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Sales New Hires Training 2008
Bart Peeters
Test Technology: DSP (Digital Signal Processing)
Fourier transform – Aliasing & leakage – Measurement functions
2 copyright LMS International - 2008
Lecture objectives
Understand the importance
of the Discrete Fourier
Transform (DFT)
Be able to explain aliasing
and leakage
See the advantages of
frequency-domain
measurement functions
By completing this lecture, you will:
0.00 800.00 Hz
0.00
7.70e-3
Am
plitu
de
g
3 copyright LMS International - 2008
DSP in Test.Lab
Acquisition Time?
Frequency Resolution?
4 copyright LMS International - 2008
Signals and processing
Signal: measurable quantity carrying information on some physical phenomenon
Pressure, displacement, acceleration, …
Temperature, voltage, biomedical potential (EKG, EEG, ...)
…
Information contained in the variation of the quantity over time (space, …)
This signal is measured with a sensor
This signal is what you want to analyse in view of a particular problem
Analog Signal
5 copyright LMS International - 2008
Signals and processing
Signal Processing: specific manipulations of the measured signals to:
Extract the key information
Understand the physical problem
Provide input data for specific analysis or even simulations
Modify the signal for specific applications
Digital Signal Processing: doing all this using computer-based systems
Transform the sensor signal in a stream of digital words
• Most sensors have an analog signal output
• Computers are limited to analysing finite datasets
Discretisation in time and in amplitude
6 copyright LMS International - 2008
System
TransferReceiver
Road
Wheel & TireSteering Wheel
Shake
Seat Vibration
Rearview mirror
vibration
Engine
Signals everywhere …
X =
Gearbox and
Transmission
Turbomachinery
Accessories
RotorCockpit vibration &
noise
Cabin comfort
Noise at Driver’s &
Passenger’s Ears
Structural Integrity
Environmental
sources
Source
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… and they can look … hmm … interesting
Ariane 5 launch and …
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Joseph did help us a lot …
Joseph Fourier (º1768 - †1830)
Théorie analytique de la chaleur
(1822)
Fourier’s law of heat conduction
Analyzed in terms of infinite
mathematical series
2
2
2
2
y
u
x
uk
t
u-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4
-3
-2
-1
0
1
2
3
4
Any signal can be described as a combination of sine waves of different frequencies
Useful by-product
9 copyright LMS International - 2008
Fourier transform
To go from time to frequency domain and back
Fourier integral:
Supported by modern signal analysers‖Spectrum analysers‖
Basic function in all our software
XtxF txXF 1
deXtx
dtetxX
tj
tj
2
1
For mathematicians …
For humans …
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4
-3
-2
-1
0
1
2
3
4
f [Hz]10 20 40
Detect sine waves in signal Draw line at frequency of sine wave
10 copyright LMS International - 2008
Some definitions
t [s]
f [Hz]
[rad/s]
T0
f0
0
Time domain
Frequency domain
Period: T0 [s]
Frequency: f0 = 1/T0 [Hz]
Pulsation / circular frequency:
0 = 2 f0 = 2 /T0 [rad/s]
1 rad
2
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Frequency spectrum – Time history
Selection of domain, depending on the application aims
Equivalence of time and frequency domain: no loss of information
Time TimeFrequency Frequency
f
f
f
f
f
ft
t
t
t
t
t
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Examples – Fourier transform
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Bridge Vibrations
t
t
f
t
Traffic
Shaker
Drop
weight
Time domain Frequency domain
14 copyright LMS International - 2008
There exist more domains
Representation of signals for analysis
t
A
f
A2/ f
A
P
f
A2/ f
A
P
t
A
Time domain:
The time history x(t)
Frequency domain:
The signal spectrum X( )
Amplitude domain:
The probability distribution P(A)
Gaussian
distribution
Uniform
distribution
15 copyright LMS International - 2008
Nice theory … but we must do it on a computer
Sampled signals
Discrete time history
Discrete frequency spectrum
Finite signal segments
Limited number time samples
Limited number of frequency lines
Numerical representation
Discrete number of possible amplitude
values
XtxF txXF 1
deXtx
dtetxX
tj
tj
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1.5
-1
-0.5
0
0.5
1
1.5
Consequences ?
16 copyright LMS International - 2008
Discretisation Effects:
Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannon’s Theorem
• Remedy
Use band-limited signals
Use low-pass filtering
The sampled function is not transient and not periodic
• Remedy
Use periodic signals
Apply windowing (errors remain!)
max2 ffssf
ALIASING
LEAKAGE
17 copyright LMS International - 2008
Sampling
Sine wave of 10 Hz, sampled at 100 Hz
Digital representation looks like a perfect sine
Following slides:
Reducing sampling frequency
0.2 0.4 0.6 0.8 1-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
amp
litu
de
sampling frequency = 1000 Hz
10 Hz harmonic function
T=N t
18 copyright LMS International - 2008
0 2 4 6 8 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ude
sampling frequency = 100 Hz
10 Hz harmonic function
4 4.2 4.4 4.6 4.8 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 100 Hz.
