# Complex-Valued Signal Processing

Using complex numbers to describe signals, e.g., in communication systems, 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. For example, optimization in the context of complex valued functions as well as taking derivatives with respect to complex valued parameters requires an extension of the well-known real valued mathematical concepts. A particularly remarkable aspect 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. For such signals, an appropriate treatment, e.g., by means of so-called widely linear filtering, is necessary, and in certain situations, it is desirable to artificially introduce improper signals in a system to exploit the full system performance. For analytical derivations, algorithm design, and numerical studies in the context of complex signals, it is necessary to combine results from statistical signal processing, linear algebra, and matrix theory as well as to switch between real-valued descriptions and complex descriptions of signals and systems.

Complex-valued signal processing is used in the following fields of research at our institute:

To learn more about complex-valued signal processing we kindly refer to the following lectures offered at our institute: