Performance of Concatenated Channel Codes

Transcription

1 1 Performance of Concatenated Channel Codes and Orthogonal Space-Tme Block Codes Harsh Shah, Ahmadreza Hedayat, and Ara Nosratna Department of Electrcal Engneerng, Unversty of Texas at Dallas, Rchardson, TX Emal:

2 2 Abstract In ths paper we analyze the performance of an mportant class of MIMO systems, that of orthogonal space-tme block codes concatenated wth channel codng. Ths system confguraton s attractve because of ts smple and well-understood desgn crtera. We study spatally ndependent fadng as well as correlated fadng that may arse from the proxmty of transmt or receve antennas. We consder the effects of tme correlaton and present a general analyss for the case where both spatal and temporal correlatons exst n the system. We present analyses for a varety of channel codes, ncludng convolutonal codes, turbo codes, TCM, and MTCM, under quas-statc and fast Raylegh fadng as well as Rcan fadng. Smulatons verfy the valdty of our analyss. I. INTRODUCTION Space-tme block codes (STBC) provde dversty but have lttle or no codng gan, therefore n practce t s often necessary to use them n conjuncton wth channel codng (see Fgure 1). Unlke space-tme trells codes [1], where codng gan and dversty are provded by the same code, the concatenated approach dvdes the overall problem such that two dstnct codes are used: the STBC provdes dversty, whle the channel code provdes codng gan. The overall decoder conssts of two separable decodng operatons, where the space-tme component has a smple, lnear form, leadng to a less complex soluton. Ths structure has been accepted, e.g., n the WCDMA standard [2]. In ths paper, we analyze the performance of concatenated channel codes and orthogonal space-tme block codes. A summary of past work n ths area s as follows. Borran et al. [3] and Gong and Ben Letaef [4] dscuss desgn ssues of concatenated channel codes and orthogonal space-tme block codes. Bauch and Hagenauer [5] gve analytcal evaluaton of error probablty wthout consderng the effect of block fadng (whch s typcally assumed for STBC decodng). Uysal and Georghades [6] gve error bounds for MTCM-STBC under Rcan fadng, but nterleavng does not appear n ther error analyss. Schulze [7] gves unon bounds for channel codes and Alamout sgnalng for..d and temporally correlated channels, wthout consderng block fadng. None of the above mentoned works dscusses spatally correlated fadng. La and Mandyam [8] smulate concatenated convolutonal/turbo codes wth two temporally and spatally correlated transmt antennas (one receve antenna) n the framework of WCDMA, but they do not present bounds or analytcal results.

3 3 Channel Encoder Interleaver Space-Tme Block Encoder Space-Tme Block Decoder Denterleaver Channel Decoder Fg. 1. Concatenated channel code and space-tme block code Our ntent s to present an analyss that s general and comprehensve. We extend exstng results as well as present new ones. In partcular, (1) we nclude the effects of block fadng n our analyss, (2) the analyss s general wth respect to the number of transmt and receve antennas, (3) we derve error bounds for the case of..d. fadng as well as spatally correlated or temporally correlated channels, (4) we present results for a varety of channel codes ncludng convolutonal codes, turbo codes, TCM and MTCM, and (5) we analyze n the presence of fast fadng and quas-statc fadng n Raylegh and Rcan channels. Our methodology s to construct a SISO equvalent representaton of the channel, resultng n a block fadng channel. We employ the moment generatng functon approach of [9] to derve bounds on the bt- and frame-error probabltes. We use a random (unform) nterleaver to overcome the problem of nterleaver-dependent error probabltes [10], [11]. The organzaton of the paper s as follows. We brefly explan the orthogonal space-tme block codes and the resultant equvalent channel n Secton II. In Secton III, we develop the tools for our boundng technque and analyze the concatenated system under..d. fadng, spatally correlated fadng, and spato-temporally correlated fadng. Secton IV gves analytcal bounds for non-bnary modulatons (TCM and MTCM). In Secton V we extend the analytcal bounds for the case of Rcan channel. In Secton VI we provde smulatons to gauge the accuracy of the bounds and to demonstrate the effect of correlatons. II. SYSTEM MODEL We consder a codng-dversty scheme where a channel code and a STBC are used as shown n Fgure 1. The channel code can be a sngle code or a concatenated code. The channel encoder converts a sequence of k nformaton bts to n coded bts. Each coded bt s modulated by a sgnal wth unt energy. Ths s further encoded by the space tme block encoder wth n T transmt antennas. The recever employs n R receve antennas and combnes ther output optmally. We

