Upper Bounds on the BER Performance of MTCM-STBC Schemes over Shadowed Rician Fading Channels

Transcription

1 EURASIP Journal on Applied Signal Proceing 2004:9, c 2004 Hindawi Publihing Corporation Upper Bound on the BER Performance of MTCM-STBC Scheme over Shadowed Rician Fading Channel M. Uyal Department of Electrical and Computer Engineering, Univerity of Waterloo, Waterloo, ON, Canada N2L 3G muyal@ece.uwaterloo.ca C. N. Georghiade Department of Electrical Engineering, Texa A&M Univerity, College Station, TX , USA georghiade@ee.tamu.edu Received 7 May 2003; Revied 2 October 2003 Space-time block coding STBC) provide ubtantial diverity advantage with a low decoding complexity. However, thee code are not deigned to achieve coding gain. Outer code hould be concatenated with STBC to provide additional coding gain. In thi paper, we analyze the performance of concatenated trelli-coded STBC cheme over hadowed Rician frequency-flat fading channel. We derive an exact pairwie error probability PEP) expreion that reveal the dominant factor affecting performance. Baed on the derived PEP, in conjunction with the tranfer function technique, we alo preent upper bound on the bit error rate BER), which are further hown to be tight through a Monte-Carlo imulation tudy. Keyword and phrae: pace-time block coding, trelli-coded modulation, Rician fading channel, hadowing, pairwie error probability.. INTRODUCTION Space-time trelli coding wa introduced in []aaneffective tranmit diverity technique to combat fading. Thee code were deigned to achieve maximum diverity gain. However, for a fixed number of tranmit antenna, their decoding complexity increae exponentially with the tranmiion rate. Space-time block coding STBC) [2] wa propoed a an attractive alternative to it trelli counterpart with a much lower decoding complexity. The work in [2] wa inpired by Alamouti early work [3], where a imple two-branch tranmit diverity cheme wa preented and hown to provide the ame diverity order a maximal-ratio receiver combining with two receive antenna. Alamouti cheme i appealing in term of it performance and implicity. Auming the channel i known at the receiver, it require a imple maximumlikelihood decoding algorithm baed only on linear proceing at the receiver. STBC generalize Alamouti cheme to an arbitrary number of tranmit antenna and i able to provide the full diverity promied by the tranmit and receive antenna. However, thee code are not deigned to achieve a coding gain. Therefore, outer code hould be concatenated with STBC to achieve additional coding gain. A pioneering work toward thi end i preented in [4] where concatenation of trelli-coded modulation TCM) with STBC i conidered. In [4], it i hown that the free ditance of the trelli code dominate performance; therefore, the optimal trelli code deigned for additive white Gauian noie AWGN) are alo optimum for concatenated TCM-STBC over quaitatic Rayleigh fading channel. We tudied the ame concatenated cheme combined with an interleaver in [5] over Rician fading channel. In thi paper, we generalize our work to hadowed Rician channel. The hadowed Rician channel [6] i a generalization of the Rician model, where the line-of-ight LOS) path i ubjected to a lognormal tranformation due to foliage attenuation or blockage, alo referred to a hadowing. Specifically,wederiveanexactpairwieerrorprobability PEP) for concatenated TCM-STBC cheme. Our exact evaluation of PEP i baed on the moment-generating function technique [7, 8], which ha been uccefully applied to the analyi of digital communication ytem over fading channel. Uing the claical tranfer function technique baed on the exact PEP, we obtain upper bound on bit error rate BER) performance, which are further verified through imulation. Our analyi alo reveal the election criteria for trelli code which hould be ued in conjunction with STBC. The organization of the paper i a follow. In Section 2 we explain our ytem model, where the concatenated TCM- STBC i decribed and the channel model under conideration i introduced. In Section 3 an exact expreion for PEP i derived for the TCM-STBC cheme uing the MGF approach. Baed on the derived PEP, we dicu the election criteria

