Signal Processing and Diagnostics

Jiří Tůma

VSB-TU Ostrava, Czech Republic

2012

Outline of a course on Signal processing and diagnostics

• Introduction to diagnostics and basic terms

• Principles of signal processing (SP)

• Transformations for signal processing (Fourier, FFT, etc)

• Basic methods and algorithms for SP

• Diagnostics principles and devices

• Noise, sound and vibration measurement

• Experimental measurement on Diagnostic Laboratory

2

Application areas for the knowledge gained in the course

of signal processing

• Signal analysis

• Sound and speech analysis

• Noise and vibration signal analysis

• Condition monitoring of machines during their technical life

• Condition base maintenance (CBM)

• Quality control of new products

• Research in experimental dynamics

• Design of new products emitting low level noise

• Environmental noise and vibration control

3

Introduction

Lecture 1

4

Signal processing tools

• Analysis of signals in the time and frequency domains

• Frequency and order analysis

• Averaging in the frequency and time domains

• Envelope analysis

• Phase demodulation

Methods

Plots

• Time Domain

• Time history

• Orbit

• Frequency Domain

• Frequency spectra (2D)

• Time-frequency analysis (3D - waterfall, spectral maps)

• Trend

5

Hardware

• Microphones

• Accelerometers, sensors of velocity, proximity probe

• Sensors for angular vibration measurements

• Sensors for RPM measurements

• Laser Doppler vibrometers

Sensors - transducers

Multipexers and A/D convertors

• Signal analysers

• Multifunction cards

• Multifunction cards for dynamic measurements

Computers

• Personal computers

• Work station

6

Sensors - transducers

Microphones

Accelerometers

RPM measurements

Laser Doppler vibrometers

for linear vibration

Laser Doppler vibrometers

for torsional vibration

7

Signal analyzers

8

• CPB (Constant Percentage Band) analyzers, alternatively designated as Real Time or 1/3-octave or 1/1-octave analyzers. The principle is based on a bank of frequency filters with the percentage bandwidth which is equal to a constant.

• FFT analyzers. The principle is based on the Fast Fourier Transform

• Vector analyzers. – It is an analyzer for the measurement of the amplitude and phase of the input signal at a single frequency within an intermediate frequency bandwidth of a heterodyne receiver.

… according to the principle of operation

… according to the application area

• Laboratory analyzers

• Analysers for machine diagnostics

• Analyzers for noise and vibration signals

• Analyzers for electrical signals (frequency range of MHz)

List of types of signal analyzers

Signal types

Lecture 2

9

Signal types

Deterministic Random (stochastic)

Periodic Non-periodic Stationary Nonstationary

Sinusoidal Complex

periodic

(harmonic)

Almost

periodic

Transient Ergodic Non-ergodic Special classification

Deterministic signals are defined as a function of time while random signals can be defined

in terms of statistical properties.

Deterministické signals can be predicted as opposed to random signals whose instantaneous

values are not predictable only statistical properties can be predicted.

Deterministic vs. random signals

Periodic vs. non-periodic signals

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,5 1,0

Time [s]

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,5 1,0

Time [s]

10

11

Random variables, random signals

Names of random variables and processes: ξ, ξ(t), ε, ε(t), …

xxpxxxP

Probability that a random variable ξ belongs to the interval of values greater than x and less

than x+Δx is proportional to the interval of the length Δx

t 1

t

Realization of the random

process

Realization of the random

variable in time t x Independent variable 1

Realization of random variables and processes: x, x(t), y, y(t), …

The coefficient of proportionality is denoted as a probability density function (pdf).

xp

tx1

tx2

tx3

t

Probability density function

12

Statistical properties of a random variable

Continuous random variable < > Discrete random variable

x

xxp ,

2exp

2

12

2

Probability density function for

the normal (Gaussian) distribution

Gaussian function (bell curve) Probability density function (pdf)

Two-dimensional pdf

13

Stationary and non-stationary signals

A stationary signal (or strict(ly) or strong(ly) stationary signal) is a stochastic process whose joint

probability distribution does not change when shifted in time or space. As a result, parameters such

as the mean and variance, if they exist, also do not change over time.

-3

-2

-1

0

1

2

3

0,0 0,5 1,0

Time [s]

-1

0

1

2

3

0,0 0,5 1,0

Time [s]

Stationary signal Non-stationary signal

xp txp ,

Probability density function

is not a function of time Probability density function

is a function of time

x x

t1

t2

t3

t1 t2 t3

The basic properties of stationary continuous signals x(t) is as follows

21212121 ,,,,, ttxxpttxxp

111, xptxp ………… ………… one-dimensional probability density function (pdf)

……… two-dimensional pdf

14

Mean value, variance and correlation function of a random

variable

xxpx dE

xxpxM kk

k dE

0E,E xxxx

xxpxmkk

k dE

2

2D m

2121212211221121 dd,,,E, xxttxxptxtxttttRxx

Centered variable ….…

Mean value ……….….

The nth central moment

The nth central moment

about a mean value μ

Variance ….……..…….

Correlation function …..

2121 ,,, ttxxpThe two-dimensional probability density function

Expected value (moment in physics)

For a random variable ξ, it is introduced

For a random signal (process) ξ(t), it is defined

valuePeakfactorCrestCrest factor ……..…….

Standard deviation ……. D

xxxxxx RttRttR 1221,If 21212121 ,,,,, ttxxpttxxp then

15

Stationary and ergodic signals

For example the parameters (mean and

variance) can be computed from values

corresponding to random signal

realizations in time t1.

If the realizations are sections of the long record and ΔT

tends to zero or to the interval, which is small enough,

them the values across a group of identical processes can

be replaced by the samples of the time record.

time time

An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time

average of a conforming process to equal the ensemble average. In practice this means that statistical

sampling can be performed at one instant across a group of identical processes or sampled over time

on a single process with no change in the measured result.

Random variables in time t1 t1 ΔT+t1 2ΔT+t1 3ΔT+t1

4 signal realizations

0 ΔT t1

-3

0

3

1

3

4

2

0 ΔT 2ΔT 3ΔT 4ΔT -3

0

3 1

2

3

4

For ergodic signals, it is assumed that the mean value can be replaced by the time average

2

2

d1

limd

T

TT

ttxT

xxpxx

16

Mean value and standard deviation (RMS) of a sine signal

0d1

0

T

ttxT

x

-1,2

-0,8

-0,4

0,0

0,4

0,8

1,2

0,0 0,5 1,0

Time [s]

t

TAtx

2cos

2d

22cos1

2

1

d2

cos1

d1

2

0

2

0

2

0

22

Att

T

A

T

ttT

AT

ttxT

T

TT

2

A

Sinusoidal signal ……

Mean value …………

The computation of RMS for the sinusoidal signal

results in the formula

The value of RMS is approximately equal to 70%

of the harmonic signal amplitude. The amplitude

of this signal is the 1.4-multiple of RMS.

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,5 1,0

Time [s]

x(t)

(x(t))2

If the signal is harmonic (sinusoidal) than we can calculate

17

Signals with continuous and discrete time

The analog signal x(t) is a real or complex function of continuous time t. The other definition

points to the fact that signal contains information. But the white noise does not formally contain

any information which is in fact information.

Analog and digital signals

Process of converting of an analog signal into a digital signal is called digitalisation. There are

two issues in digitalisation

• sampling

• quantizing.

First, we notice sampling. Sampling of an analog signal produces a time series which is a

sequence of samples in the discrete time n. The sequence of samples may be denoted either as an

indexed variable or as a function of an integer number x(n).

x(t)

x0

x1

x2

x3 x4

x5

x6

x7

0 1 2 3 4 5 6 7 where TS is a sampling interval for the uniformly sampled data.

The sampling frequency (rate) fS in hertz or in samples per

second is the reciprocal value of the sampling interval.

TS

Txxxtx ,...,,:Sampling 210

txnTtxnTttxn

Sn

n

S t

n

Sampling may be considered as a mapping

The time continuous signal is related to the time series and can be

substituted by the following way

18

Stationary and ergodic signals – continuous and discrete

time

For ergodic signals, it is assumed that

the mean value can be replaced by the

time average

2

2

d1

lim

T

TT

ttxT

x

2

2

2 d1

lim

T

TT

txtxT

s

2

2

d1

lim

T

TT

xx ttxtxT

R

The correlation function of ergodic signals

depends only on the lag

Root Mean

Square =

RMS

Mean value

Standard

deviation

Auto-correlation

Cross-correlation

1

0

1 N

i

ixN

x

1

0

21 N

i

xixN

s

2,...,2,1,0,1 1

0

NiyixN

RN

i

xy

2,...,2,1,0,1 1

0

NixixN

RN

i

xx

For discrete ergodic random signals

(x(i), i = 0, 1, 2, …), the formulas take the form

2221

0

22 1xRMSxix

Ns

N

i

1

0

21 N

i

ixN

RMS

Time and frequency domains

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0,0 0,2 0,4 0,6 0,8 1,0

U

Time [s]

6 H

z

9 H

z

23

Hz

37

Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 20 40 60 80 100

RM

S

Frequency [Hz]

The time signal Autospectrum

An example

The time domain The frequency domain

20

Signal to Noise Ratio and Nyquist-

Shannon Sampling Theorem

Lecture 3

21

Sampling a continuous signal to a discrete signal

Sampling Continuous time t

x(t)

DAC

Discrete time n

ADC

xn

Quantizing noise

0

1

-1 -0,5

p(x) 0,5

x

Quantizing

0

12

1dd

5.0

5.0

222

xxxxpx

Variance of the quantizing noise

1 2

3 4 5 6

Let x(t) be a continuous signal

Discrete time n

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

p(x) – uniform probability

density

12

1

Standard deviation of the quantizing noise

Quantizing is a part of AD conversion resulting in the attribution

of values x(nTS) to specified discrete values xn

TS sampling interval

Quantization

• Harmonic distortion – typically characterized by a family of harmonic components resulting from non-linearity in the analog signal conditioning

• Cross-talk – signals caused by inter-channel coupling

• Spurious – signals caused by various phenomena such as power supply imperfection, clock circuits, bus communication and EMC coupling between circuits

• ADC Resolution (given by the number of bits) and non-linearity

• Aliasing originating from signal components of frequencies higher than the Nyquist frequency

Digital output Analog input

Quantizer

A

D Clipping level

Clipping level

Imperfections

22

Signal-to-Noise Ratio

6log2

12log12062log20

22log20

Noise

Signallog20 1

1

mNS mm

RMS

RMS

dB

dBmNSdB

76,102,6

Number of ADC bits 4 8 10 12 14 16 18 20 22 24

Maximum of S/N dB 26 50 60 72 84 96 108 120 132 144

Standard deviation of quantizing noise

RMS value of a full-scale sine wave

m2

12 m

Full scale ADC resolution:

m – number of ADC bits

Signal analyzers The newest models

PWR

PWR

dBNS

Noise

Signallog10

Signal-to-Noise Ratio is defined as the ratio of a signal power to the noise power, which is corrupting

the signal

23

Historic overview of ADC specifications

Year ADC resolutions (bits) Signal to Noise Ratio

1970 10-12 60 dB

1980 14-16 70 dB

1990 16 80 dB

2000 24 100 dB

2005 24 110 dB

See [Bruel & Kjaer Technical Review No.1 2006]

A high quality noise and vibration transducers, including preamplifiers, can deliver

a practically noise-, spurious- and distortion-free signal over a dynamic range of 120 to 130 dB.

24

25

Effect of filtering on the S/N ratio

f

fs /2 BW

0

S(f)

sPWR fBW22Noise 2

s

f

ffSdffSs

2

2

0

2 2 Total power:

BW … bandwidth in Hz

Quantizing noise (white noise) PSD spectrum of white noise

is independent on the frequency sf

22

The frequency filtration reduces the quantizing noise power and consequently RMS of noise.

Narrowing the bandwidth reduces

the quantizing noise power

The term improving

the S/N ratio

Low pass or band pass filtration improves the S/N ratio

BWffBW

NS SPWR

S

PWR

PWR

PWR

dB2log10

Signallog10

2

Signallog10

Noise

Signallog10

22

S/N ratio for full

frequency range

26

FFT noise floor

Process gain

N-point FFT

Effective number of bits (ENOB) … 02.6

76.1SINADENOB

SINAD

FFT noise floor

m = 12 bits

N = 4096

74 dB = 6.02 m + 1.76 dB

SINAD is the abbreviation for ‘Si’gnal – to – ’N’oise – 'A'nd – ‘D’istortion

-74 dB

Bin spacing N

fS2

Sf

RMS quantization level

-107 dB

dB 0

20

40

60

80

100

120

ADC full scale

0 dB

2log10dB33 10

N

[ADC full scale] – [FFT noise floor] – [Process gain] = SINAD ENOB

The example assumes

that m = ENOB

Generaly m > ENOB

Effect of the background noise on the frequency spectrum

Autospectrum : 60 ; 80 ; 100 ; 120 SNR dB0 dB

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

0 100 200 300 400 500 600 700 800

Frequency [Hz]

RM

S d

B/r

ef 1

N = 2048

60 S/N dB 80 S/N dB

100 S/N dB 120 S/N dB

Process gain ... 10 log(N/2) = 10 log(2048/2) = 30.1 dB

Signal 1 VRMS 200 Hz

White Noise

0.001 VRMS

0.0001 VRMS

0.00001 VRMS

0.000001 VRMS

Ref level 1 VRMS

Background Noise

Hz2048Sf

27

Electronic noise

Noise is a random process, characterized by power spectral density (PSD), which is measured in

watts per hertz (W/Hz). Since the power in a resistive element is proportional to the square of the

voltage across it, noise voltage (density) can be described by taking the square root of the noise

power density, resulting in volts per root hertz ( ). HzV

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 0.2 0.4 0.6 0.8 1

time

sig

na

l

28

Nyquist-Shannon sampling theorem

Acording to the Nyquist-Shannon theorem, sampling the 10 Hz continuous sinusoidal signal

requires the sampling frequency, which is equal to 20 Hz at least

The sampling frequency, which is satisfying to the mentioned sampling theorem, results in the

fact that the sampled signal crosses the zero at the same frequency as the sampled signal

The sampling frequency 12 Hz causes that the sampled signal crosses the zero at the frequency

12 – 10 = 2 Hz. This phenomenon is called as an aliasing.

No-satisfactory sampling

Satisfactory sampling

29

Aliasing as a result of the improper sampling frequency

Nyquist

frequency

Sampling

frequency

20 SS fff

Shannon-Nyquist sampling theorem:

(Shannon-Kotelnik)

20 Sff

56.20 Sff BK signal analyzers:

f

fs 0

aliasing

S(f)

fs / 2

fo fs - fo

symmetry

1

-80 dB

0

Analog

low pass filter

A/D

Sample rate

~ 80% of fS/2

Sf

MAXS ff

Sf

MAXf

2Sf

MAXCUTOFF ff

MAXSSTOP fff

0

Sf

Stop band

(14-bit ADC)

Analog filter

(transient range)

Measurement range

Analog input Digital output

m bits

The analog low pass filter (antialiasing filter) faces

to the aliasing effect.

