Mathematical Method Of Signal Transmission For Erasures

In the process of signal transmission, data climbing and data loss can not be avoided. The coding and decoding signals based on the framework can reduce the error resulting from that effectively. The concept of the framework was put forward by R. J. Duffin and A. C. Schaeffer in 1952, in the study with irregularly spaced sampling to reconstruction of a band-limited signal, they found that the theoretical framework possess useful natures that including completeness, stability and redundancy in the study of discrete signals. After 1980 s, due to the wide application of the frame theory, a large number of research results with ideal effective were obtained by many scholars after more than ten years study. However, there is still a large space for research.We all know that it is very difficult to fully restore the original signal by using the orthogonal basis in the process of signal transmission of erasures. While, the redundancy of the frame is good for this problem, we can use mathematical method based on the frame to recover the signal. J. Lopez, D. Han and J. S. Leng and other scholar analysis the optimal dual frame problem without considering the quantization noise. The main job of this thesis is as following: we join the quantization noise conditions, and proved the optimal dual frame for data loss problem is inevitable under the condition of quantization noise. When the encoding frame is consistent frame for the condition of quantization noise, we get that the canonical dual frame is the unique optimal dual frame for 1-erasures. Then, we discussed another case that coding frame is not a consistent frame for the condition of quantization noise. We obtained that canonical dual frame is not the unique optimal dual frame for 1-erasures, and the sufficient condition under which the canonical dual frame is not the optimal dual frame for 1-erasures. Finally, we give a numerical example which under the condition of quantization noise and the coding frame is consistent frame, we can clearly see that the canonical dual frame is the best decoding frame in 1-erasures.The problem of signal transmission in real life, we may require real-time signal transmission in real application. At this time, we should not only as much as possible to recover the signal, but also to improve the transmission efficiency. For this type of problem, the coding and decoding signals based on the framework can reduce the error resulting from that effectively. Firstly, we obtained several useful properties of fusion frame by combining the theory of operator. Secondly, we summarized how to select the optimal decoding frame by combining the reconstruction formula of fusion frame. Finally, we proved that finding the optimal dual frame can conversion for the general frame of dual frame problem in fact during the fusion frame in the signal transmission of the data loss problem. We just need to structure each closed subspace of the local frame for Parserval frame.