Time-Domain Reconstruction of Wide-Bandwidth Chirp

Richard T. Lord and Michael R. Inggs
Radar Remote Sensing Group, Dept of Electrical Engineering, University of Cape Town
Private Bag, Rondebosch 7701, South Africa
Tel: +27 21 650 2799   Fax: +27 21 650 3465
Email:  rlord-avoidspam@ebe.uct.ac.za
 
 
 
Note --- This document is a short summary of work that has already been published [1, 2].  Some of the mathematical notation used in the published papers has changed in this document, in order to clarify the analogy between this time-domain method and the frequency-domain method which reconstructs the target reflectivity function.
 

I. Introduction

This document describes a time-domain method to combine a group of n narrow-bandwidth linear FM chirp pulses stepped in frequency in order to construct a wide-bandwidth signal to achieve higher range resolution.  An advantage of this method is that it does not introduce any "ghost images'' in the high-resolution range profile.
 

II. Waveform Modelling

A linear FM chirp waveform at baseband can be described by

equation6

where A is the amplitude, tex2html_wrap_inline196 the pulse length and tex2html_wrap_inline198 the chirp rate.  The spectrum is approximately rectangular with bandwidth tex2html_wrap_inline200 .

The transmitted RF signal is

equation15

where tex2html_wrap_inline202 is the centre frequency.  For a stepped-frequency system the carrier frequency of pulse i, where i = 0, ..., (n - 1), is given by

equation20

where tex2html_wrap_inline208 is the centre frequency of the reconstructed wide-bandwidth chirp pulse, and tex2html_wrap_inline210 is the frequency shift associated with pulse i, given by

equation29

The transmitted pulses belonging to one burst can thus be described by

equation34

The received signal from a single scatterer at a distance tex2html_wrap_inline214 is given by

displaymath192

After coherent demodulation, the signal at baseband is given by

equation43

This signal is then sampled at a complex sampling rate of tex2html_wrap_inline216 .
 

III. Reconstruction of Wide-Bandwidth Chirp

It is now proposed to reconstruct a wide-bandwidth chirp pulse in the time-domain by coherently adding together several narrow-bandwith chirp pulses, each obtained by separate transmission and reception of pulses of bandwidth tex2html_wrap_inline218 , but stepped appropriately in frequency by appropriate choice of the carrier frequency.  If the frequency step tex2html_wrap_inline220 , the desired reconstruction is possible, otherwise there will be gaps or overlaps in the corresponding wide-bandwidth spectrum.  Figure 1 demonstrates the reconstruction process, showing two narrow-bandwidth chirp pulses which are combined to form one wide-bandwidth chirp pulse.  The required signal processing steps are described below.
 
 
 
figure54
 
 
Figure 1: Reconstruction of wide-bandwidth chirp for n = 2 transmitted pulses.
 
 

Upsampling

Unless the narrow-bandwidth pulses are already oversampled at a rate tex2html_wrap_inline226 , they have to be upsampled before applying the frequency-shift described below.  Usually the time-domain signals have to be upsampled by a factor of n, where n is the number of pulses used to synthesise the wide bandwidth.  The upsampling operation can be achieved by applying an FFT, zero-padding the spectrum and then applying an IFFT.
 

Frequency shift

The amount of frequency-shift associated with pulse i is given by tex2html_wrap_inline210 , which can be achieved in the time-domain by muliplication with the factor tex2html_wrap_inline236 .  The frequency-shifted pulses are thus described by

equation66
 

Phase correction

The need to add a phase-correcting term to each pulse can be explained with the aid of Figure 2. In the figure, the phase versus time plot of the wide-bandwidth signal and of the second narrow-bandwidth signal (from a burst of 4) is shown.  The narrow-bandwidth signal has already been shifted in frequency and time, but a phase correcting term given by

equation74

has to be added in order to avoid phase discontinuities in the wide-bandwith signal.
 
 
 

 figure82
 
Figure 2: Addition of phase-correcting term to narrow-bandwidth signals.
 
 
This phase term may be added to the narrow-bandwidth signals before they are upsampled, thus requiring fewer multiplications and leading to a faster implementation of the procedure.
 

Time shift

Before adding the narrow-bandwidth pulses together, they have to be shifted in the time domain.  The necessary time-shift is given by

equation89

Since the signals are sampled at the A/D rate tex2html_wrap_inline242 (after they have been upsampled), the time-shift given in terms of number of samples is

equation97

It is important that tex2html_wrap_inline244 is an integer, because fractional values would have to be rounded, leading to errors in the reconstructed wide-bandwidth signal.  This requirement can be achieved by adjusting either the pulse length tex2html_wrap_inline196 or the A/D sampling rate tex2html_wrap_inline242 .
 

Combining narrow-bandwidth chirps

The reconstructed wide-bandwidth signal v'(t) is obtained by coherently adding the processed narrow-bandwidth signals.  Observing that tex2html_wrap_inline252 one obtains the following expression for v'(t) after simplification:

eqnarray112

This is simply the expression of a wide-bandwidth chirp pulse with a bandwidth of tex2html_wrap_inline256 , with tex2html_wrap_inline258 .  Range compression may be achieved by convolving this signal with a matched filter, which is given by the conjugate of the wide-bandwidth pulse.
 

References

  1. R.T. Lord and M.R. Inggs, "High Resolution SAR Processing Using Stepped-Frequencies,'' Proc. IEEE Geoscience Remote Sensing Symp., IGARSS'97, Singapore, vol. 1, pp. 490-492, August 1997. igarss97.pdf
  2. R.T. Lord and M.R. Inggs, "High Range Resolution Radar using Narrowband Linear Chirps offset in Frequency,'' Proc. IEEE South African Symp. on Communications and Signal Processing, COMSIG'97, Grahamstown, South Africa, pp. 9-12, September 1997. comsig97.pdf
  3. J.A. Scheer and J.L. Kurtz, Coherent Radar Performance Estimation, Norwood, MA 02062: Artech House, 1993.
  4. D.R. Wehner, High-Resolution Radar, Second Edition, Norwood, MA 02062: Artech House, 1995.
 
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