Let fo be satisfying

DC

1-2

Measuring chain

• BNC connector

• BNC connector for Microdot

• Lemo 7-pin connector for mic

• TNC connector for Microdot

Transducer Signal

conditioning Cables

Connector for signal analyzers

A/D

convertor

Direct (voltage input)

Microphone

Accelerometer

Digital data to

be analysed

Coaxial cables

• Signal analyzers

• Multifunction cards

A/D convertors Front-end of the signal analyzer

PULSE with connectors

(BNC and Lemo)

Microdot

Multichannel analyzers

Multichannel signal analyzers have parallel inputs - each measuring channel has its

own A / D convertor. All the convertors are triggered simultaneously in the same

time moment.

Munction cards work with a multiplexor which connects signals to a A/D convertor.

There is only one A / D converter on the multifunction card. If the sample-and-hold

circuit is not used, then between the successively measured samples is a time lag.

Multifunction cards of low price are not equipped by an antialiasing filter except

cards for dynamic measurements (higher price).

Input analog signal

is sampled

Sample-and-hold circuit

AI – analog

input

AO – analog

output

C – a control signal

Output voltage is maintained at a

constant value during conversion

Analog memory

31

Power spectrum

Power spectrum

32

S(f )

PWR = RMS2

Δf

White noise

Energy of signal ………..

x(t) … signal with the arbitrary unit U (e.g. Pa, m/s, m/s2)

x(t)2… signal power with the unit U2 (e.g. Pa2, (m/s)2, (m/s2)2)

Σ x(t)2… energy U2s t

(For instance, the electric power P = R i2 is proportional to the square

of the current i as a function of time while R is only a scale factor)

td......2

n

2......

f

frequency

PWRd

ffS

S(f )

f

constfS

Pink noise S(f )

f

21 ffS

0

0

0

(PoWeR)

fSd

Quantizing noise can be considered as white noise

„area“

Spectrum shaping

33

Power spectrum

of input signal

f

Sxx(f )

0

Frequency response

|G(f )|2

Syy(f )

0

0

f

f

Power spectrum

of output signal

fSfGfS xxyy

2

fGfGfGfGfG *2

System Input Output

Frequency response fG

It can be proved that

where

The transformation of the power spectrum of the input signal of a system to the output

signal power spectrum can be called a spectrum shaping

A linear dynamic system

defines relationship between an

amplitude and phase of a harmonic

signal at the input and its response

of the same single frequency at the

output of the given system.

Nyquist rate A/D convertors

Analog

input

Digital

output ADC

DC

0 f

fSV

2Sf Sf

DC

0

fSV

2Sf Sf

f

2SfK SfK

Digital low pass filter

Quantizing noise 12q

LSB1q

DC

0

fSV

2Sf Sf

f

2SfK SfK

Sf

Effect of integration

and low pass filtration

ADC

SfK

LP filter Dec

Sf

ΣΔ

MOD

SfK

LP filter Dec

Sf

Decimation

K-times

Area is proportional

to the noise power 2

dBmNSREFdB

76,102,6

KNSNSREFdBdB log10

Frequency spectrum of quantizing noise and

methods of its decreasing

Frequency spectrum of quantizing noise

Sampling frequency is the same at the

input and output

Sampling frequency is greater at the

input than at the output

The use of ΣΔ modulation

Signal to noise ratio

Oversampled A/D convertors

34

Shaped spectrum of white noise

White noise

Improve S/N

Nyquist Rate convertor (succesive aproximation)

Transmitters on the principle of successive approximation . This

converter is assumed as the type for which the input (the first stage

of conversion) and output frequency (second stage of conversion) is

of the same rate.

DAC

Logic

Analog input Digital output

ΔU U0 = A ΔU

R0

UREF

I 1

R

2

Rk

R

2

D/A convertor

LSB MSB

The accuracy of a convertor

depends on the accuracy of

resistors, max 12-bit ENOB

(accuracy 0.002%)

bits

0 1 0

12

12

1

Comparator

Comparator

(ENOB – Effective Number Of Bits)

ΔU

+U0

-U0

bit „1“

bit „0“

Process of succesive aproximation

Example of a 4-bit A/D convertor

The most significant bit

The less significant bit

1000 or

0000 ?

1100 or

1000 ?

1010 or

1000 ?

1001 or

1000 ?

1st Comparison step by step : 2nd 3rd 4th bit

time

(convertor otput)

(clock)

(digital output)

Start Stop

Full scale

Input signal DAC Output

Comparator result

LSB

Clock

MSB

(comparator output)

36

Flash convertors

Analog input

Reference

voltage

Accuracy 8 to 10 bits

1 GHz sampling

frequency and more

1

0

R R

R

R

R

2

0

1

1

Digital output

Voltage divider

Input voltage is less than

the output voltage of voltage

divider

Input voltage is greated than

the output voltage of voltage

divider

Latch

Latch

Latch

Comparators

Latch is a flip-flop circuit with

two stable states, memory of a bit

37

Oversampled convertors

1-bit ADC – comparator

Convertors based on principle of delta-sigma modulation

DAC

Analog input Digital output

ADC dt...

1-bit DAC

1

0

Analog

output Digital output

+Vref

-Vref

1

Analog and

reference

inputs

Digital

input

1

1

1

Convertors of this type are referred to as single-bit, even if their accuracy reaches 24 bits

Block diagram of a first order digital delta sigma modulator

difference

Integrator

38

The second order modulator

1- bit ADC – comparator

The second order modulator

DAC

Analog input Digital output

ADC dt...

1-bit DAC

1

0 0

Analog

output

Digital

output

+Vref

-Vref

1 Analog

input Digital

input

dt...

5222 5KEE 2K reduces noise by 2.5 bits

The converter of the type AD 1847 produced by Analog Devices Company contains

a 6th order modulator

16

39

Digital decimation and filtration process

Motorola Digital Signal Processors, Principles of Sigma-Delta Modulation for Analog-to-

Digital Converters by Sangil Park, Ph. D

Decimation ratio Number of bits

Sampling rate

16 samples is replaced by a sum of them

Block diagram of sigma delta convertotr of the first order

Digital decimation process

x(t)

y(t) 1bit

D/A

dt...Σ

Digital Decimation Filter

y(n)

FS 1 16

First order Σ-Δ loop

6,4 MHz

(1 bit)

100 kHz

(16 bit)

FS -> fS

+

-

x(n)

Analog

Input Σ-Δ loops

1 16 16 16 : 1 Comb Filter

4 : 1 FIR Filter

Digital

Output

6,4 MHz

(1 bit)

100 kHz

(16 bit)

400 kHz

(12 bit) Resolution

40

The resolution of a bitstream at the sampling rate 6,4 MHz is a Bit per a sample while the resolution of a data stream at the low sampling rate 100 kHz is increased to 16 Bits per a sample.

y(n) is a bitstream at the sampling frequency FS

x(n) is a data stream at the sampling frequency fS

Signal diagram for conversion of a constant voltage

Block diagram of sigma delta convertor of the first order

x(t)

y(t) 1-bit

D/A

dt...Σ

Digital Decimation Filter 1 16

6,4 MHz

(1 bit)

100 kHz

(16 bit)

FS -> fS

+

-

x(n)

41

0

x(t)

Comparator output

Integrator output

1-bit D/A output

Total sum of 16 voltages: x(n) = 5 x (-Vref) + 11 x (+Vref) = 6 x (+Vref)

D Q

Clock

y(n) D-type latch

0 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1

Clock …………....

Latch ……………

Difference ………

y(n), D

Integrator

y(t)

Clock

Q

x(t) - y(t)

+Vref

-Vref

Signal diagram for conversion of a constant voltage

The Q output of the latch

of the D-type changes

state on the leading edge

of the clock

Time

Input

voltage

0

0

0

0

0

0

Comparator

Dynamic range limitations due to measurement

inaccuracies of the measuring chain elements

Limitation

Useful dynamic

range

of measurement

False

treshold

Microphone with

a preamplifier

A/D

convertor

Filtration

42

BK Dyn-X Technology - Single Range from 0 to 160 dB

Dyn-X

convertors

Low noise

sensor

Frequency

range of

analysis

6 kHz

Switched ranges

Measurement ranges (standard)

1E+01

103 d

B

112 d

B

119 d

B

120 d

B

127 d

B

127 d

B

128 d

B

128 d

B

162

dB

161 d

B

1E+00

1E-01

1E-02

1E-03

1E-04

1E-05

1E-06

1E-07

1E-08

Dy

nam

ic O

per

atin

g R

ange

[Vrm

s]

Narrowband 6 Hz

10 Vp

(Dyn-X)

High Quality

Transducer

Super wide range analog input

ADC 1 24 bit

ADC 2 24 bit

Real-time DSP

24 bit 24 bit Σ

43

Summary of the basic properties of A/D convertors

Typ ADC Number of bits Antialiasing filter Delay

Succesive

approximation max 12 (14) required

Low (depends on the bit

number)

Sigma-delta up to 24 not required (a part of

the converter)

Large (depends on LP

filter order)

Flash 8 (10) required negligible

Bandwidth of the analyzer = number of channels (inputs) x measurement bandwidth

Set of converters with synchronized A / D conversions forms a signal analyzer. The

main parameters of the analyzer in terms of number of used converters are

simultaneously measured channels and sampling frequency

Frequency range of measurements – theoreticaly:

– in comercial signal analyzer:

2

SR

ff

Sf

56,2

SR

ff

44

History

1980 - PRODERA – Modal analysis systems of the France origin

Since the sixties, PRODERA has been manufacturing marketing world-wide complete

systems aimed at Modal Analysis. PRODERA was the first company in the world to use

microprocessors in its modal analysis systems.

The name of the computer was INTERTECHNIQUE IN-110.

f(t)

g(t)

z(t)

r

1878 – Kelvin’s harmonic analyzer

45

Brüel & Kjær signal analyzers for laboratory measurements

1996, PULSE™,

multichannel analyzer

of the BK 3560 type

with an external DSP

and connection of a

front-endu to PC

using LAN.

1983 – the first dual channel analyzers of the BK 2032 and 2034 type offers FFT and

Hilbert transform

1992, the first

multichannel

analyzer of the

BK 3550 type

End of 90th, PULSE™,

multichannel analyzer of

the BK 3560 type

without an external DSP

and connection of a

front-end to PC using

LAN.

Present days,

PULSE™

Before 1980 – High Resolution (narrow band) Signal Analyzer BK 3031 and 2033

(single channel) with a desk-top calculator

46

SKF Portable signal analyzers for diagnostic practice

SKF Microlog Vibration

Spectrum Analyzer and

Data Collector

SKF Microlog

CMVA 65

Data Collector and

FFT analyser

Microlog GX series Microlog MX series Microlog CMXA 51-IS

SKF Microlog Analyzer

AX

Data Collector and FFT

analyser

Condition monitoring

47

Multifunction cards

8 analog inputs (14-bit, 48 kS/s)

2 analog outputs (12-bit, 150 S/s); 12 digital I/O; 32-bits counter

Connection via USB

NI-DAQmx driver software and NI LabVIEW SignalExpress LE interactive data-logging software

Low price multifunction cards NI USB-6009

Products of National Instruments, PCI a PXI.

A card for dynamic measurements, NI USB-4432

5 analog inputs (24bits ΔΣ - convertors, sampling 102.4 kS/s

per channel), AC a DC coupling, powering of acc (IEPE)

Input voltage ±40 V,

48

49

Fourier Transform Lecture 5

Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides. (1807)

...2

5cos2

3cos2

cos

y

ay

ay

ay

2

12cosy

k

1

1

d2

12cos yy

kyak

In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. [http://en.wikipedia.org/wiki/Fourier_series]

Multiplying both sides by and then integrating from y = -1 to y = +1 yields:

50

One-dimensional Fourier Transform

Signal processing

51

Spectral analysis in machine diagnostics

Centrifugal air compressor

Motor

Gear Compressor

Frequency

Vibration and noise

spectra produced by

machines are composed

of tonal components,

which are originated by

gears, bearings, motors,

etc.

The amplitude of the

tonal components

reflects the technical

state of the mentioned

machine parts.

The spectral analysis is

a useful tool for

machine diagnostics.

Bearing

52

The one-dimensional Fourier Transform - summary

dttjtxX exp

dtjXtx exp2

1

1

0

2expN

n

nk NnkjxX

1

0

2exp1 N

k

kn NnkjXN

x

Continuous signal

Let x(t) be a function of time on the

infinite time interval

Dir

ect

Discrete time

Let N be a length of a period of a discrete signal

(finite time interval)

Inv

erse

1...,,1,0 Nn

1...,,1,0 Nk

,, ttx ,...2,1,0, nxx Nnn

The Discrete Fourier Transform (DFT) of N input samples produces the same number of the output complex values

A function x(t) has a direct and inverse Fourier transform provided that

dttx

exists and there is a finite number of discontinuities.

1

0

Nx

x

x

NX

X

1

X

53

Fourier transform of a cosine signal

,,cos 0 ttAtx 000 dexpcos

AttjtAX

Let x(t) be a cosine signal of the infinite duration δ(ω) … Dirac delta function,

+ω

X(ω)

Aπδ(ω-ω0)

-ω

x(t)

+t -t

TTttx

TTttAtx

,,0

,,cos 0

TTAT

ttjtAX

T

T

00

0

SincSinc

dexpcos

x(t)

+t -t

For the cosine signal limited to the finite time interval

+T -T

A

-ω0 +ω0

+ω

X(ω)

-ω -ω0 +ω0

A AT Sinc((ω-ω0)T)

Sinc(x) = sin(x)/x

54

Discrete Fourier transform of the sine burst signal

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

0,0 0,2 0,4 0,6 0,8 1,0

x(t

)

Time History : 100 Hz - 1/16 s

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,2 0,4 0,6 0,8 1,0

x(t

)

Time History : 100 Hz - 1/4 s

0

50

100

-400 -300 -200 -100 0 100 200 300 400

512 *

Am

pl

FFT : 100 Hz - 1/16 s

0

50

100

150

-400 -300 -200 -100 0 100 200 300 400

512

* A

mp

l

FFT : 100 Hz - 1/4 s

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

0,0 0,2 0,4 0,6 0,8 1,0

x(t

)

Time [s]

Time History : 10 Hz - 1 s

0

200

400

600

-400 -200 0 200 400

51

2 *

Am

pl

Frequency [Hz]

FFT : 100 Hz - 1 s

Example no.1 – Discrete Fourier transform of a constant

15...,,1,0,1 nxn

0162exp15

0

n

k nkjX

15...,,0k

W 16 0

-2 π/16

W 16 1

2 π/16

1 160exp

15

0

0 n

X

0,0

0,5

1,0

1,5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

xn

Index n

0

4

8

12

16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Xk

Index k

Let xn be a sequence of constant samples

55

Notice the fact that X0 is equal to the N multiple of the sample value while Xk is equal to zero for

k > 0 .

56

Discrete Fourier transform of the cosine signal

with the integer number of waves per record

1...,,1,0,2cos NnNMnAxn

2expexpcos jxjxx

.2exp2exp2

2exp2exp2exp2

1

0

1

0

N

n

N

n

k

NnkMjNnkMjA

NnkjNMnjNMnjA

X

1...,,1,0,22exp2exp NnNMnjNMnjAxn

MNMkXjN

AXjN

AX kMNM ,,0,exp2

,exp2

Let

It is assumed that N, M are integer numbers 20 NM

Using the formula the cosine function is changed into the exp function

After substituting into the discrete Fourier transform formula

we obtain

M is equal to the number of waves

per record of the length N. be a signal to be transformed.

