April 2008
Intermediate
832 pages
26h 2m
English
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CHAPTER 4 Orthogonality Principle
Before examining the properties of the solution(s) Ko of the normal equations KoRy = Rxy, we consider some illustrative examples in the context of symbol estimation and channel equalization. Additional examples and applications are discussed in Chapter 5.
4.1 DESIGN EXAMPLES
Example 4.1 (Noisy measurements of a binary signal)
We reconsider Ex. 1.2 of a BPSK signal x that assumes the values ±1 with probability 1/2. The measurement y is y = x + υ, where x and the disturbance υ are independent of each other, with υ being zero-mean Gaussian of unit variance.
Both x and y have zero means so that, according to Thm. 3.1, the optimal linear estimator of x is 
, where the (now scalar) coefficient ko is obtained from solving 
. We therefore need to determine the quantities 
. Now since {x, v} are independent we have
Moreover,
so that ko = 1/2, and the optimal linear estimator is . That is, we simply scale the received signal by 1/2. In contrast, the optimal estimator ...
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