This paper explores the fundamental limits of unsourced random access (URA) with a random and unknown number$\mathrm{K}_{a}$of active users in MIMO quasi-static Rayleigh fading channels. First, we derive an upper bound on the probability of incorrectly estimating the number of active users. We prove that it exponentially decays with the number of receive antennas and eventually vanishes, whereas reaches a plateau as the power and blocklength increase. Then, we derive non-asymptotic achievability and converse bounds on the minimum energy-per-bit required by each active user to reliably transmit$J$bits with blocklength$n$. Numerical results verify the tightness of our bounds, suggesting that they provide benchmarks to evaluate existing schemes. The extra required energy-per-bit due to the uncertainty of the number of active users decreases as$\mathbb{E}[\mathrm{K}_{a}]$increases. Compared to random access with individual codebooks, the URA paradigm achieves higher spectral and energy efficiency. Moreover, using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.

In wireless communications, bit-interleaved coded modulation (BICM) with Gray labeling is commonly utilized for the transmission with higher-order modulation. In case of polar codes, BICM performance can be further improved if the coded bits are grouped according to their reliability levels and the interleaving is performed independently at each level. This improvement is achieved due to the faster polarization.In this paper, we propose an efficient divide-and-conquer method to construct polar codes for BICM with quadrature amplitude modulation (QAM). At the first step we decompose the polar design for QAM into several polar code designs for quadrature phase shift keying (QPSK). Next, given the number of information bits to be assigned, we recursively compute their allocations to the polarized subchannels until some predefined length is reached. This allows to utilize the reliability sequences optimized for additive white Gaussian noise (AWGN) channel with QPSK modulation such as 5G New Radio (NR) universal reliability sequence, and enables a low-complexity approach to construct polar codes for several modulation orders. Hence we can take an advantage of the existing designs while providing the performance boost by utilizing the QAM structure in the unified manner.

This paper considers the massive random access problem in multiple-input multiple-output (MIMO) quasi-static Rayleigh fading channels. Specifically, we derive achievability and converse bounds on the minimum energy-per-bit required for each active user to transmit$J$bits with blocklength$n$, power$P$, and$L$receive antennas under a per-user probability of error (PUPE) constraint, in the cases with and without a priori channel state information at the receiver (CSIR and no-CSI). In the case of no-CSI, we consider both the settings with and without the knowledge of the number$K_{a}$of active users at the receiver. Numerical evaluation shows that the gap between achievability and converse bounds is less than 2.5 dB for the CSIR case and less than 4 dB for the no-CSI case in most considered regimes. Under the condition that the distribution of$K_{a}$is known in advance, the uncertainty of the exact value of$K_{a}$entails only a small penalty in terms of energy efficiency. Our results show the significance of MIMO for the massive random access problem. As an example, we show that the spectral efficiency grows approximately linearly with the number of receive antennas in the case of CSIR, whereas the growth rate decreases in the case of no-CSI. Moreover, in the case of no-CSI, we demonstrate the suboptimality of the pilot-assisted scheme, especially when the number of active users is large. Building on non-asymptotic results, assuming all users are active and$J=\Theta (1)$, we obtain scaling laws of the number of supported users as follows: when$L = \Theta \left ({n^{2}}\right)$and$P=\Theta \left ({\frac {1}{n^{2}}}\right)$, one can reliably serve$K = \mathcal {O}(n^{2})$users in the case of no-CSI; under mild conditions in the case of CSIR, the PUPE requirement is satisfied if and only if$\frac {nL\ln KP}{K}=\Omega \left ({1}\right)$.

In this paper, we study multiple-antenna wireless communication networks, where a large number of devices simultaneously communicate with an access point. The capacity region of multiple-input multiple-output massive multiple access channels (MIMO mMAC) is investigated. While joint typicality decoding is utilized to establish the achievability of capacity region for conventional MAC with fixed number of the users, the technique is not directly applicable for the MIMO mMAC. Instead, an information-theoretic approach based on Gallager's error exponent analysis is exploited to characterize the finite dimension region of the MIMO mMAC. Theoretical results reveal that the region is dominated by the sum rate constraint only, and the individual user rates are dominated by specific factors that correspond to the allocation of the sum rate. The rate in conventional MAC is not achievable when the number of users is comparable with codelength, which is due to the fact that successive interference cancellation cannot guarantee an arbitrary small error decoding probability for MIMO mMAC. The results further imply that, asymptotically, the individual user rate is independent of the number of transmit antennas, and channel hardening makes the individual user rate close to that when only statistic knowledge of channel is available at transmitter. The finite dimension region of MIMO mMAC is a generalization of the symmetric rate in Chen et al. (2017).

