Radar Detection - MATLAB Simulations for Radar Systems Design - page 77

where

x
∫

1
2π

2

– ζ ⁄ 2

----------

F () = e d

(2.16)

– ∞

The function can be found tabulated in most mathematical formula refer-
ence topic. Note that for the case of noise alone (

F ()

A = 0 ), Eq. (2.15) collapses

to a uniform pdf over the interval { 02π , } .

One excellent approximation for the function F () is

 

1 1

2π

x 2

– ⁄ 2

-------------------------------------------------------------- ---------- e

F () 1 = –   x ≥ 0

(2.17)



x 2



0.661 x + 0.339 + 5.51

and for negative values of x

Fx –() 1 = – F ()

(2.18)

MATLAB Function Ðque_func.mÑ

The function Ðque_func.mÑ computes using Eqs. (2.17) and (2.18) and
is given in Listing 2.2 in Section 2.11. The syntax is as follows:

F ()

fofx = que_func (x)

2.2. Probability of False Alarm

The probability of false alarm P fa is defined as the probability that a sample

R of the signal r () will exceed the threshold voltage V T when noise alone is

present in the radar,

∞
∫

V 2



r 2
2ψ 2

  

–
2ψ 2

r
ψ 2

------ ---------

P fa = exp – --------- d = exp

   

(2.19a)

   

V T

1
P fa



2ψ 2

------ 

V T = ln

(2.19b)

Fig. 2.3 s hows a plot of the normalized threshold versus the probability of false
alarm. It is evident from this figure that is very sensitive to small changes
in the threshold value. This figure can be reproduced using MATLAB program
Ðfig2_3.mÑ given in Listing 2.3 in Section 2.11.

P fa

The false alarm time T fa is related to the probability of false alarm by