Bayesian detection of radar interference in radio astronomy

Authors


Abstract

[1] L-Band observations at the Green Bank Telescope (GBT) and other radio observatories are often made in frequency bands allocated to aviation pulsed radar transmissions. It is possible to mitigate radar contamination of the astronomical signal by time blanking data containing these pulses. However, even when strong direct path pulses and nearby fixed clutter echoes are removed there are still undetected weaker aircraft echoes present which can corrupt the data. In a previous paper we presented an algorithm to improve real-time echo blanking by forming a Kalman filter tracker to follow the path of a sequence of echoes observed on successive radar antenna sweeps. The tracker builds a history which can be used to predict the location of upcoming echoes. We now present details of a new Bayesian detection algorithm which uses this prediction information to enable more sensitive weak pulse acquisition. The developed track information is used to form a spatial prior probability distribution for the presence of the next echoes. Regions with higher probability are processed with a lower detection threshold to pull out low level pulses without increasing the overall probability of false alarm detection. The ultimate result is more complete removal, by blanking the detected pulse, of radar corruption in astronomical observations.

1. Introduction

[2] Radio astronomers must often share spectrum with other licensed radio services when observing deep space objects. For example, though the neutral Hydrogen emission line at 1,420.4 MHz lies in a protected band, moderate red shifting seen in distant sources can move the signal into the 1230- to 1350-MHz range where many high power air surveillance radar transmitters operate. These impulsive sources are ever present. Even when the telescope is located in a radio quiet zone surrounded by mountains blocking the direct line of sight signal path, echoes from aircraft hundreds of kilometers away can corrupt sensitive observations [Ellingson and Hampson, 2003; Ellingson, 2002; Zhang et al., 2003].

[3] This paper reports on efforts to mitigate radar interference at the NRAO Green Bank observatory in West Virginia. Pulses from the ARSR-3 surveillance radar located 104 km to the south-southeast have corrupted Green Bank Telescope (GBT) observations at 1292 MHz even when signals arrive in the telescope deep sidelobes. Since radar transmissions are impulsive with a relatively low duty cycle, the customary mitigation approach is to “blank” or excise data samples containing corrupting echo pulses. The remaining data are still suitable for long integration power spectrum analysis. The challenge lies in detecting interfering pulses so that they can be removed.

[4] In a previous companion paper [Dong et al., 2005a] we proposed a Kalman tracker to follow a sequence of aircraft echo detections seen at a radio telescope. The track history was used to guide real-time predictive blanking. It was also suggested briefly that track information could be used to improve detection performance. In this present paper we further develop that concept, present a detection theoretic basis for a new Bayesian detection method, and analyze detection performance improvement using Monte Carlo simulation. We refer the reader to Dong et al. [2005a] for a more detailed description of the interfering radar signal characteristics and Kalman tracker implementation.

[5] The principal advantage of the proposed Bayesian method is that it enables detection of weaker echoes than is possible using conventional methods. This can improve track quality by reducing the number of missed detections arising from normal random fluctuations in aircraft radar cross section (and thus echo intensity). Furthermore, since in radio astronomy we are interested in signal levels below the noise floor, weak undetected echoes can still have higher instantaneous power levels than the signal of interest, and can thus bias observations. Any method that improves weak echo detection can potentially reduce the amount of corrupted, unblanked data used in signal analysis.

[6] In the proposed detection algorithm, Kalman track histories consisting of a series of detected echoes from each aircraft are used to form a spatially dependent prior probability model for echo arrival. This prior distribution enables the Bayesian detector approach which increases probability of echo detection (PD) without increasing the effective probability of false alarm (PFA, i.e., the probability of declaring an echo detection when none is present).

[7] The sampled radio telescope signal is first processed by a conventional radar digital matched filter receiver to produce a complex baseband time series, b[n]. Details for the radar receiver design are given by Dong [2004] and Dong et al. [2005a]. Receiver output b[n] is reordered for detection processing to form a 2-D range-azimuth map b(R, θ) as shown in Figure 1. Each column of pixels contains receiver output samples from one transmitter pulse interval. Each time sample n maps to a corresponding point in space p located at the center of a range-azimuth bin (or pixel). The midpoint of bin p has Cartesian coordinates (xp, yp) and polar coordinates (Rp, θp). In expressions where geometrical position is more meaningful than time sample index, bp will be used interchangeably with b[n], and serves as a shorthand notation for bin p in the range-azimuth map, b(Rp, θp).

Figure 1.

A typical range-azimuth map, b(R, θ), of radar data collected at the GBT. This map covers approximately 60° of azimuth θ along the 3.5-s span of the horizontal axis. The bright region at about 1.8 s and 0–100 μs delay corresponds to the transmitter beam passing overhead at the GBT and generating strong reflections from nearby mountain peaks. Several aircraft echoes are visible. Note the horizontal extent of the bright echoes due to the sidelobe directional response pattern of the radar transmit antenna. The echo extent or region of support for a single aircraft is denoted as footprint F.