4 4 consder a frequency non-selectve fadng channel, where the output of the channel s gven by y = H s + n, where y s n R 1 receved sgnal vector, s s the modulated n T 1 vector transmtted over n T transmt antennas, and n s n R 1..d. Gaussan nose at the receve antennas. The channel matrx s represented by H whose elements h j are the complex Gaussan channel coeffcents for the par of transmt antenna and receve antenna j. A space-tme block code s desgned such that the components of each codeword, despte mutual nterference over the wreless channel, can be separated at the decoder [12]. For example, n the case of two-transmt and one-receve antenna orgnally ntroduced n [13], the unmodulated sequence s arranged n blocks of two symbols, (c 0, c 1 ), and modulated as follows s(c 0) s(c 1 ), s (c 1 ) s (c 0 ) where the columns correspond to transmt antennas and the rows correspond to tme. Assumng that channel coeffcents h 11 and h 21, known to the recever, are constant over the transmsson of one block, t s possble to recover the transmtted symbols wthout nterference at the recever va lnear decodng and obtan a dversty order of two: s(c ) = ( h h 12 2 )s(c ) + n [13]. In a STBC wth n T transmt antennas, t s assumed that the channel coeffcents h j reman fxed through n T consecutve ntervals [13], [12]. Hence, the channel s block fadng wth block length n T. The outer code n our concatenated system also sees a block fadng channel, although the statstcs of the channel have been altered by the mappng nduced by the STBC encoder and decoder. We now proceed to dscuss ths equvalent channel n more detal. The multple-nput multple output (MIMO) channel, drven by an orthogonal STBC, can be represented by an equvalent sngle-nput sngle-output (SISO) channel. Assumng the recever combnes the receved sgnals from n R antennas optmally, the MIMO channel can be represented as an equvalent SISO block fadng channel wth fadng coeffcent: h eq = 1 n T n R h j n 2. (1) T Alternatvely, we can wrte the equvalent SNR =1 j=1 γ = γ H 2, (2)

5 5 No Interleaver Best Interleaver Unform Interleaver Parwse Error Probablty E /N (db) b 0 Fg. 2. Domnant event (d=5) of a convolutonal code, Alamout sgnalng wth 1-Rx antenna, n block..d. Raylegh fadng where denotes the Frobenus norm, γ = 1 R ce b n T N 0 s the average SNR per nformaton bt per transmt antenna, and R c s the code rate. If the nose components of the actual channel are ndependent, so are the nose components of the equvalent channel [5], [3]. The transmtted power s scaled by the number of transmt antennas to keep the total transmtted power constant. The equvalent fadng coeffcent follows a generalzed Raylegh dstrbuton [14]. The resultant nstantaneous SNR per bt, γ, follows ch-square dstrbuton wth degree of freedom 2n T n R [5]. The problem s now reduced to the analyss of a block fadng SISO channel whch s no longer Raylegh, but rather follows a generalzed Raylegh dstrbuton. Spatally correlated and temporally correlated channels, whch we also consder n ths paper, further modfy the probablty dstrbuton. Here t s approprate to make a note on nterleavng. Some coded space-tme transmsson systems, e.g. [4], have been proposed that do not nclude nterleavng between the outer and nner codes. However, our smulatons show that the codes of [4] can be mproved by over 2.5dB at FER= wth an nterleaver (see Secton VI, Fgure 8). In vew of these gans and the relatvely low cost of nterleavng, t s mportant to nclude nterleavng n the analyss of coded space-tme systems. Interleavng, however, necesstates a cumbersome book-keepng for calculatng parwse error probabltes. To manage ths complexty and to avod nterleaver-dependent probabltes, we use the concept of a unform random nterleaver. To demonstrate the effcacy of ths approach,

