Generalized Cardinal Polishing Splines Signal Reconstruction

Sampling and reconstruction are indispensable processes in signal processing, and appropriate foundations are crucial for spline reconstruction models. Generalized cardinal polishing splines (GCP-splines) are a class of high-precision explicit splines with pretty properties. We propose the theory of GCP-splines for signal reconstruction and differential signaling to improve signal reconstruction accuracy. First, the elementary theory of the GCP-splines signal processing is proposed, and it mainly includes Fourier transformation and Z-transformation of the GCP-splines. Then, a GCP-splines filter that can be used to reconstruct the output signal from the input discrete signal is proposed. Next, we propose differential signal reconstruction based on the GCP-splines and the sampled original signal values to obtain information on the signal change rate. Numerical experiments demonstrate that the signal reconstruction based on the GCP-splines yields lower approximation errors and better performance than the linear interpolation filter and cardinal B-spline interpolation filter.