Introduction to FFT Analysis

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

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7.70e-3

Am

plitu

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g


DSP in Test.Lab

Acquisition Time?

Frequency Resolution?


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


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


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


… and they can look … hmm … interesting

Ariane 5 launch and …


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

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2

y

u

x

uk

t

u-2.5

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Any signal can be described as a combination of sine waves of different frequencies

Useful by-product


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 …

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f [Hz]10 20 40

Detect sine waves in signal Draw line at frequency of sine wave


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


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


Examples – Fourier transform


Bridge Vibrations

t

t

f

t

Traffic

Shaker

Drop

weight

Time domain Frequency domain


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


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

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1

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1

1.5

Consequences ?


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


Sampling

Sine wave of 10 Hz, sampled at 100 Hz

Digital representation looks like a perfect sine

Following slides:

Reducing sampling frequency

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time - seconds

amp

litu

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sampling frequency = 1000 Hz

10 Hz harmonic function

T=N t


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time - seconds

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plit

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sampling frequency = 100 Hz


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sampling frequency = 100 Hz.

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tNT


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T N t=


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T N t=

20 20.2 20.4 20.6 20.8 21-2

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time - seconds

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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)


Sampling = only look from time to time …

Different interpretations possible … ???

-1.5

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1.5

t tf s

1

t tf s

1


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

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ff ff

Correct Observed

20 201.3


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


Example

Alias-free

Frequency range

suffering from aliasing


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)






• Remedy




• Remedy



max2 ffssf

ALIASING

LEAKAGE


Finite Observation Length

Limited observation

Discrete Spectrum Periodicity Assumed

Complete original signal

We are NOT analysing

the original signal !!


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Finite Observation – Side Effect

Adverse effects

Wrong amplitudes

Smearing of the

spectrum

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Leakage

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( m/s

2)

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( m/s

2)

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dB

( m/s

2)

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-60.00

0.00 dB

( m/s

2)

Linear scale

Log scale

Linear scale

Log scale

Expected spectrum of a

pure sine wave


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 ”


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


Window Types – Specific CharacteristicsT

ime d

om

ain

Fre

q. d

om

ain

Rectangular, uniform Hanning Flat top


Example 1

Periodically observed sine

Rectangular window

Hanning window

Non-periodically observed sine

Rectangular window

Hanning window

0.00 100.00 Hz

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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


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






• Remedy




• Remedy



And perhaps a 3rd one:

Amplitude discretisation (e.g. 16/24 bit ADC)

max2 ffssf

ALIASING

LEAKAGE


Amplitude discretisation – problem

Small variations are not

detected

Amplitudes are

approximated

Small signals look ―bad‖

6

7

5

4

3

2

1

0


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


So we need assistance for

Filtering

Several possible sample frequencies

Windowing

Amplification

Sufficient possible amplitude values


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 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 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


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


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


5 : Belgian blocks

1 : runups

2 : ramps

3 : asphalt

4 : ramps

Road load data analysis


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

C:/Documents and Settings/jdc/Desktop/application cases pdf/Fiat Research Center Predicts Fatigue Life up-front in Product Development.pdf


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


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



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

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-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



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



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


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:



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:



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


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