Performance Analysis of EGC Combining over Correlated Nakagami-m Fading Channels

Transcription

1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 9, No., June, 93-3 UDK: 6.39 DOI:.98/SJEE93M Pefoance Analysis of EGC Cobining ove Coelated Nakagai- Fading Channels Dejan Milić, Selena Stanojčić, Zoan Popović, Dušan Stefanović 3, Ivana Petović 4 Abstact: In this pape, pefoance analysis of divesity technique with equal gain cobining ethod (EGC) with two banches fo the detection of signals in wieless counication systes is pesented. In the following analysis, it is assued that the fading via channels is Nakagai- coelated. The fist ode statistical chaacteistics of the syste ae analysed. Useful foulae fo the pobability density function (pdf) and cuulative distibution function (cdf) of EGC output SIR ae deived, and the effects of the fading seveity on the output signal ae obseved. Keywods: Nakagai- distibution, Coelated fading; EGC cobining. Intoduction Syste pefoances of wieless counications could be seiosly degaded by ultipath fading. Theefoe, vaious techniques fo educing fading effect ae used []. The goal of divesity techniques is to upgade tansission eliability without inceasing tansission powe and bandwidth and to incease channel capacity. Space divesity is an efficient ethod fo ipoveent syste s quality of sevice (QoS) when ultiple eceive antennas ae used []. Thee ae seveal cobining techniques that can be pefoed, depending on coplexity estiction put on the counication syste and aount of channel state infoation available at the eceive. Divesity techniques fo educing the fading effects ae bette if the signals ae independent, that is, if thee is no coelation between the syste banches. Most of the papes assue independent fading between the eception banches [3 5]. Howeve, since the distance between antennas is sall, it is ealistic to take into consideation the signal coelation [5]. Faculty of Electonic Engineeing, Univesity of Nis, Aleksanda Medvedeva 4, 8 Nis, Sebia; E-ail: dejan.ilic@elfak.ni.ac.s Faculty of Electonic Engineeing, Univesity of Nis, Aleksanda Medvedeva 4, 8 Nis, Sebia; E-ail: pop@tfc.kg.ac.s 3 High Technical School, Aleksanda Medvedeva, 8 Nis, Sebia; E-ail: dusan.stefanovic@itcenta.s 4 High School of Electonic Engineeing and Copute Science, Vojvode Stepe 3, Beogad, Sebia; E-ail: petovicvanja@yahoo.co 93

2 D. Milić, S. Stanojčić, Z. Popović, D. Stefanović, I. Petović It has been found expeientally, that the Nakagai- distibution offes a bette fit fo a wide ange of fading conditions in wieless counications [6, 7]. Seveal coelation odels have been poposed and used in the liteatue fo evaluating pefoance of divesity systes. The constant coelation odel coesponds to a scenaio with closely placed divesity antennas and cicula syetic antenna aays [8 ]. An analysis of signal cobining fo Nakagai- distibuted with constant coelation odel of fading has been given [], but assuing total independence between intefeences eceived on any pai of inputs of the cobine. In oe geneal case, the abitay coelation is pesent between the signals and intefeences. In this pape, analytical study of dual banch EGC cobining involving coelated Nakagai- fading with abitay coelation paaete between banches is analysed. We assue that two signals undegoing effects of fading channels have Nakagai- joint pobability density function. At the output of the EGC cobine, signal is equal to the su of the two input signals. To find that su, the input signals have to be put in phase which. EGC cobining povides good esults but, since it could be coplicated fo ipleentation. Syste Model EGC divesity cobining povides inteediate solution in tes of pefoance and ipleentation coplexity. In EGC eceive, each signal banch is ultiplied by a coplex weight and added up. Those coplex weights ae actually a phase coections that povide the signal aplitudes to add up coheently, while noise is added incoheently. Each banch is eal aplitude weighted with the sae facto, iespective of the signal aplitude. In this pape, using convenient tansfoation of the joint pobability density function, pobability density function of the signal at the cobine output has been deived. Integation of this signal, povides the cuulative distibution function to be obtained. Also, chaacteistic function of the output signal and n-th ode oent at the output of the EGC cobine, have been calculated. Fo n, signal output aveage is pesented, and fo n, aveage of the output signal quadatic value could be obtained, which is used fo finding the vaiance of the output signal. 3 Fist Ode Statistics The pefoance of the two banches EGC can be caied out by consideing, as in [3], the insufficient antennae spacing, both desied and intefeing signal envelopes expeience coelative Nakagai- fading with coesponding joint distibutions. We ae consideing constant coelation Nakagi- odel of distibution. 94

