On the Optimal Solution for BER Performance Improvement in Dual-Hop OFDM Relay Systems

Transcription

1 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 On the Optmal Soluton for BER Performance Improvement n Dual-Hop OFD Relay Systems Ens Kocan, lca Peanovc-Dursc Faculty of Electrcal Engneerng Unversty of ontenegro Podgorca, ontenegro ensk@ac.me, mlca@ac.me George K. Karagannds Department of Electrcal & Computer Engneerng Arstotle Unversty of Thessalonk Thessalonk, Greece geokarag@eee.org Abstract In ths paper bt error rate (BER) performance of DPSK modulated dual-hop OFD based relay systems mplementng ordered subcarrer mappng (SC) at relay staton (R) s analyzed. We propose a new soluton, where n the OFD amplfy-and-forward (AF) relay system wth fxed gan (FG) at the R staton, one or more subcarrers havng the lowest sgnal-to-nose ratos (SNRs) on both hops are frst omtted, and then the SC s performed. Analytcal and smulaton results have shown that n ths way sgnfcant BER performance mprovements are attaned n the regons of medum and hgh SNRs on both hops. Then, we compare BER performances of the most common OFD relay systems wth SC,.e. the systems applyng decode-and-forward (DF), AF varable gan (AF VG) relayng, and AF FG relayng, where the same soluton based on omttng the worst subcarrers on both hops s assumed. Through these comparsons, an optmal soluton for BER performance mprovement n the gven channels condtons, s defned. Keywords OFD relay system; BER; subcarrer mappng; AF FG; AF VG; DF; optmal soluton I. INTRODUCTION Relay systems as a soluton for coverage extenson and performance mprovement of wreless communcatons systems have attracted a great research nterest n the last decade. Partcular attenton s pad to them n moble cellular systems, where a relay staton (R) takes p 1 art n communcaton process between the base staton (source S) and end-user (D destnaton) n downlnk communcatons scenaros, where drect lnk between S and D cannot be establshed. Such a scenaro, denoted as dual-hop scenaro, s adopted n both standards for IT-Advanced systems, LTE- Advanced and WrelessAN-Advanced systems, [1]. Accordng to these standards, R staton n the next generaton moble cellular systems wll be OFD (Orthogonal Frequency Dvson ultplex) based and wll perform decode-andforward (DF) relayng. DF relayng assumes that R staton fully decodes the sgnal receved from S, and then agan reencodes t before forwardng toward D. Another consdered soluton for OFD based R statons n moble cellular systems s amplfy-and-forward (AF) relayng, where the sgnal receved from S s only amplfed by R and forwarded to D, [1]-[3]. Ths knd of relyng ntroduces smaller delay /14/$ IEEE than DF, and s easer to mplement. In such a system, the sgnal receved by R may be amplfed by fxed gan (FG), where the gan s constant regardless of the varatons of S-R channel, or by varable gan (VG), where the gan factor depends on the state of S-R channel. AF VG relayng method requres S-R channel state nformaton (CSI). OFD based AF relay systems have been subect of ntensve research nterest n recent years, and t s expected that ths knd of soluton should become a part of some future specfcatons of moble cellular systems, [3]. Advantages of OFD as a physcal layer soluton for relay systems are well recognzed and used n a soluton for capacty enhancement and/or bt error rate (BER) performance mprovement through ordered subcarrer mappng (SC) at R staton, [3]-[9]. Namely, t s proven n [4] that ergodc capacty n dual-hop OFD relay systems s maxmzed f subcarrers from the frst hop (S-R lnk) are ncreasngly ordered n accordance to ther nstantaneous sgnal-to-nose ratos (SNRs) at R staton and then are mapped to approprate subcarrers from the second hop (R-D lnk), whch are also ncreasngly ordered wth respect to ther nstantaneous SNRs. Ths knd of SC scheme s denoted as best-to-best SC (BTB SC) scheme. However, when BER s performance of nterest, authors have shown n [5], usng maorzaton theory of nequaltes, that n OFD AF dual-hop relay systems BTB SC mproves performance