Spectrum Analysis 082702

Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 1 Copyright 2001

Mechanical Engineering - 22.403 ME Lab II

ME 22.403 Mechanical Lab II

Basics ofSpectrum Analysis/Measurements

andthe FFT Analyzer



Many times a transformation is performed to provide a better orclearer understanding of a phenomena. The time representationof a sine wave may be difficult to interpret. By using a Fourierseries representation, the original time signal can be easilytransformed and much better understood.

Transformations are also performedto respresent the same data withsignificantly less information.Notice that the original time signalwas defined by many discrete timepoints (ie, 1024, 2048, 4096 )whereas the equivalent Fourierrepresentation only requires 4amplitudes and 4 frequencies.

Transformation of Time to Frequency



The FFT Analyzer can be broken downinto several pieces which involve thedigitization, filtering, transformationand processing of a signal.

Several items are important here: Digitization and Sampling Quantization of Signal Aliasing Effects Leakage Distortion Windows Weighting Functions The Fourier Transform Measurement Formulation

DISPLAY ADC

ANALOG SIGNAL

ANALOG FILTER

DIGITAL FILTER

DISCRETE DATA

FFT

The Anatomy of the FFT Analyzer



Actual time signals

INPUT OUTPUT

OUTPUT INPUT

FREQUENCY RESPONSE FUNCTION COHERENCE FUNCTION

ANTIALIASING FILTERS

ADC DIGITIZES SIGNALS

INPUT OUTPUT

ANALOG SIGNALS

APPLY WINDOWS

COMPUTE FFT LINEAR SPECTRA

AUTORANGE ANALYZER

AVERAGING OF SAMPLES

INPUT/OUTPUT/CROSS POWER SPECTRA COMPUTATION OF AVERAGED

INPUT SPECTRUM

LINEAR OUTPUT

SPECTRUM

LINEAR

INPUT

SPECTRUM POWER

OUTPUT

SPECTRUM POWER CROSS

SPECTRUM POWER

COMPUTATION OF FRF AND COHERENCE

Analog anti-alias filter

Digitized time signals

Windowed time signals

Compute FFT of signal

Average auto/cross spectra

Compute FRF and Coherence

The Anatomy of the FFT Process



ALIASED SIGNAL

ACTUAL SIGNAL

Aliasing results when the sampling does not occur fast enough.

Sampling must occur faster than twice the highest frequency to bemeasured in the data - sampling of 10 to 20 times the signal issufficient for most time representations of varying signals

However, in order to accurately represent a signal in thefrequency domain, sampling need only occur at greater than twicethe frequency of interest

Anti-aliasing filters are used to prevent aliasing

These are typically Low Pass Analog Filters

Aliasing (Wrap-Around Error)

OBSERVED ACTUAL

f max

WRAP-AROUND



Anti-aliasing filters are typically specified with a cut-offfrequency. The roll-off of the filter will determine how quicklythe signal will be attenuated and is specified in dB/octave

FILTER ROLLOFF

f c

in

out1010dB V

Vlog20Glog20G ==

The cut-off frequency is usually specified at the 3 dB down point(which is where the filter attenuates 3 dB of signal).

Butterworth, Chebyshev, elliptic, Bessel are common filters

Anti-Aliasing Filters



Sampling rate of the ADC is specified as a maximum that ispossible. Basically, the digitizer is taking a series of snapshotsat a very fast rate as time progresses

Analog Signal DigitalRepresentation

ADC

Digitization of a Signal



Each sample is spaced delta t seconds apart. Sufficient samplingis needed in order to assure that the entire event is captured.The maximum observable frequency is inversely proportional to thedelta time step used

Digital Sample

fmax = 1 / 2 DtRayleigh Criteria

Dt spacing

Sampling



In order to extract valid frequency information, digitization of theanalog signal must occur at a certain rate.

Shannon's Sampling Theorem states fs > 2 fmax

That is, the sampling rate must be at least twice the desiredfrequency to be measured.

For a time record of T seconds, the lowest frequency componentmeasurable is DDf = 1 / T

With these two properties above, the sampling parameters can besummarized as fmax = 1 / 2 DDt

DDt = 1 / 2 fmax

Sampling Theory



Sampling Parameters

T=N t DD

Due to the Rayleigh Criteria and Shannons Sampling Theorum, thefollowing sampling parameters must be observed.



PICK THEN AND

DDt fmax = 1 / (2 DDt) T = N DDt

fmax DDt = 1 / (2 fmax ) DDf = 1/(N DDt)

DDf T = 1 / DDf DDt = T / N

T DDf =1 / T fmax = N DDf / 2

Due to the Rayleigh Criteria and Shannons Sampling Theorum, thefollowing sampling parameters must be observed.

Sampling Parameters

If we choose Df = 5 Hz and N = 1024Then T = 1 / Df = 1 / 5 Hz = 0.2 sec

fs = N Df = (1024) (5 Hz) = 5120 Hzfmax = fs = (5120 Hz) / 2 = 2560 Hz



An inverse relationship between time and frequency exists

TIME DOMAIN FREQUENCY DOMAIN

T BW

Given delta t = .0019531 and N = 1024 time points,then T = 2 sec and BW= 256 Hz and delta f = 0.5 Hz


BWTGiven delta t = .000976563 and N = 1024 time points,then T = 1sec sec and BW = 512 Hz and delta f = 1 Hz


BWTGiven delta t = .0019531 and N = 512 time points,then T = 1 sec and BW = 256 Hz and delta f = 1 Hz

Sampling Relationship



Sampling refers to the rate at which the signal is collected.Quantization refers to the amplitude description of the signal.

A 4 bit ADC has 24 or 16 possible values



ADC BIT STEPS

Quantization Error



Quantization errors refer to the accuracy of the amplitudemeasured. The 6 bit ADC represents the signal shown muchbetter than a 4 bit ADC

A D C

M A X

R A

N G E

A D C

M A X

R A

N G E

Quantization Error



Underloading of the ADC causes amplitude errors in the signal

All of the availabledynamic range of theanalog to digitalconverter is not usedeffectively

0.5 volt signal

This causes amplitudeand phase distortionof the measuredsignal in both the timeand frequency domains

10 voltrangeon

ADC

Quantization Error



A large DC bias can cause amplitude errors in the alternating partof the signal. AC coupling uses a high pass filter to remove theDC component from the signal

All of the availabledynamic range of theanalog to digitalconverter is dominatedby the DC signal

The alternating part ofthe signal suffers fromquantization error

This causes amplitudeand phase distortion ofthe measured signal

10 voltrangeon

ADC

AC Coupling



Overloading of the ADC causes severe errors also

The ADC range is settoo low for the signalto be measured andcauses clipping of thesignal

1.5 volt signal

This causes amplitudeand phase distortionof the measuredsignal in both the timeand frequency domains

1 voltrangeon

ADC

A D C

M A X

R A

N G

E

Clipping and Overloading



The Fourier Transform

+

-

p-= dte)t(x)f(S ft2jx

+

-

p= dfe)f(S)t(x ft2jx

and Inverse Fourier Transform

Forward Fourier Transform



Discrete Fourier TransformEven though the actual time signal is continuous, the signal isdiscretized and the transformation at discrete points is

+

-

Dp-=D dte)t(x)fm(S tfm2jx

This integral is evaluated as

However, if only a finite sample is available (which is generally thecase), then the transformation becomes

+

-=

DDp-DDDn

tnfm2jx e)tn(xt)fm(S

-

=

DDp-DDD1N

0n

tnfm2jx e)tn(xt)fm(S



T

ACTUAL

DATA

CAPTURED

DATA

RECONTRUCTED

DATA

Fourier Transform and FFT

Actual TimeSignal

Captured TimeSignal

ReconstructedTime Signal

Frequency Spectrum



Leakage

T

ACTUAL

DATA

CAPTURED

DATA

RECONTRUCTED

DATA

T

ACTUAL

DATA

CAPTURED

DATA

RECONTRUCTED

DATA

T

TIME

FREQ

UENCY

Periodic Signal

Non-Periodic Signal

Leakage due tosignal distortion



Leakage

When the measured signal is not periodic in the sample interval,incorrect estimates of the amplitude and frequency occur. Thiserror is referred to as leakage.

Basically, the actual energy distribution is smeared across thefrequency spectrum and energy leaks from a particular DDf intoadjacent DDf s.

Leakage is probably the most common and most serious digitalsignal processing error. Unlike aliasing, the effects of leakagecan not be eliminated.



Windows - Minimize LeakageIn order to better satisfy the periodicity requirement of the FFTprocess, time weighting functions, called windows, are used.Essentially, these weighting functions attempt to heavily weight thebeginning and end of the sample record to zero - the middle of thesample is heavily weighted towards unity

T

ACTUAL

DATA

CAPTURED

DATA

RECONTRUCTED

DATA

T

ACTUAL

DATA

CAPTURED

DATA

RECONTRUCTED

DATA

TIME

FREQ

UENCY

Periodic Signal

Non-Periodic Signal



Windows - Rectangular/Hanning/FlattopIn order to better satisfy the periodicity requirement of the FFTprocess, time weighting functions, called windows, are used.Essentially, these weighting functions attempt to heavily weight thebeginning and end of the sample record to zero - the middle of thesample is heavilty weighted towards unity

RectangularRectangular - Unity gain applied to entire sample interval; thiswindow can have up to 36% amplitude error if the signal is notperiodic in the sample interval; good for signals that inherentlysatisfy the periodicity requirement of the FFT process

HanningHanning - Cosine bell shaped weighting which heavily weights thebeginning and end of the sample interval to zero; this window canhave up to 16% amplitude error; the main frequency will showsome adjacent side band frequencies but then quickly attenuates;good for general purpose signal applications

Flat TopFlat Top - Multi-sine weighting function; this window has excellentamplitude characteristics (0.1% error) but very poor frequencyresolution; very good for calibration purposes with discrete sine



Windows - Rectangular/Hanning/Flattop

Time weighting functionsare applied to minimizethe effects of leakage

Rectangular

Hanning

Flat Top

and many others

Windows DO NOT eliminate leakage !!!

AMPLITUDE

ROLLOFF

WIDTH

General windowfrequency characteristics



Windows - Rectangular

-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

Amplitude

-3 -2 -1 0 1 2-2.5 -1.5 -0.5 0.5 1.5 2.5

The rectangular window function is shown below. The main lobe is narrow, but the side lobes are very largeand roll off quite slowly. The main lobe is quite rounded and can introduce large measurement errors. Therectangular window can have amplitude errors as large as 36%.



Amplitude

-3 -2 -1 0 1 2-2.5 -1.5 -0.5 0.5 1.5 2.5-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

The hanning window function is shown below. The first few side lobes are rather large, but a 60 dB/octaveroll-off rate is helpful. This window is most useful for searching operations where good frequencyresolution is needed, but amplitude accuracy is not important; the hanning window will have amplitude errorsof as much as 16%.

Windows - Hanning



Amplitude

-3 -2 -1 0 1 2-2.5 -1.5 -0.5 0.5 1.5 2.5-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

The flat top window function is shown below. The main lobe is very flat and spreads over several frequencybins. While this window suffers from frequency resolution, the amplitude can be measured very accuratelyto 0.1%.

Windows - Flat Top



Windows

-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

-16 15.9375-10 0 10-14 -12 -8 -6 -4 -2 2 4 6 8 12 14-100

0

-70

- 90

- 80

-60

-50

-40

-30

-20

-10

dB

Rectangular Hanning Flat Top



Windows - Force/Exponential for Impact TestingSpecial windows are used for impact testing

Forcewindow



Windows - Force/Exponential for Impact TestingSpecial windows are used for impact testing

Exponentialwindow



Hammer Tips for Impact Testing

Real

-976.5625us 123.9624ms

dB Mag

0Hz 6.4kHz

METAL TIP

TIME PULSE

FREQUENCY SPECTRUM

TIME PULSE

FREQUENCY SPECTRUM

Real

-976.5625us 123.9624ms

dB Mag

0Hz 6.4kHz

HARD PLASTIC TIP

TIME PULSE

FREQUENCY SPECTRUM

Real

-976.5625us 123.9624ms

dB Mag

0Hz 6.4kHz

SOFT PLASTIC TIP

TIME PULSE

FREQUENCY SPECTRUM

Real

-976.5625us 123.9624ms

dB Mag

0Hz 6.4kHz

RUBBER TIP



Pretrigger Delay and Double Impacts

NO PRETRIGGER USED

PRETRIGGER SPECIFIED

t = 0

t = 0

DOUBLE IMPACT

TIME PULSE

FREQUENCY SPECTRUM

Real

-976.5625us 998.53516ms

dB Mag

0Hz 800Hz

DOUBLE IMPACT

TIME PULSE

FREQUENCY SPECTRUM

Real

-976.5625us 998.53516ms

dB Mag

0Hz 800Hz

Pretrigger delay used toreduce the amount offrequency spectrum distortion

Double impacts should be avoided due tothe distortion of the frequency spectrumand force dropout that can occur



Exponential Window

If the signal does notnaturally decay within thesample interval, then anexponentially decaying windowmay be necessary.

However, many times changingthe signal processingparameters such as bandwidthand number of spectral linesmay produce a signal whichrequires less window weighting T = N t DD

T = N t DD



Measurement - Linear Spectra

SYSTEMINPUT OUTPUT

x(t) h(t) y(t)

Sx(f) H(f) Sy(f)

x(t) - time domain input to the system

y(t) - time domain output to the system

Sx(f) - linear Fourier spectrum of x(t)

Sy(f) - linear Fourier spectrum of y(t)

H(f) - system transfer function

h(t) - system impulse response

TIME

FREQUENCY

FFT & IFT



Measurement - Linear Spectra

+

-

p= dfe)f(S)t(x ft2jx

+

-

p= dfe)f(S)t(y ft2jy

+

-

p= dfe)f(H)t(h ft2j

+

-

p-= dte)t(x)f(S ft2jx

+

-

p-= dte)t(y)f(S ft2jy

+

-

p-= dte)t(h)f(H ft2j

Note: Sx and Sy are complex valued functions



Measurement - Power Spectra

SYSTEMINPUT OUTPUT

TIME

FREQUENCY

FFT & IFT

Rxx(t) Ryx(t) Ryy(t)

Gxx(f) Gxy(f) Gyy(f)

Rxx(t) - autocorrelation of the input signal x(t)

Ryy(t) - autocorrelation of the output signal y(t)

Ryx(t) - cross correlation of y(t) and x(t)

Gxx(f) - autopower spectrum of x(t) G f S f S fxx x x( ) ( ) ( )*=

Gyy(f) - autopower spectrum of y(t) G f S f S fyy y y( ) ( ) ( )*=

Gyx(f) - cross power spectrum of y(t) and x(t) G f S f S fyx y x( ) ( ) ( )*=



Measurement - Power Spectra

)f(S)f(Sde)(R)f(G

dt)t(x)t(yT1

Tlim

)]t(x),t(y[E)(R

)f(S)f(Sde)(R)f(G

dt)t(y)t(yT1

Tlim

)]t(y),t(y[E)(R

)f(S)f(Sde)(R)f(G

dt)t(x)t(xT1

Tlim

)]t(x),t(x[E)(R

*xy

ft2jyxyx

Tyx

*yy

ft2jyyyy

Tyy

*xx

ft2jxxxx

Txx

=tt=

t+

=t+=t

=tt=

t+

=t+=t

=tt=

t+

=t+=t

+

-

p-

+

-

p-

+

-

p-



xy SHS =H1 formulation

- susceptible to noise on the input- underestimates the actual H of the system

*xx

*xy SSHSS =

xx

yx*xx

*xy

G

G

SS

SSH =

=

2

1

xyyy

xxyx*yy

*xx

*yx

*xy2

xy HH

G/G

G/G

)SS)(SS(

)SS)(SS(==

=g

COHERENCE

Otherformulationsfor H exist

The Frequency Response Function and Coherence



12 Dr. Peter AvitabileModal Analysis & Controls LaboratoryMeasurement Definitions

Measurements - Auto Power Spectrum

INPUT FORCE

AVERAGED INPUT

POWER SPECTRUM

x(t)

G (f)xx

OUTPUT RESPONSE

AVERAGED OUTPUT

POWER SPECTRUM

y(t)

G (f)yy


Measurements - Cross Power Spectrum

AVERAGED INPUT

POWER SPECTRUM

AVERAGED CROSS

POWER SPECTRUM

AVERAGED OUTPUT

POWER SPECTRUM

G (f)xx G (f)yy

G (f)yx


Measurements - Frequency Response Function

AVERAGED INPUT

POWER SPECTRUM

AVERAGED CROSS

POWER SPECTRUM

AVERAGED OUTPUT

POWER SPECTRUM

FREQUENCY RESPONSE FUNCTION

G (f)xx G (f)yyG (f)yx

H(f)


Measurements - FRF & Coherence

Freq Resp

40

-60

dB Mag

0Hz 200HzAVG: 5

Coherence1

0

Real

0Hz 200HzAVG: 5


COHERENCE

Typical Measurements



Hammers and Tips

40

-60

dB Mag

0Hz 800Hz

COHERENCE

INPUT POWER SPECTRUM

FRF

40

-60

dB Mag

0Hz 200Hz

COHERENCE

INPUT POWER SPECTRUM

FRF



Leakage and Windows for Impact Testing

ACTUAL TIME SIGNAL

SAMPLED SIGNAL

WINDOW WEIGHTING

WINDOWED TIME SIGNAL



Simple time-frequency response relationship

FORCE

RESPONSE

time

increasing rate of oscillation

frequency

WOW !!!



Sine Dwell to Obtain Mode Shape Characteristics

MODE 1

MODE 2

MODE3

MODE 4



DOF # 1

DOF #2

DOF # 3

MODE # 1

MODE # 2

MODE # 3

Mode Shape Characteristics for a Simple Beam



1

2

3

4

5

6

MODE 1

1

2

3

4

5

6

MODE 2

Mode Shape Characteristics for a Simple Plate



Why and How Do Structures Vibrate?

INPUT TIME FORCE

INPUT SPECTRUM

OUTPUT TIME RESPONSE

OUTPUT SPECTRUM

f(t)

FFT

y(t)

IFT

f(j )ww y(j )wwh(j )ww


Motor or diskunbalance

can cause unwantedvibrations or worse

OOOPS !!!

OUCH !!!



HP 35660 FFT Dual Channel Analyzer

Spectrum Analysis 082702

Documents

mechanical lab

mechanical engineering

transformation of time

lab iithe fft analyzer

time representationof

signal issufficient

fft process

lab iime