4 4.2 4.4 4.6 4.8 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 100 Hz.
tNT
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0 5 10 15 20 25-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 40 Hz.
10 Hz harmonic function
10 10.2 10.4 10.6 10.8 11-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 40 Hz.
10 10.2 10.4 10.6 10.8 11-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 40 Hz.
T N t=
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0 10 20 30 40 50-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 20 Hz.
10 Hz harmonic function
T N t=
20 20.2 20.4 20.6 20.8 21-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 20 Hz.
20 20.2 20.4 20.6 20.8 21-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
am
plit
ud
e
sampling frequency = 20 Hz.
21 copyright LMS International - 2008
Sampling: exploring the limits …
Sampling frequency = sine wave
frequency
fs = fsine
Observed frequency = 0 Hz (DC)
Sampling frequency = 2 x sine wave
frequency
fs = 2 x fsine
Observed frequency is correct, but it is
borderline (sampling frequency cannot be
lowered)
22 copyright LMS International - 2008
Sampling = only look from time to time …
Different interpretations possible … ???
-1.5
-1
-0.5
0
0.5
1
1.5
t tf s
1
t tf s
1
23 copyright LMS International - 2008
Sampling – Potential source of trouble
20 Hz signal, sampled at 21.3 Hz, shows up as a 1.3 Hz signal “Aliasing”
fs 2fs 3fsfs/20
True
frequencies
“Sampled”
frequencies fs/2
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
ff ff
Correct Observed
20 201.3
24 copyright LMS International - 2008
Aliasing Protection
Low-Pass Filter
Make sure the signal does not contain frequencies above half the sample frequency fs
Do this by applying a sufficient performing low-pass filter
Be aware that the amplitude of the last portion of the spectrum is attenuated by the filter Alias-free
Automatically done in good data acquisition hardware
25 copyright LMS International - 2008
Example
Alias-free
Frequency range
suffering from aliasing
26 copyright LMS International - 2008
Aliasing – sometimes positive
Something strange?
Glass vibrates at 608
Hz, while we see it
vibrating at 2 Hz!
Sampling by
stroboscope at 101 Hz
(Operating range is 0 –
120 Hz)
6 x 101 Hz = 606 Hz
For the human eye: 101
Hz = analog (we don’t
see the samples)
27 copyright LMS International - 2008
Discretisation Effects:
Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannon’s Theorem
• Remedy
Use band-limited signals
Use low-pass filtering
The sampled function is not transient and not periodic
• Remedy
Use periodic signals
Apply windowing (errors remain!)
max2 ffssf
ALIASING
LEAKAGE
28 copyright LMS International - 2008
Finite Observation Length
Limited observation
Discrete Spectrum Periodicity Assumed
Complete original signal
We are NOT analysing
the original signal !!
29 copyright LMS International - 2008
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
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-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
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-2
-1.5
-1
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0.5
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1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Finite Observation – Side Effect
Adverse effects
Wrong amplitudes
Smearing of the
spectrum
-2.5
-2
-1.5
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0
0.5
1
1.5
2
2.5
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-1
-0.5
0
0.5
1
1.5
2
2.5
Leakage
0.00 100.00 Hz
0.00
1.00
Am
plit
ude
( m/s
2)
0.00 100.00 Hz
0.00
1.00
Am
plit
ude
( m/s
2)
0.00 100.00 Hz
-60.00
0.00
dB
( m/s
2)
0.00 100.00 Hz
-60.00
0.00 dB
( m/s
2)
Linear scale
Log scale
Linear scale
Log scale
Expected spectrum of a
pure sine wave
30 copyright LMS International - 2008
Leakage – Amplitude Uncertainty
Periodic observation
100% of amplitude
A-periodic observation
63% of amplitude
“ Boss, this 100.000$ system is giving me
something between 6 and 10g ”
31 copyright LMS International - 2008
Reducing Leakage by Applying Time Windows
Leakage originates from finite observation
(discontinuity-error at edges)
Original signal properties are best
represented in the middle of the observation
period : enhance information
Practical implementation : multiplication
by window-function (time domain) to reduce
discontinuities
Effects :
Improved amplitude estimate ( flatten
central lobe)
Reduce frequency range of smearing
( lower side lobes)
Local smearing of spectral energy due
to wider central lobe effective
spectral resolution decreases
32 copyright LMS International - 2008
Window Types – Specific CharacteristicsT
ime d
om
ain
Fre
q. d
om
ain
Rectangular, uniform Hanning Flat top
34 copyright LMS International - 2008
Example 1
Periodically observed sine
Rectangular window
Hanning window
Non-periodically observed sine
Rectangular window
Hanning window
0.00 100.00 Hz
-100.00
0.00
dB
( m/s
2)
AutoPow er_Per_Hann
AutoPow er_Per_Rect
0.00 100.00 Hz
-100.00
0.00
dB
( m/s
2)
AutoPow er_Nonper_Hann
AutoPow er_Nonper_Rect
35 copyright LMS International - 2008
Example 2
2 sines which are non-periodic within the measurement period. The amplitude of
the second sine is 100 lower than the amplitude of the dominant sine.
