Least-Squares Multi-User CMArray: A New Algorithm for Blind Adaptive Beamforming

This paper addresses the problem of blind adaptive beamforming for the recovery of K synchronous signals incident on an M element antenna array, as would be found in a communication system rich in co-channel interference. Least-Squares Multi-user CMArray (LS-MU-CMA), a block Constant Modulus type algorithm that efficiently performs blind spatial CO-channel interference mitigation to recover up to M unique signals, is proposed. The performance of the algorithm with respect to rate of convergence and distance from optimality is analyzed with computer simulations.

This paper is interested in the problem of jointly recovering K CO-channel sources incident upon an M element antenna array, where K ¾ M. We shall assume that the transmitted signals are uncorrelated with each other. For the signals considered, the bandwidth shall be small relative to the carrier, so that propagation delays across the array can be modeled as phase shifts. Under these assumptions, each transmitted signal,

has associated with it a 1xM array response vector,

, that relates the angle of arrival, 3.fm

, to the phasing of the signal across the antenna array. The received baseband sampled array signal is then given by equation 1

where

is a spatially and temporally white noise vector.

Given this model, Mx1 weight vectors,

, can then be applied to

in parallel to produce output signals. The pth beamformer port output is given by

To adjust these beamforming weights, a block CMA algorithm can be used. The block CMA processing concept is that a constant modulus desired signal can be formed as

Given a block of input data

the autocorrelation matrix can be estimated as

and the cross-correlation vector as

A least squares estimate of the weights can then be generated as

[3]. This type of technique provides faster convergence than steepest descent forms of CMA.

The proposed LS-MU-CMArray algorithm is characterized by a parallel structure. Since parallel structures apply the same input to all beamformer ports, they do not suffer from the performance degradation exhibited by succeeding stages of cascaded structures, such as the CM/MOP or cascaded CMArray algorithm [1, 2]. However, these parallel structures are faced with the difficult uniqueness problem. If these parallel sets of beamforming weights are allowed to adapt independently, each will converge to recover the same signal assuming common initial conditions. What is desirable however, is that each set of weights extract a unique signal. This is the goal of the LS-MU-CMA algorithm. To ensure that each port has locked onto a unique signal, LS-MU-CMA constrains the port outputs to be uncorrelated.

Algorithm provides a conceptual description of the algorithm, including motivation. It also details each step of the algorithm. Algorithm Derivation provides a detailed derivation. Computer Simulations provides a performance evaluation of the algorithm via computer simulations. Conclusions summarizes the conclusions of the paper.

The developed algorithm baselines off the cost function first proposed by Papadias and Paulraj [4], which added the cross-correlation of the output ports of a space-time processing system to the CMArray cost function to provide an added constraint that will drive each port away from every other port. The LS-MU-CMA approach will be to consider a spatial only processing system that processes blocks of data, rather than using a steepest descent algorithm.

LS-MU-CMArray monitors the cross-correlation,

, between the port of interest, p, and lower numbered ports, l. This cross-correlation can be efficiently calculated according to equation 6.

where

is an estimate of the array response vector associated with the lth signal and is given by

is the variance of the port output. The cross-correlation is normalized by the auto-correlation of the port output. If this exceeds a threshold, then it adds the outer product of the 8.fm, from the offending ports to the autocorrelation matrix before it is inverted.

is now an augmented autocorrelation matrix that emphasizes the correlated ports. It can be inverted and multiplied with

to obtain the new set of beamformer weights for that port.

This will cause the new set of weights to further suppress the signal that has already been extracted by the correlated port.

Given an autocorrelation matrix, the steps in the algorithm are as follows for each block for each port, p

Each correlation that is added creates an additional constraint that could result in an overconstrained situation. This has the effect of injecting unneeded noise into the system. For this reason, this constraint will only be added to a given port’s cost function if the correlation is large enough. Up to P-1 constraints can be added to the cost function.

A detailed derivation of the LS-MU-CMArray algorithm follows. This derivation assumes a constant modulus desired signal has been formed. The MU-CMA cost function for the pth port is given by

The first term represents the CMA cost function while the latter is the port correlation cost function.

Separating and expanding the expectation yields

If we make the following definitions

then the cost function reduces down to

where

. Taking the gradient with respect to the weights yields

From this, we arrive at the LS-MU-CMArray weight update equation

A sample scenario was created, with an 8 antenna element receiver and 7 transmitters. A uniform linear array with half wavelength spacing was used. Table 1 provides the per element SNR of each signal is given along with the angle of arrival.

Given this scenario, a Monte Carlo simulation was performed using the LS-MU-CMArray beamformer. The results are summarized in Table 2 . For comparison, the maximum attainable SINR for each signal is provided.

Table 2. Benchmark Results

For all signals, LS-MU-CMA is within 0.5 dB of the optimum. The standard deviation shows that the results are very consistent across the trials.

To illustrate the nulling performed by the beamformer weights on each port, beamplots, which show the gain applied to signals arriving from each angle, were generated. These plots are shown in Figure 1. Rays have been drawn on the beamplots to indicate the 7 arriving signals.

Figure 1. Port weight vector beamplots

They show that port 1 has locked onto signal 6, port 2 onto signal 5, port 3 onto signal 1, port 4 onto signal 2, port 5 onto signal 4, port 6 onto signal 7, and port 7 onto signal 3.

The output SINR of each of the beamformer ports was recorded while the beamformer was acquiring the environment. This resulted in convergence curves, which are shown in Figure 2.

Figure 2. Convergence curves

In all instances, the beamformer has completed convergence within 400 samples, or 6 iterations.

A novel adaptive beamforming approach has been proposed. It has been shown to rapidly approach the optimum beamforming solution for each port. At the same time, it addresses the problem of maintaining lock on a unique signal with each port. The algorithm is relatively simple to implement and is easily scalable.