UT 03 TMI Signal Recovery

8/17/2019 UT 03 TMI Signal Recovery

1/238

Signal recovery from noise in

electronic inst rum entation

T H

Wilmshurst

Department o Electronics and Computer Science

University o Southampton

Second Edition

Institute

of

Physics Pub lishing

Bristol and Philadelphia

Copyright © 1990 IOP Publishing Ltd.


2/238

OP Publishing Ltd 1990

All rights reserved. N o part of this publication may be reproduced stored

in a retrieval system or transmitted in any fo rm o r by any means elec-

tronic mechanical photocopying recording or otherwise without the

prior permission of the publisher. Multiple copying is only permitted under

the terms of the agreement between the Committee of Vice-Chancellors and

Principals and the Copyright Licensing Agency.

ritish Library Cataloguing in Publication Data

Wilmshurst T. H .

Signal recovery from noise in electronic instrumentation.-

2nd. ed.

1.

Instrumentation systems. Electrical noise

I .

Title

620.0044

ISBN 0-7503-0059-0

ISBN 0-7503-0058-2 pbk

Library o Congress

Cataloging-in-Publication

Data are available

First edition published 1985 by A da m Hilger L td

Second edition published 1990 by I O P Publishing L td

Reprinted 995

Published by Institu te of Physics Publishing wholly ow ne d by T h e Ins titute

of Physics London

Institute of Physics Publishing Tec hno H ou se Redcliffe W ay

Bristol BS1 6NX U K

U S E ditorial Office: Inst i tute of Physics Publishing T h e Public Le dg er

Building Suite 1035 Ind epende nce Squ are Philadelphia P A 19106 U SA

Printed in Great Britain by

J

W Arrow smith Ltd Bristol



3/238


4/238

vi

CONTENTS

4.7 Shot noise amp litude

4.8 Therm al noise amplitude

5.1 General properties

5.2 Measuremen t time independence

5.3 Response to combined ‘noise’

5 4 Experimental result

6.1 Low-pass filter

6.2 Running average

6.3 Visual averaging

6.4 Baseline offset correc tion

6.5 Sloping baseline correction

6.6 Step measu rement

6.7 Spa tial frequency com ponents of

l f

noise

6.8 Frequency response

of

multiple time averaging

Frequency domain view of the phase sensitive detector

7.1 Analogue multiplier

PSD

7.2 White noise erro r

7.3 Reversing

PSD

7.4 Chopping

PSD

7.5 Comp romise arrangem ents

7.6 PSD mperfections

7.7 Non-phase-sensitive detec tors

7.8 Recovery

of

a signal submerged in noise

8.1 Add itional noise du e t o digitisation

8.2 Analogue-to-digital converters

8.3 Suitability of real ADC circuits for taking average

samples

8.4 Point sampling gate

8.5

Aliasing

Magnitude determination

for

transient signals of known

shape and timing

9.1 Flash spectrome ter

9.2 Limit setting

9.3 Weighted pulse sampling

9.4 Matched filtering

9.5 Boxcar sampling gate

9.6 Offset drift an d

l f

noise

9.7 Pe ak an d valley detectors

9.8 Software realisation of weighted pulse sampling

5

l f noise

Frequency response calculations

7

8

Digitisation and noise

56

58

60

60

62

65

66

69

70

72

73

74

80

81

83

85

91

92

96

98

100

103

1 4

105

109

111

111

113

117

118

121

126

126

128

131

136

138

140

152

156



5/238

CONTENTS

vii

10

Measurement of the time of Occurrence of a signal

transient

10.1

Method for timing a transient with minimum white

noise erro r

10.2 Signal pulse timing

10.3 Signal ste p timing

10.4 Co m pa riso n of the step an d pulse method s

10.5 Offset drift and l f noise

10.6 Lock-in amplifier

10.7 Autocorrelation methods

159

160

164

176

178

180

184

185

11 Frequency measurement

188

11.1 Period ic transien ts of fixed width 189

11.2 Ratem eter 193

11.3 Phase-locked loo p 196

11.4 Periodic transients of width pro po rtio na l to period 199

1

I

.6 Single transien ts 209

11.8 Noise-optimised averaging of frequency measurem ents 217

11.9 Frequency -domain methods 219

Appe ndix: Fourier analysis 223

Tu tori al questions 225

Answers to tutorial questions 230

Index 23 1

11.5 Frequenc y tracker 202

11.7 Correlation matching 212



6/238

Preface to the second edition

Th e second edition add s a furthe r chapter t o cover the subject of fre-

quency measurem ent. Also included ar e a set

of

tutorial q uestions with the

numeric answers given. These cover the material in all chapters and will

consolidate understanding of the material in the book.

T

H Wilmshurst

July

1990



7/238

Preface

to

the

first edition

The subject of this book is the recovery of signals from noise in electronic

instrumentation. Th e term ‘electronic instrumentation’ is sometimes reserved

for instruments such as the oscilloscope and testmeter which are used to

measure specifically electrical variables. It should therefore be m ade clear that

the scope of the book goes much beyond this an d covers instrume ntation for

the measurement of any variable whether electrical o r not. The term

‘electronic’simply implies that electronic techniques a re used in the process of

measurement

The term ‘noise’ also requires clarification. Until recently it was used mainly

to refer to rand om fluctuations such as white and

l f

noise. Now however the

term is used to refer to alm ost any kind of unw anted signal in a n electronic

system. It is in the b road er sense th at the term is used in the title since the

recovery

of

the required signal from many kind s of unw anted signal is covered.

These comprise offset drift random noise comprising white an d l f noise and

interference. However to avoid confusion the term ‘noise’ will be reserved for

rand om noise. Th e other types of unw anted signal will be referred t o by their

specific nam es.

There are broadly two ways of reducing the errors du e to unw anted signals

in an electronic system. The first is to prevent the u nwanted signal from being

introduced . This is a m atter of low-noise amplifier design screening

decoupling etc and is reasonably well covered by existing texts. Th e second

app roac h which is taken up when the first has been exploited as far as possible

is to devise ‘signal recovery’ techniques which distinguish the required signal

as well as possible from whatever unw anted signals may remain. This second

app roac h is less well covered a t present a nd therefore represents the subject

matter of the book.

Th e text is intended for anyone w ho is to be involved in the development

of



8/238

xii

PREFACE

electronic instrumentation a t grad ua te level o r beyond. In or de r to indicate

better the intended readership it will be helpful to consider briefly the way in

which electronic instruments are o r should be developed.

A

number of

different disciplines is nearly always involved. Consider for exam ple such a

system a s the laser anem om eter. This is a device which uses bo th electronic and

optical techniques for the measurement of fluid velocity. Thus the

development of this instrument would require experts in optics electronics

an d fluid dynamics. When such a team is formed it is impo rtan t that each

mem ber sho uld atte m pt to maste r all aspects of the system no t just those of his

own discipline. This is as true for the electronic aspects of the system as for any

othe r. Thus the book ha s to be acceptable to two rather different classes of

reader. The first is the undergraduate electronics engineer who intends to

specialise in instrumentation. He will meet the subject probably in the third

year of his undergra duate studies an d will need to obta in a good grasp of the

entire contents. T he o the r class of reader will be the non-electronics specialist

who will probably come to the subject after graduation and at the time of

joining t he team . He will usually need t o stud y in de pth a limited selection of

topics from the book. In orde r to help this class of reader attem pts are made to

cover the material in descriptive a s well as analytical m anne r. Also rather

more extensive subsection titling than usual has been adopted in order to

facilitate ‘random access’.

The book stems from two courses that are given in the Department of

Electronics of the University of S ou tham pton . The first has run for over a

decade and is intended fo r graduates in disciplines other th an e lectronics who

ar e engaged in the development of electronic instrum entation o r sometimes

just in its use. These may be studen ts working for higher degrees postdo ctoral

research fellows technicians and sometimes academic staff.

The second course is mo re recent an d is intended primarily for third-year

undergraduates in electronic engineering. It covers that material outside the

‘core’ electronics sub jects tha t is needed for a gra du ate w ho is to specialise in

electronic instrument development. The course is also followed by students

tak ing ‘with-electronics’ courses such a s ‘physics-with-electronics’ ‘chemistry-

with-electronics’ etc.