[8] Detection processor outputs are used both to guide blanking and as inputs to the Kalman tracker [see Dong et al., 2005a]. The tracker creates detection histories and predicts locations for the next anticipated echoes. These prediction points are utilized in turn by the Bayesian detector as prior probability inputs to adjust detection sensitivity as a function of spatial position.

[9] Sections 2 and 3 present the underlying probability models and theoretical algorithm development respectively for the proposed Bayesian detector. Experimental results demonstrating performance are found in section 4, followed by conclusions in section 5. Prior conference publications providing partial results are those by Dong et al. [2005b] and Dong and Jeffs [2004].

2. Probability Models for the Bayesian-Kalman Detector

[10] This section develops statistical models for the prior probability of the presence of an aircraft radar echo given the Kalman tracker history. These models will be exploited in section 2 to form a new algorithm which improves the probability of detecting weak echo pulses for a given fixed rate of false alarms. The approach is founded on fundamental statistical decision theory principles and provides a unified framework for incorporating information from a Kalman tracker state space estimate into a constant false alarm rate detector. A new detection statistic, “total false alarm probability” (PTFA), will be introduced and used as an optimization criterion to derive the algorithm. The method is general and can be applied to any sequential detection problem possessing a dynamical state space model suitable for a Kalman tracker. The discussion here however will focus on the application of locating radar aircraft echoes in radio astronomy data.

[11] Within a single range-azimuth bin (Rp, θp) consider the problem of deciding whether to accept the hypothesis that random event Hp: “an aircraft echo is present in bin p” or its complement equation imagep: “no echo is seen in bin p,” has occurred. The associated event probabilities are P(Hp) and P(equation imagep) = 1 − P(Hp). The classical radar detector makes this decision using a Neyman-Pearson (N-P) likelihood ratio test by comparing bp with a fixed threshold to achieve maximum probability of detection (PD) while enforcing a specified constraint probability of false alarm (PFA) [Van Trees, 2001b]. This is a reasonable approach in the usual case when it is difficult to assign meaningful costs to each decision or when prior probabilities for Hp are unavailable.

[12] However, the Kalman track state history contains information that can be interpreted, if an appropriate model can be found, as a prior probability for presence of an echo. Echoes are more likely in a region �� surrounding a point where the tracker predicts an echo will occur, than in the rest of the range-azimuth map. The classical Bayes decision criterion seems a likely candidate to exploit this knowledge, but again we do not have the required cost measures in our radar detection problem [Van Trees, 2001a]. The proposed algorithm addresses this issue and is structurally related to both the Bayes and Neyman-Pearson criteria.

2.1. Kalman Tracker Notation

[13] This section reviews a few equations and notation from Dong et al. [2005a] which are necessary in developing the Bayesian detector. For simplicity the following discussion is limited to detections from a single track formed by a series of echo detections from one aircraft. However, as implemented the system manages multiple simultaneous tracks. Assume the kth radar transmit antenna rotational pass over the radio telescope occurs nominally at time t[k]. This yields an aircraft echo detection, referred to as “snapshot k.” All quantities that are updated on a once-per-snapshot basis will be represented with a [k] argument. For example, the sampled position for a single aircraft detected in snapshot k is expressed as (R, θ)[k] or (x, y)[k].

[14] The desired tracker outputs at snapshot k are: (1) a prediction point, (equation image, equation image)[k + 1∣k], where the next detection is expected, and (2) shape parameters for region �� centered on this point (see Figure 2 and section 2.2). �� specifies an area likely to contain the aircraft echo during succeeding snapshot k + 1. The “hat” notation, equation image, will be used throughout to indicate an estimated or predicted quantity. An argument [k + 1∣k] indicates a predicted value for snapshot k + 1 based on observations up to and including k, while [kk] indicates a “correction” or smoothed estimate of the true value at time k based on observations through snapshot k. The size of �� depends on the quality of the track and gets larger with an increase in observation noise, missed snapshot detections, or rapid acceleration of the target. �� is also used to define the blanked region for predictive real-time blanking, to determine which track each new detection is associated with, and as the boundary for the region of increased prior probability for an arriving echo pulse used in the Bayesian detection scheme.

Figure 2.

An example of Kalman tracking performance for data acquired at the GBT. Four aircraft tracks have been automatically established and plotted, including a pair of crossing tracks. Data from five snapshots are shown. In order to exaggerate the motion between snapshots and reduce computational demand, only every other antenna pass was used, resulting in a snapshot interval ≈24s. The final point plotted for each track is the prediction point, (equation image, equation image)[k + 1∣k]. Prediction regions, ��, shown by the dashed ellipses vary in size according to track quality. Note that the center track has a large �� due to a missed detection. The entire area covered by the figure is represented by Ω. Predictive real-time blanking is accomplished by excising the prediction region data. For this example, off-diagonal terms in P[k + 1∣k] were assumed to be zero, which forces the �� ellipses to align to the x and y axes.

