Deck 15: Fourier Analysis Techniques

Question 1

Find the trigonometric Fourier series for the voltage waveform in Fig. E15.4.

Figure E15.4

Answer 1

Given figure is of half wave summitry. So,

Question 2

Determine the first three terms of the steady-state current i ( t ) in Fig. E15.14 if the input voltage is given by


Figure E15.14

Answer 2

Refer to Figure E15.14 in the textbook.
The input voltage is,
The input signal expression can be expanded as,
The equivalent resistance of the network is,
The Fourier transform of the current,
is expressed as,
Substitute the values of
and
in the equation of current.
…… (1)
The components of
due to the frequency components present in the input voltage are as follows:
There is a dc component present in the voltage signal. At zero frequency that is, at
, the value of current is,
Substitute 10 in place of
in equation (1).
Substitute 20 in place of
in equation (1).
Hence, the first three terms of the steady state current
are,
.

Question 3

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.22.

Figure P15.22

Answer 3

Refer to periodic waveform in Figure P15.22 in the textbook.
The voltage expression is,
The period is,
The fundamental frequency is,
The trigonometric Fourier series coefficients are,
Calculate the trigonometric Fourier series coefficient
.
Calculate the trigonometric Fourier series coefficient
.
It is known that,
Therefore,
Substitute
for
.
Calculate the trigonometric Fourier series coefficient
.
It is known that,
Therefore,
Substitute
for
.
Therefore, the trigonometric Fourier series coefficients are,
.

Question 4

The discrete line spectrum for a periodic function f ( t ) is shown in Fig. P15.38. Determine the expression for f ( t ).

Figure P15.38

Question 5

Derive the Fourier transform for the following functions:
(a) f ( t ) = e 2 t cos 4 tu ( t )
(b) f ( t ) = e 2 t sin 4 tu ( t )

Question 6

Find the Fourier coefficients for the waveform in Fig. E15.1.

Figure E15.1

Question 7

Show that the exponential Fourier series for the periodic function f ( t ) = e t , 1 t 1 with period 2, can be expressed as

Question 8

Derive the exponential Fourier series for the function v ( t ) = A| sin t | shown in Fig. P15.14.

Figure P15.14

Question 9

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.23.

Figure P15.23

Question 10

Plot the first four terms of the amplitude and phase spectra for the signal

Question 11

Show that

Question 12

Given the waveform in Fig. 15PFE-1, determine if the trigonometric Fourier coefficient a n has zero value or nonzero value and why.

Figure 15PFE-1
a. a n = 0 for n even due to half-wave symmetry
b. a n = 0 for all n due to odd symmetry
c. a n is finite and nonzero for all n
d. a n is finite and nonzero for n even

Question 13

Determine the trigonometric Fourier series for the waveform shown in Fig. E15.3.

Figure E15.3

Question 14

Find the average power absorbed by the network in Fig. E15.15 if


Figure E15.15

Question 15

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.24.

Figure P15.24

Question 16

Determine the steady-state response of the current i o ( t ) in the circuit shown in Fig. P15.40 if the input voltage is described by the waveform shown in Problem 15.25.

Figure P15.40
Problem 15.25
Derive the trigonometric Fourier series for the waveform shown in Fig. P15.25.

Figure P15.25

Question 17

Use the transform technique to find v o ( t ) in the network in Fig. P15.43a if (a) i ( t ) = 4( e t e 2 t )u ( t ) A and (b) i ( t ) = 12 cos 4 t A.

Question 18

Find the exponential Fourier series for the periodic signal shown in Fig. P15.1.

Figure P15.1

Question 19

Find the exponential Fourier series for the voltage waveform shown in Fig. P15.7.

Figure P15.7

Question 20

Sketch the missing section in the periodic function shown in Fig. P15.15 between T /2 and T , which will make (a) a 0 = 0, a n = 0 and (b) b n = 0.

Figure P15.15

Question 21

Derive the trigonometric Fourier series for the waveform shown in Fig. P15.25.

Figure P15.25

Question 22

Determine the first three terms of the steady-state voltage v o ( t ) in Fig. P15.41 if the input voltage is a periodic signal of the form


Figure P15.41

Question 23

The input signal to a network is v i ( t ) = e 3 u ( t ) V. The transfer function of the network is H ( j ) = 1 / ( j + 4). Find the output of the network v o ( t ) if the initial conditions are zero.

Question 24

Find the Fourier coefficients for the waveform in Fig. E15.2.

Figure E15.2

Question 25

If the waveform in Fig. E15.1 is time-delayed 1 is, we obtain the waveform in Fig. E15.8. Compute the exponential Fourier coefficients for the waveform in Fig. E15.8 and show that they differ from the coefficients for the waveform in Fig. E15.1 by an angle n (180°).

Figure E15.8

Figure E15.1

Question 26

If f ( t ) = sin 0 t , find F ( ).

Question 27

Find the trigonometric Fourier series for the waveform shown in Fig. P15.26.

Figure P15.26

Question 28

Find the average power absorbed by the network in Fig. P15.42 if
v ( t ) = 12 + 6 cos (377 t 10°) + 4 cos (754 t 60°)V
i ( t ) = 0.2 + 0.4 cos (377 t 150°) 0.2 cos (754 t 80°) + 0.1 cos (1131 t 60°) A

Figure P15.42

Question 29

Determine v o ( t ) in the circuit shown in Fig. P15.58 using the Fourier transform if the input signal is i s ( t ) = ( e 2 t + cos t ) u ( t ) A.

Figure P15.58

Question 30

Given the waveform in Fig. 15PFE-2, describe the type of symmetry and its impact on the trigonometric Fourier coefficient b n.

Figure 15PFE-2
a. b n = 0 for n even due to odd symmetry; b n is nonzero for n odd
b. b n is nonzero for all n
c. b n = 0 for all n due to half-wave symmetry
d. b n = 0 for n even due to half-wave symmetry; b n is nonzero for n odd

Question 31

If the waveform in Problem 15.7 is time-delayed 1 second, we obtain the waveform shown in Fig. P15.8. Compute the exponential Fourier coefficients for this latter waveform and show that they differ from those for the waveform in Problem 15.7 by n (90°).

Figure P15.8
Problem 15.7
Find the exponential Fourier series for the voltage waveform shown in Fig. P15.7.

Figure P15.7

Question 32

Determine which of the Fourier coefficients are zero for f ( t ) shown in Fig. P15.16.

Figure P15.16

Question 33

Derive the trigonometric Fourier series for the function shown in Fig. P15.27.

Figure P15.27

Question 34

Determine the steady-state voltage v o ( t ) in the network in Fig. P15.43a if the input current is given in Fig. P15.43b.


Figure P15.43

Question 35

Use the Fourier transform to find i ( t ) in the network in Fig. P15.59 if v i ( t ) = 2 e t u ( t ).

Figure P15.59

Question 36

Find the exponential Fourier series for the signal shown in Fig. P15.2.

Figure P15.2

Question 37

Two periodic waveforms are shown in Fig. E15.9. Compute the exponential Fourier series for each waveform, and then add the results to obtain the Fourier series for the waveform in Fig. E15.2.

Figure E15.9

Figure. E15.2

Question 38

Determine the output v o ( t ) in Fig. E15.17 if the signal v i ( t ) = e t u ( t ) V, the network impulse response h ( t ) = e 2 t u ( t ), and all initial conditions are zero.

Figure E15.17

Question 39

Derive the trigonometric Fourier series of the waveform shown in Fig. P15.28.

Figure P15.28

Question 40

Determine the steady-state voltage v o ( t ) in the circuit shown in Fig. P15.44a if the input signal is shown in Fig. P15.44b.


Figure P15.44

Question 41

Use the transform technique to find v o ( t ) in the network in Fig. P15.60 if (a) v i ( t ) = 4 e t u (t) V and (b) v i ( t ) = 4( e 2 t + 2 e 4 t ) u ( t ) V.

Figure P15.60

Question 42

Find the exponential Fourier series for the waveform shown in Fig. E15.3.

Figure E15.3

Question 43

What type of symmetry is exhibited by the two waveforms in Fig. P15.9

Figure P15.9

Question 44

Find the trigonometric Fourier series for the voltage waveform shown in Fig. P15.17.

Figure P15.17

Question 45

Derive the trigonometric Fourier series for the function v ( t ) = A|sin t| as shown in Fig. P15.29.

Figure P15.29

Question 46

Find the average power absorbed by the network in Fig. P15.45 if
v ( t ) = 60 + 36 cos (377 t + 45°) + 24 cos (754 t 60°) V

Figure P15.45

Question 47

Determine the total 1- energy content of the signal v ( t ) = ( e t e 2 t ) u ( t ).

Question 48

Determine the first three nonzero terms of the voltage v o ( t ) in the circuit in Fig. 15PFE-3 if the input voltage v s ( t ) is given by the expression


Figure 15PFE-3
a. 8.54 cos (2 t + 26.57°) + 4.63 cos (4 t + 14.04°) + 3.14 cos (6 t + 9.46°) +... V
b. 10.82 cos(2 t + 35.63°) + 6.25cos(4 t + 18.02°) + 2.16 cos (6 t + 30.27°) +... V
c. 4.95cos(2 t 25.43°) + 3.19cos(4 t + 60.34°) + 1.78 cos (6 t 20.19°) +... V
d. 7.35cos(2 t + 50.12°) + 4.61cos(4 t + 21.24°) + 2.28 cos (6 t 10.61°) +... V

Question 49

Determine the trigonometric Fourier series for the voltage waveform in Fig. E15.10 and plot the first four terms of the amplitude and phase spectra for this signal.