Alternatively: measure longer!
Rectangular
Flat top
Hanning
Kaiser-
Bessel
36 copyright LMS International - 2008
Discretisation Effects:
Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannon’s Theorem
• Remedy
Use band-limited signals
Use low-pass filtering
The sampled function is not transient and not periodic
• Remedy
Use periodic signals
Apply windowing (errors remain!)
And perhaps a 3rd one:
Amplitude discretisation (e.g. 16/24 bit ADC)
max2 ffssf
ALIASING
LEAKAGE
37 copyright LMS International - 2008
Amplitude discretisation – problem
Small variations are not
detected
Amplitudes are
approximated
Small signals look ―bad‖
6
7
5
4
3
2
1
0
38 copyright LMS International - 2008
Amplitude discretisation – solution
Amplify signal to cover optimally
available input range
Many bits in ADC to provide many
possible values
So we can descrive accurately
small variations
Currently 24 bit ADC
6
7
5
4
3
2
1
0
MAXIMUM VOLTAGE
MINIMUM VOLTAGE
More in next lecture:
The measurement chain
39 copyright LMS International - 2008
So we need assistance for
Filtering
Several possible sample frequencies
Windowing
Amplification
Sufficient possible amplitude values
40 copyright LMS International - 2008
8
…
8
…
8
…
8
…
Analog sensor signal Fourier transform (infinite integral)
Sampled signal Discrete-time Fourier transform (DTFT)
Finite observation length Discrete Fourier transform (DFT)
Repetition of time blocks Sampled freq. domain (“spectral lines”)
Repetition of spectraSampled time domain
Fourier & Co
41 copyright LMS International - 2008
DFT Parameters
Block size N
Sampling interval t = 1/fs
Observation time T = N t
Sampling frequency fs = 1/ t
Nyquist frequency (bandwidth) fN = fs/2
Spectral lines Ns = N/2
Frequency resolution f = 1/T = fs /N
Time domain Frequency domain
t f
fN fs0f
t
T
42 copyright LMS International - 2008
DSP in Test.Lab
Spectral test specification:
Maximal signal frequency of
interest
Bandwidth (fmax, fN)
Sampling (fs, t)
Frequency separation
requirement
Resolution ( f)
Observation time
(T) and block size (N)
Aliasing prevention
Sample high enough +
filtering
Leakage prevention
Periodic signals,
transient signals, or
windowing
44 copyright LMS International - 2008
Signal analysis measurement functions
Time domain and frequency domain calculations to
extract specific information from the test signals
Time history
Time data segment statistics
Auto/cross correlation function
Frequency spectrum, auto/cross power spectrum
Rotating machinery tracked spectrum analysis (See
Signature Testing lecture)
Coherence and Frequency Response Function (See
Structural Testing lecture)
The key issues to select a function are:
What information is needed? How is this information
best brought forward from the signal?
Averaging to enhance weak signal components
Absolute values
0.00 80.00Hz
-140
-40
dB
((m/s
2)/
N
)
22.56 41.19
s
Time w inr:61:+Z
Time w inr:62:+ZAveraging
45 copyright LMS International - 2008
23/11/2002: Bradford City – Sheffield United: 0 – 5
Data acquisition: 4 h
Sampled at 80 Hz (down-sampled to 20 Hz)
Sliding RMS value ( — )
1000 samples, 50% overlap
0.00 15000.00 s
-0.02
0.02 R
eal
(m/s
2 )
-0.20
0.20
Real
(m/s
2 )
F time_record roof:1:+X / Root Mean Square
B time_record roof:1:+X
Goal 1 Goal 2 Goal 3 Goal 4 Goal 5
Half time EmptyFilling Seated Emptying
End of game
46 copyright LMS International - 2008
5 : Belgian blocks
1 : runups
2 : ramps
3 : asphalt
4 : ramps
Road load data analysis
47 copyright LMS International - 2008
To design representative test scenarios
Accelerated durability testing cycles
Meeting 1.2 million km durability
requirement
Real tests would take 3 years
Large-scale customer data collection
5000 km Turkish public road data
Ford Lommel proving ground
Development of accelerated rig test
Target setting
Test schedule definition
Resulting test schedule 8 weeks
Test acceleration of factor 100
LMS engineers performed dedicated data collection, applied extensive load
data processing techniques and developed a 6- to 8-week test track sequence
and 4-week accelerated rig test scenario that matched the fatigue damage
generated by 1.2 million km of road driving.
1
Damage based on strain gage signals, full truck
48 copyright LMS International - 2008
Electric motor powers machinery through gear reduction drive units
Increased vibration level from wear
Gearbox geometry
Main shaft frequency: 59.7 Hz
Final shaft frequency
• 59.7*(17/55)*(20/68) = 5.43 Hz
Final gear mesh frequency
• 5.43*68 = 369 Hz
Fs = 1024 Hz
0.00 400.00 Hz
0.00
0.04
Am
plit
ude
(m/s
2 )
60.00 369.00
Applications: Electric Motor & Gear Mesh Analysis
49 copyright LMS International - 2008
0.00 400.00 Hz
10.0e-6
0.10
Log
(m/s
2
)
369 30 184 60
Main shaft frequency
Half of the main shaft frequency Harmonics of the main shaft frequency
Half of the gear mesh frequency Gear mesh frequency
Applications: Electric Motor & Gear Mesh Analysis
50 copyright LMS International - 2008
Monitor current drawn by electrical
motor
Spacing and asymmetry in the
sidebands related to defects in the
motor
Analysis
60 Hz running frequency of motor
Power line sidebands: 2.75
Hz/sideband away from 60 Hz
carrier
Motor slip sidebands: 1.25 Hz
away from 60 Hz carrier
35.00 85.00 Hz
-100.00
0.00
dBA2
N = 1024, f = 1 Hz
N = 2048, f = 0.5 Hz
N = 8192, f = 0.125 Hz
Current probe power spectra
Hanning
Applications: Electric Motor & Gear Mesh Analysis
51 copyright LMS International - 2008
Power spectra – N = 8192, f = 0.125 Hz
55.00 65.00 Hz
-100.00
0.00
dBA2
35.00 85.00 Hz
-100.00
0.00
dBA2
Zoom
Rectangular window
Hanning window
Kaiser-Bessel window
Applications: Electric Motor & Gear Mesh Analysis
52 copyright LMS International - 2008
0.00 1000.00 Hz
1.00e-6
10.0e-3
Logg
Autopower Example:
Pump Vibration Signatures
Misalignment between motor and pump
assemblies causes excessive bearing
wear
Good alignment shows up as reduced
harmonic content
Accelerometer measurement on the
motor bearing cap
Computation of vibration signatures
Power Spectra
Linear
RMS
Hanning
Amplitude correction
N = 1024
fs = 2048 Hz
Good alignment
Bad alignment
0.00 52.00 s
-0.10
0.10
Real
g
Good
Bad
53 copyright LMS International - 2008
Harmonic cursor, display limited to 800 Hz, dB amplitude scale
0.00 800.00 Hz
1.00e-6
10.0e-3
Log
g
29.73
Good alignment = reduced harmonic content
Bad alignment
Autopower Example:
Pump Vibration Signatures
54 copyright LMS International - 2008
0.00 800.00 Hz
0.00
7.70e-3
Am
plit
ude
g
Good alignment = reduced harmonic content
Bad alignment
Linear amplitude scale
Autopower Example:
Pump Vibration Signatures
55 copyright LMS International - 2008
Industrial Printer Noise Problem
Story
Industrial printer
Excessive noise level
Measure effectiveness of noise
abatement shroud
0.00 33.00 s
-0.60
1.30
Real
Pa
Before noise shroud
With noise shroud
22.39 22387.21Octave 1/3
Hz
20.00
70.00
dBPa
20.00
70.00
dB Pa
A L
25.0 20000.0
Curve 25.0 20000.0 RMS Hz
28.1 46.4 69.2 dB dB
27.7 36.5 66.9 dB dB
1/3 octave band representation
56 copyright LMS International - 2008
Course summary
Good acquisition
system: aliasing
protection
Amplitude
discretisation
DFT = Discrete
Fourier
Transform
Measurement
functions
Skilled
experimentalist:
leakage
mitigation
Sales New Hires Training 2008
Bart Peeters
Thank you
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