, $a$ is a positive real constant and $-\infty \leq t \leq \infty$
a) $f(t)=|t| e^{-a||||}$;
b) $f(t)=t^{3} e^{-a|t|}$;
c) $f(t)=e^{-a|t|} \cos \alpha_{0} t$;
d) $f(t)=e^{-a|i|} \sin \omega_{0} t$;
e) $f(t)=\delta\left(t-t_{0}\right)$.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:03

Problem 5

Derive $\mathscr{F}\left\{\sin \omega_{0} t\right\}$

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 6

If $f(t)$ is a real function of $t,$ show that the inversion integral reduces to
$$f(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty}[A(\omega) \cos \omega t-B(\omega) \sin \omega t] d \omega.$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:33

Problem 7

If $f(t)$ is a real, odd function of $t,$ show that the inversion integral reduces to
$$f(t)=-\frac{1}{2 \pi} \int_{-\infty}^{\infty} B(\omega) \sin \omega t d \omega.$$

Tyler Gaona
Tyler Gaona
Numerade Educator
05:34

Problem 8

Use the inversion integral (Eq. 17.9 ) to show that $\mathscr{F}^{-1}\{2 / j \omega\}=\operatorname{sgn}(t) .$ Hint: Use Problem 17.7.

John Connell
John Connell
Numerade Educator
04:04

Problem 9

Find $\mathscr{F}\left\{\cos \omega_{0} t\right\}$ by using the approximating function
\[f(t)=e^{-e|t|} \cos \omega_{0} t.\]
where $\epsilon$ is a positive real constant.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
04:33

Problem 10

Show that if $f(t)$ is an even function.
\[\begin{array}{l}
A(\omega)=2 \int_{0}^{\infty} f(t) \cos \omega t d t, \\
B(\omega)=0.
\end{array}\]
.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:06

Problem 11

Show that if $f(t)$ is an odd function,
\[
\begin{array}{l}
A(\omega)=0 ,\\
B(\omega)=-2 \int_{0}^{\infty} f(t) \sin \omega t d t.
\end{array}
\]

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
02:34

Problem 12

a) Show that $\mathscr{F}\{d f(t) / d t\}=j \omega F(\omega), \quad$ where
$F(\omega)=\mathscr{F}\{f(t)\} .$ Hint: Use the defining integral and integrate by parts.
b) What is the restriction on $f(t)$ if the result given in (a) is valid?
c) Show that $\mathscr{F}\left\{d^{n} f(t) / d t^{n}\right\}=(j \omega)^{n} F(\omega),$ where
$F(\omega)=\mathscr{F}\{f(t)\}$.

R M
R M
Numerade Educator
04:06

Problem 13

a) Show that
\[\mathscr{F}\left\{\int_{-\infty}^{t} f(x) d x\right\}=\frac{F(\omega)}{j \omega}.\]
where $F(\omega)=g\{f(x)\} .$ Hint: Use the defining integral and integrate by parts.
b) What is the restriction on $f(x)$ if the result given in (a) is valid?
c) If $f(x)=e^{-\operatorname{ar} u}(x),$ can the operational transform in (a) be used? Explain.

Sirat Shah
Sirat Shah
Numerade Educator
07:41

Problem 14

a) Show that
\[g\{f(a t)\}=\frac{1}{a} F\left(\frac{\omega}{a}\right), \quad a > 0.\]
b) Given that $f(a t)=e^{-a|t|}$ for $a 0,$ sketch $F(\omega)=\mathscr{F}\{f(a t)\}$ for $a=0.5,1.0,$ and $2.0 . D 0$ your sketches reflect the observation that compression in the time domain corresponds to stretching in the frequency domain?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:26

Problem 15

Derive each of the following operational transforms:
a) $\mathscr{P}\{f(t-a)\}=e^{-j \omega a} F(\omega)$;
b) $\mathscr{F}\left\{e^{j \omega d} f(t)\right\}=F\left(\omega-\omega_{0}\right)$;
c) $\mathscr{F}\left\{f(t) \cos \omega_{0} f\right\}=\frac{1}{2} F\left(\omega-\omega_{0}\right)+\frac{1}{2} F\left(\omega+\omega_{0}\right)$.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:55

