chapter 4-fourier series

CHAPTER 4 : FOURIER SERIES

4.0 Introduction

A periodic signal x(t) is periodic if x(t + T) = x (t) where T is the period and .

Sinusoid of frequency nf0 where n is positive integer is said to be the nth harmonic. If:

n = odd (odd harmonic)

n = even (even harmonic)

4.1 Trigonometric Fourier series

If x(t) is a periodic function with period T, then x(t) can be expressed as trigonometric

Fourier series:

x(t) is expressed as the sum of sinusoidal components having different frequencies

where:

and - the Fourier coefficients

- the dc value of x(t)

4.1.1 Orthogonal Fuctions for Sine and Cosine Functions

Consider a function and . According to orthogonal function properties

Where m and n are integers, T is the period and rn is some value.

Let and , then

1)

If n = 0, then

2)

Let and , then

3)

Let and , then

4)

If n = 0, then

5)

4.1.2 Determination of Trigonometric Fourier Coefficients

4.1.2.1 Determination of Fourier coefficient, :

Based on the trigonometric Fourier series expression

(4.1.2)

When Equation (4.1.2) is integrated both sides for one complete cycle, then

Using Orthogonal functions relation.

Thus,

4.1.2.2 Determination of Fourier coefficient, :

When Equation (4.1.2) is multiplied both sides with and then

integrated for one complete cycle, then

Using orthogonal function relation,

and

then,

Since m = n, then

4.1.2.3 Determination of Fourier coefficient, :

When Equation (4.1.2) is multiplied both sides with and then

integrated for one complete cycle, then

Using Orthogonal functions relation,

And

then,

Since m = n, then

Graph of sin nωt

Sin 2nπ n = even 0 0

n = odd 0Sin nπ n = even 0 0

n = odd 0

Sin

n = even 0 0n = odd 1,-1, 1,-1, 1

n = both 1, 0,-1, 1, 0,

where n = 2n-1

Graph of cos nωt

Cos 2nπ n = even 1 1n = odd 1

Cos nπ n = even 1n = odd -1

Cos

n = even -1, 1, -1, 1,

n = odd 0 0n = both 0,-1, 0, 1, 0, Cos(2n - 1)π = -1

where n = 2n-1

Example 4.1.1 Express the signal x(t) shown in Figure 4.1.1 as trigonometric

Fourier series.

Figure 4.1.1

SOLUTION:

The average value, is determined as follows:

or

= [Area under the curve for one complete cycle]

The Fourier coefficient, is determined as follows:

Since: sin nπ = 0 and sin 2nπ = 0;

The Fourier coefficient, bn is determined as follows:

Since: cos nπ = (-1)n and cos 2nπ = 1;

Thus,

or

Example 4.1.2 Express signal the x(t) shown in Figure 4.1.2 as trigonometric Fourier

series

Figure 4.1.2

SOLUTION:

The average value, is determined as follows:

or

= [Area under the curve for one complete cycle]

The Fourier coefficient, is determined as follows:

Since: sin nπ = 0 and sin 2nπ = 0;

The Fourier coefficient, bn is determined as follows:

Thus,

4.2 Symmetry Properties

i) Even Symmetry

ii) Odd Symmetry

iii) Half-Wave Symmetry

iv) Even And Half-Wave Symmetry (Half-Wave Even Symmetry)

v) Odd And Half-Wave Symmetry (Half-Wave Odd Symmetry)

vi) Hidden symmetry

4.2.1 Even Symmetry

Example 4.2.1 Consider a half-cycle signal x(t) shown in Figure 4.2.1(a) where T = 2

sec.

Figure 4.2.1(a)

The signal x(t) is said to be even symmetry if x(t) = x(-t). This-property is shown in

Figure 4.2.1(b)

Figure 4.2.1(b)

The even-symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.1(c).

Figure 4.2.1(c)

4.2.2 Odd Symmetry

Example 4.2.2 Consider a half-cycle signal x(t) shown in Figure 4.2.2(a) where T = 2

sec.

Figure 4.2.2(a)

The signal x(t) is said to be odd symmetry if x(t) = -x(-t). This property is shown in

Figure 4.2.2(b)

Figure 4.2.2(b)

1 cycle 1 cycle 1 cycle

The odd-symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.2(c)

Figure 4.2.2(c)

4.2.3 Half-Wave Symmetry

Example 4.2.3 Consider a half-cycle signal x(t) shown in Figure 4.2.3(a) where T = 2

sec.

Figure 4.2.3(a)

The signal x(t) is said to be half-wave symmetry if x(t) = -x(t + T/2). This property is

shown in Figure 4.2.3(b)

1 cycle 1 cycle 1 cycle

Figure 4.2.3(b)

The half-wave symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.3(c).

