From Sigmoid Power Control Algorithm to Hopfield-like Neural Networks: "SIR"-Balancing Sigmoid-Based Networks- Part II: Discrete Time

In the first part in [12], we present and analyse a Sigmoid-based “Signal-to-Interference Ratio, (SIR)” balancing dynamic network, called Sgm"SIR"NN, which exhibits similar properties as traditional Hopfield NN does, in continuous time. In this second part, we present the corresponding network in discrete time: We show that in the proposed discrete-time network, called D-Sgm"SIR"NN, the defined error vector approaches to zero in a finite step in both synchronous and asynchronous work modes. Our investigations show that i) Establishing an analogy to the distributed (sigmoid) power control algorithm in [10] and [11] if the defined fictitious “SIR” is equal to 1 at the converged eqiulibrium point, then it is one of the prototype vectors. ii) The D-Sgm"SIR"NN exhibits similar features as discrete-time Hopfield NN does. iii) Establishing an analogy to the traditional 1-bit fixed-step power control algorithm, the corresponding “1-bit” network, called Sign"SIR"NN network, is also presented.

💡 Research Summary

This paper presents the discrete‑time counterpart of the sigmoid‑based “Signal‑to‑Interference Ratio (SIR) balancing” neural network introduced in Part I. The continuous‑time model, called Sgm“SIR”NN, exhibits Hopfield‑like dynamics by driving a fictitious SIR variable γ_i = x_i / (∑{j≠i} w{ij}x_j + b_i) toward unity. In the discrete‑time version, named D‑Sgm“SIR”NN, the state update for each neuron i is
 x_i(k+1) = x_i(k) + α