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Atribución-NoComercial-SinDerivadas 3.0 España
Abstract:
This thesis concerns the maximum coding rate at which data can be transmitted
over a noncoherent, single-antenna, Rayleigh block-fading channel using an errorcorrecting
code of a given blocklength with a block-error probability not exceeding
a given value. This thesis concerns the maximum coding rate at which data can be transmitted
over a noncoherent, single-antenna, Rayleigh block-fading channel using an errorcorrecting
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 informationtheoretical
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 ratio (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.
Finally, this thesis includes a practical case study where we analyze the benefit of
cooperation in optical wireless communications, a promising technology that can play
an important role in the next generation of wireless communications due to the high
data rates it can achieve. Specifically, a cooperative multipoint transmission and
reception scheme is evaluated for visible light communication (VLC) in an indoor
scenario. The proposed scheme is shown to provide SNR improvements of 3 dB or
more compared to a noncooperative scheme, especially when there is non-line-of-sight
(NLOS) between the access point and the receiver.[+][-]