In treating the recorded results from detectors at different locations in space, the analysis of the time delays of signals is crucial to locating the sources of detected radiation. Because the correlation method requires the manipulation of awkward matrices to evaluate its accuracy, a solution is outlined based on minimizing the sum of the squares of signal deviations, and the algorithms for evaluating the resulting error are presented

Novak, B. L.

English

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NASA TECHNICAL MEMORANDUM NASA TM-75646
ALGORITHMS FOR THE DETERMINATION OF THE TIME
DELAYS OF THE SIGNAL WHEN USING UNEQUAL DETECTORS
B. L. Novak
Translation of "Algoritmy opredeleniya vremennykh
zapazdyvanii signala pri ispol'zovanii neodinakovykh
detektorov," USSR Academy of Sciences Institute of
Space Research, Moscow, Report PR-413, 1978, pp. 1-18.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546 JULY 1979
https://ntrs.nasa.gov/search.jsp?R=19790020740 2020-03-21T22:23:01+00:00Z
STANDARD T I T L E PAGE
1. Report No.
NASA TM-75646
2. Government Accession No. 3. Recipient 's Cotolog No.
4. Title and Subtitle
ALGORITHMS FOR THE DETERMINATION OF
THE TIME DELAYS OF THE SIGNAL WHEN
USING UNEQUAL DETECTORS
5. Report Date
July 1979
6. Performing Organization Code
7. Author(s)
B.L. Novak
USSR Academy of Sciences, Institute
of Space Research
8. Performing Organization Report No.
10. Work Unit No.
9. Performing Organization Name and Address
Leo Kanner Associates
Redwood City, California 94063
11. Contract or Grant No.
N AS W-319 9
12. Sponsoring Agency Name and Address
National Aeronautics and Space Admin-
istration, Washington, DC 20546
13. Type of Report and Period Covered
Translation
14. Sponsoring Agency Code
15. Supplementary Notes
Translation of "Algoritmy opredeleniya vremennykh
zapazdyvanii signala pri ispol'zovanii neodinakovykh
detektorov," USSR Academy of Sciences Institute of
Space Research, Moscow, Report PR-413, 1978, pp. 1-18
16. Abstract
In treating the recorded results from detectors at differ-
ent locations in space, the analysis of the time delays of
signals is crucial to locating the sources of detected
radiation. Because the correlation method requir.es the
manipulation of awkward matrices to evaluate its accuracy,
the present work outlines a solution based on minimizing
the sum of the squares of signal deviations, and presents
the algorithms for evaluating the resulting .error.
17. Key Words (Selected by Auttior(t)) 18. Distribution Statement
Unclassified-Unlimited
19. Security Clossif. (of this report)
Unclassified
20. Security Clossif. (of this poge)
Unclassified
21. No. of Pages
19
22. Price
NASA-HQ
ALGORITHMS FOR THE DETERMINATION OF THE TIME
DELAYS OF THE SIGNAL WHEN USING UNEQUAL DETECTORS
B. L. Novak,
USSR Academy of Sciences, Institute of Space Research
In determining the time delays in the reception of /2*
signals from detectors located in space, it is often advis-
able to take account of the circumstance that the detectors
differ both in sensitivity and in time resolution.
The present work contains a method for determining delays
which was developed for just such a case. The method is based
on minimizing the sum of the squares of the discordances of
the signals recorded by different detectors. The results of
an investigation of this, method are presented by the method
of maximizing the correlation of the signals. The computed
algorithms .for the evaluation of the accuracy of the solution
are also presented, once obtained by the means shown in the
work.
1. There is a whole series of well-known problems whose /3
solutions require the determination of the time delays of
signals detected by several receivers. A related problem is
the localization in the celestial sphere of the sources of
gamma-ray bursts [1].
In reference [2] the method for determining delays and
the procedure of evaluating accuracy were developed for the
case of equivalent detectors. As in [2], we will deal here
with detectors having discrete signals (counters), but, in
* Numbers in the margin indicate pagination in the foreign text.
contrast to [2], we shall take into account that the detectors
may vary in sensitivity (that is, in the probability of record-
ing) and in time resolution.
