Fundamental Limits of Linear Equalizers: Diversity, Capacity, and Complexity

Linear equalizers (LEs) have been widely adopted for practical systems due to their low computational complexity. However, it is also well known that LEs provide inferior performance relative to a maximum-likelihood equalizer (MLE) or other near-MLEs, because LEs usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information. More importantly, unlike MLE or near-MLEs, the performance of LEs has not been well quantified. This hinders more general applications of LEs in wireless systems. In this paper, we reveal a fundamental link between a channel parameter—orthogonality deficiency $(od)$ of the channel matrix—and the diversity and capacity of LEs. We identify that when the $od$ of channel matrix has an upper bound strictly less than $1$ , the same diversity order as that of MLEs is collected by LEs and the outage capacity loss relative to MLEs is also a constant over signal-to-noise ratio (SNR). These results can be applied to designing a framework for hybrid equalizers. Furthermore, by studying the statistical properties of the $od$ and comparing the complexity of different equalizers, we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix $od$. The theoretical analysis is corroborated by computer simulations.