fft basics 2008
TRANSCRIPT
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Brüel & Kjær Sound & Vibration Measurement A/S.
Copyright © 2009. All Rights Reserved.
Baron Jean Baptiste Joseph Fourier
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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The Measurement Chain
Transducer Preamplifier Detector/
AveragerFilter(s) Display/
Output
RMS
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Sources of Machine Vibration
The moving parts of machines create vibration at different frequencies.
Frequency, Hz
Vibrationlevel
Vibration at the rotational frequency (1st order) of the main shaft, caused by unbalance
Harmonics of speed of main shaft caused by misalignment
Misalignment of the shaft causes vibration at lower harmonics = orders
(rotational frequency × 1, 2, 3,<)
2nd 3rd 4th 5th 6th 7th 8th 9th1st
(Frequency, Hz)Order
Extra high level of 6th harmonic/order due to imperfect fan with 6 blades
Vibration (suborders at 42-48% of RPM) caused by oil film whirl or whip in journal bearing
Vibrations at 50 or 60 Hz (incl. harmonics) caused by electromagnetic forces or electric
noise picked up from power cablesVibration caused by worn gear
12th
Vibration amplified by structural resonances in the machine structure
Many different vibrations compose a single frequency spectrum
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Easier Than This&
Imagine trying to determine all of the previous from just this!
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The Fourier Transform
( ) ( ) dtetgfG tf2j π−∞+
∞−∫=
( ) ( ) dfefGtg tf2j π∞+
∞−∫=
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“All Complex Waves&”
All Complex Waves are the Sum of Many Sine and Cosine Waves
∑
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Fourier Transform is a Mathematical Filter!
=Π∗ ttF 2sin)( Large #
=Π∗ ttF 4sin)( Med.#
=Π∗ ttF 8sin)( Small.#
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Types of Signals
Deterministic Random Continuous Transient
Non-stationary signalsStationary signals
Time Time
Frequency FrequencyFrequency Frequency
Time TimeTimeTime
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Types of Signals
TimeFrequency
Sine wave
Time Frequency
Square
wave
Time
Frequency
Transient
Time
Frequency
Ideal
Impulse
Infinite duration
in Time
Finite bandwidth
in Frequency
Finite duration in
Time
Infinite bandwidth
in Frequency
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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Filter Types
B = x Hz
B = 31,6 Hz
B = 10 Hz
B = 3,16 Hz
B = 1 octave
B = 1/3 octave
B = 3%
Constant Bandwidth Constant Percentage Bandwidth (CPB)or Relative Bandwidth
B = y% =y × f0100
0 40 8020 60 20010050 70 150LinearFrequency
LogarithmicFrequency
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What is the Fast Fourier Transform?
� An algorithm for increasing the speed of the computer
calculation of the Discrete Fourier transform.
– Reduces the number of multiplications from N2 to
(N/2)log2N
– Computation speed increased by a factor of 372 for an
800 line FFT
� Block analysis of time data samples to provide
equivalent frequency domain description
� Analysis with constant bandwidth filters
� “Rediscovered” in 1962 by Bell Lab scientists Cooley
and Tukey
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FFT Analyzer
Display/
Output
TransducerPreamplifier
A/DTime
Weighting
Memory
1
Memory
2
200 400 600 800 1k 1,2k 1,4k 1,6k 1,8k 2k HzLinearFrequency0
Amplitude
FFT Analysis
Squaring/
Averaging
GAAFFT
Auto-
spectrum
Fourier
Spectrum
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FFT Time Assumption
Rectangular or Uniform weighting
Hanning weighting
� Input signal
� Analysed signal
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FFT Fundamentals
dt = 488.3 µs
df = 1 Hz
Frequency (Hz)
Time
T = 1 s
Freq. Span
� Lines = resolution
� Span = upper freq. range
� df = Span/Lines
� T = 1/df
� dt = 1/(Span * 2.56)
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Most important Law in Frequency Analysis
B = bandwidth
T = measurement time
B ×××× T ≥≥≥≥ 1
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Uncertainty Principle Example
If the FFT
Analyzer is:
Then the lowest
freq. is 1 Hz
1 Hz 800 Hz
T = 1 s
To satisfy BT ≥≥≥≥ 1,
then 1 Hz / 1 = 1s
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Uncertainty Principle – Stationary Signals
If BT = 1 then we are ‘Certain’<but then accuracy is not very high. If we obtain more cycles then accuracy will improve greatly.
