Fft Basics 2008

Baron Jean Baptiste Joseph Fourier 

www.bksv.com Brüel & Kjær Sound & Vibration Measurement A/S. Copyright © 2009. All Rights Reserved.

FFT Analysis 101

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

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

The Measurement Chain Transducer

Preamplifier 

Filter(s)

Detector/ Averager 

RMS

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Display/ Output

Sources of Machine Vibration The moving parts of machines create vibration at different d ifferent frequencies.

Vibration level

1st 2nd 3rd 4th 5th 6th 7th 8th 9th

12th

Order  Frequency, Hz (Frequency, Hz)

Misalignment shaft vibration at Vibrations at of 50the or 60 Hzcauses (incl. harmonics) th st single Vibration Vibration Vibration Harmonics Extra Manyhigh different at (suborders amplified the level of rotational speed vibrations of by 6 at of structural 42-48% harmonic/order main frequency compose shaft of resonances RPM) caused (1 a due order) caused by to inof  harmonics orders causedVibration bylower electromagnetic caused by=worn forces gear  or electric bythe oil main film imperfect the whirl frequency shaft, machine misalignment orfan caused whip with spectrum structure in 6 by journal blades unbalance bearing noise picked up from power (rotational frequency 3,) × 1, 2,cables

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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

G(f ) = g(t ) =

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∫ ∫

+∞

g(t ) e − j2 π f t dt

−∞ +∞

−∞

G(f ) e j2 π f t df 

“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!

 F (t )

 F (t )

 F (t )

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∗ sin 2Πt  = Large #

∗ sin 4Πt  = Med.#

∗ sin 8Πt  = Small.#

Types of Signals Stationary signals

Non-stationary signals

Time

Deterministic Time

Frequency www.bksv.com, 10

Time

Random

Continuous Time

Frequency

Time

Frequency

Transient Time

Frequency

Types of Signals Sine wave Time

Frequency

Infinite duration in Time Finite bandwidth in Frequency

Square wave Time

Frequency

Transient Time Frequency

Ideal Impulse

Time Frequency

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Finite duration in Time Infinite bandwidth in Frequency

FFT Analysis 101

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

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

Filter Types Constant Bandwidth

Constant Percentage Bandwidth (CPB) or Relative Bandwidth y × f 0 B = y% = 100

B = x Hz

0

20

40

60

80

Linear  Frequency

B = 31,6 Hz B = 10 Hz B = 3,16 Hz

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50

70

100

150 200

Logarithmic Frequency

B = 1 octave B = 1/3 octave B = 3%

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  FFT Analysis Transducer 

Memory 1

Preamplifier 

Time Weighting

A/D

Memory 2

Squaring/ Averaging

FFT

GAA

Fourier  Spectrum

Autospectrum

Display/ Output

Amplitude

0

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200 400 600 800

1k 1,2k 1,4k 1,6k 1,8k

Linear  2k Hz Frequency

FFT Time Assumption 

Input signal



Analysed signal Rectangular or Uniform weighting

Hanning weighting

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FFT Fundamentals T=1s

Time



Lines = resolution



Span = upper freq. range



df = Span/Lines



T = 1/df 



dt = 1/(Span * 2.56)

dt = 488.3 µs

df = 1 Hz

Freq. Span

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Frequency (Hz)

Most important Law in Frequency Analysis

B×T≥1 B = bandwidth T = measurement time

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Uncertainty Principle Example If the FFT Analyzer is:

800 Hz

1 Hz

To satisfy BT ≥ 1, then 1 Hz / 1 = 1s

T=1s Then the lowest freq. is 1 Hz

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Uncertainty Principle – Stationary Signals g(t)

G(f) Time Freq.

g(t)

G(f) Time Freq.

g(t)

G(f) Time Freq.

If BT = 1 then we are ‘Certain’but then accuracy is not very high. If we obtain more cycles then accuracy will improve greatly. www.bksv.com, 20

Uncertainty Principle – Non Stationary Signals g(t)

G(f)

∆f = 1/∆t

Time

∆t

Freq.

h(t)

H(f)

2/τ

Time

τ

Freq.

h(t)

H(f)

2/τ

Time

τ

Freq.

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

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

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

Pitfalls in DFT

Time

Sampling

gives

Aliasing

Time limitation

gives

Leakage

Periodic repetition

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Frequency

gives Picket Fence Effect

ALIASING 

Insufficient sampling allows a high frequency signal to masquerade under a low frequency “alias”

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Aliasing Time f s

0 Hz

Freq.