Let Wj l } «=i,2 ; y = 1 , 2,. . ,Afy represent the sequence
M^ of readings of the ith detector with a time resolution
At'1' . We will interpret this sequence as generated by a
random Poisson process with several variable intensities
x(i)(.) which are unknown beforehand. It is clear that the
results produced by the functions A^1^*) assumed above will
differ for different i in displacement of the argument and
in factors; that is, there also exist a certain range of T
and B such that all values fulfill the equality
)(") • • (1)
Frequently the range of T can be limited to a certain region
! '.
ITI < T0 , (2)
where TO is a known value. We shall assume the existence of /4
just such a situation.
The problem of finding the time delays therefore reduces
to determining the values of T satisfying equation (1) by using
the measurements {•n/'M for an unknown 3 and fulfilling
equation (2).
Let us examine the possible approaches to the solution to
the stated problem. We notice that any method must be based
on a comparison of the measurements I7*/'?} and {"/ j for dif-
ferent values of the indices, and this reduces to the range
of those values H* of the index £ at which the sequences |ny J-.
and .;{//.'+£J correlate, in some sense, best with each other.
Here the value of the delay which is sought is obviously
determined exactly at the value At'^) by the expression T* =
We will not yet take into account the possibility of
different time resolutions in the detectors, but assume here
that At<2) = Atd) = At, MI = M2 = M, k = J -£jl (where [•]
designates the whole integer part of any number) , and m =
M - 2k.
Since equation (1) contains the unknown factor 3, it
is natural to use the method of maximum correlation over the
range of T. Let the index £ vary from 1 to 2k + 1. Desig-
nate the coefficient of correlation corresponding to a given
£ by r£. Then
(3)
If max r., = r~, then the solution to the problem using /5
the method of maximum correlation is T = £At.
The requirement of evaluating the accuracy of the solution
arrived at, however, complicates the use of this method. In
fact, since rfl is the realization of a random value, in orderJC
to evaluate the accuracy of the solution one must.know its
probability value. Analytically obtaining such characteristics
is a very complicated problem, however. A study of the distrib-
ution of r? by means of statistical modeling is also obviously
unrealistic because of the large (2M) number of parameters to
be defined. These circumstances force us to turn to other
methods of locating T.
If we try to find the solution to the problem by the
method of least squares, this reduces to the minimization
in the given case of the expression
(4)
with respect to £ and: 6 . Let us first prove that the solution
to the problem (1), (2) by the method of least squares is
equivalent to that of the method of maximum correlation.
That is, the same value of the index & represents the maximum
and minimum of the respective expressions (3) and (4).
At the maximum of the range of z(;/ )'• the .condition-is
rn
" '"" * - o .
VJhence
t> = nt
E
r »
Substituting this expression into (4), we arrive at the problem /6
rn
m
or
n i "V t _»" -*«*/ 1 f ~V * _ »" —'*'
•n
«l
m ' TO
The latter is equivalent to the problem
m
m
Sn
Comparing this expression with (3) , we may conclude that the
value of the index H which represents a minimum in (4) is/\
equal to £.
The use of the method of least squares makes it possible
to evaluate the accuracy of the solution in the following
manner: replace ' .ii, in (4) with the expression
ni
m
and base the subsequent procedure of evaluating accuracy on
the assumption that
,C2>° /7
We note that it .is also reasonable to represent 6 with
$"/"the expression -&' „> . Hereafter we will assume that one
- £"'•
of these substitutions has been performed.
Let us now consider the effect of differences in the time
resolutions of the detectors on the procedure of determining
the delays and the evaluation of the accuracy. We assume
that At'^' = rAt'2) (5), where r is an integer. We note
that the more general case, where r is not an integer/: can be
reduced to the procedure indicated above. Henceforth MI
shall be designated M and Mi~2k as in. We designate ^ ' »[*'
as ' nt. for j = 1,2,..., m. f=G-Or>'
It is . clear that, in order to satisfy relationship (5)
and to use the method of least squares, the procedures indi-
cated above must be based on the minimization of, rather
than the expression (4) , of the expression
m
2. We now pass to the problem of evaluating the accu-
racy of the solution. For subsequent analysis it is desirable
to have the normalized random values which, as in reference
[2], we obtain by normalizing the sequences { rt/0} and.\ { n
 e
by multiplying by l/ |/$/° and ; i / V ^ f j i respectively.
