Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

9.13 
Problems
1. Is the signal v(t) 
= 
sin(21t) 
+ 
sin(15t 
- 
75
°
) periodic? If it is, what is the period?
2. Is the signal v(t) 
= 
sin(
√
2t) 
+ 
sin(t 
- 
75
°
) periodic? If it is, what is the period? 
3. Show that (i) the product of two odd functions of t will be an even function of t, (ii) the product 
of two even functions of t will be an even function of t and (iii) the product of an odd function of 
t and an even function of t will be an odd function of t.
4. v(t) in Fig. 9.13-1 has 
w
 
= 
1 rad/s. It is shifted by t
0
to produce v
1
(t) 
= 
v(t 
- 
t
0
). (i) What is t
0 
if v
1
(t) 
has odd symmetry ? (ii) Is there any t
0
such that v
1
(t) is even on t ? 
v
(
t
)
1
2
π
2
π
–3
–
/2
/2
π
3 /2
π
π
–
π
π
/2
π
(rad)
t
ω
t
ω
Fig. 9.13-1 

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9.48
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
5. The waveform in Fig. 9.13-2 has 
w
 
= 
1 rad/s.
v
(
t
)
1
–1
–3
–
/4 /2
π
3 /4
π
π
–
–
π
/2
π
/4
π
π
/4
π
(rad)
t
ω
Fig. 9.13-2 
It is time-shifted by t
0
to produce v
1
(t) 
= 
v(t 
- 
t
0
). (i) Can a value for t
0 
be found such that v
1
(t) has 
odd symmetry ? (ii) Is it possible to find value for t
0
such that v
1
(t) is even on t ?
6. The waveform in Fig. 9.13-3 has T 
= 
1 s. (i) Does it possess odd or even symmetry? (ii) If not, will 
its time-shifted version, v
1
(t) 
= 
v(t 
-
 t
0
) have odd or even symmetry for some value of t
0
? (iii) Find 
and plot the even part and odd part of this waveform. 
v
(
t
)
1
–0.5 s
0.5 s
–1
Fig. 9.13-3 
7. A waveform v(t) 
= 
2sin(
w
t 
+ 
p
/7)
+ 
0.5sin(5
w
t 
+ 
f
) is known to possess odd symmetry when it is 
delayed by 1/14 s. Find 
w
and one possible value of 
f
.
8. Show that if v(t) is a periodic waveform with 
w
o
as its fundamental frequency, the exponential 
Fourier series coefficients of 
dv t
dt
( )
are given by 
jn
v
w
o n

where 

v
n
are its exponential Fourier series 
coefficients.
9. Show that the power spectral components of output voltage of a circuit is given by 
H j
(
)
w
2
×
power spectral component of input where 
H j
(
)
w
is the frequency response function of 
the circuit.
10. If v(t) is a periodic waveform with a period of T s and v
1
(t) 
= 
2v(t 
- 
0.5T) 
+ 
7, find the relationship 
between the trigonometric Fourier series coefficients of v(t) and v
1
(t) ?
11. v(t) is a distorted sinusoidal waveform with a fundamental frequency of 
w
o
rad/s and zero DC 
content. v
1
(t) 
= 
dv t
dt
( )
and v
2
(t) 
= 
v t dt
t
( )
−∞
∫
. Will the THD value of v
1
(t) and v
2
(t) be less than, 
equal or greater than that of v(t) ? How does the answer depend on 
w
o
?
12. A battery of open circuit voltage V and internal resistance R is delivering a load current i(t) 
= 
I
t
sin
2
w
A. Power is measured by connecting a voltmeter across the battery terminals and an 
ammeter in series with the battery and multiplying the readings. Calculate the percentage error 
(ignore meter errors) in measured power if meters are of (i) moving coil type (ii) moving iron type?
13. One cycle of a waveform v(t) is shown in Fig. 9.13-4. It is a symmetrically clipped sinusoid. 
(i) Obtain a time-shifted version of this waveform such that the resulting waveform has odd 
symmetry. (ii) Find the trigonometric Fourier series of the shifted version and thereby obtain the 
discrete Fourier spectrum for v(t).

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Problems 
9.49
–3
–
/4 /2
π
3 /4
π
π
–
–
π
/2
π
/4
π
π
/4
π
v
(
t
)
t 
(s)
1
0.5
–0.5
–1
Fig. 9.13-4 
Clipped sinusoidal waveform
14. One cycle of a periodic impulse train is shown in Fig. 9.13-5. Find its exponential Fourier series 
and plot the two-sided spectra.
(s)
t
–3 /4
π
3 /4
π
π
–
– /2
π
/2
π
π
/4
–4/
π
4/
π
4/
π
8/
π
–8/
π
–4/
π
π
/4
π
v
(
t
)
–
Fig. 9.13-5 
15. One cycle of v(t) is shown in Fig. 9.13-6. (i) What is the relationship between this waveform and 
the one in Fig. 9.13-5? (ii) Obtain the exponential Fourier series of this waveform by using this 
relationship. (iii) Obtain the trigonometric Fourier series of this waveform and plot the one-sided 
spectrum of this waveform.
–3 /4
π
–3 /4
π
–
–
–
π
/2
π
/2
π
/4
π
π
–4/
π
1.5
0.5
–0.5
–1.5
–1
1
4/
π
4/
π
–4/
π
/4
π
v
(
t
)
t
(s)
Fig. 9.13-6 
16. One cycle of a periodic impulse train is shown in Fig. 9.13-7. (i) What is the relationship between 
this waveform and the one in Fig. 9.13-6? (ii) Obtain the exponential Fourier series of this 
waveform by using this relationship. (iii) Obtain the trigonometric Fourier series of this waveform 
and plot the one-sided spectrum of this waveform.

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9.50
Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
–3 /4
/2
π
3 /4
π
π
π
/2
π
/4
π
π
/4
π