Digital Signal Processing Digital Signal Processing

ERASMUS Teaching (2008), Technische Universität Berlin
Digital Signal ProcessingDigital Signal Processing
Assoc. Prof. Dr. Pelin Gundes Bakir
gundesbakir@yahoo.com
References
ERASMUS Teaching (2008), Technische Universität Berlin
References
•
MitraSK
‘
DigitalSignalProcessing
’
AComputerbasedapproachMc
GrawHill
•
Mitra

S
.
K
.,
Digital

Signal

Processing
,
A

Computer

based

approach
,
Mc
-
Graw

Hill
,
3rd Edition, 2006.
•Heylen W., Lammens S. And Sas P., ‘Modal Analysis Theory and Testing’, Katholieke
UniversiteitLeuven1997
Universiteit

Leuven
,
1997
.
•Keith Worden ‘Signal Processing and Instrumentation’, Lecture Notes,
http://www.dynamics.group.shef.ac.uk/people/keith/mec409.htm
•"The Scientist and Engineer's Guide to Digital Signal Processing, copyright ©1997-
1998 by Steven W. Smith. For more information visit the book's website at:
www.DSPguide.com"
•Boore, D. M. and J. J. Bommer (2005). Processing of strong-motion accelerograms:
Needs, options and consequences, Soil Dynamics and Earthquake Engineering
25,93
-
115
•Boore, D. M. (2005). On pads and filters: Processing strong-motion data, Bull. Seism.
Soc. Am. 95,745-750.
P. Gundes Bakir, Vibration based structural health monitoring
2
ERASMUS Teaching (2008), Technische Universität Berlin
SignalsSignals
DefinitionDefinition
ERASMUS Teaching (2008), Technische Universität Berlin
Definition
Definition
•A signal is a function of independent variables such as
time, distance, position, temperature and pressure.
•A signal carries information, and the objective of signal
i
itttfliftiidbth
process
i
ng
i
s
t
o ex
t
rac
t
use
f
u
l

i
n
f
orma
ti
on carr
i
e
d

b
y
th
e
signal.
•Signal processingis concerned with the mathematical
re
p
resentation of the si
g
nal and the al
g
orithmic o
p
eration
pggp
carried out on it to extract the information present.
P. Gundes Bakir, Vibration based structural health monitoring
4
DefinitionDefinition
ERASMUS Teaching (2008), Technische Universität Berlin
Definition
Definition
•For most purposes of description and analysis, a signal can be defined
simply as a mathematical function,
)
(
x
f
y
where xis the independent variable which specifies the domain of the
il
)
(
x
f
y
=
s
i
杮g
l
⁥⹧⸺
•y=sin(
ω
t)is a function of a variable in the time domainand is thus a time
si
g
nal
;
g;
•X(
ω
)=1/(-m
ω
2+ic
ω
+k) is a frequency domain signal;
•An image I(x,y) is in the spatial domain.
P. Gundes Bakir, Vibration based structural health monitoring
5
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
Fildlhbidfiiii
•
F
or a s
i
mp
l
e pen
d
u
l
um as s
h
own,
b
as
i
c
d
e
fi
n
i
t
i
on
i
s:
where
θ
m
is the peak amplitude of the motion
and ω=√l/gwith lthe length of the pendulum
and gthe acceleration due to gravity.
•As the system has a constant amplitude (we assume no damping for
now
),
a constant fre
q
uenc
y

(
dictated b
y

p
h
y
sics
)
and an initial
),qy(ypy)
condition (
θ
=0 when t=0), we know the value of
θ
(t) for all time.
.
P. Gundes Bakir, Vibration based structural health monitoring
6
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
Altidtildlldf
θ
θ
t
t
0illhth
•
Al
so,
t
wo
id
en
ti
ca
l
pen
d
u
l
a re
l
ease
d

f
rom
θ
㴠
θ
0
a
t

t
=
0
, w
ill

h
ave
th
e same
motions at all time. There is no place for uncertainty here.
•If we can uniquely specify the value of
θ
for all time, i.e.,we know the
underlying functional relationship between tand
θ
, the motion is
deterministic
orpredictableInotherwordsa
signalthatcanbeuniquely
deterministic
or

predictable
.
In

other

words
,
a
signal

that

can

be

uniquely

determined by a well defined process such as a mathematical expression or
rule is called a deterministic signal.
•The opposite situation occurs if we know all the physics there is to know,
bu
t
s
till
c
ann
o
t
s
a
y

w
hat th
e

s
i
g
nal
w
ill
be
at th
e
n
e
xt tim
e
in
s
tant-th
e
n th
e

buscosywesgwbeeees
ee
signal is randomor probabilistic. In other words,asignal that is generated
in a random fashion and can not be predicted ahead of time is called a
randomsignal
P. Gundes Bakir, Vibration based structural health monitoring
7
random

signal
.
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
•Typical examples to deterministic signals are sine chirp and digital stepped
sine.
P. Gundes Bakir, Vibration based structural health monitoring
8
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
•Typical examples to random signals are random and burst random.
P. Gundes Bakir, Vibration based structural health monitoring
9
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
•Random signals are characterized by having many frequency components
present over a wide range of frequencies.
•The am
p
litude versus time a
pp
ears to var
y
ra
p
idl
y
and unsteadil
y
with
pppypyy
time.
•The ‘shhhh’ sound is a good example that is rather easy to observe using a
microphoneandoscillloscopeIfthesoundintensityisconstantwithtime
microphone

and

oscillloscope
.
If

the

sound

intensity

is

constant

with

time
,
the random signal is stationary, while if the sound intensity varies with time
the signal is nonstationary. One can easily see and hear this variation while
kih‘hhhh’d
ma
ki
ng t
h
e
‘
s
hhhh’
soun
d
.
P. Gundes Bakir, Vibration based structural health monitoring
10
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
•Random signals are characterized by analyzing the statistical
characteristics across an ensemble of records. Then, if the process is
ergodic, the time (temporal) statistical characteristics are the same as the
ensemble statistical characteristics. The word temporal means that a time
average definition is used in place of an ensemble statistical definition.
P. Gundes Bakir, Vibration based structural health monitoring
11
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
•Transient signalsmay be defined as signals that exist for a finite
range of time as shown in the figure. Typical examples are hammer
excitationofsystemsexplosionandshockloadingetc
excitation

of

systems
,
explosion

and

shock

loading

etc
.
P. Gundes Bakir, Vibration based structural health monitoring
12
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
Ailihiii
idi
il
•
A
s
i
gna
l
w
i
t
h
a t
i
me vary
i
ng mean
i
s an aper
i
o
di
cs
i
gna
l
.
P. Gundes Bakir, Vibration based structural health monitoring
13
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
Ihldbdhidiidil
•
I
t s
h
ou
ld

b
e note
d
t
h
at per
i
o
di
c
i
ty
d
oes not necessar
il
y mean a
sinusoidal signal as shown in the figure.
•For a simple pendulum as shown, if we define the period τby
thenforthependulum
,
then

for

the

pendulum
,
and such signals are defined as periodic.
P. Gundes Bakir, Vibration based structural health monitoring
14
SignaltypesSignaltypes
ERASMUS Teaching (2008), Technische Universität Berlin
Signal

typesSignal

types
Aidiilihilfiidi
•
A
per
i
o
di
c s
i
gna
l

i
s one t
h
at repeats
i
tse
lf

i
n t
i
me an
d

i
s a
reasonable model for many real processes, especially those
associated with constant speed machinery.
•Stationary signalsare those whose average properties do not
changewithtime
Stationarysignals
haveconstantparametersto
change

with

time
.
Stationary

signals

have

constant

parameters

to

describe their behaviour.
•Nonstationary signals have time dependent parameters. In an
engine excited vibration where the engines speed varies with time;
thefundamentalperiodchangeswithtimeaswellaswiththe
the

fundamental

period

changes

with

time

as

well

as

with

the

corresponding dynamic loads that cause vibration.
P. Gundes Bakir, Vibration based structural health monitoring
15
DeterministicvsRandomsignalsDeterministicvsRandomsignals
ERASMUS Teaching (2008), Technische Universität Berlin
Deterministic

vs

Random

signalsDeterministic

vs

Random

signals
Thesignalscanbefurtherclassifiedas
monofrequency
•
The

signals

can

be

further

classified

as

monofrequency

(sinusoidal) signals and multifrequencysignals such as
the square wave which has a functional form made up of
ifiititifdifftiith
an
i
n
fi
n
it
e superpos
iti
on o
f

