By Principe J., Liu W., Haykin S.

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Also, depending on the precise meanings of Gain(i) and e(i), the algorithm can take many different forms. We explore this in detail in the subsequent chapters. This amazing feature is achieved with the underlying linear structure of the reproducing kernel Hilbert space where the algorithms exist, as is discussed next. 4 REPRODUCING KERNEL HILBERT SPACES A pre-Hilbert space is an inner product space that has an orthonormal basis {x k }k∞=1. Let H be the largest and most inclusive space of vectors for which the ∞ infinite set {x k }k =1 is a basis.

The original paper on the standard RLS algorithm is that of Plackett [1950], although many other researchers are believed to have derived and rederived various versions 24 BACKGROUND AND PREVIEW of the RLS algorithm. In 1974, Godard first used Kalman filter theory successfully to solve adaptive filtering problems, which is known in the literature as the Godard algorithm. Then, Sayed and Kailath [1994] established an exact relationship between the RLS algorithm and Kalman filter theory, thereby laying the groundwork for how to exploit the vast literature on Kalman filters for solving linear adaptive filtering problems.

For the initialization of the algorithm, it is customary to set the initial value of the weight vector equal to zero. Algorithm 1. 5) However, this criterion is too weak to be of any practical value, because a sequence of zero-mean, but otherwise arbitrary random, vectors converges in this sense. 8) i =1 In typical applications of the LMS algorithm, knowledge of ςmax is not available. To overcome this difficulty, the trace of Ru may be taken as a conservative estimate for ςmax. 10) where J(∞) is the limiting constant of the mean square error E[e(i)2] as i goes to ∞ and Jmin is the irreducible error power caused by model mismatch and/or noise in the observations.

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