Notice the fact that the absolute value of XM and XN-M is equal to the N / 2 multiple

of the harmonic signal amplitude while Xk is equal to zero for k <> M, N - M .

57

Example no.2 – Discrete Fourier transform of the periodic

signal with the integer number of waves per record

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

xn

Index n

15...,,0k

-16

-8

0

8

16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Re(Xk)

Index k

-16

-8

0

8

16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Im(Xk)

Index k

M = 3

N - M = 13

symmetry

15,...,0,1632sin nnxn

13,30

13,2exp8

3,2exp8

k

kj

kj

X k

Let

The Fourier transform is as follows

3 waves per record

The Fourier transform of a harmonic function having an integer number of waves per record for FFT

be a signal to be transformed.

0,ImImandReRe MXXXX MNMMNM

Notice the complex conjugate symmetry of the Fourier transform

periodic

extension

smooth

continuation

58

Example no.3 - Fourier transform of the periodic signal

with the fractional number of waves per record

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

0 2 4 6 8 10 12 14 16

xn

Index n

0

2

4

6

8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

|Xk|

Index k

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48

xn

Index n

15,...,0,165.32sin nnxn Notice the fractional number of waves per record (3.5)

which contains 16 samples.

Period of 16 samples for

evaluation FT

Peaks are smeared

irregularities in the sequence of the adjacent periods

Let

Fourier transform of a harmonic function having a non-integer number of waves per record for FFT

be a signal. 3.5 waves per record

The fractional number of the waves in a record for computing FT causes smearing of the spectrum.

59

Positive and negative frequency

0 1 2 3 4 15 14 13 8

M N-1 N-M

0 1 2 3 4 -1 -2 -3 8

M -1 -M

-1 -3 -2 -4 -M

-5 -6 -8 -7

|XN-M|

|XN-M|

|XM|

|XM|

|X-M| |Xk|

|Xk|

fS

fS

fS /2

fS /2

- fS /2

Negative

frequency

components

frequency

Positive angle

Positive

frequency

components

index

frequency

index

frequency

index

Negative angle

fS ....... sampling frequency

fS /2 ... Nyquist frequency

N = 16, k = 0, 1, 2, ..., 15

symmetry

symmetry

Im

Re

15,...,0,1632sin nnxn

2-side spectrum

Let be the sinusoidal signal to be transformed by the use of the FT.

For a real signal the Fourier transform results in the spectrum with amplitude which is symmetric about 8 (fS/2).

If the signal to be transformed is a complex, then the spectrum may be non-symmetric.

0,ImImorReRe MXXXX MNMMNM

60

tj

P eA

X2

A real harmonic signal as a sum of the complex-conjugate

phasor pair rotating in the opposite direction

NP XXX

tj

N eA

X2

Re

Im

NP XXtjtj

AtAtx

2

expexpcos

t … time

A ... amplitude

ω … angular velocity

φ … initial phase

Φ = ωt + φ … phase

Complex

plane

Euler’s formula

+ω

-ω

Phasors XP and XN are

complex conjugates

Notice the symmetry of phasors for real harmonic signals. It can be

concluded that, generally, the following can be applied to any real signal

NPNP XXXX ImImandReRe

61

Ptj

PP eAX

A complex harmonic signal as a sum of the non-complex-

conjugate phasor pair rotating in the opposite direction

NP XXX

Ntj

NN eAX

Re

Im

t …………………………………… time

AP and AN ......................................... amplitude of phasors

ω ………………………………….. angular velocity

φP and φN ………………...………. initial phase of phasors

ΦP = ωt+ φP and ΦN = -(ωt+ φN) … phase of phasors

Complex

plane

If the phasors XP and XN are not complex conjugate phasors, then the signal is complex

+ω

-ω

Subscripts P and N indicate that the phasor is rotating in positive (P) or negative (N) direction.

Re

Im

PX

NP XXX

NX

t

The locus of all endpoints of

the phasor X is an ellipse.

One-side and two-side spectrum

-1,5

0,0

1,5

-1,5 0,0 1,5

Im(x

)

Re(x)

-1,5

0,0

1,5

0,0 1,0

Re

(x),

Im

(x)

Time [s]

Re(x)

Im(x)

0

10

20

30

40

50

60

70

-5 -4 -3 -2 -1 0 1 2 3 4 5

|X|

Frequency [Hz]

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,5 1,0

Re(x

)

Time [s]

0

8

16

24

32

40

-5 -4 -3 -2 -1 0 1 2 3 4 5

|X|

Frequency [Hz]

0

8

16

24

32

40

0 1 2 3 4 5

|X|

Frequency [Hz]

Two-side Fourier transform One-side Fourier transform

non-symmetry

FT

FT

62

63...,,1,0,642sin nnxnLet be a real signal to be transformed using the Fourier transform

63...,,1,0,3642sin642sin nnjnxnLet be a complex signal to be transformed

Time history Spectrum is symmetric about 0 Hz Symmetric part is omitted

Two-side Fourier transform Time history Spectrum is non-symmetric Orbit plot

63

Inverse Fourier transform (IFT)

Im

Re

-ω0 +ω0

-3ω0

+3ω0

-5ω0

+5ω0

+ω0

+3ω0

+5ω0

+ω

+t

+t

+t

+t

x1(t)

x3(t)

x5(t)

x(t)

0

-jω0 n Δt, n = 0, 1, …,15

j

1

-jωt

j Continuous time t

continuous rotation

Discrete time

nΔt – stepwise rotation

N = 16 samples per record T

ω0 = 2π/T

The periodic signal as a sum of the phasor pairs which are rotating in the opposite direction

exp(-jω0 t)

exp(+jω0 t)

T

T

T

T

0

0

0

0

64

Inverse discrete Fourier transform (IDFT)

The complex conjugate of the complex number is given by the formula jbaX

jbaX *

**

nn xx Let complex conjugate operation be applied two times then because yields

The inverse discrete Fourier transform may be written in the form

Im

Re

X

X*

Notice the fact that the inverse Fourier transform does not require a special algorithm.

The formula for calculation of the inverse Fourier transform

.1...,,1,0,2exp1 1

0

NnNnkjXN

xN

k

kn

.1...,,1,0,2exp2exp1

*1

0

**

1

0

***

NnNnkjN

XNnkjX

Nxx

N

k

kN

k

knn

.1...,,1,0,DFT

**

NnN

Xx k

n

65

Effect of integration and differentiation

on the amplitude of harmonic signals

tDtd cos

tA

ttAtv

sindcos

tDt

tdtv sin

d

d

tDt

td

t

tvta cos

d

d

d

d 2

2

2

tAta cos

tA

tttAttA

tD

cosddcosdsin21

Displacement

Velocity

Acceleration

Filtration in the frequency domain

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time History : Sine1+Sine2

0 100 200 300 400 500 600

0 200 400 600 800 1000

0,0

0,5

1,0

1,5

0 200 400 600 800 1000

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time : Generator 2 : Sine1+Sine2 - 2

0 100 200 300 400 500 600

0 200 400 600 800 1000

Frequency [Hz]

FFT

IFFT

fS /2 fS

FFT

txfXfXtx FF

IFFT Filter

tx fX

txF fX F

symmetry

weighting

function

66

67

Effect of the plot scale on the diagram appearance

0,0001

0,0010

0,0100

0,1000

0 200 400 600 800 1000

Frequency [Hz]

Integration

Logarithmic scales

symmetry jG 1

0,0001

0,0010

0,0100

0,1000

1 10 100 1000

Frequency [Hz]

10

100

1000

10000

1 10 100 1000

Frequency [Hz]

0

1000

2000

3000

4000

0 200 400 600 800 1000

Frequency [Hz]

jG Differentiation

Linear scales

Log

arit

hm

ic s

cale

s

+20 dB/dec

-20 dB/dec

Lo

gar

ith

mic

sca

le

Lin

ear

scal

e

Nyquist frequency Nyquist frequency

The roll-off of the frequency response is -20 dB/dec for integration and +20 dB/dec for differentiation

Integration in the frequency domain

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time History : Sine1+Sine2

0 100 200 300 400 500 600

0 200 400 600 800 1000

0,0001

0,0010

0,0100

0,1000

0 200 400 600 800 1000

0,0

0,5

1,0

1,5

2,0

0 200 400 600 800 1000

Frequency [Hz]

FFT

IFFT

fS /2 fS

FFT

txffjfX

fjfX

ff

ffI

SS

S

22

22

:2

:2 IFFT

Integration

tx fX

dttx fX I

symmetry

weighting

function

f 2f21 ffS 21

fXffXfXffX ISIS** ,

fXtx 2

-5,E-03

-3,E-03

7,E-18

3,E-03

5,E-03

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time : Generator 2 : Sine1+Sine2 - 2

68

Differentiation in the frequency domain

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time History : Sine1+Sine2

0 100 200 300 400 500 600

0 200 400 600 800 1000

0

1000

2000

3000

4000

0 200 400 600 800 1000

-1000

-500

0

500

1000

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time : Generator 2 : Sine1+Sine2 - 2

0,E+00 5,E+04 1,E+05 2,E+05 2,E+05 3,E+05 3,E+05

0 200 400 600 800 1000

Frequency [Hz]

FFT

IFFT

fS /2 fS

IFFT Differentiation

tx fX

dttdx fX D

symmetry

weighting function

f 2 f2 ffS 2

fXffXfXffX DSDS

** ,

FFT

txfXffj

fXfj

ff

ffI

SS

S

22

22

:2

:2 fXtx 2

69

70

Fast Fourier Transform

Lecture 6

James W. Cooley and John W. Tukey,

An algorithm for the machine calculation of complex Fourier series,

Math. Comput. 19, 297–301 (1965).

71

Number of complex additions and multiplications

NjNNjWN 2sin2cos2exp

1...,,2,1,0for,2exp1

0

NkNikjixixXN

i

Nk

.Nk

N

k

N WW 2Nk

N

k

N WW

The N-point discrete Fourier transform require N 2 complex multiplications and additions (operation)

Periodicity properties of the W term in the FT formula Twiddle factor

NkjW k

N 2exp

, then it can be proved that Let N be a power of 2 ( ) mN 2

k

NW2Nk

NW

2N steps

k

NW

Nk

NW

N steps

10 NW

1

NW

N2

See [Oppenheim & Schafer]

History Note - Milestones

• DIT - Decimation in time

• DIF - Decimation in frequency

This algorithm, including its recursive application, was invented around 1805 by Carl Friedrich

Gauss, who used it to interpolate the trajectories of the asteroids Pallas and Juno.

Computational methods for the Fast Fourier Transform

James W. Cooley and John W. Tukey,

An algorithm for the machine calculation of complex Fourier series,

Math. Comput. 19, 297–301 (1965).

Runge in 1903 which essentially described the FFT. Danielson and Lanczos in 1942 recognized

certain symmetries and periodicities which reduced the number of operations.

FFTs for N = 2m became popular after J. W. Cooley of IBM and John W. Tukey of Princeton

published a paper:

The fast Fourier transform belongs to the 10 algorithms with the greatest influence

on the development and practice of science and engineering in the 20th century

72

See [Oppenheim & Schafer]

73

Principle

1,...,1,0,2exp1

0

1

0

NkWxNknjxX kn

N

N

n

n

N

n

nk

Let N be a composite N = N1 N2 in the N-point discrete Fourier transform (DFT)

then we can rewrite the previous formula by letting indexes

121 Nkkk 221 nNnn

We then have

1,...,2,1,0, 121

1

0

1

0

22

2

2

2

21

1

1

11

1221121

NNkkWWWxXnk

N

N

n

nkN

N

n

nk

NnNnNkk

multiplies the result by the so-called twiddle factors WNk1n2 , and finally computes N1 DFTs

of the size N2 (the outer sum).

The term radix is used to describe N1 and N2. If N is a power of two then the small DFT of the

radix is called a butterfly.

...and1,...,2,1,0, 122

1

0

1

1

1

11

122122112

kNnWxxFTkYN

n

nk

NnNnnNnNn

The algorithm computes N2 DFTs of size N1 (the inner sum),

This decomposition is then continued recursively.

...and1,...,2,1,0, 211

1

0

11222

2

2

2

222121

kNnWkYkYFTkXnk

N

N

n

nnNNkk

74

Decimation in time, the radix-2 DIT case

.22exp2exp22exp

2exp

12

0

12

12

0

2

1

0

NkjxNkjNkjx

NknjxX

NN

N

n

nk

12...,,1,0for,2exp NkHNkjGX kkk

,,, 222

kN

NkNkNkkNk WWHHGG

12...,,1,0for,2exp2 NkHNkjGX kkNk

12...,,1,0for,2

NkHWGX

HWGX

kk

kNk

kk

kk

Even-indexed samples Odd-indexed samples

Summary

Using periodicity properties we obtain the values Xk ,

where k > N/2-1

Let N-point DFT, where N = 2m, be decomposed

into

22FT xG N 122FT xH NTwo N/2-point DFTs: NkjW k

N 2exp

Radix-2 DIT divides a DFT of size N into two

interleaved DFTs (hence the name "radix-2")

of size N/2 with each recursive stage.

The values Xk , where k < N/2

Speeding up the calculation of FT is based on splitting the sequence into two parts.

75

Radix-2 butterfly diagram

a … gain

Rules for creating signal-flow graph

Appearance of a butterfly

.,

,,

11011101

11001100

CYSXYYSYCXXX

CYSXYYSYCXXX

Addition and multiplication of complex numbers

addition

y x

y

x1

y = x1 + x2

x2

y = a x a

+

+

Signal diagrams

00 jYX

11 jYX

00 YjX

11 YjX 1

x y xy

x y xay a

1x

2x

y1a

2a

b

2211 xaxaby

76

Signal diagram for the 8-point DFT

x(0)

x(2)

x(4)

x(6)

DFT

N/2

x(1)

x(3)

x(5)

x(7)

DFT

N/2

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

G (0)

G(1)

G(2)

G(3)

H(0)

H(1)

H(2)

H(3)

-1

-1

-1

-1

W 0

W 1

W 2

W 3

x(0)

x(4)

x(2)

x(6)

DFT N/4

G(0)

G(1)

G(2)

G(3)

GG(0)

GG(1)

HG(0)

HG(1) -1

-1

DFT N/4

W0

W2

x(0)

x(4)

GG(0)

GG(1) -1

.212exp

,202exp

40401

40400

xxxjxGG

xxxjxGG

N-point DFT … N 2 operations

Two N/2-point DFT …

(N/2)2+ (N/2)2 = N2/2 operations

.22exp2exp22exp12

0

12

12

0

2 NkjxNkjNkjxXNN

k

Gk Wk Hk

Three-stage algorithm

Number of operations:

1st stage: 1 x 8

2nd stage: 2 x 4

3rd stage: 4 x 2

Total: 24

1 operation =

1 complex multiplication

and 1 complex addition

Xk = + *

77

Visual Basic code for FFT, recursive DIT case

Public Type complex

re As Double: im As Double

End Type

Public Sub MyDITFFT(z() As complex, zz() As complex)

Dim N As Long, N2 As Long, k As Long, fi As Double, w As complex

Dim z0() As complex, z1() As complex, G() As complex, H() As complex

N = UBound(z) + 1

ReDim zz(N - 1) As complex

If N = 2 Then

zz(0).re = z(0).re + z(1).re: zz(0).im = z(0).im + z(1).im

zz(1).re = z(0).re - z(1).re: zz(1).im = z(0).im - z(1).im

Else

N2 = N / 2

ReDim z0(N2 - 1) As complex, z1(N2 - 1) As complex

ReDim G(N2 - 1) As complex, H(N2 - 1) As complex

For k = 0 To N2 - 1

z0(k) = z(2 * k): z1(k) = z(2 * k + 1) ‘ Even- and odd-indexed samples

Next k

MyDITFFT z0, G ‘ Recursive calling

MyDITFFT z1, H ‘ Recursive calling

For k = 0 To N2 - 1

fi = 2 * pi / N * k: w.re = Cos(fi): w.im = -Sin(fi)

zz(k).re = G(k).re + w.re * H(k).re - w.im * H(k).im

zz(k).im = G(k).im + w.re * H(k).im + w.im * H(k).re

zz(k + N2).re = G(k).re - w.re * H(k).re + w.im * H(k).im

zz(k + N2).im = G(k).im - w.re * H(k).im - w.im * H(k).re

Next k

End If

End Sub

Recursive calls

78

Complete signal diagram for the 8-point DFT

x(0) X(0)

x(4) X(1)

x(2) X(2)

x(6) X(3)

x(1) X(4)

x(5) X(5)

x(3) X(6)

x(7) X(7)

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1 W N

0

W N 0

W N 0

W N 0

W N 0

W N 0

W N 2

W N 2

W N 2

W N 1

W N 3

W N 0

1 st STAGE 2

nd STAGE 3 rd STAGE

Reverse

bit order

input

Notice of the sequence sample index

1st stage: 4 x 2 additions and multiplications

2nd stage: 2 x 4 additions and multiplications

3rd stage: 1 x 8 additions and multiplications

79

Reverse order of bits

Initial index

order (Decimal

numbers)

Binary numbers Reversing bit

order in binary

numbers

Final index

order (Decimal

numbers)

0 000 000 0

1 001 100 4

2 010 010 2

3 011 110 6

4 100 001 1

5 101 101 5

6 110 011 3

7 111 111 7

Data reordering performs the index register of the Digital Signal Processors (DSP)

000

100

010

110

001

101

011

111

80

Pascal routine for FFT, non-recursive DIT case

PROCEDURE DIT(p,VAR f);

LOCAL Bp,Np,Np',P,b,k,BaseT,BaseB,top,bot;

BEGIN {DIT}

{initialise pass parameters}

Bp:=1<<(p-1);{No. of blocks}

Np:=2; {No. of points in each block}

{perform p passes}

FOR P:=0 TO (p-1) DO

BEGIN {pass loop}

Np':=Np>>1; {No. of butterflies}

BaseT:=0; {Reset even base index}

FOR b:=0 TO (Bp-1) DO

BEGIN {block loop}

BaseB:=BaseT+Np'; {calc odd base index}

FOR k:=0 TO (Np'-1) DO

BEGIN {butterfly loop}

top:=f[BaseT+k];

bot:=f[BaseB+k]*T(Np,k); {twiddle the odd n results}

f[BaseT+k]:= top+bot; {top subset}

f[BaseB+k]:= top-bot; {bottom subset}

END; {butterfly loop}

BaseT:=BaseT+Np; {start of next block}

END; {block loop}

{calc parameters for next pass}

Bp:=Bp>>1; {half as many blocks}

Np:=Np<<1; {twice as many points in each block}

END; {pass loop}

END; {DIT}

http://www.engineeringproductivitytools.com/stuff/T0001/PT04.HTM#Head317

{Perform in place DIT of 2^p points (=size

of f) N.B. The input array f is in bit

reversed order!

So all the 'even' input samples are in the

'top' half, all the 'odd' input samples are in

the 'bottom' half..etc (recursively). }

81

Decimation in frequency, the radix-2 DIF case

12...,,1,0,2, NkX k

.22exp

222exp22exp

22exp22exp

12

0

2

12

0

2

12

0

1

0

1

0

2

Nnjxx

NNnjxNnjx

NnjxNnjxX

N

n

Nnn

N

n

Nn

N

n

n

N

n

n

N

n

n

.22exp2exp

22exp2exp

122exp

12

0

2

1

0

1

0

12

NnjNnjxx

NnjNnjx

NnjxX

N

n

Nnn

N

n

n

N

n

n

12...,,1,0,12, NkX k

Even-indexed output

Odd-indexed output 12...,,1,0,

,

2

2

Nn

sWxx

rxx

n

n

Nnn

nNnn

.22exp

,22exp

12

0

12

12

0

2

NnjsX

NnjrX

N

n

n

N

n

n

Resulting formulas

Let

be used for substitution then

82

8-point DFT using the DIF method

22exp12

0

12 NnjsXN

n

n

x(0)

x(2)

x(4)

x(6)

DFT

N/2

x(1)

x(3)

x(5)

x(7)

DFT

N/2

X(0)

X(2)

X(4)

X(6)

X(1)

X(3)

X(5)

X(7)

r(0)

r(1)

r(2)

r(3)

s(0)

s(1)

s(2)

s(3)

-1

-1

-1

-1

W 0

W 1

W 2

W 3

22exp12

0

2 NnjrXN

n

n

nNnn rxx 2

nn

Nnn sWxx 2

Rev

erse

bit

ord

er o

utp

ut

Rev

erse

bit

ord

er i

np

ut

83

Visual Basic code for FFT, recursive DIF case

Public Type complex

re As Double: im As Double

End Type

Public Sub MyDIFFFT(z() As complex, zz() As complex)

Dim N As Long, N2 As Long, k As Long, fi As Double, w As complex

Dim r() As complex, s() As complex, G() As complex, H() As complex

N = UBound(z) + 1

ReDim zz(N - 1) As complex

If N = 2 Then

zz(0).re = z(0).re + z(1).re: zz(0).im = z(0).im + z(1).im

zz(1).re = z(0).re - z(1).re: zz(1).im = z(0).im - z(1).im

Else

N2 = N / 2

ReDim z0(N2 - 1) As complex, z1(N2 - 1) As complex, G(N2 - 1) As complex

ReDim H(N2 - 1) As complex, r(N2 - 1) As complex, s(N2 - 1) As complex

For k = 0 To N2 - 1

fi = 2 * pi / N * k: w.re = Cos(fi): w.im = -Sin(fi)

z0(k) = z(2 * k): z1(k) = z(2 * k + 1) ‘ Even- and odd-indexed samples

r(k).re = z(k).re + z(k + N2).re: r(k).im = z(k).im + z(k + N2).im

s(k).re = (z(k).re - z(k + N2).re) * w.re - (z(k).im - z(k + N2).im) * w.im

s(k).im = (z(k).re - z(k + N2).re) * w.im - (z(k).im - z(k + N2).im) * w.re

Next k

MyDIFFFT r, G ‘ Recursive calling

MyDIFFFT s, H ‘ Recursive calling

For k = 0 To N2 - 1

zz(k) = G(2 * k): zz(k) = H(2 * k + 1) ‘ Even- and odd-indexed output values

Next k

End If

End Sub

Recursive calls

84

Pascal routine for FFT, non-recursive DIF case PROCEDURE DIF(p,VAR f);

LOCAL Bp,Np,Np',P,b,n,BaseE,BaseO,e,o;

BEGIN {DIF}

{initialise pass parameters}

Bp:=1; {No. of blocks}

Np:=1<<p; {No. of points in each block}

{perform p passes}

FOR P:=0 TO (p-1) DO

BEGIN {pass loop}

Np':=Np>>1; {No. of butterflies}

BaseE:=0; {Reset even base index}

FOR b:=0 TO (Bp-1) DO

BEGIN {block loop}

BaseO:=BaseE+Np'; {calc odd base index}

FOR n:=0 TO (Np'-1) DO

BEGIN {butterfly loop}

e:= f[BaseE+n]+f[BaseO+n];

o:=(f[BaseE+n]-f[BaseO+n])*T(Np,n);

f[BaseE+n]:=e;

f[BaseO+n]:=o;

END; {butterfly loop}

BaseE:=BaseE+Np; {start of next block}

END; {block loop}

{calc parameters for next pass}

Bp:=Bp<<1; {twice as many blocks}

Np:=Np>>1; {half as many points in each block}

END; {pass loop}

END; {DIF}

See [http://www.engineeringproductivitytools.com/stuff/T0001/PT03.HTM#Head134]

{Perform in place DIF of 2^p points (=size of f)}

85

Comparison of the computional operation number with

the use of DFT and FFT

The Discrete Fourier Transform, where N is an arbitrary number, requires N 2 complex

multiplications and additions

The Cooley-Tukey algorithm for the Fast Fourier Transform, based on N = 2m, requires only

Nm = N log2 N complex multiplications and additions. It is said that the algorithm is performed in

O(N log N) time.

An example

Let N be the teth power of two (N = 1024 = 210)

DFT … approximately 1 000 000 operations FFT … 10 240 operations

Each stage of the FFT algorithm, which is divided into two algorithms applied to half the number

of points, requires (N/2)2 + (N/2)2 = N 2 /2 complex multiplications and additions.

N-point

Other FFT algorithms

• radix-4

• N = N1N2 with coprime numbers N1 and N2 (they have no common factor other than 1)

• …

86

FFT in Matlab

We can find out in the Manlab function references that the FFT functions (fft, fft2, fftn, ifft, ifft2,

ifftn) are based on a library called FFTW [3], [4]. To compute an N-point DFT when N is composite

(that is when N = N1N2), the FFTW (the fastest Fourier transform in the West http://www.fftw.org)

library decomposes the problem using the Cooley-Tukey algorithm [1], which first computes N1

transforms of size N2, and then computes N2 transforms of size N1. The decomposition is applied

recursively to both the N1- and N2-point DFTs until the problem can be solved using one of several

machine-generated fixed-size "codelets". The codelets in turn use several algorithms in combination,

including a variation of Cooley-Tukey [5], a prime factor algorithm [6], and a split-radix algorithm

[2]. The particular factorization of N is chosen heuristically.

When N is a prime number, the FFTW library first decomposes an N-point problem into three

(N-1)-point problems using Rader's algorithm [7]. It then uses the Cooley-Tukey decomposition

described above to compute the (N-1)-point DFTs.

The execution time for fft depends on the length of the transform. It is the fastest for powers of two.

It is almost so fast - as for the length - that it equals to powers of two if the prime factorization of the

length has only small prime factors. It is typically several times slower compared to the previous

case if the length is a prime or has large prime factors.

[1] Cooley, J. W. and J. W. Tukey, "An Algorithm for the Machine Computation of the Complex Fourier Series,"

Mathematics of Computation, Vol. 19, April 1965, pp. 297-301.

[2] Duhamel, P. and M. Vetterli, "Fast Fourier Transforms: A Tutorial Review and a State of the Art," Signal

Processing, Vol. 19, April 1990, pp. 259-299.

[3] Frigo, M. and S. G. Johnson, "FFTW: An Adaptive Software Architecture for the FFT," Proceedings of the

International Conference on Acoustics, Speech, and Signal Processing, Vol. 3, 1998, pp. 1381-1384

[4] Oppenheim, A. V. and R. W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989, p. 611 and 619.

87

Authors of the Fastest Fourier Transform in the West

Matteo Frigo received his Ph. D. in 1999 from the Dept. of Electrical

Engineering and Computer Science at the Massachusetts Institute of

Technology (MIT). Besides FFTW, his research interests include the theory

and implementation of Cilk (a multithreaded system for parallel

programming), cache-oblivious algorithms, and software radios.

Joint recipient, with Steven G. Johnson, of the 1999 J. H. Wilkinson Prize for

Numerical Software, in recognition of their work on FFTW.

Steven G. Johnson joined the faculty of Applied Mathematics at MIT in 2004.

He received his Ph. D. in 2001 from the Dept. of Physics at MIT, where he also

received undergraduate degrees in computer science and mathematics. His recent

work, besides FFTW, has focused on the theory of photonic crystals:

electromagnetism in nano-structured media.

See [http://www.fftw.org/]

88

FFTW speed

0

1

23

4

5

6

7

8

910

11

12

512 576 640 704 768 832 896 960 1024

Number of points for calculation FFT

Re

lative

sp

ee

d

NNNNNNNNNNNN 222222 log2log12log2log22log2log:2

The fastest FFT algorithm is for the record length that is a power-of-two. Comparison of the relative

computational speeds for a large range of log2(N) related to speed for N = 256 is shown in left figure.

0,01

0,10

1,00

10,00

100,00

1000,00

10000,00

1 3 5 7 9 11 13 15 17

k = log2(N ) = log2(2k)

Re

lative

sp

ee

d

N = 28 = 256

N log2(N )

Computer

experiment

s

The FFTW algorithm is designed for any record length including non-power-of-two. The relative

computational speeds for the record lengths in between N = 512 and 1024 are shown in right figure.

The fact that doubling the record length results in doubling the computational time can be proved by

89

Zero padding As it was noted some efficient FFT algorithms require the record length that is a power-of-two.

When applying these algorithms to non-power-of-two length signal records, you have to zero

pad or truncate the record to a power-of-two length. The record of the length M is padded with

zeros to the length N as follows

1...,,1,0,2exp2exp1

0

1

0

NkxGMNMkijxNikjyyF iNMk

M

i

i

N

i

iik

1...,,1,,0

1...,,1,0,

NMMi

Mixy

i

i

The Fourier transform of the record yi , i = 0, 1, 2, …, N -1 is related to the transform of the record

xi , i = 0, 1, 2, …, M -1 according to the equation

If kM is an integer multiple of N then iNMkik xGyF and are identical.

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

0

5

10

15

20

0 2 4 6 8 10 12 14 16

Frequency [Hz]

0

510

15

20

0 2 4 6 8 10 12 14 16

Frequency [Hz]

32 samples, 3 Hz signal

Hanning window (M = 32)

Magnitude of the FT transform

32-point record (N = 32)

Padded with zeros to128-point

record (N = 128)

Example

90

ZOOM FFT

The frequency resolution of the FFT spectrum in the baseband range is given by the formula

Δf = 1/T, where T = N / fS , N is the length of the input record for calculation FFT and fS =1/ Δt

is the sampling frequency. If the finer resolution over a limited portion of the spectrum is required

then the so-called Zoom-FFT may be applied.

Zooming-in on the mentioned portion of the frequency spectrum is defined by the central frequency

fC and the bandwidth ΔF, which is containing the number of lines as the normal baseband spectrum.

The method, called Real-time zoom, is based on three consecutive steps.

...,2,1,0,2exp itifjxy Cii

1,...,2,1,0,2exp1

,2

exp1 1

0

1

0

Nif

f

N

kijF

Nyik

NjF

Nx

N

k S

Cki

N

k

ki

1) The shift of the frequency origin to the center of the zoom band is the first step of the algorithm.

The real samples xi , i = 0, 1, 2, … are multiplied by a rotating unit vector exp(-j2π fC Δt )

The record of the real samples xi , i = 0, 1, 2, … is transferred into the record of the complex samples

yi , i = 0, 1, 2, … The index k corresponding to the center of the zoom band is given by the formula

k = N fC / fS .

Multiplication by the rotating vector may cause aliasing in the negative frequency band.

91

ZOOM FFT cont’d

3) The last step is computing the FFT spectrum. Because the input record is composed of the

complex values, the two-side spectrum is computed and presented.

ΔF

fC fN fS -fN -fS 0 -fN -fC -fN +fC

aliasing

2) To prevent aliasing the real and imaginary part of the complex record is low pass filtered. The

filter cut-off frequency is equal to ΔF/2. The zoom factor is a ratio fS /ΔF = (fN /ΔF/2). The reduced

frequency range makes it possible to reduce the sampling frequency by a factor of zoom. The

sequence of the complex samples yi , i = 0, 1, 2, … is decimated by the mentioned zoom factor.

Decimation is performed in cascaded octave steps, what required the zoom factor to be powers of 2.

For the required number of spectral lines it is needed record the corresponding number of samples.

fN = fS /2 … Nyquist frequency

See [Randall]

In addition to the method described above, there is another method, called Non-destructive zoom,

for which the FFT calculation of a long record is split into repeated calculation of shorter records.

Overlap-add method in convolution

92

1

00

M

m

mnm

m

mnmnnn xhxhxhy

The convolution of the input sequence with the finite impulse response hm, m = 0 to M -1 is a formula

where hm = 0 for m outside the interval [0, M -1]. For an arbitrary segment length, it yields

otherwise,0

1...,,1,0, Lnxx

kLn

n

k k

n

k

n xx

k

n

k

k

n

k

n

k

n

k

nnnn yxhxhxhy

where kyn is zero outside the interval [0, L+M - 2] of the index n. For any parameter

where the upperleft index k dentes the segment order. The convolution takes the form

1 MLN

it is equivalent to the N-point circular convolution in the interval [0, N -1]. The advantage is that the

circular convolution can be computed very efficiently as follows, according to the circular convolution

theorem.

where FFT and IFFT refer to the fast Fourier transform and inverse fast Fourier transform,

respectively, evaluated over N = 2L discrete points (kxn is padded by L zeros and hn is padded by 2L-M

zeros). Long signals can be break into smaller segments for easier processing. FFT convolution uses

the overlap-add method combined with the Fast Fourier transform, which makes it easy to compute

convolution. The Fourier transforms of the input signal and the filter impulse response are multiplied

together.

nn

k

n

k hxFFTy FFTIFFT

Circular convolution

93

For a periodic function xT(t), with period T, the convolution with another function, h(t), is also

periodic, and can be expressed in terms of integration over a finite interval as follows

http://en.wikipedia.org/wiki/Circular_convolution

For discrete sequences xN, n = xN, n+N

m k

kNmnNm

m

mnNmnnN xhxhhx ,,,

Tt

t

TTTT tthtxtthtxthtx0

0

dd

where t0 is an arbitrary parameter and hT(t) is a periodic extension of h(t) defined by

kk

T kTthkTthth

When xT(t) is expressed as the periodic extension of another function, x, this convolution is

sometimes referred to as a circular convolution of functions h and x.

Overlap-add algorithm

94

1st segment: N = 2L samples in total, L samples of 0xn padded by L

zeros

2nd segment: L samples of 1xn padded by L zeros

L

0xn 0

L

0 +

1xn

nx 0xn = xn

1st segment: N = 2L samples in total, L+M -1 samples equal to yi

and L -M+1 zeros

2nd segment

L

+

ny

Input

Convolution of a segment-by-segment

0xn 0

Output is produced piecewise by an overlap-add algorithm

hn 0 0

L M -1

M

0yn

0 0yn

0 1yn

2L-M

=

=

= L+M -1 samples of 0yn padded

by L-M+1 zeros

N=2L

L L

1xn = xn+L 2xn = xn+2L

Convolution with the use of FFT

95

Circular convolution of segment-by-segment

0xn 0

hn 0 0

M

0Yn

2L-M

For filter orders larger than about 64 points, FFT convolution is faster than standard convolution,

while producing exactly the same result.

0Xn

Hn

FFT

FFT

N = 2L

2L

Multiplication element-by-element

0 0yn

2L

L L

IFFT

nn

k

n

k hxFFTy FFTIFFT

Generally

FFT Convolution, example

0

2000

4000

0 8 16 24 - 200

0

200

0 2 4 6

200

- 200

0

200

0 2 4 6 0

2000

4000

0 8 16 24 0

2000

4000

0 8 16 24 - 200

0

200

0 2 4 6

- 200

0

200

0 2 4 6 - 200

0

200

0 2 4 6

Segments FFT Filter IFFT

Low Pass

Low

Pass

+ +

= =

(FFT Convolution for smoothing

discontinuities of consecutive

records)

Rev

0

2000

4000

0 8 16 24 - 200

0

0 2 4 6

Rev Ord

Rev Rev

Rev Rev Ord

96

97

Autospectrum - Cross-Spectrum

Windowing - Averaging

Lecture 7

98

Computation of autospectrum

Time History : Generator 1 : Sine1 - 1

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

FFT : Generator 1 : Sine1 - 1

0

200

400

600

0 100 200 300 400 500

Frequency [Hz]

51

2 *

Am

pl

Autospectrum : Generator 1 : Sine1 - 1

0,0

0,2

0,4

0,6

0,8

0 100 200 300 400

Frequency [Hz]

RM

S

A*N/2,

(A*N/2)*2/N/√2 = A /√2

A, amplitude

N = 1024

The RMS autospectrum requires

change of the spectrum scale

Root Mean Square

A harmonic signal to be analyzed

The magnitude of spectrum

compoment as a result of FFT

99

Autospectrum scales

Let the Fourier transfor of a signal

RMS: Root Mean Square

RMS spectrum

Autospectrum

.12...,,2,1,22

,0,0

NkNkX

kNXkRMS

.12...,,2,1,22

,0,02

2

NkNkX

kNXkPWR

.12...,,2,1,2

,0,00*

*

NkNkXNkX

kNXNXkPWR

.12...,,2,1,2

,0,02

2

NkNkXT

kNXTkPSD

12...,,2,1,0, NknxFTkX

1...,,2,1,0, Nnnxsampling frequency

SS Tf 1

PWR = RMS2 : Power

Power spectrum

PSD = PWR / Δf :

Power Spectral Density

(diagnostics)

(acoustics)

(random signals)

Unit: U, e.g. [m/s2]

Unit: U2, e.g. [(m/s2)2], [Pa2]

Unit: U2/Hz, e.g. [(m/s2)2/Hz]

TNTNff

Nfkf

SS

Sk

111

frequency of components

kPSD Unit V/√ Hz (Noise in electronic circuits)

be denoted by

100

PWR, RMS, and PSD, dB scale

12

0

12

0

Nk

k

Nk

k

fkPSDkPWRPWRRMS

PSD

0

0

f

fS / 2

PWR = RMS2

Δf

Power Spectral Density

Δf = 1 / T The frequency difference

fTNTNff SS 111

According to the Parseval’s theorem

12

0

12

0

2Nk

k

Nk

k

fkPSDkPWRRMSPWR

sound velocity acceleration force voltage

ref 2x10-5 Pa 1x10-6 mm/s 1x10-6 m/s2 1x10-6 N 1 V

Reference values

dB scale

ref

yref log20dB

101

Electrical power

Power is a product of quantities as follows:

voltage x electric current, force x velocity

Let 02,cos,cos kTtItitUtu

The instantaneous power is as

follows

The mean value of the power

per a period T

cos

cos22

cos2cos2

1

00

RMSRMS

TT

IU

IUdtt

T

UIdttp

TP

kU

kI *

cos*

RMSRMSR IUkIkU

kI

kI I kIR

sin*

RMSRMSI IUkIkU

Real part (active power W)

Imaginary part (reactive power VAR)

Let

(apparent power VA)

UIURMSURMS jIkIjUkU exp,exp

RMSRMS IU

kU kI

Im

Re

coscos2

cos2cos tUI

tUItitutp

Complex power

be position vectors at t = 0

Phasors in complex plane

be electrical voltage and current.

102

Mechanical power

F(t) force

v(t)

kF

kv*

kv kvkF *

kF

kv*

kdjkv

kd

Energy dissipation

02,cos,cos kTtVtvtFtF

velocity

Vectors

d(t)

kvkF *

kF kv*

j

kakv

ka

kvkF *

acceleration a(t)

F(t) force F(t) force

...,,, tdtvtatF

...,,, kdkvkakF

vectors, harmonic (sine, cosine) functions of time

Phasors (position vectors in complex plane) at time t = 0 and rotating

at the angular frequency 0 kPhasors

Differentiate between vectors (e.g. force, velocity and displacement defined by both the magnitude

and direction) and phasors (rotating position vectors in the complex plane creating a harmonic

signal and determining the phase of this signal).

displacement

active power

103

Cross-spectrum

12...,,2,1,2

0,002*

2*

NkNkYkXT

kNYXTCROSSk

.12...,,2,1,2

,0,00*

*

NkNkXNkX

kNXNXkPWR

kX

kX *

kXkX *

kX

kY *

kYkX R

*

kY

kYI kYR

kYkX I

*

Power as a real value (equivalent P = R I2)

Power as a complex value (equivalent P = U I*)

Real part

Imaginary part (reactive power)

(active power)

Real part

(active power)

Let

12...,,2,1,0,, NknyFTkYnxFTkX

1...,,2,1,0,, Nnnynx sampling frequency SS Tf 1be two signals.

The Fourier transform of these signals is as follows

TNTNffNfkf SSSk 11, 1

frequency of components

.12...,,2,1,2

,0,02

2

NkNkX

kNXkPWR

or

104

Weighting of a cosine signal with the use of the rectangular

time window

TTw WXWXttjtwtxX2

1d

2

1dexp

0000 2d2

1

TTTw WWAWAX

2Sinc2

2sindexp

2

2

TTT

TTttjW

T

T

T

tAtx cos

2,2,0

2,2,1

TTt

TTttw

000 dexpcos

AttjtAX

2Sinc2Sinc2 00 TTTAXw

Convolution of the Fourier transforms in the frequency domain

twtx

tt 2T 2T

The Fourier transform of the continuous harmonic signal

The Fourier transform of the rectangular time window as the continuous time signal

- 0,4 - 0,2

0 0,2 0,4 0,6 0,8

1

1,2

- 30 - 20 - 10 0 10 20 30

2Sinc T

2T

Sinc function

105

Rectangular (uniform) time window

T

2

0

0

k0

0

,...3,2,1,0k

Continuous frequency

k0

Period T

srad

Hzf0

0

Angular frequency

Frequency

Zeros

Discrete frequency

T

2

Discrete frequency

k0

0

k0

Tf

1

Zeros

Frequency

step (k = 1)

Frequency

step (k = 1) Frequency axis

The rectangular window is suitable only for signals composed from harmonics of the 1/T frequency

?

2Sinc TTWT

relative error of up to

%1002

106

Overview of the basic time window types

0,0

0,5

1,0

1,5

2,0

2,5

0,00 7,81 15,63

Time [ms]

0,0

0,2

0,4

0,6

0,8

1344 1472 1600 1728 1856

Frequency [Hz]

RM

S

-1

0

1

2

3

0,00 7,81 15,63

Time [ms]

0,0

0,2

0,4

0,6

0,8

1344 1472 1600 1728 1856

Frequency [Hz]

RM

S

-2

0

2

4

6

0,00 7,81 15,63

Time [ms]

0,0

0,2

0,4

0,6

0,8

1344 1472 1600 1728 1856

Frequency [Hz]

RM

S

127...,,1,0

,128**2cos1

n

nnw

127...,,1,0

,128**8cos0322,0

128**6cos388,0

128**4cos29,1

128**2cos98,11

n

n

n

n

nnw

Hanning window

Kaiser-Bessel window

Flat Top window Flat Top

127...,,1,0

,128**6cos00305,0

128**4cos244,1

128**2cos29,11

n

n

n

nnw

Time domain Frequency domain

Calibration

procedure

Signals with

unknown

frequency

spectrum

The use of the

time windows

:

107

Hanning time window

Hanning : Sine1 - 50 Hz ; Sine1 - 50,5 Hz

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Sine1 - 50 Hz Sine1 - 50,5 Hz

Hanning : Sine1 - 50 Hz ; Sine1 - 50,5 Hz

0,0001

0,0010

0,0100

0,1000

1,0000

35 40 45 50 55 60 65

Frequency [Hz]

RM

S [U

]

Sine1 - 50 Hz Sine1 - 50,5 Hz

The Hanning window applied to the harmonic signal of the unity amplitude

108

Flat top time window

Flat Top : Sine1 - 50 Hz ; Sine1 - 50,5 Hz

-6

-4

-2

0

2

4

6

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Sine1 - 50 Hz Sine1 - 50,5 Hz

Flat Top : Sine1 - 50 Hz ; Sine1 - 50,5 Hz

0,0001

0,0010

0,0100

0,1000

1,0000

35 40 45 50 55 60 65

Frequency [Hz]

RM

S

Sine1 - 50 Hz Sine1 - 50,5 Hz

Flat top The Flat Top window applied to the harmonic signal of the unity amplitude

109

Time window bandwidth

Autospect - Hanning : Sine 50 Hz ; Sine 50,5 Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

40 45 50 55 60

Frequency [Hz]

PS

D [U

]^2

/Hz

Autospect - Rect : Sine 50 Hz ; Sine 50,5 Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

40 45 50 55 60

Frequency [Hz]

PS

D [U

]^2

/Hz

PWRfkPSDBWfkPSDBWPWRNk

k

Nk

k

12

0

12

0

Sine 50 Hz:

Sine 50.5 Hz:

Hanning

12

0

Nk

k

fkPSD BWPWR

0.5

0.5

0.500

0.499

1

1

Sine 50 Hz:

Sine 50.5 Hz:

12

0

Nk

k

fkPSD BWPWR

0.5

0.5

0.750

0.750

1.5

1.5

Rectangular

Rectangular

Hanning

Amplitude = 1 => PWR = 0.5

Time Window

Rectangular

Hanning …….

Kaiser-Bessel

Flat Top ……..

Bandwidth BW

1

1.5

1.8

3.77

110

Random signals

Flicker Noise 2 : Signal (Alpha 0)

-5

0

5

0,0 0,5 1,0

Time [s]

Flicker Noise : Signal (Alpha 1,99)

-20

0

20

40

60

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Flicker Noise : Signal (Alpha 1)

-6-30369

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

White noise

Pink noise

Flicker noise

(1 over f noise)

111

Averaging the PWR or PSD spectra

0,00

0,01

0,02

0 1000 2000 3000

1

2

1 pwrrms

2

2

2 pwrrms

100

2

100 pwrrms

PWRPWRRMS

12,...,2,1,0

1

1

Nk

kpwrM

kPWRMm

m

mM

12,...,2,1,0

111

Nk

kpwrM

kPWRM

M

kPWR

mM

M

Recursive formula

Mean value

Frequency [Hz]

PWR spectra

Averaging results in

smoothing of spectrum

Mean value

Only the PWR or PSD spectra can be used

as an input data for averaging

The Welch’s method is based on averaging the frequency spectra, which reduces the spectrum

variance.

This method, known as Welch's method, is used for estimating the power of a signal vs. frequency.

112

Overlapping records for averaging

No overlap

0 % overlap

66 % overlap

50 % overlap

A record for computing FFT

Gaps, these signal sections do not affect the result of analysis

pwr spectrum

pwr spectrum

0,0

1,0

2,0

3,0

4,0

0 32 64 96 128 160 192 224 256 n

w n

0 1/2 2/3

113

Optimal overlap for the Hanning window

0 % overlap

2/3 or 66 % overlap

1/2 or 50 % overlap

Resulting effect of weighting and averaging

Weighting functions Overlaps

The signal sections, where the weighting function is close

to zero, do not almost influence the result of the frequency

analysis

Overlapping the records of a

signal

From the viewpoint of the weighting function uniformity, the optimal value for overlap ratio:

n /(n+1) for n = 2, 3, …

Effect of the number of averages on the random signal

frequency spectrum

02

50

50

0

75

0

10

00

12

50

15

00

17

50

20

00

22

50

25

00

27

50

30

00

125102050100

0,000,010,020,030,040,040,050,060,070,080,090,100,110,110,120,130,14

Frequency [Hz]

Averages

-4

-2

0

2

4

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

Time [s]

Averages Time interval [s]

1 0.128

2 0.171

5 0.299

10 0.512

20 0.939

50 2.219

100 4.325

The length of the time record for

spectrum analysis

White noise signal

114

115

Estimation of the spectrum variance

The length of the record

for FFT ………………. T

TB

1

A

rBT2

1

Mr

2

1

MTTA

Bandwidth …………..

Total length of record

Autospectrum : White Noise

0.0

0.1

0.2

0.3

0.4

0.5

0 4 8 12 16 20 24

Frequency [Hz]

RM

S

1

10

100

1000

Number

of averages

Relative spectrum

variance ……………..

Number of averages … M

The effect of the number of averages

on the estimation error of the resulting spectrum

determines the formula

Multispectra - Waterfall plot

Frequency

Amplitude

RPM or Time

Gearbox noise

116

Multispectra - Contour plot

Frequency

RPM or Time

Gearbox noise

Color scale

Autospectrum : Gearbox Noise

Frequency

RMS dB(A)/

ref. 2E-5 [Pa]

117

118

Campbell diagram

If this was a 4 cylinder 4

stroke engine 2E would

equate to the firing frequency.

E is an integer multiple

of the engine speed.

1000 2000 3000 4000 5000 6000

0

200

400

1000

600

800

1E

2E

3E

4E

5E

6E

7E

8E

9E 10E

35 dB 60 dB

Engine speed

RPM

Fre

qu

ency

Hz

119

Constant Percentage Band Spectrum

Lecture 9

CPB : pave 25 km/h - Axis Z ISRI 2 : PAI25 : Seat pan ; Cab

floor

80

90

100

110

120

130

140

0,80

1,25

2,00

3,15

5,00

8,00

12,5

0

20,0

0

31,5

0

50,0

0

80,0

0

Frequency [Hz]

RM

S d

B/r

ef

1E

-6

[m/s

^2

]

SEAT PAN [m/s^2]

CAB FLOOR [m/s^2]

120

Chromatic scale for piano

A4: 440 Hz = 2 x 220 Hz A5: 880 Hz = 2 x 440 Hz

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

Note that A5 has a frequency of 880 Hz. The A5

key is thus one octave higher than A4 since it has

twice the frequency.

A3: 220 Hz

121

1/n-octave bands of the PCB analyzers

dB 0

Frequency

band

Filter gain

HLC fff … central frequency

12 kfkf Cn

C

n = 1 … 1/1-octave filter

n = 3 … 1/3-octave filter

n = 6, 12, …

LH fff … bandwidth

Cf

f … relative bandwidth

The central frequency forms a geometric

sequence

k = 1, 2, 3, …

Filter bank

log f fL fC fH

Creating a bank of filters

1. A group of the IIR or FIR filters

2. FFT algorithm

N 512 1024 2048 4096 8192

1/1 6 7 8 9 10

1/3 18 21 24 27 30

Number of the 1/n bands for the N-point FFT

The FFT analyzers are based on the use of the FFT algorithm. Signal processing is organized

by repeating computations after aquiring a record of samples. The PCB analyzers are working

on the principle of a bank of the band pass filters.

The filter bandwidth

122

Central frequencies of the octave and 1/3-octave pass band

0.1 .125 0.16 0.2 0.25 .315 0.4 0.5 0.63 0.8

1 1.25 1.6 2 2.5 3.15 4 5 6.3 8

10 12.5 16 20 25 31.5 40 50 63 80

100 125 160 200 250 315 400 500 630 800

1k 1k25 1k6 2k 2k5 3k15 4k 5k 6k3 8k

10k 12k5 16k 20k

- - - - - - - .125 0.25 0.5

1 2 4 8 16 31.5 63 125 250 500

1k 2k 4k 8k 16k - - - - -

1/1-octave filters

1/3-octave filters

110 103 kfkf CC

10131 102 Approximation …

110 101 kfkf CC

The central frequencies of the 1/3-octave pass band filters forms a geometric sequence with

a quotient , which is equal to the 1/3rd power of two (1.25992). To obtain pretty numbers

in the sequence the quotient is slightly changed to the value of the 1/10th power of ten (1.25892).

123

CPB analyzer

Real Time Analyzer

Root

Low pass Filter

Squarer Band pass filter

x

x

x 2

x 2 f c 1

Input

signal

f c k CPB

spectrum

Square Root

...

...(RMS)

RMS

Mean Square

CPB spectrum example - Ride comfort

CPB : pave 25 km/h - osa Z : PA025 : Seat ; PAI25 : Seat ; PAM25 : Seat

80

90

100

110

120

130

140

0,80

1,00

1,25

1,60

2,00

2,50

3,15

4,00

5,00

6,30

8,00

10,0

0

12,5

0

16,0

0

20,0

0

25,0

0

31,5

0

40,0

0

50,0

0

63,0

0

80,0

0

Frequency [Hz]

RM

S d

B/r

ef 1

E-6

[m

/s^2

]

124

Spectrum examples

CPB : SF128 : Vibrace

0,000

0,010

0,020

0,030

0,040

0,050

125 250 500 1k 2k 4k 8k

Frequency [Hz]

RM

S

Autospectrum : SF128 : Vibrace

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0 2000 4000 6000 8000 10000

Frequency [Hz]

RM

S

CPB : SF128 : Vibrace

0,000

0,010

0,020

0,030

0,040

10

01

25

16

02

00

25

03

15

40

05

00

63

08

00

1k

1k2

1k6

2k

2k5

3k1

4k

5k

6k3

8k

10

k

Frequency [Hz]

RM

S

Autospectrum 1 : SF128 : Resampling (Vibrace)

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0 50 100 150 200 250 300 350 400

Order [-]

RM

S

Time History : SF128 : Vibrace

-0,3

0,0

0,3

0 1 2 3 4 5 6 7 8 9Time [s]

1/1-octave

1/3-octave

FFT - Hz

FFT - order

Logarithmic scale Linear scale

125

126

Frequency Weighting

See [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/soucon.html#soucon]

127

Annotated equal loudness curves

Equal loudness in phones The curves represent equal loudness as perceived by the average human ear

The ear is less sensitive to low frequencies, and this discrimination against lows becomes steeper for softer sounds.

The maximum sensitivity region for human hearing is around 3-4 kHz and is associated with the resonance of the auditory canal.

Sound intensity in decibels does not directly reflect the changes in the ear’s sensitivity with frequency and with sound level

Curve for the average threshold of hearing

100

10

20

30

40

50

60

70

80

90

100

110

1000 10000

120

100

80

60

40

Frequency [Hz]

Inte

nsi

ty i

n d

ecib

els]

See [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/eqloud.html#c1]

120

REF

RMS

p

pdB log20

Pa00002.0REFp

128

Filter contours

The A-contour filters out significantly more bass than the others, and is designed to approximate the ear at around the 40 phon level. It is very useful for eliminating inaudible low frequencies.

The intermediate B-contour approximates the ear for medium loud sounds. It is rarely use. The C-contour does not filter out as much of the lows and highs as the other contours. It

approximates the ear at very high sound levels and has been used for traffic noise surveys in noisy areas.

4

2

13

2

12

2

11

2

1

22

1

afafafaf

fKfH A

See [http://jenshee.dk/]

006315772.0

,5560957.0,06316174.0

,8404.148,3742.187

4

32

1

a

aa

aK

10001 ff

fSfHfS AA

S(f) is the unweighted RMS spectrum

SA(f) is the A-weighted RMS spectrum

129

Order Analysis

Lecture 11

Encoders

Pinion

2

1

r2

Wheel

130

Rotational speed measurements

Complete revolution

Tacho impulses

3600

1 impulse per revolution

1st revolution 2nd revolution 3rd revolution

Time interval for vibration

and noise analysis Tacho probe output

Contact-free detection of

reflective objects (rotary or

reciprocatory machine parts)

Photoelectric Tachometer

Probe

500 impulses per

revolution

1024 impulses per

revolution

ERN 460-500 type ERN 460-1024 type

Heidenhain products

131

Spectral analysis of non-uniformly rotational machines

780

790

800

810

820

0 50 100 150 200

Index

RPM

RPM variation at idle speed

Car engines Gearboxes

Many machines operates in the cyclic fashion, for example an IC engine with the constant number

of strokes per revolution or a gear train with the constant number of mesh cycles per revolution.

RPM slow variation

results in ………… uncertain number of strokes per second

uncertain number of mesh cycles per second

Non-uniform rotation

results in smearing

spectrum components as

a result of the frequency

modulation effects.

Sampling signals at the constant angle rotation increments can prevent spectrum smearing.

132

Time vs. angular domain

ADC

Clock

Time domain sampling

ADC

Encoder

Angular domain sampling

Analog

signal

Digital

signal

Analog

signal

Digital

signal

Impulses are generated at

the constant time frequency

Impulses are generated after the constant

rotation angle of the shaft rotation

Analog to digital conversions (ADC) are started by an edge of the synchronization signal

Time

Time interval TS

Angle

Rotation by the 360/N angle in degrees

Samples

Source of the

sampling frequency

(The encoder produces N impulses per a complete revolution)

It is impossible to detect angular vibration during rotations.

133

Digital order tracking – computed order tracking

Sampling at the constant

frequency in Hz

Resampling according

to rotational frequency

Time or frequency

analysis

Upsampling by 2 or 4

Low pass filtration

Interpolation 0 3600

3600

0

Nominal rotation angle

Actual rotation angle

Constant angular velocity

Angular vibration

Equidistantly spaced samples

Algorithm of resampling

Resampling to the unified number of samples is limited either by one complete revolution or by

arbitrary complete revolutions.

• Externaly controled sampling frequency by the IRC sensor

• Frequency multiplier (BK 5050) for multiplying the tacho signal frequency

• Digital order analysis based on resampling of the signal

Techniques for order analysis

Resampling signals for order analysis

Time History : Sine

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Continuous time signal

4.333 Hz sampling frequency

4 times upsampled (17.333 Hz)

3-step resampling:

Upsampling by the integer factor (2 or 4 times)

Low pass filtration to reduce the frequency range

to the value before upsampling (2 or 4 times)

Interpolation in between adjacent samples

(resampling by a fractional factor)

Interpolation

Original samples

New samples

The use of the Lagrange or Newton

interpolation polynomial

Methods:

An example demonstrating algorithm of resampling

134

Upsampling by the integer factor - Interpolation

Time History :Sine1

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,00 0,25 0,50 0,75 1,00

Time [s]

Time History : upsample(x,4)

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,00 0,25 0,50 0,75 1,00

Time [s]

Autospectrum : Sine1 - 1

0,0

0,2

0,4

0,6

0,8

0 1 2 3 4

Frequency [Hz]

RM

S

Autospectrum of upsample(x,4)

0,0

0,1

0,2

0,3

0,4

0 4 8 12 16

Frequency [Hz]

RM

S

Interpolated signal

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,00 0,25 0,50 0,75 1,00

Time [s]

Adding 3 zeros in

between samples

Upsampling by 4 Low pass filtration,

4 Hz cut-off frequency

Autospectrum : Interpolated signal

-360

-300

-240

-180

-120

-60

0

0 2 4 6 8 10 12

Frequency [Hz]

RM

S d

B/r

ef 1 Interpolation error Frequency range

4 Hz multiplied by 4 4 Hz

135

136

Effect of downsampling by an integer factor on spectrum

Let the frequency sampling of a signal be equal to fS

fS / 2 0 Δ f

Frequency spectrum before

downsampling

fS / 2 0 Δ f FS / 2

(M = 2)

Frequency spectrum after decimation

0 Δ f FS / 2

Downsampling reduce the sample count and the maximum signal

frequency and safe space between the spectrum components

Frequency spectrum after filtration

FIR low pass filter can be employed

to evaluate only output samples

Now it is possible to reject M-1 samples after each Mth one in the initial signal (samples decimation)

Let the sampling frequency fS be reduced to FS = fS / M. To prevent aliasing after downsampling the

signal frequency range has to be reduced by the antiasiasing low pass filter to the frequency FS /2

137

Order spectra

Spectral

analysis

Order

analysis

Time

domain

Time

[s]

Revolution

[-]

Frequency

domain

Frequency

[Hz]

Order

[-]

Frequency [Hz] 0 T

1

T

2

T

3

T

4

T

5

Order [-] 0 1 2 3 4 5

Reciprocal value of the time

interval length for FFT

Multiples of the tacho impulse frequency,

1 impulse per revolution

Order [-] 0

Multiples of the tacho impulse frequency,

1 impulse per n revolutions

n

21

n

1

n

3

n

n 1

Fractional orders

DC

DC

DC

The frequency in orders is the same

as the multiples or harmonics of the

machine basic frequency.

The basic frequency is the rotation

frequency of some machine part

(gearbox input shaft, countershaft,

or output shaft).

Revolution and order are

dimensionless quantities.

DC = Direct Current

138

Resampling signal according to the tacho frequency

Time History : Sine 150,5 Hz

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Signal

Tacho

Autospectrum : Sine 150,5 Hz

0,0

0,1

0,2

0,3

0,4

0,5

120 140 160 180

Frequency [Hz]

RM

S

Resampled Signal : Sine 150,5 Hz

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

0,0 0,2 0,4 0,6 0,8 1,0

Nominal revolution [-]

Resampled

Tacho

Autospectrum : Sine 150,5 Hz

0,0

0,2

0,4

0,6

0,8

80 100 120

Order [-]R

MS

The tacho signal frequency is 1.505 Hz

Analyzed signal frequency is 150.5 Hz = 100 x 1.505 Hz, amplitude 1 (RMS 0.707)

139

A simple example to resampling a non-stationary signal

Input signal

20 samples 15 samples 10 samples

Resampled signal

10 samples 10 samples 10 samples

Resampling to the unified number of sample per revolution during run-up and coast down enables

to perform order analysis of signals or to prepare a strip plot in the form revolution-by-revolution

records containing the same number of samples.

Constant sampling

frequency

Sampling frequency

is proportional to

rotational frequency

140

Frequency and order multispectra

0 RPM

Noise caused by resonances,

the frequency of excited components

(paralel lines) is independent on RPM

Excited noise, the frequency of excited components

(radial lines) is proportional to RPM

Gearbox noise

during run-up

Excited noise at frequency,

which is proportional to RPM

Resonance noise,

constant frequency in Hz

Frequency in Hz Frequency in order

141

Order spectra in machine diagnostics

COUPLE UNBALANCE

ANGULAR MISALIGNMENT

PARALLEL MISALIGNMENT

ROLLING BEARINGS

GEARBOXES

0 1 2

Acc

ord

0 1 3

Acc

ord 2

3

4

0 1 3

Acc

ord 2 4

0 1 2

Acc

ord 3

CAR

ENGINE

BPFO BPFI BPF - Ball Pass

Frequency

I – inner race

O – outer race

4

0

Acc

ord

GMF - Gear Mesh Frequency

Number

of Teeth

GMF

1

0 2 6

Acc

ord 4 8

4 strokes

4 cylinders

Rotational frequency 1 ord, acceleration signals

Synchronized averaging as a tool for filtration signals

-0,3

0,0

0,3

0,00 0,20 0,40 0,60 0,80 1,00

Nominal Revolution [-]

[m/s

2]

-0,3

0,0

0,3

0,00 0,20 0,40 0,60 0,80 1,00

Nominal Revolution [-]

[m/s

2]

-0,3

0,0

0,3

0,00 0,20 0,40 0,60 0,80 1,00

Nominal Revolution [-]

[m/s

2]

-0,1

0,0

0,1

0,00 0,20 0,40 0,60 0,80 1,00

Nominal Revolution [-]

[m/s

2]

Time History : SF128 : Acceleration

-0,3

0,0

0,3

0 2 4 6 8

Time [s]

[m/s

2]

Mean value

0,00

0,01

0,02

0,03

0 27 54 81 108 135 162 189

Order [-]

RM

S [

m/s

2]

Tachometer : SF128 : Pulse positions

1490

14921494

1496

0 2 4 6 8 10Time [s]

RP

M

The same number

of samples

Resampled records

142

143

Synchronized averaging as a comb filter

1

0

1 M

m

mTtxM

ty

1

0

1 M

m

mTtM

tg

sT

sMT

MmsT

MdtsttgsG

M

m

exp1

exp11exp

1exp

1

00

2sin

2sin

2

1exp

1

T

MT

TMj

MjG

2sin

2sin

1

T

MT

MjG

Mathematical model of averaging

(moving average)

Impulse response (x(t) = δ(t)) ….

Laplace transfer function ……….

Frequency transfer function …….

Magnitude of the frequency

transfer function ………..……….

144

Frequency response function of the synchronized

averaging

0

0

0sin

sin1

f

f

f

fM

Mf

fjG

0sin0

f

fM Z

,...3,2,1,00

k

f

fGk

f

fG

0,0

0,2

0,4

0,6

0,8

1,0

0 1 2 3 f / f o

|G( f / f o )|

Periodicity

Zeros ……

Magnitude of the comb filter frequency response

Tf

10

Dimensionless frequency

Synchronized averaging acts as a comb filter for spectrum components, which are an integer

multiple of the synchronized frequency. The component corresponding to the non-integer

(fractional) multiple are considerably attenuated.

kf

fM Z

0

,...2,1,0,0

kM

k

f

fZ

There is M zeros of the

frequency response in

between two adjacent integer

multiple of the dimensionless

frequency f / f0

145

A comparison of various approaches

to spectral analysis

Examples

146

Truck gearbox vibration spectra

Autospectrum : SF128 : Vibrace

0,0140,014

0,028

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0 1000 2000 3000Frequency [Hz]

RM

S [

m/s

2]

Autospectrum : SF128 : Resampling (Vibrace)

0,0130,013

0,027

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0 27 54 81Order [-]

RM

S [

m/s

2]

46 teeth

759 rpm

27 teeth

1293 RPM

GMF = 582 Hz

Tachometer

Notice differences in sidebands

of the GMF components

GMF = 27 ord

Frequency and order spectrum of a gearbox vibration

A simple gear train

Frequency analysis of the acceleration signal

Autospectrum 2 : FILE017 : Acc

0,000

0,005

0,010

0,015

0,020

0,025

0 2500 5000 7500 10000 12500

Frequency [Hz]

RM

S m

/s2

CPB : FILE017 : Acc

0,00

0,01

0,02

0,03

0,04

0,05

10

01

25

16

02

00

25

03

15

40

05

00

63

08

00

1k

1k2

1k6

2k

2k5

3k1

4k

5k

6k3

8k

10

k

Frequency [Hz]

RM

S m

/s2

Frequency spectra

FFT

CPB

electric motor

gearbox

accelerometer

belt

spindle

25T

tacho probe

Input

shaft

Intermediate

shaft

An example showing FFT and CPB signal

processing methods employed to analyze

rattling noise of a gear train

147

148

149

Harmonic Signal Modulation

Lecture 12

Time History : Generator : Sine1/AM - 1

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

[U]

150

Modulation of harmonic signals

• Carrying component …………………..

(harmonic signal without modulation)

Amplitude A

Phase

Initial phase

• Amplitude modulation signal ………...

• Phase modulation signal ………….…...

• Mixed modulation (amplitude and phase)

Phase

Modulation signals

Amplitude

Modulated signal Carrying component phase

PMPMPMAMAMAM tnttAtx coscoscos1 00

000 cos tAtx

AMAMAMA ttx cos

PMPMPMP ttx cos

00 tt

0

Nomenclature

See [Tůma, 1997]

151

Amplitude modulation of harmonic signals

Time History : Generator : Sine1/AM - 1

-2-1012

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Modulation signal

ttAtx AMAMAM 0coscos1

Modulated signal Carrying component

AM

AMAM f 2

00 2 f

Autospectrum : Generator : Sine1/AM - 1

0,00,20,4

0,60,8

0 100 200 300 400

Frequency [Hz]

RM

S [U

]

modulation frequency

modulation index

carrying frequency

Sideband components f0+fAM, f0-fAM Carrying component

152

Spectrum of the amplitude-modulated signal

AMAMAMAMAM

AMAMAMP

AMAMAM

ttAtA

ttAtA

ttAtx

000

0

0

coscos2cos

coscoscos

coscos1

AMAMAMAMAMAM tAtAtAtx 000 cos2cos2sin

Upper sideband component,

frequency f0 + fAM, amplitude AβAM /2

Decomposition of the modulated signal on the carrying component and its sidebands

Lower sideband component,

frequency f0 - fAM, amplitude AβAM /2

The carrying component,

frequency f0, amplitude A

Phasor model

+f0

-f0

AβAM /4

Re

Im

-f

+f

A /2

f0 + fAM f0 - fAM

Rotational

frequency

153

Phase modulation of harmonic signals

Time History : Generator : Sine1/PM

-2-1012

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

PMPMPM ttAtx coscos 0

PMPM f 2

00 2 f

Autospectrum : Generator : Sine1/PM

0,0

0,1

0,2

0,3

0 100 200 300 400

Frequency [Hz]

RM

S

modulation frequency

modulation index

carrying frequency

Modulation signal Modulated signal Carrying component

AM

The family of the sideband components f0+fPM, …, f0 –

fPM, …

Carrying component

154

Spectrum of the phase-modulated signal

tptptx

cosexpexp2

1cosexp

2

100 PMPM jtjtjtp

itjitjjJtjJtp i

i

PMiPM 00

1

00 expexpexp2

1

0

2

!!

1

2k

kik

PMPMi

ikkJ

PMi

i

PMi JJ 1

tsttx PM 0cos ,cos PMPM ts PMPM t Let be a phase-modulated signal, where

where Ji(β) is the Bessel function of the first kind, for integer orders i = 0, 1, 2, …

+f0

-f0

Re

Im

-f

+f f0 + fPM

f0 - fPM

Rotational

frequency

Phasor model

-0,5

0

0,5

1

0 2 4 6 8 10 βPM

J 0 (βPM)

J1 (βPM) J2(.) J3(.) J4(.) J5(.)

(βPM) Ji

20 PMJ

21 PMJ

Autospectrum : Sine1/PM - Beta = 5

0,0

0,1

0,2

0,3

0 50 100 150 200

Frequency [Hz]

RM

S

Effect of the phase modulation index on the sidebands

Autospectrum : Sine1/PM - Beta = 0.1

0,0

0,2

0,4

0,6

0,8

0 50 100 150 200

Frequency [Hz]

RM

S U

βPM = 5

The carrying component 100 Hz, amplitude 1, modulation signal frequency 5 Hz

Autospectrum : Sine1/PM - Beta = 1

0,0

0,2

0,4

0,6

0 50 100 150 200

Frequency [Hz]

RM

S

Autospectrum : Sine1/PM - Beta = 0.5

0,0

0,2

0,4

0,6

0,8

0 50 100 150 200

Frequency [Hz]

RM

S U

βPM = 1

βPM = 0.5

βPM = 0.1

155

156

Amplitude and phase modulation I

Time History : Generator : Sine1/PM+AM

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Modulation signals

MMPMMMAM tttAtx sinsinsin1 0

Modulated signal

PMAM ,

MM f 2

00 2 fAutospectrum : Generator : Sine1/PM+AM

0,0

0,1

0,2

0,3

0 100 200 300 400

Frequency [Hz]

RM

S

modulation frequencies

modulation

indexes

carrying frequency

Carrying component

Symmetric sidebands

Modulation signals are in phase

157

Amplitude and phase modulation II

Time History : Generator : Sine1/PM+AM

-2

-1

0

1

2

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Modulation signals

2sinsinsin1 0 MMPMMMPM tttAtx

Modulated signal

PMAM ,

00 2 f

00 2 fAutospectrum : Generator : Sine1/PM+AM

0,0

0,2

0,4

0,6

0 100 200 300 400

Frequency [Hz]

RM

S

modulation frequency

modulation

indexes

carrying frequency

Carrying component

Modulation signals are out of phase

Non-symmetric sidebands

158

159

Analytic Signals and Hilbert Transform

Lecture 13

160

Analytic signals

fP 2

Real harmonic signal Complex analytic signal

The analytic signal is a complex signal with an imaginary part, which is the Hilbert transform

of the signal real part. The decomposition of a real signal into harmonic components results

in the sum of harmonic functions. Each of this function can be decomposed into the pair

of the phasors, which are rotating in the opposite direction. The analytic signal creates

the phasors rotating in the positive direction.

The analytic signal is a tool for amplitude and phase demodulation of the modulated

harmonic signals.

To obtain the analytical signal the phasor XN has to be removed and the phasor XP has to be

multiplied by 2.

161

Analytic signals in the 3D-space

fP 2

Helix

The position vector is rotating in the complex plane. If the 2D space is extended to 3D space

with the third axis as a time axis then the vector end point moves on the helix trajectory.

162

Analytic signal

Analytic signals and the Hilbert transform

Hilbert transform

PXZ 2

PP XjY

NN XjY NNN XXjjYj

NP XXX

PXNX

2

2

2

= + j Time signal

• Fast Fourier Transform (FFT)

• Digital filters

Evaluation of the Hilbert transform

using …

P

NPNP

NPNP

X

XXXX

YYjXXZ

2

NP XXX

NP

NP

NP

XXj

XjXj

YYY

PPP XXjjYj

The complex position vectors as phasors, which are corresponding to a harmonic signal.

The phasors Y associated to the Hilbert transform of a pair of phasors X are obtained by rotation

these phasors by the angle of +/- π/2 radians.

The complex plane

163

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

-15 -10 -5 0 5 10 15

Definition of the Hilbert transform as the Cauchy

principal value integral

The Hilbert Transform can be defined as the principal value integral

The Cauchy principal value (P.V.) expands the class of certain improper integrals for which the

finite integral exists as for example the integral

b

a

xxfxxf ddlim0

where

b

a

xxfxxfba d,d,,

d..

1

t

xVPty

Let x(t) be an impulse Dirac function δ(t), then the Hilbert transformer impulse response is as

follows

ttVPtg

1d..

1

tg

t

164

The Hilbert transform as a transfer function

To turn the non-decaying function to the decaying function, let the frequency transfer function

be extended

0,

0,

j

j

jX

jYjHHT

22

00

0

0

11

2d

2

dd2

1d

2

1

t

te

jte

jt

jee

j

ejejejHtg

jtjttjtj

tjtjtj

The impulse response is an inverse Fourier transform of the frequency transfer function

See [http://w3.msi.vxu.se/exarb/mj_ex.pdf]

jHjHje

jejH HT

0lim

0,

0,

tt

ttgtgHT

1limlim

2200

The Hilbert transform of a time function to another time function can be described by the standard

transfer function in the frequency domain. Let X(jω) and Y(jω) be the Fourier transform of the

original continuous time signal x(t) into the same signal y(t), respectively.

165

Analytic signals and the Hilbert transform of some signals

Real part Imaginary part Envelope Phase

tx ty tE t

tA sin tA cos

tA cos tA sin t

2t

A

A

Signal Hilbert

Transform

tx ty

tsin tcos

tcos tsin

11 2 t 12 tt

ttsin ttcos1

t t1

The envelope and phase of the harmonic signals Hiltert transform

166

The use of FFT for computing the Hilbert transform

2

2

jYIFFTky jYjX kxFFTjX

NN XjY PP XjY

PP

PPP

XjX

XjXjY

ReIm

ImRe

NN

NNN

XjX

XjXjY

ReIm

ImRe

i. The Fast Fourier Transform (FFT) of the real input time record to obtain phasors rotating in

positive and negative directions

ii. Rotation the phasor XN in the positive direction by the angle of + π/2 radians and the phasor XP

in the negative direction by the angle of - π/2 radians (exchanging the real and imaginary parts)

iii. The Inverse Fast Fourier Transform (FFT) of the rotated phasors YP and YN to obtain the Hilbert

transform of the input record.

The algorithm for computing the Hilbert transform is broken down into three steps

Diagram showing how to transform the phasors XP and XN to the phasors YP and YN

i) ii) iii)

Exchanging the real and imag parts Exchanging the real and imag parts

167

The use of digital filters for computing the Hilbert

transform

Frequency response function

0,

0,

S

S

Tj

TjTj

HTTj

Tj

eX

eYeG

S

S

S

12,2

2,0

d2

1

knn

kn

TeeGng S

nTjTj

HTHTSS

Impulse response

Let X(e jωTS) and Y(e jωTS) be the Fourier transform of the original sample sequence xt into the

sample sequence yt, respectively.

-1,0

-0,5

0,0

0,5

1,0

-50 -40 -30 -20 -10 0 10 20 30 40 50

Index n

gH

T

Impulse Response of the Ideal Hilbert transformer

The nonzero response for the negative index n means that the impulse response corresponds to

a non-causal system. Response precedes the change at the system input.

168

Hilbert transformer filters

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

-20 -16 -12 -8 -4 0 4 8 12 16 20

Index n

gH

T

Kaiser Ideal

Frequency response function of the Hilbert

transformer

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0

Normalised Frequency [-]

Ma

gn

itu

de

Kaiser Ideal

The 160-order FIR filter with the finite

impulse response n = -80,…,+80

Hilbert Transformer

The impulse response of FIR filters is the same as these filters non-zero coefficients. If the infinite

impulse response is shorten to a finite number of non-zero samples then the this response will

corresponds to a FIR filter. Due to the linearity of the filter phase the symmetric or anti-symmetric

coefficients are preferred. As in the case of FIR filter the impulse response has to be delayed in

such a way that the impulse response of the non-causal system is changed to the response of the

causal system. The filter is called as a Hilbert transformer or a 90-degree phase shifter.

The digital filter acts as a Hilbert transformer only for a frequency band in which the magnitude of

the frequency response function is equal to unit. The impulse response which is corrected with the

use of the Kaiser window smooths the frequency response function.

Windowing Windowing

169

Amplitude and phase

demodulation

P w

w

170

Analytic signals and amplitude modulation

ttmAtx M 0coscos1

Sideband

components

Carrying

component

Analytic signal Let x(t) be a real amplitude modulated harmonic

signal described by envelope-and-phase form

Modulation signal

-2

-1

0

1

2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1Time [s]

ω0+ωM

ω0-ωM

ω0

Analytic signals and phase modulation

Let x(t) be a real phase modulated harmonic

signal described by envelope-and-phase form

0

-2

-1

0

1

2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Time [s]

Phase Modulation signal

Sideband

components

Carrying

component

Analytic signal

MtAtx 0cos

Mt 0

171

172

Demodulation of the modulated harmonic signal

Firstly the carrier component and its adjacent sidebands have to be filtered using the band pass

filter. The output signal is designated by x(t)

Secondly the Hilbert transform y(t) of the x(t) signal has to be evaluated using either the FFT

transform or the Hilbert transformer to create the analytic signal

The amplitude modulation signal, referred to as envelope, is as follows

tyjtxtz

22tytxtztA

The principal value of the phase modulation signal is as follows

txtytzt arctanArgP.V.

The phase in radians can be computed by the previous formula while taking into the count the value

sign of x(t) and y(t). The result will be in the wrapped form which is limiting the angle to the

interval tP.V.

To finish the phase demodulation process the wrapped phase has to be unwrapped into

tntztz 2Argarg

where n(t) is a sequence of integer numbers, which depends on time t, for that arg(t) is without

discontinuities larger than a permissible value.

173

The Shannon–Nyquist theorem for sampling of the phase

S

nnf

ftft

221

2

22

2

12

SS

Sf

f

f

fff

Sf

f2

Let the phase difference during the sampling interval be written in the form

The Shannon – Nyquist theorem requires

It can be concluded that the phase change during the sampling interval has to be less then π radians

It is assumed sampling a continuous harmonic signal

tfttx 2coscos

The phase of the mentioned harmonic signal is as follows

tftt 2

This phase change property is basic for unwraping phase signal.

174

samplff2

Unwrapping phase and removing the linear trend

Removing discontinuities

2

2,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 0,2 0,4 0,6 0,8 1

Nominal Revolution

rad

-4

0

4

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Nominal Revolution

rad

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1

Nominal Revolution

rad

Algorithm of the phase unwrapping is based on the phase sampling theorem

The phase demodulation results in in the following signal, which is of the sawtooth wave form.

Unwrapping algorithm

175

An alternative procedure for computing instantaneous

frequency

tytx

dt

tdytxty

dt

tdx

dt

tx

tyd

dt

tdt

22

arctan

tx

tyt arctan

tytxte 22

Phase ……………....

Angular frequency …

Envelope ..…………

dtt

0

Phase ………………

It is not always necessary to calculate the unwrapped phase. To calculate the instantaneous

frequency of the modulated harmonic signal it is possible to use the following formulas

Generator : SweptSine - 1

-2-10

12

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Frequency : SweptSine - 1

0

10

20

30

40

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time : Generator 1 : SweptSine - 1

0

10

20

30

40

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Let the frequency of the swept sine signal be running-up from 10 to 30 Hz

Rectangular window Hanning window

176

Envelope analysis

Time History : Sine / AM

-2

-1

0

1

2

0,0 0,5 1,0 Time [s]

Real

Envelope

Decaying vibration

-10

-5

0

5

10

0,0 0,5

Time [s]

Real

Envelope

Decaying vibration

-10

-5

0

5

10

0,0 0,5

Time [s]

Real

Envelope

Decaying vibration

0,0 0,2 0,4 0,6 0,8 1,0 1,2

0 20 40

Frequency [Hz]

RM

S

Original

Filtered

The amplitude demodulation is referred to as envelope analysis. The examples shows

computing the envelope for a broadband signal or signal zoomed around a resonance frequency

with the use of the bandpass filter.

The envelope is computed for narrow

band part of frequency spectrum The envelope is computed for the full

frequency spectrum

177

Phase demodulation

Autospectrum : Sine 20 Hz /PM 2 Hz

0,

0

0,

1

0,

2

0,

3

0 10 20 30 40 50

Frequency

[Hz]

RM

S

Time History : Sine 20 Hz / PM 2 Hz

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

Time : Phase : Sine 20 Hz / PM 2 Hz

-4

-2

0

2

4

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

[rad

]

Time : Unwrapped Phase : Sine 20Hz / PM 2Hz

0

50

100

150

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

[rad

]

Time : Detrended Phase : Sine 20 Hz / PM 2 Hz

- 5 - 3 - 1

1 3 5

0,0 0,2 0,4 0,6 0,8 1 ,0

Time [s]

[rad

]

Wrapped phase

(principal values

of phase])

Unwrapped phase

Detrend phase

Frequency spectrum of the

modulated signal

Phase modulated signal

Conversion of the frequency modulated impulse signal

to the harmonic signal

Time : Expanded Time(Encoder1)

-2

0

2

4

6

0,000 0,001 0,002 0,003

Time [s]

Timer : Time: Real (Expanded Time(Encoder1))

-4-2024

0,000 0,001 0,002 0,003Time [s]

The original impulse signal

The resulting harmonic signal

Band pass filtration

Autospectrum

-60

-40

-20

0

20

0 10000 20000

Frequency [Hz]

dB

/re

f 1

V

Autospectrum

40

90

140

0 10000 20000

Frequency [Hz]

dB

/re

f 1

E-6

Frequency responsem

-60

-40

-20

0

20

0 10000 20000

Frequency [Hz]

dB Band pass filter

178

179

Correlation Function

Lecture 14

Pink Noise

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

-1,00 -0,50 0,00 0,50

Lag [s]

U^

2

180

Correlation function and power spectral density

*2 11XX

TX

TSxx

T

ttjtxtxX0

dexpFT

T

xx ttxtxT

R0

d1

The Power Spectral Density of a signal x(t) over a time interval T is as follows

where X(ω) is the Fourier transform of the signal x(t)

The autocorrelation function of an stationary signal x(t) at time lag (delay) τ, which is the length of

the signal, over which each correlation is calculated, is defined as an expectance (mean value).

TIt is assumed that the length of the time interval T grows without bound

txtxERxx

For processes that are also ergodic, the expectation can be replaced by the limit of a time

average. The autocorrelation of an ergodic process is defined as

The normalised autocorrelation function takes the form

0xxxxxx RR

See [Orfanidis]

181

Wiener–Khinchin theorem

*2 11XX

TX

TSxx

T

ttjtxtxX0

dexpFT

T

T

T

T TTT

xx

jdttxtxT

ttttjtxtxT

ttjtxttjtxT

S

dexp1

ddexp1

dexpdexp1

0

0

2

0

12112

0

222

0

111

dexp2

1

dexp

jSR

jRS

xxxx

xxxx

is the Fourier transform of the corresponding autocorrelation function.

After substitution

Autocorrelation function

xxxx SS

ffSSttxT

R xxxx

T

TT

xx dd2

1d

2

10

2

lim

The power spectral density of a wide-sense-stationary random process

The Wiener–Khinchin theorem relates the autocorrelation function to the power spectral density

via the Fourier transform

tttt 121 ,

Cross-correlation function

dexp jSR xyxy

dexp jRS xyxy

yxyxyxxyxy SjRjRjRS 111 dexpdexpdexp

0xyS

For stationary random signals x(t), y(t), the cross-correlation function is defined by

tytxERxy

T

xy ttytxT

R0

d1

For uncorrelated random signals

The Fourier direct and inverse transform of the cross-correlation function is as follows

For stationary random signals x(t), y(t) , which are also ergodic, the definition turns to

It can be proved that

0xyR

182

183

Properties of the autocorrelation function

yyxxyxxyyyxx

TTTT

T

xx

RRRRRR

ttxtyT

ttytxT

ttytyT

ttxtxT

ttytxtytxT

R

0000

0

d1

d1

d1

d1

d1

The autocorrelation is an even function of the time-lag τ

The autocorrelation function reaches its peak value at the origin, where it takes a real value, i.e. for

any lag (delay) τ,

The autocorrelation of a periodic function is, itself, periodic with the very same period

The autocorrelation of the sum of two completely uncorrelated functions x(t) and y(t) (the cross-

correlation Rxy(τ) is zero for all τ) is the sum of the autocorrelations of each function separately

xxxx RR

xxxx RR 0

0

The autocorrelation of a continuous-time white noise signal will have a strong peak (the Dirac

delta function) at τ = 0 and will be absolutely 0 for all other τ.

xxR

For τ = 0 it is possible to get

ffSSttxT

R xxxx

T

xx dd2

1d

10

0

2

184

Estimation of the correlation function

txtxERxx

Tt

Ttt

;0,0

;0,1

xx

TT

xx RT

ttxtxEttT

ttxtxttT

ERE 1d1

d1

00

TRxx 1

0 T

The definition of the autocorrelation function assumes that the signal x(t) continues from minus

infinity to plus infinity

In fact, the signal measurement takes a limited time interval

The expected values of the autocorrelation function estimation is as follows

The unbiased autocorrelation function can be obtained by the use of the formula

xxR

Computation of results in a bias estimation of xxR

→ t

txt

This formula is sometime called Bow-tie correction (BK signal analyzers)

185

Auto- and cross-correlation function of the sampled signal

2,...,2,1,0,1 1

0

NiyixN

RN

i

xy

2,...,2,1,0,1 1

0

NixixN

RN

i

xxAutocorrelation

function

Cross-correlation

function

The correlations are a function of the time lag τ, that is the length of signal over which each

correlation is calculated

ix

ix

i 0 1 2 N-1

* * *

N - τ τ

*

Unbiased estimation

iy

Calculation ……

or

Let xn , i = 0, 1, 2, …, N-1 be a sampled ergodic signal, where N is the sample count

186

Autocorrelation function of a continuous harmonic signal

Biased estimation

-1,0

-0,50,0

0,5

1,0

-1,0 0,0 1,0

Lag [s]

U^2

tAtx cos

cos22

22sincos

2

1lim

dcos22cos2

1lim

dcoscos1

lim

2

0

2

0

2

0

2

AtAT

T

ttA

T

tttAT

R

T

T

T

T

T

Txx

Let be a harmonic signal. The autocorrelation function is as follows

Unbiased estimation

-1,0

-0,5

0,0

0,5

1,0

-1,0 0,0 1,0

Lag [s]

U^2

Cosine function of lag

187

Examples no.1 – Random signals

White Noise

-0,5

0,0

0,5

1,0

1,5

-1,0 -0,5 0,0 0,5

Lag [s]

U^2

Pink Noise

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

-1,00 -0,50 0,00 0,50

Lag [s]

U^2

Noisy harmonic signal

-4,0

-2,0

0,0

2,0

4,0

0,0 0,5 1,0

Time [s]

U

Noisy harmonic signal

-1,0-0,50,00,51,01,52,0

-1,0 -0,5 0,0 0,5 1,0

Lag [s]U

^2

2

xxxS 2

xxxR

22

22

y

yyS exp2

yyyR

Examples no.2 – Tonal signals

-2,0

-1,0

0,0

1,0

2,0

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

U

0,0

0,1

0,2

0,3

0,4

0,5

0 50 100

Frequency [Hz]

RM

S U

-0,4

-0,2

0,0

0,2

0,4

-1,0 -0,5 0,0 0,5

Lag [s]

U^2

-0,4

-0,2

0,0

0,2

0,4

-1,0 -0,5 0,0 0,5

Lag [s]

U^2

Time domain signal Autospectrum

Autocorrelation (biased) Autocorrelation (unbiased)

qualitatively difference

in appearance

188

Examples no.3 – Detecting an echo in a signal

SweptSine - Chirp

-1,5-1,0-0,50,00,51,01,5

0,000 0,002 0,004

Time [s]

SweptSine -frequency

0

5000

10000

15000

20000

0,000 0,002 0,004

Time [s]

Hz

Repeated chirp

-2

-1

0

1

2

0 1 2

Time [s]

Repeated chirp & Echo

-1,5-1,0-0,50,00,51,01,5

0 1 2

Time [s]

Autocorellation

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

-0,008 0,000 0,008

Lag [s]

U^2

Autocorrelation

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

-0,008 0,000 0,008

Lag [s]

U^2

Envelope of autocorellation

function

0,000

0,004

0,008

0,012

0,016

-0,008 0,000 0,008

Lag [s]

U^2 150

samples

150 samples

String of the chirp signals String of the chirp signals

affected by the 10%-echo,

which is delayed by 150

samples

Effect of echo

kx 1501.0 kxkx

Chirp String of chirps

Autocorrelation Frequency

Autocorrelation

fSampl = 44100 Hz

189

Examples no.4 – Impulse response

,ddd1

dd1

d1

111

0

111

0

111

0

ttgtRtttgttxtxT

tttgttxtxT

ttytxT

R

xx

T

TT

xy

The Wiener-Hopfov equation

G(jω)

g(t)

tx ty

Input Output

The input-output cross-correlation function

Let x(t) be white noise, 2

xxxR

gttgtttgtRR xxxxxy

2

111

2

111 dd

Cross-correlation function

-0,2

-0,1

0,0

0,1

0,2

0,3

-0,50 -0,25 0,00 0,25 0,50

Lag [s]

Impulse response

-0,2

-0,1

0,0

0,1

0,2

0,3

0,00 0,25 0,50 0,75 1,00

Time [s]

[-]

Filtration effect of δ(t)

190

Examples no.5 – Frequency response function

G(jω)

g(t)

tx ty

Input Output

Fourier Transform

-0,5

0,0

0,5

1,0

1,5

0,0 0,2 0,4 0,6 0,8 1,0

Time [s]

x(t

), y

(t)

x(t)

y(t)

1,E-10

1,E-09

1,E-081,E-07

1,E-06

1,E-05

1,E-04

1 10 100 1000

Frequency [Hz]

PS

D [-]

^2

/Hz

Sxx

Syy

Correlation

-0,0015-0,0010

-0,00050,0000

0,00050,0010

0,0015

-0,5 -0,3 0,0 0,3 0,5

Time [s]

Rxx

Ryy

FFT

1,E-06

1,E-04

1,E-02

1,E+00

1 10 100 1000

Frequency [Hz]

10

24

* A

mp

l -

FFT(Ryy)

FFT(Rxx)

dexp jRS

191