diff
eren
t
s
i
ne waves w
ith

periods τ,τ/2,τ/3,…
•1 D signalsare a function of a single independent
variable. The speech signal is an example of a 1 D
signalwheretheindependentvariableistime
signal

where

the

independent

variable

is

time
.
•2D si
g
nalsare a function of two inde
p
endent variables.
g
p
A
n image signal such as a photograph is an example of
a 2D signal where the two independent variables are the
two s
p
atial variables.
P. Gundes Bakir, Vibration based structural health monitoring
16
p
ClassificationofsignalsClassificationofsignals
ERASMUS Teaching (2008), Technische Universität Berlin
Classification

of

signalsClassification

of

signals
•
Thevalueofasignalataspecificvalueoftheindependentvariable
•
The

value

of

a

signal

at

a

specific

value

of

the

independent

variable

is called its amplitude.
•
Thevariationoftheamplitudeasafunctionoftheindependent
•
The

variation

of

the

amplitude

as

a

function

of

the

independent

variable is called its waveform.
•
Fora1Dsignaltheindependentvariableisusuallylabelledas
•
For

a

1

D

signal
,
the

independent

variable

is

usually

labelled

as

time. If the independent variable is continuous, the signal is called a
continuous-time signal. A continuous time signal is defined at
ever
y
instant of time.
y
•If the independent variable is discrete, the signal is called a
discrete-time si
g
nal. A discrete time si
g
nal takes certain numerical
g
g
values at specified discrete instants of time, and between these
specified instants of time, the signal is not defined. Hence, a discrete
time signal is basically a sequence of numbers.
P. Gundes Bakir, Vibration based structural health monitoring
17
ClassificationofsignalsClassificationofsignals
ERASMUS Teaching (2008), Technische Universität Berlin
Classification

of

signalsClassification

of

signals
•A continuous-time signal with a
continuous amplitude is usually
calledan
analogsignal
A
called

an

analog

signal
.
A

speech signal is an example of
an analog signal.
•A discrete time signal with
discrete valued amplitudes
re
p
resented b
y
a finite number
py
of digits is referred to as a
digital signal.
P. Gundes Bakir, Vibration based structural health monitoring
18
ClassificationofsignalsClassificationofsignals
ERASMUS Teaching (2008), Technische Universität Berlin
Classification

of

signalsClassification

of

signals
Adittiilith
•
A

di
scre
t
e
ti
me s
i
gna
l
w
ith

continuous valued amplitudes is
called a sampled-data signal. A
digitalsignalisthsaqantied
digital

signal

is

th
u
s

a

q
u
anti
z
ed

sampled-data signal.
•A continuous-time signal with
discrete valued amplitudes has
been referred to as a quantized
boxcar signal. This type of signal
occurs in digital electronic circuits
where the signal is kept at fixed
level (usually one of two values)
between two instants of clocking.
P. Gundes Bakir, Vibration based structural health monitoring
19
CLASSIFICATIONSOFSIGNALSCLASSIFICATIONSOFSIGNALS
ERASMUS Teaching (2008), Technische Universität Berlin
CLASSIFICATIONS

OF

SIGNALSCLASSIFICATIONS

OF

SIGNALS
Stationary
Non-Stationary
1 D
signals
1 D
signals
2 D
signals
Dtiiti
Random
D
e
t
erm
i
n
i
s
ti
c
Transient
Continuous
Periodic
Aperiodic
Monofrequency
Multi
-
Tit
Ifiitidi
P. Gundes Bakir, Vibration based structural health monitoring
20
Monofrequency
(sinuzoidal)
Multi
frequency
T
rans
i
en
t
I
n
fi
n
it
e aper
i
o
di
c
ERASMUS Teaching (2008), Technische Universität Berlin
TypicalSignalProcessingTypicalSignalProcessing
Typical

Signal

Processing

Typical

Signal

Processing

O
p
erationsO
p
erations
p
p
TypicalsignalprocessingTypicalsignalprocessing
ERASMUS Teaching (2008), Technische Universität Berlin
Typical

signal

processing

Typical

signal

processing

o
p
erationso
p
erations
p
p
•In the case of analog signals, most signal processing
o
p
erations are usuall
y
carried out in the time domain.
py
•In the case of discrete time si
g
nals, both time domain
g
and frequency domainapplications are employed.
Iihhdidiild
•
I
n e
i
t
h
er case, t
h
e
d
es
i
re
d
operat
i
ons are
i
mp
l
emente
d

by a combination of some elementary operationssuch
as:
as:
–Simple time domain operations
–Filtering
Alitddlti
P. Gundes Bakir, Vibration based structural health monitoring
22
–
A
mp
lit
u
d
e mo
d
u
l
a
ti
on
SimpleTimeDomainOperationsSimpleTimeDomainOperations
ERASMUS Teaching (2008), Technische Universität Berlin
Simple

Time

Domain

OperationsSimple

Time

Domain

Operations
The three most basic time-domain signal operations are:
•Scaling
•
Delay
Delay
•Addition

Scaling
Scalingis simply the multiplication of a signal by a positive or a negative
constant. In the case of analog signals, this operation is usually called
amplification
ifthemagnitudeofthemultiplyingconstantcalled
gain
is
amplification
if

the

magnitude

of

the

multiplying

constant
,
called

gain
,
is

greater than one. If the magnitude of the multiplying constant is less than
one, the operation is called attenuation.Thus, if x(t) is an analog signal,
thescalingoperationgeneratesasignal
y(t)=
α
x(t)
where
α
楳瑨≥
瑨≥
=
獣慬楮s
=
潰敲慴楯→
=
来湥牡瑥g
=
a
=
獩杮慬
=
y(t)=
α
x(t)
,
where

α
楳
=
瑨≥
=
浵汴楰汹楮朠捯湳瑡湴⸠
P. Gundes Bakir, Vibration based structural health monitoring
23
SimpleTimeDomainOperationsSimpleTimeDomainOperations
ERASMUS Teaching (2008), Technische Universität Berlin
Simple

Time

Domain

OperationsSimple

Time

Domain

Operations
The three most basic time-domain signal operations are:
•Scaling
•Delay
•Addition

Delay
Delay operation generates a signal that is delayed replica of
the original signal.For an analog signal
x
(t), y(t)=x(
t
-
t
0) is the
signal obtained by delaying x(t)by the amount t0, which is
assumedtobeapositivenumberIf
t
isnegativethenitisan
assumed

to

be

a

positive

number
.
If

t
0
is

negative
,
then

it

is

an

advanceoperation.
P. Gundes Bakir, Vibration based structural health monitoring
24
SimpleTimeDomainOperationsSimpleTimeDomainOperations
ERASMUS Teaching (2008), Technische Universität Berlin
Simple

Time

Domain

OperationsSimple

Time

Domain

Operations
The three most basic time-domain signal operations are:
•Scaling
•Delay
••Addition
Addition

Addition
Addition operation generates a new signal by the addition of
signals. For instance, y(t)=x1(t)+x2(t)-
x
3(t) is the signal
generated by the addition of the three analog signals x1(t), x
2(t)
and
x
(t)
and

x
3
(t)
.
P. Gundes Bakir, Vibration based structural health monitoring
25
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier transformsFourier transforms
This chapter focuses on Fourier-series expansion, the
discrete Fourier transform, properties of Fourier
TransformsandFastFourierTransform
Transforms

and

Fast

Fourier

Transform
FouriertransformsFouriertransforms
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

transformsFourier

transforms
Forieranalsisisafamilofmathematicaltechniqesallbasedon
•
Fo
u
rier

anal
y
sis

is

a

famil
y
of

mathematical

techniq
u
es
,
all

based

on

decomposing signals into sinusoids.
•
ThediscreteFouriertransform(DFT)isthefamilymemberusedwith
•
The

discrete

Fourier

transform

(DFT)

is

the

family

member

used

with

digitized signals.
•
Whyaresinusoidsused?Asinusoidalinputtoasystemisguaranteedto
•
Why

are

sinusoids

used?

