Publications

Simultaneous measurements were made of the backscatter cross section and the bistatic scattering cross section of rain and thin turbulent layers. The radar measurements were made at a frequency of 1.3 GHz using the Millstone Hill Radar. The bistatic scattering measurements were made using CW transmission at 7.7 GHz with a 145-km separation between transmitter and receiver. The receive station was the Westford Communication Terminal with a 60-foot antenna. The transmitter was van-mounted and used either a 6-foot antenna or a standard gain horn. Stable frequency sources were used to allow Doppler shift measurements on the bistatic scattering link. The measurements were made by fixing the pointing angles of the transmit antenna and scanning both the receive antenna and the radar to investigate the dependence of the scattered signals both on scattering angle and on the location of the scatterers. The measurements of the scattering cross section of the thin turbulent layers were made in the near forward direction, the measurements of rain at a large number of scattering angles. System sensitivities allowed the measurement of scattering from turbulent layers at a 10-km height with a thickness, Cn^2 product of 10^-13 N^2 m^1/3 and from rain with a 0.1 mm/hr. rate. Comparisons between the radar and bistatic measurements were in good agreement with the appropriate scattering theories.

This is the first report in the Quarterly Technical Summary series covering the Air Traffic Control activities at Lincoln Laboratory. The previous work on ATC was included in the General Research Quarterly Technical Summary. Because the allowable effort on ATC is comparatively small, it has been focused on only one facet of the problem; namely, on the data acquisition and communications task. The new group has started to make significant progress in several study aspects of the problem and has also obtained experimental L-band multipath data from an experimental airground test system. When additional support is received, the program will be expanded to include over-all system design studies and the investigation of radar improvements and multilateration systems, both ground- and satellite-based.

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.

Measurements were made of the scattering properties of thin turbulent layers at and above the tropopause. The Millstone Hill L-band radar was used to measure the backscatter cross section per unit volume of these layers as a function of time and space. An X-band forward scatter link was set up between Wallops Island, Virginia and Westford, Massachusetts to observe scattering from these layers. Although the radar could not provide observations of the common volume of the forward scatter link, for days where no clouds were observed in the vicinity of the tropopause, the radar observations of layers near the tropopause showed horizontal uniformity of height and backscatter cross section, and the radiosonde data taken near the radar and near the common volume showed similar wind and temperature structure near the tropopause, the signal strength on the forward scatter link and its dependence on scattering angle behaved in accordance with the prediction of turbulent scattering theory using the radar data as an input. The radar observations have shown that on each day measurements were made, layers were detected near and above the tropopause. Turbulent layers in the stratosphere have been detected at heights up to 22 km. These layers provide one of the mechanisms for weak, long-distance troposcatter propagation.

Simultaneous measurements of the backscatter cross section per unit volume and the sky temperature were made for limited volumes of rain showers using an L-band radar and an X-band radiometer. The object of the measurements was to provide data to validate the method used to compute attenuation and sky temperature given weather radar data as an input and to investigate the spatial changes in rainfall intensity and in the attenuation cross section per unit volume. The sky temperature was calculated using the radiative transfer equation and the distribution of attenuation cross section per unit volume estimated from the weather radar data. An empirical relationship between attenuation and backscatter cross sections was used based upon the results of a large number of Mie theory computations using measured raindrop size distributions. The results of the comparisons between calculated and measured sky temperature show good agreement. The discrepancies between the measured and calculated values are due to the difference in the antenna beamwidths for the two systems (0.6° for the L-band radar, 0.07° for the X-band radiometer). From these discrepancies the spatial distances over which the attenuation cross section can change significantly can be estimated. The results show that for the rain showers investigated, the attenuation cross section per unit volume can change an order of magnitude in 400 meters and the integrated attenuation along a horizontal line-of-sight can change an order of magnitude for a 1. 5km horizontal translation of the path.

The major propagation problem confronting the use of millimeter waves for line-of-sight communication links operating through the atmosphere is hydrometeor scattering. Rain, hail, sleet, snow, and fog all can cause severe attenuation at millimeter wave frequencies. The severest problem is that of attenuation by rain. Attenuations in excess of 1 db/km are computed for frequencies in excess of 45 GHz (6.7 mm) and rain rates in excess of 0.1"/hr (2.5 mm/hr). Light rain of this intensity occurs on the average of 80 hrs/year in the New York area and is generally wide spread, completely covering typical line-of-sight ground link distances. This means that for a ground link of 50 km extent the attenuation would exceed 50 db 0.9 percent of the time for frequencies above 45 GHz.

The problem of scattering of electromagnetic waves by a small number of closely spaced dielectric spheres is considered as a boundary value problem. The solution to this problem is obtained in a series form using partial spherical vector waves. An approximate solution is also obtained for spheres separated sufficiently far for waves scattered by one sphere and incident on another to be considered plane waves with an amplitude given by the solution to the single scattering problem. The use of both solutions is discussed.

The problem of received signal degradation for coherent pulse transmission through a rain scattering volume was investigated for large bandwidth transmission at 4.0, 8.0, 15.5 and 34. 86GHz. Calculations of pulse length and total pulse energy were made for different path lengths through the rain volume. The calculations were made for models of heavy and extreme rainfall, using rain rates 49.0 and 196.3 mm/hr. The results of the computations show that for the rain rates considered, the dominant cause of signal degradation is attenuation. Negligible pulse lengthening was noted at 34.86 GHz. For rain rates above 196 mm/hr. and frequencies of 15.5 and 8.0 GHz, measurable values of pulse lengthening were calculated for bandwidths above 2.0 GHz. At 4.0 GHz, measurable values of pulse lengthening were obtained for both rain rates considered.