Figure E15.10

Question 50

Use the transform technique to find v o ( t ) in Fig. E15.18 if u ( t ) = 15 cos 10 t V.

Figure E15.18

Question 51

The amplitude and phase spectra for a periodic function v ( t ) that has only a small number of terms is shown in Fig. P15.30. Determine the expression for v ( t ) if T 0 = 0.1 s.

Figure P15.30

Question 52

Find the average power absorbed by the 12- resistor in the network in Fig. P15.45 if
v ( t ) = 50 + 25 cos (377 t + 45°) + 12.5 cos (754 t 45°) V.

Question 53

Determine the 1- energy content of the signal in Problem 15.61 in the frequency band from 0 to 1 rad/s.
Problem 15.61
Determine the total 1- energy content of the signal v ( t ) = ( e t e 2 t ) u ( t ).

Question 54

Find the exponential Fourier series for the periodic pulse train shown in Fig. P15.3.

Figure P15.3

Question 55

Find the trigonometric Fourier series for the waveform shown in Fig. P15.10.

Figure P15.10

Question 56

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.18.

Figure P15.18

Question 57

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.31.

Figure P15.31

Question 58

Determine the Fourier transform of the waveform shown in Fig. P15.47.

Figure P15.47

Question 59

Determine the relationship between the 1- energy at the input and output of the network shown in Fig. P15.63.

Figure P15.63

Question 60

Determine the type of symmetry exhibited by the waveform in Figs. E15.2 and E15.4.

Figure E15.4

Figure E15.2

Question 61

The discrete line spectrum for a periodic function is shown in Fig. E15.11. Determine the expression for f ( t ).

Figure E15.11

Question 62

Compute the total 1- energy content of the signal v i ( t ) = e 2 t u ( t ) V using both the time-domain and frequency-domain approaches.

Question 63

Plot the first four terms of the amplitude and phase spectra for the following signal.

Question 64

Find the Fourier transform of the function f ( t ) = sin 0 t.

Question 65

The input signal for the network in Fig. P15.64 is v i ( t ) = 10 e 5 t u ( t ) V. Determine the total 1- energy content of the output v o ( t ).

Figure P15.64

Question 66

Find the average power absorbed by the network in Fig. 15PFE-4 if
v s ( t ) = 20 + 10 cos (377 t + 60°) + 4 cos (1131 t + 45°) V

Figure 15PFE-4
a. 175.25 W
b. 205.61 W
c. 150.36 W
d. 218.83 W

Question 67

Compute the exponential Fourier series for the waveform that is the sum of the two waveforms in Fig. P15.11 by computing the exponential Fourier series of the two waveforms and adding them.

Figure P15.11

Question 68

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.19.

Figure P15.19

Question 69

Compute the first four terms of the amplitude and phase spectra for the periodic signal defined in Problem 15.17.
Problem 15.17
Find the trigonometric Fourier series for the voltage waveform shown in Fig. P15.17.

Figure P15.17

Question 70

Derive the following properties of the Fourier transform.
(a) [ e j 0 t f ( t )] = F ( 0 )
(b)

Question 71

Compute the 1- energy content of the signal v o ( t ) in Fig. P15.64 in the frequency range from = 2 to = 4 rad/s.

Fig. P15.64

Question 72

Find the exponential Fourier series for the signal shown in Fig. P15.4.

Figure P15.4

Question 73

Determine the expression for the steady-state current i ( t ) in Fig. E15.12 if the input voltage v S ( t ) is given by the expression


Figure E15.12

Question 74

Compute the 1- energy content of the signal v i ( t ) = e 2 t u ( t ) V in the frequency range from 0 to 1 rad/s.

Question 75

For the amplitude and phase spectra shown in Fig. P15.34, express the function as a sum of sinusoidal functions using the Fourier series coefficients a n and b n.

Figure P15.34

Question 76

Find the Fourier transform of the function f ( t ) = te at u ( t ).

Question 77

Determine the 1- energy content of the signal v o ( t ) in Fig. P15.64 in the frequency range from 0 to 1 rad/s.

Fig. P15.64

Question 78

Find the trigonometric Fourier series for the voltage waveform in Fig. E15.2.
Fig. E15.2

Question 79

Given the waveform in Fig. P15.12, determine the type of symmetry that exists if the origin is selected at (a) l 1 and (b) l 2.

Figure P15.12

Question 80

Find the trigonometric Fourier series for the periodic waveform shown in Fig. P15.20.

Figure P15.20