Page 158 - Radar Technology Encyclopedia

148 distribution, Nakagami distribution, Weinstock

and probability distribution function: clutter, and so forth. When a = 0, the Rician distribution
æ mx 2 ö evolves into the Rayleigh distribution. See CLUTTER
G m --------- ÷ (amplitude) distribution. AIL
ç
;
2
è s ø
F () ------------------------- x >, 0 Ref.: Skolnik (1980), p. 50; Tikhonov (1980), p. 40.
x =
N G m
()
The Student distribution is the distribution of random mag-
where G(×) is the gamma function, and G(××s the incomplete nitude x with probability density function:
i
)
gamma function.
A Nakagami distribution is used in radar detection G n 2 ¤ ) x 2 – n 2 ¤
(
theory. AIL f x () --------------------------------------------------------- 1 += æ ------------ ö
S
è
(
p n – 1 )Gn – 1 ) 2 ¤ ] n – 1 ø
[
(
Ref.: Bakut (1984), p. 341.
and probability distribution function:
normal distribution (see Gaussian distribution).
x
The Poisson distribution is the distribution of discrete ran- G n 2 ¤ ) t 2 – n 2 ¤
(
F x () --------------------------------------------------------- ò æ 1 + ------------ ö t d
=
dom magnitudes. The random magnitude x is considered to be S è n – 1 ø
(
(
[
p n – 1 )Gn – 1 )2 ¤ ]
distributed according to Poisson’s law if the probability that it – ¥
will take on specific value k is expressed as
where G(×) is the gamma function, and n is the distribution
k
m – m
=
P x () ------e parameter.
k k!
A Student distribution finds use when matching results of
where m = mean value of random magnitude x. an evaluation of radar parameters in radar tests. AIL
A Poisson distribution is used in radar signal detection
The uniform distribution in the interval (a, b) is the distribu-
theory and when calculating reliability and the required num-
tion of a random magnitude x with probability density func-
ber of spares for radar maintenance. AIL
tion:
Ref.: Korn (1961), p. 18.8.1; Tikhonov (1980), p. 30.
1
x
=
The Rayleigh distribution is the distribution of random mag- fx () ------------ , a ££ b
b – a
nitude x with probability density function:
= 0 , otherwise
x æ x 2 ö
f x () ------exp – ç --------- ÷x ³, 0 that is, it is constant within the specified interval. This distri-
=
R 2 2
s è 2s ø bution is often used for statistical evaluation of random phase,
and probability distribution function: which is considered to have uniform distribution within the
interval (0, 2p). SAL
æ x 2 ö
x
F () 1 –= exp – ç --------- ÷x ³, 0
R 2 The Weibull distribution is the distribution of random mag-
è 2s ø
nitude x with probability density function:
where s is the distribution parameter.
A Rayleigh distribution is used in a statistical description b – 1 b
– (
x =
f () abx exp ax ) a >, 0 b >, 0
W
of radar receiver noise and of the echo area of certain types of
targets, as well as in signal detection theory. See CLUTTER and probability distribution function:
(amplitude) distribution. AIL
b
x
Ref.: Lawson (1950), p. 53; Bakut (1984), p. 341. F () 1 –= exp – ( ax ) x >, 0
W
The Rician distribution is distribution of random magnitude where a and b are the distribution functions.
x with probability density function: A Weibull distribution is used in statistical description of
receiver noise, clutter, and reliability (malfunctions in radar
2
2
x æ x + a ö æ ax ö
=
f x () ------ exp – ç ----------------- ÷ I 0 è ------ , x ³ 0 assemblies and units). See CLUTTER (amplitude) distribu-
0
2 ø
2
2
s è 2s ø s tion. AIL
Ref.: Bakut (1984), p. 341; Schleher (1991), p. 34; Currie (1992), p. 22.
and probability distribution function:
The Weinstock distribution is a chi-square distribution for
¥
æ a 2 ö x æ a 2 öæ x 2 ö positive k values less than unity. Weinstock was the first to
2 å
;
=
ç
F x () exp – ç --------- ÷ ------------ --------- ÷ G a + 1 --------- ÷x ³, 0
ç
0
2
2
2
k! è
è 2s ø () 2s øè 2s ø observe that fluctuations of some simple targets (e.g., cylin-
k = 0 ders) can be described with values of k between 0.3 and 2, as
where I (×) is the Bessel function of the zero order, G(×;×) is the in previous radar practice the target RCS fluctuations were
0
incomplete gamma function, and a and c are distribution described by Swerling’s models, using k values of one or
parameters. greater. SAL
A Rician distribution sometimes is referred to as a gener- Ref.: Currie (1989), p. 236.
alized Rayleigh distribution and finds use in a statistical
divergence factor (see REFLECTION from the spherical
description of receiver noise, in signal detection theory, radar
earth).