By Paulo Sergio Ramirez DINIZ (auth.)
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Additional resources for Adaptive Filtering: Algorithms and Practical Implementation
4. , RH = R, its eigenvalues are real and equal to or greater than zero. Proof: First note that given an arbitrary complex vector w, (wHRw)H wHR H (wH)H = wHRw. Therefore, wHRw is areal number. , Rqi = Aiqi· By premultiplying this equation by qr, it follows that qfRqi = Aiqfqi = Aillqil1 2 where the operation lIall 2 = laol 2 + lad 2 + ... + laNI 2 is the Euclidean norm squared of the vector a, that is always real. Since the term on the left hand is also real and R is positive semidefinite, we can conclude that Ai is real and nonnegative.
This is the principle behind the concept of smart antennas, where a set of adaptive array processors filter the signals coming from the array, and direct the beam to several different directions where a potential communication is required. For example, in a wireless communication system we are able to form a beam for each subscriber depending on its position, ultimately leading to minimization of noise from the environment and interference from other subscribers. In order to develop the theory of linearly-constrained optimal filters, let us consider the particular application of a narrowband beamformer required to pass without distortion all signals arriving at 90° with respect to the array of antennas.
In practice the parameters that determine the MSE surface shape are not available. What is left is to directly or indirectly estimate these parameters using the available data and to develop adaptive algorithms that use these estimates to search the MSE surface, such that the adaptive filter coefficients converge to the Wiener solution in some sense. The starting point to obtain an estimation procedure is to investigate the convenience of using the classical searchP. S. Ramirez, Adaptive Filtering © Springer Science+Business Media New York 2002 16 CHAPTER 2 ing methods of optimization theory [1}-[2J to adaptive filtering.
Adaptive Filtering: Algorithms and Practical Implementation by Paulo Sergio Ramirez DINIZ (auth.)