How it works--technical details

Basic adaptive filter theory

An adaptive interference canceling system for use on a radio telescope is illustrated in the Figure . All signals are digitized and have a constant sampling period with discrete time sequences indexed by n. The telescope radiometer (or primary channel) receives both the astronomical signal entering the main beam and the interference entering the sidelobes . The primary input is the sum of these two signals, . A low-gain antenna is connected to a second receiver (the reference channel) whose input is only the interference . The reference antenna is pointed towards the interferer (at the satellite or toward the horizon). The collecting area of the reference antenna is much smaller than the primary telescope, and therefore the astronomical signal would be essentially absent in the reference input on the filter's adaptation timescale. The interference in the reference channel is correlated in some unknown way with the interference in the primary channel , and the job of the adaptive filter is to estimate this correlation as a function of time. The adaptive algorithm compares the previous solution to current information and sends updated coefficients to the digital filter. The digital filter uses these coefficients to alter , producing which closely resembles the interference in the primary channel. is subtracted from the primary input to produce the system output :

,                              (1)

and then is sent to the telescope's spectrometer. No prior knowledge is required of , , , or of their interrelationships.

The signal path through the adaptive filter shown in the previous Figure illustrates the iterative nature of the system. The adaptive algorithm finds new coefficients by comparing with using a least mean squares algorithm (LMS) that minimizes the total power. The power is the square the system output:

.                  (2)

Since is uncorrelated with the interference in the primary and reference channels, the cross-terms vanish, and so the expectation value (time averaged) of the system output is:

.                        (3)

As the filter adjusts the coefficients to minimize , the power in the astronomical signal is unaffected, and so reaches a power minimum:

.                  (4)

Since the astronomical signal is constant, minimizing the total output power minimizes the output interference power, and therefore maximizes the output signal-to-interference ratio.

In a stationary environment, once the filter finds the weighting coefficients so that is a best least squares estimate of the interference in the primary channel, those coefficients are fixed. More specifically, even though the interference signal will have a non-zero bandwidth due to modulation, as long as the statistics of the waveform (i.e., mean variance and autocorrelation) in both the primary and reference channels remain the same, the coefficients found initially will suffice. However, in more realistic conditions with a radio telescope, the weighting coefficients will quickly become obsolete if propagation effects cause a relative change in what the reference and primary channels see. Significant relative changes that require updated coefficients can be caused by reflection, dispersion and telescope slewing; the timescale of these effects is on the order of hundreds of milliseconds. In an adaptive scheme, the coefficients used to weight are updated; in our prototype receiver, coefficients are updated every 2 microseconds.

Two special cases of interest can arise. First, if the reference input is perfectly correlated with the interference in the primary channel, the output signal will be completely free of interference, since . In the second case, if the reference channel is completely uncorrelated with the interference in the primary channel (e.g., if for some reason the RFI does not appear in the primary channel), goes to zero then (3) becomes

,                              (5)

and the filter turns off (in practical terms, the weighting coefficients are set to zero).

Adaptive filters and the Wiener Solution

In stationary conditions, the adaptive system converges to the optimal Wiener filter. To minimize the output power, the algorithm needs to find the minimum of an error surface, defined below. This surface is a multi-dimensional hyperboloid bowl with a unique minimum.

Once the adaptive filter finds the set of coefficients corresponding to the bottom of the bowl, those coefficients are fixed and are not updated or changed (the Wiener solution). However, the adaptive version continuously tracks changing statistics between the reference and primary channels and adjusts the coefficients in real-time.

In a realistic, non-stationary environment, the bottom of the hyperboloid bowl is slowly moving around, not by large amounts but significantly, as the sides of the bowl change shape. The adaptive system finds the Wiener solution by locking onto the minimum point and then tracking the minimum as the bowl moves around in multi-dimensional space. Since the basis of adaptive interference canceling is the Wiener solution, we will examine the framework of the Wiener filter and formulate performance expectations. In this section, we discuss the characteristics of a Wiener filter in the presence of random noise and formulate performance expectations.

The Optimal Transfer Function—Finding the Bottom of the Bowl

Since is not an exact duplicate of , we process with an adjustable weight filter with coefficients to produce , which is a close replica of . A schematic of the filter is shown in the Figure ; it is constructed using a tapped-delay line (or transversal filter) in the reference channel and a linear combiner. The unit delay is one sample time (the unit delay is the generalized discrete Fourier transform , or z-transform, of the unit sequence delayed by one sample, see Oppenheim et al. 1989).

The tapped delay line is a finite impulse response (FIR) filter, used in nearly all digital interference canceling applications due to its inherent stability. The taps are scaled by weighting coefficients, and then summed to form the FIR filter whose output is . Finally, is subtracted from the primary input to form the combiner output, . As described by (1), is the difference between the primary output and the processed output of the reference channel. Note that no filtering is done in the primary channel.

