A Method to Enhance Noise Reduction for Data Generated from a Known Differential Equation

In this paper we propose a method to enhance noise reduction for data generated from a known differential equation. We develop a theoretical basis for the procedure and then illustrate it in the case of some data generated using Duffing’s equation. This method consists of embedding the data in higher dimensions and then transforming the data into a lower dimension using a singular matrix. We show that the singular matrix squeezes out some of the noise and leaves the true signal intact. Finally, using a nonlinear function, we reverse the effect of the singular matrix to get closer to the original data. In practice, it turns out that often noise contaminates chaotic signals. This leads to great difficulty in accessing the information carried by the chaotic signal. Thus the problem of cleaning a chaotic signal from external noise is of great interest in many applications. A number of linear and nonlinear noise reduction methods have been proposed. 1)–6) In this paper we demonstrate a method to enhance the noise reduction for data generated from a known differential equation. We assume that the noise is additive. The method consists of embedding data in higher dimensions and then transforming the data into a lower dimension using a singular matrix. We show that the singular matrix squeezes out some of the noise and leaves the true signal intact. Then, using a nonlinear function, we reverse the effect of the singular matrix to get closer to the original data. This function is independent of the initial conditions and can be empirically determined via numerical experiments using the known differential equation.