6 6 Fgure 2 shows the parwse error probablty (PEP) of the domnant error event of a convolutonal code concatenated wth Alamout sgnalng (Hammng dstance 5). The (averaged) unform nterleaver gves a good approxmaton to the best nterleaver n realstc sgnal-to-nose ratos. 1 The usage of random unform nterleavng was frst proposed by Benedetto and Montors [10] for the analyss of turbo codes and has also been used by Zummo and Stark [11] to explore the effects of channel nterleavers. III. PERFORMANCE ANALYSIS The unon bound for the bt error rate of a lnear block code C(n, k) s P b k n w d and the unon bound for the frame error rate s P e k w d w n A w,dp (d) (3) n A w,d P (d), where A w,d s the multplcty of the codewords wth nput weght w and output weght d. The coeffcents A w,d form the nput-output weght enumeratng functon (IOWEF) of C. The parwse error probablty (PEP), P (d), s defned as the probablty of an error event wth Hammng weght d. To calculate bt- and block-error probabltes for ndvdual channels, we need to obtan correspondng PEP s P (d), to whch the remander of ths paper s dedcated. The performance of channel codes n block fadng envronments s studed n [15], [16], [11]. The orgnal analyss n [16] requres a generalzed weght enumeratng functon of the channel code (or generalzed transfer functon for convolutonal codes), whch depends on the order of transmtted bts of a codeword and s therefore affected by nterleavng. To address ths ssue we apply the concept of random (unform) nterleavng, n a manner closely followng Zummo and Stark [11]. The length of the coded sequence (frame length) s denoted by n. The length of a fadng block s l, thus the number of fadng blocks n each coded frame s F = n/l. We now need to determne how the error bts are dstrbuted among dfferent blocks,.e., how much error weght s present n each fadng block. To characterze that, we buld a hstogram of weghts 1 At very large SNR bad nterleavers wll domnate, hence averagng over all nterleavers wll eventually dverge from the best nterleaver at arbtrarly large SNR. Nevertheless, t s an excellent approxmaton at moderate SNR.

7 7 as follows: assume the number of blocks that have weght m s f m, and consder the vector f = (f 0,..., f w ) where w = mn(l, d). A gven vector f s a vald hstogram f f m = F and mfm = d. Now, usng the unform nterleavng concept, one may average the PEP over all vald error patterns (hstograms) [11] P (d) = E f [P (d f)] = F F/2 f 1 =1 f 2 =1... F/w f w=1 P (d f)p(f), where p(f) s the weght of occurrence of the pattern f, and E s the expectaton operator. A. PEP based on Moment Generatng Functons Assumng the all-zero codeword s transmtted, the PEP of a codeword wth weght d gven the pattern f of the fadng blocks, s w P (d f, γ) = Q 2 m m=1 f m γ m, =1. (4) Here we have collected terms correspondng to blocks wth equal weght patterns. Thus γ m, s the SNR for the -th block that has weght m (there are a total of f m blocks wth weght m). 2 Representng Q-functon n ts alternatve form [9], the PEP condtoned on the block fadng pattern f s P (d f, γ) = 1 π π 2 0 exp ( 1 sn 2 θ w m m=1 f m =1 γ m, ) dθ. Averagng the above condtonal PEP over the nstantaneous SNR γ, and assumng γ m, are ndependent, P (d f) = E γ [P (d f, γ)] = 1 π π 2 0 w m=1 =1 f m 0 ( exp mγ m, sn 2 θ ) p γ (γ m, ) dγ m, dθ. The nner ntegral s the moment generatng functon (MGF) of γ, Φ(s) = E[e sγ ], evaluated at s = m/ sn 2 θ. Snce γ m, follow the same statstcs we can wrte P (d f) = 1 π π 2 0 w [ ( Φ m )] fm dθ. (5) sn 2 θ m=1 2 Note that the subscrpts of h j refer to transmtter and recever j, whereas the subscrpts of γ m, are temporal subscrpts wth reference to the block fadng patterns. Each case should be clear from the context.

