Teached as one semester course at Technical University of Munich since 2020.
Lecture Notes on Machine Learning for Communications
Teached as one semester course at Technical University of Munich from 2013-2015.
Lecture Notes on Channel Coding.
Teached as short course Fundamentals of Bit-Interleaved Coded Modulation at NCTU, Taiwan. Teached as one semester course Coded Modulation at Technical University of Munich since 2014.
Principles of Coded Modulation.
Teached as short course Kodierung mit variabler Länge: Datenkompression und Verteilungsanpassung at the Chinesisch-Deutsches Hochschulkolleg in Shanghai, China.
In German: Kodierung mit variabler Länge: Datenkompression und Verteilungsanpassung.
In English: Lecture Notes on Variable Length Coding (in German).
]]>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
Given is a vector \(x\) with non-negative entries. If additionally the entries sum up to one, than \(x\) is a PMF. The objective is to minimize
\[\mathrm{D}(p\Vert x)=\sum_i p_i \log \frac{p_i}{x_i}\]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
\(x'=\begin{cases} x_{m-1},&\text{if }x_{m-1}\geq 4x_m\\ 2\sqrt{x_{m-1}x_{m}},&\text{if }x_{m-1}<4x_m \end{cases}\).
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.
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.
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.
example2.m
:
Output:
]]>combn.m
is provided in halfhc.tar.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.
]]>G Böcherer, P Schulte, F Steiner, Probabilistic Shaping and Forward Error Correction for Fiber-Optic Communication Systems, J. Lightw. Technol., vol. 37, no. 2, pp. 230–244, Jan. 2019.
F Steiner, P Schulte, G Böcherer, Approaching waterfilling capacity of parallel channels by higher order modulation and probabilistic amplitude shaping, 52nd Annual Conference on Information Sciences and Systems (CISS), 2018.
G. Böcherer, Tobias Prinz, Peihong Yuan, Fabian Steiner, Efficient Polar Code Construction for Higher-Order Modulation, presented at WCNC 2017.
F. Steiner, G. Böcherer, Comparison of Geometric and Probabilistic Shaping with Application to ATSC 3.0, presented at SCC 2017.
F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, W. Idler, Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration, J. Lightw. Technol., vol. 34, no. 7, 2016.
G. Böcherer, B. C. Geiger, Optimal Quantization for Distribution Synthesis, IEEE Trans. Inf. Theory, vol. 62, no. 11, pp. 6162-6172, 2016.
F. Steiner, G. Böcherer, G. Liva, Protograph-Based LDPC Code Design for Shaped Bit-Metric Decoding, IEEE J. Sel. Areas Commun., vol. 34, no. 2, pp. 397-407, 2016.
P. Schulte, G. Böcherer, Constant Composition Distribution Matching, IEEE Trans. Inf. Theory, vol. 62, no. 1, 2016.
G. Böcherer, F. Steiner, P. Schulte, Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation, IEEE Trans. Commun., vol. 63, no. 12, pp. 4651-4665, 2015.
G. Böcherer, Labeling Non-Square QAM Constellations for One-Dimensional Bit-Metric Decoding, IEEE Commun. Lett., vol. 18, no. 9, pp. 1515-1518, 2014.
G. Böcherer and R. A. Amjad, Informational Divergence and Entropy Rate on Rooted Trees with Probabilities, presented at ISIT 2014.
R. A. Amjad and G. Böcherer, Fixed-to-Variable Length Distribution Matching, presented at ISIT 2013.
G. Böcherer and R. Mathar, Operating LDPC Codes with Zero Shaping Gap, presented at ITW 2011, Paraty
G. Böcherer and R. Mathar, Matching Dyadic Distributions to Channels, presented at DCC 2011, Snowbird.
G. Böcherer, V.C. da Rocha Jr., C. Pimentel, and R. Mathar, On the Capacity of Constrained Systems, presented at SCC 2010, Siegen.