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A Jupyter Notebook for Line Coding in Access Networks (LB)

Beschreibung

For the access network case, the spectrum of the transmit signal has to be adapted to the channel properties. This can either be achieved by choosing suitable transmit pulse shapes or by encoding the (redundancy free) source symbols [1].

The students task is to implement a demonstration of two line coding schemes in Python [2] (Jupyter Notebook) and visualize the results. Additionally, the student also has to arrange code and surrounding text, such that the content becomes self-explanatory.

[1] Skript "Physical Layer Methods“

[2] "Python in 30 minutes" (https://www.programiz.com/python-programming/tutorial)

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A Jupyter Notebook for Equalization Methods (LB)

Beschreibung

Depending on the channel properties, the receive signal in a communication system can be severely distorted, causing inter-symbol interference. To mitigate these interferences, several approaches for equalization can be taken [1].

The students task is to implement a demonstration of several equalization schemes in Python [2] (Jupyter Notebook) and visualize the results. Additionally, the student also has to arrange code and surrounding text, such that the content becomes self-explanatory.

[1] Skript "Physical Layer Methods“

[2] "Python in 30 minutes" (https://www.programiz.com/python-programming/tutorial)

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Computation of Identification Function Over Faulty Channels

Stichworte:
Identification via channels, identification function, complexity of distributive computing, one/two-way probabilistic communications

Beschreibung

Identification problem (Ahlswede and Dueck, 1989) over the Binary Symmetric Channel (BSC) is linked to the communication complexity problem (one/two way communications) and interpreted in that context.

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Quasi-Linear Multiplexing in NFDM Systems with Purely Discrete Nonlinear Spectrum

Beschreibung

As the achievable rates of modern transmission systems seem to saturate, while the bandwidth demand is steadily growing, it is necessary to consider alternative approaches for fiber optic data transmission. In recent years, many publications have explored possibilities to overcome the phenomenon, commonly known as 'capacity crunch', by using the nonlinear Fourier transform (NFT).

In a special case of nonlinear frequency division multiplexing (NFDM) only discrete eigenvalues of the Zakharov-Shabat-system are used for multiplexing data streams in the nonlinear Fourier domain. While, at first, this seems appealing, because the resulting signal pulses are N-soliton breathers and thus are not affected by chromatic dispersion in the same way as e.g. wave division multiplexing (WDM) signals, the time-bandwidth product of such pulses is rather high when compared to pulses used in WDM systems. This results in a low modulation efficiency for such discrete nonlinear spectrum NFDM systems.

One possible option to increase the spectral efficiency of the previously mentioned NFDM systems is to extend the modulated linear frequency range by linearly multiplexing such NFDM signals in a WDM fashion. Since the transformations used in NFDM systems are quite involved, it is not clear what the analog of this in nonlinear domain would be.

The task of the student would be to first get familiar with the necessary preliminaries regarding NFDM systems. Subsequently, the simulation of the system described above has to be implemented (certain parts of the system will be made available to the student by the supervisor) and evaluated in terms of some performance metrics.

While specific literature will be recommended to the student over the course of the thesis, some basic literature discussing the 'capacity peak' of WDM modulated optical fiber systems [1] and the basics of the nonlinear Fourier transform (NFT) [2-4], which is a central concept in NFDM systems, are given below. Note that, since NFDM is a very broad topic it is not necessary to fully understand every notion in [1-4]. Nonetheless, these publications give the reader a solid foundation for the further study of this topic.

[1] Essiambre, René-Jean, et al. "Capacity limits of optical fiber networks."

[2] Yousefi, Mansoor I., and Frank R. Kschischang. "Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools."

[3] Yousefi, Mansoor I., and Frank R. Kschischang. "Information transmission using the nonlinear Fourier transform, Part II: Numerical methods."

[4] Yousefi, Mansoor I., and Frank R. Kschischang. "Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation."

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[identification] Implementation of identification with algebraic-geometry (Goppa) codes

Stichworte:
goppa algebraic geometry codes identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.

• https://ieeexplore.ieee.org/document/42172
• https://ieeexplore.ieee.org/document/42173

The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Identification codes can be achieved by first removing the errors from the channel with regular transmission channel coding, and then sending a challenge though the corrected channel. For every identity i, The channenge is generated by picking a random input m and computing the corresponding output T_i(m) using a function T_i that depends on the identity. The challenge is then the pair m,T_i(m) and the receiver wanting to verify an identity j will verify whether j=i by testing the challenge. This is done by recomputing the output with T_j and verifying whether T_j(m)= T_i(m). The errors are reduced by ensuring that the various functions collide on a small fraction of the possible inputs.

It turns out that choosing good sets of funtions {T_i} is the same as choosing error-correction codes {c_i} with large distance, where now each codeword c_i defines a function by mapping positions m (sometimes called code locators) to symbols c_im of the codeword.
We can thus construct identification codes by choosing error-correction codes where we are only interested in the performance of the error correction encoders (we are not interested in the error-correction decoder or error-correction codes).

Your task will be implementing identification with Goppa codes, aiming at the fastest implementation, and testing their performance in comperison to other current implementations. The reference articles for this implementation are:

• https://ieeexplore.ieee.org/document/1056879
• https://ieeexplore.ieee.org/document/1057344

The coding will be in Python/Sagemath.
The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Implementation of identification with universal hash functions

Stichworte:
universal hash identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.
The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Your task will be implementing the identification codes described in https://ieeexplore.ieee.org/abstract/document/782144, aiming at the fastest implementation, and testing their performance in comperison to other current implementations.
The coding will be in Python/Sagemath

The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Implementation of identification with Polar codes

Stichworte:
polar codes identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.

