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A theory of multiple antenna AMTI radar

Published in:
MIT Lincoln Laboratory Report TN-1971-21

Summary

This note presents a detailed mathematical analysis of a multiple-antenna AMTI radar system capable of detecting moving targets over a significantly wider velocity range than is achievable with a single-antenna system. The general system configuration and signaling strategy is defined, and relationships among system and signaling parameters are investigated. A deterministic model for the target return and a statistical model for the clutter and noise returns are obtained, and an optimum processor for target detection is derived. A performance measure applicable to a large class of processors, including the optimum processor, is defined and some of its analytical properties investigated. It is shown that an easily implementable sub-optimum processor, based on two-dimensional spectral analysis, performs nearly as well as the optimum processor. The resolution and ambiguity properties of this sub-optimum processor are studied and a detailed numerical investigation of system performance is presented, including a study of how performance varies with basic system parameters such as the number of antennas.
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Summary

This note presents a detailed mathematical analysis of a multiple-antenna AMTI radar system capable of detecting moving targets over a significantly wider velocity range than is achievable with a single-antenna system. The general system configuration and signaling strategy is defined, and relationships among system and signaling parameters are investigated. A...

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Predictive coding in a homomorphic vocoder

Published in:
IEEE Trans. Audio Electroacoust., Vol. AU-19, No. 3 September 1971, pp. 243-248.

Summary

Application of a type of predictive coding to the channel signals of a homomorphic vocoder has produced sizable bit rate reductions. With only slight degradation in speech quality, reduction (for the spectral envelope information) from 7800 to 4000 bits/s was achieved. A technique for obtaining the formant frequencies from the predictive coding parameters is described; this approach promises further bit rate reductions. As a byproduct of this study of predictive coding, direct and cascade form speech synthesizers are compared on the basis of differing quantization effects.
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Summary

Application of a type of predictive coding to the channel signals of a homomorphic vocoder has produced sizable bit rate reductions. With only slight degradation in speech quality, reduction (for the spectral envelope information) from 7800 to 4000 bits/s was achieved. A technique for obtaining the formant frequencies from the...

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Quantization effects in digital filters

Published in:
MIT Lincoln Laboratory Report TR-468

Summary

When a digital filter is implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. These quantization effects must be considered, both in deciding what register length is needed for a given filter implementation and in choosing between several possible implementations of the same filter design, which will be affected differently by quantization. Quantization effects in digital filters can be divided into four main categories: quantization of system coefficients, errors due to analog-digital (A-D) conversion, errors due to roundoffs in the arithmetic, and a constraint on signal level due to the requirement that overflow be prevented in the computation. The effects of these errors and constraints will vary, depending on the type of arithmetic used. Fixed point, floating point, and block floating point are three alternate types of arithmetic often employed in digital filtering. A very large portion of the computation performed in digital filtering is composed of two basic algorithms the first- or second-order, linear, constant coefficient, recursive difference equation; and computation of the discrete Fourier transform (DFT) by means of the fast Fourier transform (FFT). These algorithms serve as building blocks from which the most complicated digital filtering systems can be constructed. The effects of quantization on implementations of these basic algorithms are studied in some detail. Sensitivity formulas are presented for the effects of coefficient quantization on the poles of simple recursive filters. The mean-squared error in a computed DFT, due to coefficient quantization in the FFT, is estimated. For both recursions and the FFT, the differing effects of fixed and floating point coefficients are investigated. Statistical models for roundoff errors and A-D conversion errors, and linear system noise theory, are employed to estimate output noise variance in simple recursive filters and in the FFT. By considering the overflow constraint in conjunction with these noise analyses, output noise-to-signal ratios are derived. Noise-to-signal ratio analyses are carried out for fixed, floating, and block floating point arithmetic, and the results are compared. All the noise analyses are based on simple statistical models for roundoff errors (and A-D conversion errors). Of course, somewhat different models are applied for the different types of arithmetic. These models cannot in general be verified theoretically, and thus one must resort to experimental noise measurements to support the predictions obtained via the models. A good deal of experimental data on noise measurements is presented here, and the empirical results are generally in good agreement with the predictions based on the statistical models. The ideas developed in the study of simple recursive filters and the FFTare applied to analyze quantization effects in two more complicated types of digital filters frequency sampling and FFT filters. The frequency sampling filter is realized by means of a comb filter and a bank of second-order recursive filters; while an FFT filter implements a convolution via an FFT, a multiplication in the frequency domain, and an inverse FFT. Any finite duration impulse response filter can be realized by either of these methods. The effects of coefficient quantization, roundoff noise, and the overflow constraint are investigated for these two filter types. Through use of a specific example, realizations of the same filter design, by means of the frequency sampling and FFT methods, are compared on the basis of differing quantization effects.
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Summary

When a digital filter is implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. These quantization effects must be considered, both in deciding what register length is needed for a given filter implementation and in choosing between several possible implementations of...

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Roundoff noise in floating point fast Fourier transform computation

Published in:
IEEE Trans. Audio Electroacoust., Vol. AU-17, No. 3, September 1969, pp. 209-215.

Summary

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 essentially as v = log2 N 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 v2. 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 v). Also, for truncation, a greater than linear increase with v of the output noise-to-signal ratio is observed; the empirical results seem to be proportional to v2 rather than to v.
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Summary

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 essentially as v...

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A comparison of roundoff noise in floating point and fixed point digital filter realizations

Published in:
Proc. IEEE, Vol. 57, No. 6, June 1969, pp. 1181-1183.

Summary

A statistical model for roundoff noise in floating point digital filters, proposed by Kanoko 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 point curves will cross, for equal total register lengths.
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Summary

A statistical model for roundoff noise in floating point digital filters, proposed by Kanoko 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...

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