Figure 4.2.3(c)

4.2.4 Even and Half-Wave Symmetry (Half-Wave Even Symmetry)

Example 4.2.4 Consider a half-cycle signal x(t) shown in Figure 4.2.4(a) where T =

4 sec.

Figure 4.2.4(a)

For one complete cycle/the shape of signal x(t) is the same for both properties. This is

shown in Figure 4.2.4(b).

1 cycle 1 cycle 1 cycle

Figure 4.2.4(b)

Thus, the half-wave even symmetry signal x(t) for 3 complete cycles is shown in Figure

4.2.4(c).

Figure 4.2.4(c)

4.2.4 Odd and Half-Wave Symmetry (Half-Wave Odd Symmetry)

Example 4.2.5 Consider a half-cycle signal x(t) shown in Figure 4.2.5(a) where T

= 4 sec.

Figure 4.2.5(a)

1 cycle 1 cycle 1 cycle

For one complete cycle, the shape of signal x(t) is the same for both properties.This is

shown in Figure 4.2.5(b).

Figure 4.2.5(b)

Thus, the half-wave even symmetry signal x(t) for 3 complete cycles is shown in Figure

4.2.5 (c).

Figure 4.2.5 (c)

Example 4.2.6 The first half-cycle of a periodic signal y(t) is shown in Figure

4.2.6(a) and the period sec. Sketch y(t) clearly for.3

complete cycles if:

i) y(t) is an even-symmetric signal

ii) y(t) is an odd-symmetric signal

iii) y(t) is a half-wave symmetric signal

1 cycle 1 cycle 1 cycle

Figure 4.2.6(a)

SOLUTION:

The signal y(t) for all cases are given in Figure 4.2.6(b).

Figure 4.2.6(b)

4.3 Effects of Symmetry

i) If x(t) is an even symmetric signal, then its trigonometric Fourier series

expression is as follows:

Its Fourier series consists of a constant and cosine terms only where:

ii) If x(t) is an odd symmetric signal, then its trigonometric Fourier series

expression is as follows:

Its Fourier series consists of sine terms only where:

and

iii) If x(t) is a half-wave symmetric signal, then its trigonometric Fourier expression

is as follows:

Its Fourier series consists of odd harmonic of cosine and sine terms only where:

, and

iv) If x(t) is an even symmetric and also half-wave symmetric signal (half-wave

even symmetric signal), then its trigonometric Fourier series expression is as

follows:

Its Fourier series consists of a constant and cosine terms only where:

, and

v) If x(t) is an odd symmetric and also half-wave symmetric signal (half-wave odd

symmetric), then its trigonometric Fourier series expression is as follows:

Its Fourier series consists of odd harmonic of sine terms only where:

, and

The trigonometric Fourier series expressions of each symmetric signal are summarized

in Table 4.1.

Table 4.1 Fourier series Simplified Flow Techniques

Signal function

TFS Coefficients

Coefficients TFS expressions

Generala0

an

bn

Even symmetry

a0

an

bn=0

Odd symmetry

bn

a0 = an = 0

Half-wave symmetry

a0 =0

an (even) =0

bn (even) =0

Even and half-wave symmetry

a0 =0

bn =0

an (even) =0

Odd and half-wave symmetry

a0 =0

an =0

bn (even) =0

Example 4.3.1 Express signal x(t) shown in Figure 4.3.1 as trigonometric Fourier

series using symmetry property.

Figure 4.3.1

SOLUTION:

The signal x(t) is odd-symmetry and half- wave symmetry signal (half-wave odd

symmetry).

, and

The Fourier coefficient is determined as follows:

Thus,

Example 4.3.2 Express signal x(t) shown in Figure 4.3.2 as trigonometric Fourier

series using Symmetry property.

Figure 4.3.2

SOLUTION:

x(t) is even-symmetry and half- wave symmetry signal (half-wave even symmetry).

, and

The Fourier coefficient is determined as follows:

Thus,

4.4 Hidden Symmetry

Example 4.4.1 Express signal x(t) as trigonometric Fourier series.

Figure 4.4.1(a)

SOLUTION:

Signal x(t) does not posses any symmetry properties. The evaluation can be further

simplified by shifting the dc value of signal x(t). Signal g(t) is obtain from signal x(t)

where x(t) = 0.5A + g(t) and signal g(t) is shown in Figure 4.4.1 (b).