• https://ieeexplore.ieee.org/document/42172
• https://ieeexplore.ieee.org/document/42173

The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Identification codes can be achieved by first removing the errors from the channel with regular transmission channel coding, and then sending a challenge though the corrected channel. For every identity i, The channenge is generated by picking a random input m and computing the corresponding output T_i(m) using a function T_i that depends on the identity. The challenge is then the pair m,T_i(m) and the receiver wanting to verify an identity j will verify whether j=i by testing the challenge. This is done by recomputing the output with T_j and verifying whether T_j(m)= T_i(m). The errors are reduced by ensuring that the various functions collide on a small fraction of the possible inputs.

It turns out that choosing good sets of funtions {T_i} is the same as choosing error-correction codes {c_i} with large distance, where now each codeword c_i defines a function by mapping positions m (sometimes called code locators) to symbols c_im of the codeword.
We can thus construct identification codes by choosing error-correction codes where we are only interested in the performance of the error correction encoders (we are not interested in the error-correction decoder or error-correction codes).

Your task will be implementing identification with polar codes, aiming at the fastest implementation, and testing their performance in comperison to other current implementations.

The coding will be in Python/Sagemath.
The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Applications of Identification Codes in V2X Communications

Beschreibung

As part of the NewCom Project, new communication paradigms are investigated from an experimental perspective in order to construct proof-of-concept implementations that demonstrate the theoretical results obtained for Post-Shannon Communication schemes. In particular, this MSc thesis focuses on Identification Codes and their integration into a simulation environment where vehicular networks are modelled.

For this, the master student will first conduct a review of the state-of-the-art use cases for identification in the scientific literature and in form of patents, with an emphasis on V2X communications. By using an open-source V2X implementation based on LDR’s Simulation of Urban Mobility (SUMO) framework integrated with ns-3’s implementation of the ITS-G5 and LTE standards and conducting simulation in specific scenarios, the student will gain a first impression of the performance of the system using traditional transmission schemes. The integration of existing implementation of identification codes culminates this thesis, where KPIs will be defined in order to compare the advantages of using identification instead of transmission in the context of V2X communications.

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[identification] Simulation and performance improvement of identification codes

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.
The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Your task will be implementing and testing new ideas toward this goal.
The coding will be in Python/Sagemath

This work can accomodate multiple students.

The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[security] Practical implementation of physical-layer semantic security

Stichworte:
semantic, security, secrecy, programming, implementation

Beschreibung

The goal of this project is to implement in Python/Sagemath the security functions (at least one of four) described in https://arxiv.org/abs/2102.00983
Sagemath contains libraries for mosaics, BIBDs, etc, that can be used for the project.

Motivation:
There are various types of security definitions.
The mutual information based types, in increasing order of security requirement are

1. Weak secresy asks that the average mutual information of the eavesdropper I(M:E)/n goes to 0 for a uniform message M (average here means averaged over the blocklength n, an additional average over M is implicit in the mutual information)
2. Strong secrecy asks that the total mutual information I(M:E) goes to 0,
3. Semantic security asks that the total mutual informaiton I(M:E) goes to 0 for any distribution of the message M (and thus in particular for all distributions that pick any of two chosen messages with 1/2 probabilty)

Then there are the almost-equivalent respective indistiguishablity types  of security requirements (below |P-Q|_1 is the statistical distance and Exp_M is expectation value over M)

1. average indistinguishability 1/n Exp_M | P_{E|M} - P_E |_1 for a uniform message M goes to 0 (again average refers over the blocklegth n, clearly there is also the average over M)
2. total indistiguishability Exp_M | P_{E|M} - P_E |_1 for a uniform message M goes to 0
3. indistinguishability |P_{E|m} - P_{E|m'}|_1 for any two messages m and m' goes to 0.

Each of the indistiguishabilities can also be written using KL digvergence instead of statistical distance, in which case the conditions are exactly equivalent to their mutual information versions.

Strong secrecy is the standard security requirement considered in information-theoretic security, while semantic security is the minimum requirement considered in computational security.
Information-theoretic (physical-layer) security differs from computational security in that the secrecy is guaranteed irrespective of the power of the adversary, while in computational security E is computationally bounded. Computational security also assumes that the message is at least of a certain length for the schemes to work, and thus if the message to be secured is too small it needs to be padded to a larger message.

In practice, information theoretic security is expensive, because the messages that can be secured can be only as long as the keys that can be generated. However, in identification only a very small part of the message needs to be secured, which in computational security triggers padding and thus waste, but on the other side makes information-theoretic security accessible and not so expensive.

At the same time, the security of identification implicitly requires semantic security. It has been known for a while that hash functions provide information-theoretic strong secrecy. However, because the standard for information-theoretic security has been strong secrecy, before https://arxiv.org/abs/2102.00983 no efficient functions where known to provide information-theoretic semantic security.
We need an implementation of these type of functions so that we can integrate information-theoretic security into our identification project.

