Computer simulation: We assume 4DPSK transmission (M = 4) over AWGN channels and investigate BER performance improvements using computer simulations. The BER dependence on the observation symbol length L is shown in Fig. 2 at EbINo= 6dB. The VA decoding depth to output the detected sequence was set at D = 20 symbols. Also shown are the simulated BERs for 16 state ( L = 3) and 64 state ( L = 4) Viterbi DPD (note that RSVDPD equals Viterbi DPD when L = 2). For large values of L, the performance of RSVDPD approaches that of coherent phase detection with differential decoding (CPD). When L = 10, the simulated BER of the proposed scheme is 5.90 x lC3and that of CPD is 4.5 x l e 3 . The simulated BERs of RSVDPD using L = 1-10 are shown in Fig. 3 together with those of lDPD and CPD as a function of EdN,. Also in the Figure, the theoretical BER performance curves of lDPD and CPD are shown as dotted lines. It can be clearly seen in Fig. 3 that the proposed scheme outperforms 1DPD. Using L = 2 and 4 can yield performance improvements of -1.2 and 1.5dB at BER = lC3,respectively. When L = 10, performance approaches that of CPD with an EdN0 difference of only -0.2dB. Conclusion: The reduced state Viterbi DPD, called RSVDPD, was described for MDPSK signal reception. An M state Viterbi decoder is used. Its BER performance improvements were evaluated by computer simulation for 4DPSK transmission in AWGN to show that with L = 10 performance close to that of CPD is achieved with an EbINodifference of only -0.2dB. The RSVDPD decoder can simply be added to conventional lDPD receivers. It is much less complex than the Viterbi DPD [3] with respect to the number of branch metric computations per symbol, and is considered very practical. 0 IEE 1996 I5 January I996 Electronics Letters Online No: 19960365 F. Adachi (R&D Department, N T T Mobile Communications Network Incorporated, 1-2356 Take, Yokosuka-shi, Kanagawa-ken, 238 Japan) References 1 2 3 4 5 ‘Multiple-symbol differential detection of M P S K , IEEE Trans. Comrnun., 1990,38, pp. 300-308 MAKRAKIS, D., and FEHER, K.: ‘Optimal noncoherent detection Of PSK signals’, Electron. Lett., 1990, 26, pp. 398400 ADACHI, F.: ‘MLSE differential phase detection for M-ary DPSK, IEE Proc. Commun., 1994, 141, pp. 407412 DEUL-HALLEN, A., and HEEGARD, c.: ‘Delayed decision-feedback sequence estimation’, IEEE Trans. Commun., 1989,37, pp. 428436 RAHELI, R., POLYDOROS, A., and TZOU, c.-K.: ‘Per-survivor processing’, Digit. Signal Process., 1993, pp. 115-161 DIVSALAR, D., and SIMON, M.K.: omega medium, known as thie uniaxial omega medium, was introduced [2] by immersing two orthogonally-positioned ensembles of omega-shaped particles in an isotropic host medium. It has been found that the uniaxial omega medium has potential applications in the design of anti-reflecting shields. Here, a triaxial omega medium is proposed to generalise the already studied omega medium and the uniaxial omega medium. In practice, a triaxial omega medium with linear magnetoelectric interaction can be fabricated by arranging three types of omega-shaped microstructures in a host dielectric: all axes of the first type of particles are arranged along the x and y directions, those of the second type along the y and z axes, and those of the third type along the z and x axes. From a phenomenological point of view, a homogeneous triaxial omega medium can be characterised by the set of constitutive relations (the harmonic exp(iot) time dependence is adopted and suppressed hereafter) D =E. E+E.i’l B =P.H+ t .E (1) where E = E G , = p.6 are permittivity and permeability dyadics, respectively, with G = a,e,e,-tq,e,e, +a,e,e, determined by the distributions and electrical d-hensions of the omega-shaped particles. 5 = :(L&,)’/~A x I , and 5 = -i(&,pJ1/ZA x I denote the magnetoelectric pseudo-dyadics, whei-e A = crcr,+~e,+yexis determined by the physical properties of the omega-shaped inclusions. For p = y = 0, the medium reduces to the already studied uniaxial omega material [2]. Here, I = e,e,+e,e, +e,e, denotes the unit idem factor, and e, represents the unit vector in t h e j direction. 6 and p,, represent the permittivity and permeability of free space, respectively. Green dyadic: Substituting the constitutive relations eqn. 1 into the source-incorporated Maxwell1 equations, a set of field equations can be obtained (V + koA) x E 1-iwbCi. H + koA) >< H = J + iweSE. E (V (2b) where ko = u)(q,~)L/2. To simplify the analysis, we introduce a set of auxiliary fields E = e-koA,rg H = e-koA.r~l (3) where r = xe,+ye,+ze,. Then,, the E’-field vector wave equation is obtained Vx Ci-’ . V x E’ - k 2 & .E’ = -iwpekoA“J(r) (4) with kz = Ozw. To solve eqn. 4, we introduce the coordinate transformation h = ti r. Then, V = C i VI, and eqn. 4 can be represented in the h coordinate system V x x v xxE‘-k2E’ = - , j W b e k o A . f i - l . x - - l a .J(SE-l.X) (5) Noting the linearity property of eqn. 5, E’ can be constructed using the concept of the Green dyadic [3] Green dyadic and dipole radiation in triaxial omega medium E’ = -iwp lI: e k o A . a - l . q q A - A’). &-I . J ( 6 - 1 . X’)dA’ (6) The vector wave equation that the Green dyadic G(h)must satisfy can be obtained by substituting eqn. 6 in eqn. 5 D. Cheng Indexing terms: Electromagnetic$eld theory, Polymers vx x vx x G ( A ) - k 2 G ( A ) = [email protected]) (7) Using the Levine-Schwinger operational method [3], we fmd A triaxial omega medium, which can be artificially realised by embedding three types of omega-shaped particles in an isotropic host medium, is proposed. The Green dyadic and the electromagnetic field of a dipole radiator are presented, by introducing a set of auxiliary fields and coordinate transformation. where [email protected]) is the solution of V2,g(A) + k2g(A) = -6(A) (9) with Introduction: With recent advances in polymer synthesis techniques, increasing attention is being given to the analysis of interaction between electromagnetic waves and novel microwave materials to determine how to use these materials to provide better solutions to current engineering problems. Among these novel microwave materials, the omega medium should be mentioned. It was first proposed in [l] for its potential application in constructing a reciprocal phase shifter. Recently, a modification of the ELECTRONICS LE7TERS 74th March ‘1996 Vol. 32 For a,, q,a, > 0 the solution of eqn. 9 is e--zkX g(X) = with h = I h j = I G . r 1 = [(a,~)~+(a~y)’+(sz)~]’~~. No. 6 Introduction: It is well known that multilayer dielectric structures Noting the identity and the Green dyadic can subsequently be represented in the r coordinate system. Electromagnetic Jields of dipole source: For a dipole source located at the origin, J(r) = (I,e,+I,e,+I,e,)S(r). It can be rewritten in the h coordinate: J( E l . A) = (Ixex+IJ,ey+Izezj a,q,a,S(A). are excellent surface waveguides, having a fundamental surface wave mode with no cutoff. In planar transmission line applications the surface wave power leakage can cause undesirable crosstalk aud unexpected package effects. In printed circuit antenna applications the antenna performance can be severely degraded owing to improper excitation of surface waves, e.g. the reduction of antenna efficiency. So far some research work has been carried out to find the excitation characteristics of surface waves in multilayer dielectric structures. All the investigations dealt with the dielectric structures of few layers with only embedded horizontal sources, and no general expressions for surface wave power were given. In this Letter, efforts are made to explore the excitation characteristics of surface wave power in grounded or ungrounded multilayer dielectric structures, as shown in Fig. la. The sources can be any type of Hertzian dipole: horizontal electrical dipole (HED), vertical electrical dipole WED), horizontal magnetic dipole (HMD) or vertical magnetic dipole (VMD). General expressions are derived for the power carried in surface wave modes. The electric fields E(r) radiated from such a dipole source can be straightforwardly obtained from eqns. 3, 6 and 11. The electric field in the far region is e-koA r-zkX (7- e x e x ) 4nX (Iza:,cy,ez I y ~ z c y , e y+ I z ~ z c u y e z ) (15) which unifies the counterpart of the isotropic medium (a, = a, a,, and a = p = y = 0 in the constituent parameters). It can be seen from eqn. 15 that for a triaxial omega medium with a, @,y > 0, the electromagnetic waves are attenuated at the rateof&r/I&.r(. E(T) = -zap t + Concluding remarks: In the present study, a triaxial omega material is proposed. Green dyadic and electromagnetic fields radiated from a dipole source are formulated. The resulting expression indicates that the electromagnetic waves propagating in the triaxial omega medium are attenuated at the rate of exp (-k,A.r/ I G r 1, where A is the magnetoelectric pseudo-vector. This exponentiallike attenuation property of the electromagnetic waves propagating in a triaxial omega medium imply its potential applications in. antenna array to reduce the mutual coupling between antenna elements in a wideband frequency range. 3 January 1996 0 IEE 1996 Electronics Letters Online No: 19960393 D. Cheng (Wave Scattering and Remote Sensing Centre, Department of Electronic Engineering, Fudan University, Shanghai 200433, People’s Republic of China) References SAADOUN, M.M.I., and ENGHETA, N.: ‘A reciprocal phase shifter using 2 medium’, Microw. Opt. Technol. Lett., novel pseudochiral or C 1992, 5 , pp. 18k-188 2 TRETYAKOV, S A , and SOCHAVA, A.A.: ‘A proposed composite material for nonreflecting shields and antenna radomes’, Electron. Lett., 1993, 29, pp. 1048-1049 3 TAI, c.T.:‘Dyadic Green’s functions in electromagnetic theory’, 1 (Intertext, New York, 1971) ‘22F22 D22 ‘21 lJ-21 D21 dipole E11 hl D11 r’ FII a Fig. 1 Geometry of structure and equivalent transmission line models a Geometry b Transmission line models Theoretical derivation: For the case of a dipole embedded in a multilayer dielectric structure, the power generated by the source J or M is [l] J’J’J’ E . Jdw (for an electric dipole) (I) / / J ’H . Mdv (for a magnetic dipole) (2) P=P=- where E and H are the electric field and magnetic field associated with J and M, respectively. For the structure considered here, the expressions for E and H can be easily derived in the spectral domain [2]. The surface wave and space wave are easily separated from the Sommerfeld type integration of the total field: the fonner comes from the residues and the latter from the remaining part of the contour integral By applying the spectral domain dyadic Green’s functions [2], the expressions for the power carried in surface wave modes for different type of Hertzian dipoles are derived as follows: (ij HED Surface wave in multilayer dielectric structures excited by various types of ertzian dipoles Q.X. Men and S.Z. Zhu Indexing terms: Surface electromagnetic waves, Dielectric devices The surface wave excitations of deferent types of Hertzian dipoles in multilayer dielectric structures are investigated. General expressions are derived for analysis of the surface wave power produced by the dipoles in the structure. Numerical results are given for some special cases, which are in good agreement with earlier work reported by other researchers. 530 ELECTRONIC§ LETTERS 14th March 1996 Vol. 32 No. 6

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