I. INTRODUCTION
Distribution of the Envelope of
a Sum of Random Sine Waves
and Gaussian Noise
A certain narrowband process has the form
Re V0 (t)eit , with the carrier frequency and V0 (t)
the complex envelope of the process. Sampling
the envelope at time t yields the circular complex
conditionally Gaussian random variable
V = Aei£0 +
M
X
bk ei£k + N:
(1)
k=1
CARL W. HELSTROM, Life Fellow, IEEE
University of California, San Diego
The cumulative distribution of the envelope of the sum of a
sinusoidal signal, a number M of randomly phased interfering
sine waves, and, possibly, Gaussian noise is expressed as the sum
of Marcum’s Q-function and an asymptotic series of Laguerre
polynomials, much like the ordinary Edgeworth series for the
distribution of the sum of a number of independent random
variables. A test of the method with M = 20 showed that its
computation requires about 2% of the time needed for numerical
inversion of the characteristic function of the distribution.
Here A is a fixed amplitude; the M amplitudes bk
either are fixed or are independent, real random
variables; the phases £k , 0 · k · M are independently
random and uniformly distributed over (0, 2¼); and N
is a sample of the complex envelope of narrowband
Gaussian noise, whose real and imaginary parts have
expected value zero and variances equal to ¾ 2 . The
first term in (1) we call the signal, the second term
the interference. The array of coefficients bk we call
the interference spectrum, although these are not
necessarily associated with different frequencies.
That first term might represent the strong reflection
of a radar wave from a target, and the interference
might consist of random reflections or glint from
other points of the target or from nearby objects. In
a communication system the first term might represent
an information-bearing signal, with which leakage
from signals in adjacent channels interferes.
In studying such phenomena it is useful to know
the cumulative probability distribution P 0 (r) = Pr(r < r)
of a sample r = jVj = jV0 (t)j of the envelope of that
narrowband random process. Rice [1] stated that it is
given by the Fourier—Bessel transform
Z 1
0
J1 (ru)©(u) du
(2)
P (r) = Pr(r < r) = r
0
where ©(u), the characteristic function of the real and
imaginary parts of V = x + iy,
©(u) = E(eixu ) = E(eiyu )
is [1]
©(u) = J0 (Au)
M
Y
2 2
hJ0 (bk u)ie¡¾ u =2
(3)
k=1
Manuscript received September 14, 1997; revised June 22, 1998.
IEEE Log No. T-AES/35/2/04310.
Author’s address: Dept. of Electrical and Computer Engineering,
University of California, San Diego, La Jolla, CA 92093-0407.
c 1999 IEEE
0018-9251/99/$10.00 °
594
where angle brackets indicate an expected value
with respect to the distribution of the interference
amplitudes bk , if they happen to be random variables.
Here Jn (¢) is the Bessel function of order n. Rice’s
paper [1] provides references to the derivation of these
equations. Computing the distribution P 0 (r) of r by
numerically integrating (2) is time-consuming because
of the necessity of evaluating a great many Bessel
functions. Our aim here is to provide a more efficient
numerical method for determining P 0 (r).
Kluyver [2] considered the case of M randomly
phased sine waves with a uniform spectrum in the
absence of noise and gave the corresponding version
of (3), and Pearson [3] derived from it six terms of
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a series expansion for P 0 (r) in terms of Laguerre
polynomials. Their work was cited by Greenwood and
Durand [4], who determined a power series expansion
for P 0 (r) from that Laguerre series. Slack [5], treating
the same problem, presented approximations to P 0 (r)
in terms of beta functions for M · 12.
For a communications application Bennett [6]
derived a Fourier—Bessel series for the distribution
function P 0 (r) with an arbitrary, but nonrandom
interference spectrum (A = 0), and Barakat [7] applied
the same method to the distribution of the intensity in
laser speckle patterns; for both, noise was absent. A
Fourier—Bessel series for P 0 (r) that takes account of
the presence of additive Gaussian noise was developed
by Bird and George [8] and applied to determining
error probabilities in an amplitude-shift keying
system with cochannel interference. The terms of a
Fourier—Bessel series are ratios of Bessel functions
whose arguments are proportional to the zeros xk of
the Bessel function, J0 (xk ) ´ 0, 1 · k < 1.
Goldman [9] developed a series of Laguerre
polynomials for P 0 (r) that includes and indeed
requires the presence of Gaussian noise and which
permits the amplitudes of the interfering sine waves
to be random variables. Simon [10] showed how the
probability density function of the envelope in the
absence of random noise can be calculated recursively,
starting with M = 1.