[15] For an existing track, each new observation snapshot initiates an iteration of the Kalman filter which produces an updated state vector prediction, equation image[k + 1∣k], and prediction error covariance matrix P[k + 1∣k],

equation image
equation image

where

equation image

and where x and y are a single aircraft's coordinates, equation image and equation image are corresponding velocities, F is the state transition matrix, G is the process transfer matrix, Q is the process acceleration matrix, and T denotes matrix transpose. Further details are given by Dong et al. [2005a].

[16] The prediction point (equation image, equation image)[k + 1∣k] is given by the first two elements of equation image[k + 1∣k] in (1). Its accuracy is gauged using P[k + 1∣k] from (2). With this prior information, it is reasonable to assume that for the snapshot at time t[k + 1], echo detections have higher probability near the prediction point.

2.2. Echo Arrival Probability Model

[17] We propose using uncertainty region �� as the basis for a spatial prior probability density function (pdf) model for the presence of an aircraft echo in snapshot k + 1. A few terms must first be defined before this model can be introduced.

[18] Assume that for each existing track exactly one echo will occur (with probability one) during snapshot k + 1 somewhere in an annular slice region,

equation image

containing all range-azimuth bins of interest. The {(x, y)∣expression} notation defines the set of all (x, y) pairs satisfying the expression. The single echo assumption implies that as far as the probability model is concerned, an established track will always produce an aircraft echo during each new antenna pass as long as the track remains in Ω. Cases of aircraft dropping out of the airspace due to landing or other events are neglected. These contingencies are handled by track management logic which will drop a track after a series of snapshots which produce no detections [Dong et al., 2005a].

[19] Prediction region equation image is defined by the quadratic matrix form for an ellipse,

equation image

where dx,y = [xequation image, yequation image]T is the vector position of (x, y) relative to prediction point (equation image, equation image) and ∈ denotes “element of.” For notational simplicity (equation image, equation image) = (equation image, equation image)[k + 1∣k] and P = P[k + 1∣k] are used here with the snapshot dependence dropped. �� is centered on (equation image, equation image)[k + 1∣k] and has size, elongation, and orientation determined by P[k + 1∣k]. User defined parameter β controls the scale (size) of �� relative to the error variances in P.

[20] Larger prediction error variance in P[k + 1∣k] leads to a larger ��, representing increased uncertainty as to where the next radar echo will be detected. Figure 2 illustrates this behavior. The plot shows track evolution for real GBT data over five snapshots for a dense scene with multiple overlapping aircraft tracks. The ellipses represent �� for each established track. Note the variety of sizes corresponding to variations in track quality.

[21] Owing to the assumption of a single echo per track per rotational pass of the transmit antenna, we can interchangeably refer to the presence of an aircraft or its associated echo. On the basis of the antenna directional beamwidth (including sidelobes) and the transmit pulse length, shape and repetition rate, this echo spans many range-azimuth map bins covering many transmit pulses. Note that in this definition an echo is a collection of individual pulse returns, not just one pulse reflection. The echo thus has a footprint F, or region of support, due to the effective point spread function in the range-azimuth map as can be seen in Figure 1. This footprint has fixed size in (R, θ) dependent on transmit beampattern and pulse shape. The echo may or may not be detected in any of these bins, depending on echo amplitude, local noise sample statistics, and the detection algorithm. Figure 3 illustrates the geometry of ��, F, and a range-azimuth bin in (R, θ) and (x, y) coordinate systems.

Figure 3.

Geometry of a prediction region, equation image centered on (equation image, equation image). Here rx2 = β2P11[k + 1∣k] and ry2 = β2P22[k + 1∣k] where Pij[k + 1∣k] is the i, j-th matrix element in P[k + 1∣k]. Minor axis length and rotation of the ellipse are affected by off diagonal term P12[k + 1∣k]. Ap covers the range-azimuth bin centered on (xp, yp). Fq is the echo footprint for an echo centered on some other bin, q. Owing to uniform sampling in angle with spacing Δθ and because Rp > R1, the area of bin Ap is greater than that of the bin centered on (x1, y1). Note that the sizes of Ap and Fq relative to �� are exaggerated to display geometry.

[22] Let fe(x, yequation image, equation image, P) be the conditional density over (x, y) for the echo footprint centroid point (i.e., the actual aircraft position) given parameters defining equation image. The probability the centroid is contained in some arbitrary 2-D patch A is then given by ∫∫Afe(x, yequation image, equation image, P) dxdy. We propose the following pdf mixture model involving a truncated 2-D Gaussian term,

equation image

The notation equation image indicates the complement of region �� such that Ω = equation image where ∪ is set union. The notation f (·) will be used throughout the paper to denote any pdf, and subscript e here identifies this as an echo centroid distribution over space. Figure 4 presents an example realization for this pdf.