 In analogy to (1), let be the vector of delayed versions of , and be the vector of tap weights containing the set of weights . Then (1) becomes

                             (6)

where T indicates the transpose. The output power is . The filter finds by minimizing the total output power, so can also be considered as an estimation of how well the system is working; in control system theory, is called the error signal. is the error performance surface which is a multi-dimensional quadratic hyperboloid and has a unique minimum. In stationary conditions, this minimum is fixed and is described by the optimum weight vector forming the Wiener filter response. The values of are found by setting the derivative of with respect to the weights equal to zero. The expectation value for a stationary process is equivalent to the auto-correlation function (power spectrum), and so after matrix operations we arrive at:

                             (7)

where is the auto-correlation of the reference input, and is the cross-correlation function of the reference with the primary input (see Widrow & Stearns 1985 for a derivation). Taking the z-transform (the generalized Discrete Fourier transform) of (7), we

.                              (8)

This result represents the unconstrained, non-causal Wiener solution (see, e.g., Oppenheim & Schafer 1989 for details). However, any realizable physical system must be causal. In order for the performance to approach the ideal non-causal filter, a delay must be inserted in the primary input. This forces an equal delay in the response of the filter. The length of the delay is chosen to cause the peak of the impulse response to be centered along the tapped-delay line. This causal system can behave in a non-causal manner for a limited time-frame, since the solution will depend on samples at , , . A real filter has a finite number of taps, but the more taps, the closer the impulse response will be to the ideal, infinitely long filter. The number of taps for a particular digital signal processor is a cost-performance tradeoff.

Performance Expectations

The goal of any interference excision scheme is to recover the astronomical signal without distortion by the filter. To this end, the interference at the output of the canceler must be reduced down to or below the rms noise over the integration time needed for the science, and the averaged baseline noise should not be altered by the presence of the canceler.

The canceler's performance depends on a number of factors, including: random noise in each channel, quantization noise, type of algorithm used to find the optimal weights, and in the case of the adaptive system, tracking ability. One of the unique characteristics of adaptive interference cancellation is that the filtering process occurs in the reference channel, and so the astronomical signal coming through the primary channel is not distorted by the canceler. As a result, the filter is linear, and so the attenuation achieved by the filter will be entirely independent of the astronomical and interference signals in the primary channel. Linearity is preserved as long as the system is operated within the dynamic range set by the RF and IF amplifiers, quantization of the A/D converter, etc. We have used 12-bit converters in our prototype system, resulting in a dynamic range (see Ifeachor & Jervis 1993) of 72 dB.

 1) Attenuation of RFI

The canceler's performance over a given integration time is measured by how well the RFI signal is attenuated, i. e., whether the attenuation reached the rms noise, and how long one could integrate before the rms noise level would fall below the residual RFI. Conditions at the telescope will not always push the system to the edge of possible performance; if the RFI is weak to start with, a less-than maximum theoretical attenuation might still result in attenuating RFI below the level of rms noise and yield a successful observation. It is important to note that an RFI signal is never completely excised; there is always some residual RFI even when the attenuated RFI is buried in the rms noise. As the integration time gets longer, the rms noise gets smaller, and eventually the residual RFI signal could be recovered. The theoretical attenuation can be expressed as

where is the theoretical attenuation and is the interference-to-noise. A plot of the theoretical attenuation versus in the primary channel is shown in the top plot of the Figure . Both axes are in units of logarithmic relative power, decibels, defined as , if . The expression above shows the somewhat non-intuitive result that

If we assume that the canceler itself does not introduce additional noise, we can find the maximum integration time before the rms noise would fall below the residual RFI and ruin the observation. If we define to be the integration time required for RFI to appear as a 3s residual above the noise, then (see our research paper for the derivation):

which is valid for up to the digitization limit of 72 dB for our converters. For example, this expression shows that for moderately good and of 30 and 20 dB, respectively, is 60 dB and so the integration time could be as long as 3 weeks before a 3s residual would pop up above the rms noise.

 2) Random Noise Contribution by the Reference Channel

Although noise in the adaptive canceler will not distort the astronomical signal while attenuating the interference significantly, the contribution of noise from the reference channel is an important consideration. Noise added by the canceler is a result of three factors:

    quantization noise caused by digitization, a result of the digitization of the analog signal, is a small affect with appropriate signal adjustments, that is, if gains in the reference and primary channels are adjusted so that gives a noise floor significantly larger than the quantum noise floor

    residual noise outside the interference bandpass caused by the tapped-delay line can cause a baseline ripple. Since the sharpness of the digital filter is proportional to the number of taps, too few taps, can cause excess noise from outside the interference bandwidth to enter the primary channel.

     residual noise within the interference bandpass

The third noise source, residual noise within interference bandpass is introduced by the reference channel and is the most critical of the three noise sources. Random noise in the reference channel is never zero, and so incorporating a reference channel necessarily injects noise into the output spectrum. This noise will have structure in the frequency domain. A measure of this effect is the residual noise ratio (), defined as the ratio of the noise power spectra at the system output to that in the primary channel inpu.

This expression, after some algebra, can be put in terms of interference-to-noise in the primary and reference channels which are measurable quantities:

.

