4. I. Kim, J. Cha, S. Hong, J. Kim, Y.Y. Woo, C.S. Park, and B. Kim, Highly linear three-way Doherty ampliﬁer with uneven power drive for repeater system, IEEE Microwave Wirel Compon Lett 16 (2006), 176–178. 5. J.Y. Lee, J.Y. Kim, J.H. Kim, K.J. Cho, and S.P. Stapleton, A high power asymmetric Doherty ampliﬁer with improved linear dynamic range, 2006 IEEE MTT-S Int Microwave Symp Dig 3 (2006), 1348–1351. 6. M.J. Pelk, W.C.E. Neo, J.R. Gajadharsing, R.S. Pengelly, and L.C.N de Vreede, A high-efﬁciency 100-W three-way Doherty am- pliﬁer for base-station applications, IEEE Trans Microwave Theory Tech 56 (2008), 1582–1591. 7. Y. Yang, J. Cha, and B. Kim, A microwave Doherty ampliﬁer employing envelope tracking technique for high efﬁciency and lin- earity, IEEE Microw Wirel Compon Lett 13 (2003), 370–372. 8. J. Jeong and P. Asbeck, Efﬁciency enhancement of W-CDMA base-station envelope tracking power ampliﬁers via load modula- tion, Microwave Opt Tech Lett 49 (2007), 1954–1957. 9. Y.S. Lee, M.W. Lee, and Y.H. Jeong, Highly linear power tracking Doherty ampliﬁer for WCDMA repeater applications, IEEE Microw Wirel Compon Lett 18 (2008), 485–487. 10. C.H. Oxley and M.J. Uren, Measurements of unity gain cutoff fre- quency and saturation velocity of a GaN HEMT transistor, IEEE Trans Electron Devices 52 (2005), 165–169. V C 2011 Wiley Periodicals, Inc. AN INTEGRAL EQUATION MODEL FOR RADIOWAVE PROPAGATION OVER INHOMOGENEOUS SMOOTHLY IRREGULAR TERRAIN Cla ´ udio G. Batista and Ca ´ ssio G. Rego Department of Electronics Engineering, Federal University of Minas Gerais, Belo Horizonte, MG 31270-970, Brazil; Corresponding author: cassio@cpdee.ufmg.br Received 1 April 2011 ABSTRACT: This work introduces an integral equation formulation to model the radiowave propagation over inhomogeneous smoothly irregular terrain. The ground energy loss is estimated via the Leontovich boundary condition and the fast far-ﬁeld algorithm accelerating technique is applied in order to improve the computational performance when solving the integral equation. The proposed formulation was implemented in a prediction software named SPRad, which was numerically tested with two measurement campaigns performed in Denmark and Brazil. V C 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:26–31, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26473 Key words: integral equations; method of moments; radiowave propagation; fast far-field algorithm accelerating technique; prediction software 1. INTRODUCTION Accurate characterization of the propagation mechanisms of the radio channel is constantly demanded when planning wireless networks. The use of full wave analytic models is becoming more attractive due to development of cheaper and faster perso- nal computers. Also, a variety of accelerating algorithm techni- ques has been introduced over the last years which have enabled analytic models to be applied to realistic problems in an efﬁ- cient way. Among the possible approaches, techniques based on ﬁeld in- tegral equations have been used to estimate the radio link path loss over irregular terrains [1].Some simpliﬁcations were adopted in some cases: the terrain proﬁle was assumed trans- verse invariant to the direction of propagation and the ground treated as a perfect conductor. Hviid et al. [2] presented an elec- tric ﬁeld integral equation (EFIE) for grazing incidence and treated the ground as a perfect magnetic conductor (PMC). The terrain was assumed electrically smooth along the plane of inci- dence, and, this led to the neglection of back scattering and allowed the application of a forward scheme to evaluate the induced currents, avoiding a Method of Moments (MoM) full- matrix linear system treatment. In [3], Moreira derived a mag- netic ﬁeld integral equation (MFIE) for imperfect conductor (IC) ground. Teperino [4] improved the PMC ground based formula- tion of [2] by considering the phase currents with linear varia- tion when evaluating speciﬁc analytic integrals. Recently, the application on rough terrain proﬁles was possible through the development of effective techniques to deal with the MoM full- matrix linear system, such as the characteristic basis function method [5] and the spectrally accelerated forward-backward method [6]. In this work, we propose an EFIE and MFIE formulation for application in irregular inhomogeneous smoothly terrains. The ground is treated as an IC, and the electromagnetic energy loss is estimated by the Leontovich boundary condition [7]. As in [2], we discard the back scattering by assuming the terrain elec- trically smooth, once it covers many practical applications in ra- dio propagation. Unlike several works [5, 6, 8], we adopted a tri-dimensional formulation treatment with 3D Green’s function and source. The ﬁnal equations are simpliﬁed to 1D by the sta- tionary-phase method (SPM) and solved numerically. The pro- posed formulation results in a more accurate technique with faster numerical convergence than the previously investigated methods [2–4]. To increase the applicability in realistic scenar- ios, we applied the fast far-ﬁeld algorithm (FAFFA) proposed by Lu and Chew [9] and Brennan and Cullen [10], which speeds up the MoM induced currents computation. A prediction software named SPRad is presented as a com- putational implementation of the proposed integral equation based technique. Additional prediction methods were incorpo- rated for comparison reference: ITU-R Recommendation ITU-R P.1546 [11], Okumura-Hata, ﬂat earth model and Ott’s formula- tion [12] for MF/HF. The program is build with a graphic inter- face user and contains a structured database for radio links stor- age and basic statistical analysis. SPRad was initially projected as an academic tool but also has practical applications in radio frequency planning. We applied the method introduced here in two studies cases: measurement campaigns performed in Denmark by Hviid et al. [2] and performed by Mayrink and coworkers [13] at Brası ´lia, Brazil. We compare the proposed formulation with well-known prediction methods and computed some statistical parameters. Finally, the results are discussed with some comments on the computational savings reached by the accelerating technique. 2. FORMULATION 2.1. Integral Equations For equivalent electric currents ~ Jð ~ rÞ and magnetic currents ~ Mð ~ rÞ, the EFIE and MFIE equations are written as [14] ~ E ~ r ðÞ T ¼ ~ E i ~ r ðÞþ Z 0 L 1 ~ Jð ~ rÞ  þ L 2 ~ Mð ~ rÞ     ; (1) ~ H ~ r ðÞ T ¼ ~ H i ~ r ðÞþ 1 Z 0 L 1 ~ Mð ~ rÞ    L 2 ~ Jð ~ rÞ    ; (2) 26 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 1 January 2012 DOI 10.1002/mop