AN ABSTRACT OF THE THESIS OF
ROBERT JAMES BERTORELLO for the
(Name)
Electrical and
in Electronics Engineering presented
MASTER OF SCIENCE
(Degree)
on
(Major)
November 17, 1967
(Date)
Title: A ZERO - CROSSING ANALYZER FOR DISTRIBUTION -FREE
DETECTION OF A SIGNAL IN NOISE
Abstract approved:
_
Leonard ./ Weber
This thesis discusses the analysis, design, and experimental
evaluation of an instrument that can be used to detect the presence or
absence of a signal, not necessarily known, in a noisy background.
The detection principle is based on application of the sign
test of
distribution -free statistics to the stochastic process defined by the
zero -crossing intervals of a signal or signal plus noise process. It
is shown that the detector is distribution free in the sense that the
false -alarm probability can be evaluated with only a limited knowledge of the statistics of the underlying noise process.
A
theoretical discussion of the detection principle and false
alarm probability analysis is presented in conjunction with design
considerations of the circuitry used to implement the zero -crossing
analyzer technique. Results of an experimental evaluation with
narrow -band noise are presented along with a complete schematic
diagram of the analyzer. For a noise filter center frequency of 10.
kHz and with the signal frequency removed from the
3
filter center
frequency by at least 300 Hz, reliable detection can generally be obtained with a signal to noise power ratio of -8 dB.
A
Zero -Crossing Analyzer for Distribution -Free
Detection of a Signal In Noise
by
Robert James Bertorello
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
June 1968
APPROVED:
of Electrical and
Prsor
Electronics
Associate
$gineering
in charge of major
'Head of Department of`Électrical and
Electronics Engineering
Y
Dean of Graduate School
Date thesis is presented
Typed by Clover Redfern for
November
17,
1967
Robert James Bertorello
ACKNOWLEDGMENT
The author would like to acknowledge the timely
assistance and guidance that has been provided by
Associate Professor Leonard J. Weber throughout the
course of this thesis.
I would also like to thank Donald C. Amort for the
components and equipment that he supplied.
-
TABLE OF CONTENTS
Page
I.
'.'.
INTRODUCTION
1
Problem
Statement
Purpose of Study
of
II.
-
3
DETECTION PRINCIPLES
Basic Detection Problem
Distribution -Free Detector Concepts
Sign Test for Zero- Crossings
Zero- Crossing Detector False Alarm Probability
III.
ANALYZER
THE ZERO -
Principles of Operation
Key Circuits
Evaluation of the Analyzer
IV.
V.
1
TEST RESULTS
-
4
4
6
7
16
26
26
31
35
39
39
Summary of Test Parameters
Summary of Test Results
42
SUMMARY AND CONCLUSIONS
50
BIBLIOGRAPHY
APPENDIX
Appendix
Appendix
Appendix
Appendix
Appendix
I
II
III
IV
V
-
52
56
56
59
63
65
69
LIST OF FIGURES
Page
Figure
1.
Block diagram of generalized detection problem.
2.
Zero -crossings of a sinusoidal waveform where
y(t) = A sin (
11
Cumulative distribution function of the time T between
successive zero -crossings for a sinusoidal waveform.
11
(a) A possible realization from a noise process.
(b) Reduced process obtained from hard limiting.
12
Illustration of some hypothetical cumulative probability
distribution functions of the interval between zero crossings.
13
3.
4.
5.
6.
-t
Method of constructing the
S = C(T).
test statistic
S
4
where
15
Probability of false alarm versus distance between
test statistic and median.
25
8.
Block diagram of the zero -crossing analyzer.
27
9.
Timing diagram of signals in zero -crossing analyzer.
28
10,
Relative frequency response characteristics of single
pole filter used to obtain narrow -band noise process.
38
Variation of normalized statistic with signal frequency
and power for "less than- alternate" subset.
43
Variation of normalized statistic with signal frequency
and power for "greater than -alternate" subset.
44
Variation of normalized statistic with signal frequency
and power for "less than - every" subset.
45
Variation of normalized statistic with signal frequency
and power for "greater than- every" subset.
46
7.
11.
12.
13.
14.
Page
Figure
15.
16.
17.
Photograph
of
limiter output.
typical rise and fall times at the
66
Photograph of key waveforms in the zero -crossing
analyzer.
67
Single trace pictures of typical signal, noise, and
signal plus noise inputs to the analyzer.
68
:
LIST OF TABLES
Page
Table
1.
Summary of test parameters and results.
41
A
ZERO - CROSSING ANALYZER FOR DISTRIBUTION -FREE
DETECTION OF A SIGNAL IN NOISE
I.
.
INTRODUCTION
Statement of Problem
The problem of determining the presence or absence of a given
signal in a noisy background is one that arises in many different
fields. Typical examples can be found in communication systems,
radar, oceanography, learning theory, and analysis of various types
of data.
When performing the determination, the observer is pre-
sented with a mixture of signal and noise, and the task becomes one
of deciding if the
mixture contains the signal, or is composed only of
noise. Numerous procedures have been developed for making these
decisions, and except for the singular case of no noise, the possibility exists for making an error, in which case a signal might be de-
clared present, when it is absent, or vice versa. In communication
and radar terminology, the probability of declaring a signal present
when in actuality it is absent is called a false alarm, and the proba-
bility of declaring the signal present when it actually is present is
called the detection probability. In formulating an optimum structure
for making these decisions, a strategy is chosen such that in a series
of observations, the decisions are made with the
of
success. Determination
of
greatest possibility
the optimum decision structure for the
2
case of complete a priori knowledge of the statistics of the underlying
signal and noise processes has been extensively studied and numerous discussions are presented in the literature (Helstrom, 1960;
Davenport and Root, 1958; Middleton, 1960).
An item of
particular importance in realization of
a detection
device concerns the performance of the optimum device under a de-
parture of the noise statistics from the values assumed for structuring the optimum detector. An example of this might be the case
where a detector was designed to yield optimum performance for a
communication channel with Gaussian noise, but the channel is sub-
jected to impulsive disturbances. In this case, the performance of
the optimum detector might be inferior when compared to a detector
which is sub - optimum for conditions of known statistics.
This leads
to an alternate approach to the problem, wherein the decision struc-
ture is formulated with a minimum knowledge of the statistics of the
noise process. The technique utilizes the non -parametric or distri-
bution-free branch of mathematical statistics, which has appeared in
statistical literature for several years (Fraser,
1957; Kendall, 1961),
but seems to have significantly appeared in engineering literature
only in the past few years (Wolf, Thomas and Williams, 1962; Carlyle
and Thomas, 1964).
The generality of the distribution -free method
is such that useful tests of hypotheses can be made with a very limited knowledge of the noise and signal distributions.
For example, the
3
only restriction might be that the noise and signal plus noise distribu-
tions are continuous. A method of applying the distribution -free sign
test to signal detection theory is the main problem investigated in this
study.
Purpose of Study
In this study, a method is proposed for implementing the sign
test
distribution -free statistics in the analysis and design of a sig-
of
nal detection device.
The stochastic process representing the signal
plus noise is subjected to a non - linear transformation so that the in-
formation bearing element of the reduced process retains a one-toone correspondence with the zero level
crossings of the original sig-
nal plus noise process. The hypothesis proposed states that the in-
tervals between a set of zero crossings taken over some observation
period will have a given median or reference value in the case of
noise alone, but in the case of certain signals present in the noise, a
change in the interval lengths will occur and this change can be de..
tected by application of the sign test. Under this hypothesis, and for
noise processes, a detector is constructed which
is distribution -free in the sense of establishing a false alarm probaa
restricted class
of
bility without detailed knowledge of the noise process.
The primary
purpose of this study is development of the analysis and design of a
device suitable for implementing the sign test in distribution -free sig-
nal detection.
4
II.
DETECTION PRINCIPLES
Basic Detection Problem
The basic scheme that is peculiar to most detection problems
and the one of interest in this study is depicted in Figure 1.
transmitter generates a set
The
of signals which could be a simple on -off
signal or a complicated sequence; however, only the binary case will
be considered in this study.
Noise n(t)
s(t)
Transmitter
Channel
r(t)
>
Receiver
front end
v(t)
Detector
Decision
element
I>
D
Threshold
Figure
1.
n(t)
s(t)
+
So
Block diagram of generalized detection problem.
n(t)
5
transmitter output s(t) is fed into a channel which might
The
pair of wires or some other medium such as interplanetary
be a
If
s(t)
r(t)
is not subjected to a disturbance in the channel, then
would uniquely represent
s(t)
space.:
(within the constraints of arbitrary
attenuation and time delays introduced by the channel) and the re-
ceiver output y(t) would retain
a one -to -one
s(t). In the real world, however, noise
n(t)
correspondence with
is added to
as
s(t)
the signal passes through the channel and the task of the decision device becomes considerably more complicated since the noise can
mask the signal such that the decision device has the possibility of
making two types of errors. It can declare the signal present when it
is actually absent and this is called a Type I
error or a false alarm.
The decision device can also declare the signal absent when it actual-
ly is present, which is called a Type II
error or a false dismissal.
