Degrees of Freedom Region of the MIMO <formula formulatype="inline"> <tex>$X$</tex></formula> Channel

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We provide achievability as well as converse results for the degrees of freedom region of a multiple-input multiple-output (MIMO) <emphasis><formula formulatype="inline"><tex>$X$</tex></formula></emphasis> channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message. With <emphasis><formula formulatype="inline"><tex>$M=1$</tex> </formula></emphasis> antennas at each node, we find that the total (sum rate) degrees of freedom are bounded above and below as <emphasis><formula formulatype="inline"> <tex>$1 \leq \eta _{X}^{\star} \leq {{ 4}\over { 3}}$</tex></formula></emphasis>. If <emphasis><formula formulatype="inline"><tex>$M&gt;1$</tex></formula></emphasis> and channel matrices are nondegenerate then the precise degrees of freedom <emphasis><formula formulatype="inline"><tex>$\eta _{X}^{\star} = {{ 4}\over { 3}}M$</tex></formula></emphasis>. Thus, the MIMO <emphasis><formula formulatype="inline"><tex>$X$</tex></formula></emphasis> channel has noninteger degrees of freedom when <emphasis><formula formulatype="inline"> <tex>$M$</tex></formula></emphasis> is not a multiple of <emphasis emphasistype="smcaps"><formula> <tex>$3$</tex></formula></emphasis>. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the <emphasis><formula formulatype="inline"><tex>${{ 4}\over { 3}}M$</tex></formula></emphasis> degrees of freedom. If the channels vary with time/frequency then the <emphasis><formula formulatype="inline"><tex>$X$</tex></formula></emphasis> channel with single antennas <emphasis><formula formulatype="inline"><tex>$(M=1)$</tex></formula></emphasis> at all nodes has exactly <emphasis><formula><tex>${4\over 3}$</tex></formula></emphasis> degrees of freedom. The key idea for the achievability of the degrees of freedom is <emphasis emphasistype="boldital">interference alignment</emphasis>—i.e., signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired. We also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio. </para>