Low Roundoff Noise Research Articles

Orthonormal minimal structure for n -dimensional denominator-separable digital filters for multiprocessor implementation

A new structure for n-dimensional-separable digital filters is derived based on an orthonormal expansion. The new structure possesses well behaved filter parameters that are easily calculated, needs the minimum number of delays and has only lossless and passive substructures that are free from overflow oscillations and have low sensitivity and low roundoff noise. It can be mapped into a systolic multiprocessor array in a straightforward manner.

Allpass expansions of digital transfer functions

It is well known that digital allpass filters can be efficiently realised by using lossless structures which possess low sensitivity, low roundoff noise and are free from limit cycles on implementation with finite word-length registers. It is established that every rational transfer function of a stable IIR filter can be expressed as a linear combination of a set of stable first-order allpass transfer functions. This leads to two types of implementation for every stable IIR digital filter in the form of a parallel or cascade connection of first-order allpass sections.

Realization of 2-D denominator-separable digital filter transfer functions using complex arithmetic

In this paper the problem of realizing 2-D denominator-separable digital filter transfer functions is considered for processing of real sequences. The approach is based on expressing the given 2-D transfer function as a sum of two reduced-order rational transfer functions with complex coefficients. New structures are obtained for equivalent reduced-order, complex-coefficient, 2-D transfer functions. All the realizations are basically parallelform structures with minimum-norm, low round-off noise and freedom from overflow limit cycles. A comparison of the different structures is also made.

A digital filter structure with constant output roundoff noise level independent of filter bandwidth

It is shown that the roundoff noise level of an all-pass filter realized in normalized lattice structure is approximately equal to a constant value that depends only on the filter's order for both fixed and floating point arithmetics. As a consequence, the parallel all-pass realization of a digital filter employing normalized lattice all-pass sections is shown to produce very low roundoff noise at the output which is practically independent of filter bandwidth. This result is valid for all classical types of filters, including low-pass, high-pass, and bandpass filters. >

Highly molecular systolic structures for denominator-separable 2-D recursive filters

Two systolic realization structures for the implementation of 2-D denominator-separable recursive filters are presented. These structures possess a high degree of modularity and parallelism with low roundoff noise. The number of delays and multipliers is fewer than for similar realizations in the literature. The proposed structures are more suitable for special-purpose hardware and amenable to VLSI design. >

Roundoff noise reduction in IIR digital filters via parallel decomposition

The authors investigate the roundoff noise for infinite-impulse-response (IIR) digital filters realized in some canonical forms. The minimum roundoff noise realization of this class is achieved by letting the finite-impulse-response (FIR) coefficients be identical to the corresponding partial impulse response of the desired filter. When the order of the numerator of the transfer function of the desired filter is greater than or equal to that of the denominator, there is an optimal decomposition which is shown to reduce the roundoff noise without adding additional multipliers or adders. The proposed structure is applied to parallel and cascade forms, with real as well as complex arithmetic and shown to result in lower roundoff noise. By increasing the order of the FIR filter, the optimal synthesis becomes a noise reduction scheme at the cost of additional computational complexity. >

The design of wave digital filters using fully pipelined bit-level systolic arrays

Wave digital filters (WDF) based on analogue lattice and unit element cascade networks possess important properties that make them suitable for VLSI integration. These include low round-off noise even with short coefficient wordlength and a building block (two-port adaptor) with a very simple structure. Exploiting these properties leads to area efficient designs. Systolic architectures for these WDF networks give the additional advantage of very high sampling rates with potential application in sonar and video signal processing. The bit-level systolic array design of two WDFs is considered in detail, beginning at the filter specification and ending at the VLSI hardware architecture, with discussion of the expected values of the integrated circuit parameters.

