ModuleIII Lab
Signaland noise: coherenceanalysis
The coerehncemeasures the energy in the measured output caused by the
input. It is a measure of the SNR as a function of the frequency. It is
determined using average spectra for an ensemble of signals.
Two cases:
1. noise out of the range of frequency of the signal
2. noise in of the range of frequency of the signal
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
noise out of the range of frequency of the signal
demo_17
High SNRlow SNR
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d
i segnali, low SNR (high noise)
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ples, high SNR (low noise)
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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
noise out of the range of frequency of the signal (2)
The noisedoesnotinfluence
the signalcoherencyin the
rangeoffrequencieswhere
the signalisstrong
Thismeansthatwherethe
coherenceiscloseto1 the
output isrepresentativeof
the system and the noise
doesnotdisturbevenifthe
SNR islow
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coherence
signal coherence high SNR (low noise)
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frequency [Hz]
coherence
signal coherence low SNR (high noise)
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
noise in the range of frequency of the signal
The SNR islow, and the noiseiswithinthe rangeoffrequencyofthe system
demo_18
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amplitude
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amplitude
signals
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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
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coherence
signal coherence low SNR (high noise)
noise in the range of frequency of the signal (2)
The coherenceisclose(butlower)
than1 in a smallerrangeof
frequencies, wherethe output can
stillbeconsideredrepresentative
ofthe system, butthe noiseisnow
disturbingin the rangeof
frequenciesofthe system
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
A filterin the frequencydomain isa windowW thatpassesselectedfrequency
componentsX
n, rejectingthe others. Itisa pointbypointmultiplication
The trasformedY isconvertedback tothe timedomain intoy(t).
Filterscan alter the amplitudespectrum, the phasespectrumor both,
dependingon the filtercoefficientsW
n.
Filters
nnn
WXY
=
×
=
frequency
frequency
frequency
1
Y(f)
X(f)
W(f)
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
Filters(2)
A filterisasa matteroffacta LTI system, and can beimplementedin the time
domain throughitsh(t) function.
Unfortunately, the anti-transformofa box windowisa sincfunction:
Whichwouldimplytoconvolvethe originalsignalwitha veryhighlyoscillating
function(sync)
*
=
time
time
x(t)
w(t)
y(t)
time
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
Filters(3)
In ordertoavoidsuchoscillationsin the filteredsignal, smootherwindows are
usedtofilter, e.g.:
1. Hamming
2. Hanning
3. Butterworth
4. Blackmann-Harris
5. etc…
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
FilterTypes
Accordingtothe rejectedrangesoffrequencies, filtersmaybe:
1. Low pass
2. High pass
3. band-pass
4. band-reject(e.g. notch)
5. all-pass(effectsonlyon the phases)
The transitionregion“pass”to“reject”maybemore or lessgradual.
The “cutoff”frequencycorrespondstoa reductionin the signalmagnitudeof
-3dB, i.e. aboutofthe 70%.
Anotherclassificationis:
1. IIR filters(Infinite ImpulseResponseFilter): cause a nonlinearphase
distorsion
2. FIR filters(Finite ImpulseResponseFilter): cause a linearphasedistorsion
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
Zero-phasefilterimplementation
In order to eliminate phase distortion in the filtering process,special
algorithms are implemented.
E.g. the matlab“filtfilt”function uses the information in the signal at points
before and after the current point, in essence “looking into the future,”to
eliminate phase distortion.
The function filtfiltperforms zero-phase digital filtering by processing the input
data in both the forward and reverse directions. After filteringthe data in the
forward direction, filtfiltreverses the filtered sequence and runs it back
through the filter.
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
DigitalFilterTransfer Function
The transfer function is a frequency domain representation of a digital filter,
expressing the filter as a ratio of two polynomials. It is the principal
discrete-time model for filter implementation:
The constants bi
and ai
are the filter coefficients, and the ORDER OF THE
FILTER is the maximum of na
and nb.
()()()
()
zX
zazaa
zbzbb
zXzHzY
a
a
b
b
n
n
n
n

+


+

+++
+++
==
1
1
21
1
1
21
...
...
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
DigitalFilterTransfer Function(2)
When constructing a filter of given transfer function, some filter requirements
traditionally include passbandripple (Rp, in decibels), stopbandattenuation
(Rs, in decibels), and transition width (Ws-Wp, in Hertz).
This means find filter coefficients fitting specified requirements
NOTE:when constructing a filter very strict requirements, ALWAYS check
the filter response, in order to avoid unexpected filter response due to non-
convergence of parameters
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ClassicalIIR filterTypes
BUTTERWORTH FILTER
provides the best Taylor Series
approximation to the ideal lowpass
filter. Response is monotonic overall,
decreasing smoothly.
CHEBYSHEV TYPE I FILTER
minimizes the absolute difference btw
the ideal and actual frequency
response over the entire passband
by incorporating an equal ripple of
RpdB in the passband. Stopband
response is maximally flat.
The transition from passbandto
stopbandis more rapid than for the
butterworth.
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
CHEBYSHEV TYPE II FILTER
minimizes the absolute difference btw the
ideal and actual frequency response
over the entire stopband, by
incorporating an equal ripple of RsdB in
the stopband. Passbandresponse is
maximally flat.
The stopbanddoes not approach zero as
quickly as the type I filter. The absence
of ripple in the passband, however, is
often an important advantage.
ELLIPTIC FILTERS
are equiripplein both the passbandand
stopband. They generally meet filter
requirements with the lowest order of
many other filter types.
Given a filter order n, passbandripple Rpin
decibels, and stopbandripple Rsin
decibels, elliptic filters minimize
transition width.
ClassicalIIR filterTypes(2)
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
FIR Filters
have both advantages and disadvantages compared to IIR filters.
FIR filters have the following primary advantages:
1. They can have exactly linear phase.
2. They are always stable.
3. The design methods are generally linear.
4. They can be realized efficiently in hardware.
5. The filter startup transients have finite duration.
The primary disadvantage of FIR filters is that they often require a much
higher filter order than IIR filters to achieve a given level ofperformance.
Correspondingly, the delay of these filters is often much greater than for an
equal performance IIR filter.
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Filteringtoeliminate noise
demo_19
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amplitude


