Continuous Time Signals Part I Fourier Series

7/25/2019 Continuous Time Signals Part I Fourier Series

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Continuous Time Signals (Part - I)Fourier series

(a) Basics

1. Which of the following signals is/are periodic?

(a)

cos2cos3cos5(b) exp8 (c)

exp7 sin10(d) cos2cos4

[GATE 1992: 2 Marks]

Soln. (a)

cos 2 cos 3 cos 5

First term has 2Second term has 3Third term has 5

Note that ratio of any two frequencies equals p/q is rational where

p and q are integers.

Thus

is periodic

(b) exp8cos8 sin8 88 1

(c) exp7 .sin10 .sin10

, 2 210 15Due to it is decaying function, so not periodic

(d) cos 2 . cos 4 Note,

2coscoscos cos

, cos 2 cos 6


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, 26 13Rational

So, (a) , (b) and (d) are periodic.

2. The power in the signal 8 20 2 415(a)

40

(b)

41

(c)

42

(d)

82

[GATE 2005: 1 Mark]

Soln. Time average of energy of a signal = Power of Signal

l i m 1

l i m 1

||/

/

/

/

Signal power P is mean of the signal amplitude squared value of

f(t) . Rms value of signal 8 c o s 20 2 4sin15 8 si n20 4sin15 82 42 3 2 8 4 0Option (a)


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3. If a signal f(t) has energy E, the energy of the signal f(2t) is equal to

(a)

E

(b)

E/2

(c)

2E

(d)

4E

[GATE 2001: 1 Mark]

Soln. Energy of a signal is given by

Energy of the signal f(2t) is

Let 2 ,

2

2Option (b)

4.

For a periodic signal

30sin10010cos3006sin500 , the fundamentalfrequency in rad/s is(a)

100

(b)

300

(c)

500

(d)

1500

[GATE 2013: 1 Mark]


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Soln. First term has 100Second term 300Third term

500

is the fundamental frequencyis third harmonicis 5thharmonicOption (a)

5. Consider the periodic square wave in the figure shown

x

1 2 3 4 t

-1

0

The ratio of the power in the 7thharmonic to the power in the 5th

harmonic for this waveform is closest in value to ------

[GATE 2014: 1 Mark]

Soln. For a periodic square wave nthharmonic component

Thus the power in the nth harmonic component is 1 Ratio of power in 7thharmonic to 5thharmonic for the given wage form is1 71 5 2529 0.5

Answer 0.5


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6.

The waveform of a periodic signal is shown in the figure.

-4 -3 2 3

-2

-1

-3

3

1t

4

A signal

is defined by

. The average power of

is

_________ .

[GATE 2015: 1 Mark]

Soln. The equation for the given waveform can be written as 3 The period of the waveform is 3 (i.e. from -1 to +2)

. 1 13 3 3 0.

13 9. 3 01 9 3 10 0 13 93 . 01} 93 1 0

13 93 93 13 183 2

Answer 2


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7. The RMS value of a rectangular wave of period T, having a value of +V

for a duration < for the duration , equals(a)

V

(b)

(c)

(d) [GATE: 1995 1 Mark]

Soln.

-V

+V

Tt

The waveform can be drawn as per the given problem.

Period 1

1

1 . 0

1 .


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(b) Fourier series

8. The trigonometric Fourier series of an even function of time does not

have

(a)

the dc term(b)

cosine terms(c)

sine terms(d)

odd harmonic terms

[GATE 1996: 1 Mark]

Soln. For periodic even function, the trigonometric Fourier series does not

contain the sine terms (odd functions)

It has dc term and cosine terms of all harmonics.

Option (c)

9.