2 Upper Bound on the BER Performance of MTCM-STBC Scheme 239 for trelli code which hould be ued with pace-time code for optimal performance and compare them with the claical election criteria for trelli code over fading channel without tranmitter diverity. In Section 4, uing the tranfer function technique in conjunction with the derived PEP expreion, we obtain upper bound on the BER performance. Analytical performance reult are preented for two example trelli code, which are further confirmed through Monte- Carlo imulation. 2. SYSTEM MODEL We conider a wirele communication cenario where the tranmitter i equipped with M antenna and the receiver i equipped with N antenna. The binary data i firt encoded by a trelli encoder. After trelli coded ymbol are interleaved and mapped to contellation ymbol, they are fed to the STBC encoder. An STBC i defined [2] byanl M code matrix, where L repreent the number of time interval for tranmitting P ymbol, reulting in a code rate of P/L. For Tarokh et al. orthogonal pace-time block code [2], the entrie of the code matrix are choen a linear combination of the tranmiion ymbol and their conjugate. For example, the code matrix for the well-known Alamouti cheme i.e., STBC for 2 tranmit antenna) i given by [ ] x x 2 x2 x ) with M = P = L = 2. We aume that the tranmiion frame from each antenna conit of a total of FL ymbol i.e., conecutive F maller inner-frame, each of them having duration L ymbol correponding to the STBC length). The received ignal at receive antenna n n =, 2,..., N) at time interval l of the f th f =, 2,..., F) inner-frame i a uperpoition of M tranmitted ignal: M rn f l) = αm,n f xml)+η f n f l), 2) m= where x f ml) i the modulation ymbol tranmitted from the mth tranmit antenna at time interval l of the f th frame and η f n l) i additive noie, modeled a a complex Gauian random variable with zero mean and variance N 0 /2 per dimenion. α f m,n repreent the fading coefficient modeling the channel from the mth tranmit to the nth receive antenna during the f th inner frame and are aumed to be independent and identically ditributed i.i.d.). The fading coefficient i aumed to remain contant over an inner-frame period i.e., L ymbol interval). Thi aumption i neceary to make ue of the orthogonal tructure of STBC to guarantee full patial diverity. The aumption of quaitatic behavior of the channel over an inner-frame period can be jutified uing an L-ymbol interleaver over a moderately low varying channel. In our cae, the fading amplitude i decribed by the hadowed Rician fading model. In thi model, the LOS component i not contant but rather a lognormally ditributed random variable. The fading coefficient can be expreed dropping the ubcript and upercript for notational convenience) a α = µ + ξ 0 + jξ,whereξ 0 and ξ are independent Gauian random variable with zero mean and variance σ 2. Here, the LOS component i given a µ = expξ 2 ) where ξ 2 i a Gauian random variable with mean m µ and variance σµ, 2 and independent of ξ 0 and ξ. The conditional probability denity function of the fading amplitude α i ) p α µ α µ = α σ 2 exp α 2 + µ 2 ) 2σ 2 I 0 α µ σ 2 ), α 0, where I 0 ) i the zero-order modified Beel function of the firt kind, and the probability denity function of the LOS component i given by p µ µ) = 2πσµ µ exp 3) ) 2 ) ln µ mµ. 4) 2σ 2 µ The parameter σ, σ µ,andm µ in 3) and4) pecify the degree of hadowing. Denoting by C m n the vector pace of m- by-n complex matrice, and defining rn f = rn f ), rn f 2),..., rn f L) ) T C L, αn f = α,n, f α2,n, f..., αm,n) f T C M, ηn f = ηn f ), ηn f 2),..., ηn f L) ) T C L, the received ignal can be written in matrix notation a 5) rn f = X f αn f + ηn f, n =, 2,..., N, f =, 2,..., F, 6) where X f C L M conit of pace-time encoded ymbol which have been already trelli encoded) for the f th inner frame. At the receiver, firt the received ignal i paed through the pace-time decoder, which i eentially baed on linear proceing for STBC from orthogonal deign [2]. After deinterleaving, the proceed equence i fed to the trelli decoder implemented by a Viterbi algorithm. If a multiple TCM MTCM) cheme with M ymbol per branch i ued note that the number of tranmit antenna i alo given a M), the decoding tep can be combined in one tep with a proper modification of the metric employed in the Viterbi algorithm. In thi cae, the received ignal i jut deinterleaved and fed directly to the Viterbi decoder without any further proceing. 3. DERIVATION OF EXACT PEP In thi ection, we analyze the PEP of the concatenated cheme over hadowed Rician fading channel auming Throughout thi paper, we ue ) T and ) H for the tranpoe and tranpoe conjugate operation, repectively. Upper cae bold face letter repreent matrice and lower cae bold face letter repreent vector.

