The Cost of Noncoherence: Avoiding Channel Estimation Through Differential Encoding in Phase Quantized Systems

This paper proposes a novel approach that utilizes differential encoding to overcome the channel estimation problem in communication systems with low-resolution quantization receivers. For differentially encoded data, we derive the maximum likelihood detection rule for the canonical block-2 detectors, employing just two consecutive quantized observations at the channel output and without any receiver-side channel state information. We establish the optimality of this maximum likelihood detection rule within the class of block-
$L$
detectors, where
$L \geq 3$
, under the condition that
$n = \log _{2} M$
, with
$n$
and
$M$
denoting the number of quantization bits and input alphabet size, respectively. The derived detector has a simple and easily implementable structure, comparing the quantization region indices of consecutive observations to determine the transmitted message index. By leveraging the structure of the derived optimum detector, we obtain the expression for the message error probability in Rayleigh fading wireless channels. Through asymptotic analysis in the high signal-to-noise ratio regime, we reveal a crucial finding that achieving the same diversity order as infinite bit quantization with full channel knowledge requires an additional two bits at the quantizer, in addition to the minimum requirement of
$\log _{2} M$
bits. One bit compensates for the low-resolution effect, while the other addresses the lack of channel knowledge. Finally, we conduct an extensive simulation study to demonstrate the performance of the optimum detectors and quantify the performance loss resulting from the absence of channel knowledge at the receiver.