### Abstract

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

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

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

[5] We consider our method for an interferometer with *N*_{a} 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 *N*_{ch} channels. These spectra are computed from *N*_{s} digital samples.

[6] We characterize the signal received at each antenna *i* as the sum of the astronomical signal, *S*_{i}, the receiver noise, *N*_{i}, and the interference, *I*_{i}:

The autocorrelation and cross-correlation power spectra are then

[7] We assume that the source term in the cross-correlation function 〈*S*_{i}*S*_{j}*〉 goes to zero while the source term in the autocorrelation function 〈*S*_{i}*S*_{i}*〉 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

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

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 μ_{ij}, where

[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

where INR_{ref} 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

where INR_{ast} 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 *N*_{s} samples. The INR in the autocorrelation of a single sample is INR_{0}. 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

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

Substituting these results into equation (8), we find

In the limit of INR_{0} ≫ 1 and *N*_{a} ≫ 1, we find that INR_{resid} ≈ *N*_{a}^{−3/2}*N*_{s}^{−1/2} INR_{0}^{−1}.

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

where SNR_{0} is the signal-to-noise ratio of a single autocorrelation sample. Thus SIR is the ratio of equations (11)–(12). In the high INR_{0} limit discussed above, SIR ≈ *N*_{a}^{2}*N*_{s}SNR_{0}INR_{0}.

[12] We plot the SIR in Figure 1 for two cases. We assume SNR_{0} = −50 dB. In both cases the SIR increases dramatically for very large INR_{0}. 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 *N*_{a} and *N*_{s}. For a smaller *N*_{a} and *N*_{s}, the SIR dips below zero, leaving the observations potentially corrupted by weak interference.

### 3. Simulations and Practical Considerations

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

[13] We describe a method for solving for the parameters μ_{ij} described in equation (6). We use a Wiener method to calculate the best values for μ_{ij} 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].

[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.

[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.