Communication Systems 4Th Edition Simon Haykin With Solutions Manual

5.2 Geometric Represetatartha of Sign.& 3 1 5

FIGURE 5.5 Vector representations of signals s,(t) and s,(t), providing the background picture for proving the Schwarz inequality.

where we have made use of Equations (5.15), (5.13) and (5.9). Recognizing that cos 8 5 1, the Schwarz inequality of Equation (5.16) immediately follows from Equation (5.17). More- over, from the first l i e of Equation (5.17) we note that w s 8) = l if and only if s2 = cs,, that is, s2(t) = cs,(t), where cis an arbitrary constant.

The proof of the Schwarz inequality, as presented here, applies to real-valued signals. It may be readily extended to complex-valued signals, in which case Equation (5.16) is refor- mulated as

where the equality holds if and only if s2(t) = cs,(t), where cis a constant; see Problem 5.9. It is the complex form of the Schwarz inequality that was used in Chapter 4 to derive the matched filter.

Having demonstrated the elegance of the geometric representation of energy signals, how do we justify it in mathematical terms? The answer lies in the Gram-Schmidt orthogon- alization procedure, for which we need a complete orthonormal set of basis functions. To proceed with the formulation of this procedure, suppose we have a set of M energy signals denoted by s,(t), sz(t), . . . , sM(t). Starting with sl(t) chosen from this set arbitrarily, the first basis function is defined by

where El is the energy of the signal s,(t). Then, clearly, we have

where the coefficientsll = and +,(t) has unit energy, as required. Next, using the signal s2(t), we define the coefficient szl as

We may thus introduce a new intermediate function