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Test Technology: DSP (Digital Signal Processing)
Fourier transform Aliasing & leakage Measurement functions
Hong WengCustomer Service EngineerLMS - A Siemens Business
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Lecture objectives
Understand the importance
of the Discrete FourierTransform (DFT)
Be able to explain aliasingand leakage
See the advantages offrequency-domainmeasurement functions
By completing this lecture, you will:
0.00 800.00Hz
0.00
7.70e-3
Amplitude
g
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DSP in Test.Lab
Acquisition Time?Frequency Resolution?
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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
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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
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SystemTransferSystemTransfer
ReceiverReceiver
Road
Wheel & Tire Steering WheelShake
Seat Vibration
Rearview mirrorvibration
Engine
Signals everywhere
X =
Gearbox andTransmission
Turbomachinery
Accessories
RotorCockpit vibration &
noise
Cabin comfort
Noise at Drivers &Passengers Ears
Structural Integrity
Environmental
sources
SourceSource
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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 (1768 - 1830)
Thorie analytique de la chaleur(1822)
Fouriers law of heat conduction
Analyzed in terms of infinitemathematical 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 acombination of sine waves of differentfrequencies
Useful by-product
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Fourier transform
To go from time to frequency domain and back
Fourier integral:
Supported by modern signal analysersSpectrum 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
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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 = 2f0 = 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
Dropweight
Time domain Frequency domain
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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)
Gaussiandistribution
Uniform
distribution
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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 amplitudevalues
( )[ ] ( )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 ?
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Discretisation Effects:Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannons 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!)
( )max
2 ffs
s
f
ALIASING
LEAKAGE
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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
am
plitude
sampling frequency = 1000 Hz
10 Hz harmonic function
T=Nt
100 Hz
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0 2 4 6 8 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time - seconds
amplitude
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
amplitude
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
amplitude
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
amplitude
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
amplitude
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
amplitude
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
amplitude
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
amplitude
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
plitude
sampling frequency = 20 Hz.
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Sampling: exploring the limits
Sampling frequency = sine wavefrequency
fs = fsine
Observed frequency = 0 Hz (DC)
Sampling frequency = 2 x sine wavefrequency
fs = 2 x fsine
Observed frequency is correct, but it isborderline (sampling frequency cannot belowered)
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Sampling = only look from time to time
Different interpretations possible ???
-1.5
-1
-0.5
0
0.5
1
1.5
t tfs
= 1
t tfs
= 1
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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
Truefrequencies
Sampledfrequencies
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
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Aliasing ProtectionLow-Pass Filter
Make sure the signal does not containfrequencies above half the sample frequency fs
Do this by applying a sufficient performing low-pass filter
Be aware that the amplitude of the last portionof the spectrum is attenuated by the filter Alias-free
Automatically done in good data acquisition hardware
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Example
Alias-free
Frequency rangesuffering from aliasing
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Aliasing sometimes positive
Something strange?
Glass vibrates at 608Hz, while we see itvibrating at 2 Hz!
Sampling bystroboscope at 101 Hz(Operating range is 0 120 Hz)
6 x 101 Hz = 606 Hz
For the human eye: 101Hz = analog (we dontsee the samples)
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Discretisation Effects:Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannons 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!)
( )max
2 ffs sf
ALIASING
LEAKAGE
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Finite Observation Length
Limited observation
Discrete Spectrum Periodicity Assumed
Complete original signal
We are NOT analysingthe original signal !!