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[quantum] Realignment criterion and upper bounds in device-independent QKD

Beschreibung

This paper uses the partial transpose as a tool to derive upper bounds on device-independent QKD
https://arxiv.org/abs/2005.13511
In this project the goal is to try to generalize the above to the other tools like the reallignment criterion:
https://arxiv.org/abs/quant-ph/0205017
https://arxiv.org/abs/0802.2019

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[quantum] Semantic security of infinite-dimensional classical-quantum channels

Beschreibung

Generalize semantic security of classical-quantum channels to infinite dimensional channel (not necessarily gaussian)

• [1] finite dimensional classical-quantum case
  https://arxiv.org/abs/2001.05719
• finite and infinite dimensional classical case
  https://arxiv.org/abs/1811.07798
• [this subpoint can be a project by itself] the finite dimesional case needs to be recast into smooth-max information (instead than Lemma 5.7 of [1]) as the classical case does, this paper proves properties of the smooth-max-inf in finite dimension that we would need for that
  https://arxiv.org/abs/2001.05719
• papers regarding the capacity for infinite dimensional channels
  http://arxiv.org/abs/quant-ph/9912067v1
  http://arxiv.org/abs/quant-ph/0408009v3
  http://arxiv.org/abs/quant-ph/0408176v1

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[quantum] Asymptotic continuity of restricted quantum relative entropies under general channels

Stichworte:
quantum, relative entropy, Pinsker, reverse, inequality, information thoery, asymptotic, continuity

Beschreibung

Asypmtotic continuity is a property in the form of inequalities (classically known also as inequalities of the reverse-Pinker type) that is necessary to prove upper bounds on operational capacities.

The (quantum) relative entropy (also known as quantum divergence and classically also known as Kullbackt-Leibler divergence), can be used to define various entanglment measures many of which have a proven asymptotic continuity.

Of particular interest are the restricted quantum relative entropies defined by Marco Piani (https://arxiv.org/abs/0904.2705), many of which satisfy asymptotic continuity (A.S.)

• https://arxiv.org/abs/quant-ph/9910002
• https://arxiv.org/abs/quant-ph/0203107
• https://arxiv.org/abs/quant-ph/0507126
• https://arxiv.org/abs/1210.3181
• https://arxiv.org/abs/1507.07775
• https://arxiv.org/abs/1512.09047

In the above there are maybe 2-3 different proof styles.
We can group the results in the above as follows:

• A.S. for entropy, conditional entropies, mutual information, conditional mutual information
• A.S. for relative entropies with infimum over states on the second argument
• A.S. relative entropies with infimum over state *and maximization over measurement channels*

The goal of the project is to generalize the last case to asymptotic continuity for relative entropies with infimum over state and maximization over *general* channels

• Partial results toward this goal can be found in the appendix of my PhD thesis: http://web.math.ku.dk/noter/filer/phd18rf.pdf
• Such a result would have immediate applications to this paper: https://arxiv.org/abs/1801.02861
• Possible new proof directions could involve the use of Renyi α-realtive entropies with the limit α->1

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[quantum] Practical protocols for quantum synchronization in classical network

Stichworte:
quantum, network, synchronization

Beschreibung

relevant papers
https://arxiv.org/abs/1310.6043
https://arxiv.org/abs/1304.5944
https://arxiv.org/abs/1310.6045
https://arxiv.org/abs/1703.05876
https://arxiv.org/abs/1303.6357
background papers
https://ieeexplore.ieee.org/document/7509657

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[quantum] Entanglement-measures upper bounds on device-independent distillable key

Stichworte:
quantum, qkd, entanglement

Beschreibung

The goal of this work is to try to upper bound the device-independent distillable key in terms of locally restricted relative entropy of entanglement (an entanglement measure).

The following are relevant works/articles

• works toward even *a definition* of device independent distillable key
  https://arxiv.org/abs/2005.13511
  https://arxiv.org/abs/2005.12325
  https://arxiv.org/abs/1810.05627
• works relating distillable entanglement and distillable key to locally restricted relative entropy measures
  https://arxiv.org/abs/1609.04696
  https://arxiv.org/abs/1402.5927
• the first definition of restricted relative entropies
  https://arxiv.org/abs/0904.2705
• important properties of restricted relative entropies, and some overview of entanglement measures
  https://arxiv.org/abs/1210.3181
• my PhD thesis
  http://web.math.ku.dk/noter/filer/phd18rf.pdf

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Intelligent Reflecting Surface -aided Beam Alignment for mmWave Communications

Stichworte:
6G, Intelligent reflecting surface (IRS), mmWave beam alignment

Beschreibung

The future generation communication networks will be operated mainly at millimeter wave (mmWave) or even higher frequency bands in order to achieve high spectral efficiency as well as accurate localization/positioning, necessary for emerging autonomous applications. At such a high frequency bands, beamforming both at the transmitter and the receiver, or beam alignment (BA), is considered essential to compensate the high propagation and penetration loss. The design of BA achieving a good tradeoff between alignment accuracy and resource overhead has been extensively studied in the literature [1,2].  In particular, a number of recent works proposed to exploit some side information, such as location [3,4], database [5], or radar [6], at the base station side to speed up the initial acquisition time. 

Intelligent reflecting surface (IRS) consists of a large number of passive reflecting elements that can be easily controlled at a base station or access points to cooperatively beamform without the need of any radio-frequency (RF) chains. So far, a number of recent works showed that IRS can be used as an external fixed helper to increase the coverage, mitigate the interference of the network [7] and references therein). Due to its low cost together with the advanced, future user terminals can be also equipped with small IRS. This IRS-integrated device provides a new opportunity that we wish to exploit to speed up the BA protocol. 

