# New recursive digital filter structures having very low sensitivity and roundoff noise

TL;DR: For poles close to the unit circle and near z = 1, the usual realizations of recursive or IIR digital filters are highly sensitive to the coefficient quantization and have large roundoff noise as mentioned in this paper.

Abstract: For poles close to the unit circle and near z = 1 , the usual realizations of recursive or IIR digital filters are highly sensitive to the coefficient quantization and have large roundoff noise. As the sampling rate is increased the poles approach z = 1 and the problems become more severe. For these situations several new digital filter structures are presented for which the above errors remain constant and generally insignificant as the sampling rate is increased. Results on sensitivity and the roundoff errors for these new structures are presented and compared with conventional realizations. Some numerical results are also presented showing order of magnitude improvements.

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TL;DR: In this article, it is shown that the shift operator and its associated Z -transform can be replaced by delta operators and their associated transform, which is designated a Δ-transform.

Abstract: This paper examines some of the consequences of finite word lengths in digital control. It is shown that, in many cases of practical importance, the usual shift operator formulation is inferior to an alternative formulation which we designate the delta operator approach. This latter approach is shown to give better coefficient representation and less roundoff noise in many cases. We thus argue that the shift operator and its associated Z -transform can be replaced by delta operators and their associated transform which we designate a Δ-transform. An added advantage of this approach is that discrete designs and transforms converge to their continuous-time counterparts as the sampling rate is increased.

448 citations

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Motorola

^{1}TL;DR: In this paper, the authors describe a digital radio receiver which operates on a received analog signal which has been converted to a digital form after preselection at the output of the antenna.

Abstract: A digital radio receiver is described. The digital receiver of the present invention contemplates a digital radio receiver which operates on a received analog signal which has been converted to a digital form after preselection at the output of the antenna. The digital receiver of the present invention comprises a preselector, a high-speed analog-to-digital (A/D) converter, a digitally implemented intermediate-frequency (IF) selectivity section having an output signal at substantially baseband frequencies, and digital signal processor (DSP) circuit performing demodulation and audio filtering. The radio architecture of the present invention is programmably adaptable to virtually every known modulation scheme and is particularly suitable for implementation on integrated circuits.

293 citations

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TL;DR: A new technique to analog sampled data filtering is presented which can be fully integrated using MOS technology, and advantages of this new approach are reduced circuit complexity, low sensitivity to coefficient variations, and efficient utilization of silicon area.

Abstract: A new technique to analog sampled data filtering is presented which can be fully integrated using MOS technology. Advantages of this new approach are reduced circuit complexity, low sensitivity to coefficient variations, and efficient utilization of silicon area. Performance of monolithic low Q(Q=1) and high Q(Q=73) filters are presented which were implemented using NMOS technology. In implementing the high Q filter a new operational amplifier design was used which had a 14-V output range, rms noise voltage of 45 /spl mu/V, an open-loop gain of 6000, and a unity-gain bandwidth of 2 MHz.

282 citations

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01 Feb 1992

TL;DR: An attempt is made to organize and survey recent work, and to present it in a unified and accessible form, on the need for a new approach suitable for high-speed processing and the use of difference operators in numerical analysis.

Abstract: An attempt is made to organize and survey recent work, and to present it in a unified and accessible form. The need for a new approach suitable for high-speed processing is discussed in the context of several applications in control and communications, and a historical perspective of the use of difference operators in numerical analysis is presented. The general systems calculus, based on divided-different operators is introduced to unify the continuous-time and discrete-time systems theories. This calculus is then used as a framework to treat the three problems of system state estimation; system identification and time-series modeling; and control system design. Realization aspects of algorithms based on the difference operator representation, including such issues as coefficient rounding and implementation with standard hardware, are also discussed. >

276 citations

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TL;DR: A new implementation of an IIR digital filter transfer function is presented that is structurally passive and, hence, has extremely low pass-band sensitivity.

Abstract: A new implementation of an IIR digital filter transfer function is presented that is structurally passive and, hence, has extremely low pass-band sensitivity. The structure is based on a simple parallel interconnection of two all-pass sections, with each section implemented in a structurally lossless manner. The structure shares a number of properties in common with wave lattice digital filters. Computer simulation results verifying the low-sensitivity feature are included, along with results on roundoff noise/dynamic range interaction. A large number of alternatives is available for the implementation of the all-pass sections, giving rise to the well-known wave lattice digital filters as a specific instance of the implementation.