T H Wilmshurst



9/238

1 Low-pass filtering and visual

averaging

1.1 OVERVIEW

Before discussing low-pass filtering and visual averaging, the two signal

recovery m ethods t o be discussed as the main topic of the present ch ap ter, we

devote the first section to a n overview of the entire book. Here th e classes of

unw anted signal, an d also all of the signal recovery methods to be described,

will be briefly introduced.

Resistor bridge strain gauge

The first step will be to describe the operation of a simple example of an

electronic instru m en t: the resistor bridge strain gauge

of

figure 1.1. Here the

variable t o be measured is the strain (extension)

of

the mechanical com pon ent

show n, such strain usually resulting from a m echanical force (stress) applied t o

the component.

Th e principal compo nent of the strain gauge is the resistor

R,.

This is a mesh

offine wires which is atta ch ed to the mechanical com po ne nt by an adhesive.

As

the mechanical compon ent is strained,

R ,

increases; this causes the bridge to

become unbalanced, producing a voltage U , which is approximately

prop ortion al to strain.

U,

s then amplified, low-pass filtered and d isplayed o n a

chart recorder.

Th e variety of instruments of this kind is vast, the single common feature

being that some variable, which is usually non-electrical, is converted to an

electrical signal which is processed an d displayed. The device which converts

the variable of interest t o the electrical signal is called th e inp ut ‘transducer’

an d for the present system the transduce r is the resistor bridge. This conver ts

the variable of interest, the strain, into the electrical signal U , . Although the

range of input transduce r types in use may con stitute a n im po rtan t topic for


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


10/238

2 LOW-PASS

FILTERING/VISUAL AVERAGING

Resistor

bridae

Balance

Time

Chart

Amplifier

Low-pass

filter recorder

‘ R -

I ‘ Y

St re ke d mechanical

component

Figure 1.1

Resistor bridge strain gauge.

the instrument engineer, this is not the subject of the present book, and is

adequately covered elsewhere. Instead, our objective is to establish the

principles whereby the electrical signal developed by such a transducer is

recovered from the various kinds of unwanted signal.

This objective will be best served if, wherever possible in the course of

introducing the signal recovery methods, the same simple example of the strain

gauge is retained. This is in con tras t to the alternative a pp roa ch of constantly

altering the example in ord er at the sam e time to try to give some idea of the

differing transducer types. With the principles of signal recovery thus firmly

established, using one type

of

transdu cer, it will be a simple m att er to a pply the

principles to any other type of transducer that might be encountered.

Low-pass filtering

of

white noise

Th e purpose of th e low-pass filter in the stra in gauge of figure 1.1 is to reduce

the amplitude of the unwanted signal components, of which the first to be

considered is white noise. Figure

1.2

illustrates the way in which the low-pass

filter reduces such noise. Here a fixed stress is applied to the mechanical

component at time

t =0

and the requirement is to measure the difference in

recorded strain before and after application of the stress. Figure 1.2 a)shows

what might be observed a t the ou tpu t of the amplifier in this situation. Here

there is a moderately high level of white noise superimposed upon the signal

component.

In du e course it is shown th at th e effect of the filter is to produce the o utp ut

voltage U , which is the ‘runn ing average’ of the inp ut voltage U,. Th e function is

illustrated in figure 1.2(b). Here, at time t , U, is th e average of U , over the period

shown from t- T, to t . Thus T, is termed the ‘averaging time’ for the filter.

In order to construct the output function

v o ,

the averaging box in figure

1.2(b)

must be made to ‘run’ along the input function U , in a), thereby

describing the o ut pu t function

U ,

in c) .Clearly the effect is to reduce th e noise.

It is shown subsequently th at the am plitude

V

of the noise compone nt of U , is

proportional to

T,-’/’,

i.e. V;

T - ’ ’ ~ .

Th e oth er feature t ha t is clear from figure

1.2

is that it takes th e finite time

T,


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


11/238

OVERVIEW 3

T-

I

I

Noise

I

t

I

--I

a-6

6)

kl

I

I t

I I

I f: 0

Figure 1.2

Diagram showing how the low-pass filter of figure 1.1 reduces

the

white noise

amplitude

by

forming a running average

of

the filter input.

( a )

Filter

input

U,

in

figure

l . l ) ,

( b )

running average process,

( c )

filter

output U,.

for U , to respond t o the step in

uin.

Thus

time'.

is also know n as th e filter 'response

Visual averaging

The second method of signal recovery to be considered is visual averaging.

Looking a t the filtered noisy signal step in figure 1.2, he basic requirement is to

determine the difference between the signal before an d after the step. Here the

eye forms two averages, one before and one after the step, and subtracts one

from the other. Now, because there are several noise fluctuations over each

averaging period, the eye is able to determine the signal voltage t o a n accuracy

greater than tha t given by the noise amplitude, even a t the filter ou tpu t.

The reason for this improvement is that basically it is the process of

averaging the noise which reduces its effect and filter and eye are just two

different me thods of averaging. The reason that the noise am plitude

V;

at the

filter ou tpu t is proportional to

T-'l2

is that when w hite noise is averaged by

any means over a period

T,,

the value of the typical noise e rro r in the average is

proportional to T,; l i 2 . Then, since for the filter T,, = T,, the noise error is

proportional to T - l i 2 .When, however, the eye extends the averaging period

to the period TJ2 of the half step the noise err or is reduced from the value

of

say k T , - ' i Z given by the amplitude to

k(TJ2)- 12.




12/238

4

LOW-PASS FILTERING/VISUAL AVERAGING

It is the recovery of signals from white noise by low-pass filtering and by

visual averaging that constitutes the main topic of the present chapter.

How ever, before proceeding t o this detailed study we com plete the overview

by briefly introduc ing the other types of unwanted signal an d signal recovery

methods.

Drift and offset

The next two types of unwanted signal

to

be covered are offset and drift. Of

these, offset is defined as that component of the total of all unwanted signal

components which does not vary with time. For the resistor bridge strain

gauge of figure 1.1 the two main sources of offset are the transducer (the

resistor bridge) and the signal amplifier. The transducer offset arises from

initial imbalance in the bridge and is easily removed by adjusting the balance

resistor sh ow n. Similarly the DC coup led signal amplifier will also be provided

with

a

suitable balancing potentiometer. Thu s, for the present example, offset

is not a major problem. However, for some systems the offset cannot be

so

simply removed. Then an acceptable altemative is the baseline correction

method to be discussed in 1.4.

Unfortuna tely the offset, even if initially adjusted to z ero , tends to ‘drift’ as

time proceeds. This is usually because of slow changes in the temperature of

the circuit.

Now, drift error tends to increase as the time

T,

spent making a

measuremen t increases; this is directly opposed to the trend for white noise, for

which the noise error varies as T,-li2.hus, drift tends t o frustrate a ttem pts to

obtain a low white noise error by using a long measurement period, and we

shall find tha t much of the strategy in signal recovery lies in devising methods

of overcoming this problem.

Multiple time averaging

Two main m ethods are used t o reduce the drift error and so allow the low

white noise e rro r associated with a long period of measurement to be realised.

These ar e multiple time averaging (MTA) an d the phase-sensitive d etector

(PSD)

me thod. Of these

MTA

is a matter of carrying o ut the experimen t rapidly rather

than slowly, thus reducing the drift error, and then extending the averaging

time t o o bta in the low white noise err or by repeating the experiment m any

times and averaging the results. The use of

MTA

to avoid effects of drift in this

way constitutes the main topic of chapter 2.

Phase-sensitive detection

Sometimes the experimental constraints do not allow the time scale of the

experiment to be reduced. Here a suitable altemative is to modify the

experiment in such a way that the transducer produces an

A C

rather than a DC

signal. This allows a n

AC

coupled am plifier to be used which does not respond

to offset or drift.


http://0.0.0.0/http://0.0.0.0/


13/238

OVERVIEW 5

Here some kind of rectifier is usually required in order to convert the A C

amplifier ou tpu t into a DC signal suitable for display. This, then, is the function

of the phase-sensitive detector. It is the use of the P m m eth od to overcom e drift

that constitutes the main topic of chapter 3.

l f noise

Th e majo r remaining class of unwanted signal to be considered is l/f noise.

