Intro to Fourier

8/6/2019 Intro to Fourier

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Lecture 7: Basis Functions & Fourier Series

3. Basis functions (3 lectures): Concept of basisfunction. Fourier series representation of time

functions. Fourier transform and its properties.

Examples, transform of simple time functions.

Specific objectives for today:

Introduction to Fourier series (& transform)

Eigenfunctions of a system

Show sinusoidal signals are eigenfunctions of LTI

systems

Introduction to signals and basis functions

Fourier basis & coefficients of a periodic signal


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Lecture 7: Resources

Core materialSaS, O&W, C3.1-3.3

Background material

MIT Lecture 5

In this set of three lectures, were concerned withcontinuous time signals, Fourier series and Fourier

transforms only.


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Why is Fourier Theory Important (i)?

For a particular system, what signals Jk

(t) have the

property that:

Then Jk(t) is an eigenfunction with eigenvalue Pk

If an input signal can be decomposed asx(t) = 7kakJk(t)

Then the response of an LTI system isy(t) = 7kakPkJk(t)

For an LTI system, Jk(t) = estwhere sC, are

eigenfunctions.

Systemx(t) = Jk(t) y(t) = PkJk(t)


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Fourier transforms map a time-domain signal into a frequency

domain signalSimple interpretation of the frequency content of signals in the

frequency domain (as opposed to time).

Design systems to filter out high or low frequency components.

Analyse systems in frequency domain.

Why is Fourier Theory Important (ii)?

Invariant to

high

frequency

signals


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Why is Fourier Theory Important (iii)?

If F{x(t)} =X(j[) [ is the frequency

Then F{x(t)} =j[X(j[)

So solving a differential equation is transformed from a

calculus operation in the time domain into an algebraic

operation in the frequency domain (see Laplace transform)

Example

becomes

and is solved for the roots [ (N.B. complementary equations):

and we take the inverse Fourier transform for those [.

0322

2

! ydt

dy

dt

yd

0322 ! j

0)(3)(2)(2 ! jYjYjjY


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Introduction to System Eigenfunctions

Lets imagine what (basis) signals Jk(t) have the property that:

i.e. the output signal is the same as the input signal, multiplied by the

constant gain Pk (which may be complex)

For CT LTI systems, we also have that

Therefore, to make use of this theory we need:1) system identification is determined by finding {Jk,Pk}.

2) response, we also have to decomposex(t) in terms ofJk(t) bycalculating the coefficients {ak}.

This is analogous to eigenvectors/eigenvalues matrix decomposition

Systemx(t) = Jk(t) y(t) = PkJk(t)

LTI

System

x(t) = 7kakJk(t) y(t) = 7kakPkJk(t)


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Complex Exponentials are Eigenfunctions

of any CT LTI System

Consider a CT LTI system with impulse response h(t) andinput signalx(t)=J(t) = est, for any value ofsC:

Assuming that the integral on the right hand side converges

to H(s), this becomes (for any value ofsC):

Therefore J(t)=est is an eigenfunction, with eigenvalue P=H(s)

g

g

g

g

g

g

g

g

!

!

!

!

XX

XX

XX

XXX

X

X

X

dehe

deeh

deh

dtxhty

sst

sst

ts

)(

)(

)(

)()()(

)(

stesHty )()( !

g

g

! XX Xdehs s)()(


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Example 1: Time Delay & Imaginary Input

Consider a CT, LTI system where the input and output are related by

a pure time shift:

Consider a purely imaginary input signal:

Then the response is:

ej2t is an eigenfunction (as wed expect) and the associated

eigenvalue is H(j2) = e-j6.

The eigenvalue could be derived more generally. The systemimpulse response is h(t) = H(t-3), therefore:

So H(j2) = e-j6!

)3()( ! txty

tjetx 2)( !

tjjtjeeety 26)3(2)(

!!

ssedes 3)3()(

g

g

!! XXHX


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Example 1a: Phase Shift

Note that the corresponding input e-j2thas eigenvalue ej6, solets consider an input cosine signal of frequency 2 so that:

By the system LTI, eigenfunction property, the system outputis written as:

So because the eigenvalue is purely imaginary, thiscorresponds to a phase shift (time delay) in the systemsresponse. If the eigenvalue had a real component, thiswould correspond to an amplitude variation

tjtj eet 2221)2cos( !

))3(2cos(

)(

)62()62(

21

262621

!

!

