This course will discuss the theoretical, numerical, and practical foundations of Compressed Sensing (CS) which has become a very important concept in recent years in information and signal processing. It allowed an alternative approach to conventional techniques for acquiring and reconstructing sensor input signals. CS is also known as Compressive Sampling because it allows sampling of compressible analog signals with sampling rates well below the Nyquist rate. This analog setting of CS allows significant performance  improvements of analog-to-digital converters for a broad class of time  continuous signals. It is therefore possible to design universal CS based data acquisition systems with compressive sensors for analog and digital sensor input signals, even if these signals are noisy.

The compressibility of certain signals can be exploited by developing novel CS methods which, in comparison to traditional approaches like the transform coding, involve far less processing effort for data compression. On the other hand, CS requires far more effort in order to reconstruct a sensor input signal. Accordingly, CS can help to resolve data deluge in complex sensing networks, where the number and resolution of the sensors grow to a point where the performance bottleneck moves to data processing in sensors. To avoid this raw data accumulation, new designs of data acquisition systems are proposed. They combine sensing and compression in one simple operation, replacing conventional sensors with compressive sensors. Instead of acquiring a massive amount of raw data and extracting the information afterwards, compressive sensors attempt to acquire the information directly.

Data compression using CS is performed by means of a simple linear superposition, while the decompression is based on optimization algorithms for finding the unique sparse solution of an underdetermined system of linear equations. There are multiple approaches for solving this searching problem, e.g. the generic Basis Pursuit method. This fundamental CS decompression method is based on the optimization technique related to the L1-norm.

The principles of compressed sensing are difficult to comprehend from the available literature without prior special knowledge, since they encompass specific aspects and languages of many mathematical fields. The most relevant subjects to CS are high-dimensional geometries of Euclidean and Banach spaces, random matrices, information and approximation theory, linear and convex programming, harmonic analysis, and combinatorics.

This highly challenging abstract approach to compressed sensing will be in  these lectures replaced by a simpler, expressive and application-oriented approach easy understandable for engineers. The underlying principle of this new research field will be systematically explained. The lectures and exercises will illuminate the basic principles of CS using the elementary language of signal processing, linear algebra and geometry only.