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8/6/2019 Intro to Fourier
1/17
EE-2027 SaS, L7 1/16
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
8/6/2019 Intro to Fourier
2/17
EE-2027 SaS, L7 2/16
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.
8/6/2019 Intro to Fourier
3/17
EE-2027 SaS, L7 3/16
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)
8/6/2019 Intro to Fourier
4/17
EE-2027 SaS, L7 4/16
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
8/6/2019 Intro to Fourier
5/17
EE-2027 SaS, L7 5/16
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
8/6/2019 Intro to Fourier
6/17
EE-2027 SaS, L7 6/16
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)
8/6/2019 Intro to Fourier
7/17
EE-2027 SaS, L7 7/16
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
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8/6/2019 Intro to Fourier
8/17
EE-2027 SaS, L7 8/16
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
8/6/2019 Intro to Fourier
9/17
EE-2027 SaS, L7 9/16
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(
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)62()62(
21
262621
!
!
!
t
ee
eeeety
tjtj
tjjtjj
8/6/2019 Intro to Fourier
10/17
EE-2027 SaS, L7 10/16
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
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21)3(4
21)3(4
21
721
21721
21412
21412
21
!
!
!
tt
eeee
eeeeeeeety
tjtjtjtj
tjjtjjtjjtjj
8/6/2019 Intro to Fourier
11/17
EE-2027 SaS, L7 11/16
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
8/6/2019 Intro to Fourier
12/17
EE-2027 SaS, L7 12/16
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
8/6/2019 Intro to Fourier
13/17
EE-2027 SaS, L7 13/16
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
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!!k
tTjk
k
k
tjk
keaeatx )/2(0)(
T[
8/6/2019 Intro to Fourier
14/17
EE-2027 SaS, L7 14/16
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
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)(
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[
[[
[[[
Tis the fundamental
period ofx(t)
!
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tnk
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00
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)( ))sin(())cos((0 [[[
{
!!
nk
nkTdte
Ttnkj
00)( 0[
8/6/2019 Intro to Fourier
15/17
EE-2027 SaS, L7 15/16
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[
8/6/2019 Intro to Fourier
16/17
EE-2027 SaS, L7 16/16
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
8/6/2019 Intro to Fourier
17/17
EE-2027 SaS, L7 17/16
Lecture 7: Exercises
SaS, O&W, Q3.1-3.5