A New Simulation Model of Rician Fading Channel Xinxin Jin 1,2,a, Yu Zhang 1,3,b, Changyong Pan 4,c

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1 6 Iteratioal Coferece o Iformatio Egieerig ad Commuicatios Techology (IECT 6 ISB: A ew Simulatio Model of Ricia Fadig Chael Xixi Ji,,a, Yu Zhag,3,b, Chagyog Pa 4,c Tsighua atioal Laboratory for Iformatio Sciece ad Techology(TList, Tsighua Uiversity, Departmet of Electroic Egieerig, Tsighua Uiversity, Beijig 84, Chia Sciece ad Techology o Iformatio Trasmissio ad Dissemiatio i Commuicatio etworks Laboratory 3 Shezhe City Key Laboratory of Digital TV System, Shezhe 5857, Chia 4 Research Istitute of Iformatio Techology, Tsighua Uiversity, Beijig 84, Chia a jxx3@mails.tsighua.edu.c, b zhag-yu@tsighua.edu.c, c pcy@tsighua.edu.c Key words: Ricia fadig, chael simulator, secod-order statistics Abstract. I this paper, a ew simulatio model of Ricia fadig chael is proposed. Based o a existig Rayleigh fadig chael model, the ew Ricia model takes a zero-mea siusoid with radom phase as its lie-of-sight compoet. First, the statistical properties of the ew model are aalyzed. It is show that the probability desity fuctio of the Ricia fadig evelope ad phase are idepedet of time, ad they are i accordace with the referece model. The simulatio results show that the secod-order statistics of the ew model approach the referece model ad oe existig Ricia model with a small fiite umber of propagatio paths ad ay Ricia factor. Itroductio Chael fadig model simulators are always used i the laboratory to test commuicatio system, because it costs less time ad moey tha field trials. Ricia fadig chael is a importat model of the radio chaels. The Ricia fadig chael is composed of lie-of sight compoet ad Rayleigh fadig chael. Rayleigh fadig chael model is a importat part. May Rayleigh fadig chael models have bee proposed []-[8]. Jakes model [] is a simplified simulatio model of the Clarke s model [], but because of the determiistic parameters of Jakes model, it ca t reflect the radom, the geeralized statioary ad ergodicity of the chael, so it ca t be used to geerate multiple chaels. The Pop ad Beaulieu employ radom phase [3] based o the Clarke s model. I [4]-[8], Chegsha Xiao ad Yahog R. Zheg presets three Rayleigh chael models ad their statistical properties ad probability desity fuctio (PDF of fadig evelope ad phase are aalyzed i detail. Xiao ad Zheg proposed a Ricia Fadig chael model [4] which takes a zero-mea siusoid with pre-chose agle of arrival ad a radom iitial phase as lie-of-sight compoet. I the remaider of this paper a ew Ricia fadig chael model is proposed based o the Rayleigh fadig chael model i [5] ad the structure of the model i [4]. We aalyzed the PDFs of the fadig evelope ad phase, ad the statistical properties of the ew model. The we give extesive simulatios to evaluate the performace of ew model compared with the Ricia model i [4] ad the referece model which employs Clarke s model as the Rayleigh chael. ew simulatio models A Ricia Fadig chael model proposed i [4] is take as Model ad give by Z( t Z ( t jz ( t ( c s Z ( t [ X ( t K cos( f tcos ]/ K ( c c m Z ( t [ X ( t K si( f tcos ]/ K, (3 s s m

2 where K is the ratio of the specular power to scattered power, f m is the maximum radia Doppler frequece, ad ad are the pre-chose agle of arrival ad the radom iitial phase of the lie-of-sight compoet. is uiformly distributed over [,. Xc ( t ad Xs ( t are the quadrature compoets of its Rayleigh chael model [4], give by X ( t cos( f tcos (4 c m X ( t cos( f tsi, (5 s m,,,,. (6 where is the umber of propagatio paths, ad are the agle of arrival ad radom iitial phase of the th propagatio path. ad are statitically idepet ad uiformly distributed over [, for all. It is poited that whe 8, Model s PDFs of fadig evelope ad phase ad the secod-order statistics are i very good agreemet with the theoretical oes. Aother Rayleigh chael model is proposed i [5] give by M Y ( t cos( f tcos (7 c d M M Y ( t cos( f tsi (8 s d M,,,, M. (9 4M where M is oe quarter of the umber of propagatio paths., ad are statitically idepet ad uiformly distributed over [, for all. We replace the Xc ( t ad Xs ( t i Eq. ad Eq. 3 with Yc ( t ad Ys ( t, the we get a fadig sigal Z '( t as. The autocorrelatio ad cross-correlatio fuctios of the quadrature compoets, the autocorrelatio fuctios of the complex evelope ad the squared evelope of Z '( t are give by R ( [ J ( f Kcos( f cos ]/( K ( m m c c R ( [ J ( f Kcos( f cos ]/( K ( m m s s R ( Ksi( f cos / ( K ( m c s R ( Ksi( f cos / ( K (3 m s c R ( [ J ( f Kexp( j f cos ] / ( K (4 m m J(4 fm R ( { J ( fm K[ J( fmcos( fm cos ] K }/ ( K. (5 Z Z 4M where J ( is the zero-order Bessel fuctio of the first kid [9]. The proof is similar to those give i [7,8], details are omitted for brevity. The autocorrelatio ad cross-correlatio fuctios give by Eq. Eq. 4 are idepedt of M, ad they are same as the Model s. Ad whe M is ot less tha 8, the autocorrelatio of the squared evelope has good approximatio.

