Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mso.anu.edu.au/~fbriggs/bbk.pdf Äàòà èçìåíåíèÿ: Thu Jan 8 09:21:41 2004 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:49:25 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï

THE ASTRONOMICAL JOURNAL, 120 : 3351 õ 3361, 2000 December
( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

REMOVING RADIO INTERFERENCE FROM CONTAMINATED ASTRONOMICAL SPECTRA USING AN INDEPENDENT REFERENCE SIGNAL AND CLOSURE RELATIONS F. H. BRIGGS
Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV Groningen, Netherlands ; fbriggs=astro.rug.nl

AND J. F. BELL AND M. J. KESTEVEN
CSIRO Australia Telescope National Facility, P.O. Box 76, Epping, NSW 1710, Australia ; jbell=atnf.csiro.au, mkesteve=atnf.csiro.au Received 2000 June 16 ; accepted 2000 August 24

ABSTRACT The growing level of radio frequency interference (RFI) is a recognized problem for research in radio astronomy. This paper describes an intuitive but powerful RFI cancellation technique that is suitable for radio spectroscopy where time-averages are recorded. An RFI "" reference signal,îî is constructed from the cross power spectrum of the signals from the two polarizations of a reference horn pointed at the source of the RFI signal. The RFI signal paths obey simple phase and amplitude closure relations, which allows computation of the RFI contamination in the astronomical data and the corrections to be applied to the astronomical spectra. Since the method is immune to the eects of multipath scattering in both the astronomy and reference signal channels, "" clean copies îî of the RFI signal are not required. The method could be generalized (1) to interferometer arrays, (2) to correct for scattered solar radiation that causes spectral "" standing waves îî in single-dish spectroscopy, and (3) to pulsar survey and timing applications where a digital correlator plays an important role in broadband pulse dedispersion. Future large radio telescopes, such as the proposed LOFAR and SKA arrays, will require a high degree of RFI suppression and could implement the technique proposed here with the beneït of faster electronics, greater digital precision and higher data rates. Key words : instrumentation : detectors õ methods : analytical
1

. INTRODUCTION

The growing level of radio frequency interference (RFI) is a recognized problem for research in radio astronomy. Fortunately, the technological advances that are giving rise to the increasing background of radiationõthrough increased telecommunications and wide-spread use of high speed electronicsõare also providing some of the tools necessary for separating astronomical signals from undesirable RFI contamination. New radio telescopes will necessarily have RFI suppression, excision, and cancellation algorithms intrinsic to their designs. No one technical solution will make radio observations immune to interference ; successful mitigation is most likely to be a hierarchical or progressive approach throughout the telescope, combining new instrumentation and algorithms for signal conditioning and processing Ekers & Bell 2000.1 Techniques from the communication industry that are ïnding application in radio astronomy experiments include (1) adaptive beam forming with array telescopes that steer nulls of the instrument reception pattern in the directions of sources of RFI (Ellingson & Hampson 2000 ; Leshem, van der Veen, & Boonstra 2000 ; Smolders & Hampson 2000 ; Kewley et al. 2000),2 (2) parametric signal-modeling techniques, where the RFI signal is received and decoded to obtain a high signal-to-noise ratio (SNR) reference signal for subtraction from the astronomical data (Ellingson,
õõõõõõõõõõõõõõõ 1 See also the presentation by S. Ellingson on Interference mitigation techniques, available at http ://www.atnf.csiro.au/SKA/intmit/atnf/conf/. 2 See also G. Hampson et al., The Adaptive Antenna Demonstrator, at http ://www.nfra.nl/skai/archive/technical/index.shtml.

Bunton, & Bell 2000 ; Leshem & van der Veen 1999a, 1999b), and (3) adaptive ïltering using a reference horn to obtain a high SNR copy of the RFI for real-time cancellation from the signal path ahead of the standard radio astronomical backend processors (Barnbaum & Bradley 1998). This paper describes an intuitive but powerful RFI cancellation technique that is suitable for radio spectroscopy where time-averages are recorded. The method requires computation of cross power spectra between the RFI contaminated astronomical signals and high signal-to-noise ratio RFI "" reference signals îî obtained from a receiving system that senses the RFI but not the astronomical signal. The correction term that removes the unwanted RFI is computed from closure relations obeyed by the RFI signal. The test applications reported here derived the reference signal either from a separate horn antenna aimed at the RFI source or from a second feed horn at the focus of the Parkes telescope, as illustrated in Figure 1. For these experiments, we recorded digitally sampled base band signals from two polarizations for both the reference and astronomy feeds, and then we performed the cross-correlations in software o-line. However, the method could use correlation spectrometers of the sort already in use at radio observatories. With minor design enhancements, future generation correlators could incorporate this technique with the additional beneït of the faster electronics, greater digital precision, and higher data rates that technological advance promises. There are a number of advantages to performing the RFI in a "" post-correlation îî stage. Foremost is that the RFI subtraction remains an option in the data reduction path, 3351


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This paper provides a description of the post-correlation RFI cancellation technique and illustrates its success with data from the Parkes telescope. A mathematical overview shows (1) why unknown multipaths do not cause the algorithm to break down, (2) how to simply construct a suitable RFI reference spectrum, and (3) how to build an inverse ïlter to obtain immunity to low signal levels at frequencies that suer destructive interference by multipathing in the reference horn signal path.
2.