A
method of characterizing the false alarm probability for a particular
detector structure is of particular interest in this thesis.
Assume that samples of y(t)
...n
where
ti +1
-
ti
3.
=
At,
and
are taken at times
ti, i
A is chosen such that
=
1, 2,
y(ti)
are independent samples. During the time that these samples are
taken, it is also assumed that the signal is either on or off.
are interested in testing the hypothesis
K
where
H
Then we
against the alternative
6
and the noise
n(t)
H:
y(t)
=
n(t)
K:
y(t)
=
s(t)
and signal
s(t)
+
n(t)
are statistically independent.
The decision device accepts the detector output and calculates a
statistic
K
which is compared to a fixed threshold
S
is chosen and if
o
So.
o
If
S> S o
is chosen and the signal is declared
Formulation of the test statistic and selection of a
to be absent.
threshold
S < So, H
.
test
So
0
for optimum detection in the Neyman- Pearson sense
requires a knowledge of the test statistic distribution function, such
that the false alarm probability is set at some acceptable level and
the detection probability is maximized (Helstrom, 1960; Middleton,
1960).
This implies that knowledge of the distribution function of the
noise process is required to construct the optimum detector which
also implies that its performance may be largely unknown for depar-
tures from the specified noise statistics.
Distribution -Free Detector Concepts
The type of detector considered in this study is distribution -
free in the sense that the false alarm probability can be determined
with only a limited knowledge of the noise process statistics. The es-
sence of the technique is selection of a threshold and test statistic for
no- signal conditions such that the probability distribution of the test
7
statistic is known and independent of the statistics of the detection
problem. Several distribution -free tests are suitable candidates and
possible applications have been discussed in the literature (Carlyle
and Thomas, 1964; Daly and Rushforth, 1965; Hancock and Lainiotis,
1965).
Formulation
of
the tests is generally based on reduced data
obtained from transformation of the observations, such as ranking
according to polarity or magnitude, and in some cases, the ranked
variables are subjected to correlation techniques. The tests are
primarily concerned with a measure of location of the underlying distribution rather than properties concerning the shape of the distributions, and a fundamental requirement of the technique is a data samOne of the
ple obtained from noise alone.
simplest tests is the sign
test which is the one chosen for application to the problem in this
study.
Sign Test for Zero -Crossings
The sign
test generates a test statistic
by comparison of the
sample values with a reference value (Kendall and Stuart, 1961; Wilks,
1962) and observing if the sample value is
the reference value.
The
greater than or less than
resultant information is represented by the
sign of the comparison, hence the test is called the sign test.
...
example, let x., i
=
dom variable
and let
X,
1, 2,
,
n
xR
represent observed values of
For
a
ran-
denote the reference or comparison
8
value. A test statistic
served values
can be generated by subjecting the ob-
S
to the operation of Equation 1,
x.
(1)
) U(xi-xR)
S =
i=1
where the function
U(.)
is defined by the following relationship:
U(a)
=
if a>
1
ifa<0
=0
=
0
undefined if a
= 0
is called a tie, in which case, the sign test
The event that
xi
fails since
is neither greater than nor less than xR For, the
xi
=
xR
class of continuous probability distribution functions considered in
this analysis, the event of a tie has zero probability, therefore,
U(0)
is not defined and ties are excluded from consideration.
Construction of the sign test is very simple since only those
sample values which are greater than a reference value (or less than
for that matter) are considered in the test statistic. For the assump-
tion that a random variable
X
ous distribution functions, the
is a member of the class of continupth quantile is defined by
FX(xp)
=
p
(2)
9
FX() is the cumulative distribution function (c. d. f.
where
the random variable
which
100p
X.
Therefore,
x
is the value
P
x
of
)
below
percent of the distribution lies. From the basic pro-
:.
.
perties of cumulative distribution functions,
o < p <
1
and
x
locates a particular point on the distribution curve. Since the sample
values
are assumed to be statistically independent, setting xR
xi
equal to
in Equation
x
1
yields a statistic
S
which is binominal-
P
ly distributed with the number of trials equal to
of success
for each trial.
p
The
n
and probability
particular value of
p
=
1/2
corresponds to the median value of the distribution and this is the
case of primary interest in this study. Pertinent details of applying
the sign test are presented in a subsequent section, after a discussion
of the zero -crossing
principle.
The information bearing elements of
interest in this study are
the level crossings of the stochastic process at the output terminals
of the
receiver front end, and the particular level of interest is the
zero level. If there were no noise in the first stages of the receiver
and the channel was not subject to disturbance, a realization of
over an observation interval
with period
P
T
y(t)
might be a sinusoidal waveform
as depicted in Figure 2. For the case shown, the
number of zero crossings is constant. If the duration of the interval
between crossings is denoted by
usoidal waveform case,
T,
=
T.
,i
=
1, 2,
... N,
then for the sin-
P/2 for all i in the observation
10
interval
T,
where
T
=
t2
-
pared with some reference value,
do not occur, a sign
Each value of T.
t1.
test could
:
be
TR
could be com-
say, and assuming that ties
formulated for the process defined
by the distance between continuous zero- crossings. The important
point is that for a periodic waveform, the spacing of the zero- crossings is well defined and if
Ti
is considered to be a random variable,
it is described by a causal distribution function, which is depicted in
Figure
A
3.
more practical example
of
zero -crossing behavior is depict-
ed in Figure 4a, which might represent a realization from some
noise process, or perhaps a signal plus noise process.
Because of the inherent nature of the process, the length T.
of the
interval between successive zero -crossings is a random vari-
able that is described by a c. d. f, which can be denoted by
In general,
FT().
theoretical determination of this distribution function is
largely unsolved. The earliest study of the problem was undertaken
by Rice (1944, 1945) and in his classical paper he formulated an ex-
pression for the mean number of zero- crossings per second and developed a limited solution for the distribution of intervals between
adjacent zero crossings. An extension of the problem is given by
Bendat (1958), and considerable theoretical work (McFadden, 1956,
1958; Ylvisaker, 1965) and experimental work (Blotekjaer, 1958;
Rainai, 1962) has been devoted to this problem.
Figure
5
is an
11
y(t)
A
t
T
tl
Figure
2.
Figure
t2
Zero -crossings of a sinusoidal waveform where
t + cp).
y(t) = A sin
(p
3.
Cumulative distribution function of the time T
between successive zero -crossings for a sinusoidal waveform.
12
y(t)
(a)
z(t)
(b)
,. T --. T 2
-
T
N-1
possible realization from a noise process.
(b) Reduced process obtained from hard limiting.
Figure 4. (a)
A
.
t
13
FT(a)
1.0
0
a.
0
Figure
5.
Illustration of some hypothetical cumulative
probability distribution functions of the interval
between zero -crossings.
14
illustration of some possible cumulative distribution functions for the
intervals between zero crossings for the case of noise only. The
variation in form of the curves would be primarily caused by filtering
and spectral characteristics of the class of noise distributions con-
sidered. The key point to be made is that application of the random
variable
T
in a parametric
test would be very difficult, if at all
possible, because of the lack of a complete description of the probability distribution function of T. However, this would appear to be
an ideal application of the distribution -free concept.
An intuitive discussion of the zero -crossing behavior of signal
only and noise only has been presented, but the sign test concept of
signal detection depends upon the characteristics of signal plus noise.
An exact
description of the random variable
T
in the case of a
mixture of signal and noise does not appear to exist. The hypothesis
proposed in this study states that a given quantile value
distribution of
compared to
for signal plus noise. More explicitly, let
present the signal to noise ratio, and
of
T
of the
will be different for the case of signal only, when
T
T
p
F,r(; p)
as a function of the parameter
p.
represent the
re-
c. d. f.
Then, the hypothesis is
given by
H:
FT(TR; 0)
=
p
(noise only)
K:
FT(TR;
/
p
(signal plus noise)
and the alternative,
p)
p
15
where
is a quantile value of the distribution of noise only.
p
This suggests that the signal can be detected by first determining the value of a given quantile in the case where it is known that no
signal is present, then sensing a possible shift in the quantile which
would correspond to the presence of a signal. In the case where
FT(TR;
Cl)
=
FT(/#11; p),
p >
0,
the test fails which implies that cau-
tion must be used in a given application.
The
structure of the detec-
tor considered in this study is based on establishing an observation
interval and forming a counting function by counting the number of in-
tervals whose duration is less than the quantile value chosen as
reference. Principles of the scheme are depicted in Figure
quantile
Limite r
z(t) >
Interval
length
comparator
Figure
6.
6.
subset
selection
reference
y(t)
a
Gated
Counter
Method of constructing the test statistic
S = C(T).
