Impact of Channel Estimation Errors on Space Time Trellis Codes

Transcription

1 Impac of Channel Esimaion Errors on Space Time Trellis Codes by Rekha Menon Thesis submied o he faculy of Virginia Polyechnic Insiue and Sae Universiy in parial fulfillmen of he requiremens for he degree for MASTER OF SCIENCE in Elecrical Engineering Dr. R. Michael Buehrer, Chair Dr. Charles W. Bosian Dr. Jeffrey H. Reed December 5, 003 Blacksburg, Virginia Keywords: Transmi Diversiy, Rayleigh Fading Channel, Channel Esimaion, Fla Fading, Frequency Selecive Fading Copyrigh 003, Rekha Menon

2 Impac of Channel Esimaion Errors on Space Time Trellis Codes Rekha Menon Absrac Space Time Trellis Coding (STTC) is a unique echnique ha combines he use of muliple ransmi anennas wih channel coding. This scheme provides capaciy benefis in fading channels, and helps in improving he daa rae and reliabiliy of wireless communicaion. STTC schemes have been primarily designed assuming perfec channel esimaes o be available a he receiver. However, in pracical wireless sysems, his is never he case. The noisy wireless channel precludes an exac characerizaion of channel coefficiens. Even near-perfec channel esimaes can necessiae huge overhead in erms of processing or specral efficiency. This pracical concern moivaes he sudy of he impac of channel esimaion errors on he design and performance of STTC. The design crieria for STTC are validaed in he absence of perfec channel esimaes a he receiver. Analyical resuls are presened ha model he performance of STTC sysems in he presence of channel esimaion errors. Training based channel esimaion schemes are he mos popular choice for STTC sysems. The amoun of raining however, increases wih he number of ransmi anennas used, he number of muli-pah componens in he channel and a decrease in he channel coherence ime. This dependence is shown o decrease he performance gain obained when increasing he number of ransmi anennas in STTC sysems, especially in channels wih a large Doppler spread (low channel coherence ime). In frequency selecive channels, he raining overhead associaed wih increasing he number of anennas can be so large ha no benefi is shown o be obained by using STTC. The amoun of performance degradaion due o channel esimaion errors is shown o be influenced by sysem parameers such as he specific STTC code employed and he number of ransmi and receive anennas in he sysem in addiion o he magniude of he esimaion error. Hence inappropriae choice of sysem parameers is shown o significanly aler he performance paern of STTC. The viabiliy of STTC in pracical wireless sysems is hus addressed and i is shown ha ha channel esimaion could offse benefis derived from his scheme.

3 Conens. Inroducion o STTC..... Diversiy in Wireless sysems..... Moivaion for Transmi Diversiy Trellis Coded Modulaion Space-Time Trellis Codes Organizaion of Thesis STTC Performance Analysis and Design Crieria..... Sysem Model..... Design Crieria for STTC Upper Bound on Pair-wise Error Probabiliy and TSC Crieria Generalized Design Crieria Trace Crierion Code Consrucion TSC Crieria Trace Crierion Performance Resuls TSC Crieria Trace Crierion Comparison of Trace and TSC Crieria Performance Sensiiviy of STTC o Coherence Time Analyical Performance Analysis of STTC Exac PWEP for STTC wih Perfec Channel Esimaes Analyical Performance Resuls Chaper Summary Channel Esimaion Techniques for Muliple Transmi Anenna Sysems Channel Esimaion using Training Sequences Training Model and Training Sequences Complexiy Reducion Techniques for Training Sequences Informaion Theoreic Resuls for he Amoun of Training Fla Fading Scenario Frequency Selecive Scenario Blind and Semi-Blind Techniques Ieraive Channel Esimaion HMM Based Blind Channel Esimaion Chaper Summary... 6 iii

4 4. Design and Performance of STTC in he Absence of Perfec Channel Esimaes Design Crieria in he Presence of CEE TSC-RD Crieria TSC-DP Crieria Trace Crierion Analyical Performance Analysis in he Presence of Channel Esimaion Errors Exac PWEP for STTC wih Imperfec Channel Esimaes Analyical Performance Resuls Effec of Channel Esimaion Errors on STTC Performance Opimal Amoun of Training Performance Degradaion due o CEE Performance Sensiiviy o Coherence Time Performance Sensiiviy o Varying Number of Anennas Performance Comparison wih Differenial-STBC Chaper Summary STTC in Frequency Selecive Channels Design Crieria in Frequency Selecive Channels TSC-RD Design Crieria Modified RD Design Crieria New Design Crieria Effec of Channel Esimaion Errors on STTC Performance Opimal Amoun of Training Performance Degradaion due o CEE Performance Sensiiviy o Coherence Time Chaper Summary Thesis Summary and Fuure Work STTC Performance Analysis and Design Crieria Channel Esimaion Techniques for Muliple Transmi Anenna Sysems Design and Performance of STTC in he Absence of Perfec Channel Esimaes 6.4. STTC in Frequency Selecive Channels Conclusions and Direcions for Fuure work... References... 3 Via iv

5 Lis of Figures - Pariioning of he 8-PSK consellaion A TCM encoder Sysem model for a MIMO sysem employing an STTC Trellis for a 4-sae STTC QPSK symbol consellaion Feed forward encoder srucure for 4-sae STTC Trellis for 8-sae STTC Feed forward encoder srucure for 8-Sae STTC STTC feed forward encoder srucure Performance comparison of 4 sae-sttc codes in quasi-saic channel Performance comparison of 4 and 8 sae STTC codes in quasi-saic channel Performance comparison of STTC codes in fas fading channel BER comparison of STTC codes in fas fading channel BER comparison of STTC codes in spaially correlaed channel Performance comparison of STTC codes in quasi-saic channel Performance comparison of STTC codes in fas fading channel Performance of Tx-Rx 8Sae-TSC over Doppler spreads Analyical and simulaed performance comparison of Tx-Rx TSC code over fas fading channels Analyical and simulaed performance comparison of Tx-Rx TSC code over quasi-saic fading channels Analyical and simulaed performance comparison of Tx-Rx TSC code over spaially correlaed fading channels Analyical and simulaed performance comparison of Tx-Rx CYV code over quasi-saic fading channels Channel esimaion error variance for differen raining lenghs and channel SNRs [Frag] Performance of 8-sae 8-PSK STC in a Tx-Rx sysem wih opimal and sub-opimal raining sequences of lengh= [Frag3] Encoder for he 8-sae 8-PSK STTC wih ransmi anennas [Hass] Opimal amoun of raining as a funcion of channel coherence ime (N = N r = 0) [Hass]Capaciy of raining based sysem as a funcion of coherence ime (N = N r = 0) [Hass]Capaciy as a funcion of number of ransmi anennas for N r = coherence ime of 00 symbols [Hass] Capaciy as a funcion of number of ransmi anennas for N r = and coherence ime of 0 symbols Capaciy as a funcion of number of ransmi anennas for N r = and channel coherence ime of 00 symbols in he presence of muli-pah Performance of Tx-Rx TSC STTC wih ieraive raining BER performance of Tx-Rx TSC STTC wih ieraive raining v

