On Inverse Log-Digamma Difference Function for Wireless Signal Processing

The transcendental equation that represents the difference between the logarithmic and digamma functions has practical significance in wireless signal processing and wireless analytics, particularly in the applications of channel parameter estimation and channel model substitution (CMS). However, the inverse of such a crucial transcendental equation cannot be expressed in closed form using elementary or classical special functions. Therefore, in this letter, we formally define such an inverse as the inverse log-digamma difference (ILDD) function and establish it as an effective mapping inversion. We also prove the key analytical properties of the ILDD function, including its existence, uniqueness, differentiability, and monotonicity, and derive several tight bounds on it. Jointly applying the proven analytical properties and derived bounds, we further propose an accurate and robust approximation method for the ILDD function over the entire domain of definition. The effectiveness of all analytical results is verified by numerical computation via the Newton-Raphson (N-R) method and a case study of CMS.