Fo r both white and l/f noise the names refers to the spectral distribution of the

noise. For white noise the spectral density

G

is independent of frequency f

while for l/f noise G f ’. hese terms belong to the frequency dom ain view

an d will be elaborated later. F o r the present, the waveforms of figure 1.3 (a) n d

( b )will suffice to identify the difference between the two. T hese ar e specimens

of white an d l/f noise as might be observed a t the o ut pu t of the signal amplifier

in the s train gauge of figure 1.1. Clearly the

l/f

noise ha s

a

greater tendency to

‘walk away’ from th e mean. This mean s th at the ad va nta ge s of using a larger

measurement period T, ar e less th an for white noise. In fact, it will be sho wn

that the

l/f

noise error is independent of T,. This con tras ts with w hite noise,

for which the error is proportional to T , - l i 2 , nd drift, for which the error

increases with T,.

6)

a )

Figure 1.3 Types of noise seen a t the outp ut of th e filter in figure

1.1

:

a )

white noise,

( b )

l/f

noise.

It is thu s clear tha t l/f noise too frustrates any attem pt to obta in a low white

noise error by increasing T,. For low values of T, the white noise may

dominate, a nd then increasing T, will cause a reduction. H oweve r, a point will

be reached a t which t he white noise erro r falls below the l/f noise e rro r, which

does not vary with

T,.

Then any fu rther increase in T, will have

no

effect.

Fortunately, both MTA and PSD methods ar e effective in rem oving l/f noise

err or b ut, in o rde r to show how this is do ne , it is necessary to use the spectral

view of both signal and noise. This is in contrast with the rather more direct

‘time-domain’ view used in the chapters listed so far.

. Frequency-dom ain view

T he frequency-domain view is a comm only a do pte d one an d th e present text is

perhaps a little unusual in not adopting it from the start. However,

I

have

preferred the directness of the time-dom ain ap pro ach for the initial treatmen t.

Thus, in chap ter 4 the frequency-domain o r ‘spectral’ view will be introduc ed

and the ground previously covered from the time-domain viewpoint re-

covered from the frequency-domain viewpoint. Cha pte rs 5 to

7

then use the


http://0.0.0.0/http://0.0.0.0/


14/238

6

LO W-PASS FILTERING/VISUAL AVERAGING

frequency-dom ain view to show how the MTA and PSD methods remove the l/f

noise.

Digitisation

These seven chap ters com prise the first pa rt of the book , which deals with the

recovery of a con tinuously varying analogue signal from the various kinds of

unw anted signal. Th e remainder deals with one o r two closely related topics.

Frequently tod ay a n analogue signal must be digitised for com pu ter storage

an d p erhaps for further processing. In ch apter 8 the problems of digitisation

are covered. It transpires tha t the main factor of importance is tha t n o da ta

should be wasted.

Pulsed signals

Chapter

9

deals with pulsed signals o r more general types of signal transien t.

Of particular interest is the case where the shape of a transien t is know n but the

amplitude is not, being the factor to be measured. It is shown that the

processing required t o give minimum noise er ro r is what is know n as 'matched

filtering'.

Signal timing

In chapter

10

the problem is a little different. Here again the shape of th e signal

transient is known but the factor to be measured is the time of occurrence,

rather than the amplitude, of the signal. The necessary modifications to the

matched filtering in this case are discussed.

The need to use a matched filter when determining the amplitude of a

signal transient of known shape is well established. Also the modifications

needed when signal timing is to be determined are reasonably well

documented. However, such coverage is usually for white noise only. In

instrumentation, as distinct from communications applications, it is highly

likely tha t drift and l/f noise will be the predom inant unw anted signals. It is

the distinctive contribution of the present text that the case of l/f noise is

covered also.

1.2 LOW-PASS FILTERING O F SHOT NO ISE

Th e overview is now com plete an d we proceed to the main topic of the present

chapter. This is the way in which low-pass filtering and visual averaging

recover a required signal from white noise. There a re two m ain types of white

noise: thermal and shot noise. For all purposes of signal recovery these are

indistinguishable. Thus in this section the effect of low-pass filtering on shot

noise will be examined. It will be shown in particular tha t the am plitude V; of

the noise at the filter output is proportional to T,- 2 , where T, is the filter

averaging time. This relation also holds for thermal noise.



15/238

LOW-PASS FILTERING O F

SHOT

NOISE

7

Thermal noise

It will, nevertheless, be useful to indicate briefly the origins of thermal noise.

This originates mainly from the resistors in

a

circuit a nd arises from th e fact

that the mobile charge carriers in a resistor are in constant thermal motion.

Thus, at any one time the charge distribution over the resistor is not quite

uniform. Th is causes a potential difference across the ends of the device. As the

therm al motion p roceeds, the potential fluctuates in

a

random manner abou t a

mean of zero. Th is class of noise is discussed further in 4.8, with qu antitative

precision.

Shot noise

Shot noise tends to predominate in the semiconductor com pon ents in a circuit,

i.e. in the d iodes a nd the transistors. It is actually

a

manifestation of the fact

that an electric current is not a continuum but consists of discrete current

carriers, i.e. electrons and ‘holes’. Thu s the commo nly used analogy of a stream

of flowing water to represent an electric current might more properly be

replaced by a stream of grains of sand . It is then the ‘grainy’ na tu re of the flow

that constitutes the shot noise.

Shot noise model

Figure 1.4 gives a simple model for the shot noise in the amplifier of figure 1.1.

Here it is reasonable to assum e tha t all of the shot noise originates from the

first stage, because t ha t from later stages is subject t o less gain. Th e transistor

+----*-- yu p p l y

LC

t

a b )

C)

Figure 1.4 Circuit used t o sh ow the effect of th e low-pa ss filter in figure 1.1

upon the amplifier sh ot noise.

a)

First stage, ( b ) urther stages,

( c )

ow-

pass filter.

current

i

actually consists of a random train of pulses, with each pulse

corresponding to the passage of an electron from the em itter to th e collector of

the transistor. T he ar ea of each pulse will be qe, he electron charge. Th us the

area

q

of the corresponding pulse at th e ou tpu t of the amplifier A will be given


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


16/238

8

LOW-PASS FILTERING/VISUAL AVERAGING

by

4

= @ , A .

(1.1)

Figure 1.5(a) shows the sequence of these random pulses that make up the

waveform for

U,

=

U;,,

in figure 1.4. This is somewha t idealised, assuming that

the amplifier co ntains no low-pass filter, but will d o for o ur present purposes.

p T r 4

- - - I

Figure

1.5

Memory box running average approximation to filtering of

sho t noise pulse train for transistor in figure 1.4. a)Sho t noise pulse train

c,,,.

( b )

running average

c

Running average

It will next be show n t ha t the essential feature of the filter,in t he present view, is

that it forms a ‘ru nning average’ of the inpu t. Figu re

1.5

illustrates this function

once more. Here the outpu t

U ,

a t time

t

is the average of

U,,

over the period

r,

of

the mem ory box. Then the outp ut function U, is constructed as the box ‘runs’

along the input function. In mathematical terms

t.,(t)= T- ’

U, ‘)

dt‘.

1

- T,

Th e fluctuation component

U ,

of U , is the sho t noise an d arises because

U, is

pro portio nal to the num ber of pulses p in the box and

p

fluctuates because of

th e rand om placing of the pulses. T he fluctuation time is T,because this is the

time taken for all of the pulses in the box to be replaced.

Filter response

The next step is to determine the exact relation between the input U,, and

output U , of the low-pass filter and thus to show that 0, is essentially the

running average of uin .

Consider first th e ‘impulse response’

gi

of the filter, shown in figure

1.6(f).


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


17/238

LOW-PASS F IL T E R IN G O F SHOT NOISE

9

I

M e m o r y

function I + rCR

d ) f )

0 I \ \

I t

I

Figure

1.6 Relation between ou tpu t U, and input U,, for a low- pass filter. a) n p u t

U,,, ( b ) ilter,

(c)

components of outp ut U, originating from elements in a), d )

memory function, ( e ) nput impulse,

(f)

mpulse response.

This is the response of the filter to the unit impulse shown in figure 1.6(e),

which is applied a t time t

= 0.

Here th e unit impulse is a pulse of unit are a an d

of width approaching zero.

The input impulse leaves the capacitor with a charge. This decays

exponentially through the resistor

R

as shown, with the decay time C R.

Next consider the shaded element a t time t of the general inp ut waveform uin

shown in a).For 6 t small, this approxim ates to a n impulse of are a ui,(t ) 6 t .

This will produce a t the filter outp ut th e shaded transient shown in

c ) .

