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

10.4 Signal Space Analysis of Optimum Detection 527
Such a set is an orthonormal set of vectors. They capture an orthogonal vector space. Any
vector x = (xi , x2, . .. , Xn) can be represented as
X = xi <P i + X2<P2 + · · · + Xn<P n
where x; is the projection of x on the basis vector <Pk and is the kth coordinate. By using
Eq. (10.48), the kth coordinate can be obtained from
k = I , 2, . .. , n (10.49)
Since any vector in the n-dimensional space can be represented by this set of n basis vectors,
this set forms a complete orthonormal (CON) set.
10.4.2 Signal Space and Basis Signals
The concepts of vector space and basis vectors can be generalized to characterize continuous
time signals defined over a time interval 0. As described in Sec. 2.6, a set of orthonormal
signals { <p; (t)} can be defined for t E 0 if
{ . <pj (t)<pk (t) dt = {
ltE0
j -:/= k
j=k
(10.50)
If {<p;(t)} form a complete set of orthonormal basis functions of a signal space defined over 0,
then every signal x(t) in this signal space can be expressed as
x(t) = I>k'Pk (t) t E 0
k
(10.51)
where the signal component in the direction of 'Pk (t) is*
Xk = { x(t)<pk (t) dt
ltE0
(10.52)
One such example is for 0 = (-oo, oo). Based on sampling theorem, all low-pass signals
with bandwidth B Hz can be represented by
x(t) = I>k EB sine (2rrBt - krr)
k
(10.53a)
* If (,pk(t)} is complex, orthogonality implies
and Eq. (10.52) becomes
1 <f!J (t),pk (t) dt = 0
/EE)
xk = 1 x (t),pj;_ (t) dt
/EE)