original
filtered
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noisy signal
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FFT signal
frequency [Hz]
amplitude
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FFT ideal filter
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amplitude
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Filterimplementation
demo_20
NOTE: when constructing a filter very strict requirements, ALWAYS check the
filter response, in order to avoid unexpected filter response due to non-
convergence of parameters
Compare FIR1 and Butterworth
Filter response for different
Lowpassfrequencies
FIR filter are always stable
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Amplitude Response
Filter Frequency Response - lp=10Hz


Butterworth
FIR1
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Amplitude Response
Filter Frequency Response - lp=16Hz


Butterworth
FIR1
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Amplitude Response
Filter Frequency Response - lp=18Hz


Butterworth
FIR1
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Amplitude Response
Filter Frequency Response - lp=25Hz


Butterworth
FIR1
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Oftenthe filterresponseis
visualizedin the DB scale:
0 dBresponse=1
(passband)
-100 dBresponse10
-5
(stopband)
Note
FIR1 linearphase
Butterworthnonlinearphase
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Filter Frequency Response - lp=25Hz


Butterworth
FIR1
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
TimeVariationand Nonlinearity
The DFT doesnotgiveanytiming-relatedinformation drawbackfor
Nonlinear-timevariantsystems, wheretime-varying
frequency/amplitudecontent, and/or abruptchangesin the system
responsecan beencountered.
Techniquesforthe analysisofnonstationarysignalsinclude:
1. Short-TimeFourier Transform
2. band-pass filters
3. Waveletanalysis
Suchtechniquesconvertthe 1D array
in time-domainintoa 2D arrayin the
time-frequencyspacein ordertocapture
the time-varyingnature ofthe signal
numero d'onda [rad/m]
frequenza [Hz]
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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
Samplingat 40000 Hz itispossibleto
seethe frequencyincreasing
Samplingat 10000 Hz, after5000 Hz
(fnyq), fakefrequenciesare seen,
due tothe undersamplingofthe
signal(aliasingeffect)
time [s]
frequency [Hz]
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x 10
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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
STFT (Gabor, 1946)
Drawbacks in the global DFT (computed along the entire NL-TV signal) can be
decreased by extracting time windows of the original signal and analysing
each window in the frequency domain.
the k-thDFT of the k-thwindowed signal defines a matrix of
SHORT-TIME FOURIER TRANSFORMS (STFT)of the original signal
a 3D plot is obtained in which time-frequency and DFTsare plotted on the
three axes, also called SPECTROGRAM

+∞
∞−
−−=dtftjtgtxfSTFTx
)2exp()()(),(
*
πττ
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
STFT (2)
The STFT:
1. Map the signal in a 2D space time-frequency
2. Gives information on the evolution of the signal frequency content
3. Care should be done in selecting the time window length and overlapping
Windowlength: M Δt
Windowseparationq Δt
FrequencyresolutionΔfres
>≅1/M Δt
Longerwindows enhancethe frequencyresolutionbutcompriselongertime
segments, worseningthe timeresolutionand leadingtoinaccuracy.
TimeresolutionΔtres
> M Δt/2
A goodtradeoff betweentimeand frequencyresolutionissuchthat:
Δfres
Δtres
> 1/2
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Time-frequencyanalysis
demo_21
Whenthe frequencycontentofa signalvarieswithtime(nonstationary
signals, nonlinearsystems, etc…), anothertypeofrepresentationis
needed.
Letsconsider3
Harmoniccomponents
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, f=20Hz (stationary periodic signal)
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, f=50Hz (stationary periodic signal)
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signals windowing components


nonstationary
stationary
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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Time-frequencyanalysis(2)
1. Stationarysignal, frequency
contentconstantin time
2. Nonstationarysignal, frequency
contentvaryingwithtime
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s1
s2
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s2


nonstationary
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(2/5)*s
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+(2/5)*s
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3
signals in time domain


stationary
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Time-frequencyanalysis(3)
FFT withequalfrequencycontent
The frequencyanalysisdoesnotgive
anytiming information!!!
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nonstationary
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amplitude
FFT


stationary
SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
ModuleIII Lab
Time-frequencyanalysis: STFT
The STFT is able to detect the
variation of the frequency content
with time
1. Stationary signal, frequency
content constant in time
2. Nonstationarysignal, frequency
content varying with time
time [s]
frequency [Hz]


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time-frequency analysis


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SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR EUROPEAN SYNERGIES
Preparatory Course: experimental testing and theoretical background –17/19 March 2010
References
J. C. Santamarina, D. Fratta, 2005. Discrete Signals and Inverse Problems:
An Introduction for Engineers and Scientists. Wiley
L. Lo Presti, F. Neri, 1992. L'analisideisegnali. CLUT
The Mathworks, 2007. Signal processing Toolbox, User’s Guide
R.W. Clough, J. Penzien, 1993. Dynamics of Structures
J. Shao, 2003. Mathematical Statistics