A

sinusoidal

input

to

a

system

is

guaranteed

to

produce a sinusoidal output. Only the amplitude and phase of the signal can
change; the frequency and wave shape must remain the same. Sinusoids are
the onl
y
waveform that have this useful
p
ro
p
ert
y
.
yppy
•The general term Fourier transform can be broken into four categories,
resultin
g
from the four basic t
yp
es of si
g
nals that can be encountered.
gypg
P. Gundes Bakir, Vibration based structural health monitoring
27
CategoriesofFourierCategoriesofFourier
ERASMUS Teaching (2008), Technische Universität Berlin
Categories

of

Fourier

Categories

of

Fourier

TransformsTransforms
P. Gundes Bakir, Vibration based structural health monitoring
28
FouriertransformsFouriertransforms
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

transformsFourier

transforms
Thflfillltdtitidti
ifiit
Whtif
•
Th
ese
f
our c
l
asses o
f
s
i
gna
l
s a
ll
ex
t
en
d

t
o pos
iti
ve an
d
nega
ti
ve
i
n
fi
n
it
y.
Wh
a
t

if

you only have a finite number of samples stored in your computer, say a signal
formed from 1024 points?
•There isn’t a version of the Fourier transform that uses finite length signals. Sine
and cosine waves are defined as extending from negative infinity to positive
infinity.Youcannotuseagroupofinfinitelylongsignalstosynthesizesomething
infinity.

You

cannot

use

a

group

of

infinitely

long

signals

to

synthesize

something

finite in length. The way around this dilemma is to make the finite data look likean
infinite length signal. This is done by imagining that the signal has an infinite
number of samples on the left and right of the actual points. If all these “imagined”
lhlfthillk
di
d
idi
dth
samp
l
es
h
ave a va
l
ue o
f
zero,
th
e s
i
gna
l

l
oo
k
s
di
screte an
d
aper
i
o
di
c, an
d

th
e
discrete time Fourier transform applies.
Alttithiidlbdlitifthtl1024
•
A
s an a
lt
erna
ti
ve,
th
e
i
mag
i
ne
d
samp
l
es can
b
e a
d
up
li
ca
ti
on o
f

th
e ac
t
ua
l

1024

points. In this case, the signal looks discrete and periodic, with a period of 1024
samples. This calls for the discrete Fourier transform to be used.
P. Gundes Bakir, Vibration based structural health monitoring
29
TimeandfrequencydomainsTimeandfrequencydomains
ERASMUS Teaching (2008), Technische Universität Berlin
Time

and

frequency

domainsTime

and

frequency

domains
•As shown in the figure, the Discrete Fourier transformchanges an N point
in
p
ut si
g
nal into two N/2 + 1
p
oint out
p
ut si
g
nals. The in
p
ut si
g
nal contains the
pg
ppgpg
signal being decomposed, while the two output signals contain the
amplitudes of the component sine and cosine waves. The input signal is said
to be in the time domain. This is because the most common type of signal
enteringtheDFTiscomposedofsamplestakenatregularintervalsof
time.
entering

the

DFT

is

composed

of

samples

taken

at

regular

intervals

of

time.

The term “time domain” in Fourier analysis, may actually refer to samples
taken over time. The term frequency domain is used to describe the
amplitudes of the sine and cosine waves.
P. Gundes Bakir, Vibration based structural health monitoring
30
TimeandfrequencydomainsTimeandfrequencydomains
ERASMUS Teaching (2008), Technische Universität Berlin
Time

and

frequency

domainsTime

and

frequency

domains
•
The
frequencydomain
containsexactlythesameinformationasthe
timedomain
,
The

frequency

domain
contains

exactly

the

same

information

as

the

time

domain
,

just in a different form. If you know one domain, you can calculate the other.
•
Giventhetimedomainsignal,theprocessofcalculatingthefrequencydomainis
Given

the

time

domain

signal,

the

process

of

calculating

the

frequency

domain

is

called decomposition, analysis, the forward DFT, or simply, theDFT.
•If
y
ou know the fre
q
uenc
y
domain
,
calculation of the time domain is called
yqy,
synthesis, or the inverse DFT. Both synthesis and analysis can be represented in
equation form and computer algorithms.
•The number of samples in the time domain is usually represented by the variable N.
While N can be any positive integer, a power of two is usually chosen, i.e., 128,
256, 512, 1024, etc. There are two reasons for this. First, digital data storage uses
biddikifttlillthSdtht
bi
nary a
dd
ress
i
ng, ma
ki
ng powers o
f

t
wo a na
t
ura
l
s
i
gna
l

l
eng
th
.
S
econ
d
,
th
e mos
t

efficient algorithm for calculating the DFT, the Fast Fourier Transform (FFT),
usually operates with N that is a power of two. Typically, N is selected between 32
and4096.Inmostcases,thesamplesrunfrom0to
N
-
1,ratherthan1to
N.
P. Gundes Bakir, Vibration based structural health monitoring
31
and

4096.

In

most

cases,

the

samples

run

from

0

to

N
1

,

rather

than

1

to

N.
TimeandfrequencydomainsTimeandfrequencydomains
ERASMUS Teaching (2008), Technische Universität Berlin
Time

and

frequency

domainsTime

and

frequency

domains
•
Lowercaselettersrepresenttime
•
Lower

case

letters

represent

time

domain signals and upper case
letters represent frequency domain
signals.
•The figure shows an example DFT
with N = 128. The time domain signal
iscontainedinthearray:
x
[0]
to
x
is

contained

in

the

array:

x

[0]
to

x

[127].Notice that 128 points in the
time domain corresponds to 65
points in each of the frequency
domainsignalswiththefrequency
domain

signals
,
with

the

frequency

indexes running from 0 to 64.
•That is, N
p
oints in the time domain
p
corresponds to N/2 + 1 points in the
frequency domain (not N/2 points).
Forgetting about this extra point is a
common bu
g
in DFT
p
ro
g
rams.
P. Gundes Bakir, Vibration based structural health monitoring
32
gpg
TimeandfrequencydomainsTimeandfrequencydomains
ERASMUS Teaching (2008), Technische Universität Berlin
Time

and

frequency

domainsTime

and

frequency

domains
P. Gundes Bakir, Vibration based structural health monitoring
33
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•Fourier series are infinite series designed to represent general periodic
functions in terms of simple ones, namely cosines and sines.
•A function f(x) is called a periodic function if f(x) is defined for all real x and if
there is a positive number p, called a period of f(x), such that
•The graph of such a function is obtained by periodic repetition of its graph in
an
y
interval of len
g
th
p
.
)()(xfpxf
=
+
ygp
P. Gundes Bakir, Vibration based structural health monitoring
34
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•Familiar periodic functions are the cosine and sine functions. Examples of
functions that are not periodic are:
•If f(x) has period p, it also has the period 2p because the equation
xxexxx
x
ln,cosh,,,,
32
impliesthat
)()(xfpxf
=
+
implies

that
thus for any integer n=1,2,3,...
[
]
(
)
)()()2(xfpxfppxfpxf
=
+
=
+
+
=
+
)()(xfnpxf
=
+
P. Gundes Bakir, Vibration based structural health monitoring
35
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•Furthermore if f(x) and g(x) have period p, then af(x)+bg(x) with any constants
a and b also has the period p.Our problem in the first few slides will be the
representationofvariousfunctionsf(x)ofperiod2
π
楮瑥牭≥潦瑨≥s業灬i
牥灲敳敮瑡瑩潮
=
潦
=
癡物潵v
=
晵湣瑩潮∞
=
昨砩
=
潦
=
灥物潤
=
2
π
楮
=
瑥牭≥
=
潦
=
瑨≥
=
獩浰汥
=
晵湣瑩潮猠
∞
∞
nxnxxxxxsin,cos,,2sin,2cos,sin,cos,1
L
•All these
f
unctions have the period 2
π
⸠周敹.
∞
潲洠瑨攠獯⁣慬汥搠瑲楧潮潭整物挠
獹獴敭⸠周攠晩杵牥⁳桯睳⁴桥獥⁦畮捴楯湳⁡汬⁨慶攠灥物潤′πexcept for the
constant 1, which is periodic with any period.
P. Gundes Bakir, Vibration based structural health monitoring
36
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•The series to be obtained will be a trigonometric series, that is a series of the
form:
()
∑
∞
=
++=+++++
n
nno
nxbnxaaxbxaxbxaasincos2sin2cossincos
02211
L
•Here are constants called the coefficients of the series.
We see that each term has the period 2π. Hence if the coefficients are such
that the series conver
g
es, its sum will be a function of
p
eriod 2
π
.
L
,,,,,
2211
babaao
gp
•Now suppose that f(x) is a given function of period 2πand is such that it can
be represented by a series as above which converges and moreover has the
sum f
(
x
)
. Then usin
g
the e
q
ualit
y
si
g
n
,
we write:
()gqyg,
()
∑
∞
++=
n
nn
nxbnxaaxfsincos)(
0
P. Gundes Bakir, Vibration based structural health monitoring
37
=
n
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•The equation
(
)
∑
∞
+
+
=
sin
cos
)
(
nx
b
nx
a
a
x
f
iscalledtheFourierseriesoff(x).Weshallprovethatinthiscase,the
?
?
¦

?
?

1
0
sin
cos
)
(
n
nn
nx
b
nx
a
a
x
f
is

called

the

Fourier

series

of

f(x).

We

shall

prove

that

in

this

case,

the

coefficicents of the above equation are the so called Fourier coefficients of f(x)
given by the Euler formulas.
1
∫
π
L
,
2
,
1cos
)
(
1
)(
2
1
0
=
=
=
∫
∫
−
nnxdxx
f
a
dxx
f
a
n
π
π
π
L,2,1sin)(
1
,
,
)
(
==
∫
∫
−
−
nnxdxxfb
f
n
n
π
π
π
π
π
P. Gundes Bakir, Vibration based structural health monitoring
38
−
π
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•Let f(x) be periodic with period 2πand piecewise continuous in the interval
. Furthermore, let f(x) have a left hand derivative and a right hand
ditithitfthtitlThthFiif

π
π
≤
≤
−
x
d
er
i
va
ti
ve a
t
eac
h
po
i
n
t
o
f

th
a
t

i
n
t
erva
l
.
Th
en
th
e
F
our
i
er ser
i
es o
f

()
∑
∞
=
++=
1
0
sincos)(
n
nn
nxbnxaaxf
with coefficients
L
,
2
,
1
cos
)
(
1
)(
2
1
0
=
=
=
∫
∫
−
n
nxdx
x
f
a
dxxfa
π
π
π
π
L,2,1sin)(
1
,
2
,
1
cos
)
(
==
∫
∫
−
−
nnxdxxfb
n
nxdx
x
f
a
n
n
π
π
π
π
π
converges. Its sum is f(x) except at points xo where f(x) is discontinuous.
There the sum of the series is the average of the left and right limits of f(x) at
xo.
P. Gundes Bakir, Vibration based structural health monitoring
39
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•The left hand limit of f(x) at xo is defined as the limit of f(x) as x approaches xo
from the left and is commonly denoted by f(xo-h). Thus,
values.
p
ositivethrough 0as)(lim)(
0
→
−
=−
→
hh
x
f
h
x
f
oho
•The right hand limit of f(x) at xo is defined as the limit of f(x) as x approaches
xo from the right and is commonly denoted by f(xo+h). Thus,
l
iti
thh
0
)
(
li
)
(
→
+
+
h
h
f
h
f
va
l
ues.
p
os
iti
ve
th
roug
h

0
as
)
(
li
m
)
(
0
→
+
=
+
→
h
h
x
f
h
x
f
oho
P. Gundes Bakir, Vibration based structural health monitoring
40
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
FidhFiffiifhidifif()ihfi
•
Fi
n
d
t
h
e
F
our
i
er coe
ffi
c
i
ents o
f
t
h
e per
i
o
di
c
f
unct
i
on
f(
x
)

i
n t
h
e
fi
gure.
The formula is:
0
x
if
k
⎧
<
<
−
−
π
)()2(
0
0
)(
xfxfand
xifk
x
if
k
x
f
=+
⎩
⎨
⎧
<<
<
<
=
π
π
π
P. Gundes Bakir, Vibration based structural health monitoring
41
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
FidhFiffiifhidifif()ihfi
•
Fi
n
d
t
h
e
F
our
i
er coe
ffi
c
i
ents o
f
t
h
e per
i
o
di
c
f
unct
i
on
f(
x
)

i
n t
h
e
fi
gure.
The formula is:
0
x
if
k
⎧
<
<
−
−
π
)()2(
0
0
)(
xfxfand
xifk
x
if
k
x
f
=+
⎩
⎨
⎧
<<
<
<
=
π
π
π
1
S
)
(
1
)
(
1
)
(
1
)(
2
1
0
0
+
−
=
=
=
−
∫
∫
∫
∫
π
ππ
π
π
dx
k
dx
k
dx
x
f
a
dxxfa
0
2
1
2
1
)(
2
1
)(
2
1
)
(
2
)
(
2
)
(
2
0
0
0
0
=+−=+−=
+
−
=
=
−
−−
∫
∫
∫
π
π
π
π
π
π
πππ
π
π
ππ
kkkxkx
dx
k
dx
k
dx
x
f
a
•The above can also be seen without integration, since the area
dthff()bt
d
i
un
d
er
th
e curve o
f

f(
x
)

b
e
t
ween -
π
慮
≤
=π
i
s zero.
P. Gundes Bakir, Vibration based structural health monitoring
42
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
•From
cos
)
(
1
=
∫
π
nxdxx
f
a
n
cos)(cos)(
1
cos)(
1
)
(
0
0
⎥
⎦
⎤
⎢
⎣
⎡
+−==
−
−
−
∫∫∫
∫
ππ
π
π
π
π
π
π
nxdxknxdxknxdxxfa
f
n
n
0)
sin
()
sin
(
1
0
0
0
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+−=
⎦
⎣
−
π
π
π
π
π
n
nx
k
n
nx
k
:milarly1,2,....Sin allfor and ,0,-at 0sinnx because=
=
π
π
sin
)
(
1
=
∫
π
nxdxx
f
b
n
sin)(sin)(
1
sin)(
1
)
(
0
0
⎥
⎦
⎤
⎢
⎣
⎡
+−==
−
−
−
∫∫∫
∫
ππ
π
π
π
π
π
π
nxdxknxdxknxdxxfb
f
n
n
coscos1
0
0
0
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−=
⎦
⎣
−
π
π
π
π
π
n
nx
k
n
nx
k
P. Gundes Bakir, Vibration based structural health monitoring
43
:yields this1,cos0 and cos)cos(- Since
=
=
α
α
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
[
]
0
)
(
0
k
b
[
]
[]
−=
+
−
−
−
=
cos22
0
coscos
)
cos
(
0
cos
π
π
π
π
π
n
n
k
nn
n
b
n
()
−
=
=
−
=
−=
g
eneralin etc1
)
cos
(
3 1
,
cos2
,
1
)
cos
(
Now
cos1
2
π
π
π
π
π
π
n
n
k
n
⎧
⎩
⎨
⎧
=
dd
f
2
n even for 1
n oddfor 1-
cosn
g
)
(
,
,
)
(
π
⎩
⎨
⎧
=−
n even for 0
no
dd
f
or
2
cosn1 thusand
π
P. Gundes Bakir, Vibration based structural health monitoring
44
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
:
are
function
our
of
b
ts
coefficien
Fourier
the
Hence
n
:
is
of
series
Fourier
the
zero
are
the
Since
,
5
4
,0,
3
4
,0
4
:
are
function

our

of
b
ts
coefficien
Fourier

the
Hence
54321
n
f(x)
a
k
bb
k
bb,
π
k
b=====L
ππ
til
Th
5sin
5
1
3sin
3
1
sin
4
:
is
of

series
Fourier

the
zero
,
are
the
Since
n
xxx
π
k
f(x)
a
⎟
⎠
⎞
⎜
⎝
⎛
+++L
sin
4
S
:aresumspar
ti
a
l
Th
e
1
x
π
k
⎞
⎛
=
Thihtiditthtthiitdh
.3sin
3
1
sin
4
2
etcxx
π
k
S
⎟
⎠
⎞
⎜
⎝
⎛
+=
•
Th
e
i
r grap
h
seems
t
o
i
n
di
ca
t
e
th
a
t

th
e ser
i
es
i
s convergen
t
an
d

h
as
the sum f(x), the given function.
P. Gundes Bakir, Vibration based structural health monitoring
45
ExampleExample
ERASMUS Teaching (2008), Technische Universität Berlin
Example
Example
Wih0
•
W
e not
i
ce t
h
at at x=
0

and x=π, the points of
discontinuity of f(x), all
partial sums have the
value zero, the
arithmetic mean of the
limits k and –k of our
function at these
points
points
.
P. Gundes Bakir, Vibration based structural health monitoring
46
DerivationoftheEulerformulasDerivationoftheEulerformulas
ERASMUS Teaching (2008), Technische Universität Berlin
Derivation

of

the

Euler

formulasDerivation

of

the

Euler

formulas
ThkhElflihhlif
•
Th
e
k
ey to t
h
e
E
u
l
er
f
ormu
l
as
i
s t
h
e ort
h
ogona
li
ty o
f
aconceptofbasicimportanceasfollows:
nxnxxxxxsin,cos,,2sin,2cos,sin,cos,1
L
a

concept

of

basic

importance

as

follows:
THEOREM 1:The trigonometric system above is orthogonal on the
interval (hence also on or any other interval of
lth2
bfidiit)thtithitlfthdtf
π
π
≤
≤−
x
2x0
π
≤
≤
l
eng
th

