Page 110 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

98 Chapter 3

-1
T
T
Projected vector is computed as A(A A) A a

= -3.3750
2.2500
5.0000

4.2.2 Example 2

The column vectors A, B and C are displayed below.

T
A=[10 2 6 5 9 8 5 0 8 4]

B= [6 8 9 7 2 4 9 9 4 9] T

C=[43 26 44 29 32 34 35 24 35 32] T

The column vector C is related to the column vectors A and B as the
linear combination as displayed below C = m*A+ n*B. The requirement is
to find the optimal value for the scaling constant m and n.
If C is in the space spanned by the column vectors of A and B, unique m
and n values can be computed easily. But if C is not in the space spanned by
the column vectors of A and B, the constants ‘m’ and ‘n’ are the best
estimated values such that C’=m*A+n*B is in the space spanned by the
column vectors A and B.
2
Also the mean squared error (i.e) E {(C-C’) ] is minimized. This is
obtained using projection matrix as described below.
The column vector C’ is the projection of the vector ‘C’ on the space
spanned by the vectors A and B.
Representing the vectors A and B in the matrix column wise to form the
matrix ‘P’.

Thus P= 10 6
2 8
6 9
5 7
9 2
8 4
5 9
0 9
8 4
4 9