6 3- Comparison of he HMM/MAP based blind decoder performance wih ha of a decoder wih perfec channel informaion Performance of Tx-Rx TSC code over quasi-saic channel wih opimal raining Performance of Tx-Rx FVY code over fas fading channel wih opimal raining Performance of Tx-Rx CYV code over quasi-saic channel wih opimal raining Performance of Tx-Rx TSC code over quasi-saic channel wih 0dB CEE Performance of Tx-Rx TSC code over fas fading channel wih 0dB CEE Performance of Tx-Rx TSC code over quasi-saic spaially correlaed channel wih 0dB CEE Opimal raining lengh for Tx-Rx MIMO sysem over differen channel coherence imes and difference channel SNRs Performance comparison wih varying lengh of raining, for a Tx-Rx CYV Code a db channel SNR Performance of Tx-Rx CYV Code over quasi-saic channel wih varying raining lenghs Opimal raining lengh for 8Tx-Rx MIMO sysem over differen channel coherence imes and difference channel SNRs Performance comparison of CYV Code for differen number of ransmi anennas; in a sysem wih wo receive anennas and raining lengh of eigh Performance comparison of Tx-Rx CYV and TSC codes over quasi-saic channel wih raining lengh of eigh Performance comparison of Tx-Rx BBH code over quasi-saic channel wih raining lengh of eigh Performance comparison of Tx-Rx, CYV and TSC codes over varying channel coherence imes, 0dB channel SNR and opimal raining Performance comparison of Tx-Rx, CYV and TSC codes over quasi-saic channel and for differen raining lenghs Performance comparison of Tx-Rx, CYV and TSC codes over quasi-saic channel and for differen CEE SNRs Performance comparison of Tx and Tx Schemes over channel coherence ime, wih db channel SNR and perfec channel esimaes a he receiver Performance comparison of Tx-Rx CYV code over coherence ime in he presence of perfec channel esimaes, wih and wihou an inerleaver Performance comparison of differen ransmi schemes wih db channel SNR, receive anennas and opimal raining Performance comparison of differen ransmi schemes wih db channel SNR, receive anennas and opimal raining Performance of CYV code wih differen ransmi anennas, over a quasi-saic channel and CEE of 5dB Performance of CYV code wih differen receive anennas, over a quasi-saic channel and raining of lengh eigh [Taro4] Performance of Tx-Rx D-STBC scheme wih QPSK over quasi-saic channel vi

7 4-4 Performance comparison of D-STBC wih Tx-Rx CYV Code over a quasi-saic channel and differen raining lenghs Performance of TSC code in a wo-ap frequency selecive channel (Adaped from [You]) Performance comparison of 6-sae YFT code and 4-sae TSC-code in a -ap muli-pah channel Performance of he new scheme in a wo-ap frequency selecive channel FER performance comparison of Tx-Rx TSC code and he new scheme in mulipah channel BER performance comparison of Tx-Rx TSC code and he new scheme in mulipah channel Opimal raining lengh for Tx-Rx MIMO sysem over muli-pah channel wih - aps Performance comparison wih varying lengh of raining, for a Tx-Rx TSC Code a db channel SNR and over a wo-ap channel Opimal raining lengh for Tx-Rx MIMO sysem over muli-pah channel wih 3- aps Opimal raining lengh for 8Tx-Rx MIMO sysem over muli-pah channel wih 3- aps Performance of TX-Rx CYV code in a fla fading and -Tap muli-pah channel wih raining lengh of FEP Performance Comparison of Tx-Rx TSC code and he new scheme wih raining lengh eigh BER Performance Comparison of Tx-Rx TSC code and he new scheme wih raining lengh eigh Comparison of TX-Rx TSC and CYV code wih opimal raining and channel SNR of 6dB Comparison of TSC and CYV in -Tap channel wih SNR of 0dB and varying channel esimaion SNR Performance of Tx-Rx CYV over Tap channel for varying gain errors of he channel esimaes Performance of Tx-Rx CYV over Tap channel for varying phase errors of he channel esimaes Comparison of differen ransmi schemes in a -Tap channel wih SNR of 6dB and assuming perfec channel esimaes Comparison of differen ransmi schemes in a -Tap channel wih SNR of 6dB and opimal raining Comparison of differen ransmi schemes in a -Tap Channel wih SNR of 0dB and perfec channel esimaes Comparison of differen ransmi schemes in a -Tap channel wih SNR of 0dB and opimal raining... 8 vii

8 Lis of Tables - STTC codes for quasi-saic channel STTC codes for fas fading channels STTC codes for quasi-saic channel wih large diversiy order viii