Calculating the distribution P 0 (r) for r near its
maximum value
M
X
jbk j
(4)
B=
k=1
is a difficult problem addressed by Rice in [11] and
[1]. (We call B the span of the interference spectrum.)
In the former he treated a uniform spectrum (A = 0)
in the absence of noise (¾2 = 0), and in the latter he
generalized his result to an arbitrary spectrum and
showed how additive Gaussian noise could be taken
into account. In [1] Rice also treated the numerical
evaluation of the Fourier—Bessel integral (2) by means
of the trapezoidal rule.
The distribution of the envelope of the sum of two
sine waves and Gaussian noise has been treated by
Esposito and Wilson [12], Price [13], Bird [14, 15],
and the writer [16]. The method described here,
requiring the number M to be greater than about ten,
is insufficiently accurate for that problem.
Here we present asymptotic expansions for the
cumulative distribution P(R) = Pr(R = 12 r2 < R) and
its complement Q(R) = Pr(R ¸ R) that resemble
Edgeworth’s series. Edgeworth’s series consists
of corrections to the Gaussian distribution for
random variables that are sums of a large number
of independent random variables [17]. When in (1)
the number M of interfering sine waves is large, the
distribution of the envelope r = jVj has approximately
the Rayleigh form for A = 0 and the Rayleigh-Rice
form [18, eq. (3.10—11)] for A > 0. Our expansions,
which we call quasi-Edgeworth series, represent
corrections to those forms that are useful when M À
1. The key step in establishing them is to combine
the variances of the real and imaginary parts of the
sinusoidal components in (1) with those of the circular
complex Gaussian noise N; see (10) and (22) below.
II. QUASI-EDGEWORTH SERIES
We show in the Appendix that the cumulative
probability distribution P(R) = Pr(R < R) of the
squared envelope R = 12 jV0 (t)j2 = 12 jVj2 at an arbitrary
time t is given by the asymptotic expansion
1
P(R) = 1 ¡ Q(S=p, R=p) +
R ¡(S+R)=p X (¡1)m m! cm
e
p
pm
m=2
¢
1
k
X
(R=p)k X (S=p)n k+n
Lm ((S + R)=p),
(k + 1)!
n!
k=0
n=0
S = 12 A2
Z
where
1
Q(y, x) =
x
(5)
p
e¡(y+t) I0 (2 yt) dt
(6)
is a form of Marcum’s Q-function [19, 20, eq.
(C-19), p. 526], I0 (¢) being the modified Bessel
function. It can be computed as shown in [20,
sect. C.3, pp. 528—530]. Furthermore, L®m (¢) is the
associated Laguerre polynomial as in [21, sect. 10.12,
pp. 188—192].
The coefficients cm are determined in the following
way. With h¢i denoting an expected value with respect
to the distribution of the interference amplitudes bk ,
we define the coefficients
®km = h( 12 bk2 )m i,
gm = 1=m!2 ,
1·k·M
(7)
and
¯m =
M
X
¯km ,
(8)
k=1
with the ¯km calculated by the recurrence
¯k1 = g1 ®k1 = h 12 bk2 i
¯km = gm ®km ¡
1
m
m¡1
X
(9)
n¯kn gm¡n ®k,m¡n ,
m > 1:
n=1
Furthermore, the quantity p in (5) is
p = § 2 = ¾2 + ¯1 :
(10)
The coefficients cm in (5) are now given by the
recurrence
c1 = 0,
c2 = ¯2 ,
c3 = ¯3
1X
cm = ¯m +
n¯n cm¡n ,
m
m¡2
m > 3:
(11)
n=2
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The complementary cumulative distribution
Q(R) = Pr( 12 r2 ¸ R) is given by a similar asymptotic
expansion
1
X
(¡1)m m! cm
Q(R) = Q(S=p, R=p) + e¡(S+R)=p
pm
value of the last term added in, but we found that that
assumption somewhat underestimated the true error.
Generally speaking, if that absolute value is greater
than about 10¡4 times the total probability, the result
is untrustworthy.
m=2
¢
1
X
k=0
(S=p)
k!
k
k X
n=0
(R=p)n k+n¡1
((S + R)=p),
Lm
n!
S = 12 A2 :
(12)
Some advice about computing the Laguerre
polynomials is given at the end of the Appendix.