3 Whe M approaches ifiity, the evelope Z '( t is Ricia distributed ad the phase is uiformly distributed over [,, ad their PDFs are give by f z K z K K z I z K K z (6 ( ( exp[ ( ] Z ' [ ( ], f ( /, [,. (7 where I ( is the zero-order modified Bessel fuctio of the first kid [9]. The proof is similar to those give i [5], details are omitted for brevity. The fadig evelope ad the phase of are same as Model. They are statioary because their PDFs are idepet of time, ad they are idepet. Moreover the PDF of the fadig phase is uiformly distributed over [,. The PDFs of the evelope ad the phase of Ricia iclude Rayleigh fadig as a special case whe K. Simulatio We take Clarke s model as the Reyleigh chael part of the referece model whe approaches ifiity. Ad the simulatio results are carried out whe the Model ad s umbers of siusoids are fiite ( M 8, ad we choose the ormalized samplig period ft.9,, ad the Ricia factor K is chose by some typical values. All the esemble averages for simulatio results are based o radom trials. Fig. ad Fig. show the PDFs of the fadig evelope ad the phase of the proposed simulator with K, 3, 7,. It ca be see that the PDFs are i excellet agreemet with those of the referece model ad Model. Whe K, the evelope s PDF turs to Reyleigh distributed. These PDFs will have better agreemet with the desired oes whe M 8. m s.8.6 PDF of the fadig evelope, =M=8 K= Model f ( PDF of the fadig phase, =8 K= Model f Z (z K= f ( PDF of the fadig phase, M=8.8.7 K=.6 K= z Figure. PDF of the fadig evelope..5 K= Figure. PDF of the fadig phase. Fig. 3 Fig. 6 show the simulatio results of the autocorrelatios of the quadrature compoets, the cross-correlatios of the quadrature compoets, ad the autocorrelatios of the complex evelope ad squared evelope of the proposed simulator. The simulatio results show that the secod-order statistics of the ew Rricia fadig chael Model match the referece s ad Model s very well whe M is as small as 8 ad the Ricia factor K,, 7. If the value of M is icreased, the results will be eve closer.

4 Re[R (] Im[R (].5 The real compoet of evelope autocorrelatio, =M=8 K= Model -.5 K= The imagiary compoet of evelope autocorrelatio, =M=8 K= Model -.5 K= Figure 3. Autocorrelatio of the quadrature compoets of the complex evelope. R ZcZc ( R ZsZs (.5 Autocorrelatio of the real compoet, =M=8 K= Model -.5 K= Autocorrelatio of the imagiary compoet, =M=8.5 K= Model -.5 K= Figure 4. Autocorrelatio of the quadrature compoets of the fadig evelope. R ZcZs ( R ZsZc (.5 Cross-correlatio of quadrature compoets, =M=8 Model -.5 K= Cross-correlatio of quadrature compoets, =M=8 K=.5 Model K= -.5 K= Figure 5. Cross-correlatio of the quadrature compoets of the fadig evelope. ormalized R Z Z ( Autocorrelatio of squared evelope, =M=8 K= Model K= Figure 6. Autocorrelatios of the squared evelope ad. Summary I this paper, a ew Ricia fadig chael model is proposed based o the Rayleigh chael model i [5] ad the structure of the model i [4]. We take Clarke s model as the Reyleigh chael part of the referece model whe its umber of siusoids approaches ifiity, ad the Ricia model i [4] is take as Model as a compariso. The secod-order statistics ad the PDFs of the fadig evelope ad phase of the ew model are aalyzed i detail. The simulatio results show that those statistics of the ew model are i excellet greetmet with the referece model ad Model whe the umber of siusoids is as small as 8 ad whatever the Ricia factor value is. Ad is as good as Model i all cases. Therefore, the ew model ca be used as Ricia fadig chael simulator i the laboratory. Ackowledgmet This work was supported by Sciece Fud for Creative Research Groups of SFC(636, Sciece ad Techology Partership Program, Miistry of Sciece ad Techology of Chia, Stadardizatio Admiistratio of the People s Republic of Chia with AQSIQ Project DTV- ad a grat (ITD-U55/KX564 from Sciece ad Techology o Iformatio Trasmissio ad Dissemiatio i Commuicatio etworks Laboratory.

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