Multiplication-Free Polynomial-Based FIR Filters with an Adjustable Fractional Delay

J. Yli-Kaakinen and T. Saramäki, "Multiplication-free polynomial-based FIR filters with an adjustable fractional delay," Circuits Syst. Signal Process., vol. 25, no. 2, pp. 265–294, Apr. 2006.

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Electronic version of an article published as Circuits, Systems, and Signal Processing, vol. 25, no. 2, pp. 265–294, Apr. 2006. (doi: 10.1007/s00034-005-2507-3) © Copyright © 2006, Birkhäuser Boston. http://www.springer.com/birkhauser/engineering/journal/34

Abstract

An efficient coefficient quantization scheme is described for minimizing the cost of implementing fixed parallel linear-phase finite impulse response (FIR) filters in the modified Farrow structure introduced by Vesma and Saramaki for generating FIR filters with an adjustable fractional delay. The implementation costs under consideration are the minimum number of adders and subtracters when implementing these parallel subfilters as a very large-scale integration (VLSI) circuit. Two implementation costs are under consideration to meet the given criteria. In the first case, all the coefficient values are implemented independently of each other as a few signed-powers-of-two terms, whereas in the second case, the common subexpressions within all the coefficient values included in the overall implementation are properly shared in order to reduce the overall implementation cost even further. The optimum finite-precision solution is found in four steps. First, the number of filters and their (common odd) order are determined such that the given criteria are sufficiently exceeded in order to allow some coefficient quantization errors. Second, those coefficient values of the subfilters having a negligible effect on the overall system performance are fixed to be zero valued. In addition, the experimentally observed attractive connections between the coefficient values of the subfilters, after setting some coefficient values equal to zero, are utilized to reduce both the implementation cost and the parameters to be optimized even more. Third, constrained nonlinear optimization is applied to determine for the remaining infinite-precision coefficients a parameter space that includes the feasible space where the given criteria are met. The fourth step involves finding in this space the desired finite-precision coefficient values for minimizing the given implementation costs to meet the stated overall criteria. Several examples are included illustrating the efficiency of the proposed synthesis scheme.
Farrow structure
Fig: The modified Farrow structure with an adjustable fractional delay µ.
Subfilter structures
Fig: (a) Direct-form implementation for G(z) of order 2M−1 for ℓ even. (b) Direct-form implementation for G(z) of order 2M−1 for ℓ odd. The even (odd) symmetries of the impulse responses of the G(z)'s for ℓ even (odd) are exploited in these implementations. When using these direct-form structures, the number of delay elements required by the overall implementation is only 2M−1. This reduction can be achieved by properly sharing these 2M−1 delay terms between the G(z)'s.
Subfilter structures
Fig: Efficient implementation of the infinite-precision adjustable fractional delay FIR filter in Example 1 with M=14. The number of multipliers and adders for the subfilters are 28 and 46, respectively, whereas for the overall implementation the corresponding figures are 32 and 51, respectively.
Subfilter structures
Fig: Efficient implementation of the adjustable fractional delay FIR filter in Example 4. For simplifying the overall diagram, the delay elements are not shared between the subfilters with symmetrical impulse responses and the subfilters with anti-symmetrical impulse responses. For the final practical implementation, these seven delay elements can be shared, after some modifications, between the existing four branch filters.

BibTeX

@Article{ylikaaJCSSP06,
  author = {J. Yli-Kaakinen and T. Saram{\"a}ki},
  title = {Multiplication-free polynomial-based {FIR} filters with an adjustable fractional
          delay},
  journal = "J. Circuits Syst. Signal Process.",
  year = 2006,
  volume = 25,
  number = 2,
  pages = {265--294},
  month = {Apr.}
}

Citing Documents

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