Here n^ (!) .is. the/ averaged value of n^ '1^ , and no is thej j j
averaged value of n£ .
Using these normalized quantities, we then write
...... . ---- -r
x(*>= (6) /a
and adopt its use in studying probability characteristics
We shall designate random values with capital latin
letters. Let the parameter of the Poisson value N. (2) be
equal to vij'2)f the parameter of N . (1) be equal to gy^ (1)
and the parameter of Nfl be equal to y 0 . Consequently
and
Whence /
-
m
," ; a
"* "- -
-AT =.£v"
and the value x(^)/ defined by equation (6), is the realization
of the random values with a non-central chi-square distribution
with m degrees of freedom and the parameter of non-centrality:
V, =
The value of the parameter ve can serve as a measure of the
accuracy of the determination of T. Indeed, we will assume
the relationship - --
is fulfilled for H* satisfying a ,
f
Let us now examine the consequences implied by the use /9
of counting devices. For this determination we assume that
such a device is equipped with a second detector and that the
coefficient of counting is k.
We will assume. [3] that
Then
Clearly, with a counting device the expressions
tn
* ,' • . and/or
give an estimate for B^, not for 3- Therefore
(7)
just as equation (6), is the realization of the random values
with a non-central chi-square distribution with m degrees of
freedom. Thus, the use of counting devices leads merely to
the appearance of the factor 1/k in the second element of the
denominator of the coefficient in front of the summation sign
in the expression for the quantity x(^)- Henceforth, we will
use expression (7) for x(^)/ assuming k equivalent units in
the absence of a counting device.
Let us return to the problem of evaluating the accuracy /10
of the solution. As in reference [2], we will use two methods
to evaluate the accuracy of the determination of T. The pro-
cedure of the first method requires an, insignificant replacement
in the method of the reference mentioned. The interval of
confidence in v,, is determined by the formula
where
 :x^ are the upper and lower limits of confidence of
the non-centrality parameter, v.,, £a is a quantum on the order
of (2-a)/2 of the normal distribution, and 1-a is the coeffi-
cient of confidence.
We now form the system of intervals ^e=lT C^ ~4)At[ *T<(&J)A<. j
and examine , J"'^  =_lKJJj' , which is the union of all those Js
for which the condition
K ~ ti \j + { Q ^$ * H» \ y i
is fulfilled.
The error, AT, when this method.of evaluating accuracy
is used, is determined by the condition of affiliation of T to
the expression j(l).
The calculation formulas of the second method for evaluating
the accuracy of the determination of T are substantially more
complicated than those in reference [2]. We introduce the
designations:
V,.-Si.
'm*
. . . Oil ... i 100 . . . 0 ,00,..000...
_ > . . 1 00...000...0 I .
n
"*'^  \00~ ~ ~ ~00o" •"••" "060" "'" ~0~ " ""TT" 'thn"
Using these quantities, the quantity
in the form
T
can
 ^
e
 rewritten
i
In addition, we introduce the designations
n 711
o (
nJ
n
"
=J.» M
Since P(!)Q(!) = E, where E is the unit matrix, in the new
terminology
• r- TT Tr o,> m
1 0(0n
that is,
_J'
I.-*-
0(1)
n .
n
10
So we obtain for xW expression
1
1
 '*?.AtX
where
s 9 ?/•=
is the realization of the random normal vector with components
of unit dispersion. Henceforth, elements of the diagonal(a) (a)
matrix P will be designated as pn- ' = l,2,...,m, and the
/12
elements of the diagonal matrix Q
1, 2,...,M.
will be designated
The evaluation of the accuracy in determining t is then
performed using the statistic
o
n
(10)
/**
where ass'^^ and ass^ ^ are diagonal elements of the matrices
A., and A^, respectively. Let ?a be a quantum on the order of
1-a of the normal distribution, oi is the error of the second
order, and Js, as in the first method of evaluating accuracy,
is the interval
T (,S-OAt(2)«T*(,S+i)AiJ2)}
if
(ID
11
then the corresponding interval Js will be included in the
union -7"™*>VJ~3 , and the error AT is determined by the condi-
tion of affiliation of T to the expression j(2).