S
c(t)
where
The output from the limiter is a reduced representation of the
input process
y(t),
such that
z(t)
retains the zero crossing infor-
mation contained in y(t). For reasons explained in the next section,
16
alternate intervals of z(t) are analyzed by the interval length com-
parator if statistically independent samples are desired, and if the
interval is less than the reference value, the counter is incremented
by one count. If the
ith interval length is greater than the refer-
ence value, the counter will remain in its present state. In this man-
ner a counting process
simply
C(t)
is generated and the test statistic is
the state of the counter at the end of the observation
C(T),
interval. Therefore, the test statistic,
is the sum of the num-
S,
ber of successes in applying the sign test in a sequence of events
taken over an observation interval
S
T.
with a reference or threshold level
By comparing the value of
the decision element of
S ,
o
the generalized detection scheme of Figure
1
is satisfied and the
structure of a distribution -free detector can be established. Specific
characteristics
of the counting function and
detector structure are
presented in the next section.
Zero -Crossing Detector False Alarm Probability
Evaluation of the detector false alarm probability is identical to
determination
of
the test statistic distribution function for the case of
having an input process consisting of noise only.
Several key assump-
tions have been made which appear to be reasonable from an engineering viewpoint.
To simplify the
discussion, these are listed below and
further clarified in the ensuing text.
17
Key assumptions:
1.
The reference quantile can be determined either theoreti-
cally or experimentally.
2.
The noise process is stationary.
3.
The distribution of
T, FT(.; 0)
is continuous and ties do
not occur.
result from independent observations.
4.
The samples
5.
The noise process is
restricted to
tribution of N(T, 0),
unit time
T,
a
class such that the dis-
the number of zero -crossings per
can be approximated by a normal distribu-
tion.
Assumption
1
does not appear to be particularly restrictive
since in most practical cases of interest, the quantile would be unknown, but could be readily estimated by experimental methods as
was the case in this study. A possible alternative would be construc-
tion of a device which implements a learning mechanism that learns
..
the quantile of interest, in which case assumption
2
could be relaxed
for certain degrees of process nonstationarity (Groginsky, Wilson,
and Middleton, 1966).
For cases of practical interest, assumption '3 appears to be
reasonable since most noise processes encountered in the real world
are constrained by finite bandwidths which implies that the distributions of
T
is continuous and ties are excluded.
18
The assumption that the samples are independent is a necessary
condition for readily constructing the probability distribution function
of the
test statistic. Declaration
of an independent sampling scheme
would imply some knowledge of the underlying process which would
seem to contradict the distribution -free hypothesis. In an attempt to
insure that the samples are reasonably independent, the zerocrossing analyzer operates on a subset of the set of intervals
by adjacent zero -crossings.
produced
Although this is not a strict condition
for independent samples, it should be acceptable in view of a lack of
definitive information of this nature.
By
rejecting adjacent zero-
crossings it would appear that considerable information is being discarded and that an improvement in detection efficiency could be obtained if dependent sampling were allowed. The performance of a
nonparametric detector with dependent sampling has been analyzed
by Armstrong (1966), and his
results indicate that correlation be-
tween samples does improve the relative efficiency.
Assumption
5
is probably the most significant one, and also the
-
most difficult one to justify. The source of the difficulty lies in the
problem of determining the sample size. Consider the case of fixed
observation time, which implies that in Figure
a fixed value.
4, the
interval
T
is
Excluding for the moment the problem of synchroniz-
ing the beginning and ending of the sampling interval, the number of
zero -crossings per time
T, No(T, 0),
is a random variable, so
19
that consideration of the set of random variables defined by
Ti, i
=
1, 2,
...No-1,
of random variables.
is equivalent to considering a random number
In fact, if the beginning and end of the observa-
tion interval is synchronized to zero - crossings,
T
is itself a ran-
dom variable. In cases that have known distribution functions for
and
T
No,
T
the distribution function of certain linear combinations of
can be developed (Feller, 1966; Parzen, 1962; Robbins, 1948).
For the problem of interest in this study,
FT(.)
and
() are
FN
0
not readily available in exact form.
the problem of determining
Helstrom (1957) has considered
() for a Gaussian process,
FN
but the
0
results appear to be of questionable value for practical applications.
An experimental investigation was conducted by White (1958), how-
ever, his investigation was limited to less than five crossings per
unit time, which is much lower than the number of interest in this
study.
Tikhonov and Kulikov (1962) performed an experimental study
of the distribution of sample functions of noise according to the num-
ber of overshoots where the noise samples were taken at the second
detector output of a radio receiver. An overshoot is defined as a
positive overcrossing of some reference level. Results are not presented in their work for the case of zero -crossings, but from the
trends, it would appear that as the reference level tends to zero, the
envelope of the distribution tends toward one which could be approxi-
mated by the normal probability law.
20
Another possible approach would be centered around determina-
tion of the limiting distribution of the sum of the intervals of adjacent
zero -crossings. This would involve an investigation of the limiting
distribution of sums of dependent variables where the distribution of
the summands is available in approximating form and although this is
an interesting concept, this approach appears to lie outside the scope
of this thesis.
From an intuitive consideration
of the
limiting properties of re-
newal counting functions (Parzen, 1962), and from some results of
Rainai (1966), it seems reasonable to assume that the probability law
of
No(T, 0),
the number of zero-crossings in the time interval
T,
can be approximated by the normal probability law. In reality,
No(T, 0)
must be an integer valued random variable which would be
described by a probability mass function, whereas the normal density
function belongs to the class of continuous distribution functions (Parzen, 1960).
of
No(T, 0)
The absence of definitive information on the distribution
for large
T
is indicative of the need for further the-
oretical and experimental analysis of random processes, particularly
with respect to engineering applications. If it is felt that the assump-
tion of normality requires further verification, the procedure for
learning the median or quantile could be extended to learn the emper-
ical cumulative distribution function, the results of which could be
used in evaluation of the false alarm probability, since the procedure
.
and variance,
cr
2,
5
of the
normal distribution can
21
will be used, which implies that
is of sufficient generality that non -normal densities could be consid-
µ
ered. For this study, assumption
the mean
either be determined theoretically or estimated experimentally.
With fulfillment of the key assumptions, the tools are now at
hand to formulate the false alarm probability of the zero -crossing
analyzer. For independent sampling and minimization of analyzer
complexity, only those intervals corresponding to negative undershoots are considered in the analysis. In Figure 4 the odd numbered
intervals would form the basic set subjected to a sign test. The reason for selection of positive or negative overshoots is arbitrary and
the negative ones were chosen to simplify the analyzer design. Addi-
to be
tionally, the analyzer design allows the set of intervals defined by aletc,
This provision allows for limited investigation
ternate undershoots, which would be T1' T5' T9'
selected as samples.
of the relationship between the counting function and the degree of
sampling dependence.
The principle used to determine the false alarm probability
probability. Details of the analysis are
Pfa is based on conditional
given in Appendix II, but the spirit of the technique will be presented
here.
Recalling that the detection device counts the number of events
that meet a specified criterion, and then compares the count with a
22
threshold to test for the presence or absence of signal, the basic
problem is determination of the probability that
m
out of
in-
n
tervals have a length which is less than a reference value. This is
precisely the binomal problem wherein the probability
n
of
suc-
cesses in m trials is sought, and where the probability of success
at each trial is defined and is a constant. If the sample size, which
in this case is the number of negative undershoots in the observation
interval
T,
were a fixed number, then the false alarm probability
would be determined from a straight forward application of the bi-
nominal probability law. In the application considered here, the
problem is complicated by the fact that the number of events, or
equivalently the number of negative undershoots, is a random variable.
Let
Nua(T, 0)
represent the number
in the observation interval
T
of
alternate undershoots
for noise only, and let
Pfac
repre-
sent the probability of false alarm for the condition that Nua(T, 0)=n.
The random variable
Nua(T, 0)
is simply the size of the subset of
alternate undershoots taken over the interval T,
bitrary value such that
0 <
n<
co.
Then let
and
S<a(T, 0)
<a
n
is an ar-
denote the
number of alternate undershoots whose duration was less than the re-
ference quantile TR.
The conditional false
alarm probability can be
formulated more explicitly by the following equation:
23
Pfac
where
So(T)
=
P{S<a(T, 0)
So(T)\Nua(T, 0)
<
=
n]
(3)
is the test statistic threshold value.
Determination of the unconditional probability of false alarm is
then obtained by averaging
can take on, so that
Nua(T, 0)
over the possible values that
Pfac
Pfa
is given as follows:
co
{P[S<a(T' O),< So(T)\Nua(T, 0)
Pfa -
n]P[Nua(T, 0)
=
=
n]
}
(4)
n=1
Computational details of performing the operatigns of Equation. 4 are
presented in Appendix II for the case where the probability law of
is approximated by the normal probability law. The esti-
Nua(T,0)
mated mean and variance of the normal probability distribution are
represented by
Equation
Ç.
2
and
respectively. As shown in Appendix II,
reduces to the following equation:
4
S
So
1
1
J.
21r v
(2k-n)2
1
e
2
{
n
(
k=0 {n: N(n)>0
0-
}
(5)
NriTe
L.,
)
}
Actual values of these parameters depend upon the underlying process
and for a given process,
µ(T)
and
6(T)
can be determined from
theoretical analysis (Rice, 1945; Steinberg et al.
,
1955; Ylvisaker,
24
1965).