Problem 16

Given
$$y(t)=\int_{-\infty}^{\infty} x(\lambda) h(t-\lambda) d \lambda.$$
show that $Y(\omega)=\mathscr{F}\{y(t)\}=X(\omega) H(\omega),$ where
$X(\omega)=\mathscr{F}\{x(t)\}$ and $H(\omega)=\mathscr{F}\{h(t)\} .$ Hint: Use
the defining integral to write
$$\mathscr{F}\{y(t)\}=\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty} x(\lambda) h(t-\lambda) d \lambda\right] e^{-j \omega t} d t.$$
Next, reverse the order of integration and then make a change in the variable of integration; that is, let $u=t-\lambda$.

Arpit Gupta
Arpit Gupta
Numerade Educator
01:14

Problem 17

Given $f(t)=f_{1}(t) f_{2}(t),$ show that $F(\omega)=$ $(1 / 2 \pi) \int_{-\infty}^{\infty} F_{1}(u) F_{2}(\omega-u) d u .$ Hint: First, use the defining integral to express $F(\omega)$ as
$$F(\omega)=\int_{-\infty}^{\infty} f_{1}(t) f_{2}(t) e^{-j \omega t} d t.$$
Second, use the inversion integral to write
\[f_{1}(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F_{1}(u) e^{j \omega t} d u.\]
Third, substitute the expression for $f_{1}(t)$ into the defining integral and then interchange the order of integration.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:09

Problem 18

a) Show that
$$(j)^{n}\left[\frac{d^{n} F(\omega)}{d \omega^{n}}\right]=\mathscr{F}\left\{t^{n} f(t)\right\}.$$
b) Use the result of (a) to find each of the following Fourier transforms:
$$\begin{aligned}
&\mathscr{F}\left\{t e^{-a t} u(t)\right\},\\
&\begin{array}{l}
\mathscr{F}\left\{|t| e^{-a|r|}\right\}, \\
\mathscr{F}\left\{t e^{-a|t|}\right\}.
\end{array}
\end{aligned}$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:10

Problem 19

Suppose that $f(t)=f_{1}(t) f_{2}(t),$ where
$$\begin{aligned}
&f_{1}(t)=\cos \omega_{0} t,\\
&f_{2}(t)=1, \quad-\tau / 2 < t < \tau / 2;\\
&f_{2}(t)=0, \quad \text { elsewhere }.
\end{aligned}$$
a) Use convolution in the frequency domain to $\operatorname{tin} d(\omega)$.
b) What happens to $F(\omega)$ as the width of $f_{2}(t)$ increases so that $f(t)$ includes more and more cycles on $f_{1}(t) ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:56

Problem 20

a) Use the Fourier transform method to find $i_{o}(t)$ in the circuit shown in Fig. $\mathrm{P} 17.20$. The initial value of $i_{o}(t)$ is zero, and the source voltage is $125 u(t) V$.
b) Sketch $i_{o}(t)$ versus $t$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:18

Problem 21

Repeat Problem 17.20 if the input voltage $\left(v_{g}\right)$ is changed to $125 \operatorname{sgn}(t)$.

Chai Santi
Chai Santi
Numerade Educator
04:29

Problem 22

a) Use the Fourier transform method to find $v_{o}(t)$ in the circuit shown in Fig. $P 17.22$ if $v_{g}=20 \operatorname{sgn}(t) \mathrm{V}$.
b) Does your solution make sense in terms of known circuit behavior? Explain.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:09

Problem 23

Repeat Problem 17.22 except replace $v_{o}(t)$ with $i_{o}(t)$.

Arpit Gupta
Arpit Gupta
Numerade Educator
01:44

Problem 24

The voltage source in the circuit in Fig. $\mathrm{P} 17.24$ is given by the expression
$$v_{g}=15 \operatorname{sgn}(t) \mathrm{V}.$$
a) Find $v_{o}(t)$.
b) What is the value of $v_{o}\left(0^{-}\right) ?$
c) What is the value of $v_{o}\left(0^{+}\right) ?$
d) Use the Laplace transform method to find $v_{o}(t)$ for $t>0^{+}$.
e) Does the solution obtained in (d) agree with $v_{o}(t)$ for $t>0^{+}$ from $(\mathrm{a}) ?$

Manik Pulyani
Manik Pulyani
Numerade Educator
04:13

Problem 25

Repeat Problem 17.24 except replace $v_{o}(t)$ with $i_{o}(t)$.

