Discrete Linear Modular Scale Invariant System --Complete Generalized Legendre Sequence Transform

In this paper, we study a new linear system called discrete linear modular scale invariant (DLMSI) system. The feature of the DLMSI system is that the frequency magnitude response of the system is invariant to the modular scale operation of the original signal. The properties, matrix representation, and block diagram implementation of the DLMSI system are proposed. Moreover, as the relationship between the discrete Fourier transform (DFT) and the circular convolution operation, we show that the complete generalized Legendre sequence transform (CGLST) can diagonalize the DLMSI matrix and we can implement the DLMSI system with complexity O(Nlog 2 N). Finally, we apply the DLMSI on scalable data reordering and image recognition and show its advantages.