The
expression for µ(T) is essentially a relation between
the noise process autocorrelation function and its second derivative,
but
is considerably more complicated, and recourse to nu-
o-(T)
merical methods'isusually required.
and
Q
In
.
this study, the estimates
were derived from experimental data obtained in the process
of learning the median value of the undershoot
a fixed value of µ
versus D(T)
and a set of values for
interval length. Using
,
curves of Pfa
can be computed., where
D(T) =
So(T)
S<a(T, 0) -
(6)
Results of these computations are presented in Figure 7 which is one
of the main
results of this study. It should be noted that these curves
are predicated on a one -sided test and the hypothesis that
S<a(T, 0)
>
So(T),
that is the counting function will decrease as the
signal -to -noise ratio increases. This effect is discussed further in
Section IV of this study. As an example of the use of these curves,
assume that µ= 2500,
v
a value which is 12,5 counts
=
45
and the threshold
less than the median count
For these parameter values,
Pfa
is approximately
the weakening of the test with an increase in
by the spreading of
D(T),
So(T)
moo-
is set at
S<a(T, 0).
10 4.
Also,
is clearly evidenced
which is to be expected since the .uncer-
tainty in sample size would reduce the effectiveness of the test sta-
tistic.
240
1
11111
I
I
111111
I
I
IIIII
I
1
1
V
III
I
I
111111
I
1
I
IIIII
1
II111
1
1
I
I
I
IIIII
200
45
Vi
160
U
6
cd
...i
rcs
0+
= 0
120
o
.o
I
IIIII
I
I
IIIn
I
1
1
um
I
11IlII
1
1
µ
=
2500
D(T)
=
S<a(T, 0) - So(T)
(S<a(T, 0)
35
o
I
-I
i
II III
I
`\,
\\.
`\
55
H
I
80
U
cd
-+
CI)
40
=
_
these
1250 for
curves)
-
_
_
_
1,
o
I
10-13
1
1
11111
I
1012
I
1
11III
1
10-11
I
I
11111_
I
1C-10
I
I
II111
i
10 9
_:1
I
1111t
I
I
I
11111
I
I
I
Hill
I
1111111
108
10-7
10-6
105
Probability of false alarm P
1
1
111111
10-4
I
1
I
11111
10-3
I
1
1
11111
10-2
fa
Figure 7. Probability
of
false alarm versus distance between test statistic and median
1
1
111111
101
I
1
I
1lIi
1
26
III.
THE ZERO - CROSSING ANALYZER
Principles of Operation
The essential operations that must be implemented by the zero -
crossing analyzer are measurement and comparison of selected subsets of intervals defined by adjacent zero -crossings. In conjunction
with the aforementioned requirements, suitable methods are also re-
quired for synchronizing and displaying the observations.
A
block
diagram of the functional implementation of devices suitable for realization of zero - crossing analyzer requirements is shown in Figure 8.
The output from the noise source
n1(t)
is filtered by the passive
time invariant filter whose transfer function is
noise process
H(jw)
10.
3
n(t).
H(jw)
to yield a
For the case of primary interest in this study,
is determined by a single -pole filter with center frequency of
:kHz
nal process
so that
s(t)
n(t)
is a narrow -band noise process. The sig.-
is simply a periodic wave, which was a sinusoidal
wave in this study. Addition of
n(t)
and
s(t)
yields
y(t)
which
serves as the input to the hard limiter. Significant waveforms and
timing of events are presented in Figure 9. Two outputs,
QL,
QL
and
representing complementary events are available from the hard
limiter, but they are not distinguished in the bock diagram since the
complementary outputs are primarily used in logical functions.
Each time the process
y(t)
crosses zero and the derivative
of
Signal
Sour e
QL
(t)
Noise
Source
nl( t))
Filter
y(t)
HOG))
Hard
Limiter
QL
Ramp Voltage
Generator
e(t)
Reference
Counter
1 Quantile
Voltage
One-Shot
Flip- Flop
Sub - Interval
Comparator
T Voltage.
Set
Y
Sampling
Reset
Gate
QT
SIC
Reset
Y
S<a(T' E.).
Display
Register
Q
-S(t)
Y
Select
Figure
8.
Q
os
Y
Sample Space
Selector
A
Sub - Interval
Comparison
Flip- Flop
"Less than" /"Greater than"
Select
Block diagram of the zero -crossing analyzer.
eR
28
y(t)
i T1
W
®
T3
Ti
---
---
AilL
I
I
I
t
QL
eR
e(t)
TN-1
_7
/
t
7,
Qos
Q
Q
T
c
r
c
Figure
9.
Timing diagram of signals in zero -crossing analyzer.
29
the process is negative, the limiter output changes state and the ramp
voltage generator is started, and when
y(t)
returns through zero
with a positive slope, the ramp is stopped. In this manner, the ramp
voltage
e(t)
where T.
is proportional to the duration of the interval T.,i
is the interval being analyzed. Testing the magnitude of
against a reference
Ti
e(t)
TR
with a reference voltage
quantile of interest. If T.
TR,
reduces to comparing the ramp voltage
eR,
where
eR
corresponds to the
is greater than the reference value
the comparison flip -flop will be set true, and will remain true
until the end of the delay time produced by the one -shot flip -flop
In the event that
Ti is less than TR,
Q
os
.
the comparison flip -flop
will not be set true at the time of comparison,
'so
Q
c
is really a
one -bit memory which stores the outcome of the event corresponding
to a sign test of
Ti.
Since the one-shot is triggered at the end of the undershoot in-
terval, it will be true for a fraction of the overshoot interval which
follows the interval being tested. In this manner, the comparison
flip -flop can be tested and reset at a noncritical time which simplifies
counting of the number of sign tests that yielded favorable results.
As depicted in Figure 9, the comparison flip -flop
state can be
represented by a series of pulses where each pulse begins at the time
of
favorable comparison between TR
trailing edge
and
Ti
and ends at the
of the pulse produced by the one -shot flip -flop.
The
30
logical AND combination of
Q
c
and
Q
os
yields a sequence of pul-
ses where the occurrence of each pulse corresponds to the event that
Ti
>
and similarily the logicalAND combination of Qc
TR
corresponds to Ti
Qos
<
and
TR.
Mechanization of these logical operations is accomplished in
the sample space selector, which also performs the gating and selec-
tion of the
Q
c
inputs such that the observation interval and sub-
interval selection functions are implemented. The sub - interval
count-
er is simply a one -bit counter (one flip -flop) that restricts the sample
space members to alternate undershoots which provides a measure of
statistical independence as discussed in Section II. Since the detection device operates over a given time span, the sampling gate simply generates the observation period
T
which has also been dis-
cussed in Section II. Depending upon the particular mode of operation, the output of the sample space selector
pulses over an interval
T
S(t)
is a sequence of
and formulation of the detector -test sta-
tistic requires summation of the sign test results for each event in
the interval
T.
This summation is performed by the display regis-
ter, which is set to zero at the beginning of interval T. At time
t
=
T,
the display register contents represent the test statistic
for the case of most interest in this study.
which is
S<a(T, p)
S<a(T, p)
is then compared with the threshold
the detection decision.
,
S(T) to complete
31
Determination of the reference voltage was accomplished experwhich cor-
imentally in this study by searching for the value of eR
responds to the quantile of interest. For instance, assume that the
median value of TR is desired, in which case an estimate can be
formed by observing S(t)
for both cases of counting
corresponds to the median value then S<e(T,
If TR
Q
0)
and
c
=
Q
c
.
S>e(T, 0),
that is the number of intervals whose length is less than the median
will be equal to the number of intervals whose length is greater than
the median, and in this manner the median can be estimated. Other
quantile values could be estimated in a similar manner providing that
the proper relative values of
SGe(T, 0)
and
S>e(T, 0)
are used.
Key Circuits
A
large percentage of the circuitry used in the analyzer is con-
ventional, so only those parts that would appear to have some unique
or distinguishing characteristic will be discussed in this document.
A.
complete schematic diagram of the analyzer is given in Appendix
III to which the reader is
referred for identification
of
reference de-
signations mentioned in the text.
The hard limiter is comprised of a two stage differential am-
plifier which drives a tunnel diode thresholding circuit. Use
_
of a dif-
ferential amplifier input stage serves the dual purpose of providing
common mode signal rejection and prelimiting with an inherently
32
.
easy method of setting the crossing level so that the zero level of the
input waveform can be located.
The common -mode rejection feature
was not particularly important in this study. Setting of the limiting
level, or equivalently, the symmetry of the limiter is accomplished
with the zero level adjust controls R6 and R7.
With an input signal of
25
millivolts (peak -to- peak), the no load voltage at the collector of
Q3
is about 4. 4 volts (peak -to -peak) and with this input, the differ-
ential amplifier is approaching saturation.
Transistor
Q6
gives
further amplification and also drives the tunnel diode which actually
approximates the zero - crossing detector.
Theoretically, a hard
limiter using an amplifier followed by a nonlinear device would require infinite gain in the neighborhood of zero volts if abrupt transitions in the output were to be realized, otherwise an arbitrarily
small signal would cause the device to become linear and amplification rather than limiting would take place.
