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.

Effects of finite register length in digital filtering and the fast Fourier transform

Time domain and frequency domain concepts which aid in the design and efficient implementation of a digital beamformer have been described at various times in the literature. The numerous beam-former structures that result are discussed with an emphasis on hardware requirements and spectral areas of application. Time domain procedures which include delay-sum, partial-sum, interpolation and shifted-sideband beamforming, and frequency domain techniques which include the application of discrete Fourier transforms and phase shift beam-forming are considered. Hardware considerations are primarily in the areas of analog-to-digital conversion, data storage, and computational throughput requirements.

A comparison of efficient beamforming algorithms

Digital time-domain beamforming requires that samples of the sensor signals be available at a sufficient rate to realize accurate time delays for beam steering. For many applications, this input rate, which may be significantly higher than the Nyquist rate required for waveform reconstruction, places stringent requirements on A/D converter hardware and transmission cable bandwidth. Recently, a technique referred to as digital interpolation beamfonning was introduced which greatly relaxes the sampling requirement and provides substantial hardware savings through more flexible design options. In this approach, the sensor channels need only be sampled at a rate which satisfies aliasing requirements. The vernier beam-delay increments are then synthesized using digital interpolation which can be implemented at the beamformer input or output to minimize digital processing complexity. Previously, this concept was presented for the case of "low-pass" signals. This paper extends this work by examining the relationship between interpolation and beamforming for the important class of "bandpass" signals. Specifically, sampling methods are discussed whereby the original waveform can be reconstructed from samples taken at a rate consistent with the bandwidth of the bandpass signal. Beamformer implementations are presented which utilize these bandwidth-sampling techniques in conjunction with interpolation and which compute beam output points at the generally low rate dictated by the signal bandwidth. The interpolation beamformer achieves time-delay quantization (beam-steering accuracy) independent of both the input and output sampling rates. This approach generally requires less hardware than conventional procedures. Interpolation falter characteristics dictated by the bandwidth-sampling procedure are described and efficient methods of implementation employing nonrecursive digital bandpass and low-pass filters are presented.

Digital interpolation beamforming for low-pass and bandpass signals

This is a collection of articles written by members of the Underwater Acoustic Signal Processing (UASP) Technical Committee. The first article, by D. W. Tufts, deals with the history of UASP prior to 1980. In this period, initial mathematical models were developed and the first experimental investigations of underwater acoustic propagation were performed. It was also recognized during this time that there are many similarities between radar and sonar signal processing. The article by J.P. Ianniello deals with research in passive and active sonar from 1980 to the present. Work in this period included experimental verification of algorithms that had been developed in the 1960s and 1970s (e.g. for adaptive beamforming), as well as the development of new approaches, which include acoustic propagation modeling in the design of signal processing algorithms. Such processing is referred to as matched field processing. A common task in passive sonar systems is to estimate the difference in times at which different sensors receive the same signal. Time-delay estimation is a first stage that feeds into subsequent processing blocks. I. Lourtie provides a concise review of work in this field. The article by J.C. Preisig deals with underwater acoustic communications. The underwater channel has several features that make reliable communication a challenging problem. Nevertheless, progress is being made by combining results from ocean acoustic modeling, communication theory, and signal processing. The final article, by J.M.F. Moura, deals with the future of signal processing in the ocean. In addition to considering advances in detection and localization, he deals with new applications such as acoustic tomography, physical oceanography, and synthetic aperture sonar.

The past, present, and the future of underwater acoustic signal processing

It is proposed to specify a filter only in terms of upper and lower limits on the response, find the shortest filter length which allows these constraints to be met, and then find a filter of that order which is farthest from the upper and lower constraint boundaries in a minimax sense. The simplex algorithm for linear programming is used to find a best linear-phase FIR filter of minimum length, as well as to find the minimum feasible length itself. The simplex algorithm, while much slower than exchange algorithms, also allows the incorporation of more general kinds of constraints, such as concavity constraints (which can be used to achieve very flat magnitude characteristics). Examples are given to illustrate how the proposed and common approaches differ, and how the proposed approach can be used to design filters with flat passbands, filters which meet point constraints, minimum phase filters, and bandpass filters with controlled transition band behavior. >

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METEOR: a constraint-based FIR filter design program

The general Remes exchange algorithm and its implementation are discussed briefly An analog of the exchange algorithm is applied to the problem in which upper and lower constraints are imposed on the frequency response of a nonrecursive digital filter with linear phase. A typical design example is presented with rather tight constraints in the passband

Interactive minimax design of linear-phase nonrecursive digital filters subject to upper and lower function constraints

A method is presented for analyzing the effets of quantization of coefficients on the frequency response of any frequency bin of the fast Fourier transform (FFT) algorithm. Although more detail is easily obtained, we concentrate on predicting the locations and sizes of all spurious sidelobes in the frequency response which are above any specified level. Certain sidelobes are present due to the particular window or weighting function that is being used. Spurious sidelobes or artifacts are extra sidelobes which are introduced if the FFT coefficients are quantized.

H.S. Hersey

Papers

Interactive minimax design of linear-phase nonrecursive digital filters subject to upper and lower function constraints

Effects of FFT coefficient quantization on bin frequency response