The Fourier Series

1

The Fourier series is a mathematical tool used for analyzing periodic functionsby decomposing such a function into a weighted sum of much simpler sinusoidalcomponent functions.

The Fourier series is named after the French scientist and mathematician Joseph Fourier, who used them in his influential work on heat conduction.

2

)2()(

22)()(

πωω

ππω

+=

==

+=

tgtg

fT

Ttgtg

A Periodic Function

Review: Periodic Functions

0 p 2p 3p … t

Review: Series

• Sequence

• Partial summation of a sequence

• Convergence / divergence

Trigonometric Series

• Form a basis – linearly independent

• Orthogonal

Fourier Series: p=2

• Fourier Series may not converge to f(x)

Fourier Series: p=2L

Even & Odd Functions

• Even function:

• Odd function:

Key facts

• Even

• Odd

• Product of even and odd is odd

Theorem 1

• Fourier cosine series

– f(x) is an even function

– p=2L

• Fourier sine series

– f(x) is an odd function

– p=2L

Theorem 2

• Fourier coefficients of f1+f2 are the sums of the

corresponding Fourier coefficients of f1 and f2

• Fourier coefficients if cf are c times the

corresponding Fourier coefficients of f

Half-range Expansions

0 L

3

E(-x) = E(x)

O(-x) = -O(x)

Even and Odd Functions

Before looking at further examples of Fourier series it is useful

to distinguish two classes of functions for which the Euler-

Fourier formulas for the coefficients can be simplified.

The two classes are even and odd functions, which are

characterized geometrically by the property of symmetry with

respect to the y-axis and the origin, respectively.

Definition of Even and Odd Functions

Analytically, f is an even function if its domain contains the

point –x whenever it contains x, and if f (-x) = f (x) for each x

in the domain of f. See figure (a) below.

The function f is an odd function if its domain contains the

point –x whenever it contains x, and if f (-x) = - f (x) for each x

in the domain of f. See figure (b) below.

Note that f (0) = 0 for an odd function.

Examples of even functions

are 1, x2, cos x, |x|.

Examples of odd functions

are x, x3, sin x.

Arithmetic Properties

The following arithmetic properties hold:

The sum (difference) of two even functions is even.

The product (quotient) of two even functions is even.

The sum (difference) of two odd functions is odd.

The product (quotient) of two odd functions is even.

The product (quotient) of an odd and an even function is odd.

These properties can be verified directly from the definitions,

see text for details.

Integral Properties

If f is an even function, then

If f is an odd function, then

These properties can be verified directly from the definitions,

see text for details.

Cosine Series

Suppose that f and f ' are piecewise continuous on [-L, L) and

that f is an even periodic function with period 2L.

Then f(x) cos(n x/L) is even and f(x) sin(n x/L) is odd. Thus

It follows that the Fourier series of f is

Thus the Fourier series of an even function consists only of the

cosine terms (and constant term), and is called a Fourier

cosine series.

Sine Series

Suppose that f and f ' are piecewise continuous on [-L, L) and

that f is an odd periodic function with period 2L.

Then f(x) cos(n x/L) is odd and f(x) sin(n x/L) is even. Thus

It follows that the Fourier series of f is

Thus the Fourier series of an odd function consists only of the

sine terms, and is called a Fourier sine series.

Example 1: Sawtooth Wave (1 of 3)

Consider the function below.

This function represents a sawtooth wave, and is periodic with

period T = 2L. See graph of f below.

Find the Fourier series representation for this function.

Example 1: Coefficients (2 of 3)

Since f is an odd periodic function with period 2L, we have

It follows that the Fourier series of f is

Example 1: Graph of Partial Sum (3 of 3)

The graphs of the partial sum s9(x) and f are given below.

Observe that f is discontinuous at x = ±(2n +1)L, and at these

points the series converges to the average of the left and right

limits (as given by Theorem 10.3.1), which is zero.

The Gibbs phenomenon again occurs near the discontinuities.

Even Extensions

It is often useful to expand in a Fourier series of period 2L a

function f originally defined only on [0, L], as follows.

Define a function g of period 2L so that

The function g is the even periodic extension of f. Its Fourier

series, which is a cosine series, represents f on [0, L].

For example, the even periodic extension of f (x) = x on [0, 2]

is the triangular wave g(x) given below.

Odd Extensions

As before, let f be a function defined only on (0, L).

Define a function h of period 2L so that

The function h is the odd periodic extension of f. Its Fourier

series, which is a sine series, represents f on (0, L).

For example, the odd periodic extension of f (x) = x on [0, L) is

the sawtooth wave h(x) given below.

General Extensions

As before, let f be a function defined only on [0, L].

Define a function k of period 2L so that

where m(x) is a function defined in any way consistent with

Theorem 10.3.1. For example, we may define m(x) = 0.

The Fourier series for k involves both sine and cosine terms,

and represents f on [0, L], regardless of how m(x) is defined.

Thus there are infinitely many such series, all of which

converge to f on [0, L].

Example 2

Consider the function below.

As indicated previously, we can represent f either by a cosine

series or a sine series on [0, 2]. Here, L = 2.

The cosine series for f converges to the even periodic

extension of f of period 4, and this graph is given below left.

The sine series for f converges to the odd periodic extension of

f of period 4, and this graph is given below right.

4

( )0

1,2,3,...( ) cos sin

2 n nn

ag t a n t b n tω ω∞

=

= + +∑

Fourier series expression of g(t)

5

∫∫

∫∫

∫∫

==

==

==

π

π

π

ωωωπ

ω

ωωωπ

ω

ωωπ

2

00

2

00

2

000

)(.sin)(1.sin)(2

)(.cos)(1.cos)(2

)()(1)(2

tdtntgdttntgT

b

tdtntgdttntgT

a

tdtgdttgT

a

T

n

T

n

T

Fourier Constants

A Fourier series is an expansion of a

periodic function f (t) in terms of an infinite sum

of cosines and sines

Introduction

In other words, any periodic function can be

resolved as a summation of

constant value and cosine and sine functions:

The computation and study of Fourier

series is known as harmonic analysis and

is extremely useful as a way to break up

an arbitrary periodic function into a set of

simple terms that can be plugged in,

solved individually, and then recombined

to obtain the solution to the original

problem or an approximation to it to

whatever accuracy is desired or practical.

=

+ +

+ + + …

Periodic Function f(t)

t

where

*we can also use the integrals limit .

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Example 1: Fourier Series sin( 0t)

The fundamental period of sin( 0t) is 0

By inspection we can write:

So a1 = 1/2j, a-1 = -1/2j and ak = 0 otherwise

The magnitude and angle of the Fourier coefficients are:

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Example 1a: Fourier Series sin( 0t)

The Fourier coefficients can also be explicitly evaluated

When k = +1 or –1, the integrals evaluate to T and –T,

respectively. Otherwise the coefficients are zero.

Therefore a1 = 1/2j, a-1 = -1/2j

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Example 2: Additive Sinusoids

Consider the additive sinusoidal series which has a fundamental

frequency 0:

Again, the signal can be directly written as:

The Fourier series coefficients can then be visualised as:

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Example 3: Periodic Step Signal

Consider the periodic square wave, illustrated by:

and is defined over one period as:

Fourier coefficients:

NB, these

coefficients

are real

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Example 3a: Periodic Step Signal

Instead of plotting both the magnitude and the angle of

the complex coefficients, we only need to plot the value

of the coefficients.

Note we have an infinite series of non-zero coefficients

T=4T1

T=8T1

T=16T1

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Convergence of Fourier Series

Not every periodic signal can be represented as an infinite Fourier series, however just about all interesting signals can be (note that the step signal is discontinuous)

The Dirichlet conditions are necessary and sufficient conditions on the signal.

Condition 1. Over any period, x(t) must be absolutely integrable

Condition 2. In any finite interval, x(t) is of bounded variation; that is there is no more than a finite number of maxima and minima during any single period of the signal

Condition 3. In any finite interval of time, there are only a finite number of discontinuities. Further, each of these discontinuities are finite.

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Fourier Series to Fourier Transform

For periodic signals, we can represent them as linear combinations of harmonically related complex exponentials

To extend this to non-periodic signals, we need to consider aperiodic signals as periodic signals with infinite period.

