• Home
  • Textbooks
  • Fundamentals of Electric Circuits
  • Fourier Transform

Fundamentals of Electric Circuits

Charles K. Alexander, Matthew N.O. Sadiku

Chapter 17

Fourier Transform - all with Video Answers

Educators

Chapter Questions

01:59

Problem 1

Obtain the Fourier transform of the function in Fig. 17.26

Amit Srivastava
Amit Srivastava
Numerade Educator
01:07

Problem 2

What is the Fourier transform of the triangular pulse in Fig. $17.27 ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:16

Problem 3

Calculate the Fourier transform of the signal in Fig. 17.28

Amit Srivastava
Amit Srivastava
Numerade Educator
02:19

Problem 4

Find the Fourier transforms of the signals in Fig. 17.29

Amit Srivastava
Amit Srivastava
Numerade Educator
01:32

Problem 5

Determine the Fourier transforms of the functions in Fig. 17.30

Amit Srivastava
Amit Srivastava
Numerade Educator
02:34

Problem 6

Obtain the Fourier transforms of the signals shown in Fig. 17.31

Amit Srivastava
Amit Srivastava
Numerade Educator
03:12

Problem 7

Find the Fourier transform of the "sine-wave pulse" shown in Fig. 17.32

Amit Srivastava
Amit Srivastava
Numerade Educator
02:09

Problem 8

Determine the Fourier transforms of these functions.
(a) $f(t)=e^{t}[u(t)-u(t-1)]$
(b) $g(t)=t e^{-t} u(t)$
(c) $h(t)=u(t+1)-2 u(t)+u(t-1)$

Amit Srivastava
Amit Srivastava
Numerade Educator
06:38

Problem 9

Find the Fourier transforms of these functions:
(a) $f(t)=e^{-t} \cos (3 t+\pi) u(t)$
(b) $g(t)=\sin \pi t[u(t+1)-u(t-1)]$
(c) $h(t)=e^{-2 t} \cos \pi t u(t-1)$
(d) $p(t)=e^{-2 t} \sin 4 t u(-t)$
(e) $q(t)=4 \operatorname{sgn}(t-2)+3 \delta(t)-2 u(t-2)$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:06

Problem 10

Find the Fourier transforms of the following functions:
(a) $f(t)=\delta(t+3)-\delta(t-3)$
(b) $f(t)=\int_{-\infty}^{\infty} 2 \delta(t-1) d t$
(c) $f(t)=\delta(3 t)-\delta^{\prime}(2 t)$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:24

Problem 11

Determine the Fourier transforms of these functions:
(a) $f(t)=4 / t^{2}$
(b) $g(t)=8 /\left(4+t^{2}\right)$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:46

Problem 12

Find the Fourier transforms of
(a) $\cos 2 t u(t)$
(b) $\sin 10 t u(t)$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:01

Problem 13

Obtain the Fourier transform of $y(t)=e^{-t} \cos t u(t)$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:19

Problem 14

Find the Fourier transform of $f(t)=\cos 2 \pi t[u(t)-u(t-1)]$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:33

Problem 15

(a) Show that a periodic signal with exponential Fourier series
$$f(t)=\sum_{n=-\infty}^{\infty} c_{n} e^{j n \omega_{0} t}$$
has the Fourier transform
$$F(\omega)=\sum_{n=-\infty}^{\infty} c_{n} \delta\left(\omega-n \omega_{0}\right)$$
where $\omega_{0}=2 \pi / T$
(b) Find the Fourier transform of the signal in Fig. 17.33

Amit Srivastava
Amit Srivastava
Numerade Educator
01:35

Problem 16

Prove that if $F(\omega)$ is the Fourier transform of $f(t)$
$$\mathcal{F}\left[f(t) \sin \omega_{0} t\right]=\frac{j}{2}\left[F\left(\omega+\omega_{0}\right)-F\left(\omega-\omega_{0}\right)\right]$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:10

Problem 17

If the Fourier transform of $f(t)$ is
$$F(\omega)=\frac{10}{(2+j \omega)(5+j \omega)}$$ determine the transforms of the following:

(a) $f(-3 t)$
(b) $f(2 t-1)$
(c) $f(t) \cos 2 t$
(d) $\frac{d}{d t} f(t)$
(e) $\int_{-\infty}^{t} f(t) d t$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:59

Problem 18

Given that $\mathcal{F}[f(t)]=(j / \omega)\left(e^{-j \omega}-1\right),$ find the Fourier transforms of:
(a) $x(t)=f(t)+3$
(b) $y(t)=f(t-2)$
(c) $h(t)=f^{\prime}(t)$
(d) $g(t)=4 f\left(\frac{2}{3} t\right)+10 f\left(\frac{5}{3} t\right)$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:25