Nicholas Sacco
Nicholas Sacco
Numerade Educator
04:29

Problem 26

a) Use the Fourier transform to find $v_{o}$ in the cir cuit in Fig. P17.26 if $i_{g}=2 \operatorname{sgn}(t)$ A.
b) Does your solution make sense in terms of known circuit behavior? Explain.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:09

Problem 27

Repeat Problem 17.26 except replace $i_{o}$ with $v_{o}$.

Arpit Gupta
Arpit Gupta
Numerade Educator
01:15

Problem 28

a) Use the Fourier transform to find $v_{o}$ in the circuit in Fig. $\mathrm{P} 17.28$ if $v_{g}$ equals $30 e^{-5|t|} \mathrm{A}$.
b) Find $v_{o}\left(0^{-}\right)$.
c) Find $v_{o}\left(0^{+}\right)$.
d) Use the Laplace transform method to find $v_{o}$ for $t \geq 0$.
e) Does the solution obtained in (d) agree with $v_{o}$ for $t>0^{+}$ from $(\mathrm{a}) ?$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:30

Problem 29

a) Use the Fourier transform to find $i_{a}$ in the circuit in Fig. P17.28 if $v_{g}$ equals $30 e^{-||5 |}$ A.
b) Find $i_{o}(0)$.
c) Find $i_{o}\left(0^{+}\right)$.
d) Use the Laplace transform method to find $i_{0}$ for $t \geq 0$.
e) Does the solution obtained in (d) agree with $l_{0}$ for $t>0^{+}$ from $(a) ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:29

Problem 30

a) Use the Fourier transform method to find $v_{o}$ in the circuit in Fig. $P 17.30$ if $v_{g}=125 \cos 75 t \mathrm{V}$.
b) Check the answer obtained in (a) by finding the steady-state expression for $v_{o}$ using phasor domain analysis.

Amit Srivastava
Amit Srivastava
Numerade Educator
04:29

Problem 31

a) Use the Fourier transform method to find $v_{o}$ in the circuit in Fig. P17.31 when
$$i_{g}=-45 e^{400 t} u(-t)+45 e^{-400 t} u(t) \mathrm{A}.$$
b) Find $i_{L}\left(0^{-}\right)$.
c) Find $i_{L}\left(0^{+}\right)$.
d) Do the answers obtained in (b) and (c) make sense in terms of known circuit behavior? Explain.

Amit Srivastava
Amit Srivastava
Numerade Educator
04:29

Problem 32

Use the Fourier transform method to find $i_{o}$ in the circuit in Fig. $\mathrm{P} 17.32$ if $v_{g}=200 \cos 2500 t \mathrm{V}$.

Amit Srivastava
Amit Srivastava
Numerade Educator
05:04

Problem 33

The voltage source in the circuit in Fig. $P 17.33$ is generating the signal
$$v_{x}=18 e^{4 t} u(-t)-12 u(t) \mathrm{V}.$$
a) Find $v_{o}(0)$ and $v_{c}\left(0^{\circ}\right)$.
b) $\operatorname{Find} i_{o}(0)$ and $i_{o}\left(0^{+}\right)$.
c) Find $v_{o}$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
05:45

Problem 34

a) Use the Fourier transform method to find $v_{0}$ in the circuit shown in Fig. P17.34. The voltage source generates the voltage
\[v_{g}=90 e^{-400 t |} \mathrm{V}.\]
b) Calculate $v_{o}\left(0^{-}\right), v_{o}\left(0^{+}\right),$ and $v_{o}(\infty)$.
c) Find $i_{L}\left(0^{-}\right) ; i_{L}\left(0^{+}\right) ; v_{C}\left(0^{-}\right) ;$ and $v_{C}\left(0^{+}\right)$.
d) Do the results in part (b) make sense in terms of known circuit behavior? Explain.

