Publications

A new type of radar digital signal processor for use with ASR radars is described. It features spectral processing accomplished by combining a 3-pulse canceller with an 8-point, weighted, discrete Fourier transform and adaptive thresholds. This combination of circuits provides a 20-dB increase in MTI improvement factor over present ASR's and is within 2 dB of optimum processing results. An auxiliary channel is provided to allow detection of any target traveling tangentially if its return exceeds the level of ground clutter return in the occupied range/azimuth cell. The spectral processing technique provides discrimination against weather clutter if the returns from weather and from the target fall into different Doppler frequency regions. The output from this equipment is digital hit reports for transmittal to the ARTS-III IOP computer.

This report describes the elements of a simulations program that was designed to perform a realistic evaluation of a variety of reply processing techniques, antenna design parameters and receiver characteristics for a DABS sensor. The reply processing techniques are limited to the generation of information bit and monopulse off-boresight azimuth estimates for DABS downlink messages. The report describes the detailed characteristics of two elements of the simulation program; the data generator and reply processor, and the high degree of versatility incorporated within these elements to allow for a great many performance tradeoff studies.

In the report, "A Theory for Optimal MTI Digital Signal Processing. Part I. Receiver Synthesis," (1), the problem of eliminating scanning ground clutter from an aircraft surveillance radar was examined from a statistical decision theoretical point of view. An optimum processor was derived which could be approximated by a clutter filter followed by a discrete Fourier transform (DFT). In this report, additional numerical work is documented that compares the performance of the pulse cancellers, pulse cancellers with feedback and the DFT with that of the optimum processor. The issue of coherent vs incoherent integration gain is considered by comparing the filters only on their ability to reject clutter. A clutter rejection improvement factor is defined and used to compare the various filters. It is shown that the pulse cancellers can be quite effective in rejecting clutter provided the input clutter power is not too large and that additional gains are possible using the DFT.

In Part I of this report the optimum MTI receiver was derived and analyzed for the case in which the radar pulses were emitted from the transmitter equally spaced in time. For typical long range ATC surveillance radars , aliasing of the target and clutter spectra results in detection blind speeds at multiples of approximately 70 knots. It is well known operationally that these blind speeds can be eliminated by staggering the transmitter PRF. Heretofore, there has been no thorough theoretical analysis of the effect of staggered PRF on the spectral distribution of the target and clutter signals. It is shown in Part II that the clutter spectral density continues to fold over at the PRF, but that the signal spectrum becomes dispersed in frequency, somewhat like an anti-jam signal. The effect that this phenomenon has on the performance of the optimum processor is evaluated in terms of the signal-to-interference ratio (SIR) criterion that was derived in Part I. It is further noted that even when the target Doppler shifts are more than one PRF apart, the spectra are distinguishable, suggesting that unambiguous Doppler estimation may be possible. This concept is explored in detail using the MTI ambiguity function. It is shown that good SIR performance can be obtained by choosing the stagger parameters to minimize the height of the subsidiary Doppler side-lobes. The resulting design problem is noted to be similar to that of obtaining good antenna patterns for arrays having non-uniformly spaced elements.

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.

A classical problem in radar theory is the detection of moving targets in a ground clutter plus receiver noise background. Improvements in clutter rejection have recently been made by replacing analog MTI processors by their digital equivalents as this eliminates many of the problems associated with the maintenance of the analog hardware. In an attempt to determine the ultimate improvements possible using this new technology, the MTI problem was formulated as a classical detection problem and solved using the generalized likelihood ratio test. By manipulating the likelihood ratio, the receiver could be interpreted as a clutter filter in cascade with a Doppler filter bank. The performance of the optimum receiver was evaluated in terms of the output signal-to-interference ratio and compared with well-known MTI processors. It was shown that near-optimum performance can be obtained using a sliding weighted Discrete Fourier Transform (DFT). All of the results in Part I assume uniformly spaced transmitted pulses, which, for high velocity aircraft, leads to aliasing of the target and clutter spectra and detection blind speeds. In Part II the maximum likelihood method is applied using a more general model for the non-uniformly sampled target returns. This leads to an optimum receiver that is a slightly more complicated version of the sliding weighted DFT. In addition to removing the detection blind speeds, it is found that unambiguous Doppler measurements

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.

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.

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.

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.