At the

time

t ,

he output component

6 u ,

will then equa l the impulse are a multiplied by

gi(t ), .e. ui,(t )gi(t

)

6t. Then summing 6 u , for all values of uin,

Looking a t the transients in

c)

following th e various elements of uin shown

in a) ,and numbered 1 to

5 ,

it is clear th at U , consists of contr ibu tion s from U , ,

extending back from time

t

at w hich th e filter ou tp ut is observed for a period

roughly equal to

CR

This is highly reminiscent of the ru nning average, with

the averaging period T = C R . The main difference is that the 'memory

function'

g i ( t - t ' )

from equation (1.3) and shown in ( d ) has an exponential

'taper'. This co ntra sts with the a br up t cut-off of th e 'memory-box'

approximation corresponding to the running average. This property is

sometimes referred to as a 'fading' memory of the input.



18/238

10

LOW-PASS FILTERING/VISC:AL AVERAGING

Weighted running average

Clearly U, is no t a simple running average of vi over T,but a 'weighted' runn ing

average, the weighting given by the fading memory function gi(t- ) . Actually

it is only clear from eq uation (1.3) that U, is a w eighted integral of

uinr

over the

approximate period

T,.

An average, whether weighted o r otherwise, is said to

be prop erly normalised if, when th e series of values that a re being averaged a re

all the sam e, the average is equal t o an y on e of the values. Bu t, for the filter, if uin

is cons tant, then U, =

uin.

Thus U, is

a

properly normalised weighted running

average of

uin,

with the weighting given by the memory function which is the

impulse response gi reversed in time.

We now return to the main theme of the present section, which is to

calculate the amplitude

6

of the noise com ponen t U, of the filter outpu t U, when

the random train

of

impulses shown in figure 1.5 a )constitutes the input

uin.

The approximation that U, is the running average of

uin

will be assum ed, with T,

the averaging period. T he n, with

q

the pulse area, U, at any t ime

t

is given by

where

p

is the number of pulses in the box at time

t .

Probability density function and standard deviation

T he fluctuation in

U,

arises from the fluctuation in

p .

T o determine this it is first

necessary t o introduce the concepts of stan da rd deviation a nd the probability

density function.

If

the memory box is placed at a very large number of

different an d mutually exclusive locations on the inpu t pulse train,

a

histogram

of the results can be p lotted as in figure 1.7, together with th e overall mean

e.

He re the horizontal coord inate is the num ber of times, n, th at each value of

p

was obtained.

t

n,vp

Figure

1.7

Prob ability density function for the num ber of pulses

p

spanned

by the memory box of

figure 1.5.

For present purposes it is sufficient to say that the half-width

sp

of the

histogram is the 'standard deviation' and that this is given by the relation

g p=

P ) Z 1.5)




19/238

LOW-PASS

FILTERING OF SHOT NOISE

11

provided 1. The n op s a good indication of the am plitude of the fluctuation

in p . However, a little more precision may be preferred. Thus, if the count

values n in the histogram are all divided by a co mm on factor k to give v p n / k

and k is such tha t

J

v p dp = 1, the scaled histogram v p s termed th e 'probability

density function'

(PDF)

for

p .

Strictly,

v p

has the 'gaussian' form

and op s the half-width when v p alls from the central maximum by a factor

exp(- /2).

It will be noted that , unlike the rest of the text, the obse rvations m ade in this

subsection are stated rather than reasoned. They are, in fact, stand ard results of

statistical theory a nd equation (1.5), in particula r, is a 'foundation' statem ent

upon which much of wha t follows must stand . It will therefore be of value t o

reflect upon its plausibility for varying values of

P.

Notice for instance how

when e 1 the ratio ode becomes small compared with 1.

Fr om equation (1.4), he stan dard deviation U for the noise com ponent U, of

U, is given by

Then, from equation (1.5),

6

= C p q / T , * (1.7)

on

p 2q/

T,.

(1.8)

Here cc

T,.

Let be the long-term mean pulse rate. The n p YT and

U =

q(Y /T , ) ?

This can be written as

on

k T , - ' I 2

since q and r are both independent of T,.

(1.9)

(1.10)

RMS

value

The usual way to represent the amplitude of a fluctuating variable is by its

'root-mean-square' (RMS) value. F o r the general fluctuating variable

x ,

the RMS

value of x is

( ? ) I z ,

where the average is over a period very large compared

with th e fluctuation time for

x .

Th e more concise symbol x'is usually a do pte d,

i.e.

X E

(x2) ' I2 .

It is clear that the

RMS

value ?a nd the standa rd deviation nX re com parable.

In fact they a re equal an d, in this sense, the two terms a re interchangeable.

However,

x'

expresses the amplitude of the fluctuating variable, while

ox

indicates the typical deviation from the mean for just one measurement.

Since for U,, 6= U equa tion (1.10) becomes

fin =

kT,-'I2. (1.11)

Th us o ur initial objective of showing tha t 6

T - l 2

has been achieved.



20/238

12 LOW-PASS FILTERING/VISUAL AVERAGING

Experimental result

Figure

1.8(a)

to (c)shows the results of a series of measurements made upon a

signal step. These confirm tha t each time the averaging period T,, i.e. the filter

time constant

C R ,

s increased by a fac tor of

10

the noise amplitude 6 decreases

by a factor

of

It

can also be seen how the filter response time an d the

noise fluctuation time increase in proportion to T,. Note also how it is

necessary to increase the time taken to make the measurement in order to

accommodate the increase in the response time T,.

T r

20s

(C 1 id1

T 200ms

T

:IO00 s

Signal

i e )

s t e p i f

Figure 1.8 Step+white noise recorded at the output of a low-pass filter. @)-(e)

Recorded traces, (f) xperim ental system. In these traces, T, s the filter response time

and

T ,

the time taken to record the step.

Noise error

in

integral

Suppose th at, instead of the average, the integral of the shot noise is formed

over the period T,. Since the integral is the average m ultiplied by T,, an d since

the stan da rd deviation

on

f the average is prop ortiona l to

T,-

' I z , the standard

deviation ol f the integral will be proportional to T+1 2. ore generally, for



21/238

VISUAL AVERAG ING

13

an integration period

ai

=

kT,’”.

(1.12)

Amplifier filter

It should, perhaps, be noted that the noise at the output of

a

real signal

amplifier would never appear as in the idealised model of figure 1.5 a).

In

reality there would always be some sort of low-pass filter in the amplifier,

whether by accident o r design, an d this would sm oo th the train of random ly

placed impulses to prod uce

a

waveform m ore like th at of the noise in figure

1.2(a).Here the small fluctuation time will be the response time of the filter

inte m al to the amplifier. The ex ternal filter, with the m uch longer response

time T,, then extends the fluctuation time to

T,.

1 3

VISUAL AVERA GING

Visual averaging is the process whereby the averaging of a displayed noisy

signal by the eye of the observer makes the overall noise er ro r less th an th at

given by th e noise amplitude Cn. igure

1.9

shows noise a t the o utp ut of

a

filter

of averaging time T,superimposed up on a required signal U, an d displayed for

the period T,.

If

then V; =

T , - ’ ” ,

it follows that the sta nda rd deviation for the

noise averaged over the period T, will be given by

a,

where

a, = k T,- (1.13)

Then, since T ,g T,, ov< O; and a reduction is obtained.

I I

I

I T”

I

I

Figure

1.9 Noise at low-pass filter output superimposed on required DC signal t’,

and averaged by eye over the period

T,.

Another way of looking a t the process is as follows. Th e fluctuation time for

the noise is T,. Th us th e eye averages

n

= T,/T, ndependent fluctuations. The

reduction in noise erro r is by cT /on=n -I/’. This is an exam ple of the general

tru th th at, if

n

independent ran do m samples are averaged, the resulting noise

error is reduced by n-’/’.

N ote tha t the final noise erro r av s independent of the filter time con stan t T,,

althou gh the noise amplitude

Cn T - l i Z .

his is because

if

T, s reduced, say,

causing V; to rise, the num ber n=

T,/T

is increased, so that the eye, having

more fluctuations to average, can obtain a better reduction. In fact the two


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


22/238


effects exactly cance l, an d this is obvious when it is understood tha t the only

thing tha t varying T, does is to alter the division of la bo ur between th e filter

and the eye in forming the final average over the period q.

