Saddlepoint Approximations for Noncoherent Single-Antenna Rayleigh Block-Fading Channels

Proceeding of: 2019 IEEE International Symposium on Information Theory (ISIT), 7-12 July 2019, Paris, FranceThis paper presents saddlepoint approximations of state-of-the-art converse and achievability bounds for noncoherent, single-antenna, Rayleigh block-fading channels. These approximations can be calculated efficiently and are shown to be accurate for SNR values as small as 0 dB, blocklengths of 168 channel uses or more, and when the channel's coherence interval is not smaller than two. It is demonstrated that the derived approximations recover both the normal approximation and the reliability function of the channel.A. Lancho, G. Vázquez-Vilar and T. Koch have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (714161), the Spanish Ministerio de Economía y Competitividad (RYC-2014-16332, TEC2016-78434-C3-3-R (AEI/FEDER, EU) and IJCI2015-27020), the Spanish Ministerio de Educación, Cultura y Deporte (FPU014/01274), and the Comunidad de Madrid (S2103/ICE-2845). G. Durisi and J. Östman have been supported by the Swedish Research Council under Grants 2016-03293 and 2014-6066

Ultra-Reliable Short-Packet Communications: Fundamental Limits and Enabling Technologies

The paradigm shift from 4G to 5G communications, anticipated to enable ultra-reliable low-latency communications (URLLC), will enforce a radical change in the design of wireless communication systems. Unlike in 4G systems, where the main objective is to provide a large transmission rate, in URLLC, as implied by its name, the objective is to enable transmissions with low latency and, simultaneously, very high reliability. Since low latency implies the use of short data packets, the tension between blocklength and reliability is studied in URLLC.Several key enablers for URLLC communications have been designated in the literature. Of special importance are diversity-enabling technologies such as multiantenna systems and feedback protocols. Furthermore, it is not only important to introduce additional diversity by means of the above examples, one must also guarantee that thescarce number of channel uses are used in an optimal way. Therefore, it is imperative to develop design guidelines for how to enable reliable detection of incoming data, how to acquire channel-state information, and how to construct efficient short-packet channel codes. The development of such guidelines is at the heart of this thesis. This thesis focuses on the fundamental performance of URLLC-enabling technologies. Specifically, we provide converse (upper) bounds and achievability (lower) bounds on the maximum coding rate, based on finite-blocklength information theory, for systems that employ the key enablers outlined above. With focus on the wireless channel, modeled via a block-fading assumption, we are able to provide answers to questions like: howto optimally utilize spatial and frequency diversity, how far from optimal short-packet channel codes perform, how multiantenna systems should be designed to serve a given number of users, and how to design feedback schemes when the feedback link is noisy. In particular, this thesis is comprised out of four papers. In Paper A, we study the short-packet performance over the Rician block-fading channel. In particular, we present achievability bounds for pilot-assisted transmission with several different decoders that allow us to quantify the impact, on the achievable performance, of imposed pilots and mismatched decoding. Furthermore, we design short-packet channel codes that perform within 1 dB of our achievability bounds. Paper B studies multiuser massive multiple-input multiple-output systems with short packets. We provide an achievability bound on the average error probability over quasistatic spatially correlated Rayleigh-fading channels. The bound applies to arbitrary multiuser settings, pilot-assisted transmission, and mismatched decoding. This makes it suitable to assess the performance in the uplink/downlink for arbitrary linear signal processing. We show that several lessons learned from infinite-blocklength analyses carry over to the finite-blocklength regime. Furthermore, for the multicell setting with randomly placed users, pilot contamination should be avoided at all cost and minimum mean-squared error signal processing should be used to comply with the stringent requirements of URLLC.In Paper C, we consider sporadic transmissions where the task of the receiver is to both detect and decode an incoming packet. Two novel achievability bounds, and a novel converse bound are presented for joint detection-decoding strategies. It is shown that errors associated with detection deteriorates performance significantly for very short packet sizes. Numerical results also indicate that separate detection-decoding strategies are strictly suboptimal over block-fading channels.Finally, in Paper D, variable-length codes with noisy stop-feedback are studied via a novel achievability bound on the average service time and the average error probability. We use the bound to shed light on the resource allocation problem between the forward and the feedback channel. For URLLC applications, it is shown that enough resources must be assigned to the feedback link such that a NACK-to-ACK error becomes rarer than the target error probability. Furthermore, we illustrate that the variable-length stop-feedback scheme outperforms state-of-the-art fixed-length no-feedback bounds even when the stop-feedback bit is noisy

Fundamental limits of short-packet wireless communications

Mención Internacional en el título de doctorThis 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.Programa de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Joerg Widmer.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Petar Popovsk

Short Packets over Block-Memoryless Fading Channels: Pilot-Assisted or Noncoherent Transmission?