This paper investigates the maximal secret communication rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are uniformly tighter than existing bounds, and lead to the tightest bounds on the second-order coding rate for discrete memoryless and Gaussian wiretap channels. The exact second-order coding rate is established for semi-deterministic wiretap channels, which characterizes the optimal tradeoff between reliability and secrecy in the finite-blocklength regime. Underlying our achievability bounds are two new privacy amplification results, which not only refine the classic privacy amplification results, but also achieve secrecy under the stronger semantic-security metric.

In this invited paper, the concept of privacy amplification is reviewed and recent developments are discussed. Its applications in information-theoretic security problems are considered including semantic security and polar coding for privacy amplification.

It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finiteblocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman-Pearson β functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson β functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú's wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.

We investigate the maximum coding rate for a given average blocklength and error probability over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message K-user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.

We investigate the maximum coding rate achievable on a two-user broadcast channel for the case where a common message is transmitted with feedback using either fixed-blocklength codes or variable-length codes. For the fixed-blocklength-code setup, we establish nonasymptotic converse and achievability bounds. An asymptotic analysis of these bounds reveals that feedback improves the second-order term compared to the no-feedback case. In particular, for a certain class of antisymmetric broadcast channels, we show that the dispersion is halved. For the variable-length-code setup, we demonstrate that the channel dispersion is zero.

This paper studies a two-user state-dependent Gaussian multiple-access channel (MAC) with state noncausally known at one encoder. Two scenarios are considered: 1) each user wishes to communicate an independent message to the common receiver; and 2) the two encoders send a common message to the receiver and the non-cognitive encoder (i.e., the encoder that does not know the state) sends an independent individual message (this model is also known as the MAC with degraded message sets). For both scenarios, new outer bounds on the capacity region are derived, which improve uniformly over the best known outer bounds. In the first scenario, the two corner points of the capacity region as well as the sum rate capacity are established, and it is shown that a single-letter solution is adequate to achieve both the corner points and the sum rate capacity. Furthermore, the full capacity region is characterized in situations in which the sum rate capacity is equal to the capacity of the helper problem. The proof exploits the optimal-transportation idea of Polyanskiy and Wu (which was used previously to establish an outer bound on the capacity region of the interference channel) and the worst case Gaussian noise result for the case in which the input and the noise are dependent.

This paper studies a two-user state-dependent Gaussian multiple-access channel with state noncausally known at one encoder. Two new outer bounds on the capacity region are derived, which improve uniformly over the best known (genie-aided) outer bound. The two corner points of the capacity region as well as the sum rate capacity are established, and it is shown that a single-letter solution is adequate to achieve both the corner points and the sum rate capacity. Furthermore, the full capacity region is characterized in situations in which the sum rate capacity is equal to the capacity of the helper problem. The proof exploits the optimal-transportation idea of Polyanskiy and Wu (which was used previously to establish an outer bound on the capacity region of the interference channel) and the worst-case Gaussian noise result for the case in which the input and the noise are dependent.

This paper studies the maximum secrecy rate for a semi-deterministic wiretap channel, in which the channel between the transmitter and the legitimate receiver is deterministic, while that between the transmitter and the eavesdropper is a discrete memoryless channel. For a given decoding error probability and information leakage (measured by the total variation distance), the optimal second-order secrecy rate is derived. Unlike the secrecy capacity, the second-order secrecy rate characterizes the optimal tradeoff between secrecy and reliability at finite blocklength.

We investigate the maximum coding rate achievable on a two-user broadcast channel for the case where a common-message is transmitted using fixed-blocklength codes with feedback. Specifically, we focus on a family of broadcast channels composed of two antisymmetric Z-channels. For this setup, we obtain matching upper and lower bounds on the dispersion term in the asymptotic expansion of the maximum coding rate. These bounds reveal that the dispersion is halved compared to the no-feedback case.

