Gradient conversion between time and frequency domains using Wirtinger calculus
Résumé
Gradient descent algorithms are found in a variety of scientific fields, audio signal processing included. This paper presents a new method of converting any gradient of a cost function with respect to a signal into, or from, a gradient with respect to the spectrum of this signal: thus, it allows the gradient descent to be performed indiscriminately in time or frequency domain. For efficiency purposes, and because the gradient of a real function with respect to a complex signal does not formally exist, this work is performed using Wirtinger calculus. An application to sound texture synthesis then experimentally validates this gradient conversion.