Problem 19

Obtain the inverse Fourier transforms of
(a) $F(\omega)=\frac{10}{j \omega(j \omega+2)}$
(b) $F(\omega)=\frac{4-j \omega}{\omega^{2}-3 j \omega-2}$

Amit Srivastava
Amit Srivastava
Numerade Educator
06:52

Problem 20

Find the inverse Fourier transforms of the following functions:
(a) $F(\omega)=\frac{100}{j \omega(j \omega+10)}$
(b) $G(\omega)=\frac{10 j \omega}{(-j \omega+2)(\omega+3)}$
(c) $H(\omega)=\frac{60}{-\omega^{2}+j 40 \omega+1300}$
(d) $Y(\omega)=\frac{\delta(\omega)}{(j \omega+1)(j \omega+2)}$

Amit Srivastava
Amit Srivastava
Numerade Educator
05:30

Problem 21

Find the inverse Fourier transforms of:
(a) $\frac{\pi \delta(\omega)}{(5+j \omega)(2+j \omega)}$
(b) $\frac{10 \delta(\omega+2)}{j \omega(j \omega+1)}$
(c) $\frac{20 \delta(\omega-1)}{(2+j \omega)(3+j \omega)}$
(d) $\frac{5 \pi \delta(\omega)}{5+j \omega}+\frac{5}{j \omega(5+j \omega)}$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:29

Problem 22

Determine the inverse Fourier transforms of
(a) $F(\omega)=4 \delta(\omega+3)+\delta(\omega)+4 \delta(\omega-3)$
(b) $G(\omega)=4 u(\omega+2)-4 u(\omega-2)$
(c) $H(\omega)=6 \cos 2 \omega$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:26

Problem 23

Determine the functions corresponding to the following Fourier transforms:
(a) $F_{1}(\omega)=\frac{e^{j \omega}}{-j \omega+1}$
(b) $F_{2}(\omega)=2 e^{|\omega|}$
(c) $F_{3}(\omega)=\frac{1}{\left(1+\omega^{2}\right)^{2}}$
(d) $F_{4}(\omega)=\frac{\delta(\omega)}{1+j 2 \omega}$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:22

Problem 24

Find $f(t)$ if
(a) $F(\omega)=2 \sin \pi \omega[u(\omega+1)-u(\omega-1)]$
(b) $F(\omega)=\frac{1}{\omega}(\sin 2 \omega-\sin \omega)+\frac{j}{\omega}(\cos 2 \omega-\cos \omega)$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:34

Problem 25

Determine the signal $f(t)$ whose Fourier transform is shown in Fig. 17.34 (Hint: Use the duality property.)

Amit Srivastava
Amit Srivastava
Numerade Educator
03:27

Problem 26

A linear system has a transfer function
$$H(\omega)=\frac{10}{2+j \omega}$$
Determine the output $v_{o}(t)$ at $t=2$ s if the input $v_{i}(t)$ equals:
(a) $4 \delta(t)$ V
(b) $6 e^{-t} u(t)$ V
(c) $3 \cos 2 t \mathrm{V}$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:12

Problem 27

Find the transfer function $I_{o}(\omega) / I_{s}(\omega)$ for the circuit in Fig. 17.35

Amit Srivastava
Amit Srivastava
Numerade Educator
01:07

Problem 28

Obtain $v_{o}(t)$ in the circuit of Fig. 17.36 when $v_{i}(t)=u(t) \mathrm{V}$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:29

Problem 29

Determine the current $i(t)$ in the circuit of Fig. $17.37(b),$ given the voltage source shown in Fig. $17.37(\mathrm{a})$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:12

Problem 30

Obtain the current $i_{o}(t)$ in the circuit in Fig. 17.38
(a) Let $i(t)=\operatorname{sgn}(t)$ A.
(b) Let $i(t)=4[u(t)-u(t-1)]$ A.