Figure 4.

A typical realization of the two-dimensional Gaussian model for fe(x, yequation image, equation image, P) both inside and outside elliptical region ��. (xp, yp) is an arbitrary point inside ��. The x, y scale of this plot is centered relative to an arbitrary (equation image, equation image).

[23] User defined parameter σ controls how concentrated fe(x, yequation image, equation image, P) is around its central peak. The remaining terms in (4) are defined as

equation image
equation image

where ∣P∣ is the matrix determinant of P. Equation (5) was evaluated by substitution of (3) for equation image, using x = α ρ cos equation image and y = ρ sin equation image, setting (equation image, equation image) = (0, 0), and solving in polar coordinates to obtain (6). The probability the echo centroid is anywhere within �� is given by P(Hequation image) = ∫∫equation imagefe(x, y∣· dxdy = ∫∫equation imagedxdy + Pequation imagePequation image. The approximation is due to the Gaussian term in (4) dominating the expression. Outside of �� we assume each range-azimuth bin is equally likely to contain the echo centroid. This assumption is made to force a constant threshold, and thus conventional constant PFA performance outside of �� in the Bayesian detection framework introduced in the following section.

[24] The area of a bin is proportional to R, which necessitates that over equation image, fe(x, yequation image, equation image, P) must be proportional to 1/equation image = 1/R. Dividing by CΩ in (4) normalizes the integral to this bin area growth with R. Problems with division by zero are avoided since in every practical case Rmin ≫ 1. Note also that this definition insures that ∫∫Ωfe(x, yequation image, equation image, P)dxdy = 1 as required for a pdf.

[25] This model is a reasonable choice because it possesses the following necessary characteristics: (1) fe(x, yequation image, equation image, P) decreases monotonically with distance from the prediction point; (2) over �� it is a smooth function with no discontinuities; (3) the shape of fe(x, yequation image, equation image, P) depends on prediction error covariance, P[k + 1∣k], such that higher prediction error leads to lower probability density; and (4) there is a low, uniform probability across all range-azimuth bins outside the prediction region for presence of the echo centroid.

2.3. Calculating Prior Probability P(equation imagep)

[26] Given the echo centroid pdf model, fe(x, yequation image, equation image, P), it is possible to solve for the probability P(Hp) that the echo footprint intersects range-azimuth bin p. Let Ap be the 2-D patch which covers bin p and which has midpoint (xp, yp) = (Rp cosequation imagep, Rp sinθp). Uniform sampling in Rp and θp yields nonuniform spacing in (x, y), so the size and orientation of Ap depend on Rp and θp as shown in Figure 3, and must be considered when computing P(Hp). Define Fq as the footprint region for an echo with centroid at another arbitrary bin q. All bins q where Fq intersects Ap contribute to the probability P(Hp) of an echo in bin p. The probability that the echo centroid is in bin q is given by ∫∫Aqfe(x, yequation image, equation image, P) dxdy. Thus the probability that any portion of the echo footprint is contained in bin p is given by

equation image

where ϕ is the empty set and ∩ denotes intersection. Point (xp, yp) is at the approximate centroid of a region equation imagep defined as the area covered by the union of all Fq that satisfy ApFq ≠ ϕ, i.e., all Fq that overlap Ap.

[27] When equation imagep is small compared to equation image the local surface of fe(x, yequation image, equation image, P) around (xp, yp) is well approximated by a plane. We have found that equation imagep is usually smaller than �� in practice, due to the short range extent of echoes, and particularly when only the transmit beampattern mainlobe is considered to be part of the echo footprint Fq. Some very bright echoes have sidelobe extent in azimuth that could extend beyond �� (see Figure 1), but excluding these from Fq is reasonable since we do not wish to emphasize detection of sidelobes in any case. Since all bin patches Aq in this region are approximately the same size as Ap, and the average of a function of constant slope over a symmetric planar region is found at its centroid, (7) can be approximated as

equation image
equation image
equation image

where αp = ∫∫Apdxdy is the area of patch Ap and K is the number of bins, indexed by q, satisfying {qApFq ≠ ϕ}. The approximations are due to integrating over Ap rather than Aq and assuming fe(x, yequation image, equation image, P) is constant over small patch Ap in (8).

[28] Thus substituting (4) into (10) yields

equation image
equation image

[29] The specific pulse repetition and antenna sweep rates of the ARSR-3 radar as observed at the GBT, coupled with our baseband sample rate for b[n], lead to range-azimuth bin widths of ΔR ≈ 13.87 m and Δθ ≈ 0.0015 radians. This yields αp ≈ ΔθΔRRp = 0.0208Rp.

