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EE640 STOCHASTIC SYSTEMS
SPRING 2003
COMPUTER PROJECT 1
PART B: ANALYSIS(updated 4-15-03)
- 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: PlotB - 1 u^ , g^ histogram PlotB - 2 s , s , s , s , s histogram PlotB - 3 t , c histogram PlotB - 4 t^ , c^ histogram PlotB - 5 t , c histogram PlotB s histogram (^1 ) 1 2 3 4 5 I1 I I2 I I3 I (^6) int ensity
- Covariance estimate: Estimate covariance matrices (3x3) Kt from t 1 ,t 2 ,t 3 Kc from c 1 ,c 2 ,c 3 KtI from tI1,tI2,tI KcI from cI1,cI2,cI (In some context, this is called the correlation matrix) ie. where m is an N x1 vector with all elements equal to the mean value of the vector x m.
- Estimate mean vectors (3x1) from such that
t t t t
, , , 1 2 3 where
t i I i
m N
N
, ^ t^ , m
1 11/28/2020 EE640 PROJECT 1
(x - )(x- ) N K(m, n) m m^ T n n 1 1
(B- 1 )
t (^) I1 ,t (^) I2 ,t (^) I3 (B-2) (B-3) (B-4)
likewise for clutter, use to generate
c c c c
, , , 1 2 3
- Determine the peak element locations, the centroid element locations of the histograms of b^ bina ry and s^ i n t e n s i t y. 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
^ ^
M
m x h m h x f x 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 n x xn f x xn 1
Optional: Determine the time averages of b^ bin a ry and s^ i n t e n s i t yand 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 x min= a *1+ b and x max= a * M + b. So a =( x max- x min)/( M -1) and b = x min- a. The values x max and x min are the center values associated with the end bins. So we find the centroid
^ ^
M
m m h m h m f m 1 The centroid is a fractional value 11/28/2020 EE640 PROJECT 1 c (^) I1 , c (^) I2 , c (^) I3 (B-5) (B-6)