MATHEMATICAL DESCRIPTION OF THE METHOD

FIG. 1.õThree conïgurations applicable to the analysis in this paper. Top : The four base bands are recorded from the two polarizations of the Parkes telescope feed and two polarizations of a reference horn directed at the RFI source. Center : The four base bands correspond to two polarizations from each of two Parkes feeds. Bottom : A proposed system for optimal application of the RFI subtraction technique described here.

rather than a commitment made on-line and permanently. Furthermore, the correlation method is eectively a coherent subtraction, since the correlation functions retain the information describing relative phase between the RFI entering in the astronomy data stream and the RFI entering the reference antenna. As we show in this paper, this means that the RFI noise power is largely subtracted, leaving only the usual components of system noise.

In our mathematical model, we make the assumption that the RFI source emits a single signal i(t). (At the RFI source, the signal from a single power ampliïer feeds an antenna of unknown, but irrelevant, polarization.) The RFI that appears dierently in the recorded data channels at Parkes has experienced scatterings with dierent path lengths and amplitudes, so that the received signals are linear sums of time-delayed versions of the original broadcast i(t). In fact, the model is applicable to multiple interferers within the spectrometer band, provided they do not overlap in frequency. Consider for the moment a single interferer, which propagates through the four signal paths s , s , s , and s that will 123 4 be processed : there are two astronomical channels, s (t) and 1 s (t), which convey the voltages a (t) and a (t) from the 2 A B celestial sources for the two independent polarizations from the Parkes Telescope receiver along with contamination from the RFI signal i(t). Radiation from astronomical sources may be partially polarized, causing a (t) and a (t) to A B be correlated to some degree. The two reference channels carry s (t) and s (t), containing the representations of i(t) but 3 4 negligible signal from the celestial sources. For example, the measured signal in channel 1 comprises channel noise n (t), plus the convolutions of the impulse responses for 1the multiple scattering paths for the interference signal and the true astronomy signal : s (t) \ H (t) W a (t) ] H (t) W i(t) 1 1 impulseA responses for the ] n (t). Here H (t) and H (t) are A 1 A path and interference, respectively, andW 1 astronomy signal is the convolution operator. For many purposes, an intuitive picture of the multipathing results from considering the scattering sites to be achromatic mirror-like scatterers, each with relative eective areas G and G , and attaching the path delay to A each separate, jversion 1, k the astronomy and RFI signals, of s (t) \ ¸ G a (t [ q ) ] ¸ G i(t [ q ) ] n (t). The 1 j q are 1, j k 1, k 1, k 1 time delays A, j A determined by the dierent path lengths L to give q \ L /c, where c is the speed of light. 1, j the signal paths are vulnerable to corruption by sto1, j 1, j All chastic noise. The noise terms n (t), n (t), n (t), and n (t) 1 3 4 should ideally be uncorrelated among2 the dierent data paths. Unfortunately, in real astronomical systems, there is likely to be low-level coupling between the two orthogonal polarizations of a feed horn or common stray radiation pickup from spill over that will make a weakly correlated noise ÿoor in some of the cross power spectra. This will form a systematic limitation to the accuracy of the subtraction. In this experiment, the goal is to explore the usefulness of cross-correlation spectra to correct for the eects of RFI in time-averaged spectra. These spectra are products of scaled sums of the of the Fourier transforms of the astronomy signals, the RFI, and noise. In the tests with real data in ° 4,


No. 6, 2000 we compute estimates of the complex spectra

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S ( f )\g A ]g I]N , 1 AA 1 1 S ( f )\g A ]g I]N , 2 BB 2 2 S ( f )\g I]N , 3 3 3 S ( f )\g I]N , (1) 4 4 4 from Fourier transforms of ïnite length time series of discrete samples of the four signals. The transforms contain contributions from the celestial sources A ( f ) and A ( f ), A B the RFI I( f ), and the noise in each channel N ( f ), modui lated by the associated complex voltage gains, which are the Fourier transforms of the impulse responses H(t). The gains for each channel separate into dependencies on (1) the path delay L /c, which appears in a frequency dependent phase term, according to the shift theorem of Fourier transforms, and (2) a possible additional frequency dependence g( f ) of each delay path : (2) g ( f ) \ ; g ( f )ei2nfL1, k@c . 1 1, k These complex gain and delay factors are sufficiently general to include complicated scatterers and propagation through dispersive and lossy media. The real power spectra for the four data channels have the following form, once terms that average toward zero are omitted and the complex gains are assumed to be constant over the time span for which the spectra are computed : P ( f ) \ SS S*T 1 11 \ o g o2S o A o2T A A ]o g o2S o I o2T ] S o N o2T , 1 1 P ( f ) \ S S S *T 2 22 \ o g o2S o A o2T B B ]o g o2S o I o2T ] S o N o2T , 2 2 P ( f ) \ SS S*T 3 33 \ o g o2S o I o2T ] S o N o2T , 3 3 P ( f ) \ S S S *T 4 44 \ o g o2S o I o2T ] S o N o2T . (3) 4 4 We use the superscript asterisk (*) to represent complex conjugation and the S...T notation to signify averages over an integration time t ; in the tests we describe in ° 4, we int ïnd the method is eective for t as long as D1 s. We adopt a normalization where the int power levels S o A o2TB A S o I o2TBS o N o2T so that, for example, in data channel 1 1 ratio SNR Do g o2, and the interference the signal to noise A to noise ratio INR Do g o2. 1 1 cancellation will be to form estimates 1 The goal of the RFI of the o g o2S o I o2T and o g o2S o I o2T terms and then sub1 2 tract them from P ( f ) and P ( f ), while leaving the astrono1 2 mical signal (and noise) behind. This discussion has assumed that the complex gain terms are constant over the integration time t , allowing us to int separate the interference from the gain in expressions such as Sg Ig* I*T \ o g o2S o I o2T . (4) 11 1 In anticipation of the discussion of ° 4, we note that this assumption will fail for extended integration times since the scattering paths that lead the RFI signal to the telescope feed will change, resulting in loss of precision in the cancel-

