Study On Sampling In Shift-invariant Spaces And Its Application

Signal sampling and data reconstruction is a foundation of mordern electronic systems in field of testing, communication and radar. With the electronic systems frequency and band-width increasing, and the new technology system introducing, the signal's carrier frequency, band-width and complexity are promoted ahead. However, limited by the semiconductor manufacturing process of analog-digital conversion technology, signal sampling can not meet the need of morden electronic systems. Recently, the high-speed, high-resolution sampling and reconstruction technology grow to currently one of the hottest topics in the field of signal sampling and processing. The sampling in spaces, a unique theory of signal sampling transforms the traditional signal sampling into a process of weights determination, which weights the signal in spaces. The interesting point is how to achieve the prefect sampling for high-speed, wide-band signal with low-speed ADC chips. In this paper, the model, reconstruction method and application of sampling in shift-invariant spaces is studied. The main results are as follows:1. According to Shannon sampling, the geometry significance of sampling in shift-invariant spaces is introduced and extended, the over-sampling and under-sampling in shift-invariant space are defined, and the stability of the two methods are studied, a rule of judging stability based on eigenvalue of matrix is obtained.2. In order to reconstruct over-samples in shift-invariant space, the mathematical expression of correction operator and reconstruction error is obtained using generalized inverse. However, the generalized inverse is only a mathematical research tool. In order to turn correction operator into digital system, the least square method (LSM) is proposed, which turns correction operator into MIMO-FIR filter. Another, taking into account the complexity of system, a reconstruction function based on frame theory is introduced. Then the reconstruction function can be directly obtained by the frame theory, and the signal can be reconstructed by interpolation. Finally, to achieve the combination of theory and practice, over-sampling in shift-invariant space is applied to multi-band signal sampling.3. To reconstruct under-samples in shift-invariant space for sparsity signal, the minimum L0-norm, the minimum L1-norm and OMP algorithm are introduced. The reconstruct conditions and stability of three reconstruction algorithms have been studied, necessary conditions and reconstruction error bounds are obtained. It is shown that the empirical probabilities of exact recovery of minimum L0-norm is least, the reconstruction error of minimum L1-norm is largest and the reconstruction error of the three methods are all smaller than uniform over-sampling error. Finally, to achieve a shift-invariant space under-sampling in the actual application, under-sampling in shift-invariant space is applied to wide-band signal sampling, and compared with the time-interleaved sampling. It is shown that under-sampling in shift-invariant space is better than the time-interleaved system whether in structure or signal to noise ratio.4. For instantaneous signal, the adaptive sampling and reconstruction are introduced, including typical adaptive over-sampling and adaptive reconstruction of under-sampling. The relation between sampling rate and signal frequency is obtained by transformation of coordinate for adaptive over-sampling, and the time-frequency distribution of samples is analyzed, thereby, a reconstruction algorithm based on time-frequency filtering is proposed. An adaptive reconstruction model is established using short-time Fourier transforms for under-sampling, then the reconstruction filter is adjusted by OMP algorithm, thus the adaptive under-sampling is completed.Sampling in spaces is currently one of the hottest topics in the field of signal sampling and processing. This paper has studied sampling in shift-invariant space, which includes over-sampling, under-sampling for sparsity signal, adaptive sampling and reconstruction methods, and application in system, it has a certain practical value for the further study of sampling in shift-invariant space.