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This collection of papers is the result of a desire to make available reprints of articles on digital signal processing for use in a graduate course offered at MIT. The primary objective was to present reprints in an easily accessible form. At the same time, it appeared that this collection might be useful for a wider audience, and consequently it was decided to reproduce the articles (originally published between 1965 and 1969) in book form.The literature in this area is extensive, as evidenced by the bibliography included at the end of this collection. The articles were selected and the introduction prepared by the editor in collaboration with Bernard Gold and Charles M. Rader.The collection of articles divides roughly into four major categories: z-transform theory and digital filter design, the effects of finite word length, the fast Fourier transform and spectral analysis, and hardware considerations in the implementation of digital filters.
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This chapter contains sections titled: Half Title, Title, Copyright, Preface, Introduction, Contents, Half Title View full abstract»
This chapter contains sections titled: List of Symbols, Preliminaries, A Specific Isomorphism Between the Analog and Digital Signal Spaces, The Orthonormal Expansion Attached to μ, The Induced Mapping for Filters, Optimization Problems for Systems with Deterministic Signals, Random Signals and Statistical Optimization Problems, The Approximation Problem for Digital Filters, References View full abstract»
The Hilbert transform has traditionally played an important part in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform plays a similar role in digital signal processing. In this paper, the Hilbert transform relations, as they apply to sequences and their z-transforms, and also as they apply to sequences and their Discrete Fourier Transforms, will be discussed. These relations are identical only in the limit as the number of data samples taken in the Discrete Fourier Transforms becomes infinite. The implementation of the Hilbert transform operation as applied to sequences usually takes the form of digital linear networks with constant coefficients, either recursive or non-recursive, which approximate an all-pass network with 90° phase shift, or two-output digital networks which have a 90° phase difference over a wide range of frequencies. Means of implementing such phase shifting and phase splitting networks are presented. View full abstract»
It is commonly assumed that digital filters with both poles and zeros in the complex z-plane can be synthesized using only recursive techniques while filters with zeros alone can be synthesized by either direct convolution or via the discrete Fourier transform (OFT). In this letter it is shown that no such restrictions hold and that both types of filters (those wth zeros alone or those with both poles and zeros) can be synthesized using any of the three methods, namely, recursion, OFT, or direct convolution. View full abstract»
We introduce an approach to the design of low-pass (and, by extension, bandpass) digital filters containing only zeros. This approach is that of directly searching for transition values of the sampled frequency response function to reduce the sidelobe level of the response. It is shown that the problem is a linear program and a search algorithm is derived which makes it easier to obtain the experimental results. View full abstract»
When implementing a digital filter, it is important to utilize in the design a bound or estimate of the largest output value which will be obtained. Such a bound is particularly useful when fixed point arithmetic is to be used since it assists in determining register lengths necessary to prevent overflow. In this paper we consider the class of digital filters which have an impulse response of finite duration and are implemented by means of circular convolutions performed using the discrete Fourier transform. A least upper bound is obtained for the maximum possible output of a circular convolution for the general case of complex input sequences. For the case of real input sequences, a lower bound on the least upper bound is obtained. The use of these results in the implementation of this class of digital filters is discussed. View full abstract»
The literature on sampled-data filters, although extensive on design methods, has not treated adequately the important problems connected with the actual realization of the obtained filters with finite arithmetic elements. Beginning with a review of the traditional design procedures a comparison is made between the different canonical realization forms and their related computational procedures. Special attention is directed to the problems of coefficient accuracy and of rounding and trucation effects . A simple expression is derived which yields an estimate of the required coefficient accuracy and which shows clearly the relationship of this accuracy to both sampling rate and filter complexity. View full abstract»
A statistical model for roundoff noise in floating point digital filters, proposed by Kaneko and Liu, is tested experimentally for first- and second-order digital filters. Good agreement between theory and experiment is obtained. The model is used to specify a comparison between floating point and fixed point digital filter realizations on the basis of their output noise-to-signal ratio, and curves representing this comparison are presented. One can find values of the filter parameters at which the fixed and the floating View full abstract»
Recently, statistical models for the effects of roundoff noise in fixed-point and floating-point realizations of digital filters have been proposed and verified, and a comparison between these realizations presented. In this paper a structure for implementing digital filters using block-floating-point arithmetic is proposed and a statistical analysis of the effects of roundoff noise carried out. On the basis of this analysis, block-floating-point is compared to fixed-point and floating - point arithmetic with regard to roundoff noise effects. View full abstract»
This paper is concerned with the accumulation of round-off error in a floating-point digital filter. The error committed at each arithmetic operation is assumed to be an independent random variable uniformly distributed in (−2−t, 2−t) where t is the length of the mantissa. Expression for the mean square error is derived and a numerical example is given. View full 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. View full abstract»