When the interference amplitudes bk are not
random variables, but fixed, the coefficients in (5)
and (12) can be more expeditiously computed by first
determining the numbers ¯m0 by the recurrence
¯10 = g1 = 1
1X 0
n¯n gm¡n ,
m
m¡1
¯m0 = gm ¡
m>1
(13)
n=1
and then forming the coefficients
¯m = ¯m0
M
X
( 12 bk2 )m ,
m ¸ 1:
(14)
k=1
Then p = § 2 is again as in (10).
The factors (R=p)k =(k + 1)! in (5) increase with
the index k until k + 1 equals the integral part bR=pc
of R=p; beyond that they decrease to zero. The
summation over k in (5) is stopped when the absolute
value of the latest term added falls below " = 10¡8
times the absolute value of the accumulated sum,
but in order to avoid premature termination of the
summation, the index k is required to have passed
bR=pc. The summation over k in (12) is terminated
in the same manner, except that k is required to have
surpassed the integral part bS=pc.
Like Edgeworth’s series [17], (5) and (12) are
asymptotic expansions: their terms decrease for awhile
and then begin to increase, whereupon the series
begins to diverge. One therefore faces the problem of
telling one’s program when to stop its summations
over m. We used two criteria. The summation is
stopped when the absolute value of the current term
is less than 10¡8 times the absolute value of the
accumulated sum or–provided the number m of
terms is greater than ten–when the absolute value of
the current term exceeds the average of the absolute
values of the previous nine terms. Our method is
reliable only for numbers M of interference terms
equal to about ten or greater. Except in the tails of the
distribution the former criterion usually terminated the
summation.
It is difficult to estimate the error incurred by
thus stopping the summation. It is often said that the
error in an asymptotic series is less than the absolute
596
III. EXAMPLES
In Table I we list values of the cumulative
distribution P 0 (r) = Pr(r < r) and its complement
Q0 (r) = Pr(r ¸ r) for the envelope of the sum r = jVj
of M = 20 sine waves with amplitudes bk ´ 1 as
computed by evaluating the quasi-Edgeworth series
(5) and (12) and by numerical integration. The
columns headed “NT” list the number m of terms
taken in those series before termination. The left-hand
group of columns lists the distributions in the absence
of Gaussian noise (¾ 2 = 0); the right-hand columns list
them for ¾ 2 = 10. Table II lists the same distributions
with the additional presence of a single strong sine
wave of amplitude A = 20.
For the left-hand tail of the distribution of r = jVj
we numerically integrated (2). For the right-hand tail
we integrated
Z 1
d©(u)
Q0 (r) = 1 ¡ P 0 (r) = ¡
(15)
J (ru) du
du 0
0
#
"M
X b J (b u) AJ (Au)
d©(u)
k 1 k
¡
= ©(u)
+ 1
+ ¾2 u
du
J0 (bk u)
J0 (Au)
k=1
(16)
which follows by integrating (2) by parts [6]. Here we
are assuming fixed amplitudes bk for the interference.
With the quasi-Edgeworth series (5) and (12),
computing each group of values of P 0 (r) or Q0 (r)
took from 1 s to 3 s on a 66 MHz IBM PS/1. The
numerical integrations were carried out in MathCAD
2.1, with which computing each group required from
1.5 m to 4 m. It was necessary to pick an upper limit
for each integral beyond which the values of the
integrand became negligible, but not so large that
underflow occurred. All those results were confirmed
in a Sun OS5.5 computer by numerical integration in
Mathematica 3.0, which permits specifying an infinite
upper limit.
Fig. 1 displays the complementary cumulative
distribution Q0 (r) versus r2 =M for a number of
values of the number M of sine waves (bk ´ 1) in the
absence of noise and of any strong signal (¾ 2 = 0,
A = 0). The curves approach straight lines as M
increases and the distributions approach the Rayleigh
form. The ‘+’ signs for M = 10 indicate values of
Q0 (r) calculated from Rice’s series in [11, (4.2)].
With equal interference amplitudes bk ´ 1, his series
is applicable only for B ¡ 2 < r < B, B = M the span
given in (4), and when M ¸ 20 the resulting values of
Q0 (r) lie far below the range plotted in Fig. 1.
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TABLE I
Distribution of M = 20 Sine Waves
A=0
r
Edgeworth
NT
num. integ.
Edgeworth
¾2 = 0
P 0 (r)
0.01
0.05
0.1
0.5
1
2
3
4
5
6
8
10
12
14
16
18
20
22
24
bk ´ 1
NT
num. integ.
