Joint Spatial and Temporal Equalization in a Decision-Directed

Joint Spatial and Temporal Equalization in a Decision-Directed Adaptive Antenna System

Twenty-second Annual Asilomar Conference on Signals, Systems, and Concepts
October 31–November 2, 1988

This paper describes two methods for jointly optimizing the weights of a single-weight per channel adaptive diversity combiner followed by a fractionally-spaced equalizer. It is shown that such a system provides an efficient and effective means of mitigating multipath distortion and cochannel interference for a wide variety of digital communication applications. Both “blind” and decision-directed weight-update algorithms are given.

Space diversity combining* and adaptive equalization (used alone or in concert) have long been employed to mitigate the effects of propagation-induced distortion and cochannel interference on digital communication links. However, the desire for higher spectral efficiencies coupled with the advancement in digital signal processing and receiver technology, have caused new equalization and diversity techniques to be continually developed and refined. For instance, NTT Labs has recently field tested a 256-QAM digital microwave radio that required improved equalization and diversity techniques over those used in their 64-QAM radios [1].

The evolution of diversity and equalization techniques will also be evidenced in next-generation cellular radio systems where the combination of frequency reusage and digital modulation cause an increased sensitivity to cochannel interference. Even more complex distortion and interference countermeasures will be required to solve the problems encountered in developing direct digital broadcast as an alternative to commercial FM.

To date, adaptation of the diversity combiner used in digital microwave radio systems has always been performed independent of the baseband equalizer and demodulator. While several papers have discussed the subject of combiner weight control using the constant modulus algorithm or decision-directed LMS algorithm [2–5], these discussions have not made clear the concept of joint optimization of the combiner and equalizer. In this paper we illustrate why independent optimization can be grossly suboptimal compared to joint optimization. We also investigate two methods for jointly adapting the combiner and equalizer weights along with their associated strengths and weaknesses. A blind acquisition mode for each method is described and simulated.

Refer to Figure 1 for a block diagram of the structure under consideration. It consists of an M-channel antenna combiner with a single complex weight per channel followed by an N-weight FIR equalization filter. While not shown in this figure, it is assumed that the signals from each antenna have been filtered, converted to analytic signals, and sampled at exactly twice the symbol rate of the signal of interest (SOI). It should be pointed out that the signals have not been matched filtered nor has any attempt been made to make the sampling phase synchronous with maximum eye opening. Through multiplication by the complex combiner weights, the signal on each channel is phase and amplitude shifted and then summed to form the combiner output. The function of the antenna combiner is to spatially null cochannel interference and coherently combine (or null) spatially diverse multipath arrivals.

Figure 1. Combiner-equalizer structure under consideration.

It is important to point out that spatial equalization alone (performed by the antenna combiner) is generally not sufficient to permit low error rate demodulation of high-alphabet modulations such as 64 and 256-QAM. The output of the combiner must be further processed by an N-weight fractionally-spaced (T/2) equalizer. While the main objective of the equalizer is to compensate multipath and other linear distortion that the antenna combiner could not (or did not) compensate, it has two secondary functions. First, it optimizes its group delay characteristic so that the output signal (which has been decimated by two) is sampled at maximum eye opening. And second, it forms the optimal matched filter that eliminates adjacent channel interference.

The more conventional alternative to the structure shown in Figure 1 is one that incorporates an N-weight FIR filter on each channel. Obviously this alternative could accomplish the same signal conditioning as the combiner-equalizer structure under consideration, but with considerably more complexity (e.g., M×N/2 multiplies per sampling period as opposed to M+N/2). The natural question arises as to whether the more complex structure is warranted. The answer to this depends upon the degree of decorrelation between the signals on each channel. That is, are the signals simply phase shifted replicas of one another or are there frequency-selective differences between them? The answer to this question depends upon a variety of circumstances. For instance: What is the relative interference bandwidth? What is the spacing between antenna elements? Is the array in a near-field geometry? Is there frequency-selective channel mismatch in the antennas or receivers? And most importantly, what are the required null depths?