G(f)g(t)
Time
Freq.
Time
Freq.
G(f)g(t)
Time
Freq.
G(f)g(t)
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Uncertainty Principle – Non Stationary Signals
∆t
G(f) ∆f = 1/∆tg(t)
Time
Freq.
h(t) H(f) 2/τ
Time
τ Freq.
H(f)h(t)
Time
2/τ
Freq.τ
With transients things get more complicated. More resolution is not always the best<
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Are You Certain About Uncertainty (1)?
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Are You Certain About Uncertainty (2)?
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Are You Certain About Uncertainty (3)?
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What Happened? Which Is Correct?
Whichever best fit the time block!
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Ensuring Repeatable Measurements
� Think about the signal
you are measuring
– Stationary
– Transient
– Combination<
� Always report:
– Lines
– Span
– Window used
– Overlap
– Averaging Type
– # of Averages
– Start Trigger?
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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Pitfalls in DFT
Time Frequency
Sampling Aliasinggives
Time limitation Leakagegives
Periodic repetition Picket Fence Effectgives
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ALIASING
� Insufficient sampling allows a high frequency signal to masquerade
under a low frequency “alias”
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Aliasing
Time
x Hz
Freq.
x Hz
0 Hz
0 Hz
fs + x Hz
fs
Time
Freq.
Time
Freq.
Time
Freq.
?
?
fs
fs
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Anti-aliasing
filter On
Input
12
Anti-aliasing Filter
Input
Anti-aliasing
filter Off
A/D
0
2
1
Sampling Frequency
fspan fsfs/2Frequencyf1 f2 f3
f1 f2f3
f1
fspan = 2.56
fs
Example:
fs= 65 536 kHz ⇒ fspan = 25.6 kHz
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Leakage
Freq.Freq.
Freq.Freq.
TimeTime
Signal periodic with record length
Signal not periodic with record length
T T
Rectangularweighting
(no weighting)
Hanning weighting
π−
T
t2cos1
a(t) b(t)
A(f) B(f)
A(f) B(f)
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“Picket Fence” Effect
Frequency
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How to Avoid the Pitfalls of DFT
Solution:
2. Leakage:
Caused by sampling in frequency
1. Aliasing: Caused by sampling in time
Solutions:
Caused by time limitation
3. Picket fence effect:
� Use correct weighting (signals)
� Increase the frequency resolution (systems)
� Use anti-aliasing filter (fc) and
sampling rate fs>2 fc
Solutions: � Use correct weighting (signals)
� Increase the frequency resolution (systems)
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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Discontinuities in the Time Record
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Windows “smooth” the discontinuity
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Windows
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Use of Weighting Functions in Signal Analysis
Transients:
Continuous signals:
� General purpose, RTA
� Two-tone separation
� Calibration
� Pseudo random
� General purpose
� Short transient
� Long decaying transients
� Very long transients
Flat
TopTransientHanning
Rect-angular
Weighting
Expo-nential
Kaiser-
Bessel
+ overlap
+ overlap
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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Overlap Analysis with Hanning Weighting (1)
� No
overlap
Time
� 50% overlap
W(t)
Time
W(t)
� 662/3% overlap
Time
� 75% overlap
W(t)
Time
W(t)
