CLOSED-FORM CHARACTERIZATION OF THE CHANNEL CAPACITY OF MULTI-BRANCH MAXIMAL RATIO COMBINING OVER CORRELATED NAKAGAMI FADING CHANNELS

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1 CLOSED-FORM CHARACTERIZATION OF THE CHANNEL CAPACITY OF MULTI-BRANCH MAXIMAL RATIO COMBINING OVER CORRELATED NAKAGAMI FADING CHANNELS Yawgeng A. Cha and Karl Yng-Ta Hang Department of Commncaton Engneerng, Yan Ze Unversty TaoYan, 3 Tawan eeyaw@satrn.yz.ed.tw and s94864@mal.yz.ed.tw ABSTRACT The closed-form of the average channel capacty s addressed for correlated Naagam fadng channels wth the mlt-branch mamal rato combnng (MRC), and ts lower bond and pper bond are derved, where eqal and the eponental correlaton models are consdered. The lower bond and the pper bond are employed to evalate the average channel capacty. Nmercal reslts are presented to compare the channel capacty wth dfferent fadng and correlaton parameters. KEY WORDS Channel Capacty, Dversty, Mamal Rato Combnng, Correlated Naagam Fadng. Introdcton The MRC [],[] can be employed to combat the wreless mltpath fadng effect. Wth the MRC, the receved sgnals of dversty branches are lnearly weghted and smmed to yeld the mamal sgnal-to-nose rato (SNR) for sgnal detecton. The Naagam dstrbton s sally sed to characterze the mltpath fadng effect [3], and the Raylegh fadng channel s a specal case of the Naagam fadng model. When the receved sgnals of dversty branches are statstcally dependent (e.g. for spatal dversty wth closely located antennas), the correlated Naagam fadng model s sed to characterze dependent fadng channels [4]-[6]. The channel capacty of the Raylegh fadng channel has been eplored n [7] and [8]. The channel capacty of MRC scheme over the correlated Naagam fadng channel was addressed n [9] for the specal case of two dversty branches. To or best nowledge, no reslts of the average channel capacty have been dsclosed for correlated Naagam fadng channels wth the general mlt-branch MRC. In ths paper, the closed-form of the average capacty s derved for correlated Naagam fadng channels wth the mlt-branch MRC, where the eqal and the eponental correlaton models are consdered. The closed-form s dffclt to evalate drectly. Ths, the correspondng lower bond and the pper bond on the average capacty are derved. The remander of the paper s organzed n the followng way. In Secton II, the eqally and eponentally correlated Naagam fadng models wth the mlt-branch MRC are addressed. In Secton III, the closed-form of the normalzed average channel capacty and the correspondng lower bond and pper bond are derved. Then, nmercal reslts are presented n Secton IV. Conclsons are drawn n Secton V.. Correlated Naagam Fadng Models For correlated Naagam fadng channels wth L dversty branches, let be the SNR of the -th dversty branch E be the average faded SNR per symbol per and [ ] branch for,,..., L. For the channel model, s characterzed by the gamma probablty densty fncton (pdf) [3] as m m m m f ( ) ep, m m, () ( m) for,,..., L, where denotes the gamma fncton, and m represents the fadng depth, and ths the smaller the m, the deeper the fadng. The case of m n the Naagam fadng model redces to the Raylegh fadng channel. In the contet, we consder correlated Naagam fadng channels, where (,,..., L ) are characterzed by correlated L-varate gamma dstrbtons. For the dversty wth MRC, the SNR of the receved sgnal after the MRC s gven by [3] L (). Eponental Correlaton Model Wth the eponental correlaton model presented n [], the correlaton coeffcent between and has the form ρ ρ wth < ρ <, and the pdf of wth the MRC has the form