only n the regon of small values of average SNRs on both hops. For medum and hgher values of average SNRs on hops, the so called best-to-worst SC (BTW SC) scheme should be mplemented, where ncreasngly ordered subcarrers from the frst hop are mapped at R staton to decreasngly ordered subcarrers on the second hop. Analytcal evaluaton of BER performances of OFD AF FG and OFD AF VG relay systems gven n [6] and [7] have approved ths conclusons. Further BER performance mprovement n OFD relay systems wth SC can be acheved f one or more subcarrers havng the lowest SNRs (worst subcarrers) on both hops are omtted, before SC s performed, lke t s shown n [7] and [8] for OFD AF VG relay systems wth BTB SC, and for OFD DF relay systems wth BTB SC, respectvely. In ths paper we analytcally examne BER performance of DPSK (Dfferentally Phase Shft Keyng) OFD AF FG relay systems wth both BTB SC and BTW SC, where one /14/$ IEEE 276

2 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 or more the worst subcarrers from both hops are not used. DPSK scheme s chosen for analytcal convenence, and as the BER performances of the other DPSK modulated OFD based relay systems wth SC are avalable n the lterature for comparson. The nfluence of omttng the worst subcarrers on both hops n ths system has not been examned yet, and t s especally nterestng to see the level of performance mprovement wth such an soluton n case of BTW SC mplementaton. In order to dentfy the optmal soluton for BER performance mprovement n dual-hop OFD relay systems, the obtaned results are compared wth correspondng BER performances of OFD AF VG and OFD DF relay systems wth BTB SC. Raylegh fadng statstcs s assumed on both hops, for all the consdered systems. The paper s organzed as follows. Secton II descrbes the analyzed system s models. BER performance analyss of OFD AF FG relay systems wth SC s gven n Secton III. The obtaned analytcal and smulaton results are presented n Secton IV, whle Secton V gves some concludng remarks. II. SYSTE ODELS We analyze dual-hop OFD based relay systems wth ordered subcarrer mappng at the R staton, n a scenaro where the complete communcaton process s performed through the R staton,.e. there s no drect communcaton between S and D termnal. Three dfferent relayng strateges are consdered: AF FG, AF VG and DF. For OFD AF FG systems, two ordered SC schemes, BTB SC and BTW SC, known for enablng BER performance mprovement n dfferent channel condtons, are examned. It s assumed that all communcatng termnals are equpped wth a sngle antenna. Orthogonalty between the S-R and R-D channels s acheved by dvdng communcaton process nto two tme slots,.e. R operates n a half-duplex mode. It s assumed that R has perfect channel knowledge of both S-R and R-D lnks, what s mportant from the pont of choosng the approprate SC. In order to perform sgnal demodulaton t s necessary that D knows the permutaton functon performed at R. Fg. 1 presents block scheme of the OFD AF FG relay staton wth SC. After OFD demodulaton, performed through FFT (Fast Fourer Transformaton), SC s mplemented accordng to known CSI nformaton on both S- R and R-D lnks. H 1, and H 2, are the -th subcarrer channel transfer functons on the S-R lnk and R-D lnk, respectvely. s total number of subcarrers. IFFT (Inverse FFT) block for OFD modulaton follows the SC mappng, and before the transmsson, the sgnal s amplfed wth a fxed gan G. Block scheme of the OFD AF VG relay staton wth SC s gven n Fg. 2. Here, after OFD demodulaton, there s an amplfyng block n each of subcarrer branches. The gan ntroduced n the -th subcarrer branch s equal to G =1/H 1,. Such a soluton requres constant estmaton of the S-R channel and makes OFD AF VG relay ystem wth SC more complex for realzaton than t s the OFD AF FG relay system. Fg. 2. Block scheme of the OFD AF VG relay staton wth SC Fg. 3. Block scheme of the OFD DF relay staton wth SC Relay staton n the OFD DF relay system wth SC s presented n Fg. 3. In ths system, the R staton performs complete decodng of the receved sgnal, before