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Overlap Analysis with Hanning Weighting (2)
� No overlap
W(t)
Time
Time
Time
Time
W(t)
W(t)
W(t)
� 662/3% overlap
(1 -cosx)2 = 1 - 2cosx + cos2x 1/3 [ (1 -cosx)2 + (1 - cos(x -
� 50% overlap
1/2 [(1 -cosx)2 = (1 + cosx)2 ] 1 = cos2x
32π
))2
+ (1 -cos(x -34π
))2 ] = 1.5
� 75% overlap
1/4 [ (1 -cosx)2 + (1 - sinx)2 + cosx)2
+ (1 -cos(x + sinx)2 ] = 1.5
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Window Summary
Window NoiseBandwidth
Ripple HighestSidelobe
Rectangle 1.0 ∆F 3.92 dB -13 dB
Hanning 1.5∆F 1.42 dB -31 dB
Kaiser-Bessel 1.8∆F 0.98 dB -68 dB
Flat Top 3.8∆F 0.009 dB -93 dB
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FFT Analysis 101
� Introduction
� Practical Set Up of FFT Analysers
� Pitfalls of an FFT Analyser
� Real-time Analysis
� Time Weighting
� Overlap Analysis
� Signal Types and Spectrum Units
� FFT Summary
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Random Signal
Signal Types and Spectrum Units
Correct use of Units
Time Frequency
Determistic,
Periodic SignalU
Time
U
Time
Time
Transient
U
U2/Hz
U2s/Hz
Freq.B
Freq.
Freq.B
Root Mean Square
RMS
Power Spectral Density
PSD
Energy Spectral Density
ESDU
T
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FFT - Summary
� The DFT has properties very similar to the integral Fourier Transform
� The DFT has certain pitfalls: aliasing, leakage and picket fence effect
� Recording time, sampling interval, sampling frequency, frequency span and
frequency resolution are all related
The Discrete Fourier Transform:
� Many different weightings exist for different purposes
� Use of the proper weighting can reduce leakage and picket fence effect errors
� Weightings can be regarded as filters
Weighting functions, leakage and picket fence effect:
� Condition for real-time analysis: T ≥ Tcalc.
� Other weightings than rectangular may require overlap analysis to avoid loss of
data or to get a flat overall weighting function
� Many different trigger functions exist for different purposes
Real-time Analysis, Overlap Analysis and Triggering:
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CPB Advantages & Disadvantages
Higher Processor DemandsResults are consistent
Not as common as FFT in
North America
Internationally standardized
Optimized , but limited freq.
resolution
Stereotyped as an acoustics
analyzer
Can measure ‘peak’, ‘RMS’
Limited, but optimized freq.
resolution (1/1,1/3,1/12,1/24)
Excellent response time
ConsPros
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FFT Advantages & Disadvantages
“Ambiguous” resultsLower Processor Demands
No true ‘peak’ detectionGreat for ‘exact’ freq.
determinations
Not standardized
WindowsVery common
LeakageConstant bandwidth filters,
make harmonic patterns
obvious
Block time analysisWide selection of averaging
types
Poor response timeHigh freq. resolution !ZOOM!
ConsPros
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Literature for Further Reading
� Frequency Analysis by R.B.Randall
(Brüel & Kjær Theory and Application Handbook BT 0007-11)
� The Fast Fourier Transform by E. Oran Brigham
(Prentice-Hall, Inc. Englewood Cliffs, New Jersey)
� The Discrete Fourier Transform and FFT Analyzers by N. Thrane
(Brüel & Kjær Technical Review No. 1, 1979)
� Zoom-FFT by N. Thrane
(Brüel & Kjær Technical Review No. 2, 1980)
� Dual Channel FFT Analysis by H. Herlufsen
(Brüel & Kjær Technical Review No. 1 & 2, 1984)
� Windows to FFT Analysis by S. Gade, H. Herlufsen
(Brüel & Kjær Technical Review No. 3 & 4, 1987)
� Who is Fourier?
(Transnational College of LEX, April 1995)
