Structure Of The Optimal Dual Frame In Signal Transmission

In the process of signal transmission, the signal can not be transmitted accurately due to the interference of data loss, data climbing and noise. Thus, there are some errors between the received signal and the original signal, which is inevitable. If a frame is used to encode the signal and the optimal dual frame is used to decode or reconstruct the original signal vector, the error between the signal and the original signal can be minimized. Therefore, given a frame as a coder, its optimal dual frame as encoder can make the reconstruction error arising from erasure minimum.This paper made the following works based on this:Firstly, it introduce several methods of encoding and decoding signals with frame,which are the latest research achievements at home and abroad. What is more, it also summarize the advantages and disadvantages of each method.The first method is that using the same uniform Parseval frame to encode and decode the signal. The second way is that for any m dimensional error, we use the same Parseval frame to encode and decode signals. The third way is to use any given frame as the encoder, and then use the optimal dual frame to decode. The fourth way is similar to the third way, but the probability distribution of each element is considered.Afterwards, there are so many strict requirements for coder in the first method and the second method, so we propose the improved fifth method and sixth method. The improvement of the fifth method from the first method is essentially promotion from uniform Parseval frame to tight frame with some properties. The improvement of the sixth method from the second method is essentially promotion from Parseval frame to tight frame. The tight frame has very good properties, so it is often chosen as the encoder, which makes it more meaningful to improve the existed methods.Secondly, the latest research results on the structure of the optimal dual frame are summarized. Due to the particularity of the canonical dual frames’ structure and its extensive application in the process of signal transmission, we focus on that the optimal dual is the canonical dual. On this base, some new results are obtained through the analysis of the latest research results.Thirdly, a new method of constructing the optimal dual framework is proposed,that is: construct the optimal dual frame by using the eigenvalues of the frame operator.Several sufficient conditions are obtained, so that when the given frames as coder satisfy these conditions, the canonical dual frame is the unique(probability) optimal dual frame.Finally, the corresponding numerical examples and simulation results are given to prove the feasibility of the new methods.