SC s performed. After the SC block, sgnal s re-encoded and OFD modulated. Such an approach enables separaton of S- R and R-D lnks, whch means that dfferent symbol mappng schemes may be mplemented on S-R and R-D lnks, dependng on channels condtons. Ths brngs performance benefts, but at the cost of more complex realzaton compared to OFD AF relay systems wth SC. A scenaro wth ndependent and dentcally dstrbuted (..d.) Raylegh fadng among the subcarrers on S-R and R-D channels s assumed n all the systems consdered. Probablty densty functon (PDF) and cumulatve dstrbuton functon (CDF) of the SNR n each S-R subcarrer channel are gven by f (x)=λ exp(-λ x) and F (x)=1-exp(-λ x), whle the correspondng PDF and CDF of the SNR n each R-D subcarrer channel are gven by f (x)=λ exp(-λ x) and F (x)=1-exp(-λ x), respectvely. λ = 1/ and λ = 1/ denote the nverse of the average SNR on S-R and R-D lnks. Further on, BER performance for DPSK modulated OFD AF FG relay system wth SC s determned. Fg. 1. Block scheme of the OFD AF FG relay staton wth SC 277

3 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 III. BER PERFORANCE ANALYSIS The post-fft sgnal on the -th subcarrer, receved at the OFD relay staton, s gven by: Y, 1 R, = X1, H1, + N1,, (1) X s data symbol sent by S on the -th subcarrer. N 1, represents addtve whte Gausssan nose on the -th subcarrer wth varance E( N 1, 2 )=N 01, wth E( ) denotng the expectaton operator. Assumng that the SC functon ν() performs mappng of the -th subcarrer from the frst hop to the k-th subcarrer on the second hop, the frequency doman sgnal at D of OFD AF FG relay system wth SC can be wrtten as: YD, k = GH 2, kyr, υ ( ) + N2, k (2) = GH 2, k H1, X + GH 2, k N1, + N2, k, 1 k, where N 2,k s the addtve whte Gaussan nose at D, on the k- th subcarrer, wth varance E( N 2,k 2 ) =N 02. Usng (2) and the descrbed gan, the end-to-end SNR on the k-th subcarrer can be presented as: k, end =, k, k, + ρ where 2 2, = εs H1, ( N 01 ) and k, = εr H2, k ( N 02 ) denote nstantaneous SNRs on the -th subcarrer of the frst hop and k-th subcarrer of the second hop, respectvely. ε and S ε R represent average symbol energes per subcarrer transmtted by S and by R, respectvely. ρ s the coeffcent gven through 2 ρ = εr / ( G N 01). In order to evaluate BER performance of DPSK modulated OFD AF FG relay systems wth SC, moment generatng functon (GF) of the SNR for the k-th subcarrer on each hop has to be known. Thus, we use GF of the SNR for the k-th subcarrer n the system mplementng BTB SC, derved n [6]: k, end k 1 k 1 1 α α 1 = + = 0 = 0 T β ρ A, T ρ A ρ ρ A + e E1 T ( s ) β T ( s ) where E 1 ( ) represents the exponental ntegral functon defned n [10, (5.1.1)]. The coeffcents α and β are gven through: 1 k 1 α = ( 1) k 1 (4) (3) and β = + k + 1. (5) In (5), (:) denotes bnomal coeffcents. The coeffcents A and T (s) are ntroduced n (4) for the more clear presentaton of ths relaton, and they are equal to: A = β β and T = s + β /. (6) / Usng the same approach, GF of the SNR for the k-th subcarrer at D n the case of the BTW SC scheme mplemented at R s derved as, [6]: k, end k 1 k 1 α δ 1 = + = 0 = 0 T ε ρb, T ρb ρ ρb + e E1 T ( s ) ε T ( s ) In (7), the coeffcents δ and ε are equal to: 1 k δ = ( 1) k 1 whle B can be wrtten as: (7) and ε = + k, (8) B = β ε. (9) Usng the GF based approach for BER performance analyss, BER for the k-th subcarrer at D of the assumed DPSK modulated system s obtaned through, [11]: Pb, k = 0.5 (1), (10) k, end whle BER for the complete OFD AF FG relay system wth SC s derved through averagng (10) over all subcarrers: b 1 Pb, k k = 1 P =. (11) By substtutng (4) or (7) n (10) and then n relaton (11), BER performance of the DPSK modulated OFD AF FG relay system mplementng BTB SC or BTW SC s obtaned. It has been mentoned that the OFD AF FG relay system mplementng BTW SC outperforms the system wth BTB SC n terms of achevable BER performance n the regons of medum and hgh values of average SNRs on both hops. Ths motvated us to examne the level of BER performance mprovement acheved n the