Because a real device
must necessarily have finite gain, an ideal limiter cannot be realized,
but it can be approximated with sufficient accuracy by introducing am-
plification prior to the thresholding device. The amplification preceding the tunnel diode is sufficient to effectively reduce the tunnel
diode hysteresis such that full limiting is achieved for the smallest
signal of interest and the limiter outputs,
QLl
and
QL2,
are
strictly digital.
Transistor
Q7
amplifies the current transitions of tunnel diode
33
CR2 and also provides voltage
protection of the tunnel diode via the
provides further amplification and
base to emitter junction of Q7.
Q8
drives the emitter followers
and Q10. Performance of the limiter
Q9
is such that the lower threshold of limiting is about
1. 5
mV (peak-
to-peak) and solid limiting is achieved with an input signal of about
2mV (p -p), which is quite satisfactory for this study.
Figure
15
(in
--
Appendix IV) presents a photo of typical rise and fall times of the lim-
iter waveform at the collector
of
transistor
Q8.
Generation of a ramp voltage starting at the beginning of each
negative undershoot is performed by constant current charging of ca-
pacitor
C6
where the charging is gated by QLl
so that proper tim-
ing is achieved. Adjustment of C6 changes the ramp slope, with the
value of C6 selected to allow the ramp to reach its maximum value in
a time
interval which is less than the maximum length
of
interest.
Because of this, the ramp will saturate in cases where the interval is
considerably greater than the reference but no harm is done since
saturation will be after the fact.
The voltage comparator is essentially a differential amplifier
composed of Q20 and Q22, with tunnel diode CR8 used for threshold ing.
Transistors
Q17, Q18, and Q19 provide buffering between the
ramp and the comparator, since significant loading of the ramp circuit would introduce serious error. Establishment of the reference
voltage,
eR,
is obtained by the emitter follower, Q24, which is fed
34
by a simple voltage divider network.
Figure
16
A
photograph is presented in
(in Appendix IV) which shows typical waveforms of the
:.
ramp voltage, voltage comparator, comparison flip -flop, and the oneshot flip -flop.
When learning the median value of
T, eR
is adjusted until a
suitable equivalence between the -'" less than" and "greater than" 1
counting functions is achieved.
This is a relatively easy procedure,
and one could use some feed -back from the counter to make the set-
ting automatic, or implement an adaptive technique.
The one -shot, comparison, and subinterval counter flip -flops
:
are conventional (Strauss,
1960) and do not
merit any particular dis-
cussion. Resistor pair R26 and R27 introduced a 1.3
µ
sec delay in
switching of the one -shot, which eliminates transition and timing
problems in the NAND gate.
Essentially separate control
of
the on -off times of the mono-
stable device is obtained in the sampling gate by use of variable re-
sistors
R78 and R82, which
control the discharging of C23 and C25
(Tesic, 1965). This allows setting of the sampling interval
T
to a
value, which is sufficiently independent of the intersampling period
such that the counter output can be conveniently observed. Transistor
The notation "less than" is used to denote operations associated with the set of intervals whose duration was less than the reference
quantile, and similarily for "greater than ".
1
35
its associated circuitry perform a double differentiation of
Q28 and
the limiter output, such that the sampling gate switching is synchro-
nized to both positive and negative transitions of QL2 (see Figure 9).
In this manner, the main portion of the sample gate interval is deter-
mined by the resistor -capacitor network, composed of R80,
R83,
R82.,
and C25, but the actual switching of the gate is synchronized to
zero crossings, which simplified logical gating problems. The duration
T
of the gate is random, but the fluctuations
are sufficiently
small when compared to the mean duration set by the RC timing network, so that
T
is constant for practical purposes.
Selection of the sample space is accomplished in the NAND gate
composed of Q35 and its associated circuitry.
Qc
and
Qc
Switching. between
is a convenient method of counting the number of un-
dershoots whose duration was greater than the reference or less than
the reference, respectively. Selection of
QSIC
provides for sam-
pling of every undershoot or alternate undershoots, which might be
of
interest for experimental purposes.
Evaluation of the Analyzer
The overall performance of the analyzer is such that reliable
triggering and resolution are obtained to 50
kHz, which
could be ex-
tended to about 80kHz by reducing the one -shot duration from 12 t s to
about
6
µs. With an input signal of
10 kHz
and peak -to -peak input
36
voltage of 140 mV, the half cycle jitter in the limiter output is less
than 0.05 µs, and with an input of 20mV, the jitter is less than 0.15
}Is.
Input impedance measured at 15 kHz is approximately, 400 k IL ,
and the voltage swing at followers
Q9
and Q10 is from -12V to +1.1V.
The ramp voltage input to the comparator
starts from a base-
line of -11.1V and saturates at +0.8V, with the rate of rise depending
upon C6. Maximum fall time of the ramp is about
duration of
6. 25
ter in the output
8µs. For a ramp
ms, and with a comparison level of -0.2V, time jitof the
comparator is about 2µs. Similarly, for a
comparison level of -6.8V, and a ramp duration of
50 µ s,
time jitter
in the comparator output is 0.03 vs, and the jitter appeared to result
from ripple on the -12V supply. These figures indicate that the com-
parator is sufficiently sensitive for the application.
The inputs
s(t)
and
n(t)
were summed together in a sym-
metrical resistor dividing network which allowed for essentially independent adjustment of the input voltage, and easy measurement of
the voltages. A General Radio GR 1390 -B random noise generator
served as the noise source and a Hewlett - Packard 200
CD
oscillator
provided the signal. Measurements of the signal voltage can be made
with readily available instruments, but measurement of the rms noise
voltage is somewhat more difficult. The narrow -band noise was ob-
tained by filtering the wide -band noise obtained from the GR 1390 -B
operating in the
20 kHz
range, and H(jw) was determined by a
37
single pole filter whose relative voltage response for constant current
input is shown in Figure 10.
Measurement of the noise input voltage
to the analyzer was accomplished by using a probability- density-
function estimator (Senk, 1964) from which the rms voltage was cal-
culated by statistical analysis of the estimated probability density
function.
A
photograph of the estimated density function was numer-
ically integrated to compute a normalized function from which the estimated mean and variance were calculated. The calculations were
performed after the experimental data were taken and the estimated
rms voltage was 37.4 mV compared to the desired value of
but this caused no particular problems.
50 mV,
1. 0
Relative response
0. 8
0. 6
á
o
0. 4
0.
2
0. 0
-0.8
-1..2
-0.4
0
+0. 4
+1.2
+0. 8
:
Frequency deviation ¿f (kHz)
f
o
=
10.3 kHz
of=f
Figure
10.
- f
o
Relative frequency response characteristics of single pole filter used to
obtain narrow -band noise process.
m
39
TEST RESULTS
IV.
Summary of Test Parameters
test results,
As an aid to discussion of the
of the different
in Table
I
a
tabular summary
test cases and observation parameters is presented
in conjunction with references to graphical displays of the
experimental results. The signal frequency and voltage are probably
the most interesting
parameters in that the test statistic behavior can
be investigated for various relative locations of the signal in the noise
filter frequency band and also as
The signal to noise ratio
crossing analyzer
nal voltage,
s(t),
of the mean and
time
T
a function of the
signal to noise ratio.
was measured at the input to the zero
p
(y(t) in Figure 8), which was the ratio of the sig-
to the filtered noise voltage,
n(t). Estimates
-
standard deviation of the number of undershoots per
are given in the interval subset parameter columns, ex-
cepting for missing data as noted.
-
These values were calculated
from the data taken with no signal input, with a normalizing factor of
1
/nss
for the mean and
1
/(nss -1) for the variance, where
ns
is
the number of samples for each signal to noise ratio value. Selection
of an appropriate value of
n
s
was somewhat arbitrary, however,
the choice was made after preliminary reduction of some data points.
If it is assumed that the
statistics
S
and
S
are normally
distributed, which is in conformance with key assumption
5 of
Section
40
II, then confidence intervals for the mean and variance of
Spa
can be easily calculated (Bendat and Piersol, 1966).
S
-
and
<a
For ex-
ample, if a 0.95 confidence coefficient is chosen for Case I, then the
confidence interval for the mean of
S
<a
is (1225, 1265) and the
confidence interval for the standard deviation of
S
is (12.
2,
47.8). These numbers indicate that the experimental results of S<a
in Case I are reasonable. Confidence level calculations were not per-
formed for other cases, however in light of the above numbers, the
results appear satisfactory from
a
sample size viewpoint.
The values of the referenced quantile displayed in Table
1
were
calculated from the "less than- every" and "greater than -every" data
sets. The desired quantile was
0. 5,
which would correspond to a
sign test based on the median as a reference, but due to finite observation time and some instability in the zero crossing analyzer, the
median was not obtained in all cases.
