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Keywords:

  • radio frequency interference;
  • radio frequency interference mitigation;
  • interferometry

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

[1] Radio frequency interference (RFI) is a significant problem for current and future radio telescopes. We describe here a method for postcorrelation cancellation of RFI for the special case of an extended source observed with an interferometer that spatially resolves the astronomical signal. In this circumstance the astronomical signal is detected through the autocorrelations of each antenna but is not present in the cross correlations between antennas. We assume that the RFI is detected in both autocorrelations and cross correlations, which is true for many cases. The large number of cross correlations can provide a very high interference-to-noise ratio reference signal which can be adaptively subtracted from the autocorrelation signals. The residual signal is free of interference to significant levels. We discuss the application of this technique for detection of the spin-flip transition of interstellar deuterium with the Allen Telescope Array. The technique may also be of use for epoch of reionization experiments and with multibeam feeds on single-dish telescopes.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

[2] Radio frequency interference (RFI) mitigation is fast becoming a necessary aspect of radio astronomy as terrestrial and space-based radio transmitters become more widespread and more powerful, as radio telescopes become more sensitive and search for ever fainter signals, and as radio astronomers seek to observe outside of the protected radio astronomy bands. A variety of RFI mitigation methods have been developed that rely on a range of signal processing techniques [e.g., Leshem et al., 2000]. Many of these techniques are generic in their application in the sense that they are not specific to the scientific goals and the types of interferers. There is much to be gained, however, from techniques which are specific to a given type of astronomical observation or a given type of interferer [e.g., Ellingson et al., 2001].

[3] We describe here a technique developed for detection of a diffuse astronomical source with a high-resolution interferometer in the presence of point-like or partially resolved interference. The technique exploits the fact that the astronomical signal is detectable only through the autocorrelation of each antenna's signal, while the interference is detected in both the autocorrelations and cross correlations of the signals. For an interferometer with even a modest number of elements, the number of cross correlations is significantly greater than the number of autocorrelations. Thus the problem becomes similar to a reference antenna method with multiple reference antennas.

[4] In section 2, we describe the technique. In section 3, we show results of computer simulations which demonstrate its performance characteristics. In section 4, we discuss possible applications of this technique. These include detection of the interstellar deuterium spin-flip transition with an interferometer with baselines on the order of 100 m, and detection of the cosmological epoch of reionization signature. The method is also applicable for multibeam feeds on single-dish radio telescopes.

2. Cross-Correlation Subtraction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

[5] We consider our method for an interferometer with Na elements configured in a nonregular pattern. Each element receives signal from the sky and converts the signal to baseband. The baseband signals are processed by a correlator which produces autocorrelation and cross-correlation power spectra with Nch channels. These spectra are computed from Ns digital samples.

[6] We characterize the signal received at each antenna i as the sum of the astronomical signal, Si, the receiver noise, Ni, and the interference, Ii:

  • equation image

The autocorrelation and cross-correlation power spectra are then

  • equation image
  • equation image

[7] We assume that the source term in the cross-correlation function 〈SiSj*〉 goes to zero while the source term in the autocorrelation function 〈SiSi*〉 is nonzero. This assumptions holds for the case of an extended, smoothly distributed brightness distribution observed with a high-resolution interferometer. In particular, this condition holds for baselines with a fringe spacing much smaller than the angle subtended by the source on the sky. The difference between the autocorrelation and cross-correlation functions is then

  • equation image

Under the assumption that the interference is not resolved by the interferometer (i.e., that it is a point source), the second and third terms on the right hand side of the equation sum to zero. We are then left with a signal that only includes the effects of the astronomical source and the system noise. The system noise term will be dominated by the noise autocorrelation and we are left with a residual signal

  • equation image

which is simply the autocorrelation signal in the absence of interference.

[8] The cancellation of interference that takes place between the autocorrelation and cross-correlation signals as described above requires an idealized system. Under realistic conditions, a number of effects will limit the difference between the autocorrelation and cross correlation of the interference. These include multiple sources of interference, multipath propagation of interference, and variable antenna gain between two different array elements in the direction of the interferer or interferers. Adaptive cancellation methods introduce new degrees of freedom that make the realistic problem tractable. We suggest a method that minimizes the residual ε through the adjustment of the cross-correlation gains μij, where

  • equation image

[9] This method is equivalent to a reference antenna adaptive cancellation method in which the astronomy signal is the sum of the autocorrelations and the reference signal is the weighted sum of the cross correlations. We can use the results for the reference antenna adaptive cancellation technique to estimate the effectiveness of this methodology [Barnbaum and Bradley, 1998]. The attenuation of interference in the astronomy signal goes as

  • equation image

where INRref is the interference power-to-noise power ratio (INR) in the combined reference signal. Thus the residual astronomy signal will have interference at the level of

  • equation image

where INRast is the initial INR in the sum of the autocorrelations.

