Based On The Under-sampling Signal Reconstruction Algorithm

In digital signal processing applications, the first step is to convert analog signals to digital signals, then, the digital signals are processed, and finally, the processing results are converted back to analog signals. The above process involves two contents: the first one is sampling; and the other one is reconstruction of the original signals.Shannon sampling theorem resolves sampling and reconstruction problem for the band-limited signals. A continuous time signal can be exactly reconstructed from its discrete samples if the sampling rate is higher than the Nyquist rate. However, for reconstruction of the UWB signal according to Shannon sampling theorem, the required sampling rate is very high. Such high sampling rate is sometimes very difficult to achieve. For nonband-limited signals, e.g., time-limited signals, which are usually used in practice, we can not sample and reconstruct them according to the Shannon sampling theorem. In this paper, we focus on the signal sampling and reconstruction to investigate the following two problems.Firstly, we investigate two methods for sampling and reconstruction under the condition that the sampling rate is lower than the Nyquist rate, which is termed as undersampling case. The first method is constructing a impulse sampling sequence method and the second method is derivative sampling method. Theorem analysis and computation simulation results show the effectiveness of these two methods.Secondly, we investigate the sampling and reconstruction for the compactly supported signal using wavelet sampling theorem. We obtain the signal reconstruction formulation which is similar to the form of Shannon sampling theorem. We also propose to use the nonuniform sampling and derivative sampling in wavelet space to improve the disadvantages in the process. Simulation results are presented to verify the efficacy of the improved method.Finally, we investigate several methods for sampling and reconstruction of signal that related to this paper. We discuss the method to reconstruct original signal under the undersampling cases via theory analysis and examples.