# Master Projects (Masterarbeiten)

If you are interested in one of the following topics, please contact the person listed with the respective topic.

If you don't find a compelling topic in our list, we are always keen on hearing about your ideas in the area of our research interests (see the Research section of our website). You are most welcome to discuss your interests with us.

Also, we have a list of finished diploma theses on our website.

## List of Projects

- Millimeter wave RF frontend for 5G applications
- Implementation of a compressed sensing radar architecture
- Deformation sensitivity bounds for deep convolutional neural networks
- Phase transitions for matrix separation
- Estimation of fractal dimension
- Deep convolutional neural networks for scene labeling
- Line spectral estimation in the presence of structured noise

### Millimeter wave RF frontend for 5G applications (DA)

With the increasing demand for bandwidth in mobile communications applications, broad activities are undertaken in industry and academia towards a future 5G mobile communication standard [1]. 5G promises vast improvements in terms of available data rate, coverage, and latency, all based on new technologies such as advanced modulation and coding schemes, (massive) MIMO, and in particular the usage of new spectrum bands [2, 3]. The Federal Communications Commission (FCC) recently designated several new blocks of spectrum in the millimeter wave frequency bands for these new-generation wireless broadband services.

The goal of this project is to implement a portable RF frontend for up- and downconversion to allow wideband radio channel measurements in the new spectrum bands. In a second step, it is planned to extend the system to a MIMO setup. The project is carried out in collaboration with Swisscom.

Type of project: 20% theory, 80% RF hardware design and measurements

Prerequisites: Interest in RF hardware and wireless systems

Supervisor: Michael Lerjen, Ruben Merz

Professor:
Helmut Bölcskei

References:

[1]
A. Osseiran, J. F. Monserrat, and P. Marsch, "5G mobile and wireless communications technology," Cambridge University Press, 2016.

[2] T. S. Rappaport, W. Roh, and K. Cheun, "Mobile's millimeter-wave makeover," IEEE Spectrum, vol. 51, no. 9, pp. 34–58, 2014. [Link to Document]

[3] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, "Millimeter wave mobile communications for 5G cellular: It will work!," IEEE Access, vol. 1, pp. 335–349, 2013. [Link to Document]

### Implementation of a compressed sensing radar architecture (SA/DA)

Radar systems are typically modeled as linear systems that induce weighted superpositions of delayed and Doppler-shifted versions of the probing signal. Identifying the delay-Doppler shifts yields the position and relative speed of the object of interest. A similar problem arises in the identification of wireless channels, where the (unknown) delay-Doppler shifts correspond to point scatterers in the propagation environment. For suitably chosen probing signals the problem of identifying the delay-Doppler shifts can be reduced to that of recovering a sparse vector from (highly) undersampled measurements [1, 2, 3], i.e., to a compressed sensing problem. Standard approaches to solving the corresponding problem include l1-minimization or greedy algorithms such as orthogonal matching pursuit. It was shown in [3] that recovery of the delay-Doppler shifts in the radar and the system identification problem can be formulated as a multiple measurement vector (MMV) problem, which can be solved efficiently using low-complexity subspace algorithms such as, e.g. MUSIC [4].

The goal of this project is a hardware implementation of a compressed sensing radar system. Specifically, we consider a system consisting of RF instruments to generate and capture the signals and a hardware channel emulator. The subspace algorithm will be implemented in MATLAB in a first step and, time permitting, on an FPGA in a second step. The overall system shall be tested in terms of performance and practical applicability.