[10] We can determine the INRs for the reference and astronomy signals as follows. We assume that the signal is Nyquist sampled at a rate f for Ns samples. The INR in the autocorrelation of a single sample is INR0. In the case of the astronomy signal, the astronomy signal grows coherently with number of antennas and number of samples while the noise grows incoherently leading to

  • equation image

In the reference signal, the interference and noise signals grow similarly; however, the number of signals contributing to each sum grows as Na(Na − 1). This leads to

  • equation image

Substituting these results into equation (8), we find

  • equation image

In the limit of equation imageINR0 ≫ 1 and Na ≫ 1, we find that INRresidNa−3/2Ns−1/2 INR0−1.

[11] We can also compute the ratio of signal to interference (SIR). The signal-to-noise ratio in the autocorrelation sum is

  • equation image

where SNR0 is the signal-to-noise ratio of a single autocorrelation sample. Thus SIR is the ratio of equations (11)(12). In the high INR0 limit discussed above, SIR ≈ Na2NsSNR0INR0.

[12] We plot the SIR in Figure 1 for two cases. We assume SNR0 = −50 dB. In both cases the SIR increases dramatically for very large INR0. That is, powerful interferers are relatively easy to detect and subtract. Both cases also show an inflection point at which the SIR reaches a minimum. Below this point, the interference is too weak to be detected and cannot be canceled. If the minimum occurs at SIR > 0, then the technique has succeeded in protecting the signal against all interferers. In our example, this holds for the case with large Na and Ns. For a smaller Na and Ns, the SIR dips below zero, leaving the observations potentially corrupted by weak interference.

image

Figure 1. Signal-to-interference ratio in the residual astronomy signal as a function of initial interference-to-noise ratio for two cases. In both cases the signal is assumed to be 50 dB below the noise. In case 1 (solid line), with 32 antennas and 107 samples per iteration, the signal is always greater than the interference in the residual. In case 2 (dashed line), with 8 antennas and 104 samples per iteration, weak interference can corrupt the signal.

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3. Simulations and Practical Considerations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

[13] We describe a method for solving for the parameters μij described in equation (6). We use a Wiener method to calculate the best values for μij but other methods are available [Vaseghi, 2000]. The Wiener method has the advantage that it directly calculates the values; however, it is computationally expensive and may be replaced in practice by an iterative method [Ellingson et al., 2001].

[14] We recast the problem into a single astronomical signal equation image, which is the sum of the autocorrelation functions above, and Nb = Na(Na − 1)/2 reference antennas equation image. Both equation image and equation image are functions of channel number. We also add an additional reference signal which is a constant as a function of channel number to remove the effects of the band pass. For more complex band passes, multiple template functions such as a tilt or a ripple can be included. It is important that Nch > Nref, where Nref is the number of reference signals. If not, the solution is overdetermined and all signal will be removed by the algorithm. The Wiener solution is then

  • equation image

[15] In Figure 2, we show the result of a simulation of the cross-correlation method. We used an eight-element array with 65 spectral channels. The interference was a frequency comb with a spacing of 8 channels and a width of 2 channels in each peak. The gain of each antenna was modulated in each iteration separately for the interferer and for the source. The interference power was set to 20% of the noise power. The signal was characterized as a Gaussian with width 16 channels with a peak power that is 1% of the noise power. Each iteration consisted of 100 samples. We iterated for 2000 iterations. For 10 kHz of channel bandwidth, this would correspond to a 0.3 second integration. The interference is cleanly removed and the weaker signal is readily apparent in the residual spectrum with an amplitude of 1% of the noise power. The expected attenuation of the interference in the residual signal is 21 dB. The actual reduction of interference is ∼20 dB, comparable to that expected.

image

Figure 2. Results of a simulation of the cross-correlation subtraction technique. The top plot shows the astronomy signal without interference reduction. The bottom plot shows the astronomy signal with interference reduction. Details of the simulation are given in the text.