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Finite Observation Side Effect
Adverse effects
Wrong amplitudes
Smearing of thespectrum
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Leakage
0.00 100.00Hz
0.00
1.00
Amplitude
(m/s2)
0.00 100.00Hz
0.00
1.00
Amplitude
(m/s2)
0.00 100.00Hz
-60.00
0.00
dB
(m/s2)
0.00 100.00Hz
-60.00
0.00
dB
(m/s2)
Linear scale
Log scale
Linear scale
Log scale
Expected spectrum of apure sine wave
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Leakage Amplitude Uncertainty
Periodic observation100% of amplitude
A-periodic observation63% of amplitude
Boss, this 100.000$ system is giving mesomething between 6 and 10g
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Reducing Leakage by Applying Time Windows
Leakage originates from finite observation(discontinuity-error at edges)
Original signal properties are bestrepresented in the middle of the observation
period : enhance information
Practical implementation : multiplicationby window-function (time domain) to reducediscontinuities
Effects :
Improved amplitude estimate ( flattencentral lobe)
Reduce frequency range of smearing( lower side lobes)
Local smearing of spectral energy due
to wider central lobe effectivespectral resolution decreases
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Window Types Specific Characteristics
Timedoma
in
Freq
.domain
Rectangular, uniform Hanning Flat top
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Windowing Use Cases
Uniform (rectangular)
Only in leakage-free conditions
Hanning
Most commonly used for unknown signals
Compromise: amplitude relatively correct good frequency precision High side lobes may mask neighbouring frequencies with low amplitude
Kaiser-Bessel
Good selectivity (low side lobes): measure close frequencies with large amplitudedifferences
Flat top
Calibration: accurate amplitude measurement Very bad effective frequency resolution
Impact testing windows
Exponential (response)
Force-window (input signal)
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Example 1
Periodically observed sine
Rectangular window
Hanning window
Non-periodically observed sine
Rectangular window
Hanning window
0.00 100.00Hz
-100.00
0.00
dB
(m/s2)
AutoPower_Per_Hann
AutoPower_Per_Rect
0.00 100.00Hz
-100.00
0.00
dB
(m/s2)
AutoPower_Nonper_Hann
AutoPower_Nonper_Rect
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Example 2
2 sines which are non-periodic within the measurement period. The amplitude ofthe second sine is 100 lower than the amplitude of the dominant sine.
Alternatively: measure longer!
Rectangular
Flat top
Hanning
Kaiser-Bessel
Discretisation Effects:
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Discretisation Effects:Aliasing and Leakage
Two most frequently occurring problems using discretisation:
does not meet Shannons 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 3rdone:
Amplitude discretisation (e.g. 16/24 bit ADC)
( )max
2 ffs sf
ALIASING
LEAKAGE
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Amplitude discretisation problem
Small variations are notdetected
Amplitudes areapproximated
Small signals look bad
6
7
5
4
3
2
1
0
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Amplitude discretisation solution
Amplify signal to cover optimallyavailable input range
Many bits in ADC to provide manypossible values
So we can describe accuratelysmall variations
Currently 24 bit ADC
6
7
5
4
3
2
1
0
MAXIMUM VOLTAGE
MINIMUM VOLTAGE
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So we need assistance for
Filtering
Several possible sample frequencies
Windowing
Amplification
Sufficient possible amplitude values
C
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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
DFT P t
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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
DSP i T t L b
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DSP in Test.Lab
Spectral test specification:
Maximal signal frequency ofinterest
Bandwidth (fmax, fN)
Sampling (fs, t)
Frequency separationrequirement
Resolution (f)
Observation time(T) and block size (N)
Aliasing prevention
Sample high enough +filtering
Leakage prevention
Periodic signals,transient signals, orwindowing
Some histor
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Some history
Fourier series - Joseph Fourier (1822)
Origin
Discrete Fourier Transform (DFT)
Sampling + finite time
Fast Fourier Transform (FFT) Cooley & Tukey (1965)
Efficient algorithm for DFT
Power of 2 number of samples (e.g. 512, 1024, 2048, 4096, )
Fastest Fourier Transform in the West (FFTW) Frigo & Johnson (1999) Efficient algorithm for DFT for non-power-of-2 number of samples
Signal analysis measurement functions
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Signal analysis measurement functions
Time domain and frequency domain calculations toextract 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 (SeeSignature Testing lecture)
Coherence and Frequency Response Function (SeeStructural Testing lecture)
The key issues to select a function are:
What information is needed? How is this informationbest brought forward from the signal?
Averaging to enhance weak signal components
Absolute values
0.00 80.00Hz
-140
-40
dB
((m/s2)/N)
22.56 41.19
s
Time winr:61:+ZTime winr:62:+ZAveraging
23/11/2002: Bradford City Sheffield United: 0 5
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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 15000.00s
-0.02
0.02
Real
(m/s2)
-0.20
0.20
Real
(m/s2)
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
Road load data analysis
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5 : Belgian blocks
1 : runups
2 : ramps
3 : asphalt
4 : ramps
Road load data analysis
To design representative test scenarios
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To design representative test scenarios
Accelerated durability testing cycles
Meeting 1.2 million km durabilityrequirement
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 loaddata 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
Applications: Electric Motor & Gear Mesh Analysis
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Electric motor powers machinery throughgear 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
400.000.00 Hz
0.04
0.00
Amplitude
m/s2
59.76 369.00
Applications: Electric Motor & Gear Mesh Analysis
Applications: Electric Motor & Gear Mesh Analysis
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0.00 400.00Hz
10.0e-6
0.10
Log
(m/s2
)
36930 18460
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
Applications: Electric Motor & Gear Mesh Analysis
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Monitor current drawn by electricalmotor
Spacing and asymmetry in thesidebands related to defects in themotor
Analysis
60 Hz running frequency of motor Power line sidebands: 2.75
Hz/sideband away from 60 Hzcarrier
Motor slip sidebands: 1.25 Hz
away from 60 Hz carrier
35.00 85.00Hz
-100.00
0.00
d
BA2
N = 1024, f = 1 HzN = 2048, f = 0.5 Hz
N = 8192, f = 0.125 Hz
Current probe power spectraHanning
Applications: Electric Motor & Gear Mesh Analysis
Applications: Electric Motor & Gear Mesh Analysis
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Power spectra N = 8192, f = 0.125 Hz
55.00 65.00Hz
-100.00
0.00
dBA
2
35.00 85.00Hz
-100.00
0.00
dBA
2
Zoom
Rectangular windowHanning window
Kaiser-Bessel window
Applications: Electric Motor & Gear Mesh Analysis
Autopower Example:Pump Vibration Signatures
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0.00 1000.00Hz
1.00e-6
10.0e-3
Log
g
Pump Vibration Signatures
Misalignment between motor and pumpassemblies causes excessive bearingwear
Good alignment shows up as reducedharmonic content
Accelerometer measurement on themotor bearing cap
Computation of vibration signatures Power Spectra
Linear
RMS
Hanning
Amplitude correction
N = 1024
fs = 2048 Hz
Good alignmentBad alignment
0.00 52.00s
-0.10
0.10
Real
g
Autopower Example:Pump Vibration Signatures
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Harmonic cursor, display limited to 800 Hz, dB amplitude scale
0.00 800.00Hz
1.00e-6
10.0e-3
Logg
29.73
Good alignment = reduced harmonic content
Bad alignment
Pump Vibration Signatures
Autopower Example:Pump Vibration Signatures
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0.00 800.00Hz
0.00
7.70e-3
Amplitu
de
g
Good alignment = reduced harmonic contentBad alignment
Linear amplitude scale
Pump Vibration Signatures
Industrial Printer Noise Problem
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Story
Industrial printer
Excessive noise level
Measure effectiveness of noiseabatement shroud
0.00 33.00s
-0.60
1.30
Real
Pa
22.39 22387.21Octave 1/3
Hz
20.00
70.00
dBP
a
20.00
70.00
dB P
a
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
Course summary
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Good acquisitionsystem: aliasing
protection
Amplitude
discretisation
DFT = Discrete
FourierTransform
Measurement
functions
Skilledexperimentalist:
leakagemitigation
Thank you
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Questions ?
Do not hesitate to contact me or the LMS Test Support team
Test Support Phone # : 248 502 2211
Or visit www.lmsintl.com
Please fill in the survey at the end of the WebEx
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Thank you