In this mater thesis, we study IRS-aided BA and quantify how much the resource can be saved as a function of the IRS parameters. To this end, we aim to organize the work as follows:

• Understand the basic of IRS and mmWave BA.
• Study various types of IRS space-time (coding) functions to control passive reflecting elements [8].
• Study the tradeoff between the alignment accuracy and the required resource. 

References

[1] X. Song, S. Haghighatshoar, and G. Caire, ``Efficient Beam Alignment for Millimeter Wave Single-Carrier Systems With Hybrid MIMO Transceivers," IEEE Trans. Wireless Commun., vol. 18, no. 3,pp. 1518-1533, 2019.

[2] S. Chiu, N. Ronquillo, and T. Javidi, ``Active Learning and CSI Acquisition for mmWave InitialAlignment," IEEE J. Sel. Areas Commun., vol. 37, no. 11, pp. 2474-2489, 2019.

[3] V. Va, T. Shimizu, G. Bansal, and R. W. Heath, ``Position-aided millimetre wave V2I beam alignment:A learning-to-rank approach," in 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC). IEEE, 2017, pp. 1-5.

[4] A. U. Rahman and M. Biswal, ``Error-tolerant Beam Steering of mmWave Antennas by Trajectory Estimation of Highway Vehicles," in 2019 11th International Conference on Communication Systems Networks (COMSNETS), 2019, pp. 530-531.

[5] V. Va, J. Choi, T. Shimizu, G. Bansal, and R. W. Heath, ``Inverse multipath fingerprinting for millimeter wave V2I beam alignment," IEEE Trans. Veh. Technol., vol. 67, no. 5, pp. 4042-4058, 2018.

[6] Nuria Gonzalez-Prelcic, Roi Mendez-Rial, and Robert W. Heath, ``Radar aided beam alignment in mmwave V2I communications supporting antenna diversity," in Information Theory and Applications Workshop (ITA), 2016. IEEE, 2016, pp. 1-7.

[7] Qingqing Wu, Shown Zhang, Beixiong Zheng, Changsheng You, and Rui Zhang, ``Intelligent reflecting surface aided wireless communications: A tutorial," IEEE Trans. Wireless Commun., 2021.

[8] Lei Zhang et al., ``Space-time-coding digital metasurfaces,” Nature communications, vol. 9, no. 1, pp. 1-11, 2018.

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Achievable rate of Wiener and Markov phase noise channels

Beschreibung

The Nonlinear Schrödinger Equation (NLSE) describess propagation in optical channels. Its nonlinearity can be modeled as Gaussian phase noise with memory [1, 2]. A popular simplification of this model is the Wiener phase noise model, which has been used to compute capacity lower bounds [3]. However, the Wiener phase noise model is statistically inaccurate in the sense that it tends to a uniform distribution instead of a Gaussian one. We have recently proposed a Markov-Gaussian phase noise model that is statistically closer to the NLSE.

The goal of this thesis is to explore capacity bounds on the Wiener and Markov phase noise models and compare them.

[1] A. Mecozzi and R. Essiambre, "Nonlinear Shannon Limit in Pseudolinear Coherent Systems," in Journal of Lightwave Technology, vol. 30, no. 12, pp. 2011-2024, June15, 2012, doi: 10.1109/JLT.2012.2190582.

[2] Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, "Properties of nonlinear noise in long, dispersion-uncompensated fiber links," Opt. Express 21, 25685-25699 (2013)

[3] M. Secondini, E. Agrell, E. Forestieri and D. Marsella, "Fiber Nonlinearity Mitigation in WDM Systems: Strategies and Achievable Rates," 2017 European Conference on Optical Communication (ECOC), Gothenburg, 2017, pp. 1-3, doi: 10.1109/ECOC.2017.8346177.

[4] F. J. García-Gómez and G. Kramer, "Mismatched Models to Lower Bound the Capacity of Optical Fiber Channels," in Journal of Lightwave Technology, vol. 38, no. 24, pp. 6779-6787, 15 Dec.15, 2020, doi: 10.1109/JLT.2020.3021277.

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Explicit Construction of Deterministic Identification Codes

Stichworte:
Identification via channels, identification codes,

Beschreibung

In this thesis, the student after studying deterministic identification will construct the explicit codes for certain channels.

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Group testing techniques based on sparse graphs for large-scale population screening

Beschreibung

Group testing is a combinatorial technique (developed by R. Dorfman in 1943) which allows to detect infected individuals by running “pooled” tests on (blood) samples. More specifically, by merging the samples of a subset of individuals into a pool, a test is carried out to verify the positivity (or negativity) of the pool. By repeating the test on different subsets, it is eventually possible to detect the individuals carrying the infection, possibly with very few tests compared with the test population. The approach is currently scouted to speed-up the testing in the context of the COVID-19  pandemic (see https://www.nature.com/articles/d41586-020-02053-6 and [1-2]). The technique admits a description which shares several similarities with the syndrome-based error correction problem via linear block codes. It is not a surprise that major contributions in the area of group testing have been given by coding and information theorists.

