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Introduction to Fourier Series - GATE

Study Material in PDF

The previous GATE study material dealt with Linear Time Invariant Systems. In

these free GATE Notes, we will start with an introduction to Fourier Series‼ These

study material covers everything that is necessary for GATE EC and GATE EE as

well as other exams like ISRO, IES, BARC, BSNL, DRDO etc. These notes can also be

downloaded in PDF so that your exam preparation is made easy and you ace your

exam.

In these notes, we will learn what a Fourier Series is, the conditions for the existence

of a Fourier Series (also known as Dirichlet’s Conditions) as well as the different

types of Fourier Series (Trigonometric, Polar and Exponential). We will also take a

look at the Magnitude Spectrum, the Phase Spectrum and the Power Spectrum of a

Fourier Series. Then we will look at some examples.

You should probably go through the basics covered in previous articles, before

starting off with this module.

Recommended Reading –

Laplace Transforms

Limits, Continuity & Differentiability

Mean Value Theorems

Differentiation

Partial Differentiation

Maxima and Minima

Methods of Integration & Standard Integrals

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Vector Calculus

Vector Integration

Time Signals & Signal Transformation

Standard Time Signals

Signal Classification

Types of Time Systems

Introduction to Linear Time Invariant Systems

Properties of LTI Systems

What is a Fourier Series?

The concept of vectors can directly be extended to signals due to the analogy between

signals and vectors. A Fourier Series is an expansion of a periodic function f(x) in

terms of an infinite sum of sines and cosines. Fourier series makes use of

the orthogonality relationships of the sine and cosine functions.

Since infinite cosine functions and infinite sine functions are mutually orthogonal /

exclusive. So it is possible to represent any function as the sum of infinite sine and

cosine functions (or linear combination of sine and cosine functions which is known

as Fourier series representation.)

It is possible to represent a given signal in Fourier series for one period which

implies that the Fourier series is applicable for periodic signals only.

Let x(t) be a periodic signal with fundamental period T then x(t) can be represented

in Fourier Series form as -

x(t) = ∑ [an cos(nω0t) + bn sin(nω0t)]∞n=0

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an and bn are Fourier series coefficients.

Conditions for Existence of Fourier Series

The conditions for existence of Fourier series includes both necessary and sufficient

conditions. These are known as Dirichlet’s Conditions. These have been given as

below

(i) Signal should be absolutely integrable over one period

∫T |x(t)| dt < ∞

(ii) Signal must have finite number of maximas and minimas in one period.

(iii) Signal must have finite number of discontinuities in one period.

First one is the necessary condition while the remaining two are the sufficient

conditions

Forms of Fourier Series

Fourier series can be expressed into three different forms. These are been given as

follows -

A. Trigonometric Fourier series (TFS)

B. Compact Form / Polar Form Fourier series

C. Exponential Fourier series

A. Trigonometric Fourier Series

A periodic signal x(t) can be represented in the form of trigonometric Fourier series

containing sine and cosine terms –

x(t) = a0 + a1 cos ω0t + a2 cos 2ω0t + … . … . . + b1sin ω0t + b2 sin 2ω0t

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x(t) = ∑ [an cos(nω0t) + bn sin(nω0t)] ; t0 ≤ t ≤ t0 + T∞n=0

Or

x(t) = a0 + ∑ (an cos(nω0t) + bn sin(nω0t)) ; t0 ≤ t ≤ t0 + T∞n=1

Where ω0 =2π

T and a0, an, bn are coefficients of Trigonometric Fourier Series. ω0 is

fundamental frequency and 2ω0, 3ω0…. are called the harmonics of ω0.

a0 is known as DC term and its value is given by -

a0 =1

T∫ x(t)dt

T

It is clear from above equation that a0 is the average value or DC component of x(t)

over one period. Now, the coefficients an and bn are being calculated as follows

an =2

T∫ x(t) cos(nω0t) dt

T

bn =2

T∫ x(t) sin(nω0t) dt

T

a0, an and bn represent the similarity of the signal x(t) associated with DC, cosine and

sine function respectively.

Example 1:

Find harmonics and TFS coefficients of the following signals.