Figure 4.4.1 (b)

Signal g(t) posses odd-symmetry property. Thus,

Fourier series of x(t) = 0.5A + Fourier series of g(t)

For odd Symmetric signal, then

and

The Fourier coefficient, bn is determined as follows:

Thus,

Fourier series of x(t) = 0.5A + Fourier series of g(t)

Example 4.4.2 Express the signal x (t) shown in Figure 4.4.2(a) as trigonometric

Fourier series.

Figure 4.4.2(a)

SOLUTION:

Signal x(t) posses even-symmetry property. The evaluation can be further simplified by

shifting the dc value of signal x(t). Signal g(t) is obtain from signal x(t) where signal

and the signal g(t) is shown in Figure 4.4.2(b).

Figure 4.4.2(b)

Signal g(t) posses half-wave even symmetry property. Thus,

Fourier series of x(t) = + Fourier series of g(t)

Fourier series of g(t) is obtain as follows:

, and

The Fourier coefficient, an (n=odd) is determined as follows:

Fourier series of x(t) = + Fourier series of g(t)

4.5 Exponential Fourier series

The trigonometric Fourier series of signal x(t) is given as:

EULER'S IDENTITY:

The expression can be expressed as follows:

Let , and

Then,

The term can also be represented as follows:

Then,

Where:

EULER'S IDENTITY:

Then, where n = ±1, ±2, ….

Example 4.5.1 Express the signal x(t) shown in Figure 4.5.1 as an exponential

Fourier series.

Figure 4.5.1

SOLUTION:

EULER IDENTITY:

Then,

and

Thus,

Example 4.5.2 Express the signal x (t) shown in Figure 4.5.2 as an exponential

Fourier series.

Figure 4.5.2

SOLUTION:

EULER IDENTITY:

Then,

For n = 0, Cn has no meaning. Thus,

Thus,

4.6 Frequency Spectrum

Frequency spectrum consists of amplitude and phase spectrums. Amplitude spectrum

is the plot of |Cn| versus and phase spectrum is the plot of versus .

ASIDE:

To determine the amplitude and phase spectrums of Frequency spectrum, the

magnitude and phase of X are shown in Figure 4.6 and summarized in Table 4.6:

Figure 4.6

Table 4.6

Magnitude Amplitude

spectrums

Phase spectrums

1 X = a + jb |X| =

2 X = jb |X| = b

3 X = -jb |X| = b

4 X = a |X| = a

5 X = -a |X| = a

Example 4.6.1 Plot the frequency spectrum of signal x(t) shown in Figure 4.6.1(a).

Figure 4.6.1(a)

SOLUTION:

;

Im

Re

(a+jb)

a

b

X

|X|

And

Amplitude spectrum:

Since , or , it is satisfied magnitude no 3 in Table 4.6.

So the amplitude spectrum is:

and

Phase spectrum:

Since , it is satisfied magnitude no 2 and no 3 in Table 4.6.

And plot the frequency spectrum for n= 0, ±1, ±2, ±3, ±4, ±5.

The plotted amplitude and phase spectrums of signal x (t) are shown in Figure 4.6.1(b)

and in Figure 4.6.1(c).

Figure 4.6.1(b) Amplitude spectrums

n -5 -4 -3 -2 -1 1 2 3 4 5

0 0 0 0 0

0 0 0 0 0

Figure 4.6.1(c) Phase spectrums

Example 4.6.2 Plot the frequency spectrum of signal x (t) shown in Figure 4.6.2(a).

Figure 4.6.2(a)

SOLUTION:

;

And

Amplitude spectrum:

Since , or , it is satisfied magnitude no 2 in Table 4.6.

So the amplitude spectrum is:

and

Phase spectrum:

Since , it is satisfied magnitude no 2 and no 3 in Table 4.6.

And plot the frequency spectrum for n= 0, ±1, ±2, ±3, ±4, ±5.

The plotted amplitude and phase spectrums of signal x (t) are shown in Figure 4.6.2(b)

and in Figure 4.6.2(c).

Figure 4.6.2(b) Amplitude spectrums

n -5 -4 -3 -2 -1 1 2 3 4 5

Figure 4.6.2(c) Phase spectrums

4.7 Trigonometric Fourier Coefficients and Complex Fourier

Coefficients Relationship

, and

Example 4.7.1 Convert the trigonometric Fourier coefficients of signal x (t) of Figure

4.7.1 to complex Fourier coefficient.

Figure 4.7.1

; ; and

SOLUTION:

Thus,

. This is true for n= 0, ±1, ±2, ±3, …

Thus,

Example 4.7.2 Convert the trigonometric Fourier coefficients of signal x (t) of' Figure

4.7.2 to complex Fourier coefficient.

Figure 4.7.2

; ; and

SOLUTION:

Thus,

. This is true for n= 0, ±1, ±2, ±3…

Thus,