Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 7.9 Incoherent Receiver
In the example we will derive the receiver for and incoherent receiver and in Example 7.9 we will derive
and plot the performance curves for an incoherent receiver. In chapter 7.5 we derived the form of the
coherent and in this example we will do the derivation of the incoherent receiver. The techniques parallel
the coherent case but the integrals that have to be evaluated are more complex. With the help of Matlab
this can be accomplished with little difficulty.
We begin the description of the incoherent radar problem by describing the waveforms for the two
hypotheses
H 1 : s(t )  A  sin(   t   )
0t T
H0 :
s(t )  0
In the incoherent case A and  are known constant but  is a random variable from a uniform distribution
with amplitude 1 / 2  on the range 0  2   . The receiver signal is
r (t ) 
A  sin(  t   )  n(t ) if  H1
n(t )
if  H0
The likelihood function has to be modified to
2
 f r    w   d
1

0
f 0 r 
 0
The numerator of  contains the conditional density function f1(r|) which is assumed to be gaussian. If we
follow the same reasoning in going from the discrete samples to the continuous case as in Equations 7.5-3
to 7.5-6 we have for f1(r|)
 1
f1 r    F  exp 
 N0
T
 r t   A  sin   t   
0
where F is a constant which we will cancel later. Similarly f0(r) is
 1 T

2


f 0 r   F  exp 
r
t

dt


 N0 0

We first work on the integral of f1(r|) or
2

 dt 

T
 r t   A  sin   t   
2
 dt
0
Expanding the kernel
om*t+theta
syms r A om theta t u T
ker=expand((r-A*sin(u))^2)
ker =
r^2-2*A*sin(u)*r+A^2*sin(u)^2
The third integral is and letting T = 2  /  we have
ker3=simplify(int(A^2*sin(om*t+theta)^2,t,0,2*pi/om))
ker3 =
1/om*A^2*pi
subs(ker3,om,2*pi/T)
ans =
1/2*T*A^2
The middle integral becomes after incorporating 1/ N0
The
syms N0 q th0
expand(subs(2*A/N0*sin(u)*r,om*t+theta))
ans =
2*A/N0*r*sin(om*t)*cos(theta)+2*A/N0*r*cos(om*t)*sin(theta)
When we make the following substitutions in the second integral
T
T
0
0
q  cos0    r (t )  sin(  t )  dt and q  sin0    r (t )  cos(  t )  dt
we have for the second integral
Int2=maple('combine','2*A*q/N0*(cos(theta)*cos(th0) +
sin(theta)*sin(th0)),trig')
Int2 =
2*A*q/N0*cos(-theta+th0)
Using this simplification the 2nd part of numerator integral of the likelihood function becomes
I2=int(exp(2*A*q/N0*cos(-theta+th0)),theta,0,2*pi)
Warning: Explicit integral could not be found.
> In C:\MATLAB\toolbox\symbolic\@sym\int.m at line 58
I2 =
int(exp(2*A*q/N0*cos(-theta+th0)),theta = 0 .. 2*pi)
Even though this integral cannot be evaluated symbolically it can be recognized as a standard function that
has be evaluated numerically and is a function in the Matlab library
 2 A q 

I 2  I 0
N
0


This Bessel function can be plotted as a function of its argument
x=0:.2:10;
I0=besseli(0,x);
semilogy(x,I0)
4
10
3
10
2
10
1
10
0
10
0
1
2
3
4
5
6
7
8
9
10
The complete likelihood function  is found by canceling the r(t)2 term, first integral, from the numerator
and denominator and combining the third term we have the likelihood ratio test as
 A2  T 
 2 A q 
  IO
  0
exp  
 2  N0 
 N0 
or
 2 A q 
 A2  T 
  0  exp 

IO
 N0 
 2  N0 
Since the left-hand side of the inequality is a monotonic function of q we have a threshold function receiver
2
2
 where  is a constant that can be evaluated by solving
on q or q2 or q
 A2  T 
 2  A
  IO
exp  
 2  N0 
 N0

  0

for  given all the constants A, T , N0, and 
The form of the receiver can be found by expanding

q 2  q 2  cos 0   sin  0 
2
2

A diagram of the receiver is shown in the Fig. 7.5-6. The receiver is formed from the above expression by
first evaluating the terms q  cos 0  and q  sin  0  after they have been replaced by the integral
expressions and then squaring the result and adding the two squares. The result, q 2, is then thresholded to
complete the receiver.