1. x1(t) = cos(πt + 30°) + sin(2t)

𝟐. x2(t) = 10 cos2(4

5t − 450)

𝟑. x3(t) = cos(7t) − 2 sin(2t + 20°) + 5 cos (4

7t + 10°) − sin(0.2t)

4. x4(t) = –4 sin(0.8πt) + 2 cos(2πt+30°)

Solution:

1. x1(t) = cos(πt+30°) + sin2t

ω1 = π, ω2 = 2 ω1

ω2=

π

2= irrational

⇒ x1(t) is non-periodic

⇒ Fourier series does not exist for x1(t)

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𝟐. x2(t) = 10 cos2 (4

5t − 45°) = 10

(1+cos(8

5t−90°))

2

x2(t) = 5 + 5 cos (8

5t − 90°)

x2(t) = 5 + 5 sin (8

5t)

TFS –

x2(t) = a0 + ∑ (an cos (8

5nt) + bn sin (

8

5nt))∞

n=1

= a0 + a1 cos (8

5t) + a2 cos (

16

5t) +. . + b1 sin (

8

5t) b2 sin (

16

5t) +. .

By comparing, a0 = 5, a1 = 5

𝟑. x3(t) = cos(7t) − 2 sin(2t + 20°) + 5 cos (4

7t + 10°) − sin(0.2t)

ω1 = 7, ω2 = 2, ω3 =4

7, ω4 = 0.2 =

1

5

ω1

ω2=

7

2= rational,

ω2

ω3=

2×7

4= rational,

ω3

ω4=

4

7×0.2= rational

Since all the ratios are rational therefore, x3(t) is periodic with period

ω0 =GCD(7,2,4,1)

LCM(1,1,7,5)=

1

35

x3(t) = cos(35 × 7ω0t) − 2 sin(35 × 2ω0t + 20°) + 5 cos (35 ×4

7ω0t + 10°) −

sin(35 × 0.2ω0t)

x3(t) = cos(245ω0t) − 2 sin(70ω0t + 20°) + 5 cos(20ω0t + 10°) − sin(7ω0t)

x3(t) = − sin(7ω0t) + 5 cos 10° cos(20ω0t) − 2 sin 20° cos(70ω0t) + cos(245ω0t) −

5 sin 10° sin(20ω0t) − 2cos 20° sin(70ω0t)

By comparison with equation of TFS

a20 = 5 cos 10° ; a70 = −2 sin 20° , a245 = 1

b7 = −1; b20 = −5 Sin10° ; b70 = −2 cos 20°

4. x4(t) = –4 sin(0.8πt) + 2 cos(2πt+30°)

ω1 = 0.8π =4π

5, ω2 = 2π

ω1

ω2=

0.8π

2π= 0.4 = rational

⇒ x4(t) is a periodic signal with period ω0

ω0 =GCD(4π,2π)

LCM(5,1)=

2π

5

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x4(t) = −4 sin (5

2π× 0.8π × ω0t) + 2 cos (

5

2π× 2π × ω0t + 30°)

= −4 sin(2ω0t) + 2 cos(5ω0t + 30°)

= −4 sin(2ω0t) + 2 cos 30° cos(5ω0t) − 2 sin 30° sin(5ω0t)

= −4 sin(2ω0t) + 2 ×√3

2cos(5ω0t) − 2 ×

1

2sin(5ω0t)

x4(t) = −4 sin(2ω0t) + √3 cos(5ω0t) − sin(5ω0t)

By comparing it with TFS equation

b2 = −4, a5 = √3, b5 = −1

B. Polar Fourier Series This is another form of Fourier Series. It is also known as Compact form / Alternate

form / Phasor form Fourier series. Polar form is used to find the spectrums.

We know that sinθ = cos(θ - 90° ) and cosθ = sin(θ - 90° )

In Polar form, all functions are represented in terms of cosθ. This form is used to find

the magnitude and phase of various frequency components.

