# The Complex Gradient Operator and the CR-Calculus

- Published: 25 Jun 2009

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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