Multirate adaptive filtering algorithms : analysis and applications

Citation

Sathe, Vinay Padmakar (1991) Multirate adaptive filtering algorithms : analysis and applications. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ec5w-7460. https://resolver.caltech.edu/CaltechETD:etd-07122007-103754

Abstract

In this thesis, we discuss the application of multirate signal processing concepts to adaptive filtering to achieve low computational complexity and speed. To be able to analyze systems involving multirate building blocks, we have studied effects of multirate filters on the statistics of random inputs. As an example of the multirate adaptive filtering concepts, we study the problem of adaptive identification of an unknown bandlimited channel. We show that the bandlimited property can be very efficiently exploited to reduce both the speed and number of computations. The new method embeds an adaptive filter into multirate filters to reduce complexity and speed of computation.

We have applied the theoretical results obtained for the effects of multirate building blocks on stationary inputs to the adaptive identification scheme above and shown that the optimal filter is a matrix filter. We have shown through simulations that for a practical setup, a scalar adaptive filter performs almost as well if the fixed filters in the scheme are designed to have good stopband attenuation.

In a practical implementation of adaptive algorithms, computational noise is of concern. Most of the current analysis focuses on deriving the worst case upper bound on the roundoff errors. We analyze some basic signal processing steps by introducing a statistical flavor to it. This analysis answers questions such as "what is a typical value of the roundoff error?" In particular, for the case of dot product computation, we obtain expressions for the roundoff noise variance for the floating point case, and compare the results with the fixed point noise roundoff noise analysis. We also perform error variance analysis of Givens rotation and Householder transformation. These two algorithms are used in the upper triangularization of matrices. We have compared the results obtained for these cases and shown that error variance for the Householder case is lower, meaning that the Householder transformation adds lower roundoff error "on an average".

We also address the problem of bandlimited extrapolation of discrete-time signals. We have explained why the term "best solution" does not have a unique answer. Several new techniques for bandlimited extrapolation of discrete-time segments are explored. These methods apply to a wide range of situations (including multiple-burst interpolation of multiband signals). A closed form expression for the optimal solution (for a given value of the energy of extrapolated sequence) has been obtained and evaluated for various values of the final energy. The various methods are compared on the basis of out-of-band energy of the extrapolated signal, total energy of the extrapolated signal (in relation to that of the given segment), and numerical robustness.

Thesis Files