Time f s

x Hz

?

Time 0 Hz

f s

?

Time x Hz www.bksv.com, 30

Freq.

f s + x Hz

Freq.

Freq.

Anti-aliasing Filter  A/D

Input

2

1

Input 0

Anti-aliasing filter Off  Anti-aliasing filter On

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f 1 f 3

f span f s/2

f 2

f s f 3

Frequency Sampling Frequency

f 1 f 2

1

f s f span = 2.56 f 1 2

Example: f s= 65 536 kHz ⇒ f span = 25.6 kHz

Leakage Signal periodic with record length

Signal not periodic with record length

a(t)

b(t) Time

Time

T

Rectangular  weighting

T

A(f)

B(f)

(no weighting) Freq.

Hanning weighting  1 − cos 2πt    T    

A(f)

B(f)

Freq. www.bksv.com, 32

Freq.

Freq.

“Picket Fence” Effect

Frequency

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How to Avoid the Pitfalls of DFT

1. Aliasing: Solution:

2. Leakage: Solutions:

Caused by sampling in time 

Caused by time limitation  Use correct weighting (signals) 

3. Picket fence effect: Solutions:

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Use anti-aliasing filter (f c) and sampling rate f s>2 f c

Increase the frequency resolution (systems)

Caused by sampling in frequency 

Use correct weighting (signals)



Increase the frequency resolution (systems)

FFT Analysis 101

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

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

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 Weighting RectHanning angular  Transients: 

General purpose



Short transient



Long decaying transients



Very long transients

Continuous signals: 

General purpose, RTA



Two-tone separation



Calibration



Pseudo random

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+ overlap

+ overlap

Transient

Exponential

KaiserBessel

Flat Top

FFT Analysis 101

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

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

Overlap Analysis with Hanning Weighting (1) 

No overlap



662/3% overlap

W(t) W(t)

Time



50% overlap



W(t)

75% overlap

W(t)

Time

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Time

Time

Overlap Analysis with Hanning Weighting (2) 

No overlap



662/3% overlap

(1 -cosx)2 = 1 - 2cosx + cos2x W(t)

W(t)

2π 2 1/3 [ (1 -cosx)2 + (1 - cos(x )) 3 4π 2 + (1 -cos(x )) ] = 1.5 3

Time



50% overlap

Time



1/2 [(1 -cosx)2 = (1 + cosx)2 ] 1 = cos2x

75% overlap 1/4 [ (1 -cosx) 2 + (1 - sinx)2 + cosx)2 + (1 -cos(x + sinx)2 ] = 1.5

W(t)

W(t)

Time

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Time

Window Summary

Window Rectangle Hanning  Kaiser-Bessel  Flat Top

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Noise Bandwidth 1.0 ∆F 1.5∆F 1.8∆F 3.8∆F

Ripple 3.92 dB 1.42 dB 0.98 dB 0.009 dB

Highest   Sidelobe -13 dB -31 dB -68 dB -93 dB

FFT Analysis 101

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

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

Signal Types and Spectrum Units Correct use of Units

Time Determistic, Periodic Signal

Frequency U

U

Root Mean Square RMS

Time

Random Signal

U

Freq. U2/Hz

Power Spectral Density PSD

Time B

Transient

U2s/Hz

U

Freq.

Energy Spectral Density ESD

Time T www.bksv.com, 45

B

Freq.

FFT - Summary The Discrete Fourier Transform: 

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

Weighting functions, leakage and picket fence effect: 

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

Real-time Analysis, Overlap Analysis and Triggering:

≥ Tcalc.



Condition for real-time analysis: T



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

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CPB Advantages & Disadvantages Pros

Cons

Excellent response time

Limited, but optimized freq. resolution (1/1,1/3,1/12,1/24)

Can measure ‘peak’, ‘RMS’

Stereotyped as an acoustics analyzer 

Internationally standardized

Not as common as FFT in North America

Results are consistent

Higher Processor Demands

Optimized , but limited freq. resolution

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FFT Advantages & Disadvantages Pros

Cons

High freq. resolution !ZOOM!

Poor response time

Wide selection of averaging types

Block time analysis

Constant bandwidth filters, make harmonic patterns obvious

Leakage

Very common

Windows

Great for ‘exact’ freq. determinations

No true ‘peak’ detection

Lower Processor Demands

“Ambiguous” results Not standardized

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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)

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