3 240 EURASIP Journal on Applied Signal Proceing perfect channel tate information i available at the receiver. Auming equal tranmitted power at all tranmit antenna, the conditional PEP of tranmitting code matrix X which conit of X f, f =, 2,..., F) and erroneouly deciding in favor of another code matrix ˆX at the decoder i given by P X, ˆX αm,n, f µ m,n, f m =,..., M, n =,..., N, f =,..., F ) F N f ) HAf = Q αn αn f, f = n= where Q ) i the Gauian Q-function and A f i given by A f = M 7) E 2N 0 X f ˆX f ) H X f ˆX f ). 8) Here, E i the total ignal power tranmitted from all M tranmit antenna and N 0 /2 i the noie variance per dimenion. In order to find the unconditional PEP, we need to take expectation with repect to αm,n f and µ m,n. f The expectation with repect to fading coefficient can be obtained through ue of the alternative form of the Gauian Q-function [8]a P X, ˆX µ m,n, f m =,..., M, n =,..., N, f =,..., F ) = π/2 ) Φ Γ π 0 2in 2 d, 9) where Φ Γ ) i the moment generating function MGF) of F N f ) HA Γ = α f n αn. f 0) f = n= Γ i a quadratic form of complex Gauian random variable and it MGF i given a [9, 0] Φ Γ ) = F N M f = n= m= χ m exp χm d m 2 ), ) χ m where χ m are the eigenvalue of ΣA f and d m are the element of M-length vector d = µσ /2.Hereµ and Σ repreent the mean vector and the covariance matrix of αn,re- pectively. Making ue of the aumed i.i.d. propertie of the f fading channel, we obtain d m 2 = µ 2 /2σ 2. Furthermore, in our cae, A f i a diagonal matrix due to the orthogonality of STBC and the eigenvalue χ m are imply equal to the diagonal element of ΣA f, that i, E 2N 0 2σ 2 β M P xp f ˆx f p 2, 2) p= where β = form = 2andβ = 2forM>2 due to the pecial matrix tructure of STBC baed on orthogonal deign [2]. Inerting ) into 9) and uing the i.i.d. propertie for fading coefficient, we obtain P X, ˆX µ ) = [ π/2 F π 0 f = +Ω f / in 2 exp where Ω f = E 4N 0 2σ 2 β M µ2 2σ 2 Ω f / in 2 )] MN +Ω f / in 2 d, 3) P xp f ˆx f p 2. 4) p= To find the unconditional PEP, we till need to take an expectation of 3) with repect to µ, whoe ditribution i given by 4). Thi expectation yield P X, ˆX ) = π π/2 [ F +Ω f / in 2 µ=0 µ exp =0 f = 2πσµ µ2 Ω f / in 2 ) 2σ 2 +Ω f / in 2 ) 2 ) MN ln µ mµ exp dµ] d. 2σ 2 µ 5) Introducing the variable change u = ln µ m µ )/ 2σµ,5) 2 can be rewritten a P X, ˆX ) = π π/2 =0 f = [ F +Ω f / in 2 exp u= π exp u 2) Ω f / in 2 2σ 2 +Ω f / in 2 exp 2 2σ µ u +2m µ ) ) du] MN d. 6) The inner integral ha the form of exp u 2 ) f u)du, which can be expreed in term of an infinite um ee the appendix). Thi yield the final form of the exact PEP a P X, ˆX ) = π/2 F π =0 f = +Ω f / in 2 exp f ) ) + k=2 k:even k )!! ) k 2σµ k! MN k g k,d f ) ) d d, d= 7)

4 Upper Bound on the BER Performance of MTCM-STBC Scheme 24 where f ) = Ω f / in 2 2σ 2 2σ 2 +Ω f / in 2 exp ) 2m µ 8) and k )!! =.3 k [,pagexlv].thecoefficient g k,d in 7) can be computed by the recurive equation given in the appendix. It i worth noting that even conidering only the firt term in the infinite ummation in 7) giveavery good approximation for practical value of hadowing. Setting k = 2 and noting that g 2, = andg 2,2 =, we have P X, ˆX ) π/2 = π =0 F { f = +Ω f / in 2 exp f ) ) [ 2σ 2 µ f )+2σµ 2 f ) ) 2 ]} MN. 9) In our numerical reult, taking more term i.e., k>2) did not reult in a viible change in the plot. It i alo intereting to point out how 7) relate to the unhadowed cae. Auming there i no hadowing, µ i no longer a log-normal random variable, but jut given a a contant equal to it mean µ = exp2m µ ). Furthermore, inerting σ 2 µ = 0in7) and uing the relationhip σ2 = 0.5/ + K) and µ = K/ + K) in term of the