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Several Approaches to Cost Function Adaptation and Phase Approximation
Tiivistelmä:
Two main classes of cost function adaptation approaches are derived: nonrecursive and recursive. Nonrecursive adaptation of cost function can be distinguished by the fact that the error exponent is updated every iteration using only direct relationships between an approximation of the error and the power of the cost function. Alternatively, in recursive cost function adaptation, we do not need anymore to estimate the actual error of the system, the updated error exponent is computed recursively.
The idea behind the first approach relies on adjusting the error exponent parameter during the adaptation by enforcing same value of gradients for two consecutive error exponents. Two different type of algorithms are proposed: the staircase and the smooth CFA (Cost Function Adaptation) algorithms, and it is shown that they behave with comparable results.
Then the gradient approach is derived by enforcing the same direction of the gradient as in the case of nonquadratic algorithms. As a special case the linear adaptation of the power of the cost function results.
In the recursive cost function adaptation case, the new error exponent is computed from the previous one using a customary LMS (Least Mean Square) recursive equation. This method improves the sensitivity of the power of the cost function with respect to the noisy error, while the other benefits of the CFA algorithms in terms of the convergence speed and residual error remain.
The second part of the thesis deals with a special form of Hilbert transform, known as Bode gainphase relationships. Here the interest is to develop formulae for phase approximations by gain samples, equally separated in logarithmic domain. Our approach consists of the following steps:
First we discuss the most important theoretical results related to our specific subject. Sampling in logarithmic domain is pointed out as a more general rule, not necessarily only for the presented case.
Then we establish new relationship for computing the phase of the minimumphase functions from the gain derivatives, as a first advance to approach phase by gain samples. As a beginning, we show that for a given frequency the phase could be obtained from the odd derivatives of the neperian gain, evaluated for this frequency. Then we select a finite number of terms of the main formula and we derive an approximation of phase. We show that the approximations derived can be improved by taking into account the Gibbs phenomenon and the Feher kernel.
Finally we derive a completely novel relationship for approximating the phase values from the gain samples, in nepers, equally spaced in the logarithmic frequency domain. A general approximation formula is proved, then two quadrature formulae are obtained using NewtonCotes and Simpson rules. Numerical examples are also provided to emphasize the achievements of the method.