¾ 2 = 10
P 0 (r)
4.87608E-06
0.000121895
0.000487493
0.012118
0.047619
0.177588
0.356763
0.545161
0.710008
38
38
38
38
31
26
27
28
21
4.87608E-06
0.000121895
0.000487494
0.012118
0.047619
0.177588
0.356763
0.545161
0.710008
2.48414E-06
6.21017E-05
0.000248384
0.00619124
0.0245374
0.0946173
0.200472
0.328319
0.46328
19
19
19
19
19
19
19
19
19
2.48414E-06
6.21017E-05
0.000248384
0.00619124
0.0245374
0.0946173
0.200472
0.328319
0.46328
Q0 (r)
0.1661
0.0387042
0.00543121
0.000415952
1.44999E-05
1.53681E-07
32
30
26
24
31
30
0.1661
0.0387042
0.00543121
0.000415952
1.44998E-05
1.53683E-07
0.407845
0.2023
0.081751
0.0268211
0.00711355
0.00151746
0.000258832
3.50668E-05
3.74571E-06
3.12906E-07
Q0 (r)
13
13
13
12
13
11
13
13
13
13
0.407845
0.2023
0.081751
0.0268211
0.00711355
0.00151746
0.000258832
3.50668E-05
3.74571E-06
3.12906E-07
TABLE II
Distribution of M = 20 + 1 Sine Waves
A = 20
r
Edgeworth
NT
num. integ.
Edgeworth
¾2 = 0
P 0 (r)
4
8
12
16
20
24
28
32
36
40
¾ 2 = 10
P 0 (r)
3.86351E-09
2.2817E-05
0.00392284
0.088875
0.468262
34
28
27
27
24
3.86371E-09
2.2817E-05
0.00392284
0.088875
0.468262
6.15145E-05
0.00207732
0.0265673
0.154103
0.455075
19
19
19
19
19
6.15145E-05
0.00207732
0.0265673
0.154103
0.455075
Q0 (r)
0.117604
0.00632647
4.76559E-05
1.20572E-08
27
13
28
34
0.117604
0.00632647
4.76558E-05
1.20578E-08
0.214699
0.0450635
0.00459746
0.000212522
4.16938E-06
Q0 (r)
13
13
12
13
13
0.214699
0.0450635
0.00459746
0.000212522
4.16938E-06
As a final example we consider the detectability
of a sine wave of amplitude A in the presence of
nineteen interfering sine waves having a triangular
spectrum,
bk = b20¡k = 0:1k,
bk ´ 1
NT
num. integ.
1 · k · 10,
M = 19:
(17)
The receiver decides that the signal is present when
the envelope r = jVj of its input exceeds a certain
decision level r0 , which is set so that the false-alarm
probability Q0 = Pr(r > r0 j A = 0) = 10¡6 . The average
power ¯1 of the interference equals 3.35; see (8).
Fig. 2 exhibits the false-dismissal probability Q =
Pr(r · r0 j A) versus the signal-to-interference ratio
A2 =2¯1 as the curves marked “T”. The curves marked
“U” represent the false-dismissal probability Q for
a uniform spectrum of nineteen sine waves having
the same average power 12 Mbk2 ´ 3:35; bk ´ 0:59383,
and for the same false-alarm probability Q0 . For
the left-hand pair of curves, noise is absent, ¾2 = 0;
for the right-hand pair ¾ 2 = 1. The detectability of
the sinusoidal signal is greater with the triangular
interference spectrum than with the uniform one
because the span B of the former is the smaller and
a fortiori so is the decision level r0 . When noise is
present, ¾2 > 0, the dependence of the false-dismissal
probability Q on the form of the interference spectrum
is diminished.
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Fig. 1. Complementary cumulative distribution Q0 (r) of envelope
of sum of M randomly phased sine waves of unit amplitude
versus r2 =M. Curves are indexed with values of M. Values of
Q0 (r) for M = 10 calculated by Rice’s formula [11] are marked
with +.
IV. CONCLUSION
The envelope of a single sine wave in the presence
of narrowband Gaussian noise has a Rayleigh-Rice
distribution. When on the other hand there is no
strong signal, but a random process consisting of a
large number of randomly phased sine waves and
Gaussian noise, the envelope of the resultant has
nearly a Rayleigh distribution because the sums of
their in-phase and quadrature components are nearly
Gaussian random variables. In this work we have
presented series approximations to the distribution
of the envelope of an input such as (1) that span
those two extremes. They apply even in the absence
of Gaussian noise (¾2 = 0), but they require that
the number M of interfering sine waves be larger
than ten or so. Comparisons with the results of
numerically integrating (2) and (15) indicate that these
quasi-Edgeworth series are then accurate over a useful
range of probabilities.