This paper investigates the maximum coding rate over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. Specifically, upon decoding the common message, each decoder sends a stop signal to the encoder, which transmits continuously until it receives all K stop signals. We present nonasymptotic achievability and converse bounds for the maximum coding rate, which strengthen and generalize the bounds previously reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveal that-contrary to the point-to-point case-the second-order term in the asymptotic expansion of the maximum coding rate decays inversely proportional to the square root of the average blocklength. This holds for certain nontrivial common-message broadcast channels, such as the binary symmetric broadcast channel. Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order asymptotic expansion approximates accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.

A channel coding achievability bound expressed in terms of the ratio between two Neyman-Pearson β functions is proposed. This bound is the dual of a converse bound established earlier by Polyanskiy and Verdú (2014). The new bound turns out to simplify considerably the analysis in situations where the channel output distribution is not a product distribution, for example due to a cost constraint or a structural constraint (such as orthogonality or constant composition) on the channel inputs. Connections to existing bounds in the literature are discussed. The bound is then used to derive 1) the channel dispersion of additive non-Gaussian noise channels with random Gaussian codebooks, 2) the channel dispersion of an exponential-noise channel, 3) a second-order expansion for the minimum energy per bit of an additive white Gaussian noise channel, and 4) a lower bound on the maximum coding rate of a multiple-input multiple-output Rayleigh-fading channel with perfect channel state information at the receiver, which is the tightest known achievability result.

This paper investigates the maximal secrecy rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are shown to be tighter than existing bounds. The bounds also lead to the tightest second-order coding rate for discrete memoryless and Gaussian wiretap channels.

We present nonasymptotic achievability and converse bounds on the maximum coding rate (for a fixed average error probability and a fixed average blocklength) of variable-length full-feedback (VLF) and variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC). For the VLF setup, the achievability bound relies on a scheme that maps each message onto a variable-length Huffman codeword and then repeats each bit of the codeword until it is received correctly. The converse bound is inspired by the meta-converse framework by Polyanskiy, Poor, and Verdú (2010) and relies on binary sequential hypothesis testing. For the case of zero error probability, our achievability and converse bounds match. For the VLSF case, we provide achievability bounds that exploit the following feature of BEC: the decoder can assess the correctness of its estimate by verifying whether the chosen codeword is the only one that is compatible with the erasure pattern. One of these bounds is obtained by analyzing the performance of a variable-length extension of random linear fountain codes. The gap between the VLSF achievability and the VLF converse bound, when number of messages is small, is significant: 23% for 8 messages on a BEC with erasure probability 0.5. The absence of a tight VLSF converse bound does not allow us to assess whether this gap is fundamental.

Motivated by the current interest in ultra-reliable, low-latency, machine-type communication systems, we investigate the tradeoff between reliability, throughput, and latency in the transmission of information over multiple-antenna Rayleigh block-fading channels. Specifically, we obtain finite-blocklength, finite-SNR upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability. Numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths (packets of about 100 symbols). Furthermore, our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time-frequency-spatial degrees of freedom, and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom. In particular, our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time-frequency diversity branches that maximize the rate. Finally, we show that infinite-blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates.

This paper investigates the minimum energy required to transmit k information bits with a given reliability over a multiple-antenna Rayleigh block-fading channel, with and without channel state information (CSI) at the receiver. No feedback is assumed. It is well known that the ratio between the minimum energy per bit and the noise level converges to -1.59 dB as k goes to infinity, regardless of whether CSI is available at the receiver or not. This paper shows that the lack of CSI at the receiver causes a slowdown in the speed of convergence to -1.59 dB as k → ∞ compared with the case of perfect receiver CSI. Specifically, we show that, in the no-CSI case, the gap to -1.59 dB is proportional to ((log k)/k)1/3, whereas when perfect CSI is available at the receiver, this gap is proportional V to 1/√k. In both cases, the gap to -1.59 dB is independent of the number of transmit antennas and of the channel's coherence time. Numerically, we observe that, when the receiver is equipped with a single antenna, to achieve an energy per bit of -1.5 dB in the no-CSI case, one needs to transmit at least 7 × 107information bits, whereas 6 × 104bits suffice for the case of perfect CSI at the receiver.

