Design of Finite Word Length Linear-Phase FIR Filters in the Logarithmic Number System Domain

Abstract

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

Low-precision Logarithmic Number Systems

“…Basetas et al [6] show that LNS requires a reduced word length when implementing FIR and infinite impulse response filters. Alam and Gustafsson have also shown [2] that LNS improves the approximation error by around 20% when designing FIR filters as compared to a fixed-point design. Coleman proposed a 32-bit logarithmic processor and concludes that LNS provides faster execution, more accuracy and reduced architectural complexity as compared to single-precision 32-bit floating point, provided that the adder/subtractor has a latency less than that of a 32 × 32-bit multiplier.…”

“…Compared to floating point, LNS dramatically simplifies the hardware needed for multiplication, division, and square root. Multiplication and division become fixedpoint addition and subtraction of the exponents, which can be implemented with a simple integer adder [2,22]. Square root is computed by dividing the exponent by two, which can be implemented by simply dropping the least-significant bit of the fixed-point exponent.…”

“…Smaller word sizes can reduce the memory footprint of data, and the complexity of arithmetic for a variety of number systems [9,14,17,33], which will have a significant impact on the ability to deploy systems in resource constrained embedded devices deployed on the edge of networks. Further, fixed-point number systems with very short word lengths have been proposed in the literature for a variety of signal processing applications [2,24], while shorter floating-and fixedpoint numbers have also been used for neural networks [8,18,19,39,43,44].…”

“…Fully one half of LNS numbers fall between −1 and 1, and representable numbers become increasingly sparse thereafter. In applications like FIR filters, the coefficient space lies within the range of ±1 [2], while the distribution of weights and biases in DNNs is also typically clustered within a small range [25]. Thus, LNS is suitable for use in a number of different applications [2,22,30].…”

Design of Low-Power Structural FIR Filter Using Data-Driven Clock Gating and Multibit Flip-Flops

“…For low-power architectures, many techniques are used [6]. An integer linear programming (ILP) approach to design optimal finite wordlength linear-phase FIR filters in the logarithmic number system (LNS) domain has been proposed in [7], and different input wordlength and filter taps are adopted in [8]. In [9], a reduced dynamic signal representation technique is used.…”