Fig. 2. False-dismissal probability Q in detection of sinusoidal
signal of amplitude A when the 19 interfering sine waves have
triangular spectrum (17) (curves marked “T”) or uniform spectrum
(curves marked “U”). Here ¯1 is average power of interfering sine
waves. That average power ¯1 and false-alarm probability
Q0 = 10¡6 are same for both. Curves are indexed by noise
variance ¾2 .
we write
ln ©int (u) =
M
X
lnhJ0 (bk u)i
k=1
=
1
X
¯m (¡v)m ,
¯m =
M
X
¯km
k=1
in which the ¯km are calculated by the recurrence in
(9) as in [20, eq. (5-16), p. 193].
We can then write (3) as
©(u) = J0 (Au)©int (u)e¡¾
¡§ 2 u2 =2
= J0 (Au)e
2 2
u =2
"
exp
§ 2 = ¾ 2 + ¯1 = ¾ 2 +
1
X
=1+
(¡1)m (m!)¡2 hbk2m i( 12 u)2m
ln J0 (x) =
m
m=1
v = 12 u2
(18)
with gm and ®km as in (7).
First treating the interference terms,
©int (u) =
hJ0 (bk u)i
k=1
598
M
X
h 12 bk2 i:
(22)
1
X
¯m0 (¡ 14 x2 )m
(23)
m=1
gm ®km (¡v) ,
M
Y
(21)
When the interference amplitudes are nonrandom,
we can use the expansion in (18) after setting all the
bk s in that equation equal to 1, and then we can write
m=1
1
X
#
¯m (¡ 12 u2 )m
k=1
Writing out the power series for the Bessel
function J0 (¢) and averaging, we obtain
hJ0 (bk u)i = 1 +
1
X
m=2
with
APPENDIX
(20)
m=1
in which the coefficients ¯m0 are calculated by the
recurrence in (13). When we put x = bk u, we obtain
ln J0 (bk u) =
1
X
¯m0 ( 12 bk2 )m (¡v)m ,
m=1
(19)
(24)
v = 12 u2 :
Summing over k and using (14), we again obtain (21).
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Using the recurrence in (11) as in [20, eq. (5-17),
p. 193] to take the exponential function of a power
series, we can now write the characteristic function in
(3) as
"
#
1
X
2 2
©(u) = J0 (Au)e¡(1=2)§ u 1 +
cm (¡ 12 u2 )m
m=2
(25)
and this we substitute into (2).
For convenience replacing § 2 by p, we can write
(2) as
#
"
1
m
X
d
P 0 (r) = 1 +
(26)
cm m I(p)
dp
m=2
with
Z
1
0
J1 (ru)J0 (Au)e¡pu
Z
r
=
1
x dx
0
Z
=
0
Q(S=p, R=p) =
1
k
X
(S=p)k X (R=p)n ¡(S+R)=p
e
k!
n!
k=0
n=0
frm [20, eqs. (C-30)—(C-32), pp. 528—529] with M = 1
[23]. Again using (28), but now with ® = k + n ¡ 1,
we obtain (12).
For A = S = 0, (12) reduces to
#
"
1
X
(¡1)m m!
¡R=§ 2
1
2
c
L (R=§ )
1+
Q(R) = e
§ 2(m+1) m+1 m
m=1
(30)
when one uses
xL1m¡1 (x) = ¡mL¡1
m (x)
I(p) = r
Z
and into it we put
0
r
2
=2
which follows from [21, eqs. 10.12(23), 10.12(24),
p. 190]. Had we not combined the term ¯1 with ¾ 2 as
in (21), we should have obtained a result similar to
Goldman’s equation [9, eq. (27)] with n = 2.
