# Complex-Valued Signal Processing

Signals that can be conveniently modeled as complex-valued signals occur in many fields of signal processing such as in communication systems, medical imaging, audio and speech processing, analysis of meteorological or oceanographic data, and many more. Using complex numbers to describe such signals can make analytic expressions much more compact and easier to analyze and has become a de facto standard in signal processing research. However, as compared to processing real-valued signals, there are several aspects where special care must be taken when complex signals are involved.

A particularly remarkable aspect, which had for a long time been neglected by many researchers and engineers, is that complex signals can be so-called improper signals. This term is used to refer to power imbalances or certain kinds of correlations between the real and imaginary parts of complex signals. In many recent publications from various fields of signal processing, it was revealed that an appropriate treatment of such improper signals, e.g., by means of so-called widely linear filtering, can be necessary to exploit the full system performance.

One focus of our research on complex-valued signal processing is how such improper signals can be described in a way that is convenient and effective for analytical derivation, mathematical proofs, algorithm design, numerical studies, and other purposes. By combining results from statistical signal processing, linear algebra, and matrix theory, we develop new mathematical tools for describing improper signals, which complement the existing literature on the subject, and we are interested in the question, when it makes sense to switch from real-valued descriptions to complex descriptions and vice versa. Another important part of our research is studying situations in which it is desirable to have improper signals in a system---even though their name might give the false impression that such signals are somehow uncommon or inconvenient. Situations where artificially introduced impropriety can lead to performance gains can, for example, be found in communication systems with interference.