The first part of this paper presents a new measure of sensitivity specifically applicable to the realization of a linear discrete system on a digital computer. It is also shown that the sensitivity of the eigenvalues to parameter inaccuracies in the realization depends strongly on the choice of state variables. From these considerations, a realization is obtained which is “best” for a large class of systems of interest with regard to minimizing storage requirements, arithmetic operations, parameter accuracy, and eigenvalue sensitivity . The second half of the paper considers the very practical problem of determining the number of bits accuracy required in the computer-stored parameters of the system to achieve satisfactory performance . For the realization found to be a best compromise, equations are obtained for determining these bit requirements. Examples are given showing the application of this realization to the computer im - plementation of a discrete filter, and a comparison is given to other possible realizations. View full abstract»
This paper contains an analysis of the fixed-point accuracy of the powqer of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simulated fixed-point machine and their comparison with the error upper bound. View full abstract»
A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic. The result, derived for the case of white input signal, is that the ratio of mean-squared output noise to mean-squared output signal varies essentiallay as ν = log2N, where N is the number of points transformed. This predicted result is significantly lower than bounds previously derived on mean-squared output noise-to-signal ratio, which are proportional to ν2. The predictions are verified experimentally, with excellent agreement. The model applies to rounded arithmetic, and it is found experimentally that if one truncates, rather than rounds, the results of floating point additions and multiplications, the output noise increases significantly (for a given ν). Also, for truncation, a greater than linear increase with ν of the output noise-to-signal ratio is observed. View full abstract»
This collection of papers is the result of a desire to make available reprints of articles on digital signal processing for use in a graduate course offered at MIT. The primary objective was to present reprints in an easily accessible form. At the same time, it appeared that this collection might be useful for a wider audience, and consequently it was decided to reproduce the articles (originally published between 1965 and 1969) in book form.The literature in this area is extensive, as evidenced by the bibliography included at the end of this collection. The articles were selected and the introduction prepared by the editor in collaboration with Bernard Gold and Charles M. Rader.The collection of articles divides roughly into four major categories: z-transform theory and digital filter design, the effects of finite word length, the fast Fourier transform and spectral analysis, and hardware considerations in the implementation of digital filters. View full abstract»
A computational algorithm for numerically evaluating the z-transform of a sequence of N samples is discussed . This algorithm has been named the chirp z-transform (CZT) algorithm . Using the CZT algorithm one can efficiently evaluate the z-transform at M points in the z-plane which lie on circular or spiral contours beginning at any arbitrary point in the z-plane. The angular spacing of the points is an arbitrary constant, and M and N are arbitrary integers. The algorithm is based on the fact that the values of the z-transform on a circular or spiral contour can be expressed as a discrete convolution. Thus one can use well-known high-speed convolution teehniques to evaluate the transform efficiently. For M and N moderately large, the computation time is roughly proportional to (N + M) log2(N + M) as opposed to being proportional to (N ⋅ M) for direct evaluation of the z-transform at M points. View full abstract»
This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension n of the transform is factored (if possible), and n/p elementary transforms of dimension p are computed for each factor p of n. An improved method of computing a transform step corresponding to an odd factor of n is given; with this method, the number of complex multiplications for an elementary transform of dimension p is reduced from (p−1)2 to (p−1)2/4 for odd p. The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine. View full abstract»
This chapter contains sections titled: Introduction, An Algorithm Suggested by Chirp Filtering View full abstract»
An approach to the implementation of digital filters is presented that employs a small set of relatively simple digital circuits in a highly regular and modular configuration, well suited to LSI construction. Using parallel processing and serial, two's-complement arithmetic, the required arithmetic circuits (adders and multipliers ) are quite simple, as are the remaining circuits, which consist of shift registers for delay and small read-only memories for coefficient storage. The arithmetic circuits are readily multiplexed to process multiple data inputs or to effect multiple, but different, filters (or both), thus providing for efficient hardware utilization. Up to 100 filter sections can be multiplexed in audio-frequency applications using presently available digital circuits in the medium-speed range. The filters are also easily modified to realize a wide range of filter forms, transfer functions, multiplexing schemes, and round-off noise levels by changing only the contents of the read-only memory and/or the timing signals and the length of the shift-register delays. A simple analog-to-digital converter, which uses delta modulation as an intermediate encoding process is also presented for audio-frequency applications. View full abstract»
This discussion served as an introduction to the Hardware Implementations session of the IEEE Workshop on Fast Fourier Transform Processing. It introduces the problems associated with implementing the FFT algorithm in hardware and provides a frame of reference for characterizing specific implementations. Many of the design options applicable to an FFT processor are described, and a brief comparison of several machine organizations is given. View full abstract»
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