EE640 STOCHASTIC SYSTEMS
SPRING 2000
COMPUTER PROJECT 1
PART A: SYNTHESIS(updated 1-27-00)
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.
(A-2)
gi,1 = -2 ln ui,1 cos 2 ui,2
gi,2 = -2 ln ui,1 sin 2 ui,2
(A-3)
gi,1 = gi [n], gi,2 = gi [n + 1], ui,1 = ui [n], ui,2 = ui [n + 1]
(A-4)
where
where n = 1,3,5,7...,N such that adjacent but no overlapping pairs are used.
3.
Generate 6 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:
EE640 PROJECT 1
1997
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t 1 = 4 g1 + 2 g 2 + g 3
t 2 = g1 + 4 g 2 + 2 g 3
(A-7)
t 3 = 2 g1 + g 2 + 4 g 3
and the clutter vectors are:
c1 = 3 g 4 + 2 g 5 + g6
c 2 = g 4 + 3 g 5 + 2 g6
(A-8)
c 3 = 2 g 4 + g 5 + 3 g6
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.
EE640 PROJECT 1
1997
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