Radar Detection - MATLAB Simulations for Radar Systems Design - page 84

f () fx () fx ()… fx n () = • • •

(2.46)

The operator symbolically indicates convolution. The characteristic
functions for the individual pdf s can then be used to compute the joint pdf in
Eq. (2.46). The details of this development are left as an exercise. The result is

•

( n P – 1 ) ⁄ 2

2 z
n P ℜ p

1
--- n P ℜ p

   

f () = ------------- exp – z – I n P ( 2 n P z ℜ p )

(2.47)

   

– 1

is the modified Bessel function of order . Therefore, the probabil-
ity of detection is obtained by integrating from the threshold value to
infinity. Alternatively, the probability of false alarm is obtained by letting

I n P n P – 1

– 1

f ()

ℜ p

be zero and integrating the pdf from the threshold value to infinity. Closed
form solutions to these integrals are not easily available. Therefore, numerical
techniques are often utilized to generate tables for the probability of detection.

Improvement Factor and Integration Loss

Denote the SNR that is required to achieve a specific P D given a particular

P fa when n P pulses are integrated non-coherently by ( SNR ) NCI . And thus,

the single pulse SNR, ( SNR ) 1 , is less than ( SNR ) NCI . More precisely,

( SNR ) NCI = ( SNR ) 1 × In ()

(2.48)

where is called the integration improvement factor. An empirically
derived expression for the improvement factor that is accurate within

In ()

0.8 dB is

reported in Peebles 1 as

log ( 1 ⁄ P fa )

 

[ In () ] dB = 6.79 1 ( + 0.235 P D ) 1 + --------------------------- log n () (2.49)

 

46.6

) 2 ( 1 – 0.140 log n () + 0.018310 ( log n P )

Fig. 2.6a shows plots of the integration improvement factor as a function of the
number of integrated pulses with and as parameters, using Eq. (2.49).
This plot can be reproduced using the MATLAB program Ðfig2_6a.mÑ given
in Listing 2.6 in Section 2.11. Note this program uses the MATLAB function
Ðimprov_fac.mÑ , which is given in Listing 2.7 in Section 2.11.

P D P fa

MATLAB Function Ðimprov_fac.mÑ

The function Ðimprov_fac.mÑ calculates the improvement factor using Eq.
(2.49). It is given in Listing 2.7 in Section 2.11. The syntax is as follows:

[impr_of_np] = improv_fac (np, pfa, pd)

1. Peebles Jr., P. Z., Radar Principles , John Wiley & Sons, Inc., 1998.