8 8 B. Independent Fadng If the entres of the channel matrx H are ndependent, the resultng SNR s the sum of n T n R ndependent exponental varables and hence has a ch-square dstrbuton wth the pdf [9] p γ (γ) = 1 (D 1)! γ D γd 1 exp( γ/ γ), where D = n T n R. The MGF of ths pdf s gven by [9] Usng ths MGF n (5) we obtan the followng bound for P (d f) P (d f) = 1 π Φ γ (s) = (1 s γ) D. (6) 1 2 π 2 0 w m=1 ( 1 + m γ ) fmd dθ (7) sn 2 θ w (1 + m γ) fmd, m=1 where the last nequalty s the Chernoff bound. One may also obtan the correspondng result for quas-statc Raylegh fadng by the settng F = 1 whch s equvalent to m = d, f m = 1. C. Spatally Correlated Fadng If the antenna elements n ether the transmtter or the recever are not far enough apart, the fadng coeffcents h j wll be correlated. We do not assume any specfc structure for the correlaton of fadng coeffcents, and the analyss s general n that sense. We do assume that the correlaton structure s statonary (tme-nvarant). We frst present a useful model for the correlated fadng coeffcents. Ths result s not new, but we state t nevertheless to set the stage for the developments to come. Lemma 1: Consder a random matrx H wth..d. entres, and two determnstc, postve sem defnte matrces A and B of approprate sze, and assume H = A 1/2 H B 1/2. Then H can be equvalently represented thus: vec(h) = R 1/2 vec( H), (8) where vec( ) s the vectorzng operator and R = E[vec(H)vec(H) H ] = B A, the Kronecker product of B and A.

9 9 Proof: Easly follows va the denttes: (A B) T = A T B T, vec(axb) = (B T A)vec(X), (A B)(C D) = AC BD. Ths s closely related to an exstng correlaton model [17] where A and B represent the correlaton matrces at the transmtter and recever sde, respectvely. In other words, H = R 1/2 Tx H R 1/2 Rx. However, for our purposes, the followng untary transformaton s more useful vec(h) = R 1/2 s vec( H) = UΛ 1/2 vec( H), where R s = UΛU H. For future reference, t s useful to remember Λ = Λ T x Λ Rx. Now to the key decorrelatng result: Lemma 2: The moment generatng functon of γ s gven by Φ γ (s) = n T n R =1 j=1 ( 1 sλ (t) where λ (t) and are egenvalues of R Tx and R Rx respectvely. Proof: H 2 = vec(h) H vec(h) = vec( H) H Λvec( H) = n T =1 n R j=1 j γ ) 1, (9) λ (t) j h j 2. (10) From (2) and (10), γ = γ H 2 = γ The MGF of γ s Φ γ (s) = E { exp ( s γ H 2)} = n T =1 n R j=1 n T n R =1 j=1 E λ (t) j h j 2. { ( exp s γ λ (t) j h j 2 )}.