While most of the above effects can be reduced through fixed equalization filters and careful implementation, the effect of element spacing cannot. For a two-element array, it can easily be shown that the null gain is directly related to the square of the product of the relative interference bandwidth times the element spacing in wavelengths. Table 1 shows the maximum permissible spacing to achieve a 40 dB null on signals with the relative bandwidths indicated. Notice that the element spacing can be greater than 200 wavelengths for signals with less than 1% relative bandwidth. Since most signals encountered in practice (with the exception of spread spectrum signals) have relative bandwidths less than 1%, the use of a single independent weight per channel generally will not limit the nulling performance of the combiner, at least from the standpoint of element spacing. For example, digital microwave radio, digital cellular (GSM), and broadcast FM signals all have relative bandwidths of less than 0.4%.

Table 1. Effect of relative bandwidth on maximum element spacing.

	REL BW (BW/fc)	Maximum Spacing (Wavelengths)
	0%
	1%	200
	2%	100
	3%	70
	4%	50
	5%	40

It should also be pointed out that while there may be applications for which more than one independent weight per channel is warranted, there will almost always be a performance/complexity advantage of following the antenna combiner with a multi-weight equalization filter.

The purpose of the previous section was to motivate the performance/complexity advantage of the combiner-equalizer structure shown in Figure 1. In this section we address the issues related to joint adaptation of the weights of this system.

The foremost disadvantage of independent optimization of the combiner and equalizer is that it forces the combiner adaptation to be performed prior to symbol detection (pre-d) since accurate symbol decisions require assistance from the adaptive equalizer. While there are many known methods of pre-d combiner adaptation (e.g., constrained minimum output power, maximum power, maximum ratio, minimum dispersion, minimum envelop variation), these techniques can be grossly suboptimal due to the fact that their optimization criteria are not directly related to the SNR of the system output. For instance, the commonly-employed minimum dispersion combiner finds the combiner weights that result in the whitest output spectrum. This has a major deficiency; it is insensitive to uncorrelated interference such as a long-delay multipath arrival. Ideally, the combiner should null long-delay multipath. However, a minimum dispersion combiner may in fact place gain in the direction of a decorrelated multipath arrival. Similar deficiencies can be found in most other pre-d combiner techniques.

A better approach is to use the decision outputs from the equalizer as a desired response for the combiner. That is, use the variance in the error between the equalized decision output and the combiner output as the performance criterion. We refer to this as equalizer-assisted combiner adaptation. As will be shown shortly, a number of subtle issues must be taken into account when forming this error signal. Obviously this is a post-d technique since combiner adaptation is dependent upon the equalizer to generate decisions.

Aside from the symbol error rate (SER), the ultimate performance criteria is the output signal to interference plus noise ratio (SINR). For high SINRs, the decision-directed error variance (or cluster variance) provides an accurate measure of the SINR. A true joint optimization method would be one that finds the combiner and equalizer weights that minimize the cluster variance. As will soon be shown, one difficulty associated with this performance criteria is that it is a nonquadratic function of the weights and consequently may exhibit multiple local minima.

Refer to Figure 2 for a block diagram illustrating the method of equalizer-assisted combiner adaptation. This method uses decisions based on the equalizer output as the combiner desired response. For the moment, assume that the mode select switch is in the decision directed position (as shown). Notice that decision outputs are zero-stuffed and passed thru the filter C´(z) before being used as the combiner desired response. Zero-stuffing and spectral shaping are required in order to accurately compare the post-d symbol sequence with the pre-d combiner output. The filter C´(z) must compensate phase and group delay added by the fractionally-spaced equalizer H(z). If this phase shift were not compensated, the desired response vector would not be at the same phase as the combiner output and the error signal would be meaningless. The error signal generated as above is correlated with a delayed version of the antenna signals to form the complex combiner weight. Notice that the delay in the antenna signal is chosen to match the delay in the combiner output.

Figure 2. Block diagram illustrated equalizer-assisted combiner adaptation.

It should be pointed out that much of the complication associated with this approach is a result of the fractionally-spaced equalizer. The combiner output is sampled at two samples per symbol without regard to the optimum sampling phase, while the equalizer output is sampled at once per symbol. Had an inferior T-spaced equalizer been used, the combiner output would have already been matched filtered and sampled at the symbol rate. The center tap of the T-spaced equalizer could have been fixed to unity and there would have been no need to use the filter C’(z) to compensate equalizer-induced phase and group delay. The filter C’(z) however, would still be advantageous for another reason.

