New Families Of Quaternary Sequences And Its Application In Compressive Sensing

With desired properties such as long period, large linear span, low correlation, good balance, ease of implementation, Pseudorandom sequences have wide applications in radar, sonar, communication systems, cryptography systems etc.. Binary and quaternary sequences are most often used in practice because the ease of implementation in hardware and software. Binary sequences have a long history, among them the most famous are Gold sequence and m-sequence. The m-sequence is widely used in communication system. Gold sequence was proposed by R.Gold in 1967 based on m-sequence and has more preferable properties. Quaternary sequences based on Galois Ring were investigated relatively late, the Wech and Sidelnikov bounds suggest that quaternary designs exist whose maximum nontrivial correlation parameter performance better than that of the best binary sequences by a factor of 2 for a given family size M and sequence length L. At present quaternary sequences designs with desired properties are still not sufficient, so it is meaningful to design pseudorandom quaternary sequences with good properties.The Nyquist sampling theorem specifies that, to avoid losing information when capturing a signal, one must sample at least two times faster than the signal bandwidth. It has greatly restricted the ability of information processing. The appearance of compressed sensing theory breaks the traditional Nyquist sampling theorem and makes it possible to capture and represent compressible signals at a rate significantly below the Nyquist rate. The design of measurement matrix is a key procedure in compressed sensing researches relating to the signal compression and the accuracy of signal reconstruction. A popular family of sensing matrices is a random projection or a matrix of independent and identically distributed, random variables from a sub-Gaussian distribution such as Gaussian or Bernoulli. This family of measurement matrices is well known as it is universally incoherent with all other sparse bases. This universality property of a measurement matrix is important because it enables us to sense a signal directly in its original domain without significant loss of sensing efficiency and without any other prior knowledge. In addition, it can be shown that random projection approaches the optimal sensing performance of required measured value. However, compressed sensing theory will widely used in practice only if the sensing matrix is easy to hardware implementation. The main contributions of this paper are as follows:1. We construct new families of quaternary sequences based on Galois Ring with good properties. Tang has proposed a method for transforming any family of quaternary sequences with the odd period N to another family of quaternary sequences with the period 2N. The new method is to transform any family of quaternary sequences with even period 2N to another family of quaternary sequences with period 4N. As an application of the method to sequence Family B and Family U1, two new quaternary sequence families with length 4(2n-1) are obtained. We prove that the new family has low correlation and larger linear span.2. We construct new measurement matrices with good performance and easy to hardware implementation. With low correlation and ideal balance, new families can be used as sensing matrix. First, theoretic analysis shows that new matrix constructed by new family is incoherent with some sparse bases. Second, Simulation results by MATLAB verify that the new matrix can reconstruct the original signal precisely or approximately. Meanwhile, the experimental results between the new matrix and Gaussian matrix are compared.