For good performance, we want this residual noise ratio to be as close to 1 as possible; that is, we want the noise spectrum in the system output to be no greater than the noise spectrum in the primary input channel. is unity if there is no reference channel in the system. It is useful to consider the expression above in terms of the relative interference-to-noise in the reference and primary channels; in general, we expect better performance when .

We want to know how much noise is introduced into the system output for different ratios of to . If we write in terms of , so that , then (22) becomes

.

The bottom plot in the Figure the residual noise as a function of for different values of . If the interference-to-noise in the reference and primary channels are equal (), the noise in the system output (at the frequencies where the interference signal was located) will be almost twice as high as in the primary input before filtering. This is a result of a noisy interference signal in the reference channel being subtracted from the interference signal in the primary; since the noise is uncorrelated, this operation adds significant noise power to the system output. Therefore, the higher the interference-to-noise in the reference channel relative to that in the primary, the lower the residual noise in the system output.

In the case of a radio telescope, the primary channel receives RFI in the relatively weak sidelobes of the beam. The low-gain antenna for the reference channel is pointed toward the horizon in the direction of the RFI, so interference-to-noise in the reference channel can easily be higher than in the primary channel. The curves indicate that over a wide range of (especially for ) the injected noise is nearly constant. This implies that even though the interference signal level might fluctuate in the primary channel due to the telescope's slewing or to propagation effects, the amount of noise injected in place of the interference will be constant.

Adaptive filters in non-stationary environments

The results of the previous section for a Wiener filter can be applied to the adaptive filter system. In addition to the considerations for a stationary filter, an adaptive process introduces other sources of error and noise, specifically from the tracking capability.

The algorithm

The basic algorithm used to find the minimum of the error surface for the Wiener solution is also used to find the minimum in the case of an adaptive system, with the added complication of tracking the minimum as it changes. The least mean square (LMS) algorithm uses an estimate of the error surface gradient that is closely tied with the structure of the tapped-delay line, and requires a minimal amount of computing. There are other algorithms that offer improvements over LMS that would increase the performance of an adaptive system; examples include Recursive Least Squares (Haykin 1996) which uses off-line gradient estimations and Higher-Order Statistics (Shin & Nikias 1994) which is computationally complex. For our prototype receiver, we have implemented the LMS algorithm for its computational simplicity.

As conditions change in a non-stationary environment, the steepness of the sides of the error surface (hyperboloid bowl) change. For teach iteration in the adaptive system, the gradient of the error surface can be estimated from:

where is the vector of delayed versions of , is the vector of tap weights containing the coefficients , L is the number of filter tap weights, defining a direction in error space. By starting with this estimate of the gradient and using the method of gradients (see Widrow & Stearns 1985):

.

is found by tweaking . The parameter is the gain constant and is related to the step size in the search for minimum as the system tracks. The smaller the step size, the longer it takes to find the bottom of the error surface bowl. This is especially important in an adaptive filter, since the tracking time must be able to keep up with the timescale of the changing statistics, i.e., the ongoing movement of the bottom of the bowl. Speed and stability of adaptation, as well as noise in the weight vector solution, are determined by the size of ; the smaller is, the smaller the error is in , but the longer it takes for the solution to converge. A compromise between speed and introduced error is made in choosing . The optimum value of is a function of the interference power in the reference channel and is a trade-off between better adaptability and time for convergence. In our prototype system, the value of is manually adjustable.

See our research paper for a discussion of multiple reference channels.

 An Adaptive Canceling System on Radio Telescopes

Adaptive canceling shows promise as an effective means to attenuate interference in both single-dish and interferometer radio telescope systems. However, there are three basic requirements of the system and the RFI environment for success with adaptive canceling:

    the receiver must always operate in the linear regime; this assures that the RFI will not overload the receiver front-end, since distorted interference cannot be removed through a linear adaptive filter system.

    the adaptive convergence time must be finite, on the order of a few seconds; this places practical limits on the type of RFI that can be canceled effectively by this method. Signals of the continuous wave variety such as from broadcast stations or satellite downlinks and moderately long-duration signals (greater than a few seconds) from personal communication systems can be attenuated effectively since the canceler has the necessary time required to lock on and adapt. However, short bursts of interference from systems such as aviation or ship-board radar make canceling difficult without additional processing to "remember" the filter parameters from burst to burst yet turn the filter off rapidly between bursts to eliminate unwanted noise injection.

    interference-to-noise in the reference channel must be greater than that in primary channel; this places limits on where the interference source is located with respect to the main beam of the telescope. For example, if the main beam happens to be pointing directly at a satellite producing RFI, it will be impossible for the reference channel to be greater than that in the primary channel since the gain of the main beam of the telescope will be greater that the gain of the reference antenna. In contrast, if it is the sidelobes that pick up the interference, can easily be greater than since the gain of the telescope in the direction of the interference can be quite low.

When the interference source is moving, e.g., LEO satellites such as the Iridium series, the reference antenna must track the satellite across the sky to achieve good . In the case of interference arriving at the telescope from an over-the-horizon source, the adaptive filter performance is reduced as the telescope elevation is decreased in the direction of the interference, but the larger the telescope, the greater the tolerable elevation angle.