  Recently, lattice-reduction-aided detectors have been proposed for
multiple-input multiple-output (MIMO) systems to give performance with full
diversity like maximum likelihood receiver, and yet with complexity similar to
linear receivers. However, these lattice-reduction-aided detectors are based on
the traditional LLL reduction algorithm that was originally introduced for
reducing real lattice bases, in spite of the fact that the channel matrices are
inherently complex-valued. In this paper, we introduce the complex LLL
algorithm for direct application to reduce the basis of a complex lattice which
is naturally defined by a complex-valued channel matrix. We prove that complex
LLL reduction-aided detection can also achieve full diversity. Our analysis
reveals that the new complex LLL algorithm can achieve a reduction in
complexity of nearly 50% over the traditional LLL algorithm, and this is
confirmed by simulation. It is noteworthy that the complex LLL algorithm
aforementioned has nearly the same bit-error-rate performance as the
traditional LLL algorithm.