9 . Inroducion o STTC Band-limied wireless channels provide an impedimen o high daa raes and consequenly consrain aemps o improve he reliabiliy of wireless communicaions by using error correcion codes along wih high daa rae services. One mehod of increasing he capaciy of wireless channels is he use of muliple ransmi and receive anennas [Fosc]. This addiional capaciy can be hough of as channel broadening which can be used o improve sysem performance in band-limied environmens. Space Time Trellis Coding (STTC) is a unique coding scheme ha inegraes channel coding and modulaion wih he use of muliple ransmi anennas and opional receiver diversiy. This scheme offers increased performance, and helps in improving he daa rae and he reliabiliy of communicaion over fading channels. However, mos of he research on he design and performance analysis of STTC assumes he availabiliy of perfec channel esimaes a he receiver. In pracical wireless sysems his is almos impossible o achieve. This concern moivaes he sudy of he impac of channel esimaion errors on he design and performance of STTC. This hesis invesigaes he validiy of design crieria for STTC, and aemps o characerize is performance in he presence of channel esimaion errors. I also ries o resolve he facors ha influence he way channel esimaion errors affec STTC performance. In his chaper, an inroducion o Space Time Trellis Codes (STTC) and a descripion of he basic framework of a sysem ha implemens his coding echnique is presened... Diversiy in Wireless Sysems Wireless channels are prone o large flucuaions. The channel may suffer aenuaion due o desrucive addiion of muli-pah componens of he ransmied signal. This increases he burden on he receiver in deermining he ransmied signal. Muliple, possibly lessaenuaed copies of he ransmied signal, can improve he receiver decisions significanly. This is ermed diversiy and is he primary ool agains fading. Diversiy uses alernae independen or highly uncorrelaed pahs for communicaion. Hence if one of he pahs undergoes a deep fade, a differen pah may have a sronger signal and can be exploied by he receiver. Diversiy can be provided in differen dimensions as described below. Time Diversiy: This is achieved by ransmiing copies of he signal in differen ime slos. This echnique is no as effecive in slow fading channels. Channel coding is also a form of emporal diversiy. Frequency Diversiy: Transmiing he same informaion over carriers of differen frequencies achieves his. This explois he fac ha differen frequencies experience differen muli-pah fading in he propagaing media. This echnique canno be used when he delay spread is small. Equalizers exploi his ype of diversiy.

10 Spreading he Signal: A signal ha has a bandwidh much greaer han he coherence bandwidh of he channel is used. Such a signal resolves he muli-pah componens and provides he receiver wih several independen fading signal pahs. This is a form of frequency diversiy. Equalizers or Rake receivers are needed for his echnique. Anenna Diversiy: This can be achieved in differen ways. Examples are, o Polarizaion Diversiy: The independence of orhogonal polarizaions is exploied. o Space Diversiy: The anennas are spaially separaed such ha received signals are uncorrelaed. This echnique incurs no bandwidh penaly... Moivaion for Transmi Diversiy Receive diversiy can be used a he base saion or any a receiver ha permis fixed complex srucures for recepion. Bu for he downlink, receive diversiy is no as pracical. Receive diversiy is oo expensive o be implemened in handses because of he addiional processing power and RF circuiry required. I migh also no be very useful because of he elecromagneic ineracion of anenna elemens on small plaforms and he resulan correlaion beween anenna componens. These reasons form he moivaion for he sudy of ransmi diversiy schemes. Muliple ransmi anennas are used o provide he required diversiy advanage over a channel. This helps o shif he diversiy burden from he mobile o he base saion. I has been shown ha he diversiy gain provided by he use of ransmi diversiy scheme is comparable o ha provided by Maximum Raio Receive Combining (MRRC) echniques wihou he 3dB aperure gain (unless feedback is used). The same is verified in course of his hesis as well. The ransmi diversiy schemes found in he lieraure can be broadly divided ino wo caegories: hose wih and hose wihou feedback. For schemes wih feedback, ransmier sequences are weighed before ransmission o ensure maximum benefi. Weighs are chosen adapively based on he informaion feedback from he receiver. An example of his scheme is swiched diversiy proposed in [Win]. The schemes wihou feedback use linear processing a he ransmier o spread he informaion across anennas. A he receiver, informaion is obained by linear processing or Maximum Likelihood(ML) echniques. An example of his echnique is he delay diversiy scheme proposed in [Sesh]. In his scheme, copies of he same symbol are ransmied hrough muliple anennas a differen imes. A he receiver hese delays inroduce a muli-pah like disorion. This disorion can be resolved by using a Minimum Mean Square Error (MMSE) equalizer or a ML scheme o obain a diversiy gain. Space Time Block Codes (STBC) proposed in [Alam] and generalized in [Taro3] is anoher example of his echnique. These codes are defined by a mapping operaion of a block of inpu symbols ino space and ime domains creaing orhogonal sequences ha are ransmied from

11 differen ransmi anennas. The receiver uses a ML deecion rule o decode he ransmied informaion. STTC is also a ransmi diversiy scheme ha does no require feedback from he receiver. I provides coding gain in addiion o he diversiy advanage and is essenially an exension of he Trellis Coded Modulaion scheme o muliple ransmi anennas..3. Trellis Coded Modulaion Channel coding and modulaion are radiionally considered as wo separae blocks in communicaion sysem design. Channel coding adds redundancy o he informaion bis and modulaion maps hese bis ino appropriae symbols from he chosen signal consellaion. The redundancy in he daa bis can be accommodaed eiher by increasing he bandwidh of he channel or by expanding he signal se over he un-coded sysem. Increasing he bandwidh is ofen no allowable. An expansion of he signal se requires addiional power o mainain he same error rae as he Euclidean disance beween codewords is oherwise reduced. Codes wih large performance gains are required o overcome his power penaly. Trellis coded modulaion is a novel echnique ha combines coding and modulaion ino a composie operaion. A convenional convoluional code is used and he redundancy for coding is provided by using an expanded signal se. Symbols are chosen from his expanded consellaion such ha he Euclidean disance (as opposed o Hamming disance in convenional channel code design) beween code-words is maximized. This leads o significan performance gains and does no require an increase in he ransmi power for he expanded signal se. Thus neiher bandwidh nor power efficiency is compromised. The general sraegy for encoding for TCM schemes is as follows, [Wick], Add one bi of redundancy for every m source bis Expand he signal consellaion from m o m+ signals Encode he sequence using rellis code Use he m + -bi encoded source blocks o selec a signal from he expanded signal consellaion. The mos imporan par of he encoding process is he mapping of m informaion bis o m+ symbols from he expanded signal se. This mapping is achieved hrough se pariioning of he expanded signal consellaion. The pariioning is done such ha he resuling sub-consellaions have larger minimum disances han he original consellaion. Figure - shows he signal pariioning for he 8-PSK consellaion. The binary labels a he leaf nodes provide a means for pariion selecion by coded informaion bis. The binary labels are arrived a by denoing each righward branch by one and each lefward branch by a zero. 3

