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Roundoff noise in floating point fast Fourier transform computation

March 1, 1969
  |
Journal Article
Author:
Clifford J. Weinstein
Published in:
IEEE Trans. Audio Electroacoust., Vol. AU-17, No. 3, September 1969, pp. 209-215.
Topic:
signal processing
R&D area:
R&D group:

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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Roundoff noise in floating point fast Fourier transform computation

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

January 1, 1969
  |
Journal Article
Author:
Clifford J. Weinstein
…
Published in:
Proc. IEEE, Vol. 57, No. 6, June 1969, pp. 1181-1183.
Topic:
signal processing
R&D area:
R&D group:

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

Monostatic and bistatic scattering from thin turbulent layers in the atmosphere

September 18, 1968
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Technical Report
Author:
Robert K. Crane
Published in:
MIT Lincoln Laboratory Report TN-1968-34
Topic:
radar
R&D area:
R&D group:

Summary

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.
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Summary

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...

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Monostatic and bistatic scattering from thin turbulent layers in the atmosphere

Simultaneous radar and radiometer measurements of rain shower structure

September 18, 1968
  |
Technical Report
Author:
Robert K. Crane
Published in:
MIT Lincoln Laboratory Report TN-1968-33
Topic:
radar
R&D area:
R&D group:

Summary

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.
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Summary

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...

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Simultaneous radar and radiometer measurements of rain shower structure

Rain attenuation at millimeter wavelengths

January 1, 1968
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Conference Paper
Author:
Robert K. Crane
Published in:
IEEE Int. Convention Digest, 1968, p. 65.
Topic:
communications
R&D area:
R&D group:

Summary

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.
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Summary

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...

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Rain attenuation at millimeter wavelengths

Cooperative scattering by dielectric spheres

July 6, 1967
  |
Technical Report
Author:
Robert K. Crane
Published in:
MIT Lincoln Laboratory Report TN-1967-31
Topic:
RF technology
R&D area:
R&D group:

Summary

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.
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Summary

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...

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Cooperative scattering by dielectric spheres

Coherent pulse transmission through rain

December 27, 1966
  |
Technical Report
Author:
Robert K. Crane
Published in:
MIT Lincoln Laboratory Report TN-1966-63
Topic:
weather
R&D area:
R&D group:

Summary

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.
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Summary

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...

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Coherent pulse transmission through rain

Microwave scattering parameters for New England rain

October 3, 1966
  |
Technical Report
Author:
Robert K. Crane
Published in:
MIT Lincoln Laboratory Report TR-426
Topic:
radar
R&D area:
R&D group:

Summary

Scattergrams of attenuation coefficient, effective reflectivity factor, single-scattering albedo, and radio refractivity vs liquid-water content, rain rate, and reflectivity factor are presented for a raindrop temperature of 0.0 degrees celsius and frequencies of 1.29, 2.80, 8.0, 9.35, 15.5, 35.0, 70.0, and 94.0 GHz. The scattergrams were computed using Mie theory to compute the scattering parameters for single raindrops, and single-scattering theory to compute the integrated scattering effects of an ensemble of raindrops. Measured drop-size distributions were used to generate the scattergrams.
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Summary

Scattergrams of attenuation coefficient, effective reflectivity factor, single-scattering albedo, and radio refractivity vs liquid-water content, rain rate, and reflectivity factor are presented for a raindrop temperature of 0.0 degrees celsius and frequencies of 1.29, 2.80, 8.0, 9.35, 15.5, 35.0, 70.0, and 94.0 GHz. The scattergrams were computed using Mie theory...

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Microwave scattering parameters for New England rain
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