167 citations

##### References

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01 Aug 1972TL;DR: The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff, to illustrate techniques of working with particular models.

Abstract: When digital signal processing operations are implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. The main categories of finite register length effects are errors due to A/D conversion, errors due to roundoffs in the arithmetic, constraints on signal levels imposed by the need to prevent overflow, and quantization of system coefficients. The effects of finite register length on implementations of linear recursive difference equation digital filters, and the fast Fourier transform (FFT), are discussed in some detail. For these algorithms, the differing quantization effects of fixed point, floating point, and block floating point arithmetic are examined and compared. The paper is intended primarily as a tutorial review of a subject which has received considerable attention over the past few years. The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff. The analyses presented here are intended to illustrate techniques of working with particular models. Results of previous work are discussed and summarized when appropriate. Some examples are presented to indicate how the results developed for simple digital filters and the FFT can be applied to the analysis of more complicated systems which use these algorithms as building blocks.

333 citations

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TL;DR: The concept of “transpose configurations” is introduced and is found to be quite useful in digital-filter synthesis; for although such configurations have identical transfer functions, their roundoff-noise outputs and dynamic-range limitations can be quite different, in general.

Abstract: The interaction between the roundoff-noise output from a digital filter and the associated dynamic-range limitations is investigated for the case of uncorrelated rounding errors from sample to sample and from one error source to another. The required dynamic-range constraints are derived in terms of L p norms of the input-signal spectrum and the transfer responses to selected nodes within the filter. The concept of “transpose configurations” is introduced and is found to be quite useful in digital-filter synthesis; for although such configurations have identical transfer functions, their roundoff-noise outputs and dynamic-range limitations can be quite different, in general. Two transpose configurations for the direct form of a digital filter are used to illustrate these results.

287 citations

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TL;DR: In this article, the roundoff-noise outputs from two transpose configurations, each for the cascade and parallel forms of a digital filter, are analyzed for the case of uncorrelated roundoff noise and fixed dynamic range.

Abstract: The roundoff-noise outputs from two transpose configurations, each for the cascade and parallel forms of a digital filter, are analyzed for the case of uncorrelated roundoff noise and fixed dynamic range. Corresponding transpose configurations are compared on the basis of the variance, or total average power, and the peak spectral density of the output roundoff noise. In addition to providing general computational techniques to be employed in choosing an appropriate configuration for the digital filter, these results also indicate useful "rules of thumb" relating to this choice of configuration. Included are indications of good (although not necessarily optimum) sequential orderings and pole-zero pairings for the second-order sections comprising the cascade form. Computational results are presented which indicate that the analysis is quite accurate and useful.

224 citations

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TL;DR: The calculation of the statistical mean-squared error at the output of the filter is discussed in detail and some of the approaches used in investigating them are reviewed.

Abstract: The accuracy of a digital filter is limited by the finite word length used in its implementation. Techniques have been developed to analyze this problem. Good agreement between the theoretical and experimental results has been reported. This paper discusses some of these accuracy problems and reviews some of the approaches used in investigating them. The calculation of the statistical mean-squared error at the output of the filter is discussed in detail.

217 citations

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TL;DR: In this article, it was shown that quantization of a digital filter's coefficients in an actual realization can be represented by a "stray" transfer function in parallel with the corresponding ideal filter.

Abstract: The frequency response of a digital filter realized by a finite word-length machine deviates from that which would have been obtained with an infinite word-length machine. An "ideal" or "errorless" filter is defined as a realization of the required pulse transfer function by an infinite word-length machine. This paper shows that quantization of a digital filter's coefficients in an actual realization can be represented by a "stray" transfer function in parallel with the corresponding ideal filter. Also, by making certain statistical assumptions, the statistically expected mean-square difference between the real frequency responses of the actual and ideal filters can be readily evaluated by one short computer program for all widths of quantization. Furthermore, the same computations may be used to evaluate the rms value of output noise due to data quantization and multiplicative rounding errors. Experimental measurements verify the analysis in a practical case. The application of the results to the design of the digital filters is also considered.

114 citations