Adaptive Filters

CHAPTER 2 Vector-Valued Data

We have focused so far on scalar real-valued random variables. The results however can be extended in a straightforward manner, by using the convenience and power of the vector notation, to the cases of vector-valued and complex-valued random variables.

These two situations are common in applications. For example, in channel estimation problems, the quantities to be estimated are the samples of the impulse response sequence of a supposedly finite-impulse-response (FIR) channel. If we group these samples into a vector x, then we are faced with the problem of estimating a vector rather than a scalar quantity. Likewise, in quadrature amplitude modulation (QAM) or in quadrature phase-shift keying (QPSK) transmissions over a communications channel, the transmitted sym are complex-valued. The recovery of these symbols at the receiver requires that we solve an estimation problem that involves estimating complex-valued quantities.

2.1 OPTIMAL ESTIMATOR IN THE VECTOR CASE

It turns out that the optimal estimator in the general vector and complex-valued case is still given by the conditional expectation of x given y. To see this, let us start with a special case. Assume x and y are both real-valued with x a scalar and y a vector, say,

As before, let denote ...