Radar Detection - MATLAB Simulations for Radar Systems Design - page 83

from antenna
and lo w nois e

Matched
Filter

∑

Square Law
Detector

x ()

Threshold
Detector

z ()

Amp.

x ()

r ()

single pulse

Threshold V T

Figure 2.5. Simplified block diagram of a square law detector and
non-coherent integration.

The pdf for the signal was derived earlier and it is given in Eq. (2.11).
Define a new dimensionless variable

r ()

y as

r n
----

y n =

(2.40)

and also define

A 2
ψ 2

ℜ p == ------ SNR (2.41)

It follows that the pdf for the new variable is then given by

y 2

– ( + ℜ p )

f y () fr () r n

d
y n

 

= = y n I 0 ( y n ℜ p ) exp --------------------------

-------

(2.42)

 

2

n th

The output of a square law detector for the pulse is proportional to the
square of its input, which, after the change of variable in Eq. (2.40), is propor-
tional to y n . Thus, it is convenient to define a new change variable,

1

--- y 2

x n =

(2.43)

The pdf for the variable at the output of the square law detector is given by

ℜ p
2

f x () fy () y n

d

   

= = exp – x n + ------- I 0 ( 2 x n ℜ p )

-------

(2.44)

   

x n

Non-coherent integration of n p pulses is implemented as

n P
∑

z = x n

(2.45)

n = 1

Since the random variables x n are independent, the pdf for the variable z is