The relative inconsequence of the se tting of T,, regardless of its effect upon

the noise am plitude, is an effect frequently not realised. In fact,

T,

need only be

large enough to reduce the num ber of fluctuations t o a value small enough to

allow the eye to average effectively.

Experimental result

Th e sequence of measurem ents in figure 1.8(a), (d) and e)confirms this point.

Recall that the sequence from a ) to (c) corresponds to increasing the

measurement time

T,

an d the filter response time T, ogether in decade steps.

This shows the variation of

V

with T , - 1 / 2 .n con trast, the sequence a), d ) , e)

increases T, in the same way but leaves T,unaltered. Here 1.7 remains the sam e

as for (a)bu t the accuracy with which the step height ca n be determined plainly

increases, because of the increasing number of fluctuations for the eye to

average. Following the abov e contention tha t the noise erro r is independent of

T,,we assert th at , comp aring say ( b ) nd (d) (although V for

( b )

s less th an for

d)) , he increased potential fo r visual averaging in (d) reduces the final noise

error t o the same a s for (b ) .Similarly

(e)

should give the same final err or as c) .

Resolution

time

Fo r most measurements it is possible t o define a required 'resolution time' a nd

this is the period over which visual averaging is carried out. Consider, for

exam ple, the result of figure 1.10, He re the strain gauge of figure 1.1 has been

used t o plot the stress-strain curve for the mechanical comp onent. T he sample

com pone nt is placed in a testing machine which applies

a

controlled stress to

the sample. The stress is increased a t a co nsta nt rate (scanned). This is the

independent variable. Then the resulting strain (the dependent variable) is

plotted against the stress.

Here it is necessary to decide upon the number

of

independent points

needed t o define the stress-strain curve. This may be expressed as the nu m be r

npof poin ts o r the 'fractional resolution'

i

=

np- .

F o r the present rather simple

function,

np

will be fairly small. Fo r som ething more complex, such as a m ulti-

line optical spectrum,

np

would be a good deal larger.

F or the stress-strain curve example, T,,,

=

T,/n,. Th e eye averages over each

period of length T,, giving, for each time-resolved point, the noise error

IS =

T ; l l2 .

(1.14)

But

T,,= T,/np=XT,,

so

IS^

= ~(xT,) (1.15)

Since i is independent of

T,,

this means 6,

T,- 12,

and

so

the larger

T,,

the

better.


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


23/238

BASELINE SUBTRACTION

t

15

Figure

1.10 Output

U, of

the strain gauge in figure1.1 as the stress applied

to the sample is scanned at a constant rate.

1.4 BASELINE SUBTRACTION

Figure

1.11

shows an othe r method of dealing with offset. He re the example is

the stress-strain curve measurement of figure 1.10 but, instead of applying the

scanned stress immediately, zero stress is applied for the initial period G,

Figure 1.11 Baseline subtractio n: a) pplied stress,

( b )

train gauge out pu t

show ing offset

U,,,.




24/238


during which the baseline (offset)

uOb

s measured.

uob

is then subtracted from

each value of measured strain when the stress is applied.

T i

needs to be of non-zero length in ord er t o allow for noise. Th e noise err or

in the baseline measurement is equal to

kTi,- '.

Since uob is subtracted from

each time-resolved value

of

v o

for which the error is

kTe; l i ,

we require that

B T,,, if

the tot al noise err or is not to be significantly increased. No tice t ha t

the requirement is that

Ti

B T,,, not

B

T,.



25/238

2 Multiple time averaging and drift

In the overview ( 1.1)multiple time averaging MTA) was introduced as a means

of avoiding drift and so allowing the low white noise err or associated with a

long period of measurem ent t o be realised. This po int is covered in detail in the

present chapter, but first we consider a use of MTA where the object is merely to

reduce the white noise error.

Figure 2.1 shows the arrangem ent. Here the resistor bridge strain gauge of

figure 1 1is used to m on itor the oscillatory strain in a vibrating beam after the

beam has been struck by an im pacting device. If the experimen t is repeated a

num ber of times an d the results averaged using MTA, the white noise er ro r is

reduced a s the averaging time is increased.

This kind of measurement should be con trasted with one where there is an

independent variable and it is possible to vary the length of time T, used to

scan through the required range of the variable. An example of this kind of

measuremen t is the use oft he strain gauge to measure the stress-strain curve of

a samp le, as discussed in 1.3. Here the scanned independent variable is the

stress.

For such measurements, the easiest way to increase the noise averaging

time, an d thus reduce the white noise e rro r, is by increasing T,. Leaving

T

at

the original value a nd repeating the scan u p t o the full time available has

n o

advantag e over this. How ever,a s will be shown sho rtly,it is when drift (and

l/f

noise) are present that MTA can be used to advantage with the stress-strain

curve type of measu rement. Here, if a sh or t

T

is used an d MTA employed with a

repeated scan, the low drift and l/f noise associated with the short

T

are

combined with the low white noise error associated with a large averaging

period.


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


26/238

18

MULTIPLE TIME AVERAGING AND DRIFT

I 1

reco rde r

Fil ter

Figure

2.1

Use

of

a resistor bridge strain gauge to record the oscillatory

response of a vibrating beam to an impact.

2.1 MULTIPLE TIME AVERAGING METHODS

Overlay

averaging

There are two commonly occurring ways in which the results of a repeated

measuremen t ar e averaged. Th e first is shown in figure

2.2 a).

Here , each time

the beam is struck

a

new noisy transient is plotted

on

the paper

on

top

of

the

last, giving a series of ‘overlaid’ traces. These a re then averaged by eye to give

the reduced white noise error.

Figure 2.2 a) Overlay

of

three traces taken from a section

of

the output

transient from the system

of

figure

2.1.

( b ) Computer averaging of the

traces in

a).



27/238

DRIFT 19

Computer averaging

Th e second, an d m ore sophisticated, method of averaging is to use a com puter.

Here all of the values of u are input t o the compu ter, the traces are averaged by

the com puter an d the result is displayed. F o r the three comp onent traces of

figure 2.2(a), this would give the result of

(b ) .

Whichever method is used,if the num ber of traces averaged is n,, the am ount

of time available for noise ave raging is increased by the fac tor

n,.

F o r one of the

time-resolved values of strain

on

a single transient the white noise error

IT

s

given by

u n = k T e ; l i z ,

here Te, s the resolution time. Then for

n,

averaged

traces, the final erro r for one of the time-resolved po ints becomes

Thus, the averaging of n, traces gives an improvement of n;’/’.

2.2 DRIFT

Before considering how multiple time averaging can be used to reduce drift, we

first consider the subject more generally. Drift usually arises mainly from

changes in the am bien t temperature , draughts, etc. These influence the offset a t

the signal amplifier input. Th e system enclosure tends t o act as a low-pass filter

with a time cons tant of a minute o r

so

Th us the fluctuation time has this

value also.

Sloping baseline correction

Figure 2.3 shows an exam ple of such drift, a s seen at the ou tp ut of the strain

gauge. F o r an experiment such as the stress-strain curve measurement, if the

stress scan time T is somewha t less th an the drift fluctuation time a s shown ,

then the drift approximates reasonably well to a cons tant rate ram p over the

scan period.

When the drift rate is truly constant, it is possible to correct for the drift

exactly, using the method of figure 2.4. This

is

similar to the simple baseline

correc tion system of figure 1.11, but now the baseline is measured bo th before

Figure

2 3

Sample of drift of fluctu ation time T, scan time T, for

stress-strain curv e m easu rem ent.




28/238

20 M L L T I P L E T I M E AVERAGING AND DRIFT

Figure 2.4

Sloping baseline correction m etho d for drift

of

constant rate, as

applied to stress-strain curve measu reme nt. a ) Applied stress, ( b )

measured strain +drift,

(c)

reconstructe d baseline.

an d after the scan, in each case with t he applied stress zero. Fro m the m easured

values, the entire sloping baseline can be reconstructed. Then th e app ropr iat e

baseline value can be subtracted from each recorded value of measured strain

during the scan.

As before, the finite period G s needed for the baseline measurem ent. T his is

to reduce white noise error. In order for the noise err or in the baseline value to

be small compared with that in the measured strain, we again require that

q

,,, rather than & T .