We present nonasymptotic upper and lower bounds on the maximum coding rate achievable when transmitting short packets over a Rician memoryless block-fading channel for a given requirement on the packet error probability. We focus on the practically relevant scenario in which there is no \emph{a priori} channel state information available at the transmitter and at the receiver. An upper bound built upon the min-max converse is compared to two lower bounds: the first one relies on a noncoherent transmission strategy in which the fading channel is not estimated explicitly at the receiver; the second one employs pilot-assisted transmission (PAT) followed by maximum-likelihood channel estimation and scaled mismatched nearest-neighbor decoding at the receiver. Our bounds are tight enough to unveil the optimum number of diversity branches that a packet should span so that the energy per bit required to achieve a target packet error probability is minimized, for a given constraint on the code rate and the packet size. Furthermore, the bounds reveal that noncoherent transmission is more energy efficient than PAT, even when the number of pilot symbols and their power is optimized. For example, for the case when a coded packet of $168$ symbols is transmitted using a channel code of rate $0.48$ bits/channel use, over a block-fading channel with block size equal to $8$ symbols, PAT requires an additional $1.2$ dB of energy per information bit to achieve a packet error probability of $10^{-3}$ compared to a suitably designed noncoherent transmission scheme. Finally, we devise a PAT scheme based on punctured tail-biting quasi-cyclic codes and ordered statistics decoding, whose performance are close ($1$ dB gap at $10^{-3}$ packet error probability) to the ones predicted by our PAT lower bound. This shows that the PAT lower bound provides useful guidelines on the design of actual PAT schemes.Comment: 30 pages, 5 figures, journa

Normal approximations for fading channels

Proceeding of: 52nd Annual Conference on Information Sciences and Systems (CISS 2018)Capacity and outage capacity characterize the maximum coding rate at which reliable communication is feasible when there are no constraints on the packet length. Evaluated for fading channels, they are important performance benchmarks for wireless communication systems. However, the latency of a communication system is proportional to the length of the packets it exchanges, so assuming that there are no constraints on the packet length may be overly optimistic for communication systems with stringent latency constraints. Recently, there has been great interest within the information theory community in characterizing the maximum coding rate for short packet lengths. Research on this topic is often concerned with asymptotic expansions of the coding rate with respect to the packet length, which then give rise to normal approximations. In this paper, we review existing normal approximations for single-antenna Rayleigh block-fading channels and compare them with the high-SNR normal approximation we presented at the 2017 IEEE International Symposium on Information Theory (Lancho, Koch, and Durisi, 2017). We further discuss how these normal approx- imations may help to assess the performance of communication protocols.A. Lancho and T. Koch have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 714161), from the Spanish Ministerio de Economía y Competitividad under Grants TEC2013-41718-R, RYC-2014-16332 and TEC2016-78434-C3-3-R (AEI/FEDER, EU), from an FPU fellowship from the Spanish Ministerio de Educación, Cultura y Deporte under Grant FPU14/01274, and from the Comunidad de Madrid under Grant S2103/ICE-2845. G. Durisi has been supported by the Swedish Research Council under Grant and 2016-03293

Efficient evaluation of the error probability for pilot-assisted URLLC with Massive MIMO

We propose a numerically efficient method for evaluating the random-coding union bound with parameter $s$ on the error probability achievable in the finite-blocklength regime by a pilot-assisted transmission scheme employing Gaussian codebooks and operating over a memoryless block-fading channel. Our method relies on the saddlepoint approximation, which, differently from previous results reported for similar scenarios, is performed with respect to the number of fading blocks (a.k.a. diversity branches) spanned by each codeword, instead of the number of channel uses per block. This different approach avoids a costly numerical averaging of the error probability over the realizations of the fading process and of its pilot-based estimate at the receiver and results in a significant reduction of the number of channel realizations required to estimate the error probability accurately. Our numerical experiments for both single-antenna communication links and massive multiple-input multiple-output (MIMO) networks show that, when two or more diversity branches are available, the error probability can be estimated accurately with the saddlepoint approximation with respect to the number of fading blocks using a numerical method that requires about two orders of magnitude fewer Monte-Carlo samples than with the saddlepoint approximation with respect to the number of channel uses per block

Short-Packet Communications: Fundamental Performance and Key Enablers

The paradigm shift from 4G to 5G communications, predicted to enable new use cases such as ultra-reliable low-latency communications (URLLC), will enforce a radical change in the design of communication systems. Unlike in 4G systems, where the main objective is to have a large transmission rate, in URLLC, as implied by its name, the objective is to enable transmissions with low latency and, simultaneously, very high reliability. Since low latency implies the use of short data packets, the tension between blocklength and reliability is studied in URLLC.\ua0Several key enablers for URLLC communications have been designated in the literature. A non-exhaustive list contains: multiple transmit and receive antennas (MIMO), short transmission-time intervals (TTI), increased bandwidth, and feedback protocols. Furthermore, it is not only important to introduce additional diversity by means of the above examples, one must also guarantee that the scarce number of channel uses are used in an optimal way. Therefore, protocols for how to convey meta-data such as control information and pilot symbols are needed as are efficient short-packet channel codes.\ua0This thesis focuses on the performance of reliable short-packet communications. Specifically, we provide converse (upper) bounds and achievability (lower) bounds on the maximum coding rate, based on finite-blocklength information theory, for systems that employ the key enablers outlined above. With focus on the Rician and Rayleigh block-fading channels, we are able to answer, e.g., how to optimally utilize spatial and frequency diversity, how far from optimal short-packet channel codes perform, and whether feedback-based schemes are preferable over non-feedback schemes.\ua0More specifically, in Paper A, we study the performance impact of MIMO and a shortened TTI in both uplink and downlink under maximum-likelihood decoding and Rayleigh block-fading. Based on our results, we are able to study the trade-off between bandwidth, latency, spatial diversity, and error probability. Furthermore, we give an example of a pragmatic design of a pilot-assisted channel code that comes within 2.7 dB of our achievability bounds. In Paper B, we partly extend our work in Paper A to the Rician block-fading channel and to practical schemes such as pilot-assisted transmission with nearest neighbor decoding. We derive achievability bounds for pilot-assisted transmission with several different decoders that allow us to quantify the impact, on the achievable performance, of pilots and mismatched decoding. Furthermore, we design short-packet channel codes that perform within 1 dB of our achievability bounds. Paper C contains an achievability bound for a system that employs a variable-length stop-feedback (VLSF) scheme with an error-free feedback link. Based on the results in Paper C and Paper B, we are able to compare non-feedback schemes to stop-feedback schemes and assess if, and when, one is superior to the other. Specifically, we show that, for some practical scenarios, stop-feedback does significantly outperform non-feedback schemes

List Decoding of Short Codes for Communication over Unknown Fading Channels

In this paper, the advantages of list decoding for short packet transmission over fading channels with an unknown state are illustrated. The principle is applied to polar codes (under successive cancellation list decoding) and to general short binary linear block codes (under ordered-statistics decoding). The proposed decoders assume neither a-priori knowledge of the channel coefficients, nor of their statistics. The scheme relies on short pilot fields that are used only to derive an initial channel estimate. The channel estimate is required to be accurate enough to enable a good list construction, i.e., the construction of a list that contains, with high probability, the transmitted codeword. The final decision on the message is obtained by applying a list. This allows one to use very few pilots, thus reducing the the Rayleigh block-fading channel and compared to finite-length performance bounds. The proposed technique provides (in the short block length regime) gains of 1 dB with respect to a traditional pilot-aided transmission scheme

URLLC with Massive MIMO: Analysis and Design at Finite Blocklength

The fast adoption of Massive MIMO for high-throughput communications was enabled by many research contributions mostly relying on infinite-blocklength information-theoretic bounds. This makes it hard to assess the suitability of Massive MIMO for ultra-reliable low-latency communications (URLLC) operating with short blocklength codes. This paper provides a rigorous framework for the characterization and numerical evaluation (using the saddlepoint approximation) of the error probability achievable in the uplink and downlink of Massive MIMO at finite blocklength. The framework encompasses imperfect channel state information, pilot contamination, spatially correlated channels, and arbitrary linear spatial processing. In line with previous results based on infinite-blocklength bounds, we prove that, with minimum mean-square error (MMSE) processing and spatially correlated channels, the error probability at finite blocklength goes to zero as the number $M$ of antennas grows to infinity, even under pilot contamination. On the other hand, numerical results for a practical URLLC network setup involving a base station with $M=100$ antennas, show that a target error probability of $10^{-5}$ can be achieved with MMSE processing, uniformly over each cell, only if orthogonal pilot sequences are assigned to all the users in the network. Maximum ratio processing does not suffice.Comment: 30 pages, 5 figure

URLLC with Massive MIMO: Analysis and Design at Finite Blocklength

The fast adoption of Massive MIMO for high-throughput communications was enabled by many research contributions mostly relying on infinite-blocklength information-theoretic bounds. This makes it hard to assess the suitability of Massive MIMO for ultra-reliable low-latency communications (URLLC) operating with short-blocklength codes. This paper provides a rigorous framework for the characterization and numerical evaluation (using the saddlepoint approximation) of the error probability achievable in the uplink and downlink of Massive MIMO at finite blocklength. The framework encompasses imperfect channel state information, pilot contamination, spatially correlated channels, and arbitrary linear spatial processing. In line with previous results based on infinite-blocklength bounds, we prove that, with minimum mean-square error (MMSE) processing and spatially correlated channels, the error probability at finite blocklength goes to zero as the number M of antennas grows to infinity, even under pilot contamination. However, numerical results for a practical URLLC network setup involving a base station with M-100 antennas, show that a target error probability of 10^¿5 can be achieved with MMSE processing, uniformly over each cell, only if orthogonal pilot sequences are assigned to all the users in the network. Maximum ratio processing does not suffice.The work of Johan Östman, Alejandro Lancho, and Giuseppe Durisi was supported in part by the Swedish Research Council under grant 2016-03293 and in part by the Wallenberg AI, Autonomous Systems, and Software Program. The work of Luca Sanguinetti was supported in part by the Italian Ministry of Education and Research (MIUR) in the framework of the CrossLab Project (Departments of Excellence)