Page 157 - Radar Technology Encyclopedia

distribution, chi-square distribution, Nakagami 147

Chebyshev distribution (see WEIGHTING). x ö
æ
G b + 1 --- ;
The chi-square distribution is defined by è b ø
=
F x () --------------------------- x >, 0
G G b + 1 )
(
k –
k æ kX ö 1 æ – kX ö
)
i
() ---------------------- ------
WX = è ø exp è --------- ø where G(×) is the gamma function, and G(××s the incomplete
( k – 1 ) !X X X gamma function.
A gamma distribution finds use in description of noise
where X > 0.
statistics, RCS fluctuations, and in radar reliability. AIL
For description of target RCS statistics, this distribution
Ref.: Skolnik (1980), p. 50; Tikhonov (1980), p. 36.
was introduced by Swerling, using for k a value of 1 (case 1
or 2 target) or 2 (case 3 or 4 target). In the convectional theory The Gaussian distribution is the distribution of random
of statistics the parameter 2k is typically called number of magnitude x with probability density function:
degrees of freedom, which must be an integer. This distribu- ( x – m )
2
x
tion, for example, can be used to describe the effect of nonco- f x () ----------------- exp –= 1 ----------------------
g
2
herent integration of fluctuating targets plus noise. SAL s x 2p 2s x
Ref.: Currie (1989), p. 235.
and probability distribution function:
clutter amplitude distribution (see CLUTTER (ampli- æ x – m x ö
tude) distribution). F x () = F --------------- ø
è
g
s
x
The drop-size distribution (DSD) describes the size of pre- where F(× is the Laplace function, m = mean value of x, and
)
x
2
cipitation drops and is widely used in radar meteorology. The s = variance of x.
typical notation is N(D), where D is the drop diameter. One of A Gaussian distribution is widely used for describing
the most general forms of the DSD is given by the modified noise, interference, radar coordinate measurement errors, and
gamma distribution other statistical parameters. It is also called the normal distri-
m q bution. (See CLUTTER (amplitude) distribution.) AIL
() N D exp=
ND – ( LD )
0
Ref.: Barton (1969), p. 210.
where L, m, and q are the distribution parameters. When m =
The K-distribution is defined by
0 and q = 1 the DSD assumes the exponential form
n 1+ n
b u
=
() N exp – (
ND = 0 LD ) f u () ------------------------K n 1– ( bu ) u ³ 0 n 0 b >,³, 0
k
n 1–
2 Gn()
that has been used extensively to describe rain, snow, hail,
where n and b are a shape and a scale parameter, respectively,
and clouds. This distribution originally was proposed by Mar-
related to the common variance of the quadrature components
shall and Palmer, and is called the Marshall-Palmer distribu- 2 2
by s = 2n/b , and K is the modified Bessel function. DKB
tion. SAL
Ref.: Schleher (1991), p. 33.
Ref.: Meneghini (1990), p. 133; Sauvageot (1992), p. 78.
The log-normal distribution is the distribution of random
The exponential distribution is the distribution of random
magnitude x with probability density function:
magnitude x with probability density function:
2
– lx ( log x – m )
f x () = l e 1 x
e f x () -------------------- exp –= -------------------------------
l xs 2p 2
and probability distribution function: x 2s x
and probability distribution function:
– lx
F x () 1 –= e
e
æ log x – m ö
x
ç
where l is the distribution parameter. See CLUTTER F x () = F ----------------------- ÷x >, 0
l
2
(amplitude) distribution. è 2s x ø
2
An exponential distribution is widely used in evaluating where m = M (log x), s = standard deviation of log x, and
x
radar reliability, attenuation of a signal during its propagation F(×) is the Laplace function.
in the atmosphere, and so forth. AIL A log-normal distribution finds use in description of
Ref.: Skolnik (1962), p. 27; Tikhonov (1980), p. 44. radar clutter and RCS fluctuations. See CLUTTER (ampli-
tude) distribution. AIL
The gamma distribution is the distribution of random mag-
Ref.: Tikhonov (1986), p. 36; Schleher (1991), p. 33.
nitude x with probability density function:
Marshall-Palmer distribution (see drop-size distribution).
x
– ---
1 b b
=
f x () ----------------------------------x e , b > – 1 b 0 The Nakagami distribution is the distribution of random
>
,
G
b +
1
(
b G b + 1 )
magnitude x with probability density function:
2
and probability distribution function: 2 æ m ö 2m – 1 æ mx ö 1
m
x =
f () ------------- ------ x exp – ç --------- ÷ m ³, ---
G m è
N
() 2 ø
2
s è s ø 2