A Beta-Beta Achievability Bound with Applications

A channel coding achievability bound expressed in terms of the ratio between two Neyman-Pearson $\beta$ functions is proposed. This bound is the dual of a converse bound established earlier by Polyanskiy and Verd\'{u} (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) an achievability bound on the channel dispersion of additive non-Gaussian noise channels with random Gaussian codebooks, 2) the channel dispersion of the exponential-noise channel, 3) a second-order expansion for the minimum energy per bit of an AWGN 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.Comment: extended version of a paper submitted to ISIT 201

Coherent multiple-antenna block-fading channels at finite blocklength

In this paper we consider a channel model that is often used to describe the mobile wireless scenario: multiple-antenna additive white Gaussian noise channels subject to random (fading) gain with full channel state information at the receiver. Dynamics of the fading process are approximated by a piecewise-constant process (frequency non-selective isotropic block fading). This work addresses the finite blocklength fundamental limits of this channel model. Specifically, we give a formula for the channel dispersion -- a quantity governing the delay required to achieve capacity. Multiplicative nature of the fading disturbance leads to a number of interesting technical difficulties that required us to enhance traditional methods for finding channel dispersion. Alas, one difficulty remains: the converse (impossibility) part of our result holds under an extra constraint on the growth of the peak-power with blocklength. Our results demonstrate, for example, that while capacities of $n_t\times n_r$ and $n_r \times n_t$ antenna configurations coincide (under fixed received power), the coding delay can be quite sensitive to this switch. For example, at the received SNR of $20$ dB the $16\times 100$ system achieves capacity with codes of length (delay) which is only $60\%$ of the length required for the $100\times 16$ system. Another interesting implication is that for the MISO channel, the dispersion-optimal coding schemes require employing orthogonal designs such as Alamouti's scheme -- a surprising observation considering the fact that Alamouti's scheme was designed for reducing demodulation errors, not improving coding rate. Finding these dispersion-optimal coding schemes naturally gives a criteria for producing orthogonal design-like inputs in dimensions where orthogonal designs do not exist

Minimum Energy to Send kk Bits Over Multiple-Antenna Fading Channels

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 lack of CSI at the receiver causes a slowdown in the speed of convergence to $-1.59$ dB as $k\to\infty$ compared to 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 to $1/\sqrt{k}$. 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 $7\times 10^7$ information bits, whereas $6\times 10^4$ bits suffice for the case of perfect CSI at the receiver

A beta-beta achievability bound with applications

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