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Title: Error analysis of the chirp-z transform when implemented using waveform synthesizers and FFTs

Abstract

This report analyzes the effects of finite-precision arithmetic on discrete Fourier transforms (DFTs) calculated using the chirp-z transform algorithm. An introduction to the chirp-z transform is given together with a description of how the chirp-z transform is implemented in hardware. Equations for the effects of chirp rate errors, starting frequency errors, and starting phase errors on the frequency spectrum of the chirp-z transform are derived. Finally, the maximum possible errors in the chirp rate, the starting frequencies, and starting phases are calculated and used to compute the worst case effects on the amplitude and phase spectrums of the chirp-z transform. 1 ref., 6 figs.

Authors:
Publication Date:
Research Org.:
Sandia National Labs., Albuquerque, NM (USA)
Sponsoring Org.:
DOE/AD
OSTI Identifier:
6297820
Report Number(s):
SAND-90-1965
ON: DE91004939
DOE Contract Number:
AC04-76DP00789
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; FOURIER TRANSFORMATION; ALGORITHMS; ERRORS; WAVE FORMS; INTEGRAL TRANSFORMATIONS; MATHEMATICAL LOGIC; TRANSFORMATIONS; 990200* - Mathematics & Computers

Citation Formats

Bielek, T.P.. Error analysis of the chirp-z transform when implemented using waveform synthesizers and FFTs. United States: N. p., 1990. Web. doi:10.2172/6297820.
Bielek, T.P.. Error analysis of the chirp-z transform when implemented using waveform synthesizers and FFTs. United States. doi:10.2172/6297820.
Bielek, T.P.. Thu . "Error analysis of the chirp-z transform when implemented using waveform synthesizers and FFTs". United States. doi:10.2172/6297820. https://www.osti.gov/servlets/purl/6297820.
@article{osti_6297820,
title = {Error analysis of the chirp-z transform when implemented using waveform synthesizers and FFTs},
author = {Bielek, T.P.},
abstractNote = {This report analyzes the effects of finite-precision arithmetic on discrete Fourier transforms (DFTs) calculated using the chirp-z transform algorithm. An introduction to the chirp-z transform is given together with a description of how the chirp-z transform is implemented in hardware. Equations for the effects of chirp rate errors, starting frequency errors, and starting phase errors on the frequency spectrum of the chirp-z transform are derived. Finally, the maximum possible errors in the chirp rate, the starting frequencies, and starting phases are calculated and used to compute the worst case effects on the amplitude and phase spectrums of the chirp-z transform. 1 ref., 6 figs.},
doi = {10.2172/6297820},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Thu Nov 01 00:00:00 EST 1990},
month = {Thu Nov 01 00:00:00 EST 1990}
}

Technical Report:

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  • On parallel computers, the way the data elements are mapped to the processors may have a large effect on the timing performance of a given algorithm. In our previous paper, we have examined a few mapping strategies for the ordered radix-2 DIF (decimation-in-frequency) Fast Fourier Transform. In particular, we have shown how reduction of communication can be achieved by combining the order and computational phases through the use of i-cycles. A parallel methods was also presented for computing the trigonometric factors which requires neither trigonometric function evaluation nor interprocessor communication. This paper first reviews some of the experimental results onmore » the Connection Machine to demonstrate the importance of reducing communication in a parallel algorithm. The emphasis of this paper, however, is on analyzing the numerical stability of the proposed method for generating the trigonometric factors and showing how the error can be improved. 16 refs., 12 tabs.« less
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  • There exists a need to develop a method to locate underground voids, or caches. In the past, ground penetrating radar (GPR) operating in the time domain mode has been used. In this paper, we turn our attention to stepped frequency radar, capable of making frequency domain reflection coefficient measurements. We then apply the inverse Chirp-Z transform (ICZT) to this data, generating a time domain response. The scenario under consideration is that of an airborne radar passing over the surface of the earth. The radar is directed toward the surface and is capable of measuring the reflection coefficient, seen looking towardmore » the earth, as a function of frequency. The frequency domain -data in this work is simulated and is generated from a transmission line model of the problem. Using the ICZT we convert this frequency domain data to the time domain. Once in the time domain, reflections due to discontinuities appear at times indicating their relative distance from the source. The discontinuities occurring beyond the surface of the earth could be indicative of underground structures. The ICZT allows a person to zoom in on the time span of interest by specifying the starting time, the time resolution, and the number of time steps.« less
  • Wideband radar systems, especially those that operate at lower frequencies such as VHF and UHF, are often restricted from transmitting within or across specific frequency bands in order to prevent interference to other spectrum users. Herein we describe techniques for notching the transmitted spectrum of a generated and transmitted radar waveform. The notches are fully programmable as to their location, and techniques are given that control the characteristics of the notches.
  • Spectral analysis over an instantaneous bandwidth of 2.4 GHz was demonstrated, utilizing superconductive dispersive delay lines in a chirp-transform configuration. Two-tone resolution of 43 MHz and + or - 1.2 dB amplitude uniformity was achieved.