2
π
b
ecause o
f
per
i
o
di
c
it
y
)
;
th
a
t

i
s
th
e
i
n
t
egra
l
o
f

th
e pro
d
uc
t
o
f

any two functions in
over that interval is zero
,
so that for an
y
inte
g
ers n and m
,
nxnxxxxxsin,cos,,2sin,2cos,sin,cos,1
L
,yg,
∫
∫
−
≠=
π
π
π
)
(
i
i
)
(
)(0coscos)(
d
b
mnmxdxnxa
∫
∫
−
=≠=
≠
=
π
π
)(0cossin)(
)
(
0s
i
ns
i
n
)
(
mnormnmxdxnxc
mnmx
d
xnx
b
P. Gundes Bakir, Vibration based structural health monitoring
47
−
π
FourierseriesexpansionofanyFourierseriesexpansionofany
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

any

Fourier

series

expansion

of

any

period p=Tperiod p=T
•The functions considered so far had period 2π, for the simplicity of the formulas. However, we will
mostly use the variable time t and work with functions x(t) with period T. We now show that the
transition from 2πto period T is quite simple. The idea is simply to find and use a change of scale
that gives from a function f(x) of period 2
π
瑯⁡⁦畮捴楯渠潦⁰敲楯搠吮θ
•φ渠瑨攠敱畡瑩潮猠扥汯眠睥⁣慮⁷物瑥⁴桥⁣桡湧攠潦⁳捡汥⁡猺⁸㵫琠睩瑨l渠獵捨⁴桡琠瑨攠潬搠灥物潤s砽㈠
πgives for the new variable t the new period t=T. Thus
2
π
=
歔桥湣hk
=
2
π
⽔慮ax
=
歴
=
2
π
琯吮θ桩hi浰汩敳≤x
=
2
π
摴⽔睨楣w異潮獵扳瑩瑵瑩潮楮瑯
2
=
π
歔
†
桥湣h
=
欲
=
π
⽔
=
慮a
=
硫≥
=
2
π
琯吮
=
周楳
=
業灬楥i
=
摸
=
2
π
摴⽔
=
睨楣w
=
異潮
=
獵扳瑩瑵瑩潮
=
楮瑯
1
)(
2
1
0
=
∫
−
dxxfa
π
π
π
π
2
2
)(
1
2
/
2/
2/
0
=
∫
−
dttx
T
a
T
T
T
L
2
1
sin
)
(
1
,2,1cos)(
1
==
∫
∫
−
n
nxdx
x
f
b
nnxdxxfan
π
π
π
π
L
2
1
2
sin
)
(
2
,2,1
2
cos)(
2
2/
2
/
2/
==
∫
∫
−
n
dt
nt
t
x
b
ndt
T
n
t
tx
T
a
T
T
T
n
π
π
P. Gundes Bakir, Vibration based structural health monitoring
48
L,
2
,
1
sin
)
(
=
=
∫
−
n
nxdx
x
f
b
n
π
π
L,
2
,
1
sin
)
(
2/
=
=
∫
−
n
dt
T
t
x
T
b
T
n
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier series expansion
Fourier series expansion
•Since we will mostly use the variable time t and in the frequency domain
2πn/T, the equation
(
)
∑
∞
+
+
=
sin
cos
)
(
nx
b
nx
a
a
x
f
can be written as follows:
(
)
∑
=
+
+
=
1
0
sin
cos
)
(
n
nn
nx
b
nx
a
a
x
f
∑
∞
⎟
⎞
⎜
⎛
+
+
=
2
sin
2
cos
)
(
nt
b
nt
a
a
t
x
ππ
The coefficients in this case can be written as shown on the rhs rather than the
lhs.
∑
=
⎟
⎠
⎜
⎝
+
+
=
1
0
sin
cos
)
(
n
nn
T
b
T
a
a
t
x
lhs.

2
2
)(
1
2
/
2/
2/
0
=
∫
−
dxtx
T
a
T
T
T
1
)(
2
1
0
=
∫
∫
−
dxxfa
π
π
π
π
L
2
1
2
sin
)
(
2
,2,1
2
cos)(
2
2/
2
/
2/
==
∫
∫
−
n
dt
nt
t
x
b
ndt
T
n
t
tx
T
a
T
T
T
n
π
π
L
L
,
2
,
1
sin
)
(
1
,2,1cos)(
1
=
=
=
=
∫
∫
−
n
nxdx
x
f
b
nnxdxxfan
π
π
π
P. Gundes Bakir, Vibration based structural health monitoring
49
L,
2
,
1
sin
)
(
2/
=
=
∫
−
n
dt
T
t
x
T
b
T
n
,
2
,
1
sin
)
(
∫
−
n
nxdx
x
f
b
n
π
π
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
•For a function x(t)defined on the interval [-τ/2,τ/2],we have the
representation on that interval
•The coefficients are obtained as follows: Consider the integral
•
When
x(t)
issubstitutedfromthefirstequationthisintegralbreaksdowninto
•
When

x(t)
is

substituted

from

the

first

equation
,
this

integral

breaks

down

into

I
m
(1)
, Im
(2)
andI
m
(3)
.The first is:
P. Gundes Bakir, Vibration based structural health monitoring
50
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
but this is zero as sin(
π
m)=0 for all m. The second integral is:
•Assuming we can change the order of integration and summation we obtain
•Using the identity
P. Gundes Bakir, Vibration based structural health monitoring
51
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
•Now if nand mare different integers then n-mand n+mare both nonzero integers
and the sine terms in the last expression vanish. If nand mare equal, we have a
p
roblem with the second term above. We could use a limit argument but it is
simpler to go back to the first equation with n=m.
P. Gundes Bakir, Vibration based structural health monitoring
52
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
•This breaks down to:
•An orthogonality relation has been proved.
f
2
2
2
⎤
⎡
⎟
⎞
⎜
⎛
⎤
⎡
⎟
⎞
⎜
⎛
∫
d
m
n
π
π
τ
0for
f
or 0
2
cos
2
cos
2
≠
=
=
≠
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
∫
−
nm
nm
dt
t
m
t
n
τ
τ
π
τ
π
τ
0for
2
=
=
=
nm
τ
•A similar analysis for the third integral
gives
P. Gundes Bakir, Vibration based structural health monitoring
53
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
•Derives along the way the orthogonality relation
•As
•We have
0for
2

≠==nmaI
mm
τ
•Or in terms of the original expression:
0for
0
=
=
=
nmaI
m
τ
2
0for
≠
=
nm
P. Gundes Bakir, Vibration based structural health monitoring
54
FourierseriesexpansionFourierseriesexpansion
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansionFourier

series

expansion
If
0ihbli
•
If
m=n=
0

i
n t
h
e
b
e
l
ow equat
i
on:
0for
2
cos)(
2
==
⎟
⎠
⎞
⎜
⎝
⎛
=
∫
nmdtt
m
txa
m
τ
τ
τ
π
τ
•Performing the same operations using a multiplier of sin(2
π
m/
τ
)
gives:
2
⎠
⎝
−
τ
via the orthogonality relation:
where
δ
楳捡汬敤≥桥䭲潮散步K摥汴≤慮a桡h瑨≥p牯灥牴楥猺
睨敲w
=
δ
mn
is

called

the

Kronecker

delta

and

has

the

properties:
⎩
⎨
⎧
≠
=
=
mn
mn
mn
if 0
if 1
δ
P. Gundes Bakir, Vibration based structural health monitoring
55
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
Fourier Series Expansion in Exponential Form
•Recall the standard Fourier series in terms of
•Now suppose we apply de Moivres Theorem
P. Gundes Bakir, Vibration based structural health monitoring
56
FourierseriesexpansionoftheFourierseriesexpansionofthe
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansion

of

the

Fourier

series

expansion

of

the

periodic loadingperiodic loading
•This allows us to write the equation
in the following form:
where
P. Gundes Bakir, Vibration based structural health monitoring
57
FourierseriesexpansionFourierseriesexpansion
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansionFourier

series

expansion
Uithti
•
U
s
i
ng
th
e equa
ti
ons
L
2
2
,2,1
2
cos)(
2
2/
2/
2/
==
∫
∫
−
nt
ndt
T
nt
tx
T
a
T
T
T
n
π
π
in
L,2,1
2
sin)(
2
2/
=
=
∫
−
ndt
T
nt
tx
T
b
T
n
π
in
()
dt
T
nt
txidt
T
nt
tx
T
ibac
T
T
T
T
T
nnn
∫∫
+
+
−
+
−
⎞
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=−=
2
/
2/
2/
2/
2/
2
2
1
2
sin)(
2
cos)(
2
2
1
2
1
ππ
Gives:
dt
T
n
t
i
T
n
t
tx
T
T
T
∫
+
−
⎟
⎠
⎞
⎜
⎝
⎛
−=
2
/
2/
2
sin
2
cos)(
1
π
π
T
+
2
/
1
dtetx
T
c
T
T
Tnti
n
∫
+
−
−
=
2
/
2/
/2
)(
1
π
P. Gundes Bakir, Vibration based structural health monitoring
58
FourierseriesexpansionFourierseriesexpansion
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

series

expansionFourier

series

expansion
Alttiliththlitti
•
Alt
erna
ti
ve
l
y, us
i
ng
th
e or
th
ogona
lit
y equa
ti
on
0for 0
2/
≠+=
∫
mndtee
tim
T
tin
ωω
andmultiplyingtheequation
0for
2/
=
+
=
∫
−
mn
T
T
and

multiplying

the

equation

by exp(im
ω
t),and integrating directly gives:
P. Gundes Bakir, Vibration based structural health monitoring
59
FouriertransformFouriertransform
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

transformFourier

transform
Ifttlktthtlttfidiil
•
If
we wan
t

t
o
l
oo
k
a
t

th
e spec
t
ra
l
con
t
en
t
o
f
nonper
i
o
di
c s
i
gna
l
s we
have to let τ→∞as all the interval t∈[-∞, ∞] contains important
information. Recall the exponential form of the Fourier series
where
Cbiithbttii
C
om
bi
n
i
ng
th
e a
b
ove
t
wo equa
ti
ons g
i
ve:
and we now have to let
τ
→0
P. Gundes Bakir, Vibration based structural health monitoring
60
FourierTransformFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

TransformFourier

Transform
•If we su
pp
ose that the
p
osition of the t axis is ad
j
usted so that the
pppj
mean value of x(t) is zero. Then according to the first of the below
equation the coefficient a
o
will be zero.
Thiiffiitdbillilllbdifft
LL,2,1
2
sin)(
2
,2,1
2
cos)(
2
)(
1
2/
2/
2/
2/
2/
2/
0
=====
∫∫∫
−−−
ndt
T
nt
tx
T
bndt
T
nt
tx
T
adxtx
T
a
T
T
n
T
T
n
T
T
ππ
•
Th
e rema
i
n
i
ng coe
ffi
c
i
en
t
s an an
d

b
n w
ill

i
n genera
l
a
ll

b
e
diff
eren
t

and their values may be illustrated graphically as shown.
P. Gundes Bakir, Vibration based structural health monitoring
61
FourierTransformFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

TransformFourier

Transform
Recall that the spacing between the frequency lines is
so that the kthspectral line is at
From the first equation, we see that
The equation
becomes
P. Gundes Bakir, Vibration based structural health monitoring
62
FourierTransformFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

TransformFourier

Transform
A
0
h
bldlhdh
•
A
s
τ
→
0
, t
h
e
ω
n
b
散潭攠e
l
潳敲⁡→
≤
⁣
l
潳敲⁴潧整
h
敲⁡e
≤
⁴
h
攠
獵浭慴楯渠瑵牮猠楮瑯⁡渠楮瑥杲慬⁷楴栠=Δω=dω(assuming that x(t)is
appropriately well behaved. In the limit
Itfllthtifdfi
•
It

f
o
ll
ows
th
a
t

if
we
d
e
fi
ne
where
F
denotes the Fourier transform then the first equation
implies that
andthisistheinverseFouriertransform
P. Gundes Bakir, Vibration based structural health monitoring
63
and

this

is

the

inverse

Fourier

transform
.
FourierTransformFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Fourier

TransformFourier

Transform
NtthfliilittthLltfiftbti
•
N
o
t
e
th
e
f
orma
l
s
i
m
il
ar
it
y
t
o
th
e
L
ap
l
ace
t
rans
f
orm
i
n
f
ac
t
we o
bt
a
i
n
the Fourier transform by letting s= i
ω
in the Laplace transform. The
main difference between the two is the comparative simplicity of the
inverseFouriertransform
inverse

Fourier

transform
.
•{x(t),X(ω)} are a Fourier transform pair. As they are uniquely
constructable from each other they must both encode the same
information but in different domains. X(ω) expresses the frequency
content of x(t).It is another form of spectrum. However note that it
htbtiftif
idtt
h
as
t
o
b
e a con
ti
nuous
f
unc
ti
on o
f

ω
i
渠潲
≤
敲e
≥
漠牥灲敳敮
≥
潮=
灥物潤楣⁦畮捴楯湳p
P.=䝵湤敳⁂慫楲,======Vi扲慴楯渠扡獥搠獴牵捴畲慬⁨敡汴栠浯湩瑯物湧
㘴
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Ililillhilbffiidibiillb
•
I
n rea
li
ty not on
l
y w
ill
t
h
e s
i
gna
l

b
e o
f

fi
n
i
te
d
urat
i
on
b
ut
i
t w
ill

b
e
sampled. It is not possible to store a continuous function on a
computer as it would require infinite memory.
•What one usually does is take measurements from the signal at
regular intervals say t seconds apart so the signal for manipulation
takestheformofafinitevectorofNsamples
takes

the

form

of

a

finite

vector

of

N

samples
where t is a reference time. If we take t
0=0 from now on we will have
•How do we compute the spectrum of such a signal? We need the
Discrete Fourier Transform DFT.
P. Gundes Bakir, Vibration based structural health monitoring
65
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
RllhilffhFiihl
•
R
eca
ll
t
h
e exponent
i
a
l

f
orm o
f
t
h
e
F
our
i
er ser
i
es t
h
e spectra
l

coefficients are
•In keeping with our notation for the Fourier transform we will relabel
cn by Xn
from now on. Also the equation above is not in the most
convenient form for the analysis so we will modify it slightly.
•
Recallthat
x(t)
isassumedperiodicwithperiod
τ
䍯湳楤敲瑨≥
•
剥捡汬
=
瑨慴
=
x(t)
is

assumed

periodic

with

period

τ
⸠
䍯湳楤敲
=
瑨≥
=
楮瑥杲慬
P.=䝵湤敳⁂慫楲,======Vi扲慴楯渠扡獥搠獴牵捴畲慬⁨敡汴栠浯湩瑯物湧
㘶
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Ltt
′
t
thitlb
•
L
e
t

t
′
㴠
≥
+τ,
th
e
i
n
t
egra
l

b
ecomes
but x(t′)=x(t′-τ) by periodicity. Also,
τπττπ
/2/)(2

nittin
ee
−−
′
−
=
P. Gundes Bakir, Vibration based structural health monitoring
67
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
•Now, as we have only x(t) sampled at t
r
=r
Δ
t, we have to approximate the
Discrete