3. Since the dimensions of the matrices A0 and A^ in
X/ As
expression (10) - 2M - are very large numbers, operating with
these matrices immediately is difficult. In this connection
we can obtain simple expressions describing the dependence of
the statistic T? on the sequential measurements |^ ri)) and Jrj^
We designate rnin{£,£) as £j and max If,f} as £2/ so that
£2 >. £1 . Clearly,
" 713
_
,y'')r-/,/^
If the non-zero blocs of this matrix are designated A^ ' ,
A(!)T
 an<j A' ' , the matrix A. - A^. is rewritten in the formx, x,
And if we introduce the designation V for A("r>n] • and W for
:A
wTn(t\A(*r,(7} , then the quadratic form nT(A,-A^ n. from
expression (10) is obviously written in the form WT + WWT.
The formulas for the calculation of this quadratic form depend
on the relationships among £1, £2/ M/ m and r.
Let eo = \ 3~ "Mr. where [•] is the integral part of the
number, and S = £ 2 - & i - & o .
1) If the relationship £i + rm - 1 ^  £2 is fulfilled,
then
12
W =i
'••:'•'- \2\Vv^-y )
.' 4/r/
VA2
V
 "
-fjr
rrrj.J
2 .°<«) x
e _-
'"V.« 2
( »/-*rm«5
2 ^y=f*-Or»rw5'l
(12)
(13)
2) If the relationship £1 + rm - 1 < £2 is fulfilled,
then
13
(14)
(15)
Let us pass to the calculation of «$/» (\ -Ap)-. : from expres
sion (10) . As is well known, any quadratic ma trie B of dimen-
sion n with elements &.. fulfills the condition
'
fl
'=2N'
. V"
It follows from this equation that
A/
are the elements of the matrix A., - A~,Here 6a
elements of the matrix
matrix .
(1)
and
are
are elements of the
14
We obtain from the last equation the following /16
••' *»*" / ' x- *** '.-••••••• ::; "»-4>
-«
f
 2 ("«"' . 2- . • £ < + «•• 2
. , 4-i A l.}*t(*-\)r ,#M
2
ese.*r
- 2
-
.. .
- 2
.
v - i " «
2 . '^ (16)
for
for £i + rm - 1 f
(17)
Finally, for the final factor in brackets in expression (1Q)
15
the correct formulas are /17
j» * ••• j
V «
 D«>
4
 '2 * + </-i
(18)
for. £1 + rm >. £2 and
(19)
for Jl i + rm < £
 2 •
Thus, the algorithm of the accuracy of the second methods
consists of calculating the statistic (10) for all values of
Si, using formulas (12), (13), (16) and (18) for A i + rm ,> £2
and (14), (15), (17) and (19) for £ i + rm < SL2f while checking
that condition (11) is fulfilled.
16
REFERENCES
1. Mersov, G.A., and B.L. Novak, "Localization of the
celestial sources of burst radiation by means of
several space receivers," in the book Matematicheskoe
obespechenie kosmicheskikh eksperimentov ("Mathematical
Treatment of Experiments in Space"), Moscow, Nauka, 1978.
2. Novak, B.L., "Opredelenie vremen prokhozhdeniya gamma-
vspleskom razlichnykh tochek kosmicheskogo prostranstva,"
USSR Academy of Sciences Institute of Space Research,
Moscow, Preprint, Report Pr-352, 1977.
3. Gol'danskii, V.I., A.V. Kutsenko, M.I. Podgoretskii,
Statistika otschetov pri registratsii yadernykh chastits
("The Statistics of Readings from Measurements of
Nuclear Particles"), Fizmatgiz, Moscow, 1959.
IT


Algorithms for the determination of the time delays of the signal when using unequal detectors

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