The scope of the thesis is to investigate the adoption of capacity-approaching codes based on sparse graphs (e.g., low-density parity-check codes) to attack the group testing problem. The use of sparse-graph codes for this purpose has been already envisaged in [3-4]. In this work, we will address the design of adaptive/non-adaptive and quantitative/non-quantitative group testing techniques based on sparse graphs, under various detection algorithms (belief propagation, as well as combinatorial orthogonal matching pursuit).

[1] Mutesa, Leon, et al. "A strategy for finding people infected with SARS-CoV-2: optimizing pooled testing at low prevalence." arXiv preprint arXiv:2004.14934 (2020). [2] Narayanan, Krishna R., Anoosheh Heidarzadeh, and Ramanan Laxminarayan. "On Accelerated Testing for COVID-19 Using Group Testing." arXiv preprint arXiv:2004.04785(2020). [3] K. Lee, R. Pedarsani and K. Ramchandran, "SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes," 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, 2016  - available at https://arxiv.org/abs/1508.04485  [4] Aldridge, Matthew, Oliver Johnson, and Jonathan Scarlett. "Group testing: an information theory perspective,: NOW Published, 2020 - available at https://arxiv.org/pdf/1902.06002.pdf

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Efficient Implementation of Decryption Algorithms in Rank-Based Cryptography

Beschreibung

Many Rank-based cryptosystems require decoding of Gabidulin codes in their decryption algorithm. In this work, the student should compare the theoretical complexity of different Gabidulin code decoders. Based on the theoretical complexity analysis, the students should decide on the most promising decoding algorithms. Then, the algorithms should be implemented in C and their performance should be compared on a microcontroller.

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The coin weighing problem

Beschreibung

The question of finding a small subset of defective coins from a set of regular coins in the fewest number of weighings has been a notorious problem.

Suppose there is a collection of n coins so that some of them are defective. In other words, we know that the weight of regular coins is A, and the weight of the remaining coins is B, where integers A and B are given.  The problem is to determine the weight of each coin by weighing subsets of coins in a spring scale.  The main figure of merit when studying adaptive coin weighing algorithms is the number of required weighings in the worst-case and in the average-case. In this project, we design and implement an efficient algorithm which works for two coins.

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Error Correction in DNA Storage

Stichworte:
DNA storage, Error Correction, Deletion, Insertion, Substitutions

Beschreibung

DNA storage is an uprising topic in the research field of storage systems. Due its natural longetivity, robustness, and density properties the main application would arise in high-dense long-term storage systems. The interest has become larger and larger due the large amount of data nowadays and the relative new biological advances in DNA synthesis and sequencing processes (e.g. polymerase chain reaction). In contrary to conventional storing methods, due to the nature of DNA and the involved biological processes special error patterns such as insertion, deletion, and substitution errors occur. To tackle these errors novel methods for correction have to be investigated. Moreover, the model of the DNA storage channel needs to be investigated thorougly, e.g. capacity statements.

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How to guess an n-digit number

Beschreibung

In a deductive game for two players, Alice and Bob, Alice conceals an n-digit number x, and Bob, who knows n, tries to identify x by asking a number of questions, which are answered by Alice. Each question is an n-digit number y, and each answer is the number of positions at which the corresponding symbols of x and y are different. Moreover, we require Bob to ask all questions at once. The goal of this project is to design and implement an efficient coding scheme for Bob with an asymptotically optimal number of questions.  

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DNA Storage Channel Modeling and Error Correction

Beschreibung

DNA-based data storage is a novel approach for long term data archiving.

Due to the unique nature of writing and reading DNA, the channel associated with these processes is still realtively poor understood and varies over different  synthesis (writing) and sequencing (reading) technologies. The task of the student is to analyze different sequencing methods and the associated errors and formulate associated channel models. Based on these models, error-correcting schemes shall be evaluated.

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Algebraic Coding for Distributed Storage

Beschreibung

Distributed storage systems with a large number of storage commonly rely on MDS codes, such as Reed-Solomon codes, to protect the system from data loss in the event of node failures while keeping the storage overhead low. In the event of such a failure, the reconstruction of failed nodes induces a large amount of network traffic. In recent years several solutions to this problem have been proposed, most notably locally repairable codes and regenerating codes. We investigate the properties of specific subclasses of these codes, as well as the combination of the two properties.
Further, with the increased demand for data privacy, we develop methods for protecting users' data from the eyes of curious servers in distributed storage systems.

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Homomorphic encryption

Stichworte:
Cryptography

Beschreibung

Consider that a client would like to a server to do some computations for him but he does not want to give information meaningful information to the server. The client therefore sends encrypted messages c1 = Enc(pk, m1) and c2 = Enc(pk, m2) to the server and the client would like to obtain some function f of the two plaintexts f(m1,m2). It suffices for the client to get Enc(pk, f(m1,m2)) because the client owns the secret key sk. He is able to use the decryption function Dec on the ciphertext and gets Dec(sk, Enc(pk,f(m1,m2=))) = f(m1,m2).

The goal of this internship is to analyze schemes that achieve this property based on code-based cryptography.

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Learning Aided SC Flip Decoding for Polar Codes

Beschreibung

Polar codes achieve the capacity of binary-input discrete memoryless channels asymptotically in the block length under successive cancellation (SC) decoding. Polar codes have been adopted for the control channel in 5G enhanced mobile broadband (eMBB).

