Adaptive Beamformers in Communications and Direction Finding Systems

This paper elaborates on a technique described in [1] for simultaneously receiving and estimating the directions-of-arrival of multiple cochannel signals impinging on an array of sensors. The method described therein used a combination of adaptive CM Array beamformers and signal cancellers to sequentially lock onto and remove emitters one-by-one from the sensor signals. Inherent to the signal cancellation process was the generation of a direction vector associated with the signal being canceled. In this paper, results of the above method applied to digitized snapshot data are presented. Unexpected performance degradations observed in these results are explained and two remedies proposed. One remedy involves the addition of a parallel arrangement of CM Arrays whose weight vectors are initialized based on the cascaded array weight vectors and signal canceller vectors. The other remedy involves the modification of the error signals used to adapt the signal cancellers.

Introduction

As a result of frequency reusage and spectral crowding, adaptive beamformers are becoming an increasingly important part of communication and surveillance systems. As shown in Figure 1, the objective of the adaptive beamformer in these systems is to copy or receive one (or more) of the M cochannel signals impinging on an array of sensors. This reception is accomplished by forming a linear combination of the sensor output signals in such a way that nulls are placed in the directions of all but one of the cochannel emitters. Simultaneous to providing signal copy (reception), the beamformer also performs direction finding (DF) whereby it estimates the direction-of-arrival (DOA) associated with the beamformer output signal. The beamformer copy weights and DOA estimates are both determined by a processor that has access to the beamformer output and the individual sensor outputs. The processing can be done on either a snapshot or sample-by-sample basis.

Figure 1. Generic adaptive beamformer with integrated DF capability

Two distinct approaches to the simultaneous DF and copy problem can be envisioned. The conventional approach is to first determine DOA estimates for all of the cochannel emitters, then use these estimates to generate beamformer copy weights [2]. The DOA estimation could be implemented with any one of a number of the popular “superresolution” techniques, e.g. Maximum Entropy (ME), Maximum Likelihood (ML), MUSIC, or variations thereof [3]. Given a DOA estimate and associated steering vector, the copy weights are then determined using constrained adaptation as in a Frost (or Applebaum) beamformer, or the equivalent snapshot-based ML solution. The main problem with these DOA-based copy approaches is that the copy performance is highly sensitive to errors in the DOA (steering vector) estimates [4, Sec 6.3]. Since both the DOA estimate and the steering vector are found by searching an array calibration table, the copy performance is highly dependent upon the accuracy of the calibration data. A secondary problem with the above approaches is their sensitivity to correlated sources such as arise in a multipath environment.

The alternative approach, advocated in this paper, is to first adaptively generate the beamformer copy weights, then use the beamformer output to estimate the DOA associated with the received signal. The beamformer weights can be determined using self-generated reference signal techniques [4, Ch 7] such as the constant modulus (CM) array [5, 6], the multi-mode decision-directed array [7], or the hybrid regenerative array [8]. An estimate of the direction vector associated with the beamformer output signal is then determined by cross-correlating the beamformer output with the sensor signals. Array calibration is only required to generate the DOA estimate associated with the derived direction vector. Also, the copy performance of these techniques is generally immune to the SINR degradations commonly associated with multipath-induced correlated sources.

This paper is organized into the following sections.

The concept of reference signal use in an adaptive antenna system was first introduced by Widrow in [9] where he described several pilot-signal generation techniques. One of the proposed techniques used a two mode adaptation process whereby the transmitter alternated between sending a known pilot signal and actual data. The receiver had knowledge of the pilot signal and used it as the desired response for the LMS algorithm. During actual data transmission, adaptation would be switched off and the weights would coast until the pilot signal was turned back on. While an adaptive antenna utilizing this technique was probably never constructed, the concept provided the necessary seed which eventually grew into actual implementations.

As described by Compton in [4, Ch 7], the adaptive array reference signal need not be an exact replica of the desired signal; it only needs to be correlated with the desired signal and uncorrelated with the interference. Compton goes on to describe several experimental adaptive antenna systems designed for use with spread spectrum signals where the spreading sequence provided the necessary discriminate between desired signal and interference.

