CONECCT Transmit Antenna Selection For UWB Communication System Over IEEE a Channel

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1 COECCT Transmit Antenna Seection For UWB Communication System Over IEEE a Channe Abstract In this etter, we derived the bit error rate (BER of utra-wideband (UWB wireess communication system with transmit antenna seection and maximum ratio combining (TAS/MRC scheme in the IEEE a channe mode by approximating the ognorma (L distribution to the mixture of Gamma (MG distributions. Transmit antenna seection scheme is abe to capture many of the advantages of Mutipe-Input Mutipe-Output (MIMO systems with reduced system cost and computationa compexity. Simuation and anaytica resuts shows that TAS/MRC scheme can consideraby improve the BER performance of the UWB system. This paper aso presents the approximation of L distribution in IEEE a UWB channe mode to the MG distributions and how its parameter can be determined by appying the Expectation-Maximization (EM agorithm. Coseness and accuracy of this approximation can be seen by the simuation and the mean square error (MSE of the exact L and approximate MG distributions. Index Terms Utra-wideband (UWB, Transmit Antenna seection, Maximum ratio combing, mixture of Gamma distribution. I. ITRODUCTIO ULTRA-WIDEBAD (UWB wireess communication technoogy has a great potentia to provide very high data rate transmission ( Gbps at ow cost and power consumption. However, in the year 22, Federa Communication Commission (FCC imposed the power spectra density constrain of transmitting signas of the UWB system, which again imits the transmission data rates []. Incorporating the mutipe input and mutipe output (MIMO antenna technique can resove the power imit probem up to some extent [2]. As we know that MIMO technoogy enhance the spectra efficiency, coverage area and improved the transmission quaity ineary with the number of transmit and receive antennas without increasing the transmit power and bandwidth. Aong with these advantages, MIMO technique has some chaenges ike incompatibe with the current handhed device, system cost, hardware and computationa compexity increases with number of antennas [2], [3]. Transmit Antenna seection (TAS provides a viabe soution of the above chaenges without compromising the advantages of MIMO technoogy mentioned above. TAS technique uses a subset of tota avaiabe antennas of any MIMO system at transmitter side. Seection criteria Anand Agrawa and Rakhesh Singh shetrimayum Department of Eectronics and Eectrica Engineering Indian Institute of Technoogy Guwahati-7839, India. e-mai: {anand.agrawa, krs}@iitg.ernet.in to seect this antenna subset is based on maximum Signa to oise ratio (SR as defined in [4] [7]. In this paper, we proposed mixture of Gamma (MG distributions to approximate the L distribution for evauating the performance the UWB system. In the iterature [8] [] it has been presented that MG distribution approximate accuratey the composite shadowing/fading mode and many other random variabes (RVs. In the [], [2] it has been shown that L distribution can be approximated by Gamma or MG distributions. Parameters of these distributions can be determined by matching the r th moment of both the distributions or by Expectation Maximization (EM agorithm. The rest of this paper is organized as foows. Section II introduced the system and channe mode. Section III presents the MG distributions. Performance anaysis of UWB system with TAS/MRC scheme has been derived in section IV, foowed by the numerica resuts and discussion in section V. Finay, we present concusions in section VI. II. SYSTEM AD CHAEL MODEL We consider UWB-MIMO system having t transmit antennas and r receive antennas. Each wireess ink of UWB- MIMO system are assume to be independent and modeed by the IEEE a mode. In this paper, we consider a ( t,; r TAS/MRC for UWB-MIMO system. It means at one time instant, among the t transmitting antennas ony one antenna wi be active for transmission and remaining transmit antennas wi be sient whie at the receiver side a r antenna wi be active for the reception which perform Maximum Ratio Combining (MRC diversity scheme. The seection criteria to seect this one transmit antenna is based on maximum Signa to oise ratio (SR received at the receiver is defined in [5], [3] [5]. I = argmax i t /5/$3. 