10 10 Each term n the last expresson s the moment generatng functon of an exponental random varable. Substtuton gves (9). We can now substtute n (5) to obtan P (d f) = 1 π 1 2 π 2 0 w w n T n R m=1 =1 j=1 n T n R ( m=1 =1 j=1 ( 1 + mλ(r) 1 + mλ (t) j sn 2 θ γ ) fm dθ (11) j γ ) fm. (12) Usng ths formula, t s nstructve to consder two extreme cases: uncorrelated and fully correlated channels. In the case of uncorrelated channel, λ (t) = j = 1 for all, j, and the formula reduces to (7), as expected. In the case of fully correlated channel, the correlaton matrx s rank defcent and we have λ (t) 1 = n T, 1 = n R, and all other λ (t) = j = 0. Thus the above moment generatng functon reduces to Φ γ (s) = (1 sd γ) 1, (13) whch shows no dversty, but a receve gan of D = n T n R (recall that γ = 1 n T R ce b N 0 ). D. Temporal and Spatal Correlaton For varous reasons such as long data blocks or long fadng perods, t may not be practcal to use nterleavers to remove the channel memory. In such cases, we need to analyze the system wth channel memory, a task whch we undertake n ths secton. We assume that the coherence tme s much greater than n T symbols, so that the channel remans effectvely constant over each STBC block and lnear decodng s possble. Assumng a gven error event has weght d, we must concentrate on the channel matrx at tme nstances {k 1,..., k d } where the error event has nonzero value. Let the channel matrx at tme k be denoted as H and defne H = [vec(h 1 ) vec(h 2 )... vec(h d )]. Each H may be spatally correlated; the spatal correlatons are modeled by a matrx R s as before. We assume the statstcs to be statonary (tme-nvarant), therefore only one spatal correlaton matrx suffces. We model the temporal correlaton of the channel by R t, that s, R t (, j) = E[vec(H ) H vec(h j )]. Therefore, H can be modeled as H = R 1/2 s H R 1/2 t, (14)

11 11 where, H s a (nt n R ) d matrx wth..d. elements. Usng Lemmas 1 and 2, we can wrte the MGF of d n T n R correlated exponental varables as Φ γ (s) = d n T n R k=1 =1 j=1 (1 s µ k λ (t) j ) 1, (15) where µ are egenvalues of R t. Usng ths, we can calculate the parwse error probablty P (d) = 1 ( π/2 d n T n R 1 + γµ ) kλ (t) 1 j π sn 2 dθ (16) θ d k=1 =1 j=1 k=1 =1 j=1 n T n R ( 1 + µ k λ (t) j γ ) 1. (17) It s easy to see that for the specal case of quas-statc fadng, (R t ) j = 1 for all and j, therefore µ 1 = d and all other µ k = 0, and the equaton reduces to the famlar PEP for the quas-statc fadng, where there s no tme dversty but there s a codng gan of d. IV. PERFORMANCE ANALYSIS WITH MULTILEVEL MODULATION We now proceed to the analyss of a concatenaton of TCM or MTCM wth space-tme block codes. The desgn of TCM and MTCM for space-tme block codes has been addressed n [4] and [3]. Followng the same steps as before, we need to consder error patterns f (hstograms) n a manner smlar to Secton III. Because the errors can assume multple values (more than two), the constructon of the patterns, albet straght forward, s cumbersome. In the nterest of brevty we omt the detals. In a 2 m -ary modulaton, the parwse condtonal probablty of error between the all-zero codeword and a codeword e s gven by f j P (0 e f, γ) = Q 2 γ j, α j, (18) j =1 where j s the ndex of block patterns and γ j, s the nstantaneous SNR per bt for -th block n fadng pattern j. We have defned an aggregate dstance metrc α j for each block pattern j, calculated by α j = 2 m k v j,k d 2 (s k, s 0 ),

12 12 where v j,k s the multplcty of symbol s k n the block pattern ndexed by j. Averagng over γ, we fnd the PEP expresson P (0 e f) = 1 π π/2 0 ( [Φ γ α )] fj j dθ. (19) 2 sn 2 θ j For the useful class of unform error probablty (UEP) codes, where the reference codeword can always be chosen as the all-zero codeword [9], [18], the unon bound on frame error probablty s P e e 0 π 1 2 π 0 2 (m 1) cl j ) 2 m =1 Φ γ ( v j,d 2 fj (c l s, s ) 2 sn 2 dθ, (20) θ where c l s a symbol that belongs to the frst level of set-parttonng of the 2 m -ary modulaton [18]. Note that v j, and f j depend on the error word e, but the dependence has been suppressed n the formula above for notatonal smplcty. To calculate the unon bound n the case of spatally and temporally..d. fadng, the moment generatng functon (6) s substtuted n (20). To calculate the unon bound n the case of spatally correlated fadng, we nsert the moment generatng functon (9) nto (19). The unon bound n the case of temporally correlated channel requres a lttle twst. In the prevous cases, the equvalent SNR was a functon of H only, therefore decorrelatng H smplfed the MGF expressons. However, n the case of temporal correlatons, the effectve SNR s expressed as γ = d k=1 n T =1 n R j=1 h j (k) 2 δ 2 k, (21) where δ k s the Eucldean dstance of the error event at k-th error poston. Obvously decorrelatng H no longer works. Defne D = dag(δ 1,..., δ d ) and note that γ = HD 2, where H = [vec(h 1 ) vec(h 2 )... vec(h d )]. To obtan a sum of ndependent SNR components, we must dagonalze the autocorrelaton of HD. γ = HD 2 = = d k=1 d k=1 n T =1 n T =1 n R j=1 n R j=1 h j (k) 2 δ 2 k h j (k) 2 λ (t) j ˆµ k.

13 13 Recall that the spatal and temporally correlated H s modeled as H = R 1/2 s H R 1/2 t, where H has..d. entres and R s and R t are the spatal and temporal correlaton matrces, respectvely. It follows that ˆµ k are the egenvalues of DR t D. Therefore we can stll use equatons (15) and (19) except we should substtute ˆµ k for µ k. 3 V. PERFORMANCE UNDER RICIAN FADING In ths secton we consder the Rcan fadng channels wth parameter K descrbng the rato of the energy of the lne-of-sght component to the multpath component. 4 For the uncorrelated Rcan channel, the moment generatng functon of γ = γ H 2 s gven by [9] ( ) D ( ) D 1 + K Ks γ Φ γ (s) = exp. 1 + K s γ 1 + K s γ By usng ths MGF wth equaton (19), the PEP for the fast fadng Rcan channel and multlevel modulaton s gven by P (0 e) = 1 π π/2 0 [ ( 1 + K 1 + K + α j γ exp 2 sn 2 θ j K γα j 2 sn 2 θ 1 + K + γα j 2 sn 2 θ )] D dθ. (22) For the case of spatally correlated fadng, the MGF can be once agan derved usng Lemmas 1 and 2 of Secton III-C, Φ γ (s) = n T n R =1 j=1 (1 + K) 1 + K sλ (t) exp j γ ( Ksλ (t) j γ 1 + K sλ (t) j γ ), (23) where λ (t) and are the egenvalues of transmt and receve correlaton matrces R Tx and R Rx respectvely. Expressons (23) and (20) drectly yeld the desred bounds on error probablty. In the case where temporal as well as spatal correlaton s present, t s straghtforward to show that the moment generatng functon s expressed as follows Φ γ (s) = d n T n R k=1 =1 j=1 (1 + K) 1 + K sµ k λ (t) exp j γ ( K s µk λ (t) 1 + K sµ k λ (t) j γ j γ where µ k are the egenvalues of temporal correlaton matrx R t. Once agan, n combnaton wth (20), the desred bounds are obtaned. 3 The dstncton s unnecessary n the case of bnary codes wth BPSK, because n that case δ k = 1 on all error postons. 4 As long as the sze of antenna arrays are much smaller than the dstance between transmtter and recever, physcal arguments lead to the concluson that K s constant across the entres of the channel gan matrx. ),

14 Bound 1 Tx Bound 2 Tx ρ t.0 Bound 2 Tx ρ t.7 Sm 1 Tx Sm 2 Tx ρ t.7 Sm 2 Tx ρ t.0 Bt Error Rate E b /N 0 (db) Fg. 3. Convolutonal code, block..d. Raylegh Fadng, 2-Tx and 1-Rx antennas VI. RESULTS We evaluated unon bounds for convolutonal and turbo codes wth BPSK modulaton, and for 4-state, 8-PSK TCM and MTCM codes. The STBC for two-transmt antenna used here s the Alamout scheme [13]. In all fgures, dashed lnes denote smulatons, whle sold lnes denote bounds. In the case of two antennas, R Tx and R Rx are each fully defned by a sngle correlaton coeffcent ρ t for transmt antenna and ρ r for receve antenna. To evaluate analytcal expressons, we calculated the IOWEF of the codes based on the approach of [19]. A random nterleaver s placed between the channel code and STBC. We evaluate exact expressons for parwse error probablty. We begn by presentng our results for convolutonal codes. For ths experment, we used a four-state rate 1/2 code wth generator functon G(D) = (1 + D 2, 1 + D 2 + D 3 ). Each frame contans k = 100 nformaton bts (200 BPSK symbols). Ths code s concatenated wth Alamout STBC (one receve antenna). The results are shown n Fgure 3. When there s no correlaton between antennas, the dversty s two. When the correlaton between transmt antennas s ρ t = 0.7, the dversty remans the same, but the loss n codng gan s about 1.5dB at BER= We next consder a system wth two transmt and two receve antennas wth spatal correlatons ρ t = 0.7 and ρ r = 0.5 (Fgure 4). We observe that receve dversty mproves the overall performance, but the loss due to antenna correlatons s more than 2dB at BER=10 5. In

15 15 Bt Error Rate Bound 2 Tx 2 Rx ρ t ρ r Bound 2 Tx 2 Rx ρ t.7 ρ r.5 Sm 2 Tx 2 Rx ρ t.7 ρ r.5 Sm 2 Tx 2 Rx ρ t ρ r Bound 1 Tx Sm 1 Tx E /N (db) b 0 Fg. 4. Convolutonal code, block..d. Raylegh Fadng, 2-Tx and 2-Rx antennas Bt Error Rate 10 0 Bound 2 Tx ρ t Bound 2 Tx ρ t.7 Sm 2 Tx ρ t Sm 2 Tx ρ t.7 Sm 1 Tx Bound 1 Tx E /N (db) b 0 Fg. 5. Convolutonal code, 2-Tx and 1-Rx antennas, tme correlated Raylegh Fadng, F d T s = 0.1 our fnal experment wth convolutonal codes (Fgure 5) we demonstrate the effect of temporal correlaton modeled va Bessel functons [20] wth f d T s = 0.1 (obvously wth no nterleavng). For turbo coded experments we use a rate-1/3 code wth four-state consttuent recursve convolutonal codes wth the generator functon G(D) = (1, 1+D 2 1+D+D 2 ). Fgure 6 shows the performance of turbo code concatenated wth Alamout sgnalng and one receve antenna. Each frame has 500 nformaton bts (1500 BPSK symbols). We use suboptmal teratve (MAP) decodng of turbo codes wth 12 teratons. The degradaton due to transmt antenna correlaton

16 Bound 1 Tx Bound 2 Tx ρ t Bound 2 Tx ρ t.7 Sm 1 Tx Sm 2 Tx ρ t Sm 2 Tx ρ t.7 Bt Error Rate E b /N 0 (db) Fg. 6. Turbo code, block..d. Raylegh Fadng 2-Tx and 1-Rx antennas Bound 1Tx Sm 1Tx Bound 2Tx ρ t.7 Sm 2Tx ρ t.7 Bound 2Tx ρ t sm 2Tx ρ t Bt Error Rate E /N (db) b 0 Fg. 7. TCM, 2-Tx and 1-Rx antennas, block..d. Raylegh fadng of ρ t = 0.7 s about 0.8 db at BER = We see that the unon bounds cross the smulaton curves. Ths phenomenon, whch has been prevously reported n the lterature [19], s due n part to the usage of teratve decodng nstead of ML decodng. Our TCM experments use a code from [4] whose trells s shown n Fgure 7. In ths experment, the system has two transmt and one receve antenna, and frame length s 130 symbols (260 nformaton bts). We have used partal nput-output weght enumeratng functon (IOWEF) to calculate the upper bounds; the results appear n Fgure 7. The performance loss due to an-