lation scheme and leading to substantial residuals in the corrected spectra. The complex cross power spectra for all combinations of the four data channels have the following form when the leading contributions are retained : C ( f ) \ SS S*T 12 12 \ g g*SA A*T AB A B ]g g*S o I o2T 12 ]g SIN*T ] g*SN I*T 1 2 21 ]SN N*T , 12 C ( f ) \ SS S*T ij ij \ g g*S o I o2T ij ]g SIN*T ] g*SN I*T i j j i ]SN N*T , ij for i D j, j [ 2

(5)

In order to cancel the dominant RFI terms in the power spectra equation (3), we need to compute quantities of the form o g o2S o I o2T, which can be subtracted from the measured S o1S o2T. One possibility would be a combination of 1 auto and cross power spectra of the form g g* g* g o g o2S o I o2T \ 1 3 1 3 S o I o2T 1 g* g 33 o C o2 o C o2 13 \ 13 B . (6) S o S o2T o g o2S o I o2T ] S o N o2T 3 3 3 However, the problem encountered with this approach is that S o S o2T is biased by the term S o N o2T, which averages 3 over time to the total power spectrum3of the noise in data channel 3. This might be compensated by calibrating the noise spectrum, in order to improve the estimate of o g o2S o I o2T \ S o S o2T [ S o N o2T to use in the denomina3 3 tor of equation (6). 3 An alternate combination that avoids this bias forms the estimate of o g o2S o I o2T from three cross power spectra, in 1 which the noise terms have the form SN N*T and SIN*T, j j which average toward zero as the i integration time increases : g g* g* g o g o2S o I o2T \ 1 3 1 4 S o I o2T 1 g* g 34 C C* (7) B 13 14 , C* 34 g g* g* g o g o2S o I o2T \ 2 3 2 4 S o I o2T 2 g* g 34 C C* B 23 24 , (8) C* 34 g g* g* g g g*S o I o2T \ 1 4 2 3 S o I o2T 12 g g* 34 C C* B 14 23 . (9) C 34 The expressions involving the measured cross power spectra are "" approximate,îî since the cross power spectra result from ïnite integrations, and the noise terms will limit the precision of the cancellation.


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BRIGGS, BELL, & KESTEVEN and the interference power to noise power ratios

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The occurrence of the C term in the denominator for 34 equations (7)õ(9) indicates there will be a problem in implementing a correction scheme in frequency ranges where C 34 becomes small or zero. In many situations when the signal to noise ratio for the spectra in the numerator is very high, the cross power spectra in the numerator will also be small or zero whenever C is small, so that divergence will be 34 canceled. A simple means to avoid division by small numbers in the presence of noise in this kind of situation is to create a ïlter of the following form : C C* C 13 14 34 , o g o2S o I o2TB (10) 1 t( f ) ] C C* 34 34 C C* C 23 24 34 , o g o2S o I o2TB (11) 2 t( f ) ] C C* 34 34 C C* C* 14 23 34 , g g*S o I o2TB (12) 12 t( f ) ] C C* 34 34 where t( f ) is the square of the power spectrum of the noise present in C ( f ). Whenever t( f ) becomes small compared 34 with C ( f ), the expressions equations (10)õ(12) revert to 34 (7)õ(9). When the noise exceeds the signal power equations in C , the computed correction tends to zero. In practice, 34 during the test described here, a constant t was used in 0 place of t( f ). Alternatively, division by zero can be avoided by testing the amplitude of C ( f ) for signiïcance above a 34 noise threshold and setting the correction to zero when the signiïcance criterion is not met. These corrections for the autocorrelation spectra (eqs. [7] and [8]) are expected to be real valued. Therefore a logical test for the accuracy of the correction is that phases computed for the frequencies of strong RFI signal should be close to zero. In fact, this is a statement of phase closure. Note that the denominators of equations (10)õ(12) are purely real, and the numerators, such as C C* C , form 13 logical triangles for computing closure phases. 14 34 An amplitude closure relation CC g g* g g* 13 24 \ 1 3 2 4 \ 1 (13) g g* g g* CC 23 14 2314 can also be constructed and tested. It too will suer from divergence of the quotient in frequency ranges where the cross power spectra are noisy and C and C have small 23 14 amplitude. Here we avoid including C , which typically 12 addition to the has signiïcant cross-correlated power in RFI signal, due either to polarized celestial ÿux or cross talk between the channels.
3

