EE640 STOCHASTIC SYSTEMS
Spring, 2006
COMPUTER PROJECT 1
PART A: SYNTHESIS (March 21, 2006)
Let N= 512:
1. Uniform pseudo-random numbers. Generate 6 random vectors, each with a different
seed. The vectors are all N2×1 where each element is uniformly distributed between
0 and 1. Each element is independent from the others. Mathematically refer to the
vectors as:
u1, u2, u3, u4, u5, u6,(1)
2. Prove the parametric transformation equations for converting from a uniform distri-
bution to a gaussian distribution are correct.
gi[2i+ 1] = p−2 ln ui[2n+ 1] cos 2πui[2n+ 2] (2)
gi[2i+ 2] = p−2 ln ui[2n+ 1] sin 2πui[2n+ 2] (3)
where n= 0,1,2. . . (N2/2−1)
3. Generate six N2×1 gaussian random vectors from the associated vectors in part (1).
Use the transformation developed in (2). Generate them with a 0 mean and unity
variance and store as you did in (1). Refer to them as
g1, g2, g3, g 4, g5, g6,(4)
4. Linear combinations of r.v.s. Generate five N2×1 vectors such that
s1=u1+u2
s2=u1+u2+u3
s3=u1+u2+u3+u4(5)
s4=u1+u2+u3+u4+u5
s5=u1+u2+u3+u4+u5+u6
5. Change gifrom lexicographical form to 2D matrix Gi. Put the first Nelements in
gito the first row of Gi, the N−2Nelements in gito the second row of Giand so
on. The size of giis N2×1 and the size of Giis N×N.
6. Letting N= 128: Use the 12 training images from Project 1-S: Supplemental “SYN-
THESIS OF DETERMINISTIC SYSTEM” and add zero mean white Gaussian Noise
to them such that the Signal to Noise Ratio is 0 dB. You can use randn() to generate
the noise for the 6 target and 6 clutter images. Add them to the images and subplot
12 noisy images.
1
