Fft Basics 2008

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Baron Jean Baptiste Joseph Fourier Baron Jean Baptiste Joseph Fourier

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FFT Analysis 101



 IntroductionIntroduction 

 Practical Set Up of FFT AnalysersPractical Set Up of FFT Analysers 

 Pitfalls of an FFT Analyser Pitfalls of an FFT Analyser 

 Real-time AnalysisReal-time Analysis 

 Time WeightingTime Weighting 

 Overlap AnalysisOverlap Analysis 

 Signal Types and Spectrum UnitsSignal Types and Spectrum Units 

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The Measurement Chain

T

Trraannssdduucceerr PrreP eaammpplliiffiieer r Detector/Detector/ Averager Averager Filter(s)

Filter(s) Display/Display/

Output Output

RMS RMS

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Sources of Machine Vibration

The moving parts of machines create vibration at d

The moving parts of machines create vibration at different frequencies.ifferent frequencies.

Frequency, Hz Frequency, Hz Vibration Vibration level level

Vibration at the rotational frequency (1

Vibration at the rotational frequency (1stst _{order) of}_{order) of}

the main shaft, caused by unbalance the main shaft, caused by unbalance Harmonics of speed of main shaft caused by Harmonics of speed of main shaft caused by

misalignment misalignment

Misalignment of the shaft causes vibration at Misalignment of the shaft causes vibration at

lower harmonics = orders lower harmonics = orders (rotational frequency (rotational frequency

×

1, 2, 3,)1, 2, 3,) 2 2ndnd₃₃rdrd₄₄thth₅₅thth₆₆thth₇₇thth₈₈thth₉₉thth 1 1stst (Frequency, Hz) (Frequency, Hz) Order Order

Extra high level of 6

Extra high level of 6thth _{harmonic/order due to}_{harmonic/order due to}

imperfect fan with 6 blades imperfect fan with 6 blades

Vibration (suborders at 42-48% of RPM) caused Vibration (suborders at 42-48% of RPM) caused

by oil film whirl or whip in journal bearing by oil film whirl or whip in journal bearing Vibrations at 50 or 60

Vibrations at 50 or 60 Hz (incl. harmonics)Hz (incl. harmonics) caused by electromagnetic forces or electric caused by electromagnetic forces or electric

noise picked up from power cables noise picked up from power cables

Vibration caused by worn gear Vibration caused by worn gear

12 12thth

Vibration amplified by structural resonances in Vibration amplified by structural resonances in

the machine structure the machine structure

Many different vibrations compose a single Many different vibrations compose a single

frequency spectrum 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

( )

f

g

( )

t

e

dt

G

+

∞

−

j2

π

f t

∞

−

∫

=

( )

t

G

( )

f

e

df

g

+

∞

j2

π

f t

∞

−

∫

=

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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

∗

t t F

₍

₎

_sin

₂

_{Large #}

=

Π

∗

t t F

₍

₎

_sin

₄

Med.#

=

Π

∗

t t F

₍

₎

_sin

₈

Small.#

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Types of Signals

Deterministic Random Continuous Transient

Non-stationary signals Stationary signals

Time Time

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Types of Signals

Time Frequency 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

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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

×

f 0 100 0 20 40 60 80 50 70 100 150 200 Linear Frequency Logarithmic Frequency

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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

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FFT Analyzer

Display/ Output Transducer

Preamplifier

A/D _WeightingTime

Memory 1 Memory 2 200 400 600 800 1k 1,2k 1,4k 1,6k 1,8k 2k Hz Linear Frequency 0 Amplitude FFT Analysis Squaring/ Averaging G_AA FFT Auto-spectrum Fourier Spectrum

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FFT Time Assumption

Rectangular or Uniform weighting

Hanning weighting

 Input 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

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Most important Law in Frequency Analysis

B = bandwidth

T = measurement time

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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,

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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

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/∆t g(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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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

 Time Weighting

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Pitfalls in DFT

Time Frequency

Sampling gives Aliasing

Time limitation gives Leakage

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ALIASING

 Insufficient sampling allows a high frequency signal to masquerade

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Aliasing

Time Freq. x Hz 0 Hz 0 Hz f _s Time Freq. Time Freq. Time

?

f _s f _s

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Anti-aliasing filter On Input 1 2

Anti-aliasing Filter

Input Anti-aliasing filter Off A/D 0 2 1 Sampling Frequency f _span f _s/2 f _s Frequency f ₁ f ₂ f ₃ f ₁ f ₂ f ₃ f ₁ f _span = 2.56 f _s Example: f _s= 65 536 kHz

⇒

f _span = 25.6 kHz

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Leakage

Freq. Freq. Time Time Signal periodic with record length

Signal not periodic with record length

T T Rectangular weighting (no weighting) Hanning weighting       π − T t 2 cos 1 a(t) b(t) A(f) B(f) A(f) B(f)

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“Picket Fence” Effect

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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 (f _c) and

sampling rate f _s

>

2 f _c

Solutions:  Use correct weighting (signals)

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FFT Analysis 101

 Introduction

 Time Weighting

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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 Top Transient Hanning Rect-angular Weighting

Expo-nential Kaiser-_Bessel

+ overlap + overlap

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FFT Analysis 101

 Introduction

 Time Weighting

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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 + cos}2_x _1/3_[₍₁_-cosx)2 _{+ (1 cos(x}

- 50% overlap

1/2 [(1 -cosx)2 _{= (1 + cosx)}2_{] 1 = cos}2_x

3 2π ))2 + (1 cos(x -3 4π ))2_{] = 1.5}  75% overlap

1/4 [ (1 -cosx)2 _{+ (1 - sinx)}2 _{+ cosx)}2

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Window Summary

Window

Noise

Bandwidth

Ripple

Highest

Sidelobe

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

 Time Weighting

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Random Signal

Signal Types and Spectrum Units

Correct use of Units

Time Frequency Determistic, Periodic Signal U Time U Time Time Transient U U2_/Hz U2_s/Hz Freq. Freq. Freq. B

Root Mean Square RMS

Power Spectral Density PSD

Energy Spectral Density ESD

U

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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 ≥ T_calc.

 Other weightings than rectangular may require overlap analysis to avoid loss of

data or to get a flat overall weighting function

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CPB Advantages & Disadvantages

Higher Processor Demands Results 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

Cons Pros

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FFT Advantages & Disadvantages

“Ambiguous” results Lower Processor Demands

No true ‘peak’ detection Great for ‘exact’ freq.

determinations

Not standardized Windows

Very common

Leakage Constant bandwidth filters,

make harmonic patterns obvious

Block time analysis Wide selection of averaging

types

Poor response time High freq. resolution !ZOOM!

Cons Pros

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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 Analyzersby 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)