o g o2S o I o2T B o g o2 ? 1 , INR \ 3 3 3 S o N o2T 3 o g o2S o I o2T (16) INR \ 4 B o g o2 ? 1 . 4 4 S o N o2T 4 then the correction CX for P ( f ) becomes 1 1 C C* CX B 13 14 , 1 C* 34 B o g o2S o I o2T ] 2Re[g g*SIA*T ] g SIN*T] 1 1A 1 1 g* g g g* ] A 1 SAN*T ] A 1 SA*N T 3 4 g* g* 4 3 g o g o2 g* ] 1 SN N*T ] 1 SN* N T [ 1 SN* N T . 13 14 34 g* g g* g 4 34 3 (17) The terms in equation (17) involving I all appear in the power spectrum of equation (14), so that application of this correction CX to P ( f ) leads to a result with no residual contamination1 the1RFI : by P ( f ) [ CX \ o g o2S o A o2T ] S o N o2T 1 1 A A 1 g g* ] 2Re[g SAN*T] [ A 1 SAN*T A 1 3 g* 3 g* g g* [ A 1 SA*N T [ 1 SN N*T 4 13 g* g* 4 3 g o g o2 [ 1 SN* N T ] 1 SN* N T . (18) 14 34 g g* g 4 34 The complete cancellation of the RFI terms is consistent with the concept that postcorrelation subtraction is equivalent to coherent subtraction of the RFI electric ïeld i(t) in the time domain, which should leave no trace of the RFI signal, nor an increase in noise power. Provided the INR for the reference horn channels is substantially greater than the INR for the Parkes feed channels, the noise terms due to N 3 and N will be smaller by a factor of order b D 4 (INR /INR )1@2 B o g o / o g o than the normal noise contri3 1 3 1 butions arising from N plus the astronomical signal power. 1 (18) other than o g o2S o A o2T and The terms in equation A S o N o2T, such as (g* g /g*)SA*N T P o gA o b~1t~1@2 and 1 SN N*T P b~1t~1@24 average toward zero (in A1 , 4 A (g*/g*) 13 13 inverse proportion to the square root of the integration time), provided the noise in the signal channels is independent. The next higher order terms, which are not included in equation (18), have dependencies such as SIN*TSN* N T/g S o I o2T P t~1 and g*SIN*TSI\N T/ 1 3 1 g*S o I o21 P b1 t4 , 4which converge toward zero faster T ~1 ~1 3 the dominant noise terms. than
4

. NOISE AND THE ACCURACY OF THE CANCELLATION

In this section we assess the importance of the interference-to-noise ratio. First we expand the autocorrelation spectra, keeping all cross terms, including those that average toward zero. Then P ( f ) becomes 1 P ( f ) \ o g o2S o A o2T ] o g o2S o I o2T 1 A A 1 ] 2Re[g g*SIA*T ] g SIN*T] 1A 1 1 ] 2Re[g SAN*T] ] S o N o2T . (14) A 1 1 The complex correction spectra were described by equations (7), (8), and (9). Including the cross terms and noting that when g and g ? g and g 3 4 1 2 (15)

. THE TEST DATA

The astronomical data set used in testing these algorithms is a dual linear polarization data stream from the CSIRO ATNF 64 m telescope at Parkes in Australia. One conïguration has two polarizations from the central beam of the Parkes multibeam receiver Staveley-Smith et al. 1996 and two polarizations from a reference horn aimed at an interfering source Bell et al. 2000. A second conïguration uses both polarizations from two beams of the Parkes


No. 6, 2000

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Multibeam system, which are directed at slightly dierent areas of the sky. The center frequency of the data sets was 1499 MHz in each case. The data sets we used are labeled SRT00501, SRT00502, and SRT00601 (Bell et al. 2000). The main interfering source is a NSW government digital point-to-point microwave link. Examples of the time-averaged spectrum for the RFI signal are shown in Figure 2. Further details are available in the Australian Communications Authority databases.3 The four signals were down-converted to base band and passed through 5 MHz low-pass ïlters. Each signal was then digitized with 2-bit precision at a 20 MHz sampling rate to achieve a factor 2 oversampling and recorded using the CPSR recorder (van Straten et al. 2000). The data processing for these tests simulates a radio astronomy backend by computing power spectra and cross power spectra in software. The sampled voltages are treated in 8192 sample blocks (410 ks durations). Fourier transformation of each block yields a spectrum of 4096 independent complex coefficients. Examples of 25 s averages of the power spectra are shown in Figure 3. The RFI spectrum is signiïcantly dierent in the spectra for all four data channels. Since the data are 2 times oversampled, we kept the total power and cross power spectra of the lower 2048 complex Fourier coefficients. The RFI subtraction steps were performed on these spectra, as illustrated below, and subsequently the spectra were block averaged by 4 to keep power spectra of length 512 spectral channels for further calibration and display at 9.8 kHz resolution.
õõõõõõõõõõõõõõõ 3 Available at http ://www.aca.gov.au/legal/legislation.htm. See also the presentation by J. Sarkissian on transmitter database visualization, available at http ://www.atnf.csiro.au/SKA/intmit/atnf/conf/.