12 3 4 8-PSK (0 0) ( 0) (0 ) ( ) 7 Figure -: Pariioning of he 8-PSK consellaion Xm Xk+ Xk+ Cm Ck+ Ck+ Selec Signal Wihin Pariion Xk X X Rae K/(K+) Convoluion Encoder Ck C C0 Selec Signal Pariion Figure -: A TCM encoder A TCM encoder is shown in Figure -. k bis are seleced from a block of k m informaion bis, x, x,, xm and fed o a rae encoder. The resuling k + bis ( k + ) are hen used o selec a pariion from he ( k+ ) slevel of he consellaion s pariion ree. The remaining ( m k) bis are used o selec a signal wihin he pariion. For example, consider a sysem which encodes informaion in blocks of wo bis using a TCM encoder (hus m = ). Le he symbol consellaion used be 8-PSK and le k =. The firs informaion bi is fed o he convoluional encoder and he resuling k + = coded bis are used o selec a paricular consellaion from he las pariion level of Figure -. The second informaion bi is used o choose a paricular symbol from he seleced pariion. 4

13 The resuling sequences have a rellis srucure due o convoluional coding used in he encoder. Hence he encoder is called a Trellis Coded Modulaion sysem. The rellis resrics he allowable sequences and hus increases he Euclidean disance beween code words compared o an un-coded sysem. Addiional design rules ha maximize he disance beween signals are given by: Signals in he same, lowes pariion in he pariion ree are assigned o parallel ransiions Signals in he preceding pariion are assigned o ransiions ha sar or sop in he same sae All signal are used equally ofen Parallel ransiions resul in a reducion of he Euclidean disance beween code-words and can be eliminaed by passing all m informaion bis hrough he convoluional encoder. All m+ bis are hen used o selec a symbol from a consellaion. The design rules excep for he rule dealing wih parallel ransiions remain relevan. The eliminaion of parallel ransiions also increases he achievable coding gain. As a consequence of he rellis srucure impared o TCM codes, he Vierbi decoder can be used for decoding TCM sequences. The Euclidean disance beween symbols a a paricular ransiion is used as he decision meric. Reference [Bigl] provides a more deailed sudy of TCM.4. Space-Time Trellis Codes STTC combines he advanages of ransmi diversiy and TCM in an ingenious way o obain reliable, high daa rae ransmission in wireless channels. Channel coding by using convoluional encoders adds redundancy o he informaion sequence. In TCM, he resuling coded bis are hen used o choose symbols from an expanded signal consellaion. Bu in STTC, he encoded daa is used o selec symbols from N separae bu idenical consellaions, where N is he number of ransmi anenna employed by he MIMO sysem. These N symbols are hen simulaneously ransmied from he N ransmi anennas. The choice of symbols, as in he TCM case, is made such ha he Euclidean disance beween code-words is maximized. Thus TCM principles are exploied o add channel coding wihou penalizing he informaion rae, decreasing he power efficiency or increasing he bandwidh of he sysem. The scheme concurrenly inroduces diversiy by using muliple ransmi anennas in he sysem. The coding across ransmi anennas also aids in providing maximum benefi of he offered sysem diversiy. A sysem model for a Muli Inpu Muli Oupu (MIMO) sysem employing a Space Time Trellis Code (STTC) is shown in Figure -3. The sysem has N ransmi and Nr receive anennas. Consider he signal consellaion size of he sysem o be M. An encoder is used o map k (=log M) daa bis o N separae bu idenical consellaions of 5

14 size M afer applying a convoluional code o i. The STTC encoder is dependan on he number of inpu daa bis ( k ), he number of saes of he convoluional encoder ( v ) and he number of ransmi anennas ( N ). A generaor marix of size ( k+ v) N can enirely define he STTC. I can also be represened by a rellis diagram wih v saes. Each ransiion of he rellis defines he symbols ransmied from N ransmi anennas. I also defines he beginning and end saes of he encoder. The encoder is implemened by a feed forward shif regiser srucure wih memory order v. The Calderbank-Mazo algorihm can be used ranslae he rellis diagrams ino closed form expressions [Taro], which are very useful in designing he implemenaion. SOURCE STTC ENCODER CHANNEL STTC DECODER SINK Figure -3: Sysem model for a MIMO sysem employing an STTC Some examples are used o illusrae he encoder. Consider he 4-sae rellis code shown in Figure -4 (from [Taro]). Each rellis ransiion defines he symbols o be ransmied from each ransmi anenna (wo in his case) for a paricular combinaion of he sae and inpu bis. The rellis also shows he sar and end saes afer a ransiion. I can also be noiced ha here are M ( M = 4 in his case) ransiions from each sae corresponding o all possible combinaions of k ( k = in his case) inpu bis. The symbols are chosen from he QPSK consellaion shown in Figure -5. The encoder can be represened in a closed analyical form by he equaions (from [Goza]), () = ( ) + ( ) () = () + () x a a x a a where, x () and x () are he symbols ransmied from he firs and second ransmi anennas. a(), a(), a3(), a4() are he inpu and sae bis respecively. The generaor marix of he given code is herefore given by 00 T 00 and he given code can be implemened by he feed forward srucure shown in Figure -6. 6

15 Inpu Bis A A A 3 A 4 (0 0) (0 ) ( 0) ( ) Sae Bis An An Transmied Bis Figure -4: Trellis for a 4-sae STTC Symbol (0,) Symbol (-,0) Symbol 0 (,0) Symbol 3 (0,-) Figure -5: QPSK symbol consellaion 7

