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Alexander Flaig, University of Delaware (U.S.A.)
Gonzalo Ramiro Arce, University of Delaware (U.S.A.)
Kenneth E. Barner, University of Delaware (U.S.A.)
We introduce a novel, data-adaptive, and robust filtering framework: affine order-statistic filters. Affine order-statistics relate classical order-statistics to observations in their natural order and thus inherently yield a meaningful data representation. Affine order-statistic filters exploit this notion to adaptively process nonstationary signals. Affine order-statistic filters overcome many of the limitations associated with traditional order-statistic filters, in particular: filters in this class are parsimonious in the number of filter coefficients, they are statistically efficient in a wide range of signal statistics, and they admit real-valued filter weights leading to a wide-range of filtering characteristics. The class of affine order statistic filters contains two families: the WOS affine filter class whose structure can adapt, according to the observed data, from an FIR linear filter to a WOS filter, and the FIR affine filter class whose structure can adapt from an L-filter to an FIR-filter.
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TOPDusan M. Kodek, University of Ljubljana (Slovenia)
It has been known for some time that it is not possible to meet arbitrarily severe FIR filter specifications with fixed $b$-bit wordlength by sufficiently increasing the filter length $N$. For any given non-trivial specification there is a nonzero lower bound on the approximation error, below which it is not possible to go, no matter how large the value of $N$. For practical purposes it is even more useful to know a lower bound for given $N$ and $b$. This bound represents a finite wordlength FIR filter design limit which is of theoretical importance and has not been known so far. This paper presents a method for computing this limit. The method is based on a lower bound theorem and can be used to estimate the approximation error limit in practical finite wordlength FIR design cases. It is also useful in the algorithm for the optimal finite wordlength design.
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Guo Fang Xu, University of Colorado (U.S.A.)
Tamal Bose, University of Colorado (U.S.A.)
Jim Schroeder, University of Colorado (U.S.A.)
Normal form digital filters are attractive due to their desirable properties when implemented in finite wordlength arithmetic. These filters are free from all overflow limit cycles and quantization limit cycles when magnitude truncation is used. However, when two's complement truncation (TCT) quantization is used, limit cycles can still exist. In this paper, it is shown that when block structures are used, normal form digital filters can be made free of limit cycles due to TCT quantization. It is shown that this can be done with a small block size. An algorithm is presented to find the minimum block size required for a given filter. Some examples are given to illustrate the results.
ic972153.pdf TOPLina J. Karam, ASU (U.S.A.)
This paper studies the applicability and limitations of the McClellan transformation method and, as a result, extends this method so that new types of one-dimensional filters can be transformed and new types of multi-dimensional filters can be designed. For this purpose, a new expression for the frequency response of an arbitrary one-dimensional filter is derived in terms of Chebyshev polynomials and other introduced polynomials satisfying recurrence formulae. The main objective is to identify which prototype filters can be transformed, determine what types of symmetry can be designed, and present procedures for transforming the new identified prototypes as well as rules for achieving the possible symmetries.
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Hartmut Brandenstein, University of Erlangen (Germany)
Rolf Unbehauen, University of Erlangen (Germany)
In this paper the approximation of a complex-valued specification by the frequency response of a 2-D IIR separable-denominator (SD) digital filter is considered. The approximation problem is transformed into an equivalent one, where a real-valued 2-D IIR SD digital filter with some additional characteristics has to be determined that approximates a given real-valued 2-D FIR digital filter. A theorem is presented that helps to reduce the number of parameters in the FIR-to-IIR approximation problem and a procedure to solve the problem numerically is given.
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Li Lee, MIT (U.S.A.)
Alan V. Oppenheim, MIT (U.S.A.)
It has been observed empirically that each coefficient in a Parks-McClellan filter converges to a steady state value as the filter length increases. This suggests the possibility of obtaining filters that are near optimal while "re-using" filter coefficients from shorter filters in the design of longer filters. In the context of approximate processing this then allows a filtering operation to be done in stages. This paper demonstrates this observation and examines some of its implications.
ic972165.pdf ic972165.pdf TOPMathias C. Lang, Vienna University of Technology (Austria)
This paper presents two methods for the design of FIR filters with arbitrary magnitude and phase responses according to a weighted mean squared error criterion with constraints on the resulting magnitude and phase errors. This constrained least square criterion allows for an arbitrary trade-off between pure $L_2$ filters and Chebyshev filters. The resulting nonlinear optimization problem is either converted into a standard quadratic programming problem (method 1) or exactly solved by a sequence of quadratic programs (method 2). The quadratic programming problems can be solved efficiently using standard software.
