On properties of the widely linear MSE filter and its LMS implementation

Widely linear filters have been receiving much attention lately and have been proposed for many signal processing applications where the traditional circularity assumptions on the complex data do not hold. In this paper, we study the properties of the mean-square-error (MSE) widely linear filter and its least mean squares (LMS) adaptive implementation. We show that in certain cases, widely linear filter does not provide any additional advantage compared to the linear filter even with highly noncircular data. On the other hand, we show examples of cases where it can lead to important performance gains even when the input is circular. We also show that its performance can slow down significantly with highly noncircular inputs when it is implemented using an LMS type gradient descent algorithm thus making recursive least squares type adaptive implementations more desirable for an adaptive widely linear filter.