16 a a QPSK Mapping T T a 4 a 3 QPSK Mapping Figure -6: Feed forward encoder srucure for 4-sae STTC Inpu Bis Sae Bis a 5 a 4 a a a (0,0) (0,) (,0) (,) An An Transmied Bis Figure -7: Trellis for 8-sae STTC The rellis diagram of an 8-sae code (from [Taro]) is shown in Figure -7. The encoder for he same can be represened by a closed analyical form given by () = ( ) + ( ) + ( ) () = () + () + () x a a a 5 x a a a and implemened by feed forward srucure as shown in Figure -8. The corresponding generaor marix of he code is T. 8

17 a QPSK Mapping a T a 5 T T a 4 a 3 QPSK Mapping Figure -8: Feed forward encoder srucure for 8-Sae STTC The N symbols generaed by he encoder are simulaneously ransmied from he N ransmi anennas. The ransmission occurs in frames of lengh L. The receiver fron end consiss of N r receive anennas and he signal a each receive anenna is a noisy linear superposiion of simulaneously ransmied N symbols weighed by he fade coefficiens of he channel. The noise is assumed o be Addiive Whie Gaussian Noise (AWGN). Consider ci () o be he symbol ransmied from he i h ransmi anenna a ime insan. Then, he received signal r ( ) insan is given by, a he h receive anenna a any ime N () α () () η () r = c + ; N i i r i= -3 () η is he Gaussian noise a ime insan and is modeled by zero mean complex No Gaussian random process wih variance per dimension. The channel is modeled by an N N r marix Ω (), whose enries α i ( ) represen he complex Gaussian fading coefficien from he i h ransmi anenna o he h receive anenna a ime insan. The fade coefficiens are assumed o have zero mean (for a Rayleigh fading channel) and variance of 0.5 per dimension. The receiver esimaes he channel and uses a Maximum Likelihood Sequence Esimaion (MLSE) decoder o decode he informaion bis. The MLSE decoder compues he lowes accumulaed Euclidean disance meric over an enire frame o exrac he mos likely ransmied sequence. The branch meric used by he MLSE decoder in he presence of perfec channel esimaes is given by N r N () α () () r e i i = i= -4 9

18 where e (), i =,, N is a candidae codeword. i The design of a STTC scheme seeks o maximize he Euclidean disance beween codewords ransmied from differen ransmi anennas. A deailed discussion on he design crieria and opimal code-consrucion echniques for STTC over differen fading channels is presened in he nex chaper. The basic design principles involved in STTC are seen o be similar o TCM..5. Organizaion of Thesis This hesis analyzes he influence of channel esimaion errors on he design and performance of STTC over differen Rayleigh fading channels. I specifically invesigaes any aleraions caused in he design crieria or he performance behavior due o he presence of esimaion errors. Design and code-consrucion crieria for STTC over differen fla fading channels are presened in Chaper. The analyses in his chaper assume perfec channel esimaes o be available a he receiver. The performance of various STTC schemes adhering o differen design crieria are compared via simulaions. An analyical mehod ha deermines he exac pair-wise error probabiliy of STTC assuming perfec channel esimaes and hus supplemens he simulaed performance analysis is also presened. Chaper 3 inroduces several channel esimaion schemes for muliple anenna sysems and discusses heir relaive meris. I also presens a capaciy analysis of he muli-inpu channel in he presence of channel esimaion errors. This work is he summary of several previously published resuls. Chaper 4 discusses he impac of channel esimaion errors on he performance of STTC. I reevaluaes he design crieria derived in Chaper in he absence of perfec channel esimaes a he receiver. The chaper derives an exac expression for he pair-wise error probabiliy of STTC in he presence of channel esimaion errors. I also aemps o idenify and characerize facors (such as code-choice, he number of ransmi/ receive anennas in he sysem, he amoun of raining used and he coherence ime of he channel) ha influence he exen of performance loss due o channel esimaion errors in STTC sysems. Chaper 5 analyzes he performance of STTC in frequency selecive channels. I presens a new design crierion for STTC ha ensures a beer performance han exising schemes over muli-pah channels. I also evaluaes he influence of channel esimaion errors on he performance of STTC over muli-pah channels. Chaper 6 discusses he resuls and conclusions formed by he hesis and poins o direcions for fuure work. 0

19 . STTC Performance Analysis and Design Crieria Space Time Trellis Codes (STTC) are inended o exploi diversiy in boh space and ime o improve he reliabiliy of communicaions over fading channels using muliple anennas. A comprehensive sudy and analysis of he design crieria for STTC, which draws on he exensive lieraure available on he subec, is presened in his chaper. These code-consrucion crieria aemp o allow STTC o derive maximum benefi over differen fading channels and are in general formulaed by analyzing he expressions for pair-wise error probabiliy of he codes. Some of he seminal work in his area was done by Tarokh e al in [Taro]. Their paper deal wih design crieria for STTC over slow fla fading, fas fla fading and spaially correlaed channels assuming high SNRs. Alernae crieria for STTC in channels wih large possible diversiies (larger number of ransmi/receive anennas) were derived in [Chen]. This chaper discusses hese key resuls and heir exensions. The performance of various STTC schemes adhering o differen design crieria are compared via simulaions. An analyical mehod of deermining he exac pair-wise error probabiliy of STTC derived in [Turi], which provides a ool o verify and supplemen he simulaed performance analysis, is also presened. This chaper provides he necessary conex for he main work in his hesis presened in Chaper 4... Sysem Model A generalized model for Muli Inpu Muli Oupu (MIMO) sysems employing STTC is presened in his secion. All subsequen analyses in his hesis follow some form of his basic model. A MIMO sysem wih N ransmi and N r receive anennas is considered. The STTC encoder is defined by a generaor marix G and combines he encoding and symbol mapping procedure ino a composie operaion. The generaor marix is implemened by a feed-forward shif regiser wih memory order v. According o he symbol consellaion used, he inpu bi sream is subdivided ino blocks of appropriae lengh and fed o he encoder. The inpu daa is hus encoded and modulaed ino N parallel sreams of symbols. The symbols are ransmied in frames of lengh L. The channel is modeled by an N Nrmarix Ω (), whose enry α i ( ) represens he complex fading coefficien from he i h ransmi anenna o he h receive anenna a ime insan. The receiver consiss of N r receive anennas and a Maximum Likelihood Sequence Esimaion (MLSE) decoder. The MLSE decoder compues he lowes accumulaed Euclidean disance meric over an enire frame o exrac he mos likely ransmied sequence. The signal a he receiver is a noisy superposiion of simulaneously ransmied symbols weighed by he fade coefficiens. The noise is assumed o be Addiive Whie Gaussian Noise (AWGN). Consider ( ) i c o be he symbol ransmied from he h i ransmi anenna a ime insan. To allow he validiy of he following analysis o any modulaion scheme, each symbol

