| In many applications, a uniform sampling scheme is not feasible; in others, it is not efficient. In these cases, a nonuniform sampling scheme may be used. In this work, a practical way of sampling a signal more adequately in the sense that either fewer samples still lead to the same quality of reconstruction, or the same number of samples leads to a better quality of reconstruction as in the uniform sampling case is presented. In this technique the local sample rate is determined by the absolute value of the first derivative of the function, that is, a high sample rate is used when the signal changes rapidly. An extension of the Bernstein's inequality shows a relation between the absolute value of the first derivative of a signal with its local bandwidth, which confirms that the signal should be sampled more rapidly when its changing rate is high. Furthermore, an efficient algorithm to process the nonuniformly spaced samples is presented in this work. Using this technique, algorithms used for uniform sampling only (e.g., the fast Fourier transform), can also be applied in the linear processing of the nonuniformly spaced samples. Theoretical proof that shows the validity of this algorithm is presented in this work. Some examples that show the effectiveness and the simplicity of the algorithm are given. |