An estimation method of the pairing and ordering for IIR digital filters with low roundoff noise

AbstractThis paper considers the IIR cascade digital filter based on the fixed‐point arithmetic, and presents a method for estimation of the optimal pairing and ordering to minimize the output roundoff noise. In the proposed method, the computational complexity required in the estimation is reduced by considering only the pole frequencies of the transfer function in the evaluation. In the scaling of each branch node, the general L2 ‐ and L∞‐norms are assumed.The proposed method is compared with the methods of Jackson, Lee and Liu‐Peled which are used frequently in practice. The four methods are applied actually to two kinds of filters, and the computational complexities are compared. It is shown as a result that the proposed method requires almost the same computational complexity as those of Jackson and Lee from the lower‐order to higher‐order filters, and realizes the pairing and ordering with smaller output roundoff noise. It is shown also in general that the pairing has a larger effect on the normalized output roundoff noise than the ordering.

A realization of narrow-band recursive digital low-pass filter using highly quantized coefficients

A realization of a narrowband recursive digital filter using highly quantized coefficients is presented. The realization uses an accumulator structure which has a very low roundoff noise and a high pole grid density in the neighborhood of the point z=1. These features permit the coefficients of the A matrix of a state-space realization of the transfer function to be highly quantized without causing serious pole sensitivity problems. In addition, the realization is free of overflow oscillations, and requires only two effective multiplications per output sample for a second-order section.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the design of second-order state-space digital filter sections

A simplified parametric representation is introduced for the exploration of all real-scaled state-space realizations of a second-order transfer function having complex-conjugate poles. It is found that all real-scaled realizations can be represented as a function of a limited range of two real angles. It is shown that this representation can be used to obtain computationally efficient realizations with low roundoff noise and realizations which are roundoff-noise-optimal subject to L/sub infinity /-norm scaling. This latter class of realizations, which has not previously been synthesized, is found to be very similar to the realizations obtained by simply rescaling an optimal L/sub 2/-norm-scaled realization for L/sub infinity / scaling. An algebraic technique is introduced for the synthesis of second-order state-space realizations. This technique provides realizations that are as computationally efficient as possible, subject to preserving low roundoff noise, low coefficient sensitivity, and freedom from limit cycles.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lattice implementation of two-dimensional recursive digital filters

The implementation of a quarter-plane autoregressive 2-D filter using a developed 2-D Levinson algorithm is presented. The resulting structure inherits most of the desirable characteristics of the 1-D lattice filter, such as high modularity, low coefficient sensitivities, low roundoff noise, and elimination of internal overflow. The low sensitivity of the structure is shown by numerical examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Delay replacement in direct form structures

A class of high-order low-sensitivity digital filter structures is generated by means of a direct transformation of a high-order direct-form (DF) structure. The proposed structures have low coefficient dynamic range, and exhibit low coefficient frequency-response sensitivity and roundoff noise characteristics, yet require only 3N multiplications per output sample as compared to 2N in the direct-form case. >

An algorithm for constrained roundoff noise minimization in digital filters with application to two-dimensional filters

An algorithm is presented for minimizing roundoff noise effects in state-space implementations of recursive digital filters while constraining certain coefficients to be zero. Beginning with a direct form structure, the algorithm uses the scaling and noise matrices to iteratively construct an upper or lower triangular transformation matrix to apply to the original state-space model. The result is a structure with a saving of N(N-1)/2 multiples over the minimum noise structure. The technique is of great utility in the realization of two-dimensional filters that cannot, in general, be factored into the parallel or cascade connection of lower-order subfilters. Because of the sparse nature of the local state-space structure coefficient matrices in two-dimensional direct- and controller-form realizations, transforming with an upper and lower triangular block diagonal transformation matrix retains many zeros in the structure and results in a low roundoff noise, computationally efficient state-space realization. A novel controller-form state-space structure for a general two-dimensional function is presented to which the noise minimization procedure is directly applicable. Numerical examples are included. >

