Research On Frame Theory And Its Application In Signal Coding

When we take the orthogonal basis as the coding tool during the transmission of signals,the original signal cannot be recovered when partial data erasures or noise occurs during transmission,because of the uniqueness of the orthogonal basis.Compared with the orthogonal basis,the advantage of frames are their redundancy.The redundancy contributes to reconstruct the signal and may even restore the original signal.Therefore,frame theory has a significant application in image processing,signal processing and so on.The main contributions of this thesis are summarized as follows:1.We give a series of conclusions about frame tensor product.On account of it,we advance a new algorithm signal transmission,the general algorithm complexity of signal transmission by frame can reach cubic order,but such new algorithm reduces complexity to square order.In addition,we introduce the numerical iterated algorithm and prove its convergence.2.We give a equidistant erasures model and investigate the tensor product of the optimal dual frame for equidistant erasures.And we get two sufficient and necessary con-ditions that the tensor product of the dual frame and the canonical dual frame is obtained as the tensor product of the optimal dual frame for equidistant erasures.In addition,when equidistant erasures occur to the coding coefficient data during the transmission of,we can reduce the effect of data erasures on reconstructed signal by using the tensor product of frames to encode the signal,which is based on some properties of the tensor product of frames.3.We give numerical experiments.Results of the algorithm comparison exper-iment show the comparison results of two algorithms,which means new algorithm is more efficient than general algorithm;And the iterative refactoring experiment show the reconstruction results of new algorithm in iterative form;The frame tensor product com-parison experiment is given to compare the results of signal recovery between the the tensor product of the general optimal frame and the tensor product of the optimal dual frame for equidistant erasures,which means the results of signal recovery of the tensor product of the optimal dual frame are better than the results of signal recovery between the the tensor product of the general optimal frame.