It should also be noted
that the
data were taken over a period of several days and some fluctuations
due to
drift and adjustments are not unexpected. The display register
used in collecting the experimental data contained
12
binary places,
the contents of which were visually observed and recorded for each
trial taken over the observation interval. Conversion from binary to
decimal was accomplished by punching the binary data into IBM cards,
which then served as inputs to a computer program that converted the
numbers to decimal and calculated the sample means and standard
Table 1. Summary_ of te §t parameters and
Case
T
(sec)
Signal
frequency
Signal
voltage
kHz
cr mV
rests
Noise
voltage
0- mV
Interval Subset Parameters
Calculated
reference
quantile
S >a(T, 0)
S<a(T,'0)
S<e(T,
0
mean
std.
Fig.
mean
std.
Fig.
mean
std.
Fig.
11
1241
24.5
12
2549
41.2
13
2461
19.0
14
11
note
12
note
13
note
11
1427
12
2086
10.7
13
2864
11
note
12
2234
37.7
13
2825
1
(note 3)
mean
std.
Fig.
19.5
I
0. 5
9. 0
0-60
37. 4
0.508
1245
II
0.5
9.
3
0-60
37. 4
0.508
note
III
0.5
9. 9
0-60
37.4
0. 421
1088
IV
0.5
10.0
0-60
37.4
0.441
note
V
0.5
10.2
0-70
37. 4
0.501
1263
15.9
11
1244
25.3
12
2535
42.5
13
VI
0.5
10.3
0-60
37. 4
0. 492
1232
26.3
11
1305
19.8
12
2491
37.7
13
Note
1:
Note 2:
Note 3:
1
36.7
2
S>e(T, 0)
)
1
10.3
2
1
14
1
9.8
14
4. 1
14
2519
38.1
14
2562
46.5
14
These values were not experimentally determined for o- = O. For normalization purposes
in Figures 11 , 12, 13 and 14, the values presented in case I were used.
These values were not experimentally determined for 6 = O. For normalization purposes
in Figures 11 and 12, S<a(T, 0) = 1088 and S>a(T, 0) = x1427.
These values were calculated from SGe(T,
0)
and S>e(T, 0).
42
deviations.
Estimation of the quantile reference value was somewhat
tedious because of the read out technique, however, the method was
satisfactory for demonstration
of the
analyzer principle.
Summary of Test Results
The main body of experimental results obtained in this study
are presented in Figures
11
through 14, which give a parametric dis-
play of the normalized counting function behavior with variation of
signal voltage and frequency. Each data point represents the average
of the counting function for six
trials, and the normalizing factor is
the appropriate value of the statistic
S
for the case of
p
-
0,
which is the mean number of events per unit time for noise only.
For
example, Case I has six trials for each signal to noise ratio and the
average of the results for each signal to noise ratio is normalized to
2549 for the case of
"less than-every" intervals in the sample space.
The data points for the other interval subset
parameters were reduced
in a similar manner and the results are presented in the correspond.
ing figures.
Figure
11
presents
a
typical display of the sensitivity of the
normalized counting function
tage where
ri
<a
S
<a
1 a
to the signal frequency and vol-
(T, p) /S<a (T, 0).
In the instance where the sig-
nal frequency is removed from the filter center frequency, a pronounced decrease in the normalized statistic is effected by a relatively
43
1.2
I
= 10:3
-
kHz
10.2
9.3
i
0.4
Figure
11.
-af
9.9
I
I
0..8
1.2
Signal to noise ratio
p
2. 0
Variation of normalized statistic with signal frequency and power for "less than -alternate" subset.
44
2. 0
1.8
f
=
9. 0 kHz
9.3
9.9
1.6
10.0
m
RA
F
1. 4
ti
1.0
10.
0. 8
0. 0
1
0.4
I
0.8
i
i
1.2
1.6
Signal to noise ratio
p
Figure 12. Variation of normalized statistic with signal frequency and power for "greater than- alternate"
subset.
3
2.0
45
1.2
1
1
Signal
1
1
to noise ratio
p
Figure 13. Variation of normalized statistic with signal frequency
and power for "less than- every" subset.
46
2. 0
f
s
=
9.
0
kHz
1.8
1. 6
ñ
o-
u
.4
o
1.2
1.0
0. 8
0
0. 4
0.8
1.2
Signal to noise ratio
1. 6
2. 0
p
Figure 14. Variation of normalized statistic with signal frequency,
and power for "greater than -every" subset.
47
small increase in the signal voltage. For example, if the signal frequency is 9.3 kHz, a signal to noise ratio of about 0.5 will correspond
to a value of
ri<a
which is about one -half the no signal value.
As an example of the relation of the curves in Figure
11
to the
probability of false alarm curves in Figure 7, assume that
fs
=
s
9.3 kHz and
For
=
S (T,
<a
0.27)
1010.
1010,
D(T)
=
=
0. 27, in which case
a value of
1250,
<a(T, 0)
S
p =
240,
If the
1 <a
=
r)
is about 0.81.
<a
0.81
corresponds to
`
test statistic threshold
S
which can then be used in Figure
is set to
o
7
for determin-
ation of the probability of false alarm. Since v was not actually
measured in this investigation, it will be assumed that 0
purposes of illustration. This value of
variances for
S<a(T, 0)
and
S>a(T, 0)
or-
=
32
for
is obtained by adding the
as given for Case II in
Then, Figure 7 gives a probability of false alarm which is
-13
less than 10
for D(T) = 240 and C = 32. Figure 17 (in Appendix
Table 1.
IV) depicts the signal, noise, and signal plus noise
processes for a
signal frequency of 9.3 kHz and a signal -to -noise ratio of 0. 27, in
which case, the signal presence is not readily discernible by visual
observation.
As mentioned in Section II, the detection probability has not
been determined, but a normalized statistic threshold of 0.
5
would
appear to imply a high detection probability for these particular para-
meters. However, as the signal frequency approaches the filter
48
center frequency, variation of the normalized statistic with an in-
crease in
p
becomes less pronounced, and at the center frequency,
the change is negligible, which clearly indicates that the filter and
signal frequencies must be offset if a useful level of detection is to be
achieved.
In'.
Figure
12,
results are presented which are complementary
to those of Figure 11.
The behavior of
n>a
>a
is generally comple-
mentary to that displayed in Figure 11, since
an increase in
p,
increases with
r) >a
provided the signal frequency is sufficiently re-
moved from the filter center frequency.
This pattern is to be ex-
pected since the total number of overshoots (sum of
S >a(T, p)
S<a(T, p)
is relatively constant for a given value of
crease in
S>a(T, p)
<a
p,
would be offset by a decrease in
From the composite results of Figure
11
and
and an inS <a(T,
p).
and 12, it can be implied
that the sign test as applied to zero -crossings will produce reasonable results, if the signal is sufficiently removed from the filter cen-
ter frequency, which confirms the hypothesis given in Section
Figures
13
and 14 are similar to Figures
for the sample space, in that Figures
13
11
II.
and 12, excepting
and 14 are for the case of
having the sample space composed of successive undershoots, which
does not imply statistically independent samples.
of Figure
7
The
' Pfa curves
cannot be applied in this case, however the data are pre-
sented for comparative purposes since the curves are markedly
49
similar to their counterparts in Figures
11
and 12.
This would indi-
cate that the normalized statistic is not a sensitive function of the degree of sampling dependence, however these results are not sufficient
to allow for definitive conclusions and
further investigation would be
required to ascertain the exact degree of dependence.
50
V.
SUMMARY AND CONCLUSIONS
Results of theoretical and experimental work performed in this
study indicate that the sign test can be implemented by analyzing the
zero - crossings of a restricted class of stochastic processes. An
analytical expression for the false alarm probability has been derived
under a set of assumptions which appear reasonable from an engineering viewpoint.
Confirmation of the hypothesis that the zero - crossing
analyzer produces a distribution -free detection device was not verified experimentally, although this would be a useful extension of these
considerations. Experimental verification of the false alarm proba-
bility of a detection device usually implies a large number of samples
because of the small probabilities involved, and an investigation of
this nature is beyond the scope of this thesis.
No
attempt has been made to rigorously compare the perform-
ance of this device with optimum devices such as correlators and
matched filters, since the establishment of a basis of comparison
would require determination of the detection probability and relative
sample sizes required to obtain equivalent levels of performance,
which is beyond the scope of this investigation.
For the case of nar-
row-band noise, which is probably of most interest for practical con-
sideration, the results presented in Figures
reliable detection should be obtained with
11
p >
and
12
indicate that
0.4 or about -8dB,
51
providing the signal frequency is sufficiently removed from the filter
center frequency. The inherent failure of the sign test in the case of
identical test statistics for signal versus no signal conditions is amply demonstrated in the experimental results, but this is simply a
manifestation of the limited performance of the detection scheme.
In conclusion, it seems reasonable to consider the zero- cross-
ing analyzer as one method of implementing a distribution -free detec-
tor. It should be noted that complete description of a detection structure requires characterization of the detection and false alarm proba-
bilities as a function of the decision threshold. Only the false alarm
probability has been investigated in this study which means that the
zero -crossing analyzer detection scheme requires further investigation for complete characterization.
