S.3171-3175

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by ${f_c}$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/{f_c}$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program is nearly optimal. Finally, numerical experiments illustrate our methods.

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.

The noncoherent capacity of stationary discrete-time fading channels is known to be very sensitive to the fine details of the channel model. More specifically, the measure of the support of the fading-process power spectral density (PSD) determines if noncoherent capacity grows logarithmically with the signal-to-noise ratio (SNR) or slower than logarithmically. Such a result is unsatisfactory from an engineering point of view, as the support of the PSD cannot be determined through measurements. The aim of this paper is to assess whether, for general continuous-time Rayleigh-fading channels, this sensitivity has a noticeable impact on capacity at SNR values of practical interest. To this end, we consider the general class of band-limited continuous-time Rayleigh-fading channels that satisfy the wide-sense stationary uncorrelated-scattering (WSSUS) assumption and are, in addition, under spread. We show that, for all SNR values of practical interest, the noncoherent capacity of every channel in this class is close to the capacity of an additive white Gaussian noise channel with the same SNR and bandwidth, independently of the measure of the support of the scattering function (the 2-D channel PSD). Our result is based on a lower bound on noncoherent capacity, which is built on a discretization of the channel input-output relation induced by projecting onto Weyl-Heisenberg sets. This approach is interesting in its own right as it yields a mathematically tractable way of dealing with the mutual information between certain continuous-time random signals.

We establish a lower bound on the noncoherent capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel. Our result holds for arbitrary rank Q of the channel correlation matrix, arbitrary block-length L >; Q, and arbitrary number of receive antennas R, and includes the result in Morgenshtern et al. (2010) as a special case. It is well known that the capacity pre-log for this channel in the single-input single-output (SISO) case is given by 1-Q/L, where Q/L is the penalty incurred by channel uncertainty. Our result reveals that this penalty can be reduced to 1/L by adding only one receive antenna, provided that L ≥ 2Q - 1 and the channel correlation matrix satisfies mild technical conditions. The main technical tool used to prove our result is Hironaka's celebrated theorem on resolution of singularities in algebraic geometry.

We establish a lower bound on the noncoherent capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel. Surprisingly, when the covariance matrix of the channel satisfies a certain technical condition related to the cardinality of its smallest set of linearly dependent rows, this lower bound reveals that the capacity pre-log in the SIMO case is larger than that in the single-input single-output (SISO) case.

The noncoherent capacity of stationary discrete-time fading channels is known to be very sensitive to the fine details of the channel model. More specifically, the measure of the set of harmonics where the power spectral density of the fading process is nonzero determines if capacity grows logarithmically in SNR or slower than logarithmically. An engineering-relevant problem is to characterize the SNR value at which this sensitivity starts to matter. In this paper, we consider the general class of continuous-time Rayleigh-fading channels that satisfy the wide-sense stationary uncorrelated-scattering (WSSUS) assumption and are, in addition, underspread. For this class of channels, we show that the noncoherent capacity is close to the AWGN capacity for all SNR values of practical interest, independently of whether the scattering function is compactly supported or not. As a byproduct of our analysis, we obtain an information-theoretic pulse-design criterion for orthogonal frequency-division multiplexing systems.

We analyze fading interference relay networks where$M$single-antenna source–destination terminal pairs communicate concurrently and in the same frequency band through a set of$K$single-antenna relays using half-duplex two-hop relaying. Assuming that the relays have channel state information (CSI), it is shown that in the large-$M$limit, provided$K$grows fast enough as a function of$M$, the network “decouples” in the sense that the individual source–destination terminal pair capacities are strictly positive. The corresponding required rate of growth of$K$as a function of$M$is found to be sufficient to also make the individual source–destination fading links converge to nonfading links. We say that the network“crystallizes”as it breaks up into a set of effectively isolated“wires in the air.”A large-deviations analysis is performed to characterize the “crystallization” rate, i.e., the rate (as a function of$M$,$K$) at which the decoupled links converge to nonfading links. In the course of this analysis, we develop a new technique for characterizing the large-deviations behavior of certain sums of dependent random variables. For the case of no CS

We study fading interference relay networks where M single-antenna source-destination terminal pairs communicate through a set of K relays using half-duplex two-hop relaying. Two specific protocols are considered, P1 introduced in H. Bolcskei, et al. (2004), H. Bolcskei and R.U. Nabar (2004) and P2 introduced in A.F. Dana and B. Hassibi (2003). P1 relies on the idea of relay partitioning and requires each relay terminal to know one backward and one forward fading coefficient only. P2 requires each relay terminal to know all M backward and M forward fading coefficients and does not need relay partitioning. We prove that in the large-M limit the minimum rate of growth of K for P1 to achieve a strictly positive per source-destination terminal pair capacity is K infin M3whereas in P2 it is K infin M2. The protocols P1 and P2 are thus found to trade off the number of relay terminals for channel state information (CSI) at the relays; more CSI at the relays reduces the total number of relays needed to achieve a strictly positive per source-destination terminal pair capacity in the large-M limit