B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

878 APPENDIX C: GRAM-SCHMIDT ORTHOGONALIZATION OF A VECTOR SET
Figure C.1
Gram-Schmidt
process for a
two-dimensional
case.
Hence,
= x2 -
< X1 , X2 >
---XI
llx1 11 2
< YI , X2 >
IIYI 11 2 YI (C.4)
Equations (C.3) and (C.4) yield the desired orthogonal set. Note that this set is not unique.
There is an infinite number of possible orthogonal vector sets (YI , y2) that can be generated
from (XI , x2). In our derivation, we could as well have started with y = x2 instead of YI = x1 .
This starting point would have yielded an entirely different set.
The reader can extend these results to a three-dimensional case. If vectors XI , x2, x3 form
an independent set in this space, then we form vectors YI and Y2 as in Eqs. (C.3) and (C.4). To
determine y3, we approximate X3 in terms of vectors Yl and y2. The error in this approximation
must be orthogonal to both YI and Y2 and, hence, can be taken as the third orthogonal vector
y3 . Hence,
y3 = x3 - sum of projections of x3 on YI and Y2
< Yl , X3 > < Y2, X3 >
= x3 -
IIY1 11 2 YI -
IIY2 11 2 y2 (C.5)
These results can be extended to an N-dimensional space. In general, given N independent
vectors x1 , x2, ... , XN, if we proceed along similar lines, we can obtain an orthogonal set
Y1,Y2, ... , YN, where
YI = XI
(C.6)
and
j-I
" < Yk , Xj >
YJ = Xj - t:i II Yk 11 2 Yk j = 2, 3, ... , N (C.7)
Note that this is one of the infinitely many orthogonal sets that can be formed from the set
x1, x2, ... , XN. Moreover, this set is not an orthonormal set. The orthonormal set Y1, Y2, . .. ,
y N can be obtained by normalizing the lengths of the respective vectors,
We can apply these concepts to signal space because one-to-one correspondence exists between
signals and vectors. If we have N independent signals XI (t), x2 (t), ... , XN (t), we can form a