25 IEEE { C i = r j= ε i, j } where ε i, j is the energy received by the j th receiving antenna, whie the i th antenna transmits the signa. The information of I is sent to the the transmitter under the assumption of channe state information (CSI is perfecty avaiabe at the transmitter. (

2 2 ow at the transmitter we sort the C i in the ascending order of the magnitude and arrange ike C ( C (2... C (t. According to this order statistics and seection criteria (2, we assume that t th antenna wi be seected for transmission. From [6] pdf of the C (t is defined as f (t (x= t [F (x] t f (x (2 where f (x and F (x is the pdf and cdf of C i respectivey. A. IEEE a UWB Channe Mode The IEEE a standardization group has deveoped for the evauation of utra wideband communications systems. UWB channe is sighty different from conventiona composite wireess channe (Saeh-Vaenzuea mode because of its arge spectrum bandwidth. Due to it ony few mutipath components (MPC overap within each resovabe deay bin, therefore centra imit theorem is no onger appicabe, and the MPCs fading statistics are no onger Rayeigh. Aso there may be some deay bin which has no MPCs. On the basis of UWB channe behavior some modification has been proposed in the S-V channe mode to mode the UWB channe is caed modified S-V channe or IEEE a channe mode defined in []. ow the impuse response of the IEEE a channe to the ink between the i th transmit and j th receive antenna for the UWB-MIMO system is presented as h i, j (t=x M R m= r= α r,m δ(t T m τ r,m (3 where i t, j r, α r,m is the gain coefficient of r th mutipath component of the m th custer, T m is deay of the m th custer, and τ r,m is deay of the r th mutipath component reative to the m th custer arriva time, X represents the ognorma shadow fading. In the modified S-V channe mode a MPCs and X are ognorma distributed whie arriva time of the MPCs are Poisson distributed with custer arriva rate (Λ and ray arriva rate (λ. The IEEE a mode is categorized into four different measurement environments, namey CM, CM2, CM3 and CM4. Mode parameters for a four channe environment such as custer arriva rate (Λ, ray arriva rate (λ, custer decay factor (Γ, ray decay factor (γ, standard deviation of custer (σ,ray(σ 2 and shadow fading (σ x are isted in Tabe. Tabe : IEEE a channe parameters for four different environments []. Mode Parameter CM CM2 CM3 CM4 Distance (m LOS/LOS LOS LOS LOS LOS Λ (/ns λ (/ns Γ γ σ (db σ 2 (db σ x (db B. Mixture of Gamma (MG Distributions Whie performance anaysis of IEEE a channe of UWB Communication system is very difficut because of the intractabe mathematica expression of L distribution. In the IEEE a channe mode a the mutipath gain coefficients and shadowing are og-normay distributed. [] [2] shows MG distribution most accuratey approximate the composite channe and different RVs. Whie the inear weighted sum of Gamma distribution and tractabe mathematica expression of Characteristic function (CF motivate us to approximate the L distribution by MG distributions. In this section, we present the key expressions of approximate MG distributions. ow the density function and CF of MG distribution are respectivey presented as f mg (x= = w G(x,α,β, x > (4 where G (x,α,β = βα x α e β x Γ(α is a standard Gamma distribution, denotes the number of mixture components, w denotes the mixing coefficient of the Gamma distribution subjected to w > and = w =, α,β > are the shape and rate parameter of the th Gamma distribution and Γ(α is the gamma function. ow CF can be written as ( ψ mg (ν= w + jν α (5 = β In the iterature of [] [2], [7] it has been suggested some agorithms to determine the parameter of approximate MG distributions such as Least Square Fitting criteria, r th Moment agorithm, method of Maximum ikeihood and Expectation Maximization (EM agorithm. In this paper, we adapt EM agorithm because this works very efficienty and accuratey. C. Expectation Maximization (EM Agorithm Expectation Maximization (EM is an iterative technique to estimate the Maximum-ikeihood (ML based unknown parameters (θ of the MG distributions of given data points. EM agorithm aternativey repeat two basic step caed expectation (E and maximization (M steps defined in [7]. If X =(x,x 2,...x t,...,x be a test data. ow the updated parameters of MG distributions after m + iterations of EM agorithm are And, w m+ = β m+ = αm γ(/x t,θ m x tγ(/x t,θ m γ(/x t,θ m (6 (7 α m+ = α k + ζ mg α (X,θ m (8 2

3 3 where γ(/x t,θ m w m = G ( x t,α m,βm j= wm j G (, x t,α m j,βm j A. Cacuation for f (. and F(. From ( we know the C i = r ε i, j,owthecfofc i can j = be determined as G α (X,θ m = [n(x t +n(β m I(αm ]γ(/x t,θ m, ζ m denote the step size such that, ζ m (, and I(. is the digamma or poygamma function. Proof: Pease see Appendix A. The number of mixing components can be seected on the basis of mean square error (MSE between the exact (L distribution and the approximate (MG distribution is beow the threshod eve. MSE between approximate ( f App (x and exact ( f Ext (x distribution can be cacuated as [ MSE = E ( f Ext (x f app (x 2] (9 where E is the expectation operator. III. PERFORMACE AALYSIS OF UWB SYTEM WITH TAS/MRC SCHEME In this paper, we derived the CF based BER of UWB wireess communication system over IEEE a channe with TAS/MRC scheme. From [8], we know that the conditiona error probabiity (CEP of binary signa with coherent RAE receiver is P e (γ=q( ( ρr γ ( where ρ r = or - for orthogona or antipoda signas respectivey and γ is the received SR defined as γ = E b C (t whose pdf can be determined as Form (2, ( f γ (γ= γ f E (t / b E b ( f γ (γ=χ[f(γ ] t f (γ (2 where χ = t E b, γ = γ E b /. E b denote the bit energy and is the noise spectra density. Error probabiity of the UWB communication system over IEEE a channe can be cacuated by averaging the CEP over the pdf of received SR (γ. ow the average probabiity of error of binary signa with coherent RAE receiver for UWB communication system over TAS/MRC can be evauated as P e = P e (γ f γ (xdx (3 = Q( ( ρr γ f γ (γdγ From the expression (2 and (3 now we have, ( P e = χ Q ( ρ r γ E b [F (γ ] t f (γ dγ (4 ψ Ci (ν= = r j= r j= ψ εi, j (ν e (Iνε i, j fεi, j (xdx where I = and f εi, j (x denotes the pdf of ε i, j is f εi, j (x= 2π e H a3= (5 H ϑ H ( a3 ϑh a2 x a2= x H ψ ε (xa2 H f mg a3 xa3 H (6 Ixa3 H xh a2 +(xh a2 2 +(xa3 H 2] [ Proof: Pease see Appendix B. Using the Gauss-Hermite Quadrature, (5 can be presented as ψ ci (ν= r H j= a= ϑ H a f εi, j ( x H a e (Iνx H a +(xh a 2 (7 where ϑ a H, x ak a H and H are the weights, abscissas, number ak of points of the Gauss-Hermite Quadrature formua respectivey for k =,2,3. ow the pdf and cdf of C i can be evauated as f (x= ψ Ci (νe Iνx dν (8 2π and, x F (x= f (tdt (9 IV. UMERICAL RESULTS AD DISCUSSIO For sake of the better understanding we divide this section into two subsections. In the first subsection we wi focus the L distribution and its approximation and in the second subsection we wi discuss about the BER of the UWB system with TAS/MRC scheme. We consider, 5-components MG distributions to approximate the L distribution. To make cear understanding, we take one exampe. In this we generate test data sampes of L rv with µ=, σ=6 db and set the initia parameters of 5-MG distribution are w =(.35,.5,.2,.2,., α =(5,7,9,2,7 and β =(2,3,4,.5,7. ow after m= iterations of EM agorithm we have the updated parameters as w m =(.2832,.733,.23,.55,.88, α m = (4.9968, , 8.999,.988, and β m = (2.953, ,.692,.895, Figure shows the pdf and cdf of L distribution (continuous ine and its approximation 5-component MG distribution (discrete ine formed of the above defined parameters. We noticed that pdf and cdf of 5-MG distribution is exceenty matched to the L distribution. umerica resuts cacuate the mean square error (MSE =.32 between the L distribution and its approximated (MG distribution defined 3

