Shallow Compaction Modeling and Upscaling: A One-Dimensional Analytical Solution and Upscaling

Natural depositional processes frequently give rise to the heterogeneous multilayer system, which is often overlooked but essential for the simulation of a geological process. The sediments undergo the large-strain process in shallow depth and the small-strain process in deep depth. With the transform matrix and Laplace transformation, a new method of solving multilayer small-strain (Terzaghi) and large-strain (Gibson) consolidations is proposed. The results from this work match the numerical results and other analytical solutions well. According to the method of transform matrix which can consider the integral properties of multilayer consolidation, a relevant upscaling method is developed. This method is more effective than the normally used weighted average method. Correspondingly, the upscaling results indicate that the upscaled properties of a multilayer system vary in the consolidation process.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels

Orthogonal Chirp Division Multiplexing (OCDM) is a modulation scheme which outperforms the conventional Orthogonal Frequency Division Multiplexing (OFDM) under frequency selective channels by using chirp subcarriers. However, low complexity equalization algorithms for OCDM based systems under doubly selective channels have not been investigated yet. Moreover, in OCDM, the usage of different phase matrices in modulation will lead to extra storage overhead. In this paper, we investigate an OCDM based modulation scheme termed uniform phase-Orthogonal Chirp Division Multiplexing (UP-OCDM) for high-speed communication over doubly selective channels. With uniform phase matrices equipped, UP-OCDM can reduce the storage requirement of modulation. We also prove that like OCDM, the transform matrix of UP-OCDM is circulant. Based on the circulant transform matrix, we show that the channel matrices in UP-OCDM system over doubly selective channels have special structures that (1) the equivalent frequency-domain channel matrix can be approximated as a band matrix, and (2) the transform domain channel matrix in the framework of the basis expansion model (BEM) is a sum of the product of diagonal and circulant matrices. Based on these special channel structures, two low-complexity equalization algorithms are proposed for UP-OCDM in this paper. The equalization algorithms are based on block LDL H factorization and iterative matrix inversion, respectively. Numerical simulations are finally proposed to show the performance of UP-OCDM and the validity of the proposed low complexity equalization algorithms. It is shown that when the channel is doubly selective, UP-OCDM and OCDM have similar BER performance, and both of them outperform OFDM. Moreover, the proposed low complexity equalizers for UP-OCDM both show better BER performance than their OFDM counterparts.

Discrete Cosine Transform Matrix Based SLM Algorithm for OFDM with Diminished PAPR for M-PSK over Different Subcarriers

Orthogonal frequency division multiplexing (OFDM) is the highly spectrally well-organized method that has the difficulty of excessive peak power to average power ratio (PAPR), which ultimately imposes constraints on the high-power amplifier. Many practices have been projected to lessen PAPR of the OFDM systems. Amongst all the practices, the selected mapping (SLM) method has drawn more attention because of distortion-less behaviour. This technique uses unique phase sequences. It has been learnt that phase formation for SLM is very tedious. In the proposed work, the SLM method has been used, but phase arrangement formation is based on the usage of discrete cosine transform (DCT) matrix. In this proposed work, discrete cosine transform matrix has been chosen based on the requirement of optimization so that the arrangement with lowest PAPR can be nominated for the transmission. MATLAB simulation depicts that the remarkable gain is achieved as compared with the existing technique. In the proposed work, scheming of phase sequences are very informal due to the use of a DCT matrix which has a definite structure and can be generated at the receiver side with the help of side information of the phases and communicated from the transmitter to the receiver.

A class of vector-valued subspace weak Gabor duals of type II

The theory of vector-valued frames (also called superframes) has important applications in signal multiplexing, and has interested many mathematicians in recent years. In this paper, we introduce the notion of weak Gabor duals of type II under the setting of subspaces [Formula: see text], where [Formula: see text] is a periodic subset of [Formula: see text]. Using the Zak transform matrix method, we characterize weak Gabor duals of type II and their uniqueness. An example is also presented to illustrate the generality of our results.