Notice that C´(z) is a normalized version of C(z) and that C(z) is determined by adaptively minimizing the difference between the filtered symbol sequence and the combiner output. In a loose sense, C(z) can be thought of as an approximate inverse of the equalizer H(z). More precisely, C(z) is a linear model of the convolution of the transmit filter, the propagation channel, receive filters, and the combiner response. The steady state modeling error (which is adaptively minimized) is comprised of any signals that could not be modeled. (See [6] for more details of this modeling process.)

If C(z) was provided with an infinite number of weights, the modeling error would consist only of additive interference not removed by the combiner. The combiner would then be optimized (using LMS) to minimize a mean square modeling error comprised only of additive interference and no multipath. Thus the combiner adaptation would be insensitive to multipath. For short delay multipath that the equalizer can compensate, this is desirable. However, it would be more desirable if the combiner would concentrate on nulling long-delay multipath that the equalizer cannot compensate. This is accomplished by choosing the length of C(z) to be approximately one third the length of the equalizer. In this case, the modeling error will consist of additive interference and multipath with a delay spread greater than one third the equalizer’s length. Multipath less than one third the equalizer length, will be modeled and therefore won’t be passed to the combiner where valuable degrees of freedom would have to be used to null it.

It is important to mention that C´(z) is a normalized version of C(z). This gain constraint is required to prevent the combiner and modeling filter weights from collapsing to zero. The combiner could have been gain constrained, but this would not be consistent with the blind acquisition mode described next.

The above description of equalizer-assisted combiner adaptation assumed that the system was already in decision-directed mode. In general, interference and multipath preclude the use of decision direction from the start. Instead one must “open the eye” using blind adaptation prior to switching to decision direction. (See references [5] and [7] for a discussion of this multimode process.) Placing the mode select switch in the opposite position than shown accomplishes this blind acquisition. The normalized equalizer output vector is used in place of a hard decision vector. A detailed discussion of this operation is beyond the scope of this paper. Suffice it to say that this method can be used to open the eye but sometimes suffers from interference capture.

A second method of joint combiner-equalizer adaptation is shown in Figure 3. This method attempts to find the equalizer and combiner weights that minimize the output cluster variance (maximize the SINR). Adaptation is accomplished using stochastic steepest descent. The equalizer weights are adapted using standard LMS, while the combiner weights are adapted using a variant of LMS referred to by Widrow as “filtered-X LMS” [8]. The combiner X-vector is filtered thru a copy of H(z) prior to being correlated with the decision error.

Figure 3. Block diagram illustrating a system that jointly minimizes the output cluster variance.

Through simple algebra under the assumption that the equalizer and combiner weights are commutable, it can easily be shown that the instantaneous gradient of the decision error power equals the filtered version of the element signals times the decision error. The commutability assumption requires that the combiner time constant be several times the span of the equalizer. Since the combiner time constant is typically on the order of hundreds of symbols and the equalizer length is on the order of tens of symbols, this assumption will generally be valid. Beyond this, there is no other restriction on the choice of relative time constants for the combiner and equalizer. Typically they are chosen to be equal. However, their relative sizes control the convergence path through the combiner-equalizer weight space.

An intuitive explanation for filtered-X is as follows: Assume that the equalizer filter is fixed or slowly varying so that it can be commuted with the combiner weights. Then the equalizer can be pushed through the combiner summation node to the front of each combiner weight. The x-vector filter and the H(z) can then be combined and pushed directly after each antenna. The resulting structure is a combiner adapted using standard LMS operating on filtered versions of the element signals. With this in mind, it becomes apparent that the decision error power is a quadratic function of the combiner weights with the equalizer frozen, and also is a quadratic function of the equalizer weights with the combiner frozen. The decision-error power is, however a higher than quadratic function of all the weights.

The nonquadratic nature of the performance surface points to the potential for multiple local minima. The fact that multiple local minima do in fact exist can be demonstrated quite easily by considering a signal scenario consisting of a direct path plus a single reflected path. One local minimum arises from the combiner spatially nulling the multipath, while the other local minimum results from the equalizer canceling the multipath. Without noise and with an infinite length equalizer, both of these solutions would be equally good. However in the presence of noise and a finite length equalizer, one solution is likely to be better than the other. Thus steepest descent minimization of the output cluster variance is not guaranteed to result in a globally optimum solution.