20 in he signal consellaion is conraced by average symbol energy average energy of he signal consellaion is. E s, such ha he The received signal r () a he h receive anenna a any ime insan is given by, N () α () () η () r = c E + ; N i i s r i= - where, η () is he Gaussian noise a ime insan and is modeled by zero mean No complex Gaussian random process wih variance per dimension. E s is he received energy per symbol per ransmi and receive anenna. The average signal power a each receive anenna from each ransmi anenna is assumed o be he same... Design Crieria for STTC... Upper Bound on Pair-wise Error Probabiliy and TSC Crieria Design crieria for STTC over slow frequency non-selecive, fas frequency non-selecive and spaially correlaed channels were presened in [Taro]. These crieria were derived by examining expressions for he upper bound on pair-wise error probabiliy. This analysis is presened below. The ransmied codeword is assumed o be given by c= c() c() cn () c( ) c( ) c ( ) ( ) ( ) ( ) N c L c L c N L and he MLSE decoder is assumed o decide erroneously in favor of codeword e= e e e e e e e L e L e L. () () () ( ) ( ) ( ) ( ) ( ) ( ) N N N... A. Quasi-saic Channel wih Independen Fade Coefficiens A quasi-saic channel where he fade coefficiens are consan over a frame and vary over consecuive frames is assumed. The sub-channels (from each ransmi anenna o each receive anenna) are assumed o be muually independen and are modeled as independen samples of a zero mean complex Gaussian random process wih variance 0.5 per dimension. Le α i be he fade coefficien over a frame beween ransmi anenna i and receive anenna. Assuming perfec channel esimaion, he Chernoff bound for he condiional pair-wise error probabiliy is, (, ) d c e E s P( c e αi, i=,,, N, =,,, Nr ) exp - 4N0

21 where, Equaion (-3) can be rewrien as where, Nr L N d c e i ci ei = = i= ( ) -3 (, ) = α ( ) ( ) Nr N N L * i ( )( () () i ) ' i i i ' i ' -4 (, ) αα ( ) ( ) * d c e = c e c e = i= ' i = = * x denoes he complex conugae of x. Subsiuing ( α, α,, αn ) L A (, ) ( () ())( () ()) * pq c e = cp ep cq eq, p=,,, N, q=,,, N, = N r d ( c, e) = Ω * AΩ = Ω = and -5 * By definiion, A( ce, ) is a Hermiian marix. I can be wrien as VA( c, e) V = De, where V is uniary marix, whose rows are made of he eigenvecors of Ace (, ) and D e is a diagonal marix whose elemens are given by he eigenvalues, i,,, N λ = of (, ) By consrucion, he square roo of A( ce, ) is given by he difference marix, i Ace. () ( ) ( ) ( ) e c e L c L Bce (,) = en () c () ( ) ( ) N e N L c N L -6 Hence he eigenvalues of A( ce, ) are nonnegaive real numbers. * Le( β, β,, β N ) =ΩV, hen N r N d ( c, e) λi βi = i= = -7 As α i are samples of a complex Gaussian random process wih zero mean and V is a uniary marix, β i are also independen samples of a complex Gaussian random process 3

22 wih zero mean and variance 0.5 per dimension. given by ( ) exp( ) i i i p β = β β for β 0 β i follows a Rayleigh disribuion The upper bound on he pair-wise error probabiliy is he saisical average of N r N Es λi βi = i= exp wih respec o he disribuion of βi and is given by, 4N 0 i -8 P( c e) N E s + λi i= 4N 0 Nr -9 If r is he rank of A, hen he marix A has r nonzero eigenvalues, λ i, i =,,, r. Es A high Signal o Noise Raios (SNRs), λ i >> i, Equaion (-9) can be wrien as 4N 0 N r rn r r E s λi i= 4N0 P( c e) -0 The diversiy advanage is equivalen o he power of he SNR in he denominaor of he expression. The coding advanage is a measure of he gain of he sysem over an uncoded sysem offering he same level of diversiy. I is seen from Equaion (-0) ha he diversiy advanage obained is rn r and can be increased by increasing he rank of he disance marix or he number of receive and ransmi anennas. The coding advanage N r r obained is given by λi. I can be improved by increasing he produc of i= eigenvalues of he disance marix. This leads o he following design crieria, Rank Crierion: In order o maximize he diversiy gain over he Rayleigh fading channel, he minimum rank of he disance marix Aceover (, ) all pairs of disinc code-words c and e is o be maximized. Maximum diversiy advanage of N Nris obained if he marix is full rank. Oherwise a diversiy advanage of r Nr is obained where r is he rank of he marix. 4

23 Deerminan Crierion: The minimum of he produc of eigenvalues λ i, i =,,, r aken over all pairs of disinc code-words c and e mus be maximized. For a full rank marix his is equivalen o maximizing he minimum deerminan of he disance marix Aceover, all possible code-word pairs. ( ) These will be referred o in his hesis as he TSC RD (Tarokh, Seshadri, Calderbank - Rank and Deerminan) crieria. I is seen from Equaion (-0) ha he rank of he disance marix is an exponen in he probabiliy expression. Hence he minimum rank crierion becomes more imporan han he minimum deerminan crieria in deermining he code performance.... B. Fas Fading Channel wih Independen Fade Coefficiens A fas-fading channel is assumed where he fade coefficiens vary from one symbol o he nex. The pair-wise error probabiliy is approximaed by (, ) d c e E s P( c e αi (), i =,,, N, =,,, Nr, =,,, L) exp - 4N0 where, Nr L N d c e i ci ei = = i= ( ) - (, ) = α () () () The channel coefficiens αi () are assumed o vary independenly from one symbol o he nex in a fas fading channel and are hence included in he summaion over he lengh of he frame in Equaion (-) (unlike in Equaion (-4) ). Le Ω () = ( α(), α (),, αn ( )) and C () ( () ())( () ()) * pq = cp ep cq eq From Equaion (-), N L * = = r (, ) = Ω () () Ω () d c e C -3 * Since C() is Hermiian, i can be wrien as V ( ) C( ) V ( ) = De ( ), where V() is a uniary marix whose rows are made of he eigenvecors of C( ) and De () is a diagonal marix whose elemens are given by he eigenvalues λ i( ), i =,,, N of C(). As C() is a Hermiian marix, is eigenvalues are real. 5