Type of project: 50% implementation (RF measurements, Matlab programming, hardware design), 30% simulation, 20% theory

Prerequisites: Interest in RF hardware and wireless systems, Matlab, possibly VHDL, and linear algebra

Supervisor: Michael Lerjen, Céline Aubel

Professor:
Helmut Bölcskei

References:

[1]
W. U. Bajwa, K. Gedalyahu, and Y. C. Eldar, "Identification of parametric underspread linear systems and super-resolution radar," *IEEE Trans. Signal Process.*, vol. 59, no. 6, pp. 2548–2561, Jun. 2011.
[Link to Document]

[2]
M. Herman and T. Strohmer, "High-resolution radar via compressed sensing," *IEEE Trans. Signal Process.*, vol. 57, no. 6, pp. 2275–2284, 2009.
[Link to Document]

[3]
R. Heckel and H. Bölcskei, "Identification of sparse linear operators," *IEEE Trans. Inf. Theory*, vol. 59, no. 12, pp. 7985–8000, 2013.
[Link to Document]

[4]
R. Schmidt, "Multiple emitter location and signal parameter estimation," *IEEE Trans. Ant. Propag.*, vol. 34, no. 3, pp. 276–280, Mar. 1986.
[Link to Document]

### Deformation sensitivity bounds for deep convolutional neural networks (SA/DA)

Feature extractors based on so-called deep convolutional neural networks (DCNNs) have been applied with tremendous success in a wide range of practical signal classification tasks [1] such as, e.g., in handwritten digit classification. The features to be extracted in this case correspond to the edges of the digits, and we would want these features to be robust with respect to handwriting styles. This can be accomplished by demanding that the feature extractor have limited sensitivity to certain non-linear deformations.

Recently, [2], [3] established deformation sensitivity bounds for a wide class of DCNN-based feature extractors. These bounds apply to a variety of input signal classes such as band-limited functions, cartoon functions, and Lipschitz-functions. Many signals of practical interest (such as textures) exhibit, however, sharp oscillations and are therefore not captured by these results.

The goal of this project is to use the theory of approximately time- and band-limited functions [4] to develop general deformation sensitivity bounds.

Type of project: 80%-100% Theory, 0-20% Simulations, depending on the student's preference

Prerequisites: Analysis, Linear Algebra, Signals and Systems I

Supervisor: Thomas Wiatowski

Professor:
Helmut Bölcskei

References:

[1] I. Goodfellow, Y. Bengio, and A. Courville, “Deep Learning,” *MIT Press *, 2016. [Link to Document]

[2] T. Wiatowski and H. Bölcskei, “A mathematical theory of deep convolutional neural networks for feature extraction,” *arXiv:1512.06293*, 2015. [Link to Document]

[3]
P. Grohs, T. Wiatowski, and H. Bölcskei “Deep convolutional neural networks on cartoon functions,” *Proc. IEEE Int. Symp. on
Inf. Theory (ISIT)*, pp. 1163–1167, 2016. [Link to Document]

[4]
D. Slepian, "On bandwidth," *Proc. of the IEEE*, pp. 292–300, 1976. [Link to Document]

### Phase transitions for matrix separation (DA)

A phase transition is a sharp division of the parameter space of a data processing problem into a region of success and a region of failure. This phenomenon was studied for the matrix separation problem [1,2] and for the matrix completion problem [3] under specific recovery algorithms. Information-theoretic phase transitions do not depend on specific recovery algorithms and reveal what is best possible without stating how to achieve optimum performance. Recently, information-theoretic phase transitions were obtained for the matrix completion problem [4] and the signal separation problem [5].

The first goal of this project is to build on the techniques developed in [4,5] to characterize information-theoretic phase transitions for the matrix separation problem, which in its traditional incarnation separates a low-rank matrix from a sparse matrix. The second goal of the project is to investigate pairs of matrix structures (beyond low-rank and sparse) that allow for separation and to characterize their information-theoretic phase transitions.

Type of project: 80% Theory, 20% Simulation

Prerequisites: Strong mathematical background, measure theory, probability theory

Supervisor: Erwin Riegler

Professor:
Helmut Bölcskei

References:

[1] D. Amelunxen, M. Lotz, M. B. McCoy, and J. A. Tropp, “Living on the edge: A geometric theory of phase transitions in convex optimization,” *Information and Inference*, vol. 3, no. 3, pp. 224–294, Jun. 2014.
[Link to Document]

[2] V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky, “Rank-sparsity incoherence for matrix decomposition,” *SIAM Journal on Optimization*, vol. 21, no. 2, pp. 572–596, Jun. 2011.
[Link to Document]

[3] E. J. Candés and B. Recht, “Exact matrix completion via convex optimization,” *Foundations of Computational Mathematics*, vol. 9, no. 6, pp. 717–772, Dec. 2009.
[Link to Document]

[4]
E. Riegler, D. Stotz, and H. Bölcskei “Information-theoretic limits of matrix completion,” *Proc. IEEE Int. Symp. on
Inf. Theory (ISIT)*, pp. 106–110, Jun. 2015. [Link to Document]

[5]
D. Stotz, E. Riegler, and H. Bölcskei “Almost lossless analog signal separation,” *Proc. IEEE Int. Symp. on
Inf. Theory (ISIT)*, pp. 106–110, Jul. 2013. [Link to Document]

### Estimation of fractal dimension (DA)

Fractal dimensions such as the box-counting dimension or the Rényi information dimension of a random vector often appear as a measure of the vector's description complexity. Recently, fractal dimensions were shown to characterize fundamental limits in information-theoretic problems such as analog data compression [2,3] and communication over interference channels [4-6]. The exact computation of fractal dimensions is, in general, hard, as explicit formulae are available only for certain classes of well-behaved probability distributions.

The goal of this project is to apply statistical methods to estimate the fractal dimension of probability distributions from i.i.d. samples. Relevant approaches to this problem are reviewed in [1]. The estimation procedure shall then be used to gain insight into the fractal dimension of sources with structure relevant to information-theoretic problems.

Type of project: 60% Theory, 40% Simulation

Prerequisites: Strong mathematical background, measure theory, probability theory

Supervisor: Erwin Riegler

Professor:
Helmut Bölcskei

References:

[1] C. D. Cutler, “A review of the theory and estimation of fractal dimension,” in *Dimension estimation and models*, H. Tong, Ed., Nonlinear Time Series and Chaos, vol. 1, pp. 1–107, World Scientific, 1993.

[2] Y. Wu and S. Verdú, “Rényi information dimension: Fundamental limits of almost lossless analog compression,” *IEEE
Trans. Inf. Theory*, vol. 56, no. 8, pp. 3721–3748, Aug. 2010. [Link to Document]

[3]
D. Stotz, E. Riegler, and H. Bölcskei “Almost lossless analog signal separation,” *Proc. IEEE Int. Symp. on
Inf. Theory (ISIT)*, pp. 106–110, Jul. 2013. [Link to Document]

[4]
Y. Wu, S. Shamai (Shitz), and S. Verdú, "Information dimension and the degrees of freedom of the interference channel," *IEEE
Trans. Inf. Theory*, vol. 61, no. 1, pp. 256–279, Jan. 2015. [Link to Document]

[5]
D. Stotz and H. Bölcskei, "Degrees of freedom in vector interference channels," *IEEE Transactions on Information Theory*, vol. 62, no. 7, pp. 4172–4197, Jul. 2016.
[Link to Document]

[6]
D. Stotz and H. Bölcskei, "Characterizing degrees of freedom through additive combinatorics," *IEEE Transactions on Information Theory*, vol. 62, no. 11, pp. 6423–6435, Nov. 2016.
[Link to Document]

### Deep convolutional neural networks for scene labeling (SA/DA)

Deep convolutional neural networks (DCCNs) with pre-defined filters [1,2] extract characteristic features from signals by recursively applying the composition of the following three operations: convolution with a set of filters, a non-linearity, and a sub-sampling step. These networks, combined with a classifier (such as, e.g., a support vector machine) were successfully employed in a number of practical classification tasks [1,3,6]. In contrast to traditional DCNNs which learn the filters from training data [4], DCNNs with pre-defined filters rely on wavelets [1], curvelets [2], or shearlets [2]. While the use of pre-defined structured filters leads to less flexibility, it allows for faster implementations.

This project shall explore the application of DCNNs with pre-defined filters to the problem of scene labeling, where the aim is to assign a class label such as "street", "tree", or "building" to every pixel of an image. Applications of scene labeling include situational awareness systems [4], which often demand low-complexity scene labeling.

The goal of this project is to extend the work carried out in prior semester theses on the topic. In particular, the classification stage should be improved and the existing pipeline should be optimized for speed.

Type of project: 0%-20% Theory, 80%-100% Programming, depending on the student's preference

Prerequisites: Python, C programming, linear algebra

Supervisor: Michael Tschannen, Thomas Wiatowski, Lukas Cavigelli (IIS), Michael Lerjen

Professor:
Helmut Bölcskei, Luca Benini (IIS)

References:

[1]
J. Bruna and S. Mallat, "Invariant scattering convolution networks," *IEEE Trans. Pattern
Anal. Mach. Intell.*, vol. 35, no. 8, pp. 1872-1886, 2013.
[Link to Document]

[2]
T. Wiatowski and H. Bölcskei, "A mathematical theory of deep convolutional neural networks for feature extraction," *IEEE Transactions on Information Theory*, (revised version, Aug. 2016), Dec. 2015, submitted.
[Link to Document]

[3]
J. Andén and S. Mallat, "Deep scattering spectrum," *IEEE Trans. Sig. Process.*, vol. 62, no. 16, pp. 4114-4128, 2014.
[Link to Document]

[4]
Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, "Gradient-based learning applied to document recognition," *Proc. of the IEEE*, pp. 2278-2324, 1998.
[Link to Document]

[5]
L. Cavigelli, M. Magno, and L. Benini, "Accelerating real-time embedded scene labeling with convolutional networks," *Proc. of ACM/EDAC/IEEE Design Automation Conference (DAC)*, pp. 1-6, 2015.
[Link to Document]

[6]
I. Goodfellow, Y. Bengio, and A. Courville, "Deep Learning," *MIT Press*, 2016.
[Link to Document]

### Line spectral estimation in the presence of structured noise (DA)

Line spectral estimation is the task of estimating the frequencies and amplitudes of a linear combination of complex exponentials from signal samples. The recent papers [1,2,3,4] propose convex optimization methods for line spectral estimation from band-limited noisy signal samples. The performance analysis in [1,2,3] relies on the assumption that the noise is random and its distribution satisfies certain properties. In practice, however, one often has to deal with deterministic noise, rendering the performance analysis more difficult. To overcome this problem, it is sensible to exploit the fact that the noise term often admits a sparse representation in a certain basis (e.g., signals impaired by clipping or impulse noise [4]).

The goal of this project is to first understand the theory developed in [1,2,3] and to then develop a new theory for deterministic sparse noise. Finally, the algorithms developed should be tested on audio signals.

Type of project: 90% Theory, 10% Simulation

Prerequisites: Strong mathematical background, measure theory, convex optimization theory, functional analysis

Supervisor: Céline Aubel

Professor:
Helmut Bölcskei

References:

[1]
G. Tang, B. N. Bhaskar, and B. Recht, "Near minimax line spectral estimation," *IEEE Transactions on Information Theory*, nol. 61, no. 1, 2015.
[Link to Document]

[2]
E. J. Candès and C. Fernandez-Granda, "Towards a mathematical theory of super-resolution," *Communications on Pure and Applied Mathematics*, vol. 67, no. 6, pp. 906-956, 2014.
[Link to Document]

[3]
E. J. Candès and C. Fernandez-Granda, "Super-resolution from noisy data," *Journal of Fourier Analysis and Applications*, vol. 19, no. 6, pp. 1229-1254, 2013.
[Link to Document]

[4]
C. Aubel, D. Stotz, and H. Bölcskei, "A theory of super-resolution from short-time Fourier transform measurements", *Journal of Fourier Analysis and Applications*, 2017, to appear.
[Link to Document]

[5]
C. Studer, P. Kuppinger, G. Pope, and H. Bölcskei, "Recovery of sparsely corrupted signals," *IEEE Transactions on Information Theory*, vol. 58, no. 5, pp. 3115-3130, 2012.
[Link to Document]