In this same vein, a method of reference signal generation for use with signals possessing a constant modulus was described in [5,6]. The CM Array as it was called, employed the constant modulus adaptive algorithm (CMA) [10] for weight adaptation. The reference signal used by the CM Array consisted of a normalized version of the complex array output signal. In [11], it was shown that the CM Array converges to the desired solution for a variety of constant and non-constant modulus signals as well as in the presence of correlated sources [6]. In [7], a multimode version of the CM Array was developed for application to QAM data signals. In this multimode beamformer, weight adaptation initially begins using constrained output power minimization subject to a “soft” look-direction constraint, then switches into CMA adaptation, and finally ends with decision-directed adaptation to provide rapid tracking of time-variable conditions. Recently, an approach similar to the multimode CM Array has been developed and is referred to as a regenerative hybrid array [8].

In [12], an experimental two-element CM Array designed for wideband microwave communications was described along with field test results. CM Array processing results using real snapshot data from an airborne 5-element array are presented below. Figure 2(a) shows the power spectrum of the signal from one of the array antennas. The limited resolution of this spectrum is a result of the short nature of the snapshot data; 400 time samples per sensor. The signal environment consisted of three equi-powered frequency modulated (FM) signals with slightly offset carrier frequencies. Figure 2(b) shows the “beampattern” of the converged CM Array. Data recycling and gear-shifting was used to achieve convergence due to the limited snapshot size. The term beampattern is used in quotes since the points plotted are actually the dot product of the CM Array weight vector with normalized array calibration vectors and can only be loosely interpreted as a beampattern. In any case, notice that nulls have been placed in the directions of emitters #2 and #3. The CM Array output signal spectrum is shown in Figure 2(c). Based upon the distinct carrier frequencies of the three emitters, it can be seen that the CM Array output consists mainly of emitter #1. Additionally, the CM error variance associated with the converged array was indicative of near-perfect convergence.

Figure 2. CM Array copy and DF performance for real data: (a) beamformer input, (b) converged “beampattern,” (c) beamformer output, (d) direction vector “beampattern.”

The underlying principal behind DOA estimation is illustrated in Figure 3 where a planar wavefront is shown impinging upon a two-element array. If the wavefront was parallel to the array baseline, then the signal received by both sensors would be identical. However if the wavefront arrives from a nonzero angle

as shown, then the signal output from one sensor will be delayed with respect to the signal output from the other. The delay

will be proportional to the distance d, with the proportionality constant determined by the speed of propagation. The sine of the incident angle is equal to the ratio of the distance d and the baseline separation. When the bandwidth of the signal is small compared to the inverse of

, the difference between the signals received by the two sensors will be predominantly a phase shift of the signal’s carrier. Thus, by measuring the relative phase shift between the sensor outputs, the angle-of-arrival

can be easily calculated. In precision DF systems, an array calibration table is typically used to map the measured phase difference into an angle-of-arrival estimate. The table is constructed using a test emitter which is positioned at various angles around the array.

Figure 3. DF based on interferometric techniques.

If the sensor output signals are assumed to be complex, then the phase difference can be easily measured by cross-correlating the two signals. The phase angle of the zeroeth lag of the complex cross-correlation will be the phase difference. The following deterministic interpretation of this measurement process is enlightening when dealing with short data snapshots: The two complex sensor signals are assumed to be different in phase, and possibly also in magnitude, due to mismatches between the two sensor channels. If only one sample of each signal was available, then an estimate of the phase difference could be derived from the product of the conjugate of one signal sample times the other. With more than one sample, the problem can be interpreted as that of finding the complex weight which minimizes the sum of the squared errors between the weighted signal and the unweighted signal. This least-squares approach gives the same solution as the short-term cross-correlation. The importance of this deterministic interpretation will become evident shortly.