The digital all-pass filter: a versatile signal processing building block

The properties of digital all-pass filters are reviewed and a broad overview of the diversity of applications in digital filtering is provided. Starting with the definition and basic properties of a scalar all-pass function, a variety of structures satisfying the all-pass property are assembled, with emphasis placed on the concept of structural losslessness. Applications are then outlined in notch filtering, complementary filtering and filter banks, multirate filtering, spectrum and group-delay equalization, and Hilbert transformations. In all cases, the structural losslessness property induces very robust performance in the face of multiplier coefficient quantization. Finally, the state-space manifestations of the all-pass property are explored, and it is shown that many all-pass filter structures are devoid of limit cycle behavior and feature very low roundoff noise gain. >

Implementation of high-order direct- form digital filter structures

High-order direct form structures are generally avoided in the design of digital filters because of a number of finite wordlength problems associated with these structures. However, in adaptive or timevarying filter designs the high-order direct-form structure is a natural and convenient choice. A block floating-point implementation of a high-order direct-form structure is proposed, which gives lower roundoff noise and has improved scaling properties compared to the conventional high-order implementations.

On realization of low-pass and highpass recursive filters with low sensitivity and low roundoff noise

As the poles and zeros of a sharp cutoff filter are expected to be clustered around band edges and in close proximity of the unit circle, the sensitivity problem of such filters are linked with their bandwidth. Analysis of the sensitivity characteristics of some of the well-known low sensitivity structures revealed their inadequacy to encompass the entire range of bandwidth for sharp cutoff filters. Some new structures are presented to fill the gap.

Analysis of coefficient quantization errors in state-space digital filters

In this paper, we examine the sensitivity of state-space digital filters to coefficient quantization errors. A cost function using an L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> norm criteria is used to measure the deviation in frequency response of the filter. The cost function is used to indicate the connection between roundoff noise and filter sensitivity to coefficient quantization errors. Then based on this cost function it is shown that low roundoff noise state-space digital filters, filters in balanced coordinates, and filters based on polynomials orthogonal on the unit circle have low sensitivity to coefficient quantization errors. Furthermore, the sensitivity of the filters is shown to depend on the second-order modes thereby exhibiting certain robustness and immunity to filter bandwidth.

Computationally efficient low roundoff noise second-order state-space structures

A new technique is introduced for synthesizing low roundoff noise second-order state-space structures. The technique yields structures which require the same number of multiplies as the canonical direct form structure. Thus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order realizations composed of these subfilters require only about 2 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> multiplies. The number of additions required is also low, ranging from 2 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> to 2.5 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> . The synthesis of these structures consists of simple design equations and a simple procedure. A numerical example illustrating the design process is provided.

Elliptic filter design for a class of generalized halfband filters

In multirate processing, symmetric finite impulse response halfband filters constitute an important class of filters that are useful for sampling rate alteration by a factor of two. Generalizing the notion of halfband filters to include infinite impulse response filters, a class of generalized halfband filters is examined and its properties are studied. The filters are efficient and have low roundoff noise. A filter design program has been developed to design a special class of elliptic filters taking into account restrictions due to the generalized halfband characteristics. The procedure adapts the results of elliptic filter design [10] to this special case. It is shown that if the passband and stopband edge frequencies are related, then these filters are computationally efficient even when no sampling rate change is needed.

A class of low-noise computationally efficient recursive digital filters with applications to sampling rate alterations

A new structure for multiband recursive digital filters is proposed. For meeting low-pass filter specifications, it uses fewer multiplications than conventional elliptic filter realizations. An approximation to the minimax solution is obtained numerically by minimizing the LP error norm. The analytic optimum for odd order low-pass filters of this new class turns out to be the elliptic filter itself, but in a new configuration. Analytic solution is also obtained for filters used in decimation/interpolation by a factor of 2. There are several realizations for this new structure, the choice of which depends on the location of poles and zeros. Some selected realizations always have low roundoff noise and small limit cycle bounds.