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This paper focuses on analysis and design of a class of cellular neural networks (CNNs). In particular, a discrete-time CNN model is introduced and the global asymptotic stability of its equilibrium point is analyzed. By taking into account such stability results, a novel technique for obtaining associative memories is developed. The objective is achieved by considering feedback parameters related to circulant matrices and by satisfying frequency domain stability criteria. The approach, by generating CNNs where the input data are fed via external inputs rather than initial conditions, enables both hetero-associative and auto-associative memories to be designed. A numerical example is reported in order to show the capabilities of the designed network in storing and retrieving information.