The trigonometric Fourier series of a periodic time function can have

only

(a)cosine terms

(b)sine terms

(c)cosine and sine terms

(d)dc and cosine terms

[GATE 1998: 1 Mark]

Soln. The Fourier series of a periodic function

is given by the form

cos sin = Thus the series has cosine terms of all harmonics: , 0 , 1 , 2 Where 0thharmonic = dc term (average or mean) = a0and sine terms of

all harmonics: , 1 , 2 , .10.

The Fourier series of an odd periodic function, contains only

(a)odd harmonics

(b)even harmonics

(c)cosine terms

(d)sine terms

[GATE 1994: 1 Mark]

Soln. If periodic function is odd the dc term 0and also cosine terms (evensymmetry)

It contains only sine terms

Option (d)


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11.The Fourier series of a real periodic function has only

P.

Cosine terms if it is even

Q.

Sine terms if it is even

R.

Cosine terms if it is odd

S.

Sine terms if it odd

Which of the above statements are correct?

(a)P and S

(b)

P and R

(c)Q and S

(d)

Q and R

[GATE 2009: 1 Mark]

Soln. The Fourier series for a real periodic function has only cosine terms if it

is even and sine terms if it is odd

Option (a)

12.

The trigonometric Fourier series of an even function does not have the

(a)

dc term

(b)

cosine terms

(c)

sine terms

(d)

odd harmonic terms

[GATE 2011: 1 Mark]

Soln. The trigonometric Fourier series of an even function has cosine terms

which are even functions.

It has dc term if its average value is finite and no dc term if average value

is zero

So it does not have sine terms

Option (c)

13.

Which of the following cannot be the Fourier series expansion of aperiodic signals?

(a) 2 c o s 3 c o s 3 (b)

2 c o s 7 c o s (c)

c o s 0 . 5(d)2cos1.5sin3.5

[GATE 2002: 1 Mark]


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Soln. (a) 2 c o s 3 c o s is periodic signal with fundamentalfrequency 1(b)

2 c o s 7 c o s The frequency of first term

frequency of 2

nd

term is 1 1 So is aperiodic or not periodic(c) c o s 0 . 5is a periodic function with 1(d)

2 c o s1.5sin3.5 first term has frequency

1.52nd

term has frequency 3.5 1.53.5 1.53.5 3 0 . 57 0 . 5 37So about ratio is rational number is a periodic signal, withfundamental frequency 0.5Since function in (b) is non periodic. So does not satisfy Dirictilet

condition and cannot be expanded in Fourier series

14.

Choose the function, < < ,for which a Fourier seriescannot be defined.

(a)

3sin25(b)4cos2 0 3 2sin710(c)

exp|| sin25

(d)

1

[GATE 2005: 1 Mark]

Soln. Fourier series is defined for periodic function and constant

(a)3sin25 25(b)

4cos20 3 2sin710 sum of two periodic function is alsoperiodic function

(c)|| sin 25 Due to decaying exponential decaying function it is notperiodic. So Fourier series cannot be defined for it.

(d)

Constant, Fourier series exists.Fourier series cant be defined for option (c)


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15.

A periodic signal is given by

{1, || < 0, < || <

2

The dc component of is(a)

(b)

(c)

(d)

[GATE 1998: 1 Mark]

Soln.

-T

1

Given periodic signal can be drawn having period T0

Fourier series the function can be written as = cos sin

Where dc component given by

1

1


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1 [

]

1 0 2 0 2

Option (c)

16.The Fourier series representation of an impulse train denoted by

=

( 1) exp2 /= ( 1) exp /

= ( 1) exp /

=

( 1) exp2 /

=

Soln.

0


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The given impulse train with strength of each impulse as 1 isaperiodic function with period T0

= 2

1 +

1. 0 1

17.The Fourier series expansion of a real periodic signal with fundamental

frequency f0is given by = It is given that 35 (a)

5 3

(b) 3 5

(c)

5 3

(d)3 j5[GATE 2003: 1 Mark]

Soln. Given 3 5We know that for real periodic signal

,

3 5

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Continuous Time Signals Part I Fourier Series