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Therefore, in order to establish that the equations KoRy = R,, are consistent, it is enough to prove that c*RyX= 0 for all c E N(R,). So let us assume that this latter condition does not hold. This means that there exists at least one nonzero vector c such that c'R, = 0 and c*R,, # 0. This statement leads to a contradiction. Indeed, c*R, = 0 implies that c*Ryc= c * ( E y y * ) c = 0 or, equivalently, E Ic*yI2= 0. We therefore have a zero-mean random variable c*y (because y is zero-mean itself) with zero variance. Using Remark A . l , we conclude that c*y is the zero random variable with probability one. It follows that A c*R,, = c * ( E y x * = E (c'Y) Z* = 0 ) This contradicts the assumption c*RyX# 0 and we conclude that the normal equations (4.7) are always consistent.

Solving the above system of equations, we obtain w1 = [E{x1 (k) xT (k)} + b2 E{x1 (k q) xT (k q)}] 1 E{y1 (k q)x1 (k)}b1q , 1 1q 1

0.01% Fixed(dB) 1.30

FIGURE 21-16

and address) to complex data such as pictures, sounds, or video images Storing data in a precise format enables a database management system (DBMS) like Access to turn data into useful information Tables serve as the primary data repository in an Access database Queries, forms, and reports provide access to the data, enabling a user to add or extract data, and presenting the data in useful ways Most developers add macros or Visual Basic for Applications (VBA) code to forms and reports to make their Access applications easier to use A relational database management system (RDBMS), such as Access, stores data in related tables For example, a table containing employee data (names and addresses) may be related to a table containing payroll information (pay date, pay amount, and check number).

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We therefore say that the denominator in (45.5) is a measure of the energies of the disturbances in the problem, namely, { w( .), G } , while the numerator is a measure of the resulting estimation error energies. In this way, the criterion (45.5) requires that we determine estimates {Z&} such that the ratio of energies, from disturbances to estimation errors, does not exceed y2 for any {w(.), wo}.When this property holds, we shall say that the resulting estimates are robust in the sense that bounded disturbance energies lead to bounded estimation-error energies and, in a similar vein, small disturbance energies lead to small estimation-error energies. Of course, the smaller the value of 7 ,the more robust the solution is.23Observe further that no statistical assumptions on the data are being made. In this way, the robustness studies that are carried out in this chapter allow us to highlight some features of adaptive algorithms that hold regardless of statistical considerations.