!

t

ee

eeeety

tjtj

tjjtjj


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Example 2: Time Delay & Superposition

Consider the same system (3 time delays) and now consider the inputsignalx(t) = cos(4t)+cos(7t), a superposition of two sinusoidal signals

that are not harmonically related. The response is obviously:

Considerx(t) represented using Eulers formula:

Then due to the superposition property and H(s) =e-3s

While the answer for this simple system can be directly spotted, the

superposition property allows us to apply the eigenfunction concept to

more complex LTI systems.

))3(7cos())3(4cos()( ! ttty

tjtjtjtj eeeetx 7217

214

214

21)( !

))3(7cos())3(4cos(

)(

)3(7

21)3(7

21)3(4

21)3(4

21

721

21721

21412

21412

21

!

!

!

tt

eeee

eeeeeeeety

tjtjtjtj

tjjtjjtjjtjj


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History of Fourier/Harmonic Series

The idea of using trigonometric sums

was used to predict astronomicalevents

Euler studied vibrating strings, ~1750,

which are signals where linear

displacement was preserved with

time.Fourier described how such a series

could be applied and showed that

a periodic signals can be

represented as the integrals of

sinusoids that are not allharmonically related

Now widely used to understand the

structure and frequency content of

arbitrary signals

[=1

[=2

[=3

[=4


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Fourier Series and Fourier Basis Functions

The theory derived for LTI convolution, used the concept that any

input signal can represented as a linear combination of shiftedimpulses (for either DT or CT signals)

We will now look at how (input) signals can be represented as a

linear combination ofFourier basis functions (LTI

eigenfunctions) which are purely imaginary exponentials

These are known as continuous-time Fourier series

The bases are scaled and shifted sinusoidal signals, which can

be represented as complex exponentials

x(t) = sin(t) +

0.2cos(2t) +

0.1sin(5t)

x(t)ej[t


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Periodic Signals & Fourier Series

A periodic signal has the propertyx(t) =x(t+T), Tis the

fundamental period, [0 = 2T/Tis the fundamental frequency.Two periodic signals include:

For each periodic signal, the Fourier basis the set of

harmonically related complex exponentials:

Thus the Fourier series is of the form:

k=0 is a constant

k=+/-1 are the fundamental/first harmonic components

k=+/-Nare the Nth harmonic components

For a particular signal, are the values of{ak}k?

tjetx

ttx

0)(

)cos()( 0[

[

!

!

,...2,1,0)( )/2(0 ss!!! keet tTjktjkk

T[J

g

g!

g

g!

!!k

tTjk

k

k

tjk

keaeatx )/2(0)(

T[


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Fourier Series Representation of a CT

Periodic Signal (i)

Given that a signal has a Fourier series representation, we have tofind {ak}k. Multiplying through by

Using Eulers formula for the complex exponential integral

It can be shown that

tjne 0

[

g

g!

g

g!

g

g!

!

!

!

k

Ttnkj

k

T

k

tnkj

k

Ttjn

k

tjntjk

k

tjn

dtea

dt

eadt

etx

eeaetx

0

)(

0

)(

0

0

00

000

)(

)(

[

[[

[[[

Tis the fundamental

period ofx(t)

!

TTT

tnk

j dttnkjdttnkdte0

00

00

)( ))sin(())cos((0 [[[

{

!!

nk

nkTdte

Ttnkj

00)( 0[


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Fourier Series Representation of a CT

Periodic Signal (ii)

Therefore

which allows us to determine the coefficients. Also note that this

result is the same if we integrate over any interval of length T

(not just [0,T]), denoted by

To summarise, if x(t) has a Fourier series representation, then the

pair of equations that defines the Fourier series of a periodic,

continuous-time signal:

!

Ttjn

Tndtetxa

0

1 0)([

T

!

T

tjn

Tndtetxa 0)(1

[

g

g!

g

g!

!!

!!

T

tTjk

TT

tjk

Tk

k

tTjk

k

k

tjk

k

dtetxdtetxa

eaeatx

)/2(11

)/2(

)()(

)(

0

0

T[

T[


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Lecture 7: Summary

Fourier bases, series and transforms are extremely useful for

frequency domain analysis, solving differential equations andanalysing invariance for LTI signals/systems

For an LTI system

est is an eigenfunction

H(s) is the corresponding (complex) eigenvalueThis can be used, like convolution, to calculate the output of an

LTI system once H(s) is known.

A Fourier basis is a set of harmonically related complexexponentials

Any periodic signal can be represented as an infinite sum(Fourier series) of Fourier bases, where the first harmonic isequal to the fundamental frequency

The corresponding coefficients can be evaluated


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Lecture 7: Exercises

SaS, O&W, Q3.1-3.5

Intro to Fourier

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