3. Bayesian Detector Algorithm Development

[30] This section presents details of the proposed improved detection algorithm which is based on a new optimal detection criterion, “constant total false alarm probability.”

3.1. Detection Statistic Model

[31] The goal of radar detection is to observe digital receiver output time series b[n] (equivalently bp) and make a decision at each time sample (equivalently range-azimuth bin p) whether an echo is present. For a single bin p, bp is a random variable with conditional amplitude pdfs fb(bpequation imagep) and fb(bpHp), given the absence or presence of an echo respectively. Note that while fe(x, y∣·) is a distribution over space, fb(bp∣·) describes an amplitude, or voltage distribution.

[32] Digital receiver output bp is formed as the sum of the squares of the real (in phase) and imaginary (quadrature) baseband matched filter channels, each of which contains independent additive Gaussian noise [Proakis and Salehi, 2002]. As such, bp consists of the sum of two squared Gaussian random variables, so bpη (where ση2 is the noise variance for both the real or imaginary channels) has a central chi-squared (χ22) distribution with two degrees of freedom under equation image, and noncentral χ22 under H as illustrated in Figure 5 [Trees, 2001b; Papoulis, 1984]. In many communications texts, vp = equation image is used as the detection statistic, and it is shown to have a Rice distribution,

equation image

where I0(·) is the zero-th order modified Bessel function of the first kind and noncentrality parameter s is proportional to echo signal power [Proakis and Salehi, 2002]. The following discussion uses bp as the detection statistic, but a parallel development could also be presented using vp by simply taking the square root of each side of every threshold test inequality.

Figure 5.

Comparison of detection statistics for the N-P and proposed detectors. Inside &#55349;&#56494; spatially varying τp is lower than conventional fixed τNP. This increases PD for the new algorithm since the integrated area beyond τp under the curve fb(bpHp) is larger than the area beyond τNP. However PTFAPFA because P(equation imagep) < 1.0. For this illustration P(equation imagep) = 0.4 and the signal to noise ratio at the matched filter output is 4.8 dB, which determines the noncentrality parameter s for fb(bpHp).

3.2. Related Detector Structures

[33] This section discusses some well-known detector architectures as a brief tutorial background and to motivate design choices for the new method. The intent is to show that the Bayesian constant PTFA detector introduced in equations (20) –(22) below can be interpreted as a natural extension of familiar concepts.

[34] The famous Neyman-Pearson (N-P) detector uses the following likelihood ratio test:

equation image

where the stacked inequality denotes “decide Hp if the left-hand side is greater than τ′NP, otherwise choose equation imagep.” Since the likelihood ratio is monotonic in b for these particular pdfs, (14) is equivalent to a simple direct thresholding of the receiver output:

equation image

where τNP is a function of the constant threshold τ′NP. The associated probability of false alarm (detecting an echo when none is present) is defined as

equation image

The usual approach for establishing the threshold level is to fix PFA to some suitably small constant and then solve (16) for τNP. Note that PFA is defined conditionally under equation image and that the integrand comes from the denominator of the left-hand side of (14).

[35] The simple NP test does not provide a mechanism for exploiting prior probability information obtained from the tracker. One possible alternative is the Bayes decision criterion [Van Trees, 2001a; Scharf, 1991],

equation image

where constants CI,J for I, J ∈ {equation image, H} are costs (also called Bayes loss) associated with deciding I when J is true, and C = {Cequation image, Cequation image,H, CH,equation image, CH,H}. Unfortunately, (17) has some disadvantages. We do not have a satisfactory systematic approach to specify the costs. Also it is not clear how to frame this in the desired direct thresholding form of (15).

[36] To illustrate the difficulty, note that the left hand side of (17) is still monotonic in bp and that τ(C) is a constant (though dependent on C). This suggests that a corresponding threshold test comparable to (14) would be

equation image

Now threshold τ(p, C) depends on position p since P(equation imagep) and P(Hp) are spatially varying. In this Bayesian framework the direct threshold form corresponding to (15) is found by defining the likelihood ratio function L(b) = fb(bHp)/fb(bequation imagep) and operating on the left and right-hand sides of (18) with the functional inverse L−1(·) to obtain

equation image

The problem with this formulation is that although it is in fact the desired direct test on receiver output sample bp, we do not know how to evaluate threshold τ′(p, C). In a typical scenario all the CI,J are unknown. Also L−1(·) depends on fb(τ(p, C)∣Hp) which unlike fb(τ(p, C)∣equation imagep) is hard to model because aircraft echoes have unknown random intensities. One cannot readily find the appropriate threshold value because we cannot use the constant PFA constraint of (16) to avoid these complexities and still satisfy the desired Bayesian relationships of (17).