The computations of (5) and (12) begin with
evaluation of the first Laguerre polynomials,
du
uJ0 (xu)J0 (Au)e¡pu
2
=2
du
· 2
¸
x + A2
xp¡1 exp ¡
I0 (Ax=p) dx
2p
L®0 (x) ´ 1,
® = ¡1, 0, 1
= 1 ¡ Q(S=p, R=p),
S = 12 A2 ,
in terms of Marcum’s Q-function (6). Here we have
used [21, eq. 7.7.3(25), p. 50]. Thus
P(R) = 1 ¡ Q(S=p, R=p)
+
m=2
dm
cm m [1 ¡ Q(S=p, R=p)]
dp
(27)
and into this we put
1 ¡ Q(S=p, R=p) =
(31)
[21, eq. 10.12(6), p. 188], from which the subscript
index is advanced by the recurrent relation
R = 12 r2
1
X
L®1 (x) = ® + 1 ¡ x,
L®m+1 (x) = (m + 1)¡1 [(2m + ® + 1 ¡ x)L®m (x)
¡ (m + ®)L®m¡1 (x)]
(32)
[21, eq. 10.12(8), p. 188]. For the innermost
summations in (5) and (12) one must advance
the superscript ® = k + n or ® = k + n ¡ 1 by the
recurrence
¡1
®
®¡1
L®+1
m (x) = x [(® + x)Lm (x) ¡ (® + m)Lm ] (33)
1
k
X
(R=p)k+1 X (S=p)n
k=0
(k + 1)!
n=0
n!
e¡(S+R)=p
from [20, eqs. (C-33), (C-34), pp. 529—530] with
M = 1 [22].
Here we need
dm ¡a=p ¡(®+1)
[e
p
] = (¡1)m m! p®+1+m L®m (a=p)e¡a=p
dpm
(28)
with a = S + R, ® = k + n in [21, eq. 10.12(26),
p. 190]; L®m (¢) is the associated Laguerre polynomial.
Putting this into (27) we obtain (5).
From (27) we can write down the complementary
cumulative distribution
"
#
1
X
dm
Q(R) = 1 ¡ P(R) = 1 +
cm m Q(S=p, R=p)
dp
m=2
(29)
which derives from [21, eqs. 10.12(23) and (24)],
®
®
xL®+1
m (x) = (m + ®)Lm¡1 (x) ¡ (m ¡ x)Lm (x)
®¡1
= (m + ®)[L®m (x) ¡ Lm
(x)] ¡ (m ¡ x)L®m (x):
The array of values of the Laguerre polynomials
from each value of the index k in (5) or (12) is saved
for the following value of k (m fixed), and only the
k+n
new values of Lm
(¢) or Lk+n¡1
(¢) are computed at
m
each stage by using (33).
For a uniform interference spectrum, the
coefficients ¯k in (14) are of order M. The largest
contributions to the summation in (11) for m > 3
come from the terms with n = m ¡ 2, all the ¯n s being
of the same order M. Thus c4 and c5 are of order
M 2 , and a fortiori c2s and c2s+1 are of order M s .
Because p = § 2 in (5) and (12) is of order M, the
coefficients c2s =p2s in those summations are of order
M s =M 2s = M ¡s , and the coefficients c2s+1 =p2s+1 are of
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order M s =M 2s+1 = M ¡(s+1) . Thus for integral s
c2s¡1 =p2s¡1 = O(M ¡s ),
[11]
c2s =p2s = O(M ¡s )
and terms of equal order of magnitude M ¡s form
adjacent pairs. In the usual Edgeworth series ([17,
20, p. 194]) terms of the same order in M form
interlacing triplets, complicating the computations.
The orders of magnitude of successive triplets,
furthermore, decrease only by a factor M ¡1=2 ; those
of the successive pairs of coefficients in (5) and (12)
decrease by a factor M ¡1 .
[12]
[13]
[14]
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 2
APRIL 1999
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Carl W. Helstrom (SM’61–F’70–LF’93) was born in Easton, PA, on February
22, 1925. He received the B.S. degree in engineering physics from Lehigh
University, Bethlehem, PA, in 1947 and the Ph.D. degree in physics from the
California Institute of Technology, Pasadena, in 1951.
From 1944 to 1946 he was a radio technician in the U.S. Navy. From
1951 to 1966 he worked in applied mathematics at the Westinghouse Research
Laboratories, Pittsburgh, PA. On leave during the 1963—1964 academic year,
he lectured in the Department of Engineering, University of California, Los
Angeles. He joined the University of California, San Diego in 1966, where he
is now Professor Emeritus of Electrical Engineering. From 1971 to 1973 and
1974 to 1977 he was Chairman of his department, and during the 1973—1974
and 1986—1987 academic years he was Professeur Associé at the Université de
Paris-Sud.
From 1967 to 1971 Dr. Helstrom served as Editor of the IEEE Transactions on
Information Theory. He was Cochairman of the IEEE International Symposium
on Information Theory in 1982 and Program Chairman of that Symposium in
1990. He is a member of Phi Beta Kappa and a Fellow of the Optical Society of
America.
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