17 Bound, no nterlever Sm, no nterleaver [4] Bound, S random nterleaver Sm, S random nterleaver Frame Error Rate E /N (db) b 0 Fg. 8. TCM, 2-Tx and 1-Rx antennas, block..d. Raylegh fadng Sm 2 Tx ρ t Bound 2 Tx ρ t Sm 2Tx ρ t.7 Bound 2Tx ρ t.7 Bt Error Rate E /N (db) b 0 Fg. 9. TCM, 2-Tx and 1-Rx antennas, quas-statc Raylegh fadng tenna correlaton of ρ t = 0.7 s about 1.2 db at BER = To demonstrate the mportance of nterleavng, Fgure 8 gves the frame error rate n the case of two transmt and one receve antennas under..d. fadng (condtons smlar to [4]). We use a S-random nterleaver wth S=4. Interleavng gves a gan of 2.5 db at FER=, n addton to a hgher dversty that gves rse to even more mpressve gans at hgher SNR. In Fgure 9 we repeat the experment under a quas-statc Raylegh fadng channel. The unon bounds n the case of quas-statc channels are tght only f the dversty order s hgh, and the probablty of deep fades s low [5]. To get a rel-

18 Sm 1Tx Bound 1Tx Sm 2Tx ρ t Bound 2Tx ρ t Sm 2Tx ρ t.7 Bound 2Tx ρ t.7 Bt Error Rate E /N (db) b 0 Fg. 10. MTCM, 2-Tx and 1-Rx antennas, block..d. Raylegh Fadng Sm 2Tx ρ t Bound 2Tx ρ t Sm 2Tx ρ t.7 Bound 2Tx ρ t.7 Bt Error Rate E /N (db) b 0 Fg. 11. MTCM, 2-Tx and 1-Rx antennas, quas-statc Raylegh Fadng atvely tghter bound, we use the lmt-before-averagng method of [16], but as reported n [16] the bounds are stll not as tght as the fast fadng bounds. Our MTCM experments use a 4-state, 8-PSK code from [18], concatenated wth Alamout sgnalng, wth frame length of 100 symbols (200 nformaton bts). Agan, we use only the partal IOWEF to calculate upper bounds. Fgure 10 shows that a spatal correlaton of ρ t = 0.7 results n a 1dB loss at hgh SNR. We repeat ths experment under quas-statc fadng (Fgure 11). As mentoned before, the quas-statc bounds are not as tght as the fast fadng bounds.

19 Bound 1Tx Sm 1Tx Bound 2Tx ρ t.7 Sm 2Tx ρ t.7 Bound 2Tx ρ t Sm 2Tx ρ t Bt Error Rate E /N (db) b 0 Fg. 12. TCM, 2-Tx and 1-Rx antennas, block..d. Rcan fadng K = 5dB Fnally, we show results for the performance of space-tme coded TCM n Rcan fadng (Fgure 12). The Rcan fadng parameter s K = 5dB, and there are two transmt and one receve antennas. The loss due to a transmt antenna correlaton of ρ t = 0.7 s around 1dB. VII. CONCLUSION Ths work presents performance analyss for systems consstng of concatenaton of channel codes and space-tme block codes. Such systems are of theoretcal and practcal nterest. We use the concept of a unform nterleaver n the context of block fadng channel to calculate bt error probabltes. Ths analyss s performed both for the case of spatally uncorrelated fadng, as well as spatally correlated fadng due to proxmty of transmt or receve antennas. We also consder jont spato-temporal correlaton. We gve results for wde varety of codes and several type of fadng channels. Smulatons verfy the accuracy of the analyss. REFERENCES [1] V. Tarokh, N. Seshard, and A. Calderbank, Space-tme codes for hgh data rate wreless communcaton: Performance crtera and code constructon, IEEE Trans. Inform. Theory, vol. 44, no. 2, pp , March [2] 3rd Generaton Partnershp Project 3G TS , Multplexng and channel codng (FDD). [3] M. Borran, M. Memarzadeh, and B. Aazhang, Desgn of coded modulaton schemes for orthogonal transmt dversty, submtted for publcaton n IEEE transacton on Communcatons, May [4] Y. Gong and K. B. Letaef, Concatenated space-tme block codng wth trells coded modulaton n fadng channels, IEEE Transactons on Wreless Communcatons, vol. 1, no. 4, pp , Oct 2002.

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