We identify a practical vector quantizer design problem where anyfixed-length quantizer (FLQ) yields non-zero distortion at any finite rate,while there is a variable-length quantizer (VLQ) that can achieve zerodistortion with arbitrarily low rate. The problem arises in a $t \times 1$multiple-antenna fading channel where we would like to minimize the channeloutage probability by employing beamforming via quantized channel stateinformation at the transmitter (CSIT). It is well-known that in such ascenario, finite-rate FLQs cannot achieve the full-CSIT (zero distortion)outage performance. We construct VLQs that can achieve the full-CSITperformance with finite rate. In particular, with $P$ denoting the powerconstraint of the transmitter, we show that the necessary and sufficient VLQrate that guarantees the full-CSIT performance is $\Theta(1/P)$. We alsodiscuss several extensions (e.g. to precoding) of this result.
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机译：我们确定了一个实际的矢量量化器设计问题，其中任何定长量化器（FLQ）在任何有限速率下都会产生非零失真，而可变长度量化器（VLQ）可以以任意低的速率实现零失真。问题出现在一个$ t×1 $的多天线衰落信道中，在该信道中，我们希望通过在发射机（CSIT）处通过量化信道状态信息进行波束成形，以最小化信道中断的可能性。众所周知，在这种情况下，有限速率的FLQ无法实现全CSIT（零失真）中断性能。我们构造了可以以有限速率实现完全CSIT性能的VLQ。特别是，用$ P $表示发送器的功率约束，我们证明了保证完全CSIT性能的必要和充分的VLQrate是$ \ Theta（1 / P）$。我们还讨论了该结果的几个扩展（例如，预编码）。
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