3.3. Constant PTFA Detection

[37] We propose an alternative Bayesian framework for improved detection performance to address the problems mentioned above, and which is based on a physically meaningful criterion related to specifying a fixed PFA. We define the unconditional “total probability of false alarm” (PTFA) as the joint probability that a spatially varying threshold, τp, was crossed and no echo was present,

equation image

This is the total probability (i.e., not conditioned on H or equation image) that a false alarm will occur at range-azimuth bin p given some τp.

[38] Equation (20) provides a means to solve for an appropriate threshold for a direct test in the form of (15) or (19), but which incorporates the prior information P(equation imagep) drawn from the Kalman track history through equations (11) and (12). Note that τp ≠ τ′(p, C) ≠ τNP so the detector is neither N-P, nor classical Bayes decision criterion, but depends on the Bayes relationship used in (20).

[39] Spatially varying τp is found by specifying a fixed value for the acceptable PTFA and solving (20) for τp in closed form,

equation image

where PTFA is specified by the user to be some small constant, P(equation imagep) is given by (11) and (12), and 2ση2 is the complex noise variance for b[n].

[40] This is not the same test as in (17). One obvious distinction is that costs Ci,j are not included since the threshold is set by solving a PTFA equation. Some insight is provided by noting that (20) uses the left-hand side denominator of (17) as an integrand, just as (16) integrates the denominator of (14). Also, we propose that constant PTFA detection is a rational solution given that the expected total number of false alarms will be known over the range azimuth map. This is set as a design parameter. On the other hand, a fixed PFA detector will have varying numbers of false alarms as P(equation image) varies. The computational load for a radar detection system is proportional to the number of threshold crossings, so being able to fix the average number of false alarms can be a useful constraint on system capacity.

[41] Summarizing, the constant PTFA detector (CTFA) is computed with the following steps: (1) Set PTFA = a small constant. (2) Evaluate (21) to find the spatially varying τp at each range-azimuth bin, b(xp, yp). (3) Threshold each bin,

equation image

The effect of this approach is that the threshold is lowered in the prediction regions &#55349;&#56494; where based on prior track history it is known that echoes are more likely. This leads to more reliable detection of weak aircraft echoes. By proper selection of parameters PTFA, β, and σ, the probability of detection PD = ∫τpfb(bpHp) is increased while false alarms are kept low, as shown in Figure 5. Figure 6 illustrates the variable threshold τp plotted as a function of x and y for two prediction regions from two tracks. Note that the lower quality track has a wider prediction region (right), leading to a more shallow threshold depression. This corresponds to greater uncertainty about the next echo location, so less emphasis on detection is generated here. Note, however, that in both cases the threshold reduction is relatively small (less than 0.5% in this example) as a natural consequence of the way the constant PTFA detector is defined. However, since it is applied over many bins, even such a small change in threshold can have a significant effect on detection statistics, improving the probability of finding a threshold crossing in at least one bin of the echo footprint. This is demonstrated in results presented in the following section. With this small threshold, depression PD is increased while avoiding the potential problem of forcing a detection in &#55349;&#56494; whether an echo is present or not.

Figure 6.

Spatially varying detection threshold τp = τ(xp, yp). In this example, Ω contains two prediction regions equation image from two aircraft tracks. Prediction points (equation imagep, equation imagep) correspond to the two local minima. Prior knowledge that echoes are likely over the two &#55349;&#56494; ellipses leads to lowered threshold levels, and thus the probability of detection is increased. For this plot Rmin = 104 km, Rmax = 239 km, θmin = equation imageπ, θmax = equation imageπ, σ = 0.2, PTFA = 1.0 × 10−4, ση2 = 0.001, rx = 30 km for both prediction regions, and ry = 15 km and 45 km, respectively, for the left and right regions. These parameters are typical for GBT data except that rx and ry have been increased by a factor of 10, and PTFA by 100, to make detail in the plot easy to see.

[42] As noted earlier, there are often multiple concurrent aircraft tracks in Ω, each with a prediction region and associated computed threshold levels τp. This potential conflict is handled as follows. Let τp,j be the threshold computed for the pth range-azimuth bin using &#55349;&#56494;j from the jth track. A single threshold test is made for each bin as in (22) where

equation image

This method was used in Figure 6. The set of threshold crossings is then processed with the CLEAN algorithm [Högbom, 1974] to collapse the sidelobe structure to a single point, and each cleaned detection is assigned to an existing or new track following arbitration rules described by Dong et al. [2005a].

4. Experimental Results

4.1. Performance Evaluation With Simulated Data

[43] It is very difficult to quantitatively characterize detection performance using real-world data because relevant signal properties such as signal to noise ratio (SNR) and absence or presence of an echo are unknown. We have developed a realistic echo synthesis code to generate data used in Monte Carlo simulation trials to estimate detection properties such as probability of detection (PD) and probability of false alarm (PFA) as functions of SNR and aircraft motion dynamics. The new Bayesian detector is shown to improve PD for a fixed PFA as compared with the conventional approach, thus validating the strategy of detector design based on constant PTFA.