2 f ep ( ) for, where e ( ) ( ( ) ) L ρ ρ L L ρ ρ. (4). Eqal Correlaton Model For the eqal correlaton model addressed n [], by sng Lemma n [], the correlaton coeffcent between and s gven by ρ, where < ρ < s a constant parameter. To smplfy the symbol notaton, m ρ, let ( ) m ( ) m G ρ ( ρ ρl) ( ) (5) and D( ) ( ) ρl ( ρ ρl ). Notce that the The term ln( ) n (8) can be read as [3, (9.. 6)] (3) ln ( ) F(,;; ) where F ( z) (9) α, β ; ; denotes the hypergeometrc fncton defned by [4, p.-6] F αβδ, ; ; λ α δβ δ λ ( ) d ( δ) ( β) ( δ β ). () Usng (8)-(), we have the normalzed average channel capacty read as C d f ( ). ln ( )( ) () Drect evalaton of the average channel capacty gven by () for the correlated Naagam fadng channels s dffclt. In the contet, pper and lower bonds on the average channel capacty are derved and employed to evalate the capacty. The average channel capacty gven by (8) s lower bonded by constant coeffcents defned above are fnctons of. C Ma ln f( ), Wth ths eqal correlaton model, the SNR of the ln () resltant sgnal wth the MRC has the pdf To characterze the pper bond, we rewrte the average f channel capacty gven by (8) as eq ( ) G e F m, ; D (6) C for, where F [ abz, ; ] s the degenerate ln( ) f ( ) ln( ) f ( ) ln ln. hypergeometrc fncton and can be wrtten n the seres form [3, (9.)] (3) 3 Then, (3) s pper bonded by az a( a ) z a( a )( a ) z F[ a, b; z].. C b b( b )! b( b )( b ) 3! ln( ) f ( ) ln z ( a). ln ( ) ( b) ln f f ln (4) (7) where the neqalty ln( ) ln for s sed. where ( a) ( a ) ( a ). 3. Eponental Correlaton Model 3. Evalaton of Channel Capacty When the sgnal propagates throgh the addtve whte Gassan nose (AWGN) channel, the capacty of fadng channel condtoned on fed faded SNR s C log ( ) [], where denotes the bandwdth of the receved sgnal. Then, the normalzed average channel capacty of fadng channels s [7] C log ( ) f ( ) ln( ) f ( ) ln (8) where f ( ) s the pdf of. For correlated Naagam channels wth the MRC, f ( ) s gven by (6) for the eqal correlaton model and by (3) for the eponental correlaton model. Wth (3) and (), the closed-form of the average channel capacty s gven by C log ( ) f ( ) ln( ) ( ) ( ) ( ) log e Ε 3;,, :; : ln e log e F (,; ; ) e (5) Ε p; α : q; ε : denotes the MacRobert E- where r fncton [4, p.-6] and the ntegral s 97

3 v ( α) ( β) ( αβδ, ; ; λ) v e F δ Ε ( 3; α, β, :; δ : ) (6) s employed [4, p.-6] wth ν > and >. To derve the lower bond and the pper bond on the average capacty for ths model, we sbsttte (3) nto the ntegral n () and obtan ln f ( ) ψ ln ψ ln ( ) ( ) (7) ψ s the dfferentaton of ln ( ) [3, (8.36)], where ( ) and the dentty [3, (4.35)] v v ep( ) ln ( v) ψ ( v) ln (8) s sed. Ths, the lower bond for the eponentally correlated model s C Ma log m L ψ, (9) ln Let ( ab, ) and ( ab, ) denote the lower and pper ncomplete gamma fnctons, respectvely [3, (8.35)]. For the pper bond, the frst ntegral n (4) reslts n ln ln f ln( ) e ( ) ( ) ( ) ( /, /( ln. () ln /( ) where the eqaton ln( ) n t t t e dt [ ] ( ) n 3 4 t t t t t t... e dt ( ) n ( n, ) () s sed. Then, wth some manplatons, the second and the thrd ntegrals n (4) yeld e ( ) ( ) ln f ln and ( ), () ln ln ln f ( ) ln e ( ) ( ) ( /, /( ( ) / ln ln ψ ( / ) ln ( /( ln respectvely, where the followng formla n ( ) ( ) n ep ln n, n ( n, ) ( n ) n ( )! (3) n n ( ln ) ( n, ) ( n) n n ( ln ) ( n, ) ψ ( n) ( n) ( n, ) ln n n n n n ( n) ψ ( n) ( ln ) n (4) s employed. By sng ()-(3), the pper bond gven by (4) can be evalated for the eponental correlaton model as ( ) ( ) ( ) C ( ) /, / ln /( ), ln ln ψ ln [ ] ( /, /( / ( / ) ln ( /( 3. Eqal Correlaton Model (5) Sbstttng (6) nto (), we have the closed-form of the normalzed average channel capacty for the eqal correlaton model as C log ( ) f ( ) (,; ; ) F e ( m) ( ) ( ) ( m)!ln ( ) GD F(,; ; ) e!ln GD!ln G D ( 3;,, :; : ) m Ε (6) 98