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Richard Rau, Georgia Institute of Technology (U.S.A.)
James H. McClellan, Georgia Institute of Technology (U.S.A.)
We introduce the design of polar-separable 2-D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1-D filters for the radial and the angular components are designed. Then the desired filter response is approximated on many radial slices in a weighted mean square sense. In the case of circular filters, RSA outperforms other design procedures in terms of ripple size and circularity of the passband. Examples of filters with non-constant angular functions prove the flexibility of the new method.
ic972173.pdf ic972173.pdf TOPJuha Kauraniemi, Helsinki University of Technology (Finland)
Delta operator filter structures have received interest due to good roundoff noise and coefficient sensitivity properties. However, it has been reported that delta realizations may produce limit cycles. In this paper, limit cycle problem in the direct form delta operator structure is studied by using a computer-aided test. The test determines exact amplitude and period of the maximum amplitude limit cycle, including the case where the limit cycles are absent. Using this knowledge the required wordlength to satisfy limit cycle performance specifications can be accurately determined. It is shown that with narrowband lowpass filters limit cycles, if they exist, are of much smaller amplitude than those of the traditional delay realized direct form structure.
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Yong Ching Lim, NUS (Singapore)
Seo How Low, NUS (Singapore)
The frequency response masking technique is an efficient method to realize sharp 1-D filters. This technique can synthesize sharp 1-D filters with a considerably lower complexity when compared to direct-form implementations. In this paper, we extend the frequency response masking technique to the design of 2-D diamond-shaped filters. The design procedure as well as the prototype and masking filters specifications are presented in this paper. A design example is also provided to illustrate the effectiveness of the approach.
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Mitsuhiko Yagyu, Tokyo Institute of Technology (Japan)
Akinori Nishihara, Tokyo Institute of Technology (Japan)
Nobuo Fujii, Tokyo Institute of Technology (Japan)
This paper presents a method to minimize the finite wordlength error in output signals of linear phase 2-D FIR filters. The finite wordlength errors can easily be analyzed in the frequency domain when the input signal statistics are known. In the case of white input signals, impulse responses corresponding to all levels of input impulses are optimized so as to minimize the errors. A new ROM-based filter structure is proposed in which the optimized impulse responses are stored in the ROM. The output signals are generated by superposing the impulse responses corresponding to the input levels. Many results of simulations confirm that the output signals of the proposed filters have far less errors than those of conventional filters.
ic972185.pdf ic972185.pdf TOPJosé L. Sanz-González, University of Politec. Madrid (Spain)
This paper is concerned with a linked analysis of overflow and roundoff errors in fixed-point digital filter realizations. Upper bounds for the overflow error power are obtained, having considered saturation quantizer characteristics. Also, upper bounds for the overflow probability are given in order to overflow power be lower than roundoff noise power. Finally, computer simulation results support the theoretical ones, and some of these results are presented in curves for the optimal state-space digital realizations of Butterworth, Chebyshev and elliptic filters.
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Rui Yang, Dept. EE, NUS (Singapore)
Yong Ching Lim, Dept. EE, NUS (Singapore)
Maurice Bellanger, Dept. EE, NUS (Singapore)
A novel computationally highly efficient realization of sharp symmetrical bandstop FIR filter is proposed. The new structure is derived using the frequency-response-masking technique, where the bandedge-shaping filter is derived from half-band filter by substituting each delay of the half-band filter by M delays. The masking filters are unconventional. They are quadrature filters derived from linear combinations of the masking filters in the conventional frequency-response-masking technique. Approximate expressions for the optimal value of M and the corresponding number of multipliers are derived.
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Kamen Ralev, ND (U.S.A.)
Peter Bauer, ND (U.S.A.)
Different options for block floating point filter implementation are introduced and their efficiency determined. The efficiency is quantified by the additional number of operations over those required for fixed point operation. Some of the implementations are new. It is shown that they are more efficient than the existing ones. Examples are given in which the processing time per recursion of a block floating point implementation on a fixed point processor is approximately the same as the recursion time of the corresponding fixed point implementation. Application of block floating point arithmetic to block implementations is also considered.
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