Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

G.11 COMPLEX DIFFERENTIATION

For complex-valued matrix and complex column vectors x and v with appropriate dimensions:

(G.80)

where the superscript denotes complex conjugation. The second expression is called the conjugate derivative. The derivatives used in this book involving complex quantities, such as the gradient described in Chapter 13 on beamforming, are based on the conjugate derivative. Some derivative results for complex-valued A, x, and v are given by the following:

(G.81)

(G.82)

(G.83)

Example G.6. Consider again the optimization problem in Example G.5 but with complex quantities:

(G.84)

where λ is a complex-valued Lagrange multiplier. Differentiating this expression with respect to and , and using the rules above yields

(G.85)

The remaining steps in the derivation ...