Due to the serial nature of SC decoding, an erroneous bit decision can be caused by the channel noise or previous erroneous bit estimates. The main idea of SC flip decoding is trying to correct the first erroneous bit decision by sequentially flipping the unreliable decisions. 

The optimal flipping strategy is considered difficult due to lack of an analytical solution. Alternatively, (deep) learning aided SC flip algorithm are investigated in this thesis.

[1] O. Afisiadis, A. Balatsoukas-Stimming, and A. Burg, “A low-complexity improved successive cancellation decoder for polar codes,” in Proc. 48th Asilomar Conf. Signals, Systems and Computers, pp. 2116-2120, 2014.

[2] L. Chandesris, V. Savin, and D. Declercq, “Dynamic-SCFlip Decoding of Polar Codes,” IEEE Trans. Commun., vol. 66, no. 6, pp. 2333-2345, Jun., 2018.

[3] X. Wang, et al. "Learning to Flip Successive Cancellation Decoding of Polar Codes with LSTM Networks." arXiv preprint arXiv:1902.08394 (2019).

[4] N. Doan, et al. "Neural Dynamic Successive Cancellation Flip Decoding of Polar Codes." arXiv preprint arXiv:1907.11563 (2019).

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Polar Coding with Non-Binary Kernels

Beschreibung

This thesis will focus on polar codes with non-binary kernels on GF(q). Some of the following tasks might be covered: 

• Kernel selection
• Decoder implementation
• Efficient construction
• Comparison of binary and non-binary polar codes

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[identification] Implementation of identification with algebraic-geometry (Goppa) codes

Stichworte:
goppa algebraic geometry codes identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.

• https://ieeexplore.ieee.org/document/42172
• https://ieeexplore.ieee.org/document/42173

The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Identification codes can be achieved by first removing the errors from the channel with regular transmission channel coding, and then sending a challenge though the corrected channel. For every identity i, The channenge is generated by picking a random input m and computing the corresponding output T_i(m) using a function T_i that depends on the identity. The challenge is then the pair m,T_i(m) and the receiver wanting to verify an identity j will verify whether j=i by testing the challenge. This is done by recomputing the output with T_j and verifying whether T_j(m)= T_i(m). The errors are reduced by ensuring that the various functions collide on a small fraction of the possible inputs.

It turns out that choosing good sets of funtions {T_i} is the same as choosing error-correction codes {c_i} with large distance, where now each codeword c_i defines a function by mapping positions m (sometimes called code locators) to symbols c_im of the codeword.
We can thus construct identification codes by choosing error-correction codes where we are only interested in the performance of the error correction encoders (we are not interested in the error-correction decoder or error-correction codes).

Your task will be implementing identification with Goppa codes, aiming at the fastest implementation, and testing their performance in comperison to other current implementations. The reference articles for this implementation are:

• https://ieeexplore.ieee.org/document/1056879
• https://ieeexplore.ieee.org/document/1057344

The coding will be in Python/Sagemath.
The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Implementation of identification with universal hash functions

Stichworte:
universal hash identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.
The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Your task will be implementing the identification codes described in https://ieeexplore.ieee.org/abstract/document/782144, aiming at the fastest implementation, and testing their performance in comperison to other current implementations.
The coding will be in Python/Sagemath

The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Implementation of identification with Polar codes

Stichworte:
polar codes identification

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.

• https://ieeexplore.ieee.org/document/42172
• https://ieeexplore.ieee.org/document/42173

The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Identification codes can be achieved by first removing the errors from the channel with regular transmission channel coding, and then sending a challenge though the corrected channel. For every identity i, The channenge is generated by picking a random input m and computing the corresponding output T_i(m) using a function T_i that depends on the identity. The challenge is then the pair m,T_i(m) and the receiver wanting to verify an identity j will verify whether j=i by testing the challenge. This is done by recomputing the output with T_j and verifying whether T_j(m)= T_i(m). The errors are reduced by ensuring that the various functions collide on a small fraction of the possible inputs.

It turns out that choosing good sets of funtions {T_i} is the same as choosing error-correction codes {c_i} with large distance, where now each codeword c_i defines a function by mapping positions m (sometimes called code locators) to symbols c_im of the codeword.
We can thus construct identification codes by choosing error-correction codes where we are only interested in the performance of the error correction encoders (we are not interested in the error-correction decoder or error-correction codes).

Your task will be implementing identification with polar codes, aiming at the fastest implementation, and testing their performance in comperison to other current implementations.

The coding will be in Python/Sagemath.
The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[identification] Simulation and performance improvement of identification codes

Beschreibung

Identification is a communication scheme that allows rate doubly exponential in the blocklemght, with the tradeoff that identities cannot be decoded (as messages do) but can only be verified.
The double exponential growth presents various challenges in the finite regime: there are heavy computational costs introduced at the encoder and decoder and heavy trade-offs between the error and the codes sizes.

The ultimate goal is to find a fast, reliable implementation while still achieving large code sizes.

Your task will be implementing and testing new ideas toward this goal.
The coding will be in Python/Sagemath

This work can accomodate multiple students.

The working language will be in English.

Environment: we collaborate with LTI. At LNT and LTI there is currently a lot of funding for research in identification. Therefore you will find a large group of people that might be available for discussion and collaboration.