4 whch s dffclt to evalate drectly. For ths model, we also consder the lower bond and the pper bond approach. By sng the seres epresson gven by (7), the ntegral n () can be characterzed by ln f ( ) [ ] ln, ; G e F m D [ ln ] ψ (8) m! G D m G ln ln [, ; ] ln f ln e F m D G G md ln e ln e ln ln G m m D ln e ln L! ( ) ( ) m ( m md ) D G m D Gln e.... ( )! ln ( m) ( ) ( ) (7) (, ) ln ( ) ψ ( ) ( ) where D s a fncton of defned n the prevos secton. Then, wth some manplatons, the ntegral n (7) s (3) redced to the followng form respectvely. In conseqence, sng (3)-(3) for the ln f eqal correlaton model, we obtan the pper bond as where G s defned by (5). Based on (8), the lower bond on the average capacty for the eqal correlaton model can be obtaned from () as [ ] ψ C G D m Ma ( ) ln, m ln! (9) For the pper bond of ths case, the frst ntegral n (4) can be evalated as ln ln f G ln( ) e F[ m, ; D] ln G ( ) ( ) ( m ) D ln e ( m) ( )! G ( ) ( ) ( m ) D (, ) ( ln ) ( m)! ( ) ( ) (3) Then, wth some manplatons, the second and the thrd ntegrals n (4) are gven by G [, ; ] ln f ln e F m D G ( m ) ( ) D (, ) ln m and. (3) C ( ) D (, ) ( m) ( )( ( ( ) ( m) D ( ) ( ) (, ) ( m) D ( ) G ln G ln G!(ln ), ( ) ( ψ ( ) ln ). (33) 4. Nmercal Reslts In ths secton, the normalzed average channel capacty s eamned for dfferent vales of fadng parameters and dfferent nmbers of branches. For the eponental correlaton model, the average channel capacty verss dfferent fadng parameters (.e. m) s depcted n Fg., and verss dfferent dversty branches (.e. L) n Fg.. Then, smlar plots are presented n Fg. 3 and Fg. 4 for the eqal correlaton model. In Fg. and Fg. 3, trple dversty branches are sed. Notce that ncreasng ρ (from. to.6) reslts n a neglgble decrease of the lower bond and the pper bond for both correlaton models wth dfferent fadng parameters. A larger ncrease of ρ for small fadng parameters may reslt n the sgnfcant change of the average channel capacty. On the other hand, when the fadng parameter m becomes larger or more branches are sed, the average channel capacty ncreases, whch s consstent wth the ntton. 99

5 Fgre : The average channel capacty verss m for the eponental correlaton model wth trple branches. Fgre 4: The average channel capacty for the eqal correlaton model wth m Conclson Fgre : The average channel capacty for the eponental correlaton model wth m 3. The mpact of channel correlaton on the average channel capacty s eamned for the MRC over the correlated Naagam fadng channel wth mltple dversty branches, where both eqally and eponentally correlated fadng models are consdered. The closed-form epresson of the normalzed average channel capacty s derved. To evalate the average channel capacty, lower bonds and pper bonds on the average channel capacty have been derved, whch can be sed to assess the behavor of the channel capacty for correlated fadng channels. The method sed to derve the lower bond and the pper bond can also be appled to characterze the average channel capacty for other correlated channel models. References Fgre 3: The average channel capacty verss m for the eqal correlaton model wth trple branches. [] D.G. Brennan, Lnear Dversty Combnng Technqes, Proc. IRE, vol. 47, 75- [] T.S. Rappaport, Wreless Commncaton Prncples and Practce, nd ed. (New JerseyPrentce Hall, ). [3] M.K. Smon & M.-S. Alon, Dgtal Commncaton over Fadng Channels, nd ed. ( New YorWley, 4). [4] G.K. Karagannds, etc., Performance analyss of trple selecton dversty over eponental correlated Naagam-m fadng channels, IEEE Trans. Commn., 5(8), 3, [5] M.K. Smon & M.-S. Alon, A nfed performance analyss of dgtal commncaton wth dal selectve combnng dversty over correlated Raylegh and Naagam-m fadng channels, IEEE Trans. Commn., 47, 999,

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