[44] The simulator generates a time sequence b[n] of receiver outputs with the following properties:

[45] 1. Receiver sensitivity, IF bandwidth, data sample rate, and antenna sidelobe gain response all approximately match those used to collect radar data at the GBT.

[46] 2. Transmitter pulse length, pulse shape, pulse repetition rate, antenna sweep rate, and transmit beamwidth approximate those of the ARSR-3 radar system as discussed by Fisher [2001a, 2001b]. Dong [2004], and Dong et al. [2005a].

[47] 3. Additive white Gaussian noise (AWGN) is used with variance ση2 matching sample variance estimated from real radar-echo-free GBT data.

[48] 4. In order to eliminate problems in associating new detections with their corresponding tracks, there is only one simulated echo track in each randomly generated trial observation.

[49] 5. Realistic aircraft motion is simulated using the Kalman state space model with acceleration a(k) generated as a low-pass filtered 2-D Gaussian i.i.d. random noise process. The low-pass corner frequency is adjusted to produce smoothly connected random maneuvers with turn radii consistent with commercial aircraft.

[50] 6. The pulse echo amplitude follows a Swirling IV distribution model suitable for a square law receiver [Mahafza, 2000]. Received pulses have constant mean amplitude throughout an entire scan, but are uncorrelated from pulse to pulse. The echo amplitude pdf is given by

equation image

where subscript Σ denotes the Swirling pdf, σ is the random radar cross section (RCS) value, and σav is the average RCS over all target fluctuations.

[51] A Monte-Carlo simulation was used with repeated random trials to evaluate PD vs. PFA in order to estimate the receiver operating characteristic (ROC). When comparing the new Bayesian algorithm with a conventional detector, one must carefully consider the differences in how false alarms arise. As defined in (16), PFA only depends on the noise power and a threshold τNP with fixed value over the entire range-azimuth map. However, the new algorithm improves PD at the cost of a slight local false alarm rate increase inside &#55349;&#56494;. To be fair, a comparison of PD between algorithms must be made when each has the same effective total PFA value. For Figure 7 the “Probability of False Alarm” values along the horizontal axis represent actual observed error counts across a series of range-azimuth map random realizations, divided by the number of trials and the number of bins (pixels) in a map. In order to calculate sample statistics in the realistic PFA regime between 10−8 and 10−6 it was necessary to run many thousands of random trials to observe enough of the rare false alarms to keep error variance low. A family of ROC curves is presented to demonstrate the superior performance of the new algorithm. Detection probabilities are uniformly higher for the Bayesian detector when both algorithms are producing the same number of false alarms.

Figure 7.

Detection performance comparison for the “new” Bayesian constant PTFA method and “old” conventional N-P constant PFA approach. PFA versus PD ROC curves are shown for a range of SNR values. PFA values for both algorithms are computed as the ratio of observed false alarms to the number of trials times the number of bins in Ω.

4.2. Performance With Real Data

[52] Two sets of real data were recorded at the GBT and used to test performance of the echo detection algorithm, tracking, and blanking. Set one consists of five 12-second-long data windows recorded at 1-minute intervals on 5 April 2002. Set two contains a continuous block of 10 min of data (50 radar antenna rotations) recorded in January 2003. The Kalman tracker and both the conventional N-P and new Bayesian detectors were run on all data in both sets. A more extensive presentation of these results is given by Dong [2004].

[53] In data set one, there are five snapshots containing eight distinct aircraft tracks. The Bayesian detector found three echoes that were missed by the conventional method. A missed detection was declared when no conventional threshold crossing occurred in a track's prediction region at snapshot t[k + 1], but was reacquired in snapshot t[k + 2]; that is, the tracker had to make a two-step prediction update to keep the track alive. The reacquisition at t[k + 2] implies that the aircraft echo was present in &#55349;&#56494; at t[k + 1], but was just too weak to detect. In total, there were 33 track associated detections with the conventional algorithm, and 36 with the new detector. No “false alarm” detections (i.e., new threshold crossings not associated with a track) arose with the new method.

[54] In data set two, there were four missed detections recovered by the Bayesian algorithm over 15 snapshots with six active tracks as shown in Figure 8. One more possible false alarm was made by the Bayesian detector than with the N-P approach. For this experiment both PFA and PTFA were set to 3.87 × 10−8 when calculating threshold settings.

Figure 8.

All detections for the even-numbered antenna sweeps in data set 2. Note that the conventional N-P detector missed four echoes (marked by crosses without circles) found by the new Bayesian method. This represents an 9% increase in finding true echoes when using Bayesian detection. The new detector added one false alarm, marked with a triangle. PTFA = PFA = 3.87 × 10−8.

