Introduction to FFT

8/21/2019 Introduction to FFT

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Test Technology: DSP (Digital Signal Processing)

Fourier transform Aliasing & leakage Measurement functions

Hong WengCustomer Service EngineerLMS - A Siemens Business

[email protected]


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2 copyright LMS International - 2013

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


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


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



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


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


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



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


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



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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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Remedy




Remedy



( )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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Remedy




Remedy



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




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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




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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




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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


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



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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




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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


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

[email protected]

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

Documents

Introduction to FFT