Introduction

In this chapter, several examples of how the theory of complex-valued matrix derivatives can be used as an important tool to solve research problems taken from signal processing and communications. The developed theory can be used to solve problems in areas where the unknown matrices are complex-valued matrices. Examples of such areas are signal processing and communications. Often in these areas, the objective function is a real-valued function that depends on a continuous complex-valued matrix and its complex conjugate. In Hjørungnes and Ramstad (1999) and Hjørungnes (2000), matrix derivatives were used to optimize filter banks used for source coding. The book by Vaidyanathan et al. (2010) contains material on how to optimize communication systems by means of complex-valued derivatives. Complex-valued derivatives were applied to find the Cramer-Rao lower bound for complex-valued parameters in van den Bos (1994b); and Jagannatham and Rao (2004)

The rest of this chapter is organized as follows: Section 7.2 presents a problem from signal processing on how to find the derivative and the Hessian of a real-valued function that depends on the magnitude of the Fourier transform of the complex-valued argument vector. In Section 7.3, an example from signal processing is studied in which the sums of the squared absolute values of the off-diagonal elements in a covariance matrix are minimized. This problem of minimizing the off-diagonal elements has applications in blind carrier frequency offset (CFO) estimation.