well-known Rician parameter K,weobtain P X, ˆX ) = π π/2 F =0 f = +K +K + ) E /4N 0 β/ in 2 ) P x f p= p ˆx p f K ) E /4N 0 β/ in 2 ) P x f p= p ˆx f p 2 MN exp +K + ) E /4N 0 β/ in 2 ) P x f p= p ˆx p f 2 d, 20) which wa previouly preented in [5]. It i alo intereting to note that imply by etting = π/2in7) and20), the claical Chernoff bound would be obtained for hadowed and unhadowed Rician channel, repectively. For ufficiently large ignal-to-noie ratio i.e., E /N 0 ), evaluating the integrand in 7) at = π/2, we obtain a Chernoff-type bound a P X, ˆX ) E 4N 0 ) Ψ NM Ψ β P x M P ˆx f f P 2) NM [q )] Ψ NM, σ, σµ, m µ f = P= 2) 2 where q ) σ, σ µ, m µ = 2σ 2 exp + k=2 k:even 2σ 2 exp ) ) 2m µ k )!! k! 2σµ ) k k d= g k,d 2σ 2 exp ) ) d 2m µ. 22) Here, Ψ i the et of inner frame with a length of L ymbol) at nonzero Euclidean ditance ummation and Ψ i the number of element in thi et. Thi can be compared to effective length EL) in TCM cheme [2], which i defined a the mallet number of ymbol at nonzero Euclidean ditance. Contrary to the ymbol-by-ymbol count in the definition of EL, frame-by-frame count i conidered here a a reult of the multidimenional tructure of STBC panning an interval of L ymbol. It hould alo be noted that ymbolby-ymbol interleaving i conidered for the ingle antenna cae while anl-ymbol interleaver i employed in our cae. In 2), the lope of the performance curve, which yield the diverity order, i determined by Ψ NM and it can be defined a generalized effective length GEL) for multiple antenna ytem in an analogy to the effective length for ingle antenna cae. The econd term in 2) contribute to the coding gain, which correpond to the horizontal hift in the performance curve. Recalling the definition of product ditance PD) for the ingle antenna cae which i given a the product of nonzero branch ditance along the error event), we now define the generalized product ditance GPD) Ψ f = β M P xp f ˆx f p 2) NM p= 23) which involve the product of nonzero branch ditance ummation, where the ummation i over P term baed on the STBC ued. The third term in 2) i completely characterized by channel parameter. Since maximization of diverity order i the primary deign criterion, the firt tep in good code deign i the maximization of Ψ, ince M and N are already fixed. Once diverity order i optimized, the third term become jut a contant. Thi make u conclude that the GEL and GPD are the appropriate performance criteria in the election of trelli code over hadowed Rician channel. Thi alo how that the trelli code deigned for optimum performance baed on claical effective code length and minimum product ditance) over fading channel for the ingle tranmit antenna cae are not necearily optimum for the multiple antenna cae. To derive the upper bound on bit error probability from the exact PEP, we follow the claical tranfer function approach. The upper bound i given in term of the tranfer

5 242 EURASIP Journal on Applied Signal Proceing A B C D 0, 0 0, 4 2, 2 A = 2, 6 4, 0 4, 4 6, 2 0, 2 0, 6 2, 0 B = 2, 4 4, 2 4, 6 6, 0, 3, 7 3, C = 3, 5 5, 3 5, 7 7,,, 5 3, 3 D = 3, 7 5, 5 5, 7, 7 6, 6 6, 4 7, 5 7, 3 a) 2 3 E F G H 0, 0, 5 2, 2 E = 3, 7 4, 4 5, 6, 6 0, 4, 2, 6 F = 3, 3 4, 0 5, 5 6, 2 0, 2, 7 2, 4 G = 3, 4, 6 5, 3 6, 0 2, 0 3, 5 4, 2 H = 5, 7 6, 4 7, 0, d 2 3 d4 2 d5 2 d 2 2 d 2 6 d 2 d d 2 = d2 7 = d2 2 = d2 6 = 2 d3 2 = d2 5 = 3.4 d4 2 = 4 7, 3 7, 7 7, 5, 3 6 b) c) Figure : a) Code A2, optimum for AWGN, b) Code F2, optimum for Rayleigh fading channel with one tranmit antenna, c) 8-PSK ignal contellation. function of the code TD, I)by[8, 2] π/2 P b π 0 n b I T D), I ) I= d, 24) where n b i the number of input bit per tranition and TD),I) i the modified tranfer function of the code, where D), i given in our cae, by D) = + Ω ) MN f in 2 exp MN f ) ) + k=2 k:even k )!! k! baed on the derived PEP in 7). ) k k 2σµ g k,d f ) ) d d= MN 25) 4. EXAMPLES In thi ection, we conider two different TCM cheme a outer code whoe trelli diagram are illutrated in Figure. Thee are 2-tate 8-PSK-MTCM code with 2 ymbol per branch, which are optimized for bet performance over AWGN and Rayleigh fading channel, repectively [2]. For convenience, we ummarize the important parameter of thee code from [2]. The free ditance of the code A2 i dfree 2 = It minimum EL i determined by the error event path of { 0, 4 },whichdiffer by one ymbol from the correct path the all-zero path i aumed to be the correct path baed on the uniform propertie of the code) achieving Table : Parameter for variou degree of hadowing. Parameter Light Average Heavy σ m µ σ µ EL =. The correponding PD i d4 2 = 4. On the other hand, the code F2 ha a free ditance of dfree 2 = and it achieve EL = 2 with a product ditance of d 2 d5 2 = 2, which i determined by the error event path of {, 5 }. Since EL i the primary factor affecting performance PD a a econdary factor) over fading channel, F2 i expected to have better performance than A2. A an example of the hadowed Rician model, we conider the Canadian mobile atellite channel [6]. Table how the value of hadowing parameter for thi channel, which are determined by empirical fit to meaured data within Canada. In thi table, the term light, average, and heavy are ued to repreent an increaing effect of the hadowing. The upper bound for both code with the ingle tranmit antennaareillutratedinfigure 2.NoSTBCiconideredin thi cae. A expected for the ingle tranmit antenna cae, F2 perform better than A2, where the performance i determined by the choice of EL and PD. Thi obervation hold for all conidered degree of hadowing. In Figure 3, upper bound for the concatenated cheme are illutrated. Here we ue the STBC deigned for 2-TX antenna i.e., Alamouti code). Baed on thi code, we have P = L = M = 2andβ =. Our reult demontrate that

6 Upper Bound on the BER Performance of MTCM-STBC Scheme BER BER E b /N 0 db) E b /N 0 db) A2 heavy A2 average A2 light F2 heavy F2 average F2 light A2 heavy A2 average A2 light F2 heavy F2 average F2 light Figure 2: Upper bound for code A2 and F2 with ingle tranmit antenna over hadowed Rician channel -TX and -RX antenna). Figure 3: Upper bound for concatenated MTCM-STBC cheme with code A2 and F2 a outer code over hadowed Rician channel 2-TX and -RX antenna). the concatenated cheme uing A2 and F2 a outer trelli code achieve roughly the ame performance. Thi i a reult of the fact that the dominant factor for the ingle antenna cae no longer determine performance. In the 2-TX antenna cae, both cheme achieve GEL equal to 2 and GPD equal to 4, that i, [d 2 + d 2 5)/2] 2 = 4forF2and[d d 2 4)/2] 2 = 4 for A2, baed on 23).SincebothofthemhaveequalGEL and GPD, their performance turn out to be almot identical. Thi obervation hold to be true independent of conidered degree of hadowing. Comparion between the one- and two-tranmitantenna cae alo reveal intereting point on the performance. In both figure, code F2 give a diverity order of 2 i.e., lope of the curve), regardle of antenna number. Only an additional coding gain i.e., horizontal hift in the curve) i oberved with the ue of two antenna. However, thi reult i omewhat a coincidence becaue of the particular choice of the parameter characterizing thi pecific example. For the ingle tranmit antenna cae, the code F2 ha EL = 2 and the performance curve varie with E b /N 0 ) 2. On