Trigonometric Fourier series of x(t) was given as -

x(t) = a0 + ∑ (an cos(nω0t) + bn sin(nω0t))∞n=1

This can also be written in polar form as below

x(t) = s0 + ∑ sn cos(nω0t + θn)∞n=1

Expanding the above equation, we get

x(t) = s0 + ∑ (sn cos θn cos(nω0t) − sn sin θn sin(nω0t))∞n=1

On comparing the trigonometric and polar form, we get

a0 = s0, an = sn cos θn , bn = −sn sin θn

Or s0 = a0, sn = √an2 + bn

2, θn = −tan−1 (bn

an)

Spectrum of Trigonometric Fourier Series x(t) = s0 + ∑ sn

∞n=1 cos(nω0t + θn)

= s0 + s1 cos(ω0t + θ1) + s2 cos(2ω0t + θ2) + s3 cos(3ω0t + θ3) + ⋯

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x(t) = s0 + s1∠θ1and ω = ω0 + s2∠θ2and ω = 2ω0 + s3∠θ3 and ω = 3ω0

Spectrum of TFS is one sided line spectrum or one sided discrete spectrum which is

defined at 0, ω0, 2ω0, 3ω0, ……………..

(i) Magnitude Spectrum

Magnitude spectrum of the signal is being constructed using the following terms

|s0 | → Magnitude associated with DC term. (ω=0)

|s1| → Magnitude associated with frequency ω0

|s2| → Magnitude associated with frequency 2ω0

The magnitude spectrum can be drawn as follows with the values calculated from

trigonometric Fourier series coefficients based on the formula given below.

s0 = a0, sn = √an2 + bn

2

(ii) Phase Spectrum

The phase spectrum of the Fourier series consists of the following values

ϕ0 → Phase associated with DC

ϕ1 → Phase associated with ω0

ϕn → Phase associated with nω0

The phase spectrum is drawn as below with the values calculated from the formula

given previously.

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ϕ0 = {0° ; for + ve DC

180° ; for − ve DC

ϕn = tan−1 (−bn

an)

(iii) Power Spectrum

We have the signal as x(t) = s0 + ∑ sn cos(nω0t + θn)∞n=1 .

The power spectrum can be calculated from the formulae given below and can be

drawn as following -

Ptotal = P0 + ∑ Pn∞n=1

P0 =s0

2

2, Pn =

sn2

2=

an2 +bn

2

2; n ≠ 0

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C. Exponential Fourier Series

The exponential Fourier series (EFS) is simpler and more compact. Hence this is

most widely used. Trigonometric Fourier series of x(t) is given as follows –

x(t) = a0 + ∑ (an cos(nω0t) + bn sin(nω0t))∞n=1

This can also be written in exponential form as below -

x(t) = a0 + ∑ (anejnw0t+e−jnw0t

2+ bn

ejnw0t−e−jnw0t

2j)∞

n=1

= a0 + ∑ (an−jbn

2) ejnw0t +∞

n=1 (an+jbn

2) e−jnw0t

= C0 + ∑ Cn∞n=1 ejnw0t + C−ne−jnw0t

x(t) = ∑ Cn ejnw0t∞n=−∞ → EFS and Cn are exponential Fourier series coefficients.

Where, C0 = a0;

Cn =an−jbn

2 and C−n =

an+jbn

2

x(t) = ∑ Cn∞n=−∞ ejnw0t

Where C0 =1

T∫ x(t)dt

T ; for n = 0

Cn =1

T∫ x(t)e−jnw0tdt ; for n ≠ 0

T.

The relation of coefficients of trigonometric and exponential Fourier

series are being given as follows- c0 = a0

cn =an−jbn

2

c−n =an+jbn

2

|

a0 = c0

an = cn + c−n

bn = (jcn − c−n)

cn = c−n∗

Since cn and c-n are complex. Hence exponential Fourier series is also known as

complex Fourier series (CFS) and coefficients are known as complex coefficients and

spectrum is complex spectrum.

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Spectrum of Exponential Fourier Series

For EFS, the signal has been represented as follows

x(t) = ∑ cn∞n=−∞ ejnw0t = c0 + c1ejnw0t + c2ejnw0t+. . +c−1e−jw0t + c−2e−j2w0t+..

