Radar Detection - MATLAB Simulations for Radar Systems Design - page 91

The threshold value can then be approximated by the recursive formula
used in the Newton-Raphson method. More precisely,

V T

, ( )
G ′ V Tm 1

GV Tm 1

–

V Tm = V Tm 1 – ----------------------------- ; m = 123… ,,,

(2.63)

, , –

( )

, –

The iteration is terminated when V Tm – V Tm 1 < V Tm 1 ⁄ 10000.0 . The

, , – , –

functions G and G ′ are

) n P

⁄ n fa

GV Tm ( ) = ( 0.5 – Γ I ( V T , n P ) (2.64)

,

– V T n P – 1

e V T

G ′ V Tm ( ) = – ---------------------------

(2.65)

,

( n P – 1 )!

The initial value for the recursion is

V T 0 = n P – n P + 2.3 – log P fa ( – log P fa + n P – 1 ) (2.66)

,

MATLAB Function Ðincomplete_gamma.mÑ

In general, the incomplete Gamma function for some integer N is

x
∫

– v

v N

– 1

e

Γ I ( xN , ) = ---------------------- d

(2.67)

( N – 1 )!

0

The function Ðincomplete_gamma.mÑ implements Eq. (2.67). It is given in
Listing 2.9 in Section 2.11. Note that this function uses the MATLAB function
Ðfactor.mÑ which is given in Listing 2.10. The function Ðfactor.mÑ calculates
the factorial of an integer. Fig. 2.7 shows the incomplete Gamma function for
. This figure can be reproduced using the MATLAB program
Ðfig2_7.mÑ given in Listing 2.11. The syntax for this function is as follows:

N = 13610 ,,,

[value] = incomplete_gamma (x, N)

where

Symbol Description Units Status

x units of x input

Γ I ( xN , )

variable input to

N none / integer input

variable input to

Γ I ( xN , )

value none output

Γ I ( xN , )