[55] On the other hand, Figure 9 illustrates what occurs when the N-P detection threshold is lowered (by increasing the design PFA) in an attempt to recover the lost echoes which were found by the Bayesian detector. When PFA is raised from 3.87 × 10−8 to 2.32 × 10−7, two of the four missing detections were found, but four false alarms were added. In order to detect the remaining two echoes, PFA had to be raised so high that hundreds of false alarms appeared (not shown in Figure 9). Clearly the Bayesian approach provides better false alarm management while improving weak echo detection.

Figure 9.

Detection comparison when the N-P threshold is raised in an attempt to locate some missed echoes that were found by the Bayesian detector. Boxed detections are new correct ones. Triangles indicate new N-P false alarms. PTFA = 3.87 × 10−8, and PFA = 2.32 × 10−7.

[56] To illustrate the potential scientific observational impact of blanking the previously missed echoes which are now caught by the Bayesian algorithm we processed a segment of data from data set two. This window from 187.2 to 188.4 s in the recording produced no aircraft detections using the conventional algorithm, but the new method found three. Figure 10 shows power spectral estimates accumulated over the 1.2-s window which contained ground terrain echo clutter. The upper curve includes no radar pulse blanking and exhibits a dominant spectral peak around 5.5 MHz caused by ground clutter. The lower curve shows the spectrum after fixed time window blanking synchronized to the transmit antenna sweep. This corresponds to excising all data between a delay of 0 and 150 μs in Figure 1. At this plotting scale all RFI appears to have been eliminated. All blanking was implemented by “zero-stuffing” [Zhang et al., 2003], that is, placing zeros into the time samples where radar interference is detected.

Figure 10.

Sample power spectra at the GBT showing radar RFI near 5.0 MHz. The upper curve includes no pulse blanking. The lower curve has only blanking for fixed ground echoes.

[57] However, at the expanded scale of Figure 11, a difference is seen between spectral estimates using the new Bayesian detection as compared with conventional N-P detection. The Kalman track was established from five previous antenna sweeps, which occur at 12-s intervals. The three weak aircraft echoes which were detected with the new algorithm were blanked, and the resulting lower curve spectrum shows reduced bias near 5.5 MHz corresponding to the radar pulse center frequency.

Figure 11.

Spectrum for the same 1.2-s data set seen in Figure 10, but using Kalman tracking and detected pulse blanking. Five previous antenna sweeps were used to form the track history. The upper curve resulted from conventional detection methods. The lower curve used the new Bayesian detector. Three additional aircraft were detected in this snapshot which had been missed by the conventional detector. The lower curve shows the resulting lower spectral levels in the radar RFI region near 5.0 MHz.

5. Conclusions

[58] The foregoing analysis has demonstrated the potential for a Kalman tracker based Bayesian detector to provide high performance pulsed interference removal from radio astronomical data. Even when resulting spectral estimates are improved only subtly, it is desirable in radio astronomy to remove every interference component possible to increase confidence in the affected scientific observation. These radar systems are not going away, and as interest increases for deep space observation in the radar band allocations, such a system will become increasingly necessary.

[59] Though the mathematical development presented here appears fairly complex, computer implementation of the tracker-constant PTFA detector is quite straightforward. Computational burdens are small because track state and threshold computation updates need be made only once per 12-s antenna sweep. Thus, aside from the digital receiver functions, only modest computational resources are required as long as a digitized signal stream is available. The digital radar receiver must operate at RF sample rates, but such systems are in widespread use. Any system used to blank pulses after detecting them (e.g., using N-P detection) must use the same type digital radar receiver front end. Therefore, by comparison, the additional computational burden to add Kalman tracking and constant PTFA detection could be handled by the slowest of PCs.

[60] This study suggests that the Kalman tracker Bayesian detector system is a good candidate for real-time operation and permanent implementation at the observatory. Plans are currently under consideration to pursue funding for such an obvious next step.

[61] However, even without investing in significant real-time processing resources the proposed system can be used in a post processing mode on virtually any extended prerecorded data set containing pulsed radar RFI. In a post-processing scenario it would also be possible to adapt the proposed approach to operate both backward and forward in time. One can envision a forward-backward two-pass echo Kalman tracking scheme and a Bayesian detector which draws on the entire data set (rather than just past detections) for prior probability inference. This should offer additional detection performance gains, particularly at the beginning of an aircraft track where start-up errors are large. As described above, two successive N-P detections are required before a track can be initiated. Using backward prediction it would be possible to find weak echoes which did not cross the N-P threshold and which occurred before the causal track was created. Future research plans include developing this bidirectional tracking detector. Block estimation methods that do not rely on sequential processing, either forward or backward, can also be investigated.

Acknowledgments

[62] This work was supported by the National Science Foundation grant AST-9987339.

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