Thus for Exponential Fourier series, spectrum of EFS is two sided line spectrum or

two sided discrete spectrum, which is defined at -

ω = 0, ±ω0, ±2ω0, ±3ω0, ….

There are three components to represent the spectrum. These are magnitude, phase

and power spectrum.

(i) Magnitude Spectrum

For magnitude spectrum, we have |c0| = |a0|, |cn| =1

2√an

2 + bn2 = |c−n| =

sn

2

Hence it can be drawn as

Therefore, magnitude spectrum is even function of ω.

(ii) Phase Spectrum

The Phase spectrum for this series is drawn from the following formulae

θ0 = {0° ; for + ve DC

180° ; for − ve DC

cn =an−jbn

2⇒ θn = tan−1 (

−bn

an) = − tan−1 (

bn

an)

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c−n =an+jbn

2⇒ θ−n = tan−1 (

bn

an)

θn = −θ−n

A phase spectrum will generally look like the one below

Phase spectrum of EFS is odd function of ω, i.e. anti-symmetric about ω = 0.

(iii) Power Spectrum

The power spectrum will be drawn through the following formulae

p0 = |c0|2

pn = |cn|2 =1

4(an

2 + bn2) =

sn2

4, p−n = |c−n|2 =

1

4(an

2 + bn2) =

sn2

4

pn = p−n

A typical Power Spectrum will be looking like the one below

Power spectrum is even function of ω. i. e. symmetric about ω = 0.

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Example 2:

Find the Fourier series of the following signal –

Solution:

x1(t) represents the periodic train of impulses with period T. Strength A and centered

about t=0.

x1(t) = {Aδ(t); for t = 0

0 ; for −T

2< t <

T

2 and t ≠ 0

and periodic with period T =2π

ω0

x1(t) = ∑ cnejnω0t∞n=−∞

c0 =1

T∫ x1(t) dt

T=

1

T∫ Aδ(t)

T2⁄

−T2⁄

=A

T

cn =1

T∫ x1(t)e−jnω0t

T dt =

1

T∫ Aδ(t)

T2⁄

−T2⁄

e−jnω0tdt

=A

T. e−jnω00 ∫ δ(t)dt =

A

T

T2⁄

T2⁄

cn =A

T

x1(t) =A

T∑ ejnω0nt∞

n=−∞ =A

T∑ ej

2π

Tnt∞

n=−∞ → Periodic train of impulses

Example 3:

Find the Fourier series of the following signal –

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Solution:

x2(t)represents periodic train of pulses with period T, height A, width τ and centered

about t = 0.

x2(t) = {A ; − τ

2⁄ < t < τ2⁄

0 ; − T2⁄ < t < − τ

2⁄ and τ2⁄ < t < T

2⁄

and periodic with period, T =2π

ω0

x2(t) = ∑ cnejnω0t∞n=−∞

c0 =1

T∫ x2(t)dt =

1

T∫ A dt

τ2⁄

−τ2⁄

=Aτ

TT

cn =1

T∫ x2(t)e−jnω0t dt =

1

T∫ Ae−jnω0t dt

τ2⁄

−τ2⁄T

=A

T.

1

−jnω0e−jnω0t|

−τ2⁄

τ2⁄

cn =A

T.

1

−jnω0(e−jnω0

τ2⁄ − ejnω0

τ2⁄ )

=A

T

2

−nω0

e−jnω0τ

2⁄ −ejnω0τ

2⁄

2j=

A

T .

2

nω0 . sin(nω0

τ2⁄ )

=A

T .2 × τ

2⁄ . sin(nω0

τ

2)

nω0τ

2

=Aτ

T .

sin(n×2π

T×

𝜏

2)

n×2π

T×

τ

2

= Aτ

T .

sin(nπτ

T)

nπτ

T

cn =Aτ

T . sinc (

nτ

T)

x2(t) = ∑Aτ

T .∞

n=−∞ sinc (nτ

T) . ejnω0t → Periodic train of pulses

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We will continue with the properties of Fourier series in the next article. Did you like this article on Introduction to Fourier Series? Let us know in the comments. You may also enjoy –

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