In reality th e drift rate is no t constan t. Figure 2.5 shows the err or tha t results

in the stress-strain curve measurement for bo th simple and sloping baseline

Figure

2.5 Drift err ors rem aining after simple baseline correctio n

of

figure

1.1

1

an d sloping baseline correction

of

figure 2.4. a ) Simple baseline,

( b )

sloping baseline, c ) actual drift, d ) error for

a ) ,

e ) error for ( b ) .



29/238

DRIFT

21

correc tion. Clearly, provided

T, <

z, he second method still gives a reduced

drift error, although not now zero.

Drift reduction by multiple time averaging

It is clear that for either

of

the baseline correction m ethod s offigure

2.5

the drift

error can be reduced by reducing T,. With T x 4 as shown, the simple

correction gives a drift error approxim ately propo rtional to T, an d the sloping

baseline correc tion method gives a decrease with T, which is even mo re rapid.

Unfortunately, white noise er ro r is prop ortional to T,-

’ I 2 , so

reducing T, in

this way gives an increased noise error. This is avoided by repeating the scan

over and over in the time available and averaging the results. In this way a

large measurement period can be used to give the same low w hite noise err or as

for one long scan taking the entire period, but with the actual scan time

reduced to the degree necessary to mak e the drift err or negligible.

Here it is assumed that, when a short scan is repeated and the results

averaged, the final drift error is the sam e as for th e original short scan. It is

conceivable th at the effect of repetition an d averaging would be an increase in

drift erro r with the numb er

n

of scans averaged. Figure

2.6

shows tha t this is

not so. This shows the drift error fo r a series of scans, using simple baseline

correction. Here the overall period

n,T,

is somewhat less than the drift

fluctuation time

z.

When n,T,<

the drift rate is nearly con stant over the

overall period

n,T,,

an d then the averaged drift er ro r is the sam e as for on e of

the short component scans. When, on the other ha nd ,

n,T,

> but still with

T


30/238

22

MU LTIPLE TIME AVERAGING AND DRIFT

does increase with the num ber

n

of scans averaged a nd , for con stan t rate drift,

is proportional to n .

For some measurements, offset is of no importance. For example, for the

measurem ent of a signal step the offset ca n simply be ignored. F or this class of

measurement therefore the baseline correction is not strictly required. This

con trasts with the above stress-strain curve measurem ent where offset er ro r is

important. Here, for multiple time averaging to be effective, some type of

baseline correction is required.

2.3 MULTIPLE TIME AVERAGING BY OSCILLOSCOPE

It is im po rta nt t o realise, when predicting the noise er ro r of a measurement,

tha t the norm al oscilloscope display may incorp orate a high degree of overlay

multiple time averaging of the type shown in figure 2.2(a). Here a number

n,

of

traces a re overlaid, where

n

T JIT; an d where

I T

is the oscilloscope time-base

period and

T

is a ‘memory time’ compounded from the

CRT

screen

persistence, the persistence of the observer’s eye and the observer’s visual

memory. The noise error is thus reduced from the value kT,;”’ given by

equation (1.14) by the factor (T,JIT;)-”’ to give

n k TeST,,/IT;)’”.

(2.2)

Clearly , for very large values of n

T JIT;

this is somewhat d egraded, as the

eye becomes ‘saturated‘.

2.4 MULTIPLE TIME AVERAGING BY COMPUTER

Hardware and program

Figure 2.7 indicates how a com puter is used for multiple time averaging. Here

the signal averaged is tha t from the vibrating beam of figure 2.1. T he beam is

struck periodically using the impact transducer which is drawn by the pulse

generator output A ) .Here the pulse period 6 is made somewhat larger than

the decay time for the transient oscillators of the beam, which is shown in B ) .

Th e overall program MTAV for averaging an d display is summ arised in figure

2.8 and includes the procedures ADNT,

DOT

and

DISP

(figures 2.9-2.1

1).

Th e signal transient is sampled every T, seconds, as shown in B ) ,giving a

total of n s = 500 samples between each impact. Within th e com pu ter is a d ata

buffer of

n

locations which is initially cleared. This is the first step of

MTAV.

Then, following each impact, the n samples are added to the value stored a t

each successive location in the d at a buffer. F o r o ne transient this is detailed in

the procedure ADOT This is repeated for the required nu m be r n of transien ts as

indicated in ADNT, hich is the second step of MTAV.

Then the data buffer contents represent the sum of n ransients. Thus, to


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


31/238

W



32/238

24

MUL T IPL E

TIME AVERAGING

AND

DRIFT

Output contents

of

d n t a b u f f e ra t p oint er l o ca t i o n t o

digital-to-analogue converter

MTAV

Cle ar d a t a b u f f e r

ADNT Sum

nt

reco rded t rans i en ts in to the d a ta bu f fe r

Div ide each sample in the b uf fe r by n t

D l S P Display the b uf fe r conte nts on the oscil loscope

Figure

2.8

Structure

of

wAv-main program for multiple time averaging,

using the system of figure

2.7.

ADNT

Transient counter t

A DO T: Add o ne recorde d t ra n s ie n t i n t o t h e d a t a b u f f e r

Decrement t rans ien t counter; O ?

YES

Figure

2.9

Procedure ADNT sums

n

ecorded transients into the d ata buffer.

DlSP

I

ai t for short per iod to a l low osci lloscope scan t o

complete and th en for f lybac k

igure

2.10

Procedure DISP displays the contents

of

the data buffer

continuously on the oscilloscope screen.




33/238


34/238

26

M U L T I P L E T I M E A V ER A GI NG A N D D R I F T

adjustment an d a restart, the o per ato r is not m ade aw are of this until the full

sequence of n, transients has been accumulated.

One way to obtain continuous monitoring is to run the display routine

interleaved w ith the logging rou tine, on some so rt of interrupt basis. The n for

the present arrangement the eight most significant bits from the 16-bit data

buffer word are ou tput to the

DAC

for display. Th is has th e effect of dividing the

displayed value by 2', the presen t value of n,. Then, as the sequence of 2'

transien ts builds u p in the buffer, the displayed trans ient is seen to grow from

zero to the full range of the DAC. Because the signal amplitude increases directly

with the number of transients

n,

summed to da te, while the noise am plitude

increases with n:I2, the signal appears t o grow o ut of the less rapidly increasing

noise.

The above statement concerning the increase in noise should perhaps be

justified.

In

42.1

it is shown th at , when

n,

transients are averaged, the stan da rd

deviation for th e averaged white noise is reduced by

n [ l 2 .

Thus, for the sum

formed prior to averaging, the noise amplitude will be proportional to n: .

A further advantage of continuous m onitoring is tha t if, during

a

successful

run, it is observed th at an adequately noise-free signal has been ac cumulated

before the end of the sequence of n, transients, it is possible to save time by

stopping the logging sequence early. At this stage the usual display-only mod e

will be en tered, and it may then be necessary to increase the effective gain of the

display, because fewer than the usual number of transients will have been

accumulated. The simplest method is to output to the DAC, not the most

significant eight bits of the 16-bit da ta buffer word , but a suitable sequence of

bits of lower significance.

Continuous time averaging

With o r without continuo us display, the above method of time averaging still

has the property that averaging is carried out over a given, usually

predetermined, period and then has t o sto p, for fear of overflowing the da ta

buffer. In some instances a mode of averaging more akin to the running

average formed by a low-pass filter is more suitable. Here the displayed

transient would represent the average of the last n, transients t o be recorded.

This would have the dua l advantage of giving a continuously updated account

of the signal transient, and also of not overflowing the data buffer.

T he low-pass filter is no t suitable for multiple time averaging in its norm al

form, because the values tha t it averages are consecutive in time. F or multiple

time averaging, in con tras t, the averaged values are taken fram points widely

separated in time, with the values for any one data buffer location all taken

from different input transients.

If, however, a method can be found for making the computer perform the

basic function of the low-pass filter, then the inherent flexibility of the

computer should make it an easy matter to adapt the process to the

requirements of multiple time averaging.



35/238

MTA BY

COMPUTER

27

Figure 2.12 Simple low-pass filter to be simulated

by

computer.

We therefore sta rt by determining how the com puter can be made to carry

ou t the fu nction of the low-pass filter shown in figure 2.12. F o r the filter show n,

dudd t C and

i uin- ,) /R, so

that

(2.3)

where T,

CR.

Th us, if the in itial value of

U ,

is know n, the subsequen t values are determined

by the input function uinand equation (2.3).

F o r the compu ter to carry o ut this function, the simple C R circuit must be

replaced by the ADC, the computer, and a DAC. The ADC converts the analogue

inpu t voltage uin o digital form, the com puter executes the process of equation

(2.3) an d the DAC converts the result int o the ou tpu t voltage

U .

Suppose th at u n is digitised at regular brief intervals of length

A t .

Then, to

reflect th e process of equa tion (2.3), the dig ital num ber representing

U ,

must

increase at each interval by the amount

A V ,

given by

2.4)

(2.5)

du,/dt vi,,- ,)/T,

AUJAt vi,,-

, /T,.

U ,

+ U , + vin- , ) AtlT,.

This implies the algorithm

Here T, is the period over which both

C R

filter and comp uter average uin .

F o r the co mpu ter, this will equal

n, At,

where

n,

is the num ber of digitised input

values averaged. Thus the algorithm can be expressed in the form

uo o

+

uin- o)/nt*

(2.6)

Ad aptation t o multiple time averaging is simple; all that is necessary is to

replace the simple summing algorithm in the present program by that of

equa tion (2.6). Here, uin represents the value output by the ADC, and

U ,

the

current value of the corresponding word in the data buffer.

For the present arrangement the most suitable way of executing the

continuously averaging algorithm would be as follows. First the 8-bit input

representing

u n

is regarded as the

8

most significant bits of a 16-bit word , fo r

which the remaining

8

bits are zero. Then the stored 16-bit value representing

U ,

is sub tracted from this value. Next the division by n,,here 2*, is performed by

shifting the 16-bit differencedow n by eight bits. The n the divided difference is

added to the stored value for U , .

If this metho d is used, an d the eight most significant bits of the d at a bu ffer



36/238

28 MULTIPLE TIME AVERAGING AND DRIFT

are output to the DAC as before, then the initial growth of the displayed signal

will be the sam e as for the simple summ ing algorithm. H owever, when the 16-

bit numbers representing the values

U

of the stored transient begin to

ap pro ach the effective 16-bit num bers representing t he inp ut transient

u in ,

he

rate

of

growth decreases. The rise then becomes exponential and eventually

terminates when the magnitude of the store d transient becomes equal to that

of

the input transient. Thereafter the signal transient becomes largely stable,

reflecting only changes th at are sustained for

a

period comparable with the

time taken to record n, transients, as for example m ight occur if the am plitude

of

the signal transient was gradually increasing due to some deterioration in

the structure

of

the vibrating beam.

Throughout, the noise amplitude will remain at the reduced level

corresponding to the averaging of

n,

samples.



37/238

3

Phase sensitive detector methods

It has been stated previously th at the phase-sensitive de tecto r

PSD)

method is a

useful altemative to multiple time averaging

MTA)

for avoiding errors due t o

drift and

l/f

noise, and thus th at either can be used to prevent these effects from

frustrating the low white noise error obtainable by using a long period of

measurement. Th e principal object of the last cha pt er was t o show th at MTA

can be used to reject drift in this way. In this chapter the drift rejecting

properties of the

PSD

system are explained. The discussion of the ability of

either system to reject

l/f

noise is considered later, for the PSD in chap ter 7.

Th e present chapter en ds with a nu mber of miscellaneous app lications of the

PSD method.

3.1

ADAPTATION

OF

RESISTOR BRIDGE STRAIN GAUGE

The example used t o introduce the PSD method is the usu al on e of the resistor

bridge stra in gauge of figure 1.1, an d figure

3.1

shows this ad apted to the

PSD

method. H ere the DC supp ly to the bridge is replaced by th e AC supply eg and

trans form er shown in figure 3.1. Figures 3.1 ( b ) , c) and e ) hen show how the

strain

s

is converted to the AC signal U at the amplifier output.

The phase-sensitive detector then dem odu lates the signal to give the voltage

up

in figure 3. l( f) . Th e low-pass filter then extracts the slowly varying m ean,

which is displayed as

U .

Th e filter operates by forming the runnin g average of

up over the period T,of the filter and this requires th at T ,+ f ; , where

f,,

s the

signal frequency.

Notice tha t

U

K

s

the strain . W ere the simple full-wave rectifier of figure

3.2

to be used in place of the

PSD,

the o utp ut would be as in figure

3.1 9)

and

U

would become a distorted version of

s.


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


38/238

30

PHASE-SENSITIVE

DETECTOR METHODS

U )

Isolat ing

transformer Phase-sensitive

detector

I

-- - - - - - - -

Adjustable Switch

phase driver

--+=-

ine wave sh if te r

signal generator

(frequency fol

Figure 3 1 a )

The phase-sensitive detector method as applied to the

resistor bridge strain ga uge of figure

1.1:

system diagram.

The p hase shifter in figure 3.1 is needed t o ensu re that the switching of the

PSD is correctly phased , as in figure 3.1 d ) . This is to correct for phase erro rs in

the transformer, etc, which, in som e cases, are so small as to make the phase

shifter unnecessary.

It is actually unlikely that the

PSD

switch will be mechanical as shown,

altho ugh a relay is sometimes used. The use of som e sort of tran sistor switch is

much more likely.

3.2 AC COUPLED AMPLIFIER

There are two ways in which the

PSD

system rejects drift: one is tha t a n

AC

coup led amplifier can be used in place of the usua l

DC

coup led signal amplifier

and the other

is

that the PSD itself tends t o reject drift. In this section th e drift-

rejecting p roperties of the AC coupled amplifier are considered.

Figure 3.1 shows the AC coup led amplifier replacing the

DC

coup led amplifier

of figure 1.1. Figure 3.3 show s the essential features of the

A C

coup led am plifier.

Here th e single-stage transis tor amplifiers are sepa rated by simple high-pass

C R filter sections, and the arrangement is terminated with a unity-gain

isolator to drive the

PSD

The exact arrangement would now be a little ou t of

da te a nd a feedback amplifier section, such as th at in figure 3.3(b), would

probab ly replace each single stage

of

figure 3.3(a). However, the response is the

same, and for ou r present p urpose it is convenient t o analyse system a).

offs t

The first step is to show that the AC coupled amplifier rejects offset. One way of

showing this is to consider the response of one of the C R networks to a step


http://0.0.0.0/http://0.0.0.0/


39/238


40/238

32 P HAS E- SENS ITIVE DETEC TOR M ET HOD S

:i l te r

ow-pass

Figure

3.2

Full-wave rectifier

Single- stage

t rans istor ampl i f ie r

\

A C coupling

O P

amp

Figure 3.3 AC coupled amplifier circuits. a) Cascaded single-stage (time

constant =

C R )

ransisto r amplifiers, ( b ) eedback amplifier (gain =

R J R , ,

t ime constant =

C R , ) .

applied t o the input. Figure 3.4 shows the s tep

( b )

applied to th e network a )

giving the response c). Here the o utp ut responds initially in full to th e inpu t

change, but then decays to zero a s the capacitor C charges thro ug h the resistor

R . Thu s, after the initial transient, the response t o DC applied to the input is

zero.

Square wave signal response

It is important to show that the AC coupling networks d o not attenuate the

required signal. De pend ing up on the way in which the system is ad ap ted , this

can be a square wave or a sine wave, simply depending upon whether the

generator

eg

in figure 3.1 is a sq uare o r a sine wave generator. Figure 3 5 shows

the effect when a square-wave signal is applied to one of the C R sections of

figure 3.3.It is clear that, to avoid atten uati on , we require tha t T, To,where To

is the signal period and T, is the filter time constant C R .


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


41/238

AC COUPLED AMPLIFIER

33

t ”In

bl

Figure 3.4 Step response of A C coupling element in

figure 3.3 a). a )

A C

coupling network,

( b )

nput step,

(c)

output response

U,=

Vexp(

-t/q .

Figure

3.5

Response of A C coupling element in

figure 3.3 a)

to a square

wave of period

To

omewhat less than the filter response time T,. a) nput,

( b )

output.

Drift response

In this chap ter we shall consider mainly th e ability of the AC coupled amplifier

to reject drift of cons tant rate. Drift of varying rate requires th e spectra l view of

the next chapter and is considered later, in chapter

5.

To demonstrate the constant-rate drift response, we first consider the

response of one

C R

section to the input signal of figure

3.6 a).

This is a

constant-rate ramp star ting a bruptly a t t ime t

=O

Consider first the ‘steady-

state’ response at a time well afte r the initial transient. Here the o ut pu t U is

con stant. Thus, for the capacitor voltage u c , duJdt will eq ua l du,Jdt. But the

capacitor current i

=C

duJdt,

so

i =

C

dui dt. Since U

=

iR, one can write

U = C R

dui,Jdt. But du,Jdt has the con sta nt value

V’ s

that

U

has the

constant value CRV’, as shown.




42/238

34 PHASE-SENSITIVE DETECTOR

METHODS

,

/

Figure

3.6 Response of

AC

coupling element

of

figure

3.3 a)

to a ramp

input .

a)

Input , b ) output .

It ap pe ars then th at a single high-pass filter section does not entirely reject

drift. The drift is converted to a steady output. This, of course, is easily

eliminated by add ing an ot he r high-pass filter section. This is why a typical

A C

coupled amplifier will have a t least two high-pass section s; the arrangem ent of

figure

3.3 a)

has three.

3.3

DRIm

REJECTION OF PSD

Although the

AC

coupled amplifier commonly used with the

PSD

system has

drift rejecting properties, as explained above, the use of the AC coupled

amplifier is not actually fundamental to the technique, because the

PSD

itself

tends to reject drift, and offset, an d indeed l/f noise. It is the pu rpose of this

section to show how the PSD rejects offset an d drift.

Figure 3.7 illustrates the ab ove point. H ere U represents drift and offset at

the amplifier ou tp ut , i.e. the PSD input. This slowly varying signal is converted

to a square wave

c)

with a corresponding slow variation in am plitude. The n,

because the frequency of the square wave is the signal frequency

fo,

and

because

fo f,,

where

f ,

is the cut-off frequency of the low-pass filter that

follows the PSD, the filter virtually rejects the square wave, leaving only the

small residual component d ) .

PSD imperfections

Th ere are two kinds of imperfection in a real PSD circuit which imp air its ability

to reject drift a nd offset, an d therefore mak e it desirable t o retain the AC

coupled amplifier. The first is th at the magnitudes of the g;in in the forward

an d reversed states of the switch may n ot be exactly equal. This causes

a

small

DC component at the PSD output which, unlike the output square wave

com ponen t resulting from the offset, will be accepted by the low-pass filter.

Then, naturally, if the input offset component drifts, so also will the small DC

component at the output.

Th e oth er potential imperfection of the real PSD circuit is th at , in general, the


http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/http://0.0.0.0/


43/238

WHITE NOISE ERROR 35

Figure

3.7

Waveforms showing how the PSD circuit of figure

3.1

avoids the

effects of drift and offset. a) Drift an d offset a t amplifier ou tpu t, ( b )

PSD

switch states, c )

PSD

output up d ) ow-pass filter output uo .

periods for the forward a nd reversed states of the switch will no t be exactly the

same. This will have essentially the same effect as when the gains fo r the tw o

states are unequal.

3 4 WHITE NOISE ERROR

It has now been shown that when a system is adapted to phase-sensitive

detection, drift is removed. It remains to be shown t ha t the adap tation gives no

increase in white noise error.

Figure 3.8 allows this point to be dem onstrated.

a )

how s the essentials of

the resistor bridge strain gauge of figure

1.1

and

( b )

hows an adaptat ion to

phase-sensitive detection. Here the sine-wave generator eg of figure 3.1 is

replaced by the DC source E and the reversing switch of figure 3.8 b). This

constitutes a square-wave generator a nd allows a fair comparison of a ) and

( b ) ,

because the same primary power source

E

is used in both cases.

The first point t o note is that the signal outp ut for the two systems, U in a )

and u p in ( b )are the sam e. This is because the two reversing switches in ( b )

cancel each other. The w aveforms in c)- f) confirm the po int. In c)there is an

output com ponent

uaa.

This is the signal com ponen t resulting from the strain

which unbalances the resistor bridge.

( (f)

then show tha t the PSD system

output

u p

is the same, i.e. up

= U

Waveforms 9) to j ) now show the result of adding noise. Here the

fluctuation time is determined by the response time of the signal amplifier. As

before, the signal components U are the same at each output. The noise,




44/238

36 PHASE-SENSITIVE

DETECTOR

METHODS

I

I

I

€ g a y

Amplif ier

I

I

I

I

Figure

3.8

Diagram showing how adapting the resistor bridge strain gauge

to the PSD method leaves the system response to both signal and white

noise unaltered.

a)

Simple resistor bridge strain gauge, ( b )adaptation to

PSD. c )and (g), amplifier output for a); d ) and

( h ) ,

amplifier output for

b), e) , i) PS D switch states, ( f ) , j ) PSD output.

however, differs slightly. T hat at the

PSD

ou tp ut is subject t o periodic reversals,

due to the

PSD

reversing sw itch. Fo r either system th e o ut pu t noise is passed

thr ou gh the final low-pass filter of time constan t T,(not show n). It is fairly clear

th at the noise am plitud e after this final filtering will not be influenced by these

periodic reversals, and so the final noise amplitude is the same for both

systems. This , in fact, is so an d is argued more rigorously from the frequency-

domain viewpoint in chapter

7 .



45/238

MISCELLANEOUS APPLICATIONS

37

It has now been established that both multiple time averaging and the

phase-sensitive detector method are effective in removing drift and offset

without increasing white noise error. Th us either can be used, in principle, to

allow the low white noise er ro r associated with

a

large measu rement time t o be

realised. In practice, however, the choice between the two m ay be limited. F o r

example, to use

MTA

to reduce drift when measuring a stress-strain curve

requires making the stress scan time

T,

small. In practice

a

sufficiently low

value of

T,

may not be possible.

3 5 MISCELLANEOUS APPLICATIONS

Th e arrangem ent of figure

3.1

is no t th e total solution to drift an d offset for the

resistor bridge strain gauge. While amplifier drift and offset are eliminated,

transducer drift and offset remain: if the bridge is initially unbalanced this

constitutes transducer offset; an d when the imbalance changes-say du e t o

therm al changes in the relative values of the bridge resistors-this cons titutes

transducer drift.

A possible,

if

far-fetched, solution can be found for the stress-strain curve

measu rement. Here the original

DC

supply is used to drive the resistor bridge,

instead of the

A C

supply of figure 3.1. Also, instead of scanning the applied

stress uniformly as in figure 3.9 a), the stepped scan of

( b )

is used. This

produces an A C signal compone nt of the frequency

o

shown, which is then

processed by the

AC

coupled amplifier and PSD in the usual way. For this

arrangem ent, in con tras t with th at of figure

3.1,

the transducer drift an d offset

is not modulated to produce an

AC

signal component, and thus is rejected.

a ) Normal

scan

\

' b f p 4

Figure 3.9 Stress modulation to avoid transducer offset and drift in

stress-strain curv e me asure men t.

a )

Norm al scan ,

( b )

modulated scan.

F o r this particular application, the idea is impractical, because th e inertia of

the testing machine would severely restrict the value of fo. How ever, the idea is

sound in principle a nd serves t o illustrate just on e of the grea t variety

of

ways

in which the

PSD

method can be refined to allow rejection of various kinds of

unwanted response.




46/238

38 PHASE-SENSITIVE DETECTOR METHODS

It is not really possible to lay dow n g eneral principles beyond this point.

Each system w ill present its own problems, an d it is a m att er for the ingenuity

of the experimenter t o devise a m ethod which will mo dulate the variable tha t it

is wished to measure, while leaving un mo dulated all unw anted effects. Th us we

finish by giving a few rather more practical examples of the kind of strategy

that is adopted.

Choppers

Figure 3.10 shows on e of the simplest metho ds for converting a DC signal into

an AC signal suitable for AC coupled amplification and phase-sensitive

detection. Here the vibrating switch contacts chop th e signal to produc e a

square wave of amplitude equal to the initial slowly varying

DC

voltage.

0

Chopped

output

S ~ O W I ~

arying

signal source

supply

Figure 3.10 Relay-ope rated cho ppin g circuit.

a )

Circuit, ( b )signals, (c) switch states

0

op en, c =closed).

Disadvantages of the arrangement are the obvious problems of wear

associated with a n electromechanical system, an d also th e thermoelectric EM F

th at is generated across the con tacts, even in the absence of a signal.

Both

UT 03 TMI Signal Recovery

Documents

Transcript of UT 03 TMI Signal Recovery