The Fast Algorithm Of Discrete Signal Transforms Based On The First-order Moments

Since Jianguo Liu tried to calculation the convolution, fourier transform, cosine transform and discrete sine transform through moments, the algorithm of discrete signal transforms based on the first-order moments have become a new direction of the fast algorithms. There have been many papers reporting the development with the fast discrete signal transforms and first-order moments. However, these two research threads have been developed independently and the first-order moments based method has never been used for discrete W transforms, discrete winger-vill distribution and discrete hartley transform. This paper proposed a new method about how to set up the links between discrete W transforms, discrete winger-vill distribution and discrete hartley transform, and first-order moments. The moments computation and the polynomial function calculation are essentially the same, besides the first-order moments calculation can only be realized by additions. Thus the first-order moments based method can accelerate these discrete signal transforms. We finally conclude that all the discrete signal transforms which have similar inner product form can be used the first-order moments based method. We also analyzed the advantages and disadvantages of the proposed methods.Firstly, the p-network base method for geometric moments has been introduced, and a new method for the calculation of first-order moments has been proposed, a novel fast algorithm of geometric moments base on the first-order moments was presented. We also compare it with the latest block based method.Secondly, this paper proposes a novel approach to compute DWT via computation of the first-order moments without multiplications. A scalable and efficient systolic array is designed to implement this approach. Compared with the latest split-radix methods, the proposed algorithm is simpler and more applicable.Thirdly, a novel approach to Discrete Wigner-Ville Distribution(DWVD) is proposed in this paper. Because the first-order moments based method can not be directly used to compute the DWVD, we first introduce the theory of the winger-ville distribution(WD) and get the product form of WD by the theory of it, then, the first-order moments based method can be applied in the DWVD, similarly.Fourthly, the method that based on the first-order moments can be applied in the discrete hartley transforms similarly. We find when the length of the signal is small, this algorithm is even fast. The length-N type-III discrete hartley transforms can be decomposed into several length-16 type-III discrete hartley transforms based on the radix-2 fast algorithm, and the length-16 type-III discrete hartley transforms can be computed by first-order moments. It can save a lot of arithmetic operations and the computational complexity of the algorithms is lower than some existing methods. Moreover, this algorithm can be easily implemented.In the last, the proposed algorithm has several advantages which are: the algorithm can be directly realized in the form of a systolic array; the computational structure is very simple; there are no multiplications in our method.