On Efficient Digital Filter Structure And Its Applica-tion In The Virtual Auditory Realization

Digital filters/controllers play a very important role in many areas such as multi-media signal processing, communications and control engineering. In real-time applica-tions, these well-designed systems have to be implemented with a digital device of fi-nite precision such as micro-controller unit (MCU), digital signal processor (DSP) or field programmable gate array (FPGA). This finite precision implementation will lead to the so-called finite word length (FWL) effects including round-off noise, instability and over-flow, which can degrade the system performance dramatically such that the actually im-plemented system is out of work. How to reduce these effects has been one of the most important issues.One way to reduce the FWL effects is simply to use a device of high precision. How-ever, in many applications with the increase of complexity and performance of systems, the specification requirement for the filters becomes high and high such that it is inappropriate to rely solely on increasing the word length. For example, low-power and low-cost are very important design criteria in many portable devices. In addition, the low complexity filter structure design is coincident with the current trend of green design.As well-known, there exist a number of different structures that can be used for im-plementing a given digital filter and these structures are all equivalent in infinite precision since they represent the same transfer function. The important fact is that different struc-tures have a different capability against the FWL effects. The theory of filter structures, developed during the last two decades, has been considered one of the most effective tools in reducing the FWL effects.In this theses, based on a study of the existing results in optimal FWL filter structure design a state-space normal realization method is proposed for digital filter implementa-tion. The construction procedure is also given. At the same time, it is shown that such a system is free of self-oscillation. Note that for a given filter there exist an infinite num- ber of such normal structures, the optimal round-off noise normal realization problem is formulated in terms of finding the normal realizations that minimize the round-off noise within this set of structures and is solved analytically. The corresponding round-off noise gain and error propagation gain are also obtained. Numerical examples show that the op-timal round-off noise normal realization has the advantage over those classical optimal realizations, especially in the cases of short word length.Like the classical optimal realizations, the optimal round-off noise normal realization is fully parameterized. In order to reduce the implementation complexity, a more efficient structure is derived by means of the Schur decomposition of the transit matrix via a series of Givens similarity transformations. This proposed realization is shown free of limit cy-cles and the number of trivial parameters it possesses is about 3.5 times the filter order. The corresponding round-off noise gain is also derived. This realization has the advantage over the classical realizations on the ability against the FWL effects and implementation complexity.As an application of the theoretical results developed in this thesis, a virtual auditory display system is considered, where the optimal round-off noise sparse realization is used to implement the head-related transfer function (HRTF) which is the key part of this sys-tem. The HRTF is an finite impulse response (FIR) filter of order between 200 and 500, which makes the implementation very difficult. Based on the analysis of the traditional system reduction methods, a new method that combines the Yule-Walker algorithm with the balanced model truncation (BMT) is proposed. The proposed system reduction method yields an infinite impulse response (â…¡R) of much lower order that approximates the HRTF filter very well. Based on this reduced order IIR filter, the HRTF is then successfully im-plemented with our proposed optimal round-off noise sparse structure. Simulations show that a virtual auditory display system realized in the proposed approach yields a satisfying performance.