the other hand, for the 2-TX antenna cae we have Ψ =, ince an L = 2- ymbol interleaver i ued. However, the overall diverity i determined by GEL i.e., Ψ NM = 2 = 2), reulting again in the ame lope a in the ingle tranmit antenna cae. To examine the tightne of upper bound, we alo evaluate the performance of code A2 and F2 through computer imulation, auming 2-TX antenna. Simulation reult for the code F2 are illutrated in Figure 4 with the correponding BER E b /N 0 db) Heavy Average Light Figure 4: Upper bound veru imulation reult for code F2 olid: upper bound, dahed: imulation).

7 244 EURASIP Journal on Applied Signal Proceing upper bound plotted a olid line) computed by 24) and 25). The upper bound are in very good agreement with imulation reult, demontrating the tightne of the new upper bound baed on the exact PEP. A expected baed on our previou dicuion on upper bound expreion), code A2 yield nearly identical imulation reult to thoe of code F2, which we do not include here for brevity. 5. CONCLUSION We analyzed the performance of trelli-coded STBC cheme over hadowed Rician fading channel. Our analyi i baed on the derivation of an exact PEP through the moment generating function approach. The derived expreion provide inight into the election criteria for trelli code which hould be ued in conjunction with STBC over fading channel. Our reult alo how that the trelli code deigned for optimum performance over Rician channel with ingle tranmit antenna are not necearily optimum for the multiple tranmit antenna cae. Uing tranfer function technique baed on the new PEP, we preent upper bound on the bit error probability for the concatenated cheme. We alo provide imulation reult, which eem to be in good agreement with the derived upper bound. APPENDIX Thi appendix evaluate the inner integral in 6)intermof an infinite um. Defining a = Ω f / in 2 2σ 2 +Ω f / in 2, b = 2 2σ µ, c = 2m µ, A.) we can write the inner integral in 6)a exp u 2) f u)du A.2) with f u) = exp a expbu + c)). Expanding f u) in Taylor erie, we obtain exp u 2) f f u)du k 0) = u k exp u 2) du, k! k=0 A.3) where f k 0) are the Taylor erie coefficient and, in our cae, they can be determined a f k 0) = exp a expc) ) k b k ) d, g k,d a expc) d= where g k,d can be computed by the recurive equation A.4) g k,d = dg k,d g k,d with g k, = fork =, 2,..., g k,d = 0 ford>k. A.5) Uing the integral form given by [, page 382, equation ], it can eaily be hown that the integral in A.3) i zero for the odd value of k. For even value of k,wecanue the reult [, page 382, equation ] and expre A.3) a exp u 2) f u)du = π k=0 f k 0) k )!! k! 2 k/2. A.6) Replacing A.2) bya.6) witha, b, andc value given a in A.), one can obtain the final form for the inner integral of 6) leading to 7). ACKNOWLEDGMENT Thi paper wa preented in part at IEEE Vehicular Technology Conference VTC-Fall 02), Vancouver, Canada, October REFERENCES [] V. Tarokh, N. Sehadri, and A. R. Calderbank, Space-time code for high data rate wirele communication: performance criterion and code contruction, IEEE Tranaction on Information Theory, vol. 44, no. 2, pp , 998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block code from orthogonal deign, IEEE Tranaction on Information Theory, vol. 45, no. 5, pp , 999. [3] S. M. Alamouti, A imple tranmit diverity technique for wirele communication, IEEE Journal on Selected Area in Communication, vol. 6, no. 8, pp , 998. [4] S.M.Alamouti,V.Tarokh,andP.Poon, Trelli-codedmodulation and tranmit diverity: deign criteria and performance evaluation, in Proc. IEEE 998 International Conference on Univeral Peronal Communication, vol., pp , Florence, Italy, October 998. [5] M. Uyal and C. N. Georghiade, Analyi of concatenated trelli coded STBC cheme over Rician fading channel, in Proc. 39th Annual Allerton Conference on Communication, Control and Computing, Monticello, Ill, USA, October 200. [6] C. Loo, A tatitical model for a land mobile atellite link, IEEE Tran. Vehicular Technology, vol. 34, no. 3, pp , 985. [7] M. K. Simon and M.-S. Alouini, A unified approach to the performance analyi of digital communication over generalized fading channel, Proceeding of the IEEE, vol.86,no.9, pp , 998. [8] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channel: A Unified Approach to Performance Analyi, John Wiley & Son, New York, NY, USA, [9] G. L. Turin, The characteritic function of Hermitian quadratic form in complex normal variable, Biometrika, vol. 47, no. /2, pp , 960. [0] A. M. Mathai and S. B. Provot, Quadratic Form in Random Variable: Theory and Application, Marcel Dekker, New York, NY, USA, 992. [] I. S. Gradhteyn and I. M. Rzyhik, Table of Integral, Serie and Product, Academic Pre, San Diego, Calif, USA, 5th edition, 994. [2] E. Biglieri, D. Divalar, P. J. McLane, and M. K. Simon, Introduction to Trelli-Coded Modulation with Application, Macmillan Publihing, New York, NY, USA, 99.

8 Upper Bound on the BER Performance of MTCM-STBC Scheme 245 M. Uyal waborninitanbul,turkey,in 973. He received the B.S. and the M.S. degree in electronic and communication engineering from Itanbul Technical Univerity, Itanbul, Turkey, in 995 and 998, repectively, and the Ph.D. degree in electrical engineering from Texa A&M Univerity, Texa, in 200. From 995 to 998, he worked a a Reearch and Teaching Aitant in the Communication Theory Group at Itanbul Technical Univerity. From 998 to 2002, he wa affiliated to the Wirele Communication Laboratory, Texa A&M Univerity. During the fall of 2000, he worked a a Reearch Intern at AT&T Lab-Reearch, New Jerey. In April 2002, he joined the Department of Electrical and Computer Engineering, Univerity of Waterloo, Canada, a an Aitant Profeor. Hi reearch interet lie in communication theory with pecial emphai on wirele application. Specific area include pace-time coding, diverity technique, coding for fading channel, and performance analyi over fading channel. Dr. Uyal currently erve a an Editor for IEEE Tranaction on Wirele Communication and a the Guet Coeditor for Special Iue on MIMO Communication of Wiley Journal on Wirele Communication and Mobile Computing. C. N. Georghiade received hi doctorate in electrical engineering from Wahington Univerity in May 985. Since September 985 he ha been with the Electrical Engineering Department at Texa A&M Univerity where he i a Profeor and holder of the Delbert A. Whitaker Endowed Chair. Hi general interet are in the application of information, communication, and etimation theorie to the tudy of communication ytem, with particular interet in wirele and optical ytem. Dr. Georghiade ha erved over the year in everal editorial poition with the IEEE Information Theory and Communication Societie and ha been involved in organizing a number of conference. He currently erve a Chair of the Fellow Evaluation Committee of the IEEE Information Theory Society and in the Award Committee of the IEEE Communication Society. He alo erve a General Cochair for the IEEE Information Theory Workhop in San Antonio, Texa, in October Dr. Georghiade wa the recipient of the 995 Texa A&M Univerity College of Engineering Halliburton Profeorhip and the 2002 E.D. Brockett Profeorhip. From 997 to 2002 he held the J. W. Runyon Jr. Endowed Profeorhip and in 2002 he became the inaugural recipient of the Delbert A. Whitaker Endowed Chair.