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[quantum] Realignment criterion and upper bounds in device-independent QKD

Beschreibung

This paper uses the partial transpose as a tool to derive upper bounds on device-independent QKD
https://arxiv.org/abs/2005.13511
In this project the goal is to try to generalize the above to the other tools like the reallignment criterion:
https://arxiv.org/abs/quant-ph/0205017
https://arxiv.org/abs/0802.2019

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Fano's Inequality: Applications in Communications and Beyond the Communication Problem

Beschreibung

Information theory plays an indispensable role in the development of algorithm-independent imposibility results, both for communication problems and for seemingly distinct areas such as statistics and machine learning. While numerous information-theoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread: Fano's inequality.

References:

[1]  An Introductory Guide to Fano's Inequality with Applications in Statistical Estimation.  Jonathan Scarlett, Volkan Cevher. (and references within).

https://arxiv.org/abs/1901.00555

Please contact the supervisor for possible additional references, too.

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Decoding of product codes

Beschreibung

The task of the student is to implement iterative decoding of product codes.

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BCJR Algorithm

Beschreibung

The task of the student would be to implement the Bahl, Cocke, Jelinek and Raviv (BCJR) algorithm for linear block codes and comvolutional codes

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Identification Codes via Prime Numbers

Stichworte:
Identification via channels, Prime Number Encryption

Kurzbeschreibung:
An approach for construction of identification codes for noiseless channel by means of the prime number encryption would be studied.

Beschreibung

In original scheme of identificaion via channels (Ahlswede and Dueck, 1989), a non-constructive method for coding for noiseless channel was studied. To address the explicit construction of identificaion codes, foremost Ahlswede and Verboven, 1991 provide a number theoretic approach based on the two successive prime number encryption. This method require the knowledge of first 2^n prime numbers for a block-length of n codeword. In this research internship, this method along with related prime number encryption tools and theorems would be investigated. Further, the extension of this scheme to a general DMC will be analyzed.

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On the Equivalence of Identification and Authentication

Stichworte:
Identification via channel, identification codes, authentication, authentication codes

Kurzbeschreibung:
A Certain equivalence of identification and authentication would be shown.

Beschreibung

It would be shown that under suitable formulations (preserving all salient features) the two problem of Identification (Ahlswede and Dueck, 1989) and Authentication (Simmons, G. J. 1984) are in essence very close to each other. This equivalency was conjectured first by M. S. Pinsker. In this research internship the student is expected to address this conjecture. Both problems must be studied separately and then the similar essence of them should be drawn out. In particular the identification codes and authentication codes along with theire relation will be investigated.

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Analysis of Criss-Cross Deletion in Arrays

Beschreibung

yle="font-weight: normal;" lang="de-DE" align="left">In this project we study two dimensional deletion/insertion problem introduced in https://arxiv.org/pdf/2004.14740.pdf. An array of n rows and n columns is transmitted through a specific deletion channel which for example occurs in DNA storage systems and racetrack memories. After transmission the receiver observes an array of dimension (n-1)*(n-1). The goal is to allow the receiver to know exactly which n*n array was transmitted by using error-correction techniques.

The goal of the project is to derive bounds on the deletion ball of any two-dimensional array. A deletion ball of an array X is the set of unique arrays resulting from the deletion of any possible combination of a column and a row in X. Deriving such bounds is not trivial since it depends heavily on the structure of the array. Thus, it is important to investigate useful characteristics representing the structure of an array for this particular setting. Furthermore, having this bound will be helpful to characterize the optimal redundancy of a deletion-correcting code for this setting.

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Index Coding & Coded Caching

Beschreibung

Coded caching problem has two phases, placement phase and delivery phase.

For a fixed placement scheme, designing the delivery protocol is equivalent to the index coding problem.

Transforming a coded caching problem with coded placement to the corresponding index coding problem is a hignly-interested topic and what is the optimal delivery scheme for a coded caching problem with coded placement is still an open problem.

References: 

[1] N. S. Karat, S. Samuel, and B. S. Rajan. Optimal error correcting index codes for some generalized index coding problems. IEEE Transactions on Communications, 67(2):929–942, 2019.

[2] Z. Chen, P. Fan, and K. B. Letaief. Fundamental limits of caching: improved bounds for users with small buffers. IET Communications, 10(17):2315–2318, 2016.

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Error Correction in DNA Storage

Stichworte:
DNA storage, Error Correction, Deletion, Insertion, Substitutions

Beschreibung

DNA storage is an uprising topic in the research field of storage systems. Due its natural longetivity, robustness, and density properties the main application would arise in high-dense long-term storage systems. The interest has become larger and larger due the large amount of data nowadays and the relative new biological advances in DNA synthesis and sequencing processes (e.g. polymerase chain reaction). In contrary to conventional storing methods, due to the nature of DNA and the involved biological processes special error patterns such as insertion, deletion, and substitution errors occur. To tackle these errors novel methods for correction have to be investigated. Moreover, the model of the DNA storage channel needs to be investigated thorougly, e.g. capacity statements.

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Homomorphic encryption

Stichworte:
Cryptography

Beschreibung

Consider that a client would like to a server to do some computations for him but he does not want to give information meaningful information to the server. The client therefore sends encrypted messages c1 = Enc(pk, m1) and c2 = Enc(pk, m2) to the server and the client would like to obtain some function f of the two plaintexts f(m1,m2). It suffices for the client to get Enc(pk, f(m1,m2)) because the client owns the secret key sk. He is able to use the decryption function Dec on the ciphertext and gets Dec(sk, Enc(pk,f(m1,m2=))) = f(m1,m2).

