Adaptive Filters

CHAPTER C Complex Gradients

In this chapter we explain how to differentiate a scalar-valued function g(z) with respect to a complex-valued argument, z, and its complex conjugate, z*. The argument z could be either a scalar or a vector.

C.1 CAUCHY-RIEMANN CONDITIONS

We start with a scalar argument z = x + jy, where . In this case, we can regard g(z) as a function of the two real scalar variables, x and y, say,

(C.1)

with u(., .) denoting its real part and v(., .) denoting its imaginary part. Now, from complex function theory, the derivative of g(z) at a point z_o = x_o + jy_o is defined as (see, e.g., Ahlfors (1979)):

where ∆z = ∆x + j∆y. For g(z) to be differentiable at z_o, in which case it is also said to be analytic at z_o, the above limit should exist regardless of the direction from which z approaches z_o. In particular, if we assume ∆y = 0 and ∆x → 0, then the above definition gives

(C.2)

If, on the other hand, we assume that ∆x = 0 and ∆y → 0 so that ∆z = j∆y, then the definition gives

(C.3)

The expressions (C.2) and (C.3) should coincide. Therefore, by adding them we get