4 4.9.8 Lognorma Distribution 5 component MG Distribution ( t = r =,2,3 Anaytica Simuation.7 f X (x and F X (x CDF PDF BER x SR (db Fig.. PDF, CDF of ognorma and its approximate 5-component MG RVs. Fig. 3. Performance comparison of UWB system for IEEE a CM with TAS/MRC scheme over different number of transmit and receive antennas. (9 of above defined parameters aso verify the coseness of these two distributions. x 4. 2 Anaytica Simuation.2.3 Likeihood for =5 mixture of Gamma components BER 3 4 Likeihood CM umber of iterations m Fig. 2. Likeihood of L (X m,θ m SR (db Fig. 4. Performance comparison of UWB system for IEEE a CM-4 with TAS/MRC (3,;3 scheme. Figure 2 shows the ikeihood L (X m,θ m of X is the function of iterations and we observed that initiay it increases upto 2 iterations then remains amost constant, it means after 2 iterations we get constant update parameters of MG distributions. Figure. 3 shows the simuation and anaytica BER of UWB system for IEEE a CM with TAS/MRC (2,;2, (3,;3 and without TAS/MRC (,; scheme. In this figure, we noticed the significant BER improvement in the UWB system by appying the TAS/MRC scheme. Whie the Figure. 4 shows the simuation and anaytica BER of UWB system with TAS/MRC (3,;3 for IEEE a CM-4. V. COCLUSIOS In this paper, we compute BER anaysis of UWB communication system for IEEE a channe with TAS/MRC scheme by approximating the L distribution to the MG distributions. We observed that TAS/MRC scheme can significanty improve the performance of UWB system for IEEE a channe mode. [] shows that MG distribution converges to any pdf over (, and parameters of MG distribution can be cacuated by EM agorithm. We aso noticed that whie approximating the L distribution to the MG distributions we can save a significant amount of computer simuation time. Appendix A: APPEDIX Consider the X =(x,x 2...x t...x be a test data of the MG distribution (4 and its Log-ikeihood (LLH can be written 4

5 5 as L (X,θ= = og( f mg (x og ( w G (x t,α,β = where θ =(θ,θ 2,...θ...θ k and θ {α,β,w }.Itisvery difficut to optimize the LLH because it has og of summation. We assume that test data vector (X is the independent and identicay distributed (i.i.d. incompete data. ow we put the some other data point Y =(y,y 2...y with distribution P(y t = =w. ow the compete argument data is (X,Y. ow the ikeihood function of compete argument data is L (X,Y =og [ w yt G yt (x t,α yt,β yt EM agorithm first cacuate the expected vaue of L (X,Y now we have, Z (X,θ,θ m =E [ L (X,Y /θ m,x ] where θ m is the current parameter of approximate GM distribution is used to cacuate the Z (X,θ,θ m after that EM agorithm perform second step which is to maximize the Z (X,θ,θ m to estimate the new parameters of GM distribution, can be cacuated as θ m+ =argmax θ =argmax θ [Z (X,θ,θ m ] [ = γ ( / x t,θ m] Using the Bayes s rue, ] [og(w +og(g(x t,α,β ] γ ( / x t,θ m = wm G(x t,α m,βm j= w m j G(x t,α m j,βm j Sove the Z(X,θ,θm θ =, to evauate the new parameters. We know that = w =, therefore we introduced Lagrange mutipier ( to find the expression of w w [ = γ ( / x t,θ m + [og(w +og(g(x t,α,β ] ( w = ] γ(/x t,θ m + = w = Summing both side over,wehave =. ow the update expression of w is w m+ = γ ( / x t,θ m ow we proceed to update β m+ Z (X,θ,θ m β = [ β where, = ] {og(w +og(g(x t,α,β }γ(/x t,θ m = og(g(x t,α,β = α og(β +(α og(x t x t β og(γ(α After differentiation of the above expression now we