Notice in Figure 3 that this system readily lends itself to the blind acquisition mode. In blind mode, the normalized equalizer output is used in place of the hard decision output as a desired response.

Before closing this section, one final word should be said concerning the complexity of the method shown in Figure 2. Recall that one of the main reasons for using the combiner-equalizer structure was that it was less complex than using an N-weight filter on each channel. At first glance, it appears that this complexity advantage may be gone as a result of the filtered-X processing. However, it should be pointed out that many applications do not require the weights be updated every input sample. For instance, digital microwave radio sampling rates are in the megahertz, yet required adaptation time constants are on the order tens of milliseconds. In this case, the weight update process can be decimated and the complexity associated with the filtered-X processing is no longer dominant.

The above techniques have undergone fairly extensive computer simulation. Many of the details of the equalizer-assisted technique were uncovered during this simulation process. This section presents the results of one such simulation run. As will become apparent severe conditions were chosen for this simulation in order to stress the algorithms under test.

The simulated combiner-equalizer structure was as follows: The combiner consisted of 3 omnidirectional colinear elements spaced 5 wavelengths apart. The initial weight taper was uniform. The equalizer was comprised of 33 T/2-spaced taps and the pulse shaping filter (when used) had 17 taps. The initial equalizer response was a raised cosine bandpass filter as was the pulse shaping filter.

The signal environment was chosen as follows: The signal of interest was 64-QAM with 40% square root raised cosine spectral shaping. The angle of arrival (AOA) of the direct path was 0 degrees. In addition to the direct path, two dominant multipaths were simulated. One had an intermediate delay of 3 symbol periods and an AOA of 8 degrees and the other had a longer delay of 6 symbols and an AOA of –6 degrees. Numerous short delay (less than 1 symbol) dispersed paths were also added at AOA’s less than 1 degree. A sinusoidal cochannel interferer with power equal to the SOI (0 dB SIR) was added at an AOA of 3 degrees. The signal to thermal noise ratio (SNR) on each element was 40 dB.

Figure 4 shows the converged beampattern and equalizer frequency response for the system shown in Figure 2. The vertical arrows indicate the AOA’s of the cochannel interference and the multipath. Notice that the combiner has placed deep nulls in the direction of the long delay multipath arrival and the cochannel interference. The shorter delay multipath was not nulled by the combiner but instead was compensated by the equalizer. The equalized output SINR at convergence was 27 dB. The combiner did not attack the intermediate delay multipath because the pulse shaping filter had only 17 taps and the main tap was centered. Thus, multipath delays less than 4 symbols could be modeled by the shaping filter and did not show up in the modeling error passed to the combiner.

Figure 4. Converged (a) beampattern and (b) frequency response for the system shown in Figure 2.

Figure 5 shows the converged beampattern and equalizer frequency response for the system shown in Figure 3. Again the vertical arrows indicate the AOAs of the cochannel interference and multipath. The equalizer SINR at convergence for this method was 33 dB, 6 dB better than the previous technique. This improvement can be understood by observing the difference between the beampatterns shown in Figures 4 and 5. While the combiner-equalizer system of Figure 3 chose to spatially null this multipath, the system of Figure 4 chose to let a small amount of the long-delay multipath through the combiner so that the equalizer could use it to coherently combine with the direct path and enhance the SNR. Thus, joint optimization of the combiner-equalizer system to minimize cluster variance can yield a much better solution than independent optimization of the combiner. It should be pointed out, however, that with a different set of initial conditions, the steepest descent process converged to the same solution found by the system in Figure 2. That is, the multiple local minima problem described earlier was uncovered in this simulation.

Figure 5. Converged (a) beampattern and (b) frequency response for the system shown in Figure 3.

This paper has presented two methods for jointly adapting the weights of an antenna combiner preceding a fractionally-spaced equalizer. Both blind and decision-directed algorithms were used. The merits of joint versus independent optimization of the combiner and equalizer were described. A typical computer simulation was presented illustrating the operation of the two techniques.

Further work should be carried out to characterize the local minima behaviour of the output cluster variance minimization technique. Methods should be explored to insure that the globally optimum solution is found. Additionally, the blind acquisition behaviour of both methods needs further characterization.

*For the applications addressed in this paper, a space diversity combiner and an adaptive antenna system are considered one in the same.