Wirtinger calculus

How do you differentiate real-valued functions of complex arguments? Wirtinger calculus. This is a convenient hack that works well for most practical signal processing over the complex field, especially in optimisation. It arises naturally in, for example, phase retrieval (H. Zhang and Liang 2016; Candes, Li, and Soltanolkotabi 2015; Chen and Candès 2015; Seuret and Gouaisbaut 2013). Because of its popularity, this will almost surely arise in combination also with matrix calculus.

There are various write-ups of Wirtinger calculus; this will be a literature review of those because they are of highly variable quality and I want to ensure I understand what I’m playing with here.

Bouboulis (2010) has a punchy intro as part of a paper that takes Wirtinger derivatives inside inner products:

Wirtinger’s calculus has become very popular in the signal processing community mainly in the context of complex adaptive filtering, as a means of computing, in an elegant way, gradients of real valued cost functions defined on complex domains ( $^{} $ ). Such functions, obviously, are not holomorphic and therefore the complex derivative cannot be used. Instead, if we consider that the cost function is defined on a Euclidean domain with a double dimensionality ($^{2} $), then the real derivatives may be employed. The price of this approach is that the computations become cumbersome and tedious. Wirtinger’s calculus provides an alternative equivalent formulation, that is based on simple rules and principles and which bears a great resemblance to the rules of the standard complex derivative… A common misconception …is that Wirtinger’s calculus uses an alternative definition of derivatives and therefore results in different gradient rules in minimization problems. We should emphasize that the theoretical foundation of Wirtinger’s calculus is the common definition of the real derivative. However, it turns out that when the complex structure is taken into account, the real derivatives may be described using an equivalent and more elegant formulation which bears a surprising resemblance with the complex derivative. Therefore, simple rules may be derived and the computations of the gradients, which may become tedious if the double dimensional space $^{2} $, is considered, are simplified.

The extension to Hilbert space operations is nifty.

Other favoured resources:

• Terry Tao, 246a 1: Complex differentiation
• Fischer (2005) is a short take-me-straight-there appendix.
• Adali, Schreier, and Scharf (2011) introduces more machinery for complex valued signal processing.
• Caracalla and Roebel (2017) shows an actual application in a musical signal processing optimization problem, which is simple and yet possibly the only non-trivial and also comprehensible application in any of the introductions.