The mathematical group of scattering matrices: Introduction of scattering matrix multiplication

Abstract

When the patch with the slot is added, the proposed antenna covers all operating bands. Figure 3 shows the simulated and measured VSWR for the proposed antenna. The measured bandwidth (VSWR Ͻ 2.5) is 9.78% (875-965 MHz) and 59.95% (1460( -2710 in the low and high band, respectively. The measured radiation patterns are shown in Figure 4. Since the shape of the radiation pattern is consistent for the DCS, PCS and UMTS bands, the measured radiation pattern at the centre frequency of GSM, GPS, DCS/PCS/UMTS, Bluetooth/WiFi, and SDMB bands only is presented here. It is noticed that the E component is dominant for GSM, GPS, DCS, PCS, and UMTS bands but the E component becomes more dominant for Bluetooth/WiFi and SDMB bands. The measured peak gains are listed in Table 1. It is shown that the performance of the proposed antenna can be suitable for multiband mobile handset applications. ACKNOWLEDGMENTSThis work is financially supported by the Ministry of Education and Human Resources Development (MOE), the Ministry of Commerce, Industry and Energy (MOCIE), and the Ministry of Labor (MOLAB) through the fostering project of the Lab of Excellency. Scattering parameters (S-parameters) are the most common choice to characterize high frequency networks. Significant effort has been invested in the arrangement of S-parameter in a matrix form to allow the calculation of transmitted and reflected signals using standard matrix mathematics [1][2][3]. However, these calculations are limited to matrix-vector operations, where the vectors are the incident or reflected waves. But the actual power of matrix algebra, matrix-matrix operations, is yet to be unleashed in S-parameter network analysis. Previous efforts to provide matrix-matrix operations for Scattering Parameter Matrices (S-matrices) are limited to reciprocal or lossless networks [4 -6], which are not the most common networks encountered in today's engineering environment. In the absence of matrix-matrix operations, alternative approaches were developed or implemented. Current solutions for RF network analysis are based on S-parameter measurements, so network synthesis is implemented through the conversion to other parameters (i.e. Z, Y, h, T) or through the use of semigraphical methods based on signal flow analysis and numerical simulations [7][8][9][10][11][12][13][14][15]. These approaches work well for 2-port networks, but are not easily generalized to N-port circuits. THE S-MATRIX GROUPA mathematical group is defined as a set of elements and a binary operation that satisfies the group axioms. In particular, we will define the elements of the group as the S-matrices whose elements are composed of complex numbers. We will consider that an S-matrix represents a realizable network. Formally, we will define the S-parameter Subspace (⌺ N ) as the subspace of N ϫ N complex matrices that contains only S-matrices. We will then note the binary operation as R and we will name it S-matrix multiplication. The group should satisfy the following axioms:If Sa, Sb and Sc ʦ N f͑Sa Sb͒ Sc ϭ...