It must also be noted thatthe ef-
fect of departures from the key assumptions is largely unknown and
that these assumptions are indicative of the need for further investigation of level crossing processes. Evaluation of the false alarm and
detection probabilities for the case of dependent sampling would also
be a useful extension of this study,
52
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Armstrong,
1966.
G. L.
Gordon H. Bower and Edward J. Crothers.
1965. An introducation to mathematical learning theory. New
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Atkinson, Richard C.
,
-
Bendat, Julius S.
1958.
Principles and applications of random noise
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Bendat, Julius S. And Allan G. Piersol. 1966. Measurement and
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Blotekjaer, Kjell. 1958. An experimental investigation of some properties of band -pass limited Gaussian noise. IRE Transactions
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Carlyle, J. W. and J. B. Thomas. 1964. On non- parametric signal
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Daly, R. F. and C. K. Rushforth. 1965. Nonparametric detection of
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on Information Theory IT-11:70-76.
Davenport, William B. and William L. Root, 1958. An introduction
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Feller, William. 1966. An introduction to probability theory and its
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Fraser,
D. A. S.
1957.
York, Wiley,
626 p.
Nonparametric methods in statistics. New
299 p.
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Groginsky, H. L. , L. R. Wilson and David Middleton. 1966. Adaptive detection of statistical signals in noise. IEEE Transaction
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Hancock, J. C. and D. G. Lainiotis.
On learning and
1965.
Helstrom, Carl W. 1957. The distribution of the number of crossings of a Gaussian stochastic process. IRE Transactions on
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Helstrom, Carl
W.
1960.
Statistical theory of signal detection.
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364 p.
Kanefsky, M. and J. B. Thomas. 1965. On adaptive non - parametric
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Kanefsky, M. and J. B. Thomas. 1965. On polarity detection
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Kendall, Maurice G. and Alan Stuart. 1961. The advanced theory of
statistics. Vol. 2. New York, Hafner. 676 p.
.
Levenbach, Hans. 1965. Approximate zero -crossing distributions
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on Information Theory IT-11:507-512.
Middleton, David. 1960. An introduction to statistical communication theory. New York, McGraw -Hill. 1140 p.
Middleton, David. 1965. Topics in communication theory.
York, McGraw -Hill. 126 p,
New
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a
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Parzen, Emanuel. 1962. Stochastic processes. San Francisco,
-
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324 p.
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Skolnik, Merrill I. 1962. Introduction to radar systems.
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-to -space ratio.
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Tikhonov, V. L and Y. I. Kulikov. 1962. Distribution of overshoots
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Wilks, Samuel S.
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56
APPENDIX I
List of Symbols
a dummy
variable.
a
=
D(T)
=
fs
=
eR
=
FX()
=
H(jW)
=
transfer function
j
=
complex number defined by
n(t)
=
noise process as a function of time.
No(T, 0)
=
number of zero -crossings in time interval
difference between the statistic
hold value S.
S
0)
<a(T,
and the thres-
frequency of input signal.
comparator reference voltage for the reference quantile
interval length.
generic symbol representing the cumulative distribution
function (c. d. f: ) of the random variable X.
of noise
filter.
T
for noise
only.
number of alternate undershoots in the time interval
for the signal -to -noise ratio p.
T
Nua(T, p)
=
p
=
quantile value of c. d. f of zero -crossing interval lengths.
pN(n)
=
0),
probability mass function of the distribution of Nua(T,
ua
Pd
=
probability of detection.
fa
=
probability of false alarm.
Pfac
s(t)
=
conditional probability of false alarm.
=
signal process as a function of time.
57
value of threshold level for test statistic.
So
=
S<a(T, p)
=
S)a(T, p)
=
S<e(T, p)
=
S>e(T, p)
=
T
=
T
=
TR
=
interval length
y(t)
=
sum of signal plus noise at receiver output.
A
size of the subset of the set of alternate undershoots such
that the subset contains those undershoots whose duration
was less than the reference interval, (referred to as
"less than -alternate" in text. ).
same as S<a(T, p) except the subset contains those undershoots whose duration was greater than the reference
interval, (referred as "greater than- alternate" in text. ).
same as S<a(T, p) excepting that the set of undershoots
forming the sample space contains every undershoot interval, (referred to as "less than -every" in thé.text).
N
=
o2
=
s
=
crn
=:
except the subset contains those undershoots whose duration was greater than the reference
interval, (referred to as "greater than - every" in text. ).
same as
SSe(T, p)
duration of observation interval.
random variable representing an interval between successive zero crossings.
of
reference quantile.
estimated mean value of the normal density function used
to approximate pN(n).
estimated variance of the above density.
rms value of the signal voltage at the zero crossing analyzer input.
rms value of the noise voltage at the zero crossing analyzer input.
58
Sa(T,p)
TI
normalized value of the relevant statistic.
S <a(T, O)
<a
S>a(T, p)
S>a(T, 0) '
>a
>a(
S<e(T, p)
/1<e
11>e
p
normalized value of the relevant statistic.
=
S<e(T, 0)' '
S>e(T' p)
- S>e(T, 0)
'
normalized value of the relevant statistic.
normalized value of the relevant statistic.
=
angular frequency.
_
ss
o-
n
signal to noise ratio
59
APPENDIX II
Derivation of False Alarm Probábility
Let
N
represent the number
which is equivalent to
cr
ua
equal the mean of
(T, 0), let µ
and
N,
equal the standard deviation of N. Also, let the probability
mass function of
N<
N
of undershoots per unit time,
and finally, let
represented by pN(n),
be
N
denote the number of undershoots whose duration is less than the
reference level. All of the above variables are defined on an observation interval of
0 <
t
< T.
If the outcome of a particular set of zero -crossings is known,
N
will be a fixed number and
bility law. If
p
Pfa
is given by the binominal proba-
is the probability that each undershoot duration is
less than the reference value, then
q =
- p
1
is the complementary
event probability and the conditional probability
P c that k of the
undershoots have a duration which is less than the reference can be
easily formulated. More explicitly,
Pc
._
P[N<=k'N-n]
=
n)pnqn-k
(Al)
k
By averaging over the possible values
that
N
can take on, the
following expression is obtained for the unconditional probability that
k
of the undershoots
duration:
have a duration which is less than the reference
60
P[N<
k]
(AZ)
P[N< =k1N=nJpN(n)
{n: prT(n)>0}
such that pN(n)
The sum is taken over the set of n
tive. If a threshold
bility that
So
0
is posi-
is established, one then asks for the proba.
or more explicitly:
N< < So,
PfaFN< (So)
=
(A3)
P[N< <So]
This is readily obtained by summing over
k,
such that
So
o
Pfa
n
()p
=
k=0
{n:
n n-k
q
(A4)
PN (n)
n)>0}
In accordance with key assumption
5
of Section II,
pN(n)
is
approximated by a normal density function of the form
1
1
N(n)
where
npq
µ
and
G
I 2 ne
/I
e
2
"n-µ
2( $
)
(A5)
were previously defined. For those cases where
is sufficiently large (Parzen, 1960), the probability mass func-
tion of the binominal law can be approximated as follows:
61
1
(n)Pnqn
k
kN
ti
npq
2
e
1
2 n npq
(k-np)2
(A6)
Substitution of Equations A5 and A6 into A4 yields the final ex-
pression for Pfa:
S
1
Pfa
2Tr6
o
//
k=0
1
1
p(n)>0 }
In the special case of
p
= q =
1/2,
l
So
P
Pfa
a
1
=
2
e
np
/I
1
e-
2
(k-np)2 +( n-µ
L npq
2
)
(A7)
Equation A7 reduces to
}
n-
(2k-n)2
+(
n
(A8)
k=0 {n: pN(n)>0}
Closure of this series is not readily apparent, although a great
Pfa
deal of effort was not devoted towards this. However,
can be
easily evaluated by using computer techniques as was the case in this
The limits on
study.
n<
IÑ
± 4aß
I
,
a parameter.
Figure
7
ó
.
were chosen to satisfy the relationship
and the results are presented in Figure
7
with
It is interesting to note that the curves for
approach the curve for
The curve for
As
n
e
= 0
o
=
0,
as
v
e
o^-
> 0
as
in
tends toward zero.
was obtained from the fixed sample size case.
tends to zero, the uncertainty in sample size is decreased
62
since the distribution of
N
would tend toward a causal type, so
Equation A8 is consistent in this respect.
Also, summation of Equation A8 over the range defined by
2-2 <k<2+2
which is a plus or minus four sigma range, gave a value of 0.999995
for the cumulative distribution function, which is unity for practical
purposes.
APPENDIX
COMPARISON FLIP-FLOP
ONE- SHOT FLIP-FLOP
HARD LIMITER
III
A
-12V
I2V
1243
0
Qo,
R2
08
Rt4
-I
eV
Q9
V
R29
Qu
2;R26
R16
R52
=
C4
Qo,
RI
256
251
E!