3 Pefoance Analysis of EGC Cobining ove Coelated Nakagai- Fading Channels The powe coelation coefficient ρ fo desied signal is defined as cov(r i, R j )/(va(r i )va(r j )) / We ae assuing abitay coelation coefficients between fading signals, because coelation coefficients depend on the aival angles of the contibution with the boadside diections of antennas. In a two-banches EGC divesity syste, at the EGC cobine input, the two signals ae epesented as two ando vaiables.the joint pobability density function fo the two signals and is given by [5]: + ( ) 4( ) e Ω ρ ρ Γ( ) Ω ρ y Ω ρ ρ 95 p ( ) I. () Hee, epesents the fading seveity paaete fo the desied signal. Fo the special case of we can evaluate expession fo pdf fo the Rayleigh signal. I - () epesents the odified Bessel function of the fist kind [], while Ω is the aveage signal powe. Γ() epesents the well-known Gaa function [4]. Cuulative distibution function fo two Nakagai- ando vaiables is obtained by integation of thei joint pobability density function. The closed fo of such integation is pefoed by developing the Bessell function fo the Nakagai- distibution expession into the seies and by integating the te by te. Cuulative distibution function of the two ando vaiables and is obtained by integating the joint pdf fo the expession (): d d F x x p x x x + x 4( xx ) exp Ω ( ρ) ρxx dx dx I ( ) ( )( y Ω ) ( ρ) Γ Ω ρ ρ ( ) dxdxbx x exp x x I xx i+ α + B γ i+ i+ i+ ( i+ α) γ ( i+ α) i! Γ ( i+ ) α α with coefficients B and and α defined as:,,, 4 ρ B ; α ;, (3) Γ( ) Ω ( ρ)( Ω ρ Ω ) ( ρ) Ω( ρ) while γ(a,x) denoting the lowe incoplete Gaa function []. ()

4 D. Milić, S. Stanojčić, Z. Popović, D. Stefanović, I. Petović Chaacteistic function of the and now can be pesented as: s + s M s s d de p x x ss s ( ) + s α + d d e B e I i i i s ( ) + s α + Bdde B e i+ i! Γ i+ B i+ s s i+ d d i! Γ ( i+ ) i + α e e i + α e e (4) s s B i+ + i! Γ +!! α i+ i i i+ i i ( i ) i i Γ ( i+ + i ) Γ ( i+ + i ). i+ + i α Joint oent of the and now can be pesented as: pq p q p q d d M p p q α ( + ) d d B e I B i i i p+ q+ α ( + ) Bdd e i+ i! Γ i+ i+! i+ i Γ i+ p+ + i α p+ + i α e d e d (5) i+ B i+ i+ + p i! Γ i+ α Γ ( i+ + p ) Γ ( i+ + q ). i+ + q α At the output of the EGC cobine, the two signals fo ae added: +, theefoe:. Pobability density function of the su of the signals and is: 96

5 Pefoance Analysis of EGC Cobining ove Coelated Nakagai- Fading Channels p p, d ( ) α + B e I ( ) d i i+ + i+ ( ) α + d i+ i! Γ ( i+ ) i+ i+ i α + ( ) + i+ i! Γ i+ d i+ i+ i+ i+ k α B e i+ i! i Γ i+ k k j j j α i+ + k + i γ(,α i+ + k + i ) j j B e B e i+! α i+ k α B B i, B i,, k B j e B i, j, k,,. 3 4 k j Chaacteistic function of the output signal is: i+ i i s s M( s) M ( s, s) B i+! i Γ i+ i! i! Γ ( i+ + i ) Γ ( i+ + i i+ + i ) i+ + i α α Moent of the n-th ode of the output signal is: n + n n n k k n ( ) k k i n + B i+ k k i! i Γ ( + ) ( n k) Γ i ( i k. i k ) n k + + Γ i+ + α α 4 Nueical Results (8) In the Fig., the pobability density functions fo diffeent values of fading seveity paaete, fo dual EGC cobining ove Nakagai- channels, ae pesented. 97 (6) (7)

6 D. Milić, S. Stanojčić, Z. Popović, D. Stefanović, I. Petović...8 ρ p(r) R Fig. PDF fo diffeent values of R and powe coelation coefficient ρ.5. Fig. shows the output oents fo diffeent values of the seveity paaete and ρ.5, while Fig. 3 shows the output oents fo vaious values of coelation paaete ρ. 5 Output EGC oents 4 3 st ode oent nd ode oent 3 d ode oent EGC with banches ρ Fig. Fist, second, and thid output oent fo diffeent values of the the fading seveity paaete. 98

7 Pefoance Analysis of EGC Cobining ove Coelated Nakagai- Fading Channels 9 8 st ode oent nd ode oent 3 d ode oent Output EGC oents Fig. 3 Fist, second, and thid output oent fo diffeent values of the powe coelation coefficient ρ. ρ Moent of the fist ode shows vey slow incease, while oent of the second ode incease uch faste with incease of paaete. Moent of the thid ode gows fast fo the vey beginning, that is, even fo the sall values of. Moent shows an obvious decease with the incease of the value of the coelation coefficient ρ. Decease is oe intensive fo the geate ode oents. 5 Conclusion In this pape, pefoance analysis of EGC, opeating ove coelated Nakagai- fading channels was pesented. The poposed theoetical analysis is siple, and copaed to othe analysis, this one povides acceptable solution in tes of pefoance and ipleentation coplexity. The useful fo of foulae fo pdf and cdf of EGC output SIR, wee deived. Using these foulae the effects of the fading seveity and the level of coelation to the output EGC oents, wee obseved. 6 Refeences [] G.L. Stube: Pinciples of Mobile Counication, Kluwe Acadeic Publishes, Boston, USA,. [] W.C.Y. Lee: Mobile Counications Engineeing: Theoy and Applications, Mc-Gaw- Hill, NY, USA,

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