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[16] There are a few drawbacks and additional considerations to the method outlined here. The first problem is that the mean power off source of the residual signal is reduced from the no interference case. This may lead to calibration uncertainty. This effect results from the fact that all of the reference signals are positive quantities. The second problem is that in cases where the astronomical source signal is strong, the interference is oversubtracted. Since the algorithm is attempting to minimize the total spectrum, the presence of an astronomical signal biases the subtraction. The algorithm will also remove strong astronomical point sources from the residual signal. Whether this is a problem is a function of the particular experiment being undertaken. Additionally, the effects of partial resolution of the astronomical source of interest have not been fully explored. In what SNR regime does the remaining component of the source serve to mitigate the source itself? Finally, interferers in the near field as well as those undergoing significant multipath propagation may be resolved by the interferometer. It may not be possible to remove interferers of this kind from the astronomy signal.

[17] Alternative methods of performing the subtraction may avoid some of these problems. Knowledge of the noise power spectrum obtained through careful calibration, for instance, would allow one to give each spectrum zero mean. One could also generalize the two-reference antenna method described by Briggs et al. [2000] for this multiple reference antenna case. It is also worthwhile to explore the effect of using subsets of baselines to generate the reference signal.

4. Applications

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

4.1. Interstellar Deuterium

[18] The 92 cm deuterium spin flip transition (DI) is one of the most important radio spectroscopic lines not yet detected [Weinreb, 1962; Anantharamaiah and Radhakrishnan, 1979; Blitz and Heiles, 1987; Chengalur et al., 1997]. The transition is the deuterium analog of the hydrogen 21 cm line (HI) and arises from a flip in the spin direction of the electron with respect to the nuclear spin. The deuterium-to-hydrogen ratio (N(D)/N(H)) which can be determined from detection of the DI line is an important constraint on cosmic nucleosynthesis.

[19] The signal is known to be quite weak. Blitz and Heiles [1987] place an upper limit of N(D)/N(H) < 3 × 10−5, which is consistent with ultraviolet detections at 2 × 10−5. This ratio implies a line strength toward the Galactic anticenter on the order of 1 mK, on the order of −50 dB times the system temperature of a radio telescope observing in the Galactic plane at these wavelengths. Since the DI emission traces the HI emission, the line width is expected to be quite narrow (∼10 km/s) toward the anticenter.

[20] We have simulated DI observations with the 32 element configuration of the Allen Telescope Array [DeBoer et al., 2004]. The ATA-32 employs 6.1 m paraboloids distributed in two dimensions with baseline lengths that range from 8m to ∼100m. We simulated a snap shot observation toward the galactic anticenter using MIRIAD software. The sky model assumes that the DI traces HI as observed by the Leiden Dwingeloo survey [Hartmann and Burton, 1997]. We determined the correlated amplitude as a function of baseline length (Figure 3). The source is substantially resolved on baselines longer than about 20 m. These baselines comprise more than 90% of all ATA-32 baselines. Thus the signal will primarily be detected in the autocorrelations from each antenna and will be absent from the cross correlations. The weakness of the signal, its spatially diffuse nature, and the difficult interference environment at 327 MHz make this experiment an excellent candidate for the cross-correlation subtraction method.

image

Figure 3. Correlation amplitude as a function of baseline length for deuterium as observed with the ATA-32. The signal drops off sharply with increasing baseline length, making this problem well suited to the cross-correlation subtraction method.

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4.2. Epoch of Reionization

[21] The Universe went through a phase transition at a redshift z > 6 from neutral gas to ionized gas corresponding to the appearance of the first stars, quasars or other objects which produced ultraviolet photons [Sunyaev and Zeldovich, 1972]. Prior to this epoch of reionization (EOR), all hydrogen was in an atomic or molecular state. The spin flip transition of atomic hydrogen (HI) is expected to produce emission that is essentially continuous at frequencies less than νHI/zEOR, where νHI is the rest frequency of the HI transition and zEOR is the redshift of the EOR. At frequencies above this critical frequency, the signal disappears because of the ionization of all atomic hydrogen. The expected amplitude of the signal is on the order of 10 mK, or >30 dB below foreground noise. The redshift of the EOR places the signature at frequencies where interference is a critical problem for radio telescopes.

