EE640 STOCHASTIC SYSTEMS
SPRING 2005
COMPUTER PROJECT 1
PART B: ANALYSIS(updated 4-1-05)
0.
In similar way as you did in Part A, generate independent identically distributed test
vectors. The target vectors are:
t IID1 = g 1 + 3, t IID2 = g 2 + 3
(A-11)
t IID3 = g 3 + 3
and the clutter vectors are:
c IID1 = g 4 + 1, c IID2 = g 5  1
c IID3 = g 6 + 1
(A-12)
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
KtIID from tIID1,tIID2,tIID3
KcIID from cIID1,cIID2,cIID3
(In some context, this is called the correlation matrix) ie.
1
K(m,n) 
( xm -  m )T ( xn -  n )
(B-1)
N  1
where m is an Nx1 vector with all elements equal to the mean value of the vector xm and
K(m,n) is the mth, nth element of the matrix K.
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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
(B-4)
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
Determine the time averages of b binary and s int ensity and compare with the centroid averages.
They should be close.
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Optional Method: A conceptually easier technique for implementing the centroid is the
following: The approach will first get the fractional value of the estimated bin number. Then that
value is linearly mapped to the x dimension. 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
and the mean of x is then x=m*a+b.
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