case where one or more subcarrers wth the lowest SNRs on both hops are not used, and to examne the mutual performance rato for ths two SC schemes n such a case. BER performance of the OFD AF FG relay system wth BTB SC, where m subcarrers wth the lowest SNRs from both hops are omtted, s easy to obtan analytcally from (11), f the summaton starts from k=m nstead of k=1, and averagng s done over -m subcarrers,.e.: Pb = m = 1 Pb, k. (12) k m In order to analytcally evaluate BER performance of the OFD AF G relay system wth BTW SC, wthout the worst m subcarrers from both hops, let us frst assume that the subcarrers from the frst hop are ncreasngly ordered n accordance to ther SNR, and subcarrers from the second hop 278

4 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 are decreasngly ordered wth respect to ther SNRs. If the worst m subcarrers from each of the two hops are omtted, then t means that the (m+1)st worst subcarrer from the frst hop s mapped to the best subcarrer on the second hop, thus gvng the expresson for the end-to-end SNR on the k-th subcarrer at D n the form: k + m, k, k, end =, = 1,.., k, + ρ k m (13) From (13), t s obvous that we need order statstcs for the subcarrer from the frst hop havng the (k + m )th lowest SNR. By modfyng the approach gven n [6], PDF of the SNR for ths subcarrer from the S-R lnk s obtaned as: where k 1+ m w x k + m, ( ) = λα, (14) = 0 f x e β λ 1 k 1+ m α = ( 1) and β = + k + m. (15) k 1+ m Usng the same approach as the one presented n [6], we derve GF of the SNR for the k-th subcarrer at D n the OFD AF FG relay system wth BTW SC, where m subcarrers from both hops are not used: k, end k 1+ m k 1 α δ 1 = + = 0 = 0 T ε ρb, T ρb ρ ρb + e E1, k = 1,..., m T ε T wth (16) B = β ε. (17) Substtutng (16) n (11) enables BER performance evaluaton of the system consdered. IV. RESULTS The subsequent analytcal and smulaton results assume perfectly synchronzed dual-hop OFD based relay systems wth mplemented SC. As n the real case scenaro, realzaton of the SC scheme on a subcarrer bass would requre large sgnalng overhead, adacent subcarrers should be grouped n chunks, and SC can be realzed on chunks bass, lke t s proposed n [9]. Thus, subcarrers of the OFD relay system may actually present chunks wth uncorrelated fadng. It s assumed that nose varances at R and D are equal,.e. N 01 =N 02, as well as the average symbol energes transmtted by S and by R, εs = ε. In the OFD AF FGF relay system R the so called sem-blnd scenaro s consdered, where R uses knowledge on channel state nformaton about the S-R lnk to calculate the gan G: yeldng to: G ( H1, 01) = E + ε ε N, (18) 2 2 R k S 2 εr 1/ 1 G = e E. (19) 2 1 εse H1, k Smulaton results are obtaned through onte Carlo smulatons of the part of the OFD relay system that belongs to frequency doman, what s possble and accurate approach, as we have assumed perfect synchronzaton among the communcaton termnals. The subcarrer channel transfer functons on both hops are generated as ndependent Gaussan complex random varables wth zero mean. Fg. 2 gves analytcal and smulaton results for the DPSK modulated OFD AF FG relay system mplementng both BTB SC and BTW SC, where the k=1 worst subcarrer on each hop s omtted before SC s mplemented (w/o k=1). A scenaro wth equal average SNR values on both hops s consdered. Systems wth =32 and =64 subcarrers (chunks) are analyzed. For the sake of comparson, BER performances of the ordnary (w/o SC) OFD AF FG relay system s also presented, as well as of the OFD AF FG relay system wth SC. Total matchng between the smulaton and analytcal results n Fg. 2 confrms the accuracy of the undertaken analytcal approach. The gven BER results show that omttng subcarrers wth the lowest SNRs on both hops preserves the mutual relaton n terms of BER performance between OFD AF FG relay systems mplementng BTW SC and BTB SC. Namely, n the regon of small values of average SNRs on hops, relay system mplementng BTB SC w/o k=1 acheves very small advantage compared to correspondng system wth BTW SC, whle n the regon of medum and hgh values of the average SNR on hops, relay system wth BTW SC w/o k=1 attans sgnfcantly better BER performance. Thus for example, f comparng systems wth =64 subcarrers (chunks) w/o k=1, t can be seen that, for the BER value of 10-3, the system wth BTW SC acheves SNR gan of approxmately 10dB compared to the correspondng system wth BTB SC, slghtly lower than 10dB SNR gan compared to the ordnary AF FG relay system, and about 7dB SNR gan when comparng to the AF FG relay system wth BTW SC whch uses all subcarrers. Even better BER performances are attaned n the OFD AF FG relay system wth BTW SC w/o k=1, havng less subcarrers (=32). Such a sgnfcant BER performance mprovement, obtaned by omttng ust one subcarrer wth the lowest SNR on each hop of the OFD AF FG relay system wth BTW SC, motvated us to examne f ths soluton can be consdered as the optmal one for the dual-hop OFD based relay systems wth SC,.e. to compare ts BER performance wth the performance of correspondng OFD based DF and AF VG relay systems wth SC. We used analytcal results obtaned n [7] and [8] for the competng OFD relay systems, as well as the concluson derved n the latter one, 279

5 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 that the BTB SC scheme presents optmal soluton for BER performance mprovement n OFD systems wth DF relayng for all SNR regons of nterest. out that the optmal soluton strongly depends on the channel condtons on both hops BER AF FG AF FG BTW SC AF FG BTB SC 10-4 AF FG BTB SC, w/o k=1, =64 AF FG BTB SC, w/o k=1, =32 AF FG BTW SC, w/o k=1, =64 AF FG BTW SC, w/o k=1, =32 smulaton results Average SNR per hop [db] Fg. 4. BER performance of the OFD AF FG relay systems wth SC BER performance comparson of the three dual-hop OFD relayng systems wth SC, but wthout k=6 worst subcarrers from both hops s presented n Fg. 5. A scenaro wth equal average SNRs on hops s consdered agan, and t s taken that OFD relay systems have =256 subcarrers. It can be seen that for the average SNR per hop up to 9dB, the OFD DF relay system wth BTB SC outperforms all the other consdered systems, whle for the hgher values of the average SNRs, OFD AF FG relay system wth BTW SC acheves the best BER performance. The SNR gan attaned n comparson to the OFD DF relay system wth BTB SC s a lttle bt more than 1dB for all BER values below In order to obtan more comprehensve nsght nto possble optmal soluton for BER performance mprovement n dual-hop OFD relay systems wth SC, we analyzed scenaro wth good channel condtons on the S-R lnk, as t corresponds to downlnk communcaton between the base staton (S) and the relay staton (R), where the poston of R s carefully chosen. BER performances n such a scenaro, presented as a functon of the average SNR on the R-D lnk, for two values of the average SNR on the S-R lnk, =15dB and =20dB, are shown n Fg. 6. In the former case, the system wth =128 subcarrers wthout k=6 worst subcarrers from the both hops s analyzed, whle n the latter case t s taken that =256 and k=8 worst subcarrers from each of the two hops are omtted. Results gven n Fg. 6 provde new nsght nto the soluton for BER performance mprovement n dual-hop OFD relay systems wth SC. It can be seen that the OFD DF relay system wth BTB SC slghtly prevals n BER performances, not only for very small values, but also n the regon of very hgh values of, for the both values of the analyzed average SNR on the S-R lnk. It turns BER 10-3 =256, w/o k=6 AF FG BTB SC 10-4 AF FG BTW SC AF VG BTB SC AF VG BTW SC DF BTB SC DF BTW SC Average SNR per hop [db] Fg. 5. BER performance comparson of the OFD relay systems wth SC BER =20dB, =256, w/o k=8 OFD AF FG BTW SC OFD AF VG BTB SC OFD DF BTB SC =15dB, =128, w/o k= Average SNR on R-D lnk [db] Fg. 6. BER performance of the OFD relay systems wth SC as a functon of average SNR on R-D lnk In the real case scenaro, for the greatest range of possble values of average SNRs on hops, OFD AF FG mplementng BTW SC where the worst subcarrers from each of the two hops are omtted, attans the best BER performance. The SNR gan acheved n ths regon of SNR values may be very sgnfcant. Thus for example, for =20dB and =10dB, the SNR gan that ths system attans compared to the correspondng OFD DF relay system wth BTB SC s almost 6dB, whle for the case when =20dB and =15dB, ths SNR gan s almost 5dB. On the other sde, when the channel condtons on both hops (or ust on the R-D lnk) are very bad, and n cases when s 2-3dB hgher than, the OFD DF relay system wth BTB SC not usng the worst subcarrers, represents the optmal soluton for BER performance mprovement n dual-hop OFD based relay systems. 280