Amit Srivastava
Amit Srivastava
Numerade Educator
05:42

Problem 31

Find current $i_{o}(t)$ in the circuit of Fig. 17.39

Kajal Gautam
Kajal Gautam
Numerade Educator
02:23

Problem 32

If the rectangular pulse in Fig. $17.40(\mathrm{a})$ is applied to the circuit in Fig. $17.40(\mathrm{b}),$ find $v_{o}$ at $t=1 \mathrm{s}$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:39

Problem 33

Calculate $v_{o}(t)$ in the circuit of Fig. 17.41 if $v_{s}(t)=10 e^{-|t|} \mathrm{V}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:30

Problem 34

Determine the Fourier transform of $i_{o}(t)$ in the circuit of Fig. 17.42

Amit Srivastava
Amit Srivastava
Numerade Educator
01:32

Problem 35

In the circuit of Fig. $17.43,$ let $i_{s}=4 \delta(t)$ A. Find $V_{o}(\omega)$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:49

Problem 36

Find $i_{o}(t)$ in the op amp circuit of Fig. 17.44

Amit Srivastava
Amit Srivastava
Numerade Educator
04:29

Problem 37

Use the Fourier transform method to obtain $v_{o}(t)$ in the circuit of Fig. 17.45

Amit Srivastava
Amit Srivastava
Numerade Educator
02:57

Problem 38

Determine $v_{o}(t)$ in the transformer circuit of Fig. 17.46

Amit Srivastava
Amit Srivastava
Numerade Educator
01:14

Problem 39

$$\text { For } F(\omega)=\frac{1}{3+j \omega}, \text { find } J=\int_{-\infty}^{\infty} f^{2}(t) d t$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:01

Problem 40

$$\text { If } f(t)=e^{-2|t|}, \text { find } J=\int_{-\infty}^{\infty}|F(\omega)|^{2} d \omega$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:02

Problem 41

Given the signal $f(t)=4 e^{-t} u(t),$ what is the total energy in $f(t) ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:47

Problem 42

Let $f(t)=5 e^{-(t-2)} u(t) .$ Find $F(\omega)$ and use it to find the total energy in $f(t)$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:29

Problem 43

A voltage source $v_{s}(t)=e^{-t} \sin 2 t u(t) V$ is applied to a $1-\Omega$ resistor. Calculate the energy delivered to the resistor.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:17

Problem 44

Let $i(t)=2 e^{t} u(-t)$ A. Find the total energy carried by $i(t)$ and the percentage of the $1-\Omega$ energy in the frequency range of $-5 < \omega < 5 \mathrm{rad} / \mathrm{s}$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:07

Problem 45

An AM signal is specified by
$$f(t)=10(1+4 \cos 200 \pi t) \cos \pi \times 10^{4} t$$
Determine the following:
(a) the carrier frequency,
(b) the lower sideband frequency,
(c) the upper sideband frequency.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:07

Problem 46

A carrier wave of frequency $8 \mathrm{MHz}$ is amplitude-modulated by a 5 -kHz signal. Determine the lower and upper sidebands.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:09

Problem 47

A voice signal occupying the frequency band of 0.4 to $3.5 \mathrm{kHz}$ is used to amplitude-modulate a $10-\mathrm{MHz}$ carrier. Determine the range of frequencies for the lower and upper sidebands.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:11

Problem 48

For a given locality, calculate the number of stations allowable in the AM broadcasting band $(540$ to $1600 \mathrm{kHz}$ without interference with one another

Amit Srivastava
Amit Srivastava
Numerade Educator
01:05

Problem 49

Repeat the previous problem for the FM broadcasting band $(88 \text { to } 108 \mathrm{MHz})$, assuming that the carrier frequencies are spaced $200 \mathrm{kHz}$ apart.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:01

Problem 50

The highest-frequency component of a voice signal is $3.4 \mathrm{kHz}$. What is the Nyquist rate of the sampler of the voice signal?

Amit Srivastava
Amit Srivastava
Numerade Educator
01:02

Problem 51

A TV signal is band-limited to $4.5 \mathrm{MHz}$. If samples are to be reconstructed at a distant point, what is the maximum sampling interval allowable?

Amit Srivastava
Amit Srivastava
Numerade Educator
01:27

Problem 52

Given a signal $g(t)=\operatorname{sinc}(200 \pi t),$ find the Nyquist rate and the Nyquist interval for the signal.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:49

Problem 53

The voltage signal at the input of a filter is $v(t)=50 e^{-2|t|} | \mathrm{V} .$ What percentage of the total $1-\Omega$ energy content lies in the frequency range of $1 < \omega < 5 \mathrm{rad} / \mathrm{s} ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:06

Problem 54

A signal with Fourier transform
$$F(\omega)=\frac{20}{4+j \omega}$$
is passed through a filter whose cutoff frequency is 2 $\operatorname{rad} / \mathrm{s}(\text { i.e., }, 0 < \omega < 2) .$ What fraction of the energy in the input signal is contained in the output signal?

Amit Srivastava
Amit Srivastava
Numerade Educator