As the period becomes infinite, the corresponding frequency components form a continuum and the Fourier series sum becomes an integral (like the derivation of CT convolution)

Instead of looking at the coefficients a harmonically –related Fourier series, we’ll now look at the Fourier transform which is a complex valued function in the frequency domain

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Definition of the Fourier Transform

We will be referring to functions of time and their Fourier transforms. A signal x(t) and its Fourier transform X(j ) are related by the Fourier transform synthesis and analysis equations

and

We will refer to x(t) and X(j ) as a Fourier transform pair with the notation

As previously mentioned, the transform function X() can roughly be thought of as a continuum of the previous coefficients

A similar set of Dirichlet convergence conditions exist for the Fourier transform, as for the Fourier series (T=(- , ))

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Example 1: Decaying Exponential

Consider the (non-periodic) signal

Then the Fourier transform is:

a = 1

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Example 2: Single Rectangular Pulse

Consider the non-periodic rectangular pulse at zero

The Fourier transform is:

Note, the values are real

T1 = 1

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Example 3: Impulse Signal

The Fourier transform of the impulse signal can be

calculated as follows:

Therefore, the Fourier transform of the impulse function

has a constant contribution for all frequencies

X(j )

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Example 4: Periodic Signals

A periodic signal violates condition 1 of the Dirichlet conditions for the

Fourier transform to exist

However, lets consider a Fourier transform which is a single impulse of

area 2 at a particular (harmonic) frequency = 0.

The corresponding signal can be obtained by:

which is a (complex) sinusoidal signal of frequency 0. More generally,

when

Then the corresponding (periodic) signal is

The Fourier transform of a periodic signal is a train of impulses at the

harmonic frequencies with amplitude 2 ak

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Lecture 8: Summary

Fourier series and Fourier transform is used to represent

periodic and non-periodic signals in the frequency

domain, respectively.

Looking at signals in the Fourier domain allows us to

understand the frequency response of a system and also

to design systems with a particular frequency response,

such as filtering out high frequency signals.

You’ll need to complete the exercises to work out how to

calculate the Fourier transform (and its inverse) and

evaluate the frequency content of a signal

16/16

Properties of Discrete Fourier Series • Linearity

• Shift of a Sequence

• Duality

Symmetry Properties

Symmetry Properties Cont’d

Example 1

Determine the Fourier series representation of the

following waveform.

Solution

First, determine the period & describe the one period

of the function:

T = 2

Then, obtain the coefficients a0, an and bn:

Or, since

y = f(t) over the interval [a,b], hence

is the total area below graph

Notice that n is integer which leads ,

since

Therefore, .

Notice that

Therefore,

or

Finally,

Some helpful identities

For n integers,

[Supplementary]

The sum of the Fourier series terms can

evolve (progress) into the original

waveform

From Example 1, we obtain

It can be demonstrated that the sum will

lead to the square wave:

(a) (b)

(c) (d)

(e)

(f)

Example 2

Given

Sketch the graph of f (t) such that

Then compute the Fourier series expansion of f (t).

Solution

The function is described by the following graph:

T = 2

We find that

Then we compute the coefficients:

since

Finally,

Example 3

Given

Sketch the graph of v (t) such that

Then compute the Fourier series expansion of v (t).

Solution

The function is described by the following graph:

T = 4

We find that

0 2 4 6 8 10 12 t

v (t)

2

Then we compute the coefficients:

since

Finally,

Symmetry Considerations

Symmetry functions:

(i) even symmetry

(ii) odd symmetry

Even symmetry

Any function f (t) is even if its plot is

symmetrical about the vertical axis, i.e.

Even symmetry (cont.)

The examples of even functions are:

Even symmetry (cont.)

The integral of an even function from A to

+A is twice the integral from 0 to +A

A +A

Odd symmetry

Any function f (t) is odd if its plot is

antisymmetrical about the vertical axis, i.e.

Odd symmetry (cont.)

The examples of odd functions are:

Odd symmetry (cont.)

