EE640 STOCHASTIC SYSTEMS
SPRING 2003
COMPUTER PROJECT 1
PART A: SYNTHESIS(updated 4-8-03)
Let N=1024:
1.
Uniform pseudo-random numbers. Generate 6 random vectors,each with a different seed.
The vectors are all Nx1 where each element is uniformly distributed between 0 and 1.
Each element is independent from the others. Mathematically refer to the vectors as:
(A-1)
u1 ,u 2 ,u 3 ,u 4 ,u 5 ,u6
2. Prove the parametric transformation equations for converting from a uniform distribution
to a gaussian distribution are correct.
g i 2n  1   2 ln u i 2n  1 cos 2 u i 2n  2
(A-2)
g i 2n  2   2 ln u i 2n  1 sin 2 u i 2n  2
(A-3)
where n=0,1,2, …., (N/2 -1).
3.
Generate six Nx1 gaussian random vectors from the associated vectors in part A.1. Use
the transformation developed in A.2. Generate them with a 0 mean and unity variance and
store as you did in A.1. Refer to them as
g1 , g 2 , g 3 , g 4 , g 5 , g 6
(A-5)
4.
Linear combinations of r.v.s. Generate five Nx1 vectors such that
s1 = u1 +u2
s 4 = u1 +u2 +u3 +u4 +u5
s 2 = u1 +u2 +u3
s 5 = u1 +u2 +...u6
(A-6)
s 3 = u1 +u2 +u3 +u4
5.
Generate correlated test vectors. Three vectors will represent target class data and three
vectors will represent clutter class data. The 3 target vectors are:
t 1 = 4 g 1 + 2 g 2 + g 3  34
t 2 = g 1 + 4 g 2 + 2 g 3  26
(A-7)
t 3 = 2 g 1 + g 2 + 4 g 3  38
and the clutter vectors are:
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c1 = 4 g 4 + 2 g 5 + g 6  6
c 2 = g 4 + 4 g 5 + 2 g 6  12
(A-8)
c3 = 2 g 4 + g 5 + 4 g 6  17
6.
Generate independent test vectors. The target vectors are:
t I1 = 4 g1 + 6, t I2 = g 2 + 2
t I3 = 0.1  g 3 + 6
(A-9)
and the clutter vectors are:
c I1 = 4 g 4 + 0,
c I2 = g 5
c I3 = 0.1  g6 + 4
(A-10)
7.
Generate two more vectors, a pseudo-random binary (i.e., bipolar) sequence and pseudorandom intensity sequence suth that:
(A-11)
 1 for u 1  n  0.5
b binary  n  
1 for u 1  n  0.5
and


s int ensity n  g1 n
2
(A-12)
where n=1,2,...N.
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