EE640 STOCHASTIC SYSTEMS
SPRING 2003
COMPUTER PROJECT 1
PART B: ANALYSIS(updated 4-15-03)
1.
Histogram: Design a program, or use the MATLAB function hist.m, that will estimate
the histogram of an Nx1 vector of random numbers. Have the program use specified M
bin intervals. Run the program and plot for:
u1 , g1
PlotB - 1
histogram
PlotB - 2 s1 , s 2 , s 3 , s 4 , s 5 histogram
t I1 , c I1
PlotB - 3
histogram
t I2 , c I2
PlotB - 4
histogram
t I3 , c I3
PlotB - 5
histogram
PlotB  6
s int ensity
histogram
2.
Covariance estimate: Estimate covariance matrices (3x3)
Kt from t1,t2,t3
Kc from c1,c2,c3
KtI from tI1,tI2,tI3
KcI from cI1,cI2,cI3
(In some context, this is called the correlation matrix) ie.
1
(B-1)
K(m,n) 
( xm -  m )T ( xn -  n )
N  1
where m is an Nx1 vector with all elements equal to the mean value of the vector xm.
3.
Estimate mean vectors (3x1) from
t I1 ,t I2 ,t I3
(B-2)
such that
  t ,1 


 t   t ,2 
  t ,3 
(B-3)
where
 t ,i 
1 N
 t I ,i  m 
N m 1
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likewise for clutter, use
c I1 , c I2 , c I3
(B-5)
to generate
  c,1 


 c   c,2 
  c,3 
(B-6)
4. Determine the peak element locations, the centroid element locations of the histograms of
b binary and s int ensity . The centroid is determined by “simulating” a pdf. For example, let x[n] be
a sequence and you want to approximate E{x[n]}. Let h(x) be the histogram of x[n]. First, form a
pseudo pdf as
f x x  
h x 
M
 hm
m 1
where m is the bin number of a total M bins in the histogram. The value of h(x) returns the bin
value that contains the value of x.
The centroid is then
N
 x   xn f x  xn
n 1
Optional: Determine the time averages of b binary and s int ensity and compare with the centroid
averages. They should be close.
Optional: A possibly easier technique for implementing the centroid, is the following: Given your
bin numbers for the histograms h[m] are equally spaced and vary from 1 to M. We can map bin
values to signal values with xmin=a*1+b and xmax=a*M+b. So a=(xmax-xmin)/(M-1) and b=xmin-a.
The values xmax and xmin are the center values associated with the end bins. So we find the
centroid
f m m  
hm
M
 hm
m 1
The centroid is a fractional value
M
 m   mf m m 
m 1
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and the mean of x is then x=m*a+b.
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