6 IEEE WCNC Workshop on Wreless Evoluton Beyond 2020 V. CONCLUSIONS In ths paper we frstly examned, analytcally and through smulatons, BER performances of dual-hop OFD AF FG relay system wth SC, n the case where one or more subcarrers wth the lowest SNRs on both hops are not used. It has been shown that sgnfcant BER performance mprovement s attaned wth ths soluton, especally n the regon of medum and hgh values of average SNRs on hops, where the BTW SC scheme prevals n terms of BER performances. Havng ths n mnd, we wanted to determne f such soluton can be generally consdered as the optmal one for BER performance mprovement n dual-hop OFD based relay systems. Thus, we compared ts BER performance wth BER performances of dual-hop OFD DF and OFD AF VG relay systems wth SC, where also the worst subcarrers on both hops are omtted. The obtaned results have shown that the optmal soluton depends on the channel condtons on both hops. In the regon of low values of average SNRs on hops, as well as n the cases where the average SNR on the R- D lnk s 2-3 db hgher than the average SNR on the S-R lnk, the OFD DF relay system wth BTB SC and wthout the worst subcarrers on both hops attans the best BER performances. Otherwse, the OFD AF FG relay system wth BTW SC, where the worst subcarrers from the both hops are omtted, represents the optmal soluton whch enables sgnfcant BER performance mprovements. Besdes, ths system s the smplest for mplementaton. ACKNOWLEDGENT Ths paper s a part of the research proect Advanced solutons for performance mprovement of wreless cooperatve e-servce nfrastructures funded by nstry of Scence of ontenegro. The paper s produced ontly wth the researchers from the EPFL, Lausanne, Swtzerland, n the frame of the SCOPES proect Fosterng Development of an ICT Centre of Excellence n ontenegro. REFERENCES [1]. Peng, C. Yang, Z. Zhao, W. Wang, H-H. Chen, Cooperatve network codng n relay-based IT-Advanced Systems, IEEE Comm. agazne, vol. 50, no. 4, pp , Aprl [2] E. Yang, H. HU, J. Xu, G. ao, Relay Technologes for WAX and LTE-Advanced oble Systems, IEEE Commun. agazne, pp , October [3].Peanovc-Dursc, E.Kocan, R.Prasad, OFD based relay systems for future wreless networks, Rver Publshers, Denmark, [4] I. Hammerstrom and A. Wttneb, Jont power allocaton for nonregeneratve IO-OFD relay lnks, n Proc. of the IEEE Internatonal Conference on Acoustc, Speech and Sgnal Processng, vol.4, page IV, ay [5] C. K. Ho and A. Pandharpande, BER mnmzaton n relay-asssted OFD systems by subcarrer permutaton, n Proc. of the IEEE VTC08, pp , Sngapore, [6] E. Kocan,. Peanovc Dursc, D. S. chalopoulos, G. K. Karagannds, Performance evaluaton of OFD Amplfy-and- Forward Relay System wth Subcarrer Permutaton, IEICE Trans. on Commun., Vol.E93-B, no.05, pp , ay [7] E. Kocan. Peanovc-Dursc, G. K. Karagannds, New Soluton for BER Performance Improvment of OFD AF Relay Systems, n Proc. of IEEE conf. TELFOR 2012, pp , Belgrade, Serba, November [8] E. Kocan. Peanovc-Dursc, BER Performance Improvement for OFD DF Infrastructure Relay Lnks, n Proc. of WPC13 conf., pp. 1-5, Atlantc Cty, USA, June [9]. Herdn, A chunk based OFD amplfy-and-forward relayng scheme for 4G moble rado systems, n Proc. of the IEEE ICC 2006, pp , Istanbul, Turkey, [10]. Abramovtz and I. A. Stegun, Handbook of athematcal Functons wth Formulas, Graphs, and athematcal Tables, 9th ed. New York: Dover, [11] K. Smon and.-s. Aloun, Dgtal Communcaton over Fadng Channels, 2nd ed. New York: Wley,