Complex gradient and Hessian

The gradient and Hessian are often used in analytical and numerical function optimisation complex valued parameter estimation problems. In a number of signal processing applications the function is a real function of complex variables. Then the optimisation is usually carried out with respect to the real and imaginary part of these variables; therefore, the gradient and Hessian concerned are real. The reason for this approach is to avoid difficulties with the definition and interpretation of the gradient and Hessian with respect to the complex variables. Definitions of a complex gradient and Hessian are proposed to solve these difficulties. The proposed and the real gradient and Hessian are fully compatible and are related by simple linear transformations. The results presented are an extension of a result by Brandwood (1983) concerning a complex gradient.