Fft Basics 2008
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Baron Jean Baptiste Joseph Fourier
www.bksv.com Brüel & Kjær Sound & Vibration Measurement A/S. Copyright © 2009. All Rights Reserved.
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
www.bksv.com, 3
The Measurement Chain Transducer
Preamplifier
Filter(s)
Detector/ Averager
RMS
www.bksv.com, 4
Display/ Output
Sources of Machine Vibration The moving parts of machines create vibration at different frequencies.
Vibration level
1st 2nd 3rd 4th 5th 6th 7th 8th 9th
12th
Order Frequency, Hz (Frequency, Hz)
Misalignment vibration at Vibrations at of 50the or shaft 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 journal blades unbalance bearing noise picked up from power (rotational frequency ×by 1, 2,cables 3,<) www.bksv.com, 5
Easier Than This& Imagine trying to determine all of the previous from just this!
www.bksv.com, 6
The Fourier Transform
G(f ) = ∫ g(t ) e − j2 π f t dt +∞
−∞
g(t ) = ∫ G(f ) e +∞
−∞
www.bksv.com, 7
j2 π f t
df
“All Complex Waves&” All Complex Waves are the Sum of Many Sine and Cosine Waves
∑
www.bksv.com, 8
Fourier Transform is a Mathematical Filter! F (t ) ∗ sin 2Πt = Large #
F (t ) ∗ sin 4Πt = Med.#
F (t ) ∗ sin 8Π t = Small.#
www.bksv.com, 9
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
www.bksv.com, 11
Finite duration in Time Infinite bandwidth in Frequency
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
www.bksv.com, 12
Filter Types Constant Bandwidth
Constant Percentage Bandwidth (CPB) or Relative Bandwidth
B = x Hz
0
20
40
60
B = y% =
80
Linear Frequency
B = 31,6 Hz B = 10 Hz B = 3,16 Hz www.bksv.com, 13
50
70
100
y × f0 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
www.bksv.com, 14
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 www.bksv.com, 15
200 400 600 800
Linear 1k 1,2k 1,4k 1,6k 1,8k 2k Hz Frequency
FFT Time Assumption Input signal
Analysed signal
Rectangular or Uniform weighting
Hanning weighting
www.bksv.com, 16
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
www.bksv.com, 17
Frequency (Hz)
Most important Law in Frequency Analysis
B×T≥1 B = bandwidth T = measurement time
www.bksv.com, 18
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
www.bksv.com, 19
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’
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<
www.bksv.com, 21
Are You Certain About Uncertainty (1)?
www.bksv.com, 22
Are You Certain About Uncertainty (2)?
www.bksv.com, 23
Are You Certain About Uncertainty (3)?
www.bksv.com, 24
What Happened? Which Is Correct?
Whichever best fit the time block!
www.bksv.com, 25
Ensuring Repeatable Measurements Think about the signal
you are measuring – Stationary – Transient – Combination< Always report:
– – – – – – –
www.bksv.com, 26
Lines Span Window used Overlap Averaging Type # of Averages Start Trigger?
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
www.bksv.com, 27
Pitfalls in DFT Time
Sampling
gives
Aliasing
Time limitation
gives
Leakage
Periodic repetition
www.bksv.com, 28
Frequency
gives Picket Fence Effect
ALIASING Insufficient sampling allows a high frequency signal to masquerade
under a low frequency “alias”
www.bksv.com, 29
Aliasing Time fs
0 Hz
Freq.
Time fs
x Hz
?
Time 0 Hz
fs
?
Time x Hz www.bksv.com, 30
Freq.
fs + x Hz
Freq.
Freq.
Anti-aliasing Filter A/D
Input
2
1
Input 0
Anti-aliasing filter Off Anti-aliasing filter On
www.bksv.com, 31
f1 f3
fspan fs/2
f2
fs f3
Frequency Sampling Frequency
f1 f2
1
fs fspan = 2.56 f1 2
Example: fs= 65 536 kHz ⇒ fspan = 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
A(f)
Freq. B(f)
2πt 1 − cos T
Freq. www.bksv.com, 32
Freq.
“Picket Fence” Effect
Frequency
www.bksv.com, 33
How to Avoid the Pitfalls of DFT
1. Aliasing: Solution:
Caused by sampling in time Use anti-aliasing filter (fc) and
sampling rate fs>2 fc
2. Leakage: Solutions:
3. Picket fence effect: Solutions:
Caused by time limitation Use correct weighting (signals) Increase the frequency resolution (systems) Caused by sampling in frequency Use correct weighting (signals) Increase the frequency resolution (systems)
www.bksv.com, 34
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
www.bksv.com, 35
Discontinuities in the Time Record
www.bksv.com, 36
Windows “smooth” the discontinuity
www.bksv.com, 37
Windows
www.bksv.com, 38
Use of Weighting Functions in Signal Analysis Weighting Rectangular
Hanning Transient
Transients:
General purpose
Short transient
Long decaying transients
Very long transients
Continuous signals:
General purpose, RTA
Two-tone separation
Calibration
Pseudo random
www.bksv.com, 39
+ overlap
+ overlap
Exponential
KaiserBessel
Flat Top
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
www.bksv.com, 40
Overlap Analysis with Hanning Weighting (1) 662/3% overlap
No
overlap W(t) W(t)
Time
Time
50% overlap
75% overlap
W(t)
W(t)
Time
www.bksv.com, 41
Time
Overlap Analysis with Hanning Weighting (2) 662/3% overlap
No 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 3 )) ] = 1.5
Time
50% overlap
75% overlap
1/2 [(1 -cosx)2 = (1 + cosx)2 ] 1 = cos2x W(t)
Time
www.bksv.com, 42
Time
1/4 [ (1 -cosx)2 + (1 - sinx)2 + cosx)2 + (1 -cos(x + sinx)2 ] = 1.5 W(t)
Time
Window Summary
Window Rectangle Hanning Kaiser-Bessel Flat Top
www.bksv.com, 43
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 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
www.bksv.com, 44
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:
www.bksv.com, 46
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
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
www.bksv.com, 47
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
www.bksv.com, 48
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) www.bksv.com, 49
Questions Jason Kunio Application Engineer 678-250-7684 [email protected]
www.bksv.com, 50
www.bksv.com Brüel & Kjær Sound & Vibration Measurement A/S. Copyright © 2009. All Rights Reserved.
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
www.bksv.com, 3
The Measurement Chain Transducer
Preamplifier
Filter(s)
Detector/ Averager
RMS
www.bksv.com, 4
Display/ Output
Sources of Machine Vibration The moving parts of machines create vibration at different frequencies.
Vibration level
1st 2nd 3rd 4th 5th 6th 7th 8th 9th
12th
Order Frequency, Hz (Frequency, Hz)
Misalignment vibration at Vibrations at of 50the or shaft 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 journal blades unbalance bearing noise picked up from power (rotational frequency ×by 1, 2,cables 3,<) www.bksv.com, 5
Easier Than This& Imagine trying to determine all of the previous from just this!
www.bksv.com, 6
The Fourier Transform
G(f ) = ∫ g(t ) e − j2 π f t dt +∞
−∞
g(t ) = ∫ G(f ) e +∞
−∞
www.bksv.com, 7
j2 π f t
df
“All Complex Waves&” All Complex Waves are the Sum of Many Sine and Cosine Waves
∑
www.bksv.com, 8
Fourier Transform is a Mathematical Filter! F (t ) ∗ sin 2Πt = Large #
F (t ) ∗ sin 4Πt = Med.#
F (t ) ∗ sin 8Π t = Small.#
www.bksv.com, 9
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
www.bksv.com, 11
Finite duration in Time Infinite bandwidth in Frequency
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
www.bksv.com, 12
Filter Types Constant Bandwidth
Constant Percentage Bandwidth (CPB) or Relative Bandwidth
B = x Hz
0
20
40
60
B = y% =
80
Linear Frequency
B = 31,6 Hz B = 10 Hz B = 3,16 Hz www.bksv.com, 13
50
70
100
y × f0 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
www.bksv.com, 14
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 www.bksv.com, 15
200 400 600 800
Linear 1k 1,2k 1,4k 1,6k 1,8k 2k Hz Frequency
FFT Time Assumption Input signal
Analysed signal
Rectangular or Uniform weighting
Hanning weighting
www.bksv.com, 16
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
www.bksv.com, 17
Frequency (Hz)
Most important Law in Frequency Analysis
B×T≥1 B = bandwidth T = measurement time
www.bksv.com, 18
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
www.bksv.com, 19
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’
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<
www.bksv.com, 21
Are You Certain About Uncertainty (1)?
www.bksv.com, 22
Are You Certain About Uncertainty (2)?
www.bksv.com, 23
Are You Certain About Uncertainty (3)?
www.bksv.com, 24
What Happened? Which Is Correct?
Whichever best fit the time block!
www.bksv.com, 25
Ensuring Repeatable Measurements Think about the signal
you are measuring – Stationary – Transient – Combination< Always report:
– – – – – – –
www.bksv.com, 26
Lines Span Window used Overlap Averaging Type # of Averages Start Trigger?
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
www.bksv.com, 27
Pitfalls in DFT Time
Sampling
gives
Aliasing
Time limitation
gives
Leakage
Periodic repetition
www.bksv.com, 28
Frequency
gives Picket Fence Effect
ALIASING Insufficient sampling allows a high frequency signal to masquerade
under a low frequency “alias”
www.bksv.com, 29
Aliasing Time fs
0 Hz
Freq.
Time fs
x Hz
?
Time 0 Hz
fs
?
Time x Hz www.bksv.com, 30
Freq.
fs + x Hz
Freq.
Freq.
Anti-aliasing Filter A/D
Input
2
1
Input 0
Anti-aliasing filter Off Anti-aliasing filter On
www.bksv.com, 31
f1 f3
fspan fs/2
f2
fs f3
Frequency Sampling Frequency
f1 f2
1
fs fspan = 2.56 f1 2
Example: fs= 65 536 kHz ⇒ fspan = 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
A(f)
Freq. B(f)
2πt 1 − cos T
Freq. www.bksv.com, 32
Freq.
“Picket Fence” Effect
Frequency
www.bksv.com, 33
How to Avoid the Pitfalls of DFT
1. Aliasing: Solution:
Caused by sampling in time Use anti-aliasing filter (fc) and
sampling rate fs>2 fc
2. Leakage: Solutions:
3. Picket fence effect: Solutions:
Caused by time limitation Use correct weighting (signals) Increase the frequency resolution (systems) Caused by sampling in frequency Use correct weighting (signals) Increase the frequency resolution (systems)
www.bksv.com, 34
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
www.bksv.com, 35
Discontinuities in the Time Record
www.bksv.com, 36
Windows “smooth” the discontinuity
www.bksv.com, 37
Windows
www.bksv.com, 38
Use of Weighting Functions in Signal Analysis Weighting Rectangular
Hanning Transient
Transients:
General purpose
Short transient
Long decaying transients
Very long transients
Continuous signals:
General purpose, RTA
Two-tone separation
Calibration
Pseudo random
www.bksv.com, 39
+ overlap
+ overlap
Exponential
KaiserBessel
Flat Top
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
www.bksv.com, 40
Overlap Analysis with Hanning Weighting (1) 662/3% overlap
No
overlap W(t) W(t)
Time
Time
50% overlap
75% overlap
W(t)
W(t)
Time
www.bksv.com, 41
Time
Overlap Analysis with Hanning Weighting (2) 662/3% overlap
No 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 3 )) ] = 1.5
Time
50% overlap
75% overlap
1/2 [(1 -cosx)2 = (1 + cosx)2 ] 1 = cos2x W(t)
Time
www.bksv.com, 42
Time
1/4 [ (1 -cosx)2 + (1 - sinx)2 + cosx)2 + (1 -cos(x + sinx)2 ] = 1.5 W(t)
Time
Window Summary
Window Rectangle Hanning Kaiser-Bessel Flat Top
www.bksv.com, 43
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 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
www.bksv.com, 44
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:
www.bksv.com, 46
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
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
www.bksv.com, 47
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
www.bksv.com, 48
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) www.bksv.com, 49
Questions Jason Kunio Application Engineer 678-250-7684 [email protected]
www.bksv.com, 50
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