FIG. 3.õPower spectra PKS A \ P ( f ), PKS B \ P ( f ), Ref 1 \ 1 2 P ( f ), and Ref 2 \ P ( f ) for scan SRT00502. These spectra are the aver3 4 ages of D25 s of data. A passband calibration has been applied to compensate for the gain dependence of the 5 MHz band limiting ïlters. The upper panels show the spectra both before and after cancellation. See the electronic edition of the Journal for a color version of this ïgure.

FIG. 2.õScan-averaged RFI spectra measured with the reference horn for scan SRT00502. The solid line is the cross power spectrum C ( f ) 34 deïned in eq. (5). Dashed and dotted lines indicate the autocorrelation spectra P ( f ) and P ( f ) in eq. (3). The spectra have been passband4 calibrated3using approximate gain curves for the 5 MHz ïlters. There are 512 frequency channels covering a 5 MHz band.

Examples of the cross power spectra are shown in Figure 4. Both C and C have broadband correlated power, 12 which is clear in the 34 integrated spectra both as a signiïcant non-zero amplitude and as a well deïned trend in phase across the band. The phase gradients that are clearly visible in the C ( f ), C ( f ), C ( f ), and C ( f ) spectra indicate 13 24 dierential path 14 delays. 23 delay for C ( f ), C ( f ) and The 14 for the channel numbers above 300 in C13( f ), and C ( f ) 23 amounts to 15 time steps (0.75 ks). The steeper phase24 gradient for channels below 300 in C ( f ), and C ( f ) imply 23 24 path delays of 60 time steps \ 3 ks or a path length of D900 m. Apparently, the frequency dependent reception pattern of the far side lobes of the Parkes dish, coupled with complicated scatterers in the ïeld, can lead to quite complex spectral dependence for the RFI. A strength of our method is the simplicity with which it handles this complex multipathing. Examples of the complex correction spectra deïned in equations (7)-(9) are shown in Figure 5. The spectra for correction A and correction B have zero phase over the spectral range where the signal is signiïcantly nonzero, conïrming that phase closure applies. Figure 6 shows averages of the amplitude closure quantity from equation (13), plotted as amplitude and phase across the spectrum. There is deterioration in the closure relation when the RFI signal is low, as expected. Figure 2 compares three time-averaged power spectra representing the RFI signal sensed by the reference horn : power spectra for each of the two polarizations and one cross power spectrum. For use in correction schemes, the cross power spectrum C ( f ) has the advantage that it is not contaminated by the34 positive bias of the receiver noise total power that is seen in the autocorrelation power spectra. In practice, the low-level correlated signal in C ( f ), due to actual cross correlated broadband power or 34 due to correlated quantization noise, may limit the accuracy of the cross power spectrum as an estimate of the RFI.


FIG. 4.õCross power spectra C ( f ), C ( f ), C ( f ), C ( f ), C ( f ), and C ( f ) for scan SRT00502. These spectra are the averages of D25 s of data. 23 24 34 The spectrum C ( f ) at the lower 12 right is 13 plotted with a rescaling of a factor of 100 in order to display the noise level away from the frequencies also 14 34 containing strong RFI.

FIG. 5.õComplex correction spectra derived from eqs. (10)õ(12) for scan SRT00502 for application to the Parkes spectra A, B, and A ] B. These plots show the averages of D25 s of data. The RFI subtraction was actually performed on 82 ms averages. A & B indicate the two polarizations.


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FIG. 6.õComplex closure quantity C C /C C , averaged for scan 13 24 23 14 SRT00502. L eft : full spectrum. Right : expanded scales. Reduced scale plots of the average RFI cross power spectrum from Fig. 2 are drawn in the bottom panels for comparison.

Figure 7 compares uncorrected and corrected spectra for two scans. For the purposes of display, the spectra have been crudely passband calibrated by dividing the spectra by gain templates formed from the scan average of the total power spectra of scan SRT00601, which was recorded while the sky frequency for the Parkes data channels was tuned o the RFI frequency. Since this gain template is in common to the processing for both scans SRT00501 and SRT00502, some of the common structure in the spectra in Figure 7 results from this common gain template.

FIG. 7.õComparison of two scans : SRT00501 and SRT00502. Top Panels : Uncorrected power spectra for the two Parkes polarizations. Spectra for SRT00501 are displaced vertically by 0.2 in amplitude. Bottom Panels : Corrected power spectra for the two Parkes polarizations. Spectra for SRT00501 are displaced vertically by 0.05 in amplitude. Reduced scale copies of the RFI cross power spectrum are included for reference with vertical dashed lines to indicate the minima in the RFI spectrum.