24 * ( β β β N ) =Ω( ) V ( ) Le () () (),,,, hen * () C() () λ () β () i i i= N Ω Ω = -4 As ( ) α i are samples of a complex Gaussian random variable wih mean zero and V is a uniary marix, β () are independen samples of a complex Gaussian random process i wih mean zero and variance 0.5 per dimension. ( ) given by ( ) ( βi ()) = βi () exp βi () for i ( ) 0 p β follows a Rayleigh disribuion i β -5 The upper bound on he pair-wise error probabiliy is found by averaging ( e) P c Nr L N Es λi i = = i= exp 4N0 () β () -6 wih respec o he disribuion of β ( ) i and is given by, P( c e) L N E s + λi = i= 4N 0-7 The columns of C() are muliples of ( ) c() e() = c() e(), c() e(),, cn () e () N. Hence ( ) if c() c() cn ( ) e( ) e( ) e ( ) N possible nonzero eigenvalue of D( ) is c() e() C has rank only and rank zero oherwise. Thus he only. Subsiuing in Equaion (-7), P( c e) L () () E s + c e = 4N 0-8 6

25 Le l be he number of ime insances in a frame ha c( ) e( ) 0 (-8) can be expressed as N r ln l r E s = 4N0, hen Equaion P( c e) c( ) e( ) -9 Diversiy advanage is seen o be governed byln. The design crieria follow as, Disance Crierion: The diversiy gain can be maximizing he number of ime insances for which he srings c() c() cn ( ) and e ( ) ( ) ( ) e en of any pair of disinc code-words differ during he duraion of a frame. If he code-words differ for l ime insances in a frame, a diversiy advanage of l N is obained. Produc crierion: To maximize coding gain, he minimum of he produc of he disances beween all pairs of code-words a, ime insances when he disances are no zero, should be maximized. These will be referred o as he TSC DP (Tarokh, Seshadri, Calderbank Disance and Produc) crieria.... C. Dependen Quasi-saic Fade Coefficiens The channels are assumed o be correlaed and channel coefficiens are modeled by samples of dependen zero mean complex Gaussian random variables wih variance 0.5 per dimension. Le Y( c, e) be a block diagonal marix wih dimension N Nr whose diagonal elemens are A(c,e) and le Ω= ( Ω, Ω,, Ω Nr ) probabiliy is hen given by r r. The Pair-wise error * (, ) Ω ΩY c e E s P( c e αi, i =,,, N, =,,, Nr ) exp 4N0-0 * Le Θ be he correlaion marix of Ω, Θ = E Ω Ω. Θ is assumed o be full rank. As Θ is a nonnegaive definie Hermiian marix, i has a square roo Ψ. The diagonal elemens of ω =Ω Ψ, hen * Θ are equal o uniy. Hence he rows of Ψ are also of lengh one. Le ( ) he elemens of ω are samples of uncorrelaed complex Gaussian random process wih variance of 0.5 per dimension. ω hus follows he Rayleigh disribuion. From Equaion (-0), 7

26 (, ) E ( ) * * sωψ Y c e Ψω P c e exp 4N0 - This is similar o he case of independen fading coefficiens wih Ace (, ) replaced * by Ψ Y( c, e) Ψ. N r rn r r E s λi i= 4N0 P( c e) - * Where r is he rank of Ψ Y( c, e) Ψand λ i, i =,,, r are he eigenvalues of * Ψ Y( c, e) Ψ. From Equaion (-), as Ψ is full rank, maximizing he rank * of Ψ Y( c, e) Ψ is equivalen o maximizing he rank of Y( c, e ), which is in urn equivalen o maximizing he rank of Ace. (, ) Maximizing he deerminan of * Ψ Y( c, e) Ψis also equivalen o maximizing he deerminan of A( ce, ). Thus comparing wih he firs case (Secion...A), i is seen ha he design crieria for he case of independen quasi-saic channel coefficiens also holds for he case of dependen quasisaic channel coefficiens.... Generalized Design Crieria Generalized design crieria were formulaed in [Gama] ha deermined he diversiy and coding advanage achieved by STTC in MIMO block-fading channels. By fixing he size of he blocks o appropriae values, he design crieria in [Taro] (Secion..) for quasi-saic and fas fading MIMO channels were derived as special cases of he new crieria. Consider ha each frame of size L has M blocks over which he channel coefficiens are consan. Then he fade coefficiens are consan over L consecuive symbol duraions. M h Le αi ( m) be he fading coefficien for he m fading block. The oher parameers can also be expressed accordingly over a fading block. ( ) m L ml R [ m] = r +,, r N M M L M [ m] ( ) m L ml = η +,, η M M [ ] α [ ],, α [ ] H m = m N m N L M 8

