Jekyll2023-02-27T14:26:29+01:00http://localhost:4000/feed.xmlGeorg Böcherer’s BlogOn this website, I publish blog posts about my research on telecommunications.Talks2023-02-16T00:00:00+01:002023-02-16T00:00:00+01:00http://localhost:4000/talks
Teached as short course Kodierung mit variabler Länge: Datenkompression und Verteilungsanpassung at the Chinesisch-Deutsches Hochschulkolleg in Shanghai, China.
]]>Work-in-progress: Lecture Notes on Machine Learning for Communications2022-03-01T00:00:00+01:002022-03-01T00:00:00+01:00http://localhost:4000/machine-learning-for-communicationsSince 2020, I teach a one semester course at Technical University of Munich on machine learning for communications. In this course, we design the components of a communication system using deep learning. In the tutorials, we use the PyTorch framework. The focus is on choosing architectures and loss functions suitable for communication system design. The lecture notes are a work in progress. The latest released version can be found under the link below.
]]>New JLT-Article: Probabilistic Shaping and Forward Error Correction for Fiber-Optic Communication Systems2019-02-04T00:00:00+01:002019-02-04T00:00:00+01:00http://localhost:4000/2019/02/04/probabilistic-shaping-tutorialIn this tutorial paper, we provide an in-depth explanation of a layered architecture to integrate probabilistic shaping and forward error correction (FEC). Probabilistic Amplitude Shaping (PAS) is an exemplary instance of this layered architecture.
The focus of this tutorial is on the development of a practical performance metric that allows to separately assess the practical limitations of shaping (e.g., the rate loss of a finite length distribution matcher) and the FEC code (e.g., the back-off in SNR to achieve a BER of \(10^{-15}\)). The shaping performance is characterized by the shaping set size, which can be calculated in closed form for many existing distribution matching algorithms. The FEC performance benchmark is characterized by the uncertainty, which incorporates the decoding metric in use (e.g., hard-decision, soft-decision, quantized LLRs, etc) and the channel quality (e.g., via a Gaussian noise decoding model or measurements from transmission experiments). This separation of the performance metric into shaping set size and uncertainty reflects the two layers of the probabilistic shaping architecture and makes it very useful for the design of transceivers that integrate PS and FEC.
Georg Böcherer, Fabian Steiner, and Patrick Schulte have won the third prize at the 2015 Bell Labs Prize! Our proposed Probabilistic Amplitude Shaping (PAS) is based on ideas we published in
]]>Geometric Huffman Coding2018-10-28T00:00:00+02:002018-10-28T00:00:00+02:00http://localhost:4000/ghcDyadic probability mass functions (PMFs) can be generated by parsing a stream of independent equiprobable data bits by a full prefix-free code. Formally, \(p\) is dyadic if \(p_i=2^{-\ell_i}\), \(\ell_i \in\mathbb{N}\) and \(\sum_i 2^{-\ell_i}=1\). Such a PMF \(p\) is generated by a prefix-free code with codeword lengths \(\ell_i\). To match dyadic PMFs to communication channels, the Kullback-Leibler distance between the dyadic PMF and the capacity achieving PMF of the considered channel has to be minimized. See Matching Dyadic Distributions to Channels for details. On this web site I provide implementations of algorithms that find the optimal dyadic PMF.
subject to \(p\) is a dyadic PMF. Geometric Huffman Coding (GHC) finds the dyadic PMF \(p\) that minimizes \(\mathrm{D}(p\Vert x)\). GHC constructs a Huffman tree using the following updating rule. Denote by \(x_{m}\) and \(x_{m-1}\geq x_m\) the two smallest entries of \(x\). Then
For comparison, conventional Huffman coding finds the dyadic PMF \(p\) that minimizes \(\mathrm{D}(x\Vert p)\) (minimization is here over the second argument) by using the updating rule \(x' = x_{m-1}+x_m\).
ghc.m in matching.tar is a Matlab implementation of GHC.
Normalized Geometric Huffman Coding (nGHC)
Given is a target vector \(x\) with non-negative entries and a weight vector \(w\) with strictly positive entries. The objective is to minimize the Kullback-Leibler distance \(\mathrm{D}(p\Vert x)\) normalized by the average weight \(w^Tp=\sum_i p_iw_i\), i.e., to minimize the fraction
\[\frac{\mathrm{D}(p\Vert x)}{w^Tp}\]
subject to \(p\) is a dyadic PMF. Normalized Geometric Huffman Coding (nGHC) iteratively finds the solution. nGHC supersedes the LEC Algorithm stated in Matching Dyadic Distributions to Channels.
nghc.m in matching.tar is a Matlab implementation of nGHC.
Cost Constrained Geometric Huffman Coding (ccGHC)
Given is a target vector \(x\) with non-negative entries and a weight vector \(w\) with strictly positive entries. The objective is to minimize the Kullback-Leibler distance \(\mathrm{D}(p\Vert x)\) subject to an average cost constraint \(w^Tp\leq E\) over all dyadic PMFs \(p\). Cost Constrained Geometric Huffman Coding (ccGHC) iteratively finds the solution.
ccghc.m in matching.tar is a Matlab implementation of ccGHC.
w = (11:20)';
C = fsolve(@(s) sum(exp(-w*s))-1,1);
t = exp(-w*C);
d_nghc = nghc(t,w);
d_hc = hc(t);
R_nghc = -d_nghc'*log(d_nghc)/(w'*d_nghc);
R_hc = -d_hc'*log(d_hc)/(w'*d_hc);
disp(['codeword lengths of nghc: ' num2str(-log2(d_nghc'))]);
disp(['codeword lengths of hc : ' num2str(-log2(d_hc'))]);
disp(['rate achieved by nghc:' num2str(R_nghc)]);
disp(['rate achieved by hc: ' num2str(R_hc)]);
Output:
octave:7> addpath /home/georg/matching
octave:7> example2
codeword lengths of nghc: 3 3 3 3 3 3 4 4 4 4
codeword lengths of hc : 2 3 3 3 3 4 4 4 5 5
rate achieved by nghc:0.15273
rate achieved by hc: 0.15265
octave:8>
]]>Half Huffman Coding2018-10-28T00:00:00+02:002018-10-28T00:00:00+02:00http://localhost:4000/halfhcThe ultimate objective of source coding is to transform the data into a sequence of 0s and 1s that is indistinguishable from a fair bit stream, i.e., a sequence of independent and identically distributed (iid) equiprobable bits. Accordingly, most work on channel coding starts with the assumption that the data to be transmitted comes as a fair bit stream. On this web site we provide an implementation of half Huffman coding (halfHC), an algorithm that finds optimal prefix-free source codes that produce 1s with a frequency close to 0.5.
halfhc.tar: implementation of halfHC. It is tested in Matlab R2010b. Having the Communications System Toolbox installed is advantageous. Having the Optimization Toolbox installed is partially required. combn.m is provided in halfhc.tar.
Quotes.txt: the short English text to test halfHC.
]]>ICT-Cubes RWTH Aachen2018-10-28T00:00:00+02:002018-10-28T00:00:00+02:00http://localhost:4000/ict-cubesWe developed a webservice to decode the facades of the ICT-Cubes, the new buildings of the electrical engineering faculty at the RWTH Aachen University.
Our webservice allows to decode any slat sequence fragment. The fragment needs to be provided as a string that consists of l (left slat), r (right slat), and m (middle slat). Example:
You can copy any fragment of the complete slat sequence below and append it to ict-cubes.appspot.com/?slats= and let our webservice decode it.
]]>Publications2018-10-28T00:00:00+02:002018-10-28T00:00:00+02:00http://localhost:4000/publicationsBelow a selected list of publications. For a more complete list, see google scholar. I provide slides of selected talks on a separate page.
