Unified cellular neural network cell dynamical equation using delta operator

The signal processing algorithms based on conventional shift operator tend to be ill-conditioned in situations involving fast sampling and shorter wordlength. To alleviate this problem delta operator based analysis and design has been proposed for high speed digital signal processing and control systems. The advantage for delta (δ) operator seems to come from the fact as sampling period T_s → 0, the discrete time system process resembles that of continuous time system. In this paper we develop a unified cellular neural network (CNN) cell model using the delta operator approach. The model gives a general discrete-time (DT) CNN cell dynamics in which the sampling period T_s is an explicit parameter. As T_s → 0, we get the continuous time (CT)-CNN equation. Several results connected with the stability and robustness of CT-CNN and DT-CNN can be linked using this approach. This approach highlights the similarities, rather than the differences between discrete and continuous CNNs, thus allowing continuous insights to be applied to the discrete CNN case. Further, more importantly from the implementation point of view delta operator based DT-CNN cell design can be obtained using δ^-1 as an integrator {instead of a delay (z^-1)}. The δ^-1 integrator can be realized using switched current/switched capacitor circuits. The dynamic circuit element in the DT-CNN is thus 'δ^-1'.