The Stability Theory and Design of Two-Dimensional Recursive Digital Filters and Recursive Digital Lattice Filters
Date Issued
2016
Date
2016
Author(s)
Du, Jiun-Shian
Abstract
A two-dimensional (2-D) digital allpass filter (DAF) has a property of varying only phase with constant magnitude and it has mainly been used as a phase compensator for distorted signals. It is a structure that has some desirable attributes such as low complexity and low coefficient quantization error. It also can be used to design a wide range of filtering functions. In this doctoral dissertation, we present the monotone phase-response property of a two-dimensional (2-D) causal digital allpass filter (DAF) with real coefficients or complex coefficients in the quarter-plane (QP) support region. Regarding the circumstance of real coefficients, we also prove that the previously proposed bounded-input bounded-output (BIBO) stability criterion on the viewpoint of unwrapped phase is necessary and sufficient for 2-D separable DAFs, but is only sufficient for QP DAFs. The resultant property possesses the advantage of increasing the freedom of phase design over the previously proposed one. A remarkable application of the presented property is choosing an appropriate specification for the desired phase response of a 2-D QP DAF design. A 2-D nonsymmetric half-plane (NSHP) recursive DAF possesses more general causality and performs better than a 2-D quarter-plane (QP) recursive DAF. Hence, we also present the phase-response property for the BIBO stability of a 2-D causal recursive DAF with NSHP support region. Both cases of filters with real coefficients and complex coefficients are explored. Moreover, the effect of the numerator polynomial of a 2-D NSHP DAF on stability is also considered. The presented phase-response property has several applications. A remarkable application is that it can be utilized to enforce stability for a 2-D NSHP DAF design by choosing an appropriate phase specification. The eigenfilter design of 2-D NSHP DAFs for this application is also presented. The 1-D lattice filter structure exhibits the attractive advantages of low passband sensitivity and robustness to quantization error. The modularity of this structure makes industrial application. Additionally, 1-D digital lattice filter structure requires lower computational cost than 1-D direct form digital filter. The filter coefficients of 1-D direct-form allpass filter and the reflection coefficients of 1-D lattice allpass filter have a one-to-one mapping relationship. However, 2-D lattice allpass structures always do not have this relationship. Hence, we present a lattice structure for the realization of 2-D recursive DAFs with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a NSHP support region. Two variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Then, we present a parallel-combination structure composed of the 2-D lattice DAFs for the design of 2-D recursive filters. The novelty of the 2-D recursive filter is that it not only inherits the desirable attributes of lattice filters but also possesses the advantage of better performance over the 2-D recursive NSHP filters.
Subjects
Digital allpass filter (DAF)
quarter-plane (QP)
BIBO stability
unwrapped phase response
2-D recursive filter
nonsymmetric half-plane (NSHP)
lattice structure
Type
thesis
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