We investigate the maximum coding rate achievable over a two-user broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. Specifically, upon decoding the common message, each decoder sends a stop signal to the encoder, which transmits continuously until it receives both stop signals. For the point-to-point case, Polyanskiy, Poor, and Verdú (2011) recently demonstrated that variable-length coding combined with stop feedback significantly increases the speed at which the maximum coding rate converges to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, a result a.k.a. zero dispersion. In this paper, we show that this speed-up does not necessarily occur for the broadcast channel with common message. Specifically, there exist scenarios for which variable-length stop-feedback codes yield a positive dispersion.

This paper investigates the minimum energy required to transmit, with a given reliability, k information bits over a stationary memoryless Rayleigh-fading channel, under the assumption that neither the transmitter nor the receiver have a priori channel state information (CSI). It is well known that the ratio between the minimum energy per bit and the noise level converges to -1.59 dB as k goes to infinity, regardless of whether CSI is available at the receiver or not. This paper shows that lack of CSI at the receiver causes a slowdown in the speed of convergence to -1.59 dB as k → ∞ compared to the case of perfect receiver CSI. Specifically, we show that in the noCSI case, the gap to -1.59 dB is proportional to ((log k)/k)1/3, whereas when perfect CSI is available at the receiver, this gap is '/ proportional to 1/√(k). Numerically, we observe that to achieve an energy per bit of -1.5 dB in the no-CSI case, one needs to transmit at least 7 × 107information bits, whereas 6 × 104bits suffice for the case of perfect CSI at the receiver (same number of bits as for nonfading AWGN channels). Interestingly, all results (asymptotic and numerical) are unchanged if multiple transmit antennas and/or block fading is assumed.

We provide nonasymptotic upper and lower bounds on the sum-rate capacity of Rayleigh block-fading multiple-access channels for the set up where a priori channel state information is not available. The upper bound relies on a dual formula for channel capacity and on the assumption that the users can cooperate perfectly. The lower bound is derived assuming a noncooperative scenario where each user employs unitary space-time modulation (independently from the other users). Numerical results show that the gap between the upper and the lower bound is small already at moderate SNR values. This suggests that the sum-rate capacity gains obtainable through user cooperation are minimal for the scenarios considered in the paper.

This paper investigates the maximal channel coding rate achievable at a given blocklength n and error probability ϵ, when the codewords are subjected to a long-term (i.e., averaged-over-all-codeword) power constraint. The second-order term in the large-n expansion of the maximal channel coding rate is characterized both for additive white Gaussian noise (AWGN) channels and for quasi-static fading channels with perfect channel state information available at both the transmitter and the receiver. It is shown that in both the cases, the second-order term is proportional to (n-1ln n)1/2. For the quasi-static fading case, this second-order term is achieved by truncated channel inversion, namely, by concatenating a dispersion-optimal code for an AWGN channel subject to a short-term power constraint, with a power controller that inverts the channel whenever the fading gain is above a certain threshold. Easy-to-evaluate approximations of the maximal channel coding rate are developed for both the AWGN and the quasi-static fading case.

This paper investigates the maximal channel coding rate achievable at a given blocklength n and error probability ε, when the codewords are subject to a long-term (i.e., averaged-over-all-codeword) power constraint. The second-order term in the large-n expansion of the maximal channel coding rate is characterized both for AWGN channels and for quasi-static fading channels with perfect channel state information at the transmitter and the receiver. It is shown that in both cases the second-order term is proportional to √(log n)/n.

This paper analyzes the channel dispersion of quasi-static multiple-input multiple-output fading channels with no channel state information at the transmitter. We show that the channel dispersion is zero under mild conditions on the fading distribution. The proof of our result is based on Stokes' theorem, which deals with the integration of differential forms on manifolds with boundary.

This paper investigates the maximal achievable rate for a given blocklength and error probability over quasi-static multiple-input multiple-output fading channels, with and without channel state information at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases. In the example of a particular 1 × 2 single-input multiple-output Rician fading channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared with the blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-blocklength achievability and converse bounds.

We investigate the maximal achievable rate for a given blocklength and error probability over quasi-static single-input multiple-output (SIMO) fading channels. Under mild conditions on the channel gains, it is shown that the channel dispersion is zero regardless of whether the fading realizations are available at the transmitter and/or the receiver. The result follows from computationally and analytically tractable converse and achievability bounds. Through numerical evaluation, we verify that, in some scenarios, zero dispersion indeed entails fast convergence to outage capacity as the blocklength increases. In the example of a particular 1×2 SIMO Rician channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared to the blocklength required for an AWGN channel with the same capacity.

In future wireless networks a significant number of wireless broadband users will be vehicular, i.e., they will be in public transportation vehicles like buses, trams or trains. In this paper, we show that the efficient use of relay nodes to serve vehicular users can greatly improve the energy efficiency of the network while maintaining the required quality-of-service (QoS). We consider vehicular users moving along a road within the coverage of a base station (BS). Communication can take place in a single-hop fashion (baseline case) or can be assisted by a single relay node (dual-hop), which can either be a fixed relay node (FRN) deployed at a specific position on the road or a moving relay node (MRN) mounted on top of the vehicle. We compare the required overall transmit energy for direct transmission, FRN and MRN assisted transmission in a noise limited system under Rayleigh fading while assuming an outage probability (OP) target. A lower bound is derived for the required energy of the FRN assisted transmission. It is shown that as the vehicular penetration loss (VPL) increases, both FRN and MRN assisted transmission can significantly lower the overall transmit energy compared to the conventional one-hop case. Moreover, transmission relying on an MRN outperforms the FRN assisted case when VPL is moderate to high.

We characterize the capacity of Rayleigh block-fading multiple-input multiple-output (MIMO) channels in the noncoherent setting where transmitter and receiver have no a priori knowledge of the realizations of the fading channel. We prove that unitary space-time modulation (USTM) is not capacity-achieving in the high signal-to-noise ratio (SNR) regime when the total number of antennas exceeds the coherence time of the fading channel (expressed in multiples of the symbol duration), a situation that is relevant for MIMO systems with large antenna arrays (large-MIMO systems). This result settles a conjecture by Zheng & Tse (2002) in the affirmative. The capacity-achieving input signal, which we refer to as Beta-variate space-time modulation (BSTM), turns out to be the product of a unitary isotropically distributed random matrix, and a diagonal matrix whose nonzero entries are distributed as the square-root of the eigenvalues of a Beta-distributed random matrix of appropriate size. Numerical results illustrate that using BSTM instead of USTM in large-MIMO systems yields a rate gain as large as 13% for SNR values of practical interest.

We find the capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel in the noncoherent setting. It is well known that for block-lengthLand rank of the channel covariance matrix equal toQ, the capacity pre-log in the single-input single-output (SISO) case is given by 1-Q/L. Here,Q/Lcan be interpreted as the pre-log penalty incurred by channel uncertainty. Our main result reveals that, by adding only one receive antenna, this penalty can be reduced to 1/Land can, hence, be made to vanish for the block-lengthL→∞, even ifQ/Lremains constant asL→∞. Intuitively, even though the SISO channels between the transmit antenna and the two receive antennas are statistically independent, the transmit signal induces enough statistical dependence between the corresponding receive signals for the second receive antenna to be able to resolve the uncertainty associated with the first receive antenna's channel and thereby make the overall system appear coherent. The proof of our main theorem is based on a deep result from algebraic geometry known as Hironaka's Theorem on the Resolution of Singularities.

We analyze the capacity of Rayleigh block-fading multiple-input multiple-output (MIMO) channels in the noncoherent setting and prove that unitary space-time modulation (USTM) is not capacity-achieving when the total number of antennas exceeds the coherence time of the fading channel. This situation is relevant for MIMO systems with large antenna arrays (large-MIMO systems). Our result settles a conjecture by Zheng & Tse (2002) in the affirmative. The capacity-achieving input signal, which we refer to as Beta-variate space-time modulation (BSTM), turns out to be the product of a unitary isotropically distributed random matrix, and a diagonal matrix whose nonzero entries are distributed as the square-root of the eigenvalues of a Beta-distributed random matrix of appropriate size. Numerical results illustrate that using BSTM instead of USTM in large-MIMO systems yields a rate gain as large as 13% for SNR values of practical interest.

We study the maximal achievable rate R*(n, ∈) for a given block-length n and block error probability o over Rayleigh block-fading channels in the noncoherent setting and in the finite block-length regime. Our results show that for a given block-length and error probability, R*(n, ∈) is not monotonic in the channel's coherence time, but there exists a rate maximizing coherence time that optimally trades between diversity and cost of estimating the channel.