Page 114 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

102 Chapter 3

The co-efficients are

T

c1= V *o1 = 16.8375
T
c2=V *o2 = 4.8558
T
c3 =V *o3 = 2.4333

Thus the vector V is represented as

V = c1*o1+c2*o2+c3*o3

Computing the co-efficients become easier if the basis are orthonormal to
each other and hence orthonormal basis is required.
If adding few more independent vectors increases the number of
independent vectors spanning the space, the space spanned by these vectors
also increases. The corresponding orthonormal vectors obtained using Gram-
Schmidt orthogonalization procedure shows that the orthonormal basis are
the same as that of the earlier except the inclusion of additional few
orthogonal basis corresponding to the additional independent vectors. Hence
the co-efficients of the already existing basis remains the same.
In the previous example if the vector chosen in the space spanned by the
vectors v1 and v2.

6

6 (say)
14

The vector can be represented as the linear combination of ‘o1’ and
‘o2’ with the corresponding co-efficients 16.0357 and 3.2950 respectively.
They are computed using inner product technique. Suppose the same vector
is represented as the linear combination of ‘o1’ ‘o2’ and ‘o3’, the
corresponding co-efficients are obtained as 16.0357, 3.2950 and 4.4409e-
015 respectively. Note that first two co-efficients remains unchanged and
also note that the value of the third co-efficient is almost insignificant for
representing the vector. This property is used in compression techniques in
which the significant co-efficients are stored rather than the vector itself.