Kajal Gautam
Kajal Gautam
Numerade Educator
04:29

Problem 35

a) Use the Fourier transform method to find $v_{0}$ in the circuit in Fig. $\mathrm{P} 17.35$ when
$$v_{g}=60 e^{5 t} u(-t)+900 t e^{-5 t} u(t) \mathrm{V}.$$
b) Find $v_{o}\left(0^{-}\right)$.
c) Find $v_{o}\left(0^{+}\right)$.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:08

Problem 36

When the input voltage to the system shown in Fig. $P 17.36$ is $8 u(t) \vee,$ the output voltage is
$$v_{o}=\left[60-40 e^{-5 t}+20 e^{-201}\right] u(t) \mathrm{V}.$$
What is the output voltage if $v_{i}=8 \operatorname{sgn}(t)$ V?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:26

Problem 37

It is given that $F(\omega)=e^{\omega} u(-\omega)+e^{-\omega} u(\omega)$.
a) Find $f(t)$.
b) Find the $1 \Omega$ energy associated with $f(t)$ via time-domain integration.
c) Repeat (b) using frequency-domain integration.
d) Find the value of $\omega_{1}$ if $f(t)$ has $90 \%$ of the energy in the frequency band $0 \leq \omega \leq \omega_{1}$.

Amit Srivastava
Amit Srivastava
Numerade Educator
07:17

Problem 38

The input current signal in the circuit seen in Fig. $P 17.38$ is
$$i_{g}=3 e^{-25 t} \mathrm{A}, \quad t \geq 0^{+}.$$
What percentage of the total $1 \Omega$ energy content in the output signal lies in the frequency range 0 to $10 \mathrm{rad} / \mathrm{s} ?$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:10

Problem 39

The input voltage in the circuit in Fig. $\mathrm{P} 17.39$ is $v_{g}=60 e^{-|5 t|} u(t) \mathrm{V}$.
a) Find $v_{o}(t)$.
b) Sketch $\left|V_{g}(\omega)\right|$ for $-10 \leq \omega \leq 10$ rad/s.
c) Sketch $\left|V_{o}(\omega)\right|$ for $-10 \leq \omega \leq 10$ rad/s.
d) Calculate the $1 \Omega$ energy content of $v_{g}$.
e) Calculate the $1 \Omega$ energy content of $v_{o}$.
f) What percentage of the $1 \Omega$ energy content in $v_{g}$ lies in the frequency range $0 \leq \omega \leq 10 \mathrm{rad} / \mathrm{s} ?$
g) Repeat (f) for $v_{o}$.

James Kiss
James Kiss
Numerade Educator
03:55

Problem 40

The circuit shown in Fig. $P 17.40$ is driven by the current
$$i_{R}=30 e^{-2 t} u(t) \mu \mathrm{A}$$.
What percentage of the total $1 \Omega$ energy content in the output voltage $v_{o}$ lies in the frequency range $0 \leq w \leq 4 \operatorname{rad} / \mathrm{s} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
03:17

Problem 41

The amplitude spectrum of the input voltage to the high-pass $R C$ filter in Fig. $P 17.41$ is
$$V_{i}(\omega)=\frac{200}{|\omega|}, \quad 100 \mathrm{rad} / \mathrm{s} \leq|\omega| \leq 200 \mathrm{rad} / \mathrm{s};$$
$V_{i}(\omega)=0, \quad$ elsewhere.
a) Sketch $\left|V_{i}(\omega)\right|^{2}$ for $-300 \leq \omega \leq 300$ rad/s.
b) $\operatorname{Sketch}\left|V_{o}(\omega)\right|^{2}$ for $-300 \leq \omega \leq 300$ rad/s.
c) Calculate the $1 \Omega$ energy in the signal at the input of the filter.
d) Calculate the $1 \Omega$ energy in the signal at the output of the filter.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:49

Problem 42

The input voltage to the high-pass $R C$ filter circuit in Fig. P17.42 is
$$v_{i}(t)=A e^{-a t} u(t).$$
Let $\alpha$ denote the corner frequency of the filter, that is, $\alpha=1 / R C$.
a) What percentage of the energy in the signal at the output of the filter is associated with the frequency band $0 \leq \omega \leq \alpha$ if $\alpha=a ?$
b) Repeat (a), given that $\alpha=\sqrt{3} a$.
c) Repeat (a), given that $\alpha=a / \sqrt{3}$.

Amit Srivastava
Amit Srivastava
Numerade Educator