CR9
E
CRIO
259
R53
INPUT
CR3
+12V
-I2V
R4
CRaO
04
R6
R27
010
-12V
-12V
RIO
RAMP GENERATOR
RI5
R28
--C3
COARSE ZERO LEVEL ADJUST
P30
211
VOLTAGE ÁOMPARATOP
1
OQI3CR7
Q
Q
(TO
SSA)
(TO
SIB
(TO
R68
)
)
019
5
C6
Q14
-
018
R46
234
R33
29
+12V
-12 V
+18 V
CR IA-
QI7
II
QL2
(TO
12V
RES
FINE ZERO LEVEL ADJUST
C9
C5
QI2
Q5
P44
Qos
+12V
125
®
RAR
(24
CR1
*I2V
C14
255
+12v
263
+
-12V
COMPARATOR
REFERENCE
LEVEL
Q
L
2
Q
08)
ADJUST
JUST
264
+12V
SCHEMATIC DIAGRAM OF THE
ZERO- CROSSING ANALYZER
SHEET
I
OF 2
63
64
APPENDIX III
SAMPLING GATE AND COUNTER
SAMPLING INTERVAL
SAMPLING GATE
ADJUST
REPETITION RATE
-12V
R67
C23
s
-I(R67
R91
IR78
R69
RESET
R83
+
R79
R84
C28
- I( +
-
c251
Q31
C26 R88
C18
(TO
QI6
(TO
Q27 )
Qc
(TO
Q26)
Qc
)
©Q28
Q33
Qos
R85
12
-12V
+12V
12V
SUE- INTERVAL COUNTER FLIP
290
R87
270
RT7
4.115114Q
CRIC
30
B
410)
S
QL2
273
C22
R94
R95
DISPLAY REGISTER
C29
593
INPUT
Q3S
Q,
S2
p
11292
"GREATER THAN°
ALTERNATE
Qs r c
C19
15
'LESS THAN'
A
S1
Si
A
568
-12V
+12V
OCR13
CCI
CRII
SAMPLE SPACE SELECTOR
+12V
V
C20
--
C27
1
(TO
AY REGISTER
RESET PULSE
OCR16
EVERY
POWER
+
12V
+I8V
- 12V
V
REQUIREMENTS:
40MA
7.5MA
BOMA
SCHEMATIC DIAGRAM OF THE
ZERO-CROSSING ANALYZER
SHEET
2 OF 2
65
APPENDIX IV
Photographs of Salient Waveforms
66
upper
C
upper
center
lower
center
¡
Figure
15.
Photograph of typical rise and fall times at
limiter output.
vertical
Scale:
- 10
horizontal
f
upper:
upper
center:
.
lower
center:
lower:
.
lower
traces)
as noted below
-
s
v /cm (all
=
10 kHz
Q8
collector,
20
vs/cm
Q8
collector,
10
µs /cm
Q8
collector, (rise time)
Q8
collector, (fall time)
o-
0.
s
=
50 mV
2 µ s
0. 2 µs
/cm
/cm
o-
n
=0
67
-
ramp
voltage
0
0
voltage
comparator
0
comparison
flip -flop
0
one- shot
flip -flop
Figure 16. Photograph of key waveforms in the zero crossing analyzer.
vertical
scale:
- 10
horizontal
f
ramp
voltage:
s
=
Q14
collector
Q25
collector
Q26
collector
Q15
collector
comparison
flip -flop:
one -shot
flip -flop:
- 10 p s /cm
10 kHz
voltage
comparator:
v /cm (all
traces)
(all traces)
o-s =
s
50 mV
v
n
= 0
-
0
---
..--
_
0
.
68
upper
ir 11,1
L
center
.
gdjg1LaJ
-
o
L'
Figure
scale:
,
i
,
L
Li I
ummi
,
.
,
.l
V
.'',.t
lower
q
Single trace pictures of typical signal, noise,
and signal plus noise inputs to the anlyzer.
17.
.
vertical
mV /cm
- 50
horizontal
-
Z
ms /cm
(traces were synchronized to the signal but were not recorded simultaneously)
upper:
signal voltage
center: signal
lower:
+
noise
noise voltage
o-s = 10
s
mV,
v
n
=
37. 4 mV
o
n
=
37. 4 mV
f
s
=
9.
3
kHz
69
APPENDIX V
List of Components
Unless otherwise specified, component values are as follows:
1.
All capacitance values are in picofarads,+ 10 percent,
100 VDC
2.
rating.
All resistance values are in ohms, ± 10 percent, with 1/2
watt rating.
Reference
Designator
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
R17
Component
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor (Variable)
Resistor (Variable)
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Value
10 K
47 K
4.7
K
470
470
10 K
500
4.7.K
5%
9. 1: K
470
6.2.K
1. 8. K
4.7.K
47 K
10 K
18 K 5%
1.6
K 5%
70
Reference
Designator
R18
R19
R20
R21
R22
R23
R24
R25
R26
R27
R28
R29
R30
R31
R32
R33
R34
R35
R36
R37
R38
R39
R40
R41
R42
R43
R44
R45
R46
Component
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Value
100
820
4.7
K
12 K
4.7
K
3.9
3.9
K
K
56 K
150 K
47 K
2.2
4.7
K
K
10
10 K
10 K
2..2 K
1
K
22.K 5%
22 K 5%
: 3..3 K
1.8
18
K
K 5%
22 K 5%
12 K 5%
15 K 5%
1.8 K
560
4.7
K
100 K
71
Reference
Designator
R47
R48
R49
R50
R51
R52
R53
R54
R55
R56
R57
R58
R59
R60
R61
R62
R63
R64
R65
R66
R67
R68
R69
R70
R71
R72
R73
R74
R75
Component
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor (Variable)
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Value
300
2.2
K
10 K
2.2
K
K 5%
2.2
18 K 5%
4.7
K
12 K
6. 8 K
22 K 5%
2.2
K
18 K 5%
4.7
K
220 5%
12 K
1.2 K
10 K
12 K
2.0
K
10
1
K
1
.K
2.2 K
2.2 K
22 K 5%
18 K 5%
220 5%
4.7
K
22 K 5%
72
Reference
Designator
R76
R77
R78
R79
R80
R81
R82
R83
R84
R85
R86
R87
R88
R89
R90
R91
R92
R93
R94
R95
Cl
C2
C3
C4
C5
C6
C7
C8
Component
Resistor
Resistor
Resistor (Variable)
Resistor
Resistor
Resistor
Resistor (Variable)
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Resistor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Value
18 K 5%
2.2
K
500 K
150 K
4.7
K
220 5%
50 K
100 K
1.2
K
5%
47 K
20 K
62 K
'1 K
10 K
2.2
2.2
K
K
47 K
20 K 5%
62 K
1.8 K
100 MFD, ±20 %, 25VDC
39
39
200
100 MFD, ±20 %, 25 VDC
0.0162 MFD,
220
100
4-5%, 50 VDC
73
Reference
Designator
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
C29
Value
Component
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
Capacitor
7500
140
220
220
0.02 MFD
_
1
MFD, +20 %, 25VDC
100
39
15 MFD, ±20%, 25VDC
92
39
200
200
0.015 MFD
100 MFD, ±20%, 25VDC
15
7
MFD, L 25 %, 25VDC
1000
33
1
MFD, +20 %, 25VDC
39
74
Reference
Designator
Type (Note
Component
1)
1N759 250 mw
CR2
Zener Diode
Tunnel Diode
CR3
Zener Diode
1N200-2 250 mw
CR4
Diode
WE400D
CR5
Diode
WE400D
CR6
Diode
WE400D
CR7
Diode
WE400D
CR8
Tunnel Diode
(Note 2)
CR9
Diode
WE400D
CR10
Diode
WE400D
CR11
Diode
WE400D
CR12
Diode
WE400D
CR13
Diode
WE400D
CR14
Diode
WE400D
Diode
WE400D
CR1
.
CR15
:
Diode
CR16
1N3712
WE400D
-
Use of the Western Electric diodes is optional and equivalent
types may be substituted. Typical characteristics of the WE400D axe:
Note
1:
I
= 5
ma at 0.75 V,
IBR
=
0.
1
ma at -20 V.
These components were supplied as part of the College Gift Plan of
the Western Electric Company.
Note 2: Tunnel diode CR8 was of unknown origin. The approximate
characteristics or requirements are as follows:
I
p
= 5
ma,
p
= 60
mv,
I
v
=
2.5
ma.,
V = 175 mv.
v
that reasonably approximates these values should be
an acceptable substitute.
A tunnel diode
75
Reference
Designator
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
Q10
Q11
Q12
Q13
Q14
Q15
Q16
Q17
Q18
Q19
Q20
Q21
Q22
Q23
Q24
Q25
Q26
Q27
Q28
Component
Type
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
2N3638
2N3638
2N3638
2N3638
2N3638
2N3638
2N1304
2N3605
2N3638
2N3638
2N3638
2N404
2N404
2N404
2N404
2N404
2N404
2N404
2N1305
2N1305
2N404
2N1305
2N1304
2N3638
2N3638
2N404
2N404
2N3638
76
Reference
Designator
Q29
Q30
Q31
Q32
Q33
Q34
Q35
Component
Type
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
Transistor
2N404
2N404
2N1305
2N1305
2N3638
2N3638
2N3638