A Fast Least Squares Adaptive Algorithm 
for Tracking Non-Stationary Environments

A common structure used in adaptive arrays for interference cancellation or source tracking purposes is shown in Figure 1. In this figure, the complex gains or weights w1(n) … wM(n) have to be adjusted by using an adaptive algorithm to minimise the output power continuously. This will result in the cancellation of all unknown interferences with powers larger than the receiver noise power, and will be useful in applications where the desired signal is weak or can be easily filtered out, as in radar sidelobe cancellation or spread spectrum communication systems.

Figure 1: Adaptive array for interference cancellation and source tracking.

Rather than adjusting the weights as if they were going to settle down to some constant values, faster tracking of non-stationary environments can be obtained by assuming that the weights also have unknown linearly changing components, even in the steady-state. Even though the algorithm will then be more complex, it will have better performance in tracking moving sources.

As an example, Figure 2 compares the trajectories of the phase error of the system response null for tracking a moving source for both the new and the traditional recursive least square (RLS) algorithms. In the RLS algorithm, the processing weights are taken to be constant and optimised within a certain time window. As evident in Figure 2, significant phase error variations with time occur for the RLS algorithm, and indicate that such an assumption can lead to significant tracking errors in situations where the relative motion between the system and sources is large. On the other hand, with the additional flexibility of time-varying weights incorporated, the new technique accounts for substantial changes in the environment within the time window, and thus results in much better tracking performance as can also be seen in Figure 2.

Figure 2: Comparing the performance of the new and the traditional RLS algorithm. 

Although the algorithm has been described in terms of an adaptive array with weights adjusted to minimise the output power, the concept of having weights which vary within the processing window to better track the environment is applicable to adaptive noise cancellation, mobile communications, control and other applications. In addition, the time-variation of the weights need not be constrained to be linear to result in even better performance.