27 [ ] cm ( ) m L ml c + c M M = ( m ) L ml cn + c N M M L N M Then, for m M, he received signal can be expressed as, [ ] [ ] [ ] [ ] R m = H m c m + N m -3 The pair-wise error probabiliy of ransmiing a code word c and deciding erroneously in favor of a code-word e can be approximaed by where, (, ) d c e E s P( c e α i, i =,,, N, =,,, Nr) exp -4 4N0 N M d c e H m c m e m = m= r (, ) = [ ]( [ ] [ ]) * -5 For a marix X, X = XX. Following he approach in [Taro], he upper bound on he pair-wise probabiliy of error can be wrien as where, d N M m r µ me s P( c e) m= 4N -6 0 ( ) d dm = rank ( c[ m] e[ m] ), µ m m = λ[ m] λ[ m] λd [ m] and λ [ ] [ ] [ ] m m, λ m,, λd m m are eigenvalues of Am [ ] = ( cm [ ] em [ ])( cm [ ] em [ ]) H. The diversiy order of he sysem is given by M M -7 m ( [ ] [ ]) d = d = rank c m e m m= m= and he coding gain of he sysem is given by, 9

28 M d = [ m] [ m] d [ m] m m= µ λ λ λ -8 Hence he design crierion for block fading channels is as follows. Block Fading Sum of Ranks Crierion To maximize he diversiy advanage, d is o be maximized over all pairs of disinc code-words c and e. Block Fading Produc Disance Crierion To maximize he coding gain, µ is o be maximized over all pairs of disinc codewords c and e. The design crieria for quasi-saic and fas fading channels can be obained from he block fading crieria by leing M = and M = L respecively...3. Trace Crierion I was shown in [Taro] (Secion..) ha he rank crierion is more imporan han he deerminan crierion in deermining code performance. However, he difference marix, B ( ce, ) (defined in Equaion (-6)) of dimension N L, has a maximum rank given by he min ( N, v) where v is he consrain lengh of he code. Hence a full rank value of N is no always achievable. A new design crierion for STTC was inroduced in [Chen] which did no require he difference marix o have full rank. I was shown in [Chen] ha when STTC is used in sysems wih a large produc of he number of ransmi and receive anennas (>3), he muliple fading sub-channels converge o an addiive whie Gaussian channel. The new design crierion (described below) akes advanage of his approximaion. Assume ha a maximum likelihood receiver decides erroneously in favor of a signal e= e e e e e e e L e L e L assuming ha () () N () ( ) ( ) ( ) ( ) ( ) ( ) N N () () () ( ) ( ) ( ) ( ) ( ) ( ) c= c c c c c c c L c L c L was ransmied. Le N N N r ( r N ) be he rank of he difference marix B(c, e) and le λ i be he eigenvalues of he disance marix Ace (, ). As menioned earlier, Ace (, ) is he square of marix B(c, e) by consrucion...3. A. Quasi-saic Channel The condiional pair-wise probabiliy is upper bounded by (from [Gama] and Equaion (-)), 0

29 where, Es p( c e α ) exp d ( c, e) 4 N Nr L N d (,) c e = αi, ci ei = = i= 0 ( () ()) The condiional probabiliy can also be expressed as (from [Taro] or subsiuing Equaion (-7) in Equaion (-)), p( c e α) exp E N r N s λi βi = i= 4N0-9 i β follows he cenral chi square disribuion since β i is Rayleigh. Is mean and variance is equal o (From [Gama]). For a large NN r(>3) value, according o Cenral Limi Theorem, he expression wih mean N r N = i= λ β i i approaches a Gaussian random variable D and variance N µ = N λ -30 σ D r i i= N D Nr λi i= = -3 Thus, he uncondiional pair-wise error probabiliy can be upper bounded by [Taro], E p ( c e) exp s D p( D) dd 4 0 N0-3 where, p(d) is a Gaussian disribuion. Es σ D µ D E 4 s Es N p( c e) exp σ µ D D Q 4 N0 4 N 0 σ D By using ( ) x Q x e x 0, N E ( ) exp s p c e Nr λi 4 N i=

30 The pair-wise error probabiliy can be minimized by maximizing he sum of he eigenvalues of he marix Ace (, ). For a square marix he sum of he eigenvalues equals he race of he marix. The race of marix Ace (, ) can be wrien as r(a)= N L i= = e i () c () i -35 Thus he pair wise error probabiliy can be minimized if he minimum Euclidean disance beween any wo code words is maximized. Hence i is shown ha when he diversiy order rnr 3 he maximum coding gain is governed by he Euclidean disance beween any wo code-words over all ransmi anennas. This crierion was referred o by he auhors of [Chen] as he race crierion. I is also noed ha when he number of ransmi anennas is equal o wo, i is imporan for he disance marix Ace (, ) o be full rank. If N 3, hen he full rank crierion is no necessary...3. B. Fas-Fading Channels An exension of he design crieria for sysems wih large diversiies o fas fading channels was presened in [Vuce] and is described in his secion. From he analysis for STTC over a fas fading channel discussed before, expression N i i i= * (-4), () C() () λ () β () Ω Ω =, can be wrien as i i= N * Ω () C() Ω () = c() e() β () -36 as he only possible nonzero eigen value of D( ) is c() e() over he Gaussian random variable, he PWEP can be upper bounded by,. By averaging (-36) E s 4 E s E 4 Nd s r E p( c e) exp NrD Nrd E Q NrD 4N 4 0 4N 0 4N0 D -37 L where, () () 4 de = c e and D = c() e() 4 = L =. Hence i is seen ha in he presence of a large diversiy order, he Trace crierion is valid for fas fading channels as well..3. Code Consrucion In his secion an overview of he code consrucion mehods employing he design crieria derived in he previous secions are given.

31 The STTC Encoder can be modeled by a feed-forward shif regiser wih a memory order of v. This srucural represenaion helps in code-consrucion and is described below. Consider a QPSK sysem wih N ransmi anennas. The encoder has wo branches wih memory orders v and v. A any given ime wo binary inpus, I () and I () are fed o he encoder. These inpu sreams are passed hrough heir respecive shif regiser branches and muliplied by coefficien vecors given by, a p a p, a p,, a p b q = b q, b q,, b q respecively, ( ) ( ) ( ) = ( ) ( ) N ( ) and ( ) ( ) ( ) N ( ) where, ai( p), bi( q) { 0,,,3 }, i =,,, n, p=,,, v, q =,,, v. The symbol ransmied on he i h anenna a ime is compued as v v i c = I p ai p + I q bi p p= 0 q= 0 ( ) ( ) ( ) ( ) mod 4-38 The associaed feed forward srucure is illