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

2.4 Signals Versus Vectors 29
Fi g
ure 2.8
Component
(projection) of a
vector along
another vector.
e
ex
X
Fi g ure 2.9
Approximations
of a vector in
terms of another
vector.
X
X
(a)
(b)
Fig. 2.8, the vector g can be expressed in terms of vector x as
g=ex+e (2.16)
However, this does not describe a unique way to decompose g in terms of x and e. Figure 2.9
shows two of the infinite other possibilities. From Fig. 2.9a and b, we have
(2.17)
The question is: Which is the "best" decomposition? The concept of optimality depends on
what we wish to accomplish by decomposing g into two components.
In each of these three representations, g is given in terms of x plus another vector called
the error vector. If our goal is to approximate g by ex (Fig. 2.8),
g::::: g = ex (2.18)
then the error in this approximation is the (difference) vector e = g - ex. Similarly, the errors
in approximations of Fig. 2.9a and bare e1 and e 2 , respectively. The approximation in Fig. 2.8
is unique because its error vector is the shortest (with the smallest magnitude or norm). We
can now define mathematically the component (or projection) of a vector g along vector x to
be ex, where e is chosen to minimize the magnitude of the error vector e = g - ex.
Geometrically, the magnitude of the component of g along x is I lgl I cos 0, which is also
equal to c!lxll- Therefore
cllxll = llgll cos e
Based on the definition of inner product between two vectors, multiplying both sides by I lx l I
yields
and
cllx ll 2 = llgll llx ll cos 0 = < g, x >
c = < g, X >
< X, X >
I
jjxjj 2 < g, X > (2.19)