The Complex Gradient Operator and the CR-Calculus

Preprint English OPEN
Kreutz-Delgado, Ken (2009)
  • Subject: Mathematics - Optimization and Control | Mathematics - Complex Variables
    arxiv: Physics::Physics Education

A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical Engineering graduate level course on the use of complex gradients and complex Hessian matrices, and has been successfully used in teaching at UC San Diego. Going beyond the commonly encountered treatments of the first-order complex vector calculus, second-order considerations are examined in some detail filling a gap in the pedagogic literature.
  • References (4)

    23A tangent space at the point z is the space of all differential displacements, dz, at the point z or, alternatively, the space of all velocity vectors v = ddzt at the point z. These are equivalent statements because dz and v are scaled version of each other, dz = vdt. The tangent space TzZ = Czn is a linear variety in the space Z = Cn. Specifically it is a copy of Cn affinely translated to the point z, Czn = {z} + Cn.

    24The “cogradient” is a covariant operator [22]. It is not itself a gradient, but is the co mpanion to the gradient operator defined below. [1] Optimum Array Processing, H.L. Van Trees, 2002, Wiley Interscience. [2] Elements of Signal Detection & Estimation, C.W. Helstrom, 1995, Prentice Hall. [4] Complex Variables, 2nd Edition, S. Fisher, 1990/1999, Dover Publications, New York. [7] Principles of Mobile Communication, 2nd Edition, G.L. Stuber, 2001, Kluwer, Boston. [8] Digital Communication, E. Lee & D. Messerchmitt, 1988, Kluwer, Boston. [9] Introduction to Adaptive Arrays, R. Monzingo & T. Miller, 1980, Wiley, New York. [35] Linear Operator Theory in Engineering and Science, A.W. Naylor & G.R. Sell, Springer-

    Verlag, 1982. [36] “The Constrained Total Least Squares Technique and its Application to Harmonic Super-

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