When there is more than one wavefront impinging on the array, the above cross-correlation approach obviously breaks down. Figure 4 shows the proposed approach to direction finding in a cochannel environment. The CM Array is used to lock onto one of the cochannel signals impinging on the array. The output of the CM Array beamformer is then cross-correlated with the sensor signals. Assuming the emitters are uncorrelated with one another, the contributions to the cross-correlation will be zero for all but the emitter acquired by the CM Array. Thus, this cross-correlation is identical to that which would have been generated with only the one emitter present. The cross-correlation vector is often referred to as the direction vector since it contains information concerning the DOA of the emitter. Again, a calibration table is searched to determine the angle-of-arrival associated with the direction vector.

Figure 4. DF in a cochannel environment.

The above procedure was applied to the snapshot data from the 5-element airborne array. The result is shown in Figure 2(d). Each point in this plot is the dot product of the estimated direction (cross-correlation) vector with one vector from the array calibration table. The calibration table had 7 entries for angles between –100 and –120 degrees. The dot product achieved its maximum for the entry associated with –104 degrees. No attempt was made to interpolate the calibration data, however it is simple to do so in the one-dimensional case. The actual emitter angle was –104 degrees, indicating near-perfect performance of the DF approach.

In view of the deterministic interpretation of the cross-correlation process given above, it is easy to understand why there could be a larger error in a direction vector estimated when multiple signals are present relative to the case when there is only one signal present. The additional signals are perceived as noise in the least-squares fitting process and consequently longer data snapshots would be required to achieve the same level of accuracy as if they had not been present.

Assume for the moment, that multiple CM Arrays could be used to lock onto (isolate) all of the emitters present. If such were the case, then an improved method of estimating the direction vectors might be as follows: Feed the output of each CM array (containing an isolated emitter) into a complex linear combiner. Form the difference between the linear combiner output and a sensor signal. Now choose the combiner weighting that minimizes this difference over the length of the snapshot. Perform this process for each sensor signal. The only noise in the above estimation process is that due to signals uncorrelated with the emitters, e.g. thermal noise. If the thermal noise is assumed to be zero (or negligible), then only M sensors (where M is the number of emitters) would be required to perfectly estimate the M direction vectors. Obviously the difficulty with this approach surrounds the assumption that the emitters can be isolated using multiple CM Arrays. This is the subject addressed in the next section.

To this point, we have purposefully not brought up the fact that the CM Array can capture any one of the cochannel signals impinging on the array. A solution that steers a beam in the direction of one signal and nulls in the directions of the others will be a stationary point of the adaptive algorithm. Which emitter the CM Array captures depends upon their relative strengths as well as the initial array weight vector. For more details on the capture performance of the CM Array, the reader is referred to references [5,6]. In this section, we concentrate on the problem of simultaneously copying each of the cochannel emitters.

One potential solution to this problem is to use several CM Arrays operating in parallel. Obviously if the arrays were all initialized with the same weight vector, they would all capture the same emitter. To avoid identical capture, it might be possible to let the arrays converge one at a time, using knowledge of the weight vector and output signal of the converged array to generate a new weight vector which has a null in the direction of the captured signal that could then be used to initialize the next array. This might be workable for a limited set of scenarios, but cases would undoubtedly arise where two arrays would capture the same signal.

As proposed in [1], a better solution to the capture problem is to operate the CM Arrays in the cascade architecture shown in Figure 5. Each stage captures one of the signals present at its input and produces a multichannel output with the captured signal removed. Cancellation of the captured signal from each input channel is accomplished with a single weight LMS canceller—a process tantamount to the least-squares fitting process described earlier in this paper. Interestingly, the vector of canceller weights is identical to the cross-correlation or direction vector discussed earlier. Thus the generation of a direction vector associated with the cancelled signal is inherent to the stage-by-stage cancellation process.

Figure 5. Cascaded CM Array architecture.

Figure 6 shows the direction finding results for the cascaded CM Arrays applied to snapshot data taken from an airborne array. The array was comprised of the same 5 sensors used in Figure 2. The signal environment consisted of three cochannel FM emitters where emitters #1 and #3 were of equal power and emitter #2 was 10 dB weaker. The plot shows actual and estimated bearings for each of the emitters as a function of flight time. The true bearings were derived from on-board telemetry. The estimated bearings were derived from the signal canceller vectors using the procedure associated with Figure 2(d). Notice the near-perfect tracking provided by the cascaded CM Array architecture, even when emitters #2 and #3 are separated by less than 5 degrees.

Figure 6. Direction finding results for real flight data.

The copy (SINR) performance of the cascaded CM Array architecture was also assessed for the period of time shown in Figure 6. Though the copy weights were determined from time slots in which all three emitters were transmitting, they could be checked against data in which only one of the emitters was transmitting at a time, to calculate the achieved SINR during the three emitter time slots. This is because the emitters were being switched on and off at a rapid rate (relative to the changing array-emitter geometry) to provide a check of the three emitter copy results.

An unexpected result was observed. The processed SINR for emitters #1 and #3 was expected to be nearly equivalent since they were transmitted with the same power and were nearly equidistant from the array. However, it was discovered that the SINR of the signal captured by the first CM Array stage was 5 to 10 dB greater than the second stage. The first stage could be made to capture either emitter #1 or emitter #3 by altering its initial weight vector. When doing this, the first stage consistently exhibited the higher SINR regardless of which signal was captured.

The reason for the degraded copy performance described above is thought to be attributed to imperfect cancellation of the captured signal in the first stage. The cancellation process can be interpreted as creating an artificial emitter perfectly correlated with the captured signal and originating from the same location, but 180 degrees out of phase. If the cancellation process is perfect, then the captured emitter should be annihilated. However if there is an error associated with the cancellation vector, then the cancelling process will actually generate an additional emitter, correlated with the captured emitter but originating from a slightly different direction. This places an additional burden on the second stage CM Array. It now needs to null out two emitters, highly correlated and originating from nearly the same direction. The second stage CM Array is unable to do this with only 400 samples of snapshot data. Regardless of the amount of data recycling and gear shifting, the second stage CM Array performance could not be improved upon.

The architecture shown in Figure 7 was used to improve the degraded copy performance explained above. This architecture consists of the cascade CM Array structure of Figure 5 augmented with a set of parallel CM Arrays. The parallel CM Array weight vectors are initialized with the effective weight vector associated with each stage of the cascaded architecture. The effective weight vector for a particular stage is determined by multiplying the CM Array weight vector of that stage by the transformation matrix associated with the previous stages. This generates a weight vector which has the appropriate gain in the direction of the captured emitter and nulls in the directions of all other emitters.

Figure 7. Combined series/parallel CM Array architecture.

A new set of copy results was generated using the system in Figure 7 for the flight corresponding to Figure 6. The results are presented in Figure 8. Unlike the copy results associated with the cascade architecture, the SINR performance of emitters #1 and #3 are roughly the same, and the SINR of emitter #2 is 10 dB lower. This would be expected from the relative transmitter powers. The parallel CM Arrays have done a near-perfect job of eliminating the cochannel interference.

Figure 8. Copy results for real flight data.

An alternative architecture for improving the copy results is shown in Figure 9. Recall the cross-correlation discussion of the previous section. The error in a short-term estimate of the cross-correlation (canceller weight) is related to the power of the signal uncorrelated with the captured signal, i.e. the canceller error power. In the first canceller stage, the error power is comprised of all other emitters and is consequently large. In the last canceller stage however, all emitters should have been removed so that the error power should be small. If we assume that each CM Array has captured one of the emitters, then we should be able to feed back the final canceller error signal to all previous canceller weights. This is indicated by the up position of the switch in Figure 9. Upon convergence of a stage, the canceller error of the previous stage is augmented by throwing the switch. As described in the previous section, use of the augmented error should reduce the amount of data required to achieve a given level of canceller precision and consequently improve the copy results.

Figure 9. Canceller adaptation based upon joint error.

This paper has demonstrated that adaptive beamforming with self-generated reference signals provides a viable method of cochannel signal separation without array calibration. In addition, it has demonstrated a simple yet effective method of direction finding based upon the copy beamformer output.

Further investigation is warranted in the following areas: (1) performance on short duration signals, (2) investigation of the convergence and stability properties of the joint canceller adaptation method, and (3) DF performance in the face of multipath-induced correlated sources.