Resumen de Fundamental limits of short-packet wireless communications
This thesis concerns the maximum coding rate at which data can be transmitted over a noncoherent, single-antenna, Rayleigh block-fading channel using an error-correcting code of a given blocklength with a block-error probability not exceeding a given value. This is an emerging problem originated by the next generation of wireless communications, where the understanding of the fundamental limits in the transmission of short packets is crucial. For this setting, traditional information-theoretical metrics of performance that rely on the transmission of long packets, such as capacity or outage capacity, are not good benchmarks anymore, and the study of the maximum coding rate as a function of the blocklength is needed.
For the noncoherent Rayleigh block-fading channel model, to study the maximum coding rate as a function of the blocklength, only nonasymptotic bounds that must be evaluated numerically were available in the literature. The principal drawback of the nonasymptotic bounds is their high computational cost, which increases linearly with the number of blocks (also called throughout this thesis coherence intervals) needed to transmit a given codeword. By means of different asymptotic expansions in the number of blocks, this thesis provides an alternative way of studying the maximum coding rate as a function of the blocklength for the noncoherent, single-antenna, Rayleigh block-fading channel.
The first approximation on the maximum coding rate derived in this thesis is a high-SNR normal approximation. This central-limit-theorem-based approximation becomes accurate as the signal-to-noise ration (SNR) and the number of coherence intervals L of size T tend to infinity. We show that the high-SNR normal approximation is roughly equal to the normal approximation one obtains by transmitting one pilot symbol per coherence block to estimate the fading coefficient, and by then transmitting T-1 symbols per coherence block over a coherent fading channel. This suggests that, at high SNR, one pilot symbol per coherence block suffices to achieve both the capacity and the channel dispersion. While the approximation was derived under the assumption that the number of coherence intervals and the SNR tend to infinity, numerical analyses suggest that it becomes accurate already at SNR values of 15 dB, for 10 coherence intervals or more, and probabilities of error of 10^{-3} or more.
The derived normal approximation is not only useful because it complements the nonasymptotic bounds available in the literature, but also because it lays the foundation for analytical studies that analyze the behavior of the maximum coding rate as a function of system parameters such as SNR, number of coherence intervals, or blocklength. An example of such a study concerns the optimal design of a simple slotted-ALOHA protocol, which is also given in this thesis.
Since a big amount of services and applications in the next generation of wireless communication systems will require to operate at low SNRs and small probabilities of error (for instance, SNR values of 0 dB and probabilities of error of 10^{-6}), the second half of this thesis presents saddlepoint approximations of upper and lower nonasymptotic bounds on the maximum coding rate that are accurate in that regime. Similar to the normal approximation, these approximations become accurate as the number of coherence intervals L increases, and they can be calculated efficiently. Indeed, compared to the nonasymptotic bounds, which require the evaluation of L-dimensional integrals, the saddlepoint approximations only require the evaluation of four one-dimensional integrals. Although developed under the assumption of large L, the saddlepoint approximations are shown to be accurate even for L=1 and SNR values of 0 dB or more.
The small computational cost of these approximations can be further avoided by performing high-SNR saddlepoint approximations that can be evaluated in closed form. These approximations can be applied when some conditions of convergence are satisfied and are shown to be accurate for 10 dB or more.
In our analysis, the saddlepoint method is applied to the tail probabilities appearing in the nonasymptotic bounds. These probabilities often depend on a set of parameters, such as the SNR. Existing saddlepoint expansions do not consider such dependencies. Hence, they can only characterize the behavior of the expansion error in function of the number of coherence intervals L, but not in terms of the remaining parameters. In contrast, we derive a saddlepoint expansion for random variables whose distribution depends on an extra parameter, carefully analyze the error terms, and demonstrate that they are uniform in such an extra parameter. We then apply the expansion to the Rayleigh block-fading channel and obtain approximations in which the error terms depend only on the blocklength and are uniform in the remaining parameters.
Furthermore, the proposed approximations are shown to recover the normal approximation and the reliability function of the channel, thus providing a unifying tool for the two regimes, which are usually considered separately in the literature. Specifically, we show that the high-SNR normal approximation can be recovered from the normal approximation derived from the saddlepoint approximations. By means of the error exponent analysis that recovers the reliability function of the channel, we also obtain easier-to-evaluate approximations of the saddlepoint approximations consisting of the error exponent of the channel multiplied by a subexponential factor. Numerical evidence suggests that these approximations are as accurate as the saddlepoint approximations.
