SP – Signal Processing
Christophe Schram & Miguel Mendez
von Karman Institute for Fluid Dynamics
AR & EA depts
Why sampling a signal ?
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Analog vs. digital signals
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What is signal sampling ?
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Deterministic vs. random signals
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Infinite duration vs. transient signals
Transducers and signals
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Hot wire signal in a subsonic jet
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Pointwise measurement (?)
Velocity magnitude
Laser Doppler Velocimeter signal
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Fourier transform
Irregular sampling rate
Particle Image Velocimetry signal
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Cross-correlation
“Averaging”
The measured signal is itself the result of a more or less
implicit processing (à calibration)
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Time and frequency domains
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Time domain (sometimes) better suited to
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Transient signals:
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Broadband random signal:
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White noise à no significant spectral feature
Frequency domain (sometimes) better suited to
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Steady-state linear systems
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Impulse response of a system (e.g. hot wire calibration)
Multi-frequency excitation (e.g. acoustic scattering)
Harmonic signals
Numerous data processing techniques involve both domains
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Convolution in time-domain ßà Multiplication in frequency domain
Re-sampling
Short-duration Fourier transform, wavelet analysis
Data processing methods
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Statistical
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Spectral analysis
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Mean, standard deviation, histograms
Transducer calibration (beware of non-linear calibration curve…)
Uncertainty analysis
Fourier transform, Welch’s periodogram
Correlation analysis
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Auto- and cross-correlation, coherence (≠ causality!)
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Coherent structure eduction
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Model reduction
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Mathematical reduction: Singular Value Decomposition, Proper
Orthogonal Decomposition, Dynamic Mode Decomposition, …
Model-based reduction: acoustic mode decomposition, …
Course content
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Data sampling: A/D conversion
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Statistical analysis
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Hot wire anemometer, pressure probes
Correlation analysis
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Cosine transform, single vs double-sided, normalization, units, …
Welch’s periodogram method
Wavelet analysis
Dynamic calibration
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Fundamentals of statistics, statistical convergence, confidence intervals, …
Fourier analysis
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Sample and hold, sampling error, dynamic range, …
Aliasing
Filters
Coherence, de-noising, …
Coherent structures eduction
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Q-criterion, λ2-criterion, continuous wavelet analysis
Data sampling
Sample and hold, sampling error, dynamic range, …
Aliasing
Filters
Analog-to-digital (A/D) conversion
clock
amplifier
trigger
… + anti-aliasing filter
Analog-to-digital (A/D) conversion
A/D converter
Nb bits
Nb levels
Resolution
(10 V span)
3
8
1.25 V
12
4096
2.44 mV
16
65536
0.15 mV
Analog-to-digital (A/D) conversion
Multi-channel A/D conversion
Multiplexer
Sample and Hold
A/D conversion methods
Tracking A/D
Parallel
(flash)
A/D
Successive A/D
Real A/D converter issues
Aperture time
Analog-to-digital (A/D) conversion
Aliasing
Nyquist rule: in order to avoid aliasing, a signal must be sampled at a frequency
at least equal to twice the maximum frequency contained in the signal:
fs ≥ 2 fmax
How to know this max frequency ? Filter the signal à fmax = fcut-off,low
Data acquisition checklist
l What question am I trying to answer ?
Which transducer should I use ?
l Is it measuring the desired quantity, and only this one ?
l If not: what is it measuring ? Do the desired parameters
vary ? Can the global response be calibrated ?
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l What kind of processing will I need ?
Statistical vs. Fourier analysis ?
l Do I need an anti-aliasing filter ?
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l How to setup the A/D chain ?
Maximum possible voltage ? Minimum voltage to resolve ?
l Do I need an amplifier ? With offset ?
l What is the Nyquist frequency ?
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Fourier analysis
Cosine transform, single vs double-sided, normalization, units, …
Welch’s periodogram method
Wavelet analysis
Fourier series
Baron
Joseph
Fourier
(1768-1830)
An infinite number of coefficients is generally necessary
l Cosine or sine series correspond to even or odd functions, resp.
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Fourier series of a periodic square wave
Gibbs phenomenon
Fourier series: complex form
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Fourier series:
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Introducing
we have
with
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One coefficient includes amplitude and phase information
Fourier transform
l Fourier transform of a periodic signal:
l Fourier transform of an aperiodic signal:
Power Spectral Density
l Power Spectral Density:
l Auto-correlation function:
Fourier transform and convolution
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Response of a system with transfer function h(t), subjected to
excitation x(t) : convolution product
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We have:
Fourier transform and convolution
l Convolution in time domain = product in Fourier
domain, and vice-versa
l Determination of transfer function of a system:
l Calculation of system response:
Discrete Fourier transform
Try to use Nt = 2N à Fast Fourier Transform
l Frequency axis:
f=[0:Nt-1]'/Nt*fs;
l Normalization Fourier transform:
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Fourier coefficient, double-sided FFT: X=fft(x)/Nt;
Fourier coefficient, single-sided FFT: X=2*fft(x)/Nt;
Parseval equality: X=fft(x)/fs;
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Integral equal to variance: Xs2=fft(x)/sqrt(fs*(Nt-1));
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Fourier transform averaging
Fourier transform usually yields quite noisy spectrum
l To obtain smoother spectrum à averaging (pwelch)
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FFT of full-length
signal (4096 pts)
average of 32 FFTs of signal
segments (128 pts each)
[S_pw,f_pw] =
pwelch(s,hann(Nfft),Noverlap,Nfft,fs);
Normalization of the pwelch transform
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[S_pw,f_pw]=
pwelch(s,hann(Nfft),Noverlap,Nfft,fs);
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Parseval equality: P=sum(S1_pw)*Nt/Nfft;
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Integral equal to variance:
S_pws2=
S_pw*Nt/((Nt-1)*Nfft)*fs/(f_pw(2)-f_pw(1));
Matlab demo
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Bad synchronization of pure sine à leakage
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Construction of the frequency axis
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Normalization
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Parseval’s equality
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Fourier transform averaging and pwelch function
Wavelet transform
l Principle: convolution of signal with a localized test
function instead of a periodic one
Mother
wavelet
Wavelet family
Selectivity in space and scale
Convolution:
Wavelet
Test signal
Cconv
Wavelet analysis vs. short-time
Fourier transform (Gabor filter)
Gabor’s uncertainty principle of signal processing: Δt Δω ≥ ½
l FT vs. STFT:
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Example for s = sin(2πt) + sin(2π2t) + sin(2π4t)
f (Hz)
f (Hz)
4
4
2
2
1
t (s)
STFT
1
t (s)
Wavelet
Example of continuous wavelet transform
sin(2π10t)
FFT
sin(2π50t)
white noise
narrow gaussian
CWT
Continuous vs. discrete
wavelet transforms
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Continuous wavelet transform:
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Well suited to data analysis
Infinitely redundant à inversion not straightforward
Discrete wavelet transform:
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Well suited to data compression (ex.: JPEG)
Less suited to data analysis
Signal and Data Processing
See slides of the
Introduction to Measurement Techniques
Lecture Series:
Fourier analysis applied to data acquisition
Statistics
Stationary Random Processes
l Define random processes
l Ensemble mean:
l Covariance:
Stationary
Stationary
and
Ergodic Processes
l Mean of the individual sample
l Covariance:
Ergodic
:
Ergodic
Joint Probability Density Function
l Two random variables
and
(e.g. pressure measured at two different
locations)
l Joint Probability Density Function (’and’):
Statistical Independence
l The individual probability density functions of
each variable can be obtained from the joint
function:
l If the variables
independent:
and
are statistically
Covariance
l Covariance between
and
l Particular case:
(variance)
l It can be shown that
:
Correlation Coefficient
l Correlation coefficient:
l Statistically independent random variables:
Covariance and Correlation
Functions
Properties
even
max
max
x and/or y
have zero
mean
Time-delay system
Time delay
Σ
Summary
l The ergodicity property of a stationary random
process allows to compute its statistical
properties using a single sample time sequence
instead of performing ensemble statistics
l Performing system analysis in a statistical sense is
more robust
Spectral Density Functions
l Can be obtained by 3 different means:
via finite Fourier transforms
l via Fourier transform of correlation function
l via filtering-squaring-averaging operations
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Fourier Transform of the
Correlation Functions
l If the auto-correlation and cross-correlation
functions
and
L1-norm are finite, i.e.
exist, and if their
, then:
Cosine Transform
l From the symmetry property of the auto-
correlation function:
Properties
l Double-sided spectral density functions:
Real-valued, symmetric
Complex-valued
l Single-sided spectral density functions:
Relation to Mean Square Values
l Inverse Fourier transform and Cosine transform:
l Mean square value:
Coherence function
l It can be shown that
l Coherence function
Time-delay system
Time delay
Σ
Data processing in frequency
domain
l The results obtained in frequency domain are
much stronger than in time domain, since they
apply for each frequency composing the signals
Coherent structure eduction
l Apparent order in chaos ?
l Phenomenological understanding of turbulence ?
l Reduced-order model for flow control ?
Galilean indicator functions
l Velocity field: not Galilean-invariant
l Vorticity field:
Galilean-invariant
l Signed
l Strong in shear layers as well
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l Enstrophy field:
Galilean-invariant
l Strong in shear layers as well
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Q-criterion
l Based on a decomposition of the velocity gradient
tensor
l Definition:
l Q > Qthresh à vortical motion
Reference
l J.S. Bendat and A.G. Piersol, Random Data –
Analysis and Measurement Procedures, Wiley &
Sons, Inc. (2nd Edition), 1986