Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 7.4 MSE Estimation in Two Dimensions
In the example we will make use of the results of section 7.2 to derive MS optimal processing of a two dimensional vector process. We make use of the expressions in example and extend the results to the condition where X
0
is a two dimensional process.
A linear MS estimate, x0, of the RV X0 in terms of the vector X where X t = [ x
1 x2 ... xn] is given by x
0
 a
1
 x
1
 a
2
 x
2
   a n
 x n where xi represent samples form the random processes Xi
The conditionals can also be used to express the above results
E

X X

0
 a
1
 x
1
 a
2
 x
2
   a n
 x n
E
 
X
0
 x

X


E
 
X
0
 x
0
 

0
E


X
 x

0
2
X


E


X
 x

0 0


P
When X
0
is gaussian we may express the conditional density function as f

X X

0

G
 x , P

0
The two dimensional MS estimation of X0 where X0 is a scalar and there is two samples to construct this estimate is
E

X
0 x
1
, x
2

 a
1
 x
1
 a
2
 x
2
Using the correlation expressions from Example 7.2 we have a set of equations that can be solved for a1 and a2 symbolically in terms of the correlation coefficients and we have simplified the expression by assuming that R21 = R12 syms a1 a2 R11 R12 R22 R01 R02
A=solve('R11*a1+R12*a2-R01','R12*a1+R22*a2-R02',a1,a2);
A.a1,A.a2 ans =
-(-R22*R01+R02*R12)/(R11*R22-R12^2) ans =
(R11*R02-R01*R12)/(R11*R22-R12^2)
The expression for the covariance
P

R
00
 a
1

R
01
 a
2

R
02

Substitution for a1 and a2 syms R00
P=R00-A.a1*R01-A.a2*R02;
Simplifying we obtain for P pretty(simplify(P))
2 2 2
R00 R11 R22 - R00 R12 - R22 R01 + 2 R01 R02 R12 - R11 R02
------------------------------------------------------------
2
R11 R22 - R12
The expression for the density function becomes f

X
0 x
1
, x
2


G
 a
1
 x
1
 a
2
 x
2
, P

Two Random variables X a
and X b
can be MS estimated from x1 and we can find COV
 eX a
, eX b x
1
 where eXa = Xa -aa x1 etc. Using the expressions for X a
and X b
individually we have
E

X a x
1

 a a
 x
1 with a a

R a 1
R
11 and
E

X
 b x
1
 a b
 x
1 with a b

R b 1
R
11
The covariance expressions are
P a

E
 
X a
 a a
 x
1

X a


R aa

2
R a 1
R
11
Now the
COV
P b

 eX a
, eX b x
1

=
COV
E
 
X
 eX a b
 a b
, eX b
 x
1

X b


R bb

2
R b 1
R
11
 because each error is independent of x1 so
P ab

E
 
X a
 a a

X
1

X b
 a b

X
1
 

R ab

2
R a 1
R
11

2
R a 1
R
11

R a 1

R b 1
R
11
This expression simplifies when Ra1 = Rb1. The two dimensional Gaussian density function is f

X a
, X b x
1


G
 a a
 x
1
, a b
 x
1
, P a
, P b
, P ab
