We consider the emph{effective} degrees of freedom (DoF) achieved by interference alignment when channel state information (CSI) is acquired by training and feedback. For a flat block-fading $Ktimes K$ interference channel with power $P$ per transmitter and $M$ antennas per node, we show that interference alignment achieves higher DoF than orthogonal transmission (e.g., TDMA) only if the channel coherence time is large and the capacity of feedback link is at least as $Theta(log P)$. Under this condition, to maximize the effective DoF, each receiver needs to feed back CSI via $(M^2- 1)log P$ bits per coherence interval; smaller growth rate of feedback bits will decrease the effective DoF. We also show that in the presence of training and feedback cost, $K=3$ achieves the optimal DoF for a broad range of channel characteristics; with larger number of user pairs, the DoF falls short of its optimum and beyond a certain point becomes a decreasing function of $K$.
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机译：当通过训练和反馈获取信道状态信息（CSI）时，我们考虑通过干扰对齐实现的有效的自由度（DoF）。对于具有每个发射机功率$ P $和每个节点$ M $天线功率的平坦块衰落$ Ktimes K $干扰信道，我们证明，仅当信道相干时间为时，干扰对准才能获得比正交传输（例如TDMA）更高的DoF。反馈链接的容量至少为$ Theta（log P）$。在这种情况下，为了使有效DoF最大化，每个接收器需要通过每个相干间隔通过$（M ^ 2-1）log P $位反馈CSI；反馈比特的较小增长率将降低有效自由度。我们还表明，在存在培训和反馈成本的情况下，$ K = 3 $可以在广泛的信道特征范围内获得最佳的自由度。如果用户对数量较大，则DoF不能达到其最佳值，超过一定点将成为$ K $的递减函数。
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