A Unified Approach to Gaussian Channels with Finite Blocklength

Among the emerging trends in the wireless industry, the Internet of Things (IoT) and machine-to-machine (M2M) communications are notable examples that prompt power efficient communication schemes that can communicate short packets of time-sensitive control information with very low latency. A mathematical analysis and design of networks with such stringent latency requirements, however, cannot rely on conventional information theoretic results which assume asymptotically large blocklengths. In this dissertation, we build upon a recent line of work on non-asymptotic information theory to characterize second-order approximations for channel coding rates over a variety of Gaussian settings at finite blocklength. Our results reveal several interesting insights for practical systems design. For any Gaussian channel at finite blocklength, i.i.d. Gaussian input, although capacity-achieving, is not second-order optimal. Good codes must possess certain structures to ensure power efficiency and achieve high rates. For the discrete-input Gaussian channel at low SNR, coded-PSK surprisingly outperforms coded-QAM with common random i.i.d. codes. For the non-ergodic fading channel with short blocklength, the Gaussian noise can have as significant an impact on the outage probability as fading can. And finally, when channel uses are precious, time division multiple access (TDMA) is costly, particularly when the number of users grows. On the theoretical side, our key contribution is a unified and convenient approach for proving sharp achievability results for a variety of Gaussian settings. We amend the standard method of random coding and typicality decoding to make it applicable to non-i.i.d. inputs and single-user as well as multi-user channels with cost constraints. In particular, we use a new change of measure technique and also apply a convenient yet powerful form of the Central Limit Theorem (CLT), called the CLT for functions, to analyze certain dependent information random variables. In comparison with other sharp achievability methods, our unified approach is more transparent and follows conventional lines of reasoning.