have, [ ] α x t γ(/x t,θ m = β ow we have, = α m β m+ = Simiary Z (X,θ,θ m = α = γ ( / x t,θ m x t γ ( / x t,θ m [og β +og(x t I(α m ]γ(/x t,θ m = where I ( α m = og(γ(α α denotes the digamma or poygamma function. Above expression does not give any soution. Therefore from this method we do not have any update vaue of α m but EM agorithm is gradient based agorithm. ow using gradient based agorithm we can estimate update α m as defined in (8. Appendix B: From [8] approximate energy coected by the L fingers receiver in the UWB system is presentation as ε = X 2 ε where ε is the energy coected without shadow fading. ote that X 2 and ε are statisticay independent to each other, Therefore pdf of ε can be computed as ( x f ε (x= y f X 2 f ε (ydy y where f X 2(., f ε (. are the density function of X 2 and ε respectivey. As we know that X is L distribution, so X 2 is aso be L distributed. In this paper we argue that L distribution can modeed by the MG distributions. Therefore f X 2(. = f mg (.. From [9], CF of ε can be presented as ψ ε (ν=l, (νe [ λψ ν(,l] e [ ΛJ(ν,L] Using (6 and Gauss-Hermite Quadrature formua now the pdf of ε is 5

6 6 f ε (y= 2π H a2= ϑ H a2ψ ε (x H a2e ow fina pdf expression of ε is f εi, j (x= 2π e H a3= H a2= (Iyx H a2 +(xh a2 2 ϑ H a3 ϑh a2 x H a3 ψ ε (x H a2 f mg ( x [Ix H a3 xh a2 +(xh a2 2 +(x H a3 2] x H a3 REFERECES [] J. Foerster, Channe modeing sub-committee report fina, IEEE P /49, 22. [2] T. aiser, F. Zheng, and E. Dimitrov, An overview of utra-wide-band systems with MIMO, Proc. IEEE, vo. 97, no. 2, pp , 29. [3] Y. S. Cho, J. im, W. Y. Yang, and C. G. ang, MIMO-OFDM wireess communications with MATLAB. John Wiey & Sons, pp. 35-4, 2. [4] Z. Chen, J. Yuan, and B. Vucetic, Anaysis of transmit antenna seection/maxima-ratio combining in rayeigh fading channes, IEEE Trans. Veh. Techno., vo. 54, no. 4, pp , Juy 25. [5] B. A. Bjerke, Z. Zvonar, and J. G. Proakis, Antenna diversity combining schemes for WCDMA systems in fading mutipath channes, IEEE Trans. Wireess Commun., vo. 3, pp. 97 6, Jan 24. [6] H. guyen, F. Zheng, and T. aiser, Antenna seection for time reversa mimo uwb systems, in in Proc. IEEE 69th VTC, 29, pp. 5. [7] Z. Chen, Z. Chi, Y. Li, and B. Vucetic, Error performance of maximaratio combining with transmit antenna seection in fat nakagami-m fading channes, IEEE Trans. Wireess Commun., vo. 8, no., pp , Jan 29. [8] M. Wiper, D. R. Insua, and F. Ruggeri, Mixtures of gamma distributions with appications, Journa of Computationa and Graphica Statistics, vo., no. 3, 2. [9] B. G. Lindsay, R. S. Pia, and P. Basak, Moment-based approximations of distributions using mixtures: Theory and appications, Ann. Inst. Statist. Math., vo. 52, pp , 2. [] S. Atapattu, C. Teambura, and H. Jiang, A mixture gamma distribution to mode the snr of wireess channes, IEEE Trans. Wireess Commun., vo., pp , 2. [] A. Abdi and M. aveh, On the utiity of gamma pdf in modeing shadow fading (sow fading, in Proc. IEEE Veh. Techno. Cof., Houston, TX., 999, pp [2] J. Amhana, Z. Liu, V. Chouakian, and R. McGorman, A recursive agorithm for gamma mixture modes. IEEE ICC, 26, pp [3] M. Z. Win and J. H. Winters, Anaysis of hybrid seection/maximaratio combining in rayeigh fading. IEEE ICC, 999, pp. 6. [4] A. F. Moisch and M. Z. Win, MIMO systems with antenna seection, IEEE, Microwave Magazine, vo. 5, no., pp , 24. [5] A. F. Moisch, MIMO systems with antenna seection-an overview, in in Proc. IEEE RAWCO., 23, pp [6] H. A. David and H.. agaraja, Order statistics. Wiey, 97. [7] R. A. Redner and H. F. Waker, Mixture densities, maximum ikeihood and the EM agorithm, SIAM review, vo. 26, no. 2, pp , 984. [8] L.-C. Wang and W.-C. Liu, Bit error rate anaysis in IEEE a UWB channes, IEEE Trans. Wireess Commun, vo. 9, no. 5, pp , 2. [9]. Hao and J. A. Gubner, The distribution of sums of path gains in the IEEE a UWB channe mode, IEEE Trans. Wireess Commun., vo. 6, pp. 8 86, 27. 6