The time dependence of the RFI sensed by the Parkes telescope is displayed in image format in Figure 8, side by side with the spectra after the RFI subtraction. These spectra received the same processing as described for the averages in Figure 7, with the additional step of subtracting a third order polynomial spectral baseline from each time step. This additional step was done to remove some faint

FIG. 8.õDynamic power spectra over 564 time steps of 82 ms each (Scans SRT00501 and SRT00502). There are four spectra plotted in parallel with time increasing vertically. Left : The two Parkes polarizations prior to RFI subtraction. Right : The two Parkes polarizations after RFI subtraction. All spectra are passband calibrated to compensate for the frequency dependence of the 5 MHz ïlters. A third order polynomial spectral baseline was ïtted to channels 30-235, 370-405, and 420-500 and subtracted for each time step.


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FIG. 9.õRaw and corrected Parkes A ] B cross power spectra. L eft : Raw, complex cross power spectrum. Right : Corrected complex spectrum.

the other hand, if the integration time is long, the impulse response functions that couple the astronomy and RFI signals to the receiver will vary, and the mathematics in equation (4) and equations (7)-(9) will break down. Figure 10 shows a series of tests with a range of integration times (t D 8 ms to 8 s) for application of the RFI subtraction. In int all cases, the plots give the grand averages of the entire 25 s of the scan after application of the algorithm on the shorter data segments. The parameter t has an appropriate value o in each case to produce a stable noise level and representation of spectral features in the frequency range away from the RFI contamination. For these data, the algorithm was most eective for integration times of only a D1 s or less. When treated on 8 s averages, substantial RFI remains unsubtracted.
5

variations in the total power level that occurred from one integration to the next. The cross polarized spectrum from the Parkes A ] B is shown in Figure 9. The ïgure includes the raw spectrum and the corrected spectrum after subtraction of the correction shown in Figure 5. Remaining in the corrected spectrum, there is a slow modulation of the power across the 5 MHz band, probably indicating that this power represents broad band noise signal that is scattered within the Parkes telescope structure with the same delay path lengths associated with traditional standing waves and path lengths of a few hundred feet. The integration time over which the RFI corrections are applied is a critical parameter when the signal-to-noise ratio of the reference signal data path is low. If the integration time is too short, then some of the derived correction will be noise, and this will be folded into the resulting spectrum. On

. DISCUSSION OF LIMITATIONS

The wide range of delays for the scattered RFI led to problems in our initial experiments, which simulated an FX correlator with 2048 sample transforms. The RFI illustrated in Figure 4 has two strong components that are delayed by 15 and 60 time steps. A 60 time step delay is D3% of the time block being processed, so that the cross-correlation is not being performed on fully overlapped data streams. Increasing the window to 8192 was adequate to reduce the residuals to a level that was barely visible above the noise in the corrected spectra. An additional but less signiïcant improvement resulted from applying an constant delay oset of 35 time steps to the reference signal at the input to the "" FX correlator îî to make it closer to the average of the principal delays in the Parkes data channels. A traditional time-domain lag correlation spectrometer would not encounter this problem, provided a sufficient number of lags are allocated to fully cover the range of delays experienced by the RFI. In these tests at Parkes, there is a possibility that some uncancelled signal may be present because of a second transmitter operating at these same frequencies. The reference horn was pointed at the stronger, nearby transmitter, while the second transmitter, located at approximately three times the distance, can be scattered into the Parkes Telescope signal paths without being sensed by the reference horn. The coarse digitization of the RFI reference signals will eventually form a limitation to the precision of the subtraction. Crude quantization generates an artiïcial noise ÿoor throughout the spectrum, eectively scattering power out of the narrow band RFI. Since both polarizations from the reference horn are recorded at relatively high signal to noise ratio, the quantization noise is also correlated between the two data channels, so that there is corruption of the cross power spectrum as well as for the autocorrelation spectra. (The implication is that the term SN N T in equation (5) 4 will not average to zero with increased3integration.)
6

. THE TOXICITY TEST

FIG. 10.õThe eect of varying the time interval on which algorithm described in eqs. (10)õ(12) is applied. The interval varies from 8 ms to 8 s in factors of 10. In each panel, the upper spectrum shows the corrected, calibrated, baselined spectrum ; the lower spectrum is the rms scatter about the mean spectrum for each channel as a function of time. The typical value for rms should decrease by 101@2 for each increase in factor 10 in integration time.

A crucial requirement of an RFI subtraction algorithm is that it must leave the astronomical signal of interest unaltered. To test the current method, we added simulated galaxy signals to the two Parkes input data streams. Independent Gaussian noise was ïltered with a double-horned galaxy proïle and the instrumental IF ïlter passbands and then injected into the data pipeline just before the correlation stage. Figure 11 shows a comparison between the


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FIG. 11.õSurvival of an injected synthetic galaxy signal in the astronomy channels through the RFI subtraction process. Top : Raw spectraõwith and without the synthetic galaxy signal. Center : Corrected spectraõwith and without the synthetic galaxy. Bottom : Dierence between the corrected spectra and the ratio between the input synthetic spectrum and the dierence spectrum. See the electronic edition of the Journal for a color version of this ïgure.