Fourier

TransformDiscrete

Fourier

Transform
integral by a rectangular sum,
and as τ=N
Δ
t, this becomes,
•As we started off with only N independent quantities xr
we can only derive
N
iddttllittThithlti
N
i
n
d
epen
d
en
t
spec
t
ra
l

li
nes a
t
mos
t
.
Thi
s means we mus
t

h
ave re
l
a
ti
ons
between the Xn
. The simplest one is periodicity. Consider,
P. Gundes Bakir, Vibration based structural health monitoring
68
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
•Therefore, it has been confirmed that we have at most
N
independent lines.
In fact, there must be less than this as the Xn
are the complex quantities.
Given Nreal numbers, we can only specify N/2complex numbers.
•Looking at the exponent in
if this is to be identified with the exponent i
ω
ntof the Fourier transform, we
must have
,
,
o
r
P. Gundes Bakir, Vibration based structural health monitoring
69
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
AlttilififthfiiHt
•
Alt
erna
ti
ve
l
y,
if
we spec
if
y
th
e
f
requency spac
i
ng
i
n
H
er
t
z
•When n=0, the spectral line is given by:
whichisthearithmeticmeanorDCcomponentofthesignalThereforeX
which

is

the

arithmetic

mean

or

DC

component

of

the

signal
.
Therefore

X
0
corresponds to the frequency ω=0 as we might expect.
•This means that the highest frequency that we can represent is
22
1
22
s
f
ttN
N
f
N
=
Δ
=
Δ
=Δ
where fs
is the sampling frequency. This frequency is very important in
signal processing and is called the Nyquist Frequency.
P. Gundes Bakir, Vibration based structural health monitoring
70
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Thitthtlthfithlffthtlli
•
Thi
s argumen
t
says
th
a
t
on
l
y
th
e
fi
rs
t

h
a
lf
o
f

th
e spec
t
ra
l

li
nes are
independent-so what are the second half? Consider:
N
r
N
i
N
1
)
1
(
2
1
1
1
π
−
−
−
Nriri
N
r
r
N
r
N
i
N
r
rN
eex
N
ex
N
X
/22
1
0
)
1
(
2
1
0
1
1
1
ππ
π
+−
=
−
=
−
∑
∑
==
but ris an integer, so e-i2
π
r=1, this means,
*
*
1
/
2
/
2
1
1
1
N
N
r
i
N
r
i
N
⎟
⎞
⎜
⎛
∑
∑
−−
π
π
*
1
0
/
2
/
2
0
1
1
1
X
e
x
N
e
x
N
X
r
N
r
i
r
N
r
i
r
rN
=
⎟
⎠
⎞
⎜
⎝
⎛
=
=
∑
∑
=
−
=
−
π
π
•A similar argument shows that generally,
*
k
k
N
XX=
−
P. Gundes Bakir, Vibration based structural health monitoring
71
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
•Recall that it is a property of the Fourier Transform that
)()(
*
ωω
−=XX
•This means that the spectral coefficient X
N-1
corresponds to the frequency
-Δωor more generally X
N-k corresponds to the frequency -k
Δω
. So the
arrayX
storesthefrequencyrepresentationofthesignal
x
asfollows:
array

X
n
stores

the

frequency

representation

of

the

signal

x
r
as

follows:
P. Gundes Bakir, Vibration based structural health monitoring
72
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
hhifhilfkfhih
kk?
hiilih
2k
•W
h
at
h
appens
if
t
h
ere
i
s a va
l
ue o
f

k

f
or w
hi
c
h

N
-
k
=
k?
T
hi
s
i
mp
li
es t
h
at N=
2k
so
the number of sample points is even.
•Actually this turns out to be the most usual situation, in fact it is a requirement of
the Fast Fourier Transformwhich we shall meet in the next lecture.
•In this case, we have
2/
*
2/
N
N
XX=
so this spectral line is real.
•This finally justifies our assertion that the maximum frequency represented is
N
Δω
/2.
P. Gundes Bakir, Vibration based structural health monitoring
73
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Proofoftheinversiontheorem
Proof

of

the

inversion

theorem
•Let us proove that
N
1
Nrni
r
N
r
n
eXx
/2
1
0
π
∑
−
=
=
where
Nrpi
N
p
pr
ex
N
X
/2
1
0
1
π
−
−
=
∑
=
•If the second equation is substituted in the first equation, we obtain:
⎫
⎧
NrniNrpi
N
p
p
N
r
n
eex
N
x
/2/2
1
0
1
0
1
ππ
⎭
⎬
⎫
⎩
⎨
⎧
=
−
−
=
−
=
∑∑
P. Gundes Bakir, Vibration based structural health monitoring
74
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Ifhthdfthtibti
•
If
we c
h
ange
th
e or
d
er o
f

th
e summa
ti
ons, we o
bt
a
i
n:
⎭
⎬
⎫
⎩
⎨
⎧
=
−
−−
∑
∑
NrniNrpi
N
p
N
n
eex
N
x
/2/2
11
1
ππ
whichcanberewrittenas:
¿
¯

¦
¦
r
p
p
N
00
which

can

be

rewritten

as:
⎭
⎬
⎫
⎩
⎨
⎧
=
−
−−
∑
∑
Npnri
N
p
N
n
ex
N
x
/)(2
11
1
π
which can be reexpressed as follows provided the term in brackets is
called
q
:
⎭
⎩
==
∑
∑
r
p
p
N
00
[
]
⎫
⎧
−
−
N
N
1
1
1
q
[
]
⎭
⎬
⎫
⎩
⎨
⎧
=
−
==
∑
∑
rNpni
N
r
p
N
p
n
ex
N
x
/)(2
1
0
1
0
1
π
P. Gundes Bakir, Vibration based structural health monitoring
75
=q
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
•Thus, if n≠
p
, within the e
q
uation
p
q
[]
⎭
⎬
⎫
⎩
⎨
⎧
=
−
−
=
−
=
∑∑
rNpni
N
r
p
N
p
n
ex
N
x
/)(2
1
0
1
0
1
π
=
q
we have a geometric series as follows:
...1
2
1
0
+++=
∑
−
=
qqq
N
r
r
q
If we call the left hand side of the above equation s
r, we have:
...1
2
+++=qqs
r
If we multiply both sides of the above equation by q, and subtract the
resulting equation from the above equation:
N
r
+
1
1
N
r
r
N
r
r
qqss
qqqqs
−=−
+
+
+
=
+
−
1
....
1
1
P. Gundes Bakir, Vibration based structural health monitoring
76
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
Wbtithlf
fh
•
W
e can o
bt
a
i
n
th
e va
l
ue o
f

s
r
f
rom
h
ere as:
q
s
N
−
=
1
Nowq
N
=1whichcanbeprovedas:
q
s
r
−
=
1
Now
,
q
N
=1
,
which

can

be

proved

as:
qN=ei2π(n-p)r=cos2πr(n-p)+isin2πr(n-p)=1+0=1
which results in the below result for n≠p :
1
N
0
1
1
=
−
−
=
q
q
s
N
r
P. Gundes Bakir, Vibration based structural health monitoring
77
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
F
hibkihiblb1
•
F
or n=p, t
h
e term
i
n
b
rac
k
ets
i
n t
h
e equat
i
on
b
e
l
ow
b
ecomes
1
.
[
]
⎭
⎬
⎫
⎩
⎨
⎧
=
−
−−
∑
∑
rNpni
N
p
N
n
ex
N
x
/)(2
11
1
π
•The summation of all ones r times gives N. N’s cancel each other in
[
]
⎭
⎩
==
∑
∑
r
p
p
N
00
the above equation and letting p=ngives:
N
x
N
x
∑
−
δ
1
1
•Thus
,
the above e
q
ualit
y
is
p
roved to be satisfied which conse
q
uentl
y

npn
p
p
x
N
x
N
=
∑
=
δ
0
,qypqy
proves the inversion theorem. This takes us to the final proven
Discrete Fourier Transform formulas in the next slide.
P. Gundes Bakir, Vibration based structural health monitoring
78
DiscreteFourierTransformDiscreteFourierTransform
ERASMUS Teaching (2008), Technische Universität Berlin
Discrete