The goal of this internship is to analyze schemes that achieve this property based on code-based cryptography.

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Distributed learning on serverless compute services

Stichworte:
Distributed machine learning, straggler mitigation, neural networks, coding theory

Kurzbeschreibung:
The goal of this project is to implement a neural network on thousands of serverless computing instances. We start by evaluating two existing coding techniques designed to scale stochastic gradient descent for convex loss functions. We then deviate to devise a coding scheme tailored for distributed neural network that scale for convex and non convex loss functions.

Beschreibung

Applying machine learning on large data became part of daily life. Scalability of distributed machine learning algorithms requires tolerance of slower computing nodes referred to as stragglers. To mitigate the effect of stragglers, one can add redundancy to the distributed data. However, stochastic gradient descent (SGD) is known to perform well even when computing on a part of the data. It is shown that for convex loss functions, a small redundancy or no redundancy at all is enough to guarantee good performance of SGD.

In this project, we evaluate the existent coding techniques requiring small redundancy on serverless compute functions offered by Amazon Web Services, or equivalently cloud functions offered by Google Computing Platform. Our goal is to devise a coding technique that goes beyond SGD and can be used for non-convex loss functions. The main targeted application is a full implementation of a neural network on serverless functions with straggler tolerance.

Voraussetzungen

Kontakt

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Investigation of polar and Reed-Muller codes under inactivation decoding for distributed computing

Beschreibung

Recently, channel codes are proposed to speed up computation in distributed systems by introducing redundant calculations to avoid the latency due to, e.g., straggling nodes (see, e.g., [1]). Since this problem can be casted as coding for erasure channels (where the erasure probability itself might be a random variable), there are works using inactivation decoding, an efficient way of implementing Gaussian elimination, for this problem [2]. In this internship, the task of the student is to understand the advantages of polar codes under successive cancellation (SC) decoding [3] and investigate polar codes and their variants, e.g., Reed-Muller codes, under an inactivation decoder [4] to understand their performance compared to the existing works, e.g., [2].

[1] https://arxiv.org/pdf/1512.02673.pdf

[2] https://arxiv.org/pdf/1712.08230.pdf

[3] https://arxiv.org/pdf/1901.06811.pdf

[4] https://arxiv.org/abs/2004.05969

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Learning Aided SC Flip Decoding for Polar Codes

Beschreibung

Polar codes achieve the capacity of binary-input discrete memoryless channels asymptotically in the block length under successive cancellation (SC) decoding. Polar codes have been adopted for the control channel in 5G enhanced mobile broadband (eMBB).

Due to the serial nature of SC decoding, an erroneous bit decision can be caused by the channel noise or previous erroneous bit estimates. The main idea of SC flip decoding is trying to correct the first erroneous bit decision by sequentially flipping the unreliable decisions. 

The optimal flipping strategy is considered difficult due to lack of an analytical solution. Alternatively, (deep) learning aided SC flip algorithm are investigated in this thesis.

[1] O. Afisiadis, A. Balatsoukas-Stimming, and A. Burg, “A low-complexity improved successive cancellation decoder for polar codes,” in Proc. 48th Asilomar Conf. Signals, Systems and Computers, pp. 2116-2120, 2014.

[2] L. Chandesris, V. Savin, and D. Declercq, “Dynamic-SCFlip Decoding of Polar Codes,” IEEE Trans. Commun., vol. 66, no. 6, pp. 2333-2345, Jun., 2018.

[3] X. Wang, et al. "Learning to Flip Successive Cancellation Decoding of Polar Codes with LSTM Networks." arXiv preprint arXiv:1902.08394 (2019).

[4] N. Doan, et al. "Neural Dynamic Successive Cancellation Flip Decoding of Polar Codes." arXiv preprint arXiv:1907.11563 (2019).

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Polar Coding with Non-Binary Kernels

Beschreibung

This thesis will focus on polar codes with non-binary kernels on GF(q). Some of the following tasks might be covered: 

• Kernel selection
• Decoder implementation
• Efficient construction
• Comparison of binary and non-binary polar codes

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A Jupyter Notebook for Line Coding in Access Networks (LB)

Beschreibung

For the access network case, the spectrum of the transmit signal has to be adapted to the channel properties. This can either be achieved by choosing suitable transmit pulse shapes or by encoding the (redundancy free) source symbols [1].

The students task is to implement a demonstration of two line coding schemes in Python [2] (Jupyter Notebook) and visualize the results. Additionally, the student also has to arrange code and surrounding text, such that the content becomes self-explanatory.

[1] Skript "Physical Layer Methods“

[2] "Python in 30 minutes" (https://www.programiz.com/python-programming/tutorial)

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A Jupyter Notebook for Equalization Methods (LB)

Beschreibung

Depending on the channel properties, the receive signal in a communication system can be severely distorted, causing inter-symbol interference. To mitigate these interferences, several approaches for equalization can be taken [1].

The students task is to implement a demonstration of several equalization schemes in Python [2] (Jupyter Notebook) and visualize the results. Additionally, the student also has to arrange code and surrounding text, such that the content becomes self-explanatory.

[1] Skript "Physical Layer Methods“

[2] "Python in 30 minutes" (https://www.programiz.com/python-programming/tutorial)

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