[22] The emission is expected to be globally distributed with a characteristic scale length of ∼10 arc min [Zaldarriaga et al., 2004]. Detection of the global signature is the first step toward characterization of the EOR. The cross-correlation subtraction method can be applied to this problem provided that baselines of the interferometer are sufficiently long to resolve out the global EOR signal. In practice, this implies baselines of 3 km for an attempted detection at a frequency of 100 MHz (zEOR ∼ 13). Integration times must be less than 10 seconds to prevent decorrelation of the interfering signal on the long baselines.

[23] Since such an experiment may have baselines on many intermediate baselines, it may be desirable to determine the astronomical signal from more than just the autocorrelation functions. A set of cross correlations from short baselines may also be included in the summed astronomical signal or have a separate interference signal removed from them.

4.3. Focal Plane Array Feeds

[24] Focal plane array feeds are increasingly important tools for radio astronomy [Staveley-Smith et al., 1996]. These feeds place multiple receivers in the focal plane of a single antenna. Typically, they are designed only to produce autocorrelations for each receiver. If designed to produce the full set or a partial set of cross correlations, however, these array feeds could make use of the cross-correlation subtraction method to mitigate interference. The technique may also be of use in eliminating band-pass ripples due to solar interference.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References

[25] We have described a new method for cancellation of interference in the power domain for the specific case in which the astronomy signal is apparent only in the autocorrelation signal. Provided that the RFI is not resolved by interferometer, cross-correlation signals are used to determine a high interference-to-noise ratio reference signal. Analytical results suggest that the method could prove powerful at removing very weak interference for arrays with Na greater than a few. Simulations demonstrate the basic performance of the algorithm. There are a few issues for further research. Most important among these is testing the effect of small source contributions to the cross correlations. This method highlights a critical aspect of RFI mitigation research: the techniques that work best are those that are best suited to the interferer and to the scientific goal.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Cross-Correlation Subtraction
  5. 3. Simulations and Practical Considerations
  6. 4. Applications
  7. 5. Conclusions
  8. References
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  • Barnbaum, C., and R. F. Bradley (1998), A new approach to interference excision in radio astronomy: Real-time adaptive cancellation, Astron. J., 116, 2598.
  • Blitz, L., and C. Heiles (1987), A new assault on the 92 centimeter line of DI, Astrophys. J., 313, L95.
  • Briggs, F. H., J. F. Bell, and M. J. Kesteven (2000), Removing radio interference from contaminated astronomical spectra using an independent reference signal and closure relations, Astron. J., 120, 3351.
  • Chengalur, J. N., R. Braun, and W. B. Burton (1997), DI in the outer galaxy, Astron. Astrophys., 318, L35.
  • DeBoer, D. R., et al. (2004), The Allen Telescope Array, Proc. SPIE Int. Soc. Opt. Eng., 5489, 1021.
  • Ellingson, S. W., J. D. Bunton, and J. F. Bell (2001), Removal of the GLONASS C/A signal from OH spectral line observations using a parametric modeling technique, Astrophys. J. Suppl. Ser., 135, 87.
  • Hartmann, D., and W. B. Burton (1997), Atlas of Galactic Neutral Hydrogen, Cambridge Univ. Press, New York.
  • Leshem, A., A. van der Veen, and A. Boonstra (2000), Multichannel interference mitigation techniques in radio astronomy, Astrophys. J. Suppl. Ser., 131, 355.
  • Staveley-Smith, L., et al. (1996), The Parkes 21 CM multibeam receiver, Publ. Astron. Soc. Aust., 13, 243.
  • Sunyaev, R. A., and Y. B. Zeldovich (1972), Formation of clusters of galaxies: Protocluster fragmentation and intergalactic gas heating, Astron. Astrophys., 20, 189.
  • Vaseghi, S. (2000), Advanced Digital Signal Processing and Noise Reduction, 2nd ed., John Wiley, Hoboken, N. J.
  • Weinreb, S. (1962), An attempt to measure Zeeman splitting of the galactic 21-CM hydrogen line, Astrophys. J., 136, 1149.
  • Zaldarriaga, M., S. R. Furlanetto, and L. Hernquist (2004), 21 centimeter fluctuations from cosmic gas at high redshifts, Astrophys. J., 608, 622.