The integral of an odd function from A to

+A is zero

A +A

Even and odd functions

(even) (even) = (even)

(odd) (odd) = (even)

(even) (odd) = (odd)

(odd) (even) = (odd)

The product properties of even and odd

functions are:

Symmetry consideration

From the properties of even and odd

functions, we can show that:

for even periodic function;

for odd periodic function;

How?? [Even function]

(even) (even)

| |

(even)

(even) (odd)

| |

(odd)

How?? [Odd function]

(odd) (odd)

| |

(even)

(odd) (even)

| |

(odd)

(odd)

Example 4

Given

Sketch the graph of f (t) such that

Then compute the Fourier series expansion of f (t).

Solution

The function is described by the following graph:

T = 4

We find that

0 4 6 2 4 6 t

f (t)

2

1

1

Then we compute the coefficients. Since f (t) is

an odd function, then

and

since

Finally,

Example 5

Compute the Fourier series expansion of f (t).

Solution

The function is described by

T = 3

and

T = 3

Then we compute the coefficients.

Or, since f (t) is an even function, then

Or, simply

;

Finally,

and since f (t) is an even function.

Function defines over a finite interval

Fourier series only support periodic functions

In real application, many functions are non-

periodic

The non-periodic functions are often can be

defined over finite intervals, e.g.

y = 1 y = 1

y = 2

Therefore, any non-periodic function must be

extended to a periodic function first, before

computing its Fourier series representation

Normally, we prefer symmetry (even or odd)

periodic extension instead of normal periodic

extension, since symmetry function will provide

zero coefficient of either an or bn

This can provide a simpler Fourier series

expansion

T

T

T

Periodic extension

Even periodic extension

Odd periodic extension

Non-periodic

function

Half-range Fourier series expansion

The Fourier series of the even or odd

periodic extension of a non-periodic

function is called as the half-range Fourier

series

This is due to the non-periodic function is

considered as the half-range before it is

extended as an even or an odd function

If the function is extended as an even

function, then the coefficient bn= 0, hence

which only contains the cosine harmonics.

Therefore, this approach is called as the

half-range Fourier cosine series

If the function is extended as an odd

function, then the coefficient an= 0, hence

which only contains the sine harmonics.

Therefore, this approach is called as the

half-range Fourier sine series

Example 6

Compute the half-range Fourier sine series expansion

of f (t), where

Solution

Since we want to seek the half-range sine series,

the function to is extended to be an odd function:

T = 2

0 t

f (t)

1

1

2 2 0 t

f (t)

1

Hence, the coefficients are

and

Therefore,

Example 7

Determine the half-range cosine series expansion

of the function

Sketch the graphs of both f (t) and the periodic

function represented by the series expansion for

3 < t < 3.

Solution

Since we want to seek the half-range cosine series,

the function to is extended to be an even function:

T = 2

t

f (t)

t

f (t)

Hence, the coefficients are

Therefore,

Parseval’s Theorem

Parserval’s theorem states that the

average power in a periodic signal is equal

to the sum of the average power in its DC

component and the average powers in its

harmonics

=

+ +

+ + + …

f(t)

t

Pavg

Pdc

Pa1 Pb1

Pa2 Pb2

For sinusoidal (cosine or sine) signal,

For simplicity, we often assume R = 1 ,

which yields

For sinusoidal (cosine or sine) signal,

Exponential Fourier series

Recall that, from the Euler’s identity,

yields

and

Then the Fourier series representation becomes

Here, let we name ,

Hence, and .

c0 c n cn

Then, the coefficient cn can be derived from

In fact, in many cases, the complex

Fourier series is easier to obtain rather

than the trigonometrical Fourier series

In summary, the relationship between the

complex and trigonometrical Fourier series

are:

or

Example 8

Obtain the complex Fourier series of the following

function

Since , . Hence

Solution

since

Therefore, the complex Fourier series of f (t) is

*Notes: Even though c0 can be found by substituting

cn with n = 0, sometimes it doesn’t works (as shown

in the next example). Therefore, it is always better to

calculate c0 alone.

cn is a complex term, and it depends on n .

Therefore, we may plot a graph of |cn| vs n .

In other words, we have transformed the function

f (t) in the time domain (t), to the function cn in the

frequency domain (n ).

Example 9

Obtain the complex Fourier series of the function in

Example 1.

Solution

But

Thus,

Therefore,

*Here notice that .

The plot of |cn| vs n is shown below

0.5