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

2.4 Signals Versus Vectors 33
function x(t) over an interval (t1 ::: t :::: t2)
g (t) :::::: cx(t) (2.3 1)
where g(t) and x(t) are complex functions of t. In general, both the coefficient c and the error
e(t) = g (t) - cx(t) (2.32)
are complex. Recall that the energy E x of the complex signal x( t) over an interval [t 1 , t2] is
E x =
lh t1
- lx(t) 1 2 dt
For the best approximation, we need to choose c that minimizes E e , the energy of the error
signal e(t) given by
Recall also that
E e =
lt1
h
- lg(t) - cx(t)1 2 dt
(2.33)
l u + v l 2 = (u + v)(u* + v*) = iu i 2 + lv l 2 + u*v + uv*
(2.34)
Using this result, we can, after some manipulation, express the integral E e in Eq. (2.33) as
1 /2 I
1
1 12
1
2
I
1
1 1
E e
=
lg(t) l 2 dt - ID g(t)x* (t) dt + c/Ex - ID g(t)x* (t) dt
t1 v Ex t1 v Ex t1
2 l
Since the first two terms on the right-hand side are independent of c, it is clear that E e is
minimized by choosing c such that the third term is zero. This yields the optimum coefficient
c = -
I 1 12
E x
t1
g (t)x*(t) dt
2
(2.35)
In light of the foregoing result, we need to redefine orthogonality for the complex case as
follows: complex functions (signals) x1 (t) and x2 (t) are orthogonal over an interval (t :S ti <
t 2 ) as long as
12
or 1 xf (t)x2 (t) dt = 0
11
(2.36)
In fact, either equality suffices. This is a general definition of orthogonality, which reduces to
Eq. (2.25) when the functions are real.
Similarly, the definition of inner product for complex signals over a time domain e can
be modified:
< g(t), x(t) > = { g(t)x* (t) dt
l{t:IE(-) )
(2.37)
Consequently, the norm of a signal g(t) is simply
[ ]
llg(t)II = { lg(t)l 2 dt
lr1:1EeJ
1/2
(2.38)