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Design of Finite Word Length Linear-Phase FIR Filters inthe Logarithmic Number System Domain
Linköping University, Department of Electrical Engineering, Electronics System. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-1509-9678
Linköping University, Department of Electrical Engineering, Electronics System. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-3470-3911
2014 (English)In: VLSI design (Print), ISSN 1065-514X, E-ISSN 1563-5171, Vol. 2014, no 217495Article in journal (Refereed) Published
Abstract [en]

Logarithmic number system (LNS) is an attractive alternative to realize finite-length impulse response filters because ofmultiplication in the linear domain being only addition in the logarithmic domain. In the literature, linear coefficients are directlyreplaced by the logarithmic equivalent. In this paper, an approach to directly optimize the finite word length coefficients in theLNS domain is proposed. This branch and bound algorithm is implemented based on LNS integers and several different branchingstrategies are proposed and evaluated. Optimal coefficients in the minimax sense are obtained and compared with the traditionalfinite word length representation in the linear domain as well as using rounding. Results show that the proposed method naturallyprovides smaller approximation error compared to rounding. Furthermore, they provide insights into finite word length propertiesof FIR filters coefficients in the LNS domain and show that LNS FIR filters typically provide a better approximation error comparedto a standard FIR filter.

Place, publisher, year, edition, pages
Egypt: Hindawi Publishing Corporation, 2014. Vol. 2014, no 217495
Keyword [en]
Logarithmic Number System, FIR Filter, Integer Linear Programming, Branch and Bound
National Category
Signal Processing
URN: urn:nbn:se:liu:diva-105861DOI: 10.1155/2014/217495OAI: diva2:711604
Available from: 2014-04-10 Created: 2014-04-10 Last updated: 2016-05-30Bibliographically approved
In thesis
1. Techniques for Efficient Implementation of FIR and Particle Filtering
Open this publication in new window or tab >>Techniques for Efficient Implementation of FIR and Particle Filtering
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

FIR filters occupy a central place many signal processing applications which either alter the shape, frequency or the sampling frequency of the signal. FIR filters are used because of their stability and possibility to have linear-phase but require a high filter order to achieve the same magnitude specifications as compared to IIR filters. Depending on the size of the required transition bandwidth the filter order can range from tens to hundreds to even thousands. Since the implementation of the filters in digital domain requires multipliers and adders, high filter orders translate to a large number of these arithmetic units for its implementation. Research towards reducing the complexity of FIR filters has been going on for decades and the techniques used can be roughly divided into two categories; reduction in the number of multipliers and simplification of the multiplier implementation. 

One technique to reduce the number of multipliers is to use cascaded sub-filters with lower complexity to achieve the desired specification, known as FRM. One of the sub-filters is a upsampled model filter whose band edges are an integer multiple, termed as the period L, of the target filter's band edges. Other sub-filters may include complement and masking filters which filter different parts of the spectrum to achieve the desired response. From an implementation point-of-view, time-multiplexing is beneficial because generally the allowable maximum clock frequency supported by the current state-of-the-art semiconductor technology does not correspond to the application bound sample rate. A combination of these two techniques plays a significant role towards efficient implementation of FIR filters. Part of the work presented in this dissertation is architectures for time-multiplexed FRM filters that benefit from the inherent sparsity of the periodic model filters.

These time-multiplexed FRM filters not only reduce the number of multipliers but lowers the memory usage. Although the FRM technique requires a higher number delay elements, it results in fewer memories and more energy efficient memory schemes when time-multiplexed. Different memory arrangements and memory access schemes have also been discussed and compared in terms of their efficiency when using both single and dual-port memories. An efficient pipelining scheme has been proposed which reduces the number of pipelining registers while achieving similar clock frequencies. The single optimal point where the number of multiplications is minimum for non-time-multiplexed FRM filters is shown to become a function of both the period, L and time-multiplexing factor, M. This means that the minimum number of multipliers does not always correspond to the minimum number of multiplications which also increases the flexibility of implementation. These filters are shown to achieve power reduction between 23% and 68% for the considered examples.

To simplify the multiplier, alternate number systems like the LNS have been used to implement FIR filters, which reduces the multiplications to additions. FIR filters are realized by directly designing them using ILP in the LNS domain in the minimax sense using finite word length constraints. The branch and bound algorithm, a typical algorithm to implement ILP problems, is implemented based on LNS integers and several branching strategies are proposed and evaluated. The filter coefficients thus obtained are compared with the traditional finite word length coefficients obtained in the linear domain. It is shown that LNS FIR filters provide a better approximation  error compared to a standard FIR filter for a given coefficient word length.

FIR filters also offer an opportunity in complexity reduction by implementing the multipliers using Booth or standard high-radix multiplication. Both of these multiplication schemes generate pre-computed multiples of the multiplicand which are then selected based on the encoded bits of the multiplier. In TDF FIR filters, one input data is multiplied with a number of coefficients and complexity can be reduced by sharing the pre-computation of the multiplies of the input data for all multiplications. Part of this work includes a systematic and unified approach to the design of such computation sharing multipliers and a comparison of the two forms of multiplication. It also gives closed form expressions for the cost of different parts of multiplication and gives an overview of various ways to implement the select unit with respect to the design of multiplexers.

Particle filters are used to solve problems that require estimation of a system. Improved resampling schemes for reducing the latency of the resampling stage is proposed which uses a pre-fetch technique to reduce the latency between 50% to 95%  dependent on the number of pre-fetches. Generalized division-free architectures and compact memory structures are also proposed that map to different resampling algorithms and also help in reducing the complexity of the multinomial resampling algorithm and reduce the number of memories required by up to 50%.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 109 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1716
FIR filter, Particle filter, architecture, ASIC, FPGA
National Category
Electrical Engineering, Electronic Engineering, Information Engineering
urn:nbn:se:liu:diva-124195 (URN)10.3384/diss.diva-124195 (DOI)978-91-7685-915-5 (Print) (ISBN)
Public defence
2016-02-23, Visionen, Hus B, Campus Valla, Linköping, 13:15 (English)
Available from: 2016-01-25 Created: 2016-01-21 Last updated: 2016-03-01Bibliographically approved

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