FIG. 13.õCross polarization spectra for two feeds of the Parkes multibeam system before and after RFI subtraction for scan SRT00108. An attempt was made to apply a passband calibration using the same passbands determined from the scan SRT006­01 for the reference horn experiment. The data were treated in D82 ms averages, which in turn were integrated over the 20 s duration of the scan. Dotted lines show the spectra ampliïed by a factor 10.

RFI corrected spectra both with and without the galaxy. highlight the dierence between the injected signal and output after RFI subtraction, the bottom panel shows the dierence between the output with and without

To the (1) the

added signal and (2) the ratio of this dierence to the synthetic galaxy proïle added to the input. The rms deviation of the ratio about unity is 0.005 for polarization A and 0.006 for polarization B. No systematic deviations are seen across the band, other than rises in noise level at the edges where the galaxy proïle is approaching zero at the edge of the proïle. The conclusion is that this method does no systematic harm to the astronomical signals.
7.

THE PARKES TWO FEED EXPERIMENT

The mathematical description (eqs. [3] to [12]) can be equally well applied to a case that uses a second feed from the Parkes Telescope as the "" reference horn.îî Both feeds may receive astronomical signals, but since the feeds point dierent directions in the sky, these are independent astronomical signals, which will not correlate and therefore will not be subtracted from each other by this algorithm. The two feeds do sense the same RFI signal i(t), although through dierent scattering paths. This is sufficient commonality that the cross power spectral approach should permit each feed to serve as the reference antenna for the other. Similar experiments have been reported.4 Figure 12 shows a comparison between the autocorrelation spectra measured for the Parkes two-feed experiment, before and after RFI subtraction. There is noticeable dierence among the four channels in the eectiveness of the RFI subtraction. This probably results from the bias created by the broadband polarized ÿux or correlated noise in the
FIG. 12.õAutocorrelation spectra for both polarizations of two feeds of the Parkes multibeam system before and after RFI subtraction. An attempt was made to apply a passband calibration using the same passbands determined from the scan SRT006­01 for the reference horn experiment. The data were treated in D82 ms averages, which in turn were integrated over the 20 s duration of the scan. See the electronic edition of the Journal for a color version of this ïgure. õõõõõõõõõõõõõõõ 4 See also the presentation by B. Sault on Cross-correlation approaches to interference elimination, available at http ://www.atnf.csiro.au/SKA/WS and that by L. Kewley, R. Sault, & R. Ekers on Interference excision using the Parkes multibeam receiver, available at http ://www.atnf.csiro.au/SKA/ intmit/atnf/conf/.


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cross polarized spectra 3 ] 4 that is being used as the "" reference îî for data channels 1 and 2. As shown in Figure 13, this noise ÿoor is higher in feed 2 (INR D 351) than in feed 1 (INR D 100: 1), causing the RFI template spectra derived from the cross power spectra to be less faithful, which in turn leads to larger error in the correction spectra to be subtracted from feed 1. For comparison, the reference horn spectrum C ( f ) in Figure 4 has INR D 1000: 1 34
8

. CONCLUSIONS

RFI subtraction can be performed using cross power spectra between the astronomy data channels and RFI "" reference îî channels. In principle, the reference channel can also be an astronomy channel provided it carries an astronomy signal that is uncorrelated with the astronomy in the channel that is being corrected. The tests made at Parkes demonstrate that a speciïcally designed reference sensor provided a higher signal-to-noise ratio reference signalõand consequently cleaner cancellationõthan that obtained from a second horn feed at the Parkes Telescope focus, whose principal function is to illuminate the Parkes dish. A reïnement will be to implement this scheme using two reference antennas that are spatially separated (as in the lower diagram of Fig. 1) in order to avoid correlated noise contributions, while still obtaining as clear and stable path to the RFI source as possible. The cross power spectra from the two spatially separated antennas would form an optimal RFI reference spectrum C for use in the equations (10)õ(12). To avoid problems34 with dierential delay causing loss of coherence in the reference signal, the spectrometer would need to operate with spectral resolution *f \ (*//2n)c/L > c/L , where */ is the allowable phase rotation across a spectrometer channel and L is the spatial separation of the sensors. The 2.4 kHz spectrometer resolution emulated in software for the study reported here would allow spatial separations of up to 1200 m, if */ is required to be less than 0.02n radians. There are a number of advantages to performing this type of "" post-correlation îî RFI subtraction : 1. Provided the required correlation products are recorded (i.e., the on-line system is capable of recording correlation functions with a sufficiently large number of delay lags), the RFI subtraction can be performed o-line, where it remains an option in the data reduction path, rather than a commitment made on-line and permanently. 2. The method is not vulnerable to the eects of sporadic RFI, which hurt many algorithms that have an initialization period while they acquire the RFI signal and optimize their cancellation parameters. 3. Nor is the result inÿuenced by changes in beam shape during adaptive nulling. 4. The correlation method is eectively a coherent subtraction, since the correlation functions retain the information describing relative phase between the RFI entering in the astronomy data stream and the RFI entering the reference antenna. We showed in ° 3, this means that the RFI noise power is largely subtracted, leaving only system noise. 5. Generalization of the method to an array of telescopes is straightforward but demands additional correlator capacity. If there are two reference signal sensors, labeled "" x îî and "" y,îî that sense negligible astronomical signal, then their cross power spectrum C ( f ) containing a high INR xy

signal can be used to correct any other power spectrum C ( f ) through the closure relation C C*/C* . The i, j ij ix jy xy indices can denote orthogonal or parallel polarizations drawn from any combinations of antennas in the array or auto correlation, when i \ j. 6. A modiïcation of the method can be applied to pulsar data streams in which a digital correlator replaces the narrowband ïlter bank used in compensating for pulse dispersion. One possible implementation would construct cross power "" coupling spectra îî X( f ) that are valid for the time interval t , during which the "" g factors îî of equation (1) are int stable. The coupling spectrum is g* g C* X ( f ) \ 1 4 B 14 , 13 g* g C* 34 34 where C and C are averaged for up to 1 s as appropri14 34 ate. Then the correction CX can be computed and applied 1 to cancel RFI in measurements of P ( f ) on shorter time1 scales : CX \ o g o2S o I o2T \ X g g*S o I o2TB X SS S*T , 1 1 13 1 3 13 1 3 where SS S*T \ C . Alternatively, one could construct 13 13 the corrected time sequence s (t) by subtracting the correc1 tion SX \ g I \ X* g I B X* S 1 1 13 3 13 3 from S ( f ) and inverse Fourier transforming to obtain an 1 RFI cancelled version of s (t). Computationally efficient 1 schemes could be implemented that include coherent dedispersion Hankins 1974 in the same transform operations as the RFI cancellation. 7. The method can be generalized to removal of solar radiation whose multi-path scattering eects give rise to the spectral "" standing wave îî problem. The important dierence that the Sun generates a dual polarized signal with a variable polarized component ; this will necessitate a processing path more akin to the subspace decomposition (Leshem et al. 2000),5 in order to identify orthogonal components of the solar "" RFI îî signals. 8. The present generation of digital correlators, which typically operate with single bit to 9 level precision, could implement this method for use in moderate levels of RFI for testing and astronomical observation in the near future. The disadvantages of the method are : (1) The data rates will be high, since the method requires a full-scale cross correlator, preferably with multi-bit precision to accept large SNR RFI reference signals, that must dump spectra after relatively short integrations of less than D1 s. Of course, these data rates are lower than recording the full base band, but they are substantially higher than a real-time adaptive ïlter approach, which would allow long spectral integrations once the RFI has been canceled. (2) When the interference is much stronger than the astronomical signal, a 1 or 2 bit sampler is "" captured îî so that the data stream consists primarily of ^2, and the zero crossings are determined by the phase of the interferer. Under these condiõõõõõõõõõõõõõõõ 5 See also S. Ellingson on interference mitigation techniques, available at http ://www.atnf.csiro.au/SKA/intmit/atnf/conf/.


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tions, the astronomical data will be largely lost. These applications will require correlators with greater digital precision. The case where multiple interferers occupy the same frequency band will require a greater number of sensors and more correlator capacity devoted to processing the RFI signals, as laid out in the analyses of Sault (1997) and Ellingson (1999).

The authors are grateful to W. van Stratten, M. Bailes, S. Anderson, S. Ellingson, R. Sault, P. Perillat, R. Ekers, J. Bunton, L. Kewley, M. Smith, and P. Sackett for helpful comments and discussion. F. B. is grateful to the ATNF in Epping, NSW, the Department of Astronomy at OSU, Columbus, OH, and to the IAS, Princeton, NJ, for their hospitality while this work was done.

REFERENCES Barnbaum, C., & Bradley, R. 1998, AJ, 116, 2598 Leshem, A., & van der Veen, A. J. 1999b, in IEEE Workshop on Higher Bell, J. F., et al. 2000, Proc. Astron. Soc. Australia, submitted Order Statistics (Los Alamitos, CA : IEEE Comp. Soc.), 25 Ekers, R. D., & Bell, J. F. 2000, in IAU Symp. 199, The Universe at Low Leshem, A., van der Veen, A. J., & Boonstra, A. J. 2000, ApJS, in press Radio Frequencies (San Francisco : ASP), in press Smolders, B., & Hampson, G. A. 2000, IEEE Trans. Ant. Propag., subEllingson, S. W., Bunton, J. D., & Bell, J. F. 2000, in Proc. SPIE 4015, 400 mitted Ellingson, S. W., & Hampson, G. A. 2000, IEEE Trans. Ant. Propag., Staveley-Smith, L., et al. 1996, Proc. Astron. Soc. Australia, 13, 243 submitted van Straten, W., Britton, M. C., Bailes, M., Anderson, S. B., & Kulkarni, S. Hankins, T. H. 1974, A&AS, 15, 363 2000, in ASP Conf. Ser. 202, Pulsar Astronomyõ2000 and Beyond, ed. Kewley, L., Sault, R. J., Bell, J. F., Gray, D., Kesteven, M. J., & Ekers, R. D. M. Kramer, N. Wex, & R. Wielebinski (San Franciso : ASP), 238 2000, in preparation Leshem, A., & van der Veen, A. J. 1999a, in IEEE Workshop on Signal Processing Advances in Wireless Communications 1999 (Piscataway, NJ : IEEE), 374