Günther Koliander

A lower bound on the noncoherent capacity pre-log for the MIMO channel with temporally correlated fading

We derive a lower bound on the capacity pre-log of a temporally correlated Rayleigh block-fading multiple-input multiple-output (MIMO) channel with T transmit antennas and R receive antennas in the noncoherent setting (no a priori channel knowledge at the transmitter and the receiver). In this model, the fading process changes independently across blocks of length L and is temporally correlated within each block for each transmit-receive antenna pair, with a given rank Q of the corresponding correlation matrix. Our result implies that for almost all choices of the coloring matrix that models the temporal correlation, the pre-log can be lower-bounded by T(1 - 1/L) for T ≤ (L - 1)/Q provided that R is sufficiently large. The widely used constant block-fading model is equivalent to the temporally correlated block-fading model with Q = 1 for the special case when the temporal correlation for each transmit-receive antenna pair is the same, which is unlikely to be observed in practice. For the constant block-fading model, the capacity pre-log is given by T(1 - T/L), which is smaller than our lower bound for the case Q = 1. Thus, our result suggests that the assumptions underlying the constant blockfading model lead to a pessimistic result for the capacity pre-log.

Entropy and Source Coding for Integer-Dimensional Singular Random Variables

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables-which are neither discrete nor continuous- has not been available so far. Here, we present such an extension for the practically relevant class of integer-dimensional singular random variables. The proposed entropy definition contains the entropy of discrete random variables and the differential entropy of continuous random variables as special cases. We show that it transforms in a natural manner under Lipschitz functions, and that it is invariant under unitary transformations. We define joint entropy and conditional entropy for integer-dimensional singular random variables, and we show that the proposed entropy conveys useful expressions of the mutual information. As first applications of our entropy definition, we present a result on the minimal expected codeword length of quantized integer-dimensional singular sources and a Shannon lower bound for integer-dimensional singular sources.

Oversampling Increases the Pre-Log of Noncoherent Rayleigh Fading Channels

We analyze the capacity of a continuous-time, time-selective, Rayleigh block-fading channel in the high signal-to-noise ratio (SNR) regime. The fading process is assumed stationary within each block and to change independently from block to block; furthermore, its realizations are not known a priori to the transmitter and the receiver (noncoherent setting). A common approach to analyzing the capacity of this channel is to assume that the receiver performs matched filtering followed by sampling at symbol rate (symbol matched filtering). This yields a discrete-time channel in which each transmitted symbol corresponds to one output sample. Liang & Veeravalli (2004) showed that the capacity of this discrete-time channel grows logarithmically with the SNR, with a capacity pre-log equal to 1-Q/N. Here, N is the number of symbols transmitted within one fading block, and Q is the rank of the covariance matrix of the discrete-time channel gains within each fading block. In this paper, we show that symbol matched filtering is not a capacity-achieving strategy for the underlying continuous-time channel. Specifically, we analyze the capacity pre-log of the discrete-time channel obtained by oversampling the continuous-time channel output, i.e., by sampling it faster than at symbol rate. We prove that by oversampling by a factor two one gets a capacity pre-log that is at least as large as 1-1/N. Since the capacity pre-log corresponding to symbol-rate sampling is 1-Q/N, our result implies indeed that symbol matched filtering is not capacity achieving at high SNR.

Generic correlation increases noncoherent MIMO capacity

We study the high-SNR capacity of MIMO Rayleigh block-fading channels in the noncoherent setting where neither transmitter nor receiver has a priori channel state information. We show that when the number of receive antennas is sufficiently large and the temporal correlation within each block is “generic” (in the sense used in the interference-alignment literature), the capacity pre-log is given by T(1 - 1/N) for T <; N, where T denotes the number of transmit antennas and N denotes the block length. A comparison with the widely used constant block-fading channel (where the fading is constant within each block) shows that for a large block length, generic correlation increases the capacity pre-log by a factor of about four.

Lossless linear analog compression

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors x ∈ ℝ^m from the noiseless linear measurements y = Ax with measurement matrix A ∈ ℝ^n×m. Specifically, for a random vector x ∈ ℝ^m of arbitrary distribution we show that x can be recovered with zero error probability from n > inf dim_MB(U) linear measurements, where dim_MB(·) denotes the lower modified Minkowski dimension and the infimum is over all sets U ⊆ ℝ^m with P[x ∈ U] = 1. This achievability statement holds for Lebesgue almost all measurement matrices A. We then show that s-rectifiable random vectors-a stochastic generalization of s-sparse vectors-can be recovered with zero error probability from n > s linear measurements. From classical compressed sensing theory we would expect n ≥ s to be necessary for successful recovery of x. Surprisingly, certain classes of s-rectifiable random vectors can be recovered from fewer than s measurements. Imposing an additional regularity condition on the distribution of s-rectifiable random vectors x, we do get the expected converse result of s measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as s-analytic random vectors.

Entropy for singular distributions

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables (which are neither discrete nor continuous) has not been available so far. Here, we propose such an extension for the practically relevant class of singular probability measures that are supported on a lower-dimensional subset of Euclidean space. We show that our entropy transforms in a natural manner under Lipschitz functions and that it conveys useful expressions of the mutual information. Potential applications of the proposed entropy definition include capacity calculations for the vector interference channel, compressed sensing in a probabilistic setting, and capacity bounds for block-fading channel models.

How costly is it to learn fading channels

Recent results in communication theory suggest that substantial throughput gains in wireless fading networks can be achieved by exploiting network coordination (e.g., CoMP, network MIMO, interference alignment). However, these results are often based on the simplifying assumption that each node in the network has perfect channel knowledge and ignore the channel-estimation overhead. In this tutorial paper, we take a fresh look at the problem of learning fading channels. By focusing on simple channel models, we will illustrate how to quantify rigorously the throughput loss due to channel-estimation overhead. Specifically, by exploiting that in the absence of a priori channel knowledge at the receiver, the noiseless received signal is a nonlinear function of the transmitted signals and the propagation channel, we will show how to unveil the geometric structure underlying the channel input output relation, and how to use this geometry to characterize capacity at high SNR. We will also demonstrate that this approach is useful to determine the largest rate achievable at finite SNR and finite blocklength.

Local detection and estimation of multiple objects from images with overlapping observation areas

We propose a method for detecting and estimating multiple objects from multiple noisy images with partly overlapping observation areas. The goal is to detect the objects that are “locally” present in the individual observation areas and to estimate their states. Our method is based on a new closed-form expression of the marginal posterior probability hypothesis density (PHD) and admits a distributed implementation. Simulation results demonstrate performance gains over correlation-based and PHD-based methods that do not take advantage of the overlapping observation areas.