Presentations

The winter school combines tutorials by renowned experts in coding and information theory with the attending students presenting the problems they are looking at during their Ph. D. studies. These may also be work in progress.

The following tutorials will be presented (180 minutes each):
 
Communication over noncoherent underspread fading channels
by Prof. Helmut Bölcskei (ETH Zürich)

Abstract: The tutorial starts with a review of the literature on the capacity of noncoherent fading channels. We will argue that the results available are very specific to the channel model used and often very sensitive to the fine points of the model. We will then report the results of our attempt to analyze the noncoherent capacity of (the very general class of) continuous-time wide-sense stationary uncorrelated scattering (WSSUS) channels. This will be done by starting with an in-depth review of the transfer function calculus for linear time-varying systems followed by showing how Weyl-Heisenberg frames arise naturally in discretizing underspread WSSUS channels. Virtually all wireless communication channels are (highly) underspread. After an introduction to Weyl-Heisenberg frame theory, we study the capacity behavior of noncoherent underspread WSSUS channels. In particular, we derive (tight) bounds on capacity under a peak constraint on the transmit signal. The bounds are explicit in the channel's scattering function, are useful for a large range of bandwidths, and allow to (coarsely) identify the capacity-optimal bandwidth. Furthermore, we obtain a closed-form expression for the first-order Taylor series expansion of capacity in the limit of infinite bandwidth. We conclude with a review of open problems in the area.

Recent Advances in Shannon Sampling Theory - Fundamental Limits for Digital Signal Processing
by Prof. Holger Boche (TU Berlin)

Abstract: In many theoretical and practical applications the reconstruction and approximation of signals from their samples is important. In this talk we analyze different reconstruction and approximation processes, all of which are based on sampling series, for the frequently utilized Paley-Wiener space PWπ1 of signals with absolute integrable Fourier transform, which is the largest space in the scale of Paley-Wiener spaces. Additionally, we analyze the behavior of the sampling series for certain bandlimited wide-sense stationary processes. Surprisingly, there is a close connection between the convergence of the sampling series for deterministic signals in PWπ1 and the mean-square convergence of the sampling series for bandlimited wide-sense stationary processes.
For practical applications it is important to have a stable signal reconstruction in the sense of an sampling series that is globally uniformly convergent, or at least locally uniformly convergent and globally bounded. Recently it was shown that signals in PWπ1 cannot be stably reconstructed from its samples taken equidistantly at Nyquist rate. However, if oversampling is applied, a stable reconstruction is possible and even the Shannon sampling series is a stable reconstruction process. In addition to equidistant sampling we analyze the convergence behavior of sampling series with non-equidistant sampling points that are made of the zeros of sine-type functions.
Sometimes the sole reconstruction of signals is not enough. Instead, some processed version of the signal is of interest and has to be approximated by using only the samples of the signal. We analyze sampling series representations for stable linear time invariant (LTI) systems, operating on bandlimited signals. One possible application for this sampling-based signal processing approach is in sensor networks. Surprisingly, oversampling and the design of special kernels does not improve the convergence behavior in this case.
Another important topic is the influence of non-linear operators, which are used to model imperfect conditions in practical signal processing applications, on the signal reconstruction. In particular we discuss how the good local approximation behavior of the Shannon sampling series is affected if the samples are disturbed either by the non-linear threshold operator or the quantization operator. We show that the intuition that a decreased quantization step size always leads to a reduced reconstruction error is wrong. Moreover, we discuss the influence of both operators on the sampling-based signal processing approach, i.e., on the approximation of stable LTI systems. Finally, we give a game theoretic interpretation of the problem in the setting of a game against nature and show that nature has a universal strategy to win this game.

Information Combining
by Prof. Johannes Huber (FAU Erlangen-Nürnberg)

Abstract: Consider coded transmission over a binary-input symmetric memory- less channel. The channel decoder uses the noisy observations of the code symbols to reproduce the transmitted code symbols. Thus, it combines the information about individual code symbols to obtain an overall information about each code symbol, which may be the reproduced code symbol or its a-posteriori probability. This tutorial addresses the problem of “information combining” from an information-theory point of view: the decoder combines the mutual information between channel input symbols and channel output symbols (observations) to the mutual information between one transmitted symbol and all channel output symbols. The actual value of the combined information depends on the statistical structure of the channels. However, it can be upper and lower bounded for the assumed class of channels. This tutorial first introduces the concept of mutual information profiles and revisits the well-known Jensen’s inequality. Using these tools, the bounds on information combining are derived for single parity-check codes and for repetition codes. The application of the bounds is illustrated in four examples: information processing characteristics of coding schemes, including extrinsic information transfer (EXIT) functions; design of multiple turbo codes; bounds for the decoding threshold of low-density parity-check codes; EXIT function of the accumulator.

Channel and Source Coding via Polar Coding
by Prof. Rüdiger Urbanke (EPF Lausanne).

Abstract: This tutorial is about polar codes, a coding scheme which was recently introduced by Arikan. Polar codes achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels (and even extensions thereof) under a low complexity successive decoding strategy. The idea of polar codes is based on the phenomenon of "channel polarization." More precisely, by recursively combining and splitting individual channels, some of those channels become essentially error free, whereas other channels become completely noise. Further, the fraction of essentially error-free channels tends to the capacity of the underlying binary symmetric channel. It was shown by Arikan and Telatar that the error probability for these codes decays like exp(-sqrt(N)) for sufficiently large blocklengths. (In words, such codes are said to achieve an exponent of 1/2.) Channel polarization is a common phenomenon, and the basic construction of Arikan can be generalized to a large class of code families. With a proper generalization we can achieve exponents arbitrarily close to 1, i.e., to an almost exponential decay of the error probability (Korada, Sasoglu, and U). Further, although polarization codes were originally only introduced for channel coding, they are equally useful for source coding applications; both lossless as well as lossy compression can be achieved (in an optimal manner) with polar codes and they are adaptable also for distributed scenarios (Hussami, Korada, and U). In fact, it is hard to think of a coding scenario that cannot be addressed efficiently by polar codes.

Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering
by Christian Scheunert

Optimized Error/Erasue Bounded Minimum Distance Decoding for the AWGN Channel
by Christian Senger

Efficient Image Transmission using Ultra Wideband PPM and Closed-Loop Predictive Coding
by Long Pham

Majorization for Vectors and Functions – Some Applications
by Martin Mittelbach

Distributed Storage System
by Vinodh Venkatesan