## 1. Introduction

[2] No matter what the astronomical application, radio frequency interference (RFI) is becoming an increasing problem in radio astronomy, and many methods for removing or suppressing the RFI are being proposed, evaluated and implemented. In most cases, the astronomer's targets are the correlations of signals from one or more antennas, and it is only these longer-term time averages of power that are wanted—there is no requirement to preserve voltage modulation. These applications are generally either finding the autocorrelation of signals from a single antenna (to measure the power spectrum of the astronomy signal), or the cross correlation of signals from more than one antenna (to measure the spatial coherence—or visibilities—of the astronomy signal). For details and specific examples see *Thompson et al.* [1986] and various chapters of *Taylor et al.* [1999].

[3] Suppose that a sampled voltage stream consists of an additive mixture of components that are uncorrelated with each other. Suppose also that each component is statistically stationary so that if the component happens to be present in more than one voltage stream the phase difference and the ratio of the sampled amplitudes measured at two receivers are constant. Then as described by *Briggs et al.* [2000], if one of these correlated components is undesired and hindering our ability to probe cosmic components, it is possible to cancel this RFI from the power spectra, after the voltages have been correlated. Canceling RFI from correlations, referred to as “postcorrelation canceling,” can offer many advantages over canceling the additive RFI voltage directly, particularly in regard to computational efficiency, since the canceling is performed on each correlation, tens or hundreds of times a second, rather than each voltage sample, tens or hundreds of times a millisecond. Furthermore, it is possible to implement the technique in some current arrays with no modification, albeit at reduced array performance, for example [*Kesteven*, 2002]. Before considering cancellation techniques further, the signal itself needs to be briefly described.

[4] Consider the voltage sequence sampled at an antenna as a combination of three complex additive components: receiver noise, *N*(ν, *t*); a noise-like cosmic component, *S*(ν, *t*); and interference, *I*(ν, *t*). Since each quasi-monochromatic spectral channel of the signal will be considered independently, the frequency labeling will be dropped to condense equations. After experiencing a phase shift, ϕ_{m}(*t*), due to the geometric delay of the signal relative to an arbitrary reference point, and being amplified and possibly phase shifted by a gain term, *G*(*t*), the signal at the output of antenna *m*'s sampler at time step *i*, which is only measuring the voltage in the narrow spectral band centered at frequency ν, is

where it is assumed that the signal has been amplified and delayed so that the cosmic signal is in phase with equal power at all of the receivers.

[5] In the absence of the RFI component, one could detect and measure the amount of cosmic power by comparing the voltage sequences from two antennas, *V*_{l} and *V*_{m}, since the background noise is different for the two receivers. However, the presence of the interfering signal obscures the cosmic detection. If one were to compare each main antenna voltage sequence with voltage sequences from antennas that do not measure the cosmic signal (so only comparing the RFI components of various signals), then one could attempt to model and remove the RFI component in the comparison of *V*_{l} and *V*_{m} and recover the astronomy signal. In practice reference antennas, such as parabolic reflectors pointed in the direction of a known transmitter, will measure some cosmic power through their sidelobes, but it is assumed that this is negligible compared with the receiver noise power. Assuming a negligible cosmic contribution, the reference signals are of the form

[6] For any pair of antennas, the signals are compared by correlating the two voltage sequences together, that is, multiplying one signal by the complex conjugate of the other and accumulating the product for some accumulation time (∼1 s). Uncorrelated components will multiply to give zero mean noise which will average away as the number of samples accumulated approaches infinity, while a component that is present in both signals will correlate constructively, with an amplitude proportional to its power.

[7] The correlated power terms in the obscured main antenna cross correlation (so ignoring zero mean noise terms) are

where ϕ_{lm} = ϕ_{l} − ϕ_{m}, σ_{S}^{2} and σ_{I}^{2} are the variances of the cosmic and interfering signals respectively, the asterisk superscript indicates a complex conjugation, the angular brackets represent the expectation operator which is approximated by a time average, and it has been assumed that the gain and phase terms are constant over the time interval. The cross-correlated receiver noise term, which should be zero mean, has been included to remain general, since the following techniques also apply to autocorrelations where the noise is correlated against itself, that is, when *l* = *m*.

[8] Postcorrelation cancelers, which are briefly reviewed in the following section, estimate and then subtract the RFI power, , from *P*_{lm}.