Fourier

TransformDiscrete

Fourier

Transform
DiFiTf
•
Di
screte
F
our
i
er
T
rans
f
orm
Nnri
N
e
x
X
/2
1
π
−
−
∑
=
•Inverse Discrete Fourier Transform
r
r
n
e
x
X
0=
∑
=
Nnri
N
r
n
e
X
N
x
/2
1
1
π
∑
−
=
r
r
n
N
0
∑
=
P. Gundes Bakir, Vibration based structural health monitoring
79
Gibb
’
sphenomenonGibb
’
sphenomenon
ERASMUS Teaching (2008), Technische Universität Berlin
Gibbs

phenomenonGibbs

phenomenon
•
Thefirstfigureshowsatimedomainsignalbeingsynthesizedfrom
•
The

first

figure

shows

a

time

domain

signal

being

synthesized

from

sinusoids. The signal being reconstructed is shown in the graph on the left
hand side. Since this signal is 1024 points long, there will be 513 individual
frequencies needed for a complete reconstruction. The figure on the right
shosareconstrctedsignalsingfreqencies0throgh100Thissignal
sho
w
s

a

reconstr
u
cted

signal
u
sing

freq
u
encies

0

thro
u
gh

100
.
This

signal

was created by taking the DFT of the signal on the left hand side, setting
frequencies 101 through 512 to a value of zero, and then using the Inverse
DFT to find the resulting time domain signal. The figure in the middle shows
ttdilifi0thh30
a recons
t
ruc
t
e
d
s
i
gna
l
us
i
ng
f
requenc
i
es
0

th
roug
h

30
.
P. Gundes Bakir, Vibration based structural health monitoring
80
Gibb
’
sphenomenonGibb
’
sphenomenon
ERASMUS Teaching (2008), Technische Universität Berlin
Gibbs

phenomenonGibbs

phenomenon
•
Whenonlysomeofthefrequenciesareusedinthereconstructioneachedgeshows
•
When

only

some

of

the

frequencies

are

used

in

the

reconstruction
,
each

edge

shows

overshootand ringing(decaying oscillations). This overshoot and ringing is known as
the Gibbs effect, after the mathematical physicist Josiah Gibbs, who explained the
phenomenon in 1899.
•
Thecriticalfactorinresolvingthispuzzleisthatthe
width
oftheovershootbecomes
•
The

critical

factor

in

resolving

this

puzzle

is

that

the

width
of

the

overshoot

becomes

smaller as more sinusoids are included. The overshoot is still present with an infinite
number of sinusoids, but it has zerowidth. Exactly at the discontinuity the value of the
reconstructed signal converges to the midpoint of the step. As shown by Gibbs, the
summation conver
g
es to the si
g
nal in the sense that the errorbetween the two has
gg
zero energy.
P. Gundes Bakir, Vibration based structural health monitoring
81
Gibb
’
sphenomenonGibb
’
sphenomenon
ERASMUS Teaching (2008), Technische Universität Berlin
Gibbs

phenomenonGibbs

phenomenon
•
ProblemsrelatedtotheGibbseffectarefrequentlyencounteredin
•
Problems

related

to

the

Gibbs

effect

are

frequently

encountered

in

DSP. For example, a low-pass filter is a truncationof the higher
frequencies, resulting in overshoot and ringing at the edges in the
time domain. Another common
p
rocedure is to truncate the ends of
p
a time domain signal to prevent them from extending into
neighboring periods. By duality, this distorts the edges in the
frequency domain. These issues will resurface in future chapters on
filterdesign
filter

design
.
P. Gundes Bakir, Vibration based structural health monitoring
82
Example:AnimportantDFTpairExample:AnimportantDFTpair
ERASMUS Teaching (2008), Technische Universität Berlin
Example:

An

important

DFT

pairExample:

An

important

DFT

pair
•
Figureisanexamplesignalwewishtosynthesizeanimpulseatsample
•
Figure

is

an

example

signal

we

wish

to

synthesize
,
an

impulse

at

sample

zero with anamplitude of 32. Figure b shows the frequency domain
representation of this signal. The real part of the frequency domain is a
constant value of 32. The imaginary part (not shown) is composed of all
eros
z
eros
.
•As discussed in the next chapter, this is an important DFT pair:an impulse
in the time domain corresponds to a constant value in the frequency
domain. For now
,
the im
p
ortant
p
oint is that
(
b
)
is the DFT of
(
a
),
and
(
a
)
is
,pp()
(),()
the Inverse DFT of (b).
P. Gundes Bakir, Vibration based structural health monitoring
83
BandwidthBandwidth
ERASMUS Teaching (2008), Technische Universität Berlin
Bandwidth
Bandwidth
•
Asshowninthefigurethe
•
As

shown

in

the

figure
,
the

bandwidth can be defined by
drawing dividing lines between the
samples. For instance, sample
nmber
5
occrsintheband
n
u
mber

5

occ
u
rs

in

the

band

between 4.5 and 5.5; sample
number 6 occurs in the band
between 5.5 and 6.5, etc.
Edftifth
E
xpresse
d
as a
f
rac
ti
on o
f

th
e
total bandwidth (i.e., N/2), the
bandwidth of each sample is 2/N.
An exception to this is the
lhdhihh
•DFT can be calculated by the fast
samp
l
es on eac
h
en
d
, w
hi
c
h

h
ave
one-half of this bandwidth, 1IN.
This accounts for the 2/N scaling
factor between the sinusoidal
Fourier transform (FFT), which is
an ingenious algorithm that
decomposes a DFT with N points,
i
N
DFThihil
amplitudes and frequency domain,
as well as the additional factor of
two needed for the first and last
sam
p
les.
i
nto
N

DFT
s eac
h
w
i
t
h
a s
i
ng
l
e
point.
P. Gundes Bakir, Vibration based structural health monitoring
84
p
FrequencyresponseFrequencyresponse
--
Impulse
Impulse
ERASMUS Teaching (2008), Technische Universität Berlin
Frequency

responseFrequency

response
Impulse

Impulse

responseresponse
•A system’s frequency response is the Fourier transform of its impulse
response.
•Kee
p
in
g
with standard DSP notation
,
im
p
ulse res
p
onses use lowe
r
-case
pg,pp
variables, while the corresponding frequency responses are upper case.
Since h[ ] is the common symbol for the impulse response, H[ ] is used for
thefrequencyresponseThatis
convolution
inthetimedomain
the

frequency

response
.
That

is
,
convolution

in

the

time

domain

corresponds to multiplication in the frequency domain.
P. Gundes Bakir, Vibration based structural health monitoring
85
How much resolution can
y
ou obtain in How much resolution can
y
ou obtain in
ERASMUS Teaching (2008), Technische Universität Berlin
y
y
the frequency response?
the frequency response?
•The answer is: infinitely high, if you are willing to pad the impulse response with an
infinite number of zeros. In other words, there is nothing limiting the frequency
resolution except the length of the DFT.
•Even though the impulse response is a discrete signal, the corresponding frequency
response is continuous. An N point DFT of the impulse response provides N/2 + 1
samples of this continuous curve. If you make the DFT longer, the resolution improves,
and you obtain a better idea of what the continuous curve looks like.
•This can be better understood by the discrete time Fourier transform (DTFT).
Consider an N sample signal being run through an N point DFT, producing an N/2 + 1
sam
p
le fre
q
uenc
y
domain. DFT considers the time domain si
g
nal to be in
f
initel
y
lon
g

pqyg
fyg
and periodic. That is, the N points are repeated over and over from negative to positive
infinity. Now consider what happens when we start to pad the time domain signal with
an ever increasin
g
number o
f
zeros, to obtain a finer and finer sam
p
lin
g
in the
P. Gundes Bakir, Vibration based structural health monitoring
86
g
pg
frequency domain.
How much resolution can
y
ou obtain in How much resolution can
y
ou obtain in
ERASMUS Teaching (2008), Technische Universität Berlin
y
y
the frequency response?
the frequency response?
•
Addingzerosmakestheperiodofthetimedomain
longer
whilesimultaneously
•
Adding

zeros

makes

the

period

of

the

time

domain

longer
,
while

simultaneously

making the frequency domain samples closer together.
•
Nowwewilltakethistotheextreme,byadding
an
infinite
numberofzerostothe
Now

we

will

take

this

to

the

extreme,

by

adding

an
infinite

number

of

zeros

to

the

time domain signal. This produces a different situation in two respects.
•First, the time domain signal now has an infinitely long period. In other words, it
hdi
idi
il
h
as turne
d

i
nto an aper
i
o
di
c s
i
gna
l
.
•Second, the frequency domain has achieved an infinitesimally small spacing
betweensamplesThatisithasbecomea
continuoussignal
ThisistheDTFTthe
between

samples
.
That

is
,
it

has

become

a

continuous

signal
.
This

is

the

DTFT
,
the

procedure that changes a discrete aperiodic signal in the time domain into a
frequency domain that is a continuous curve. In mathematical terms, a system's
frequency response is found by taking the DTFT of its impulse response. Since this
cannotbedoneinacomputertheDFTisusedtocalculatea
sampling
ofthetrue
cannot

be

done

in

a

computer
,
the

DFT

is

used

to

calculate

a

sampling

of

the

true

frequency response. This is the difference between what you do in a computer (the
DFT) and what you do with mathematical equations (the DTFT).
P. Gundes Bakir, Vibration based structural health monitoring
87
ClosepeaksClosepeaks
ERASMUS Teaching (2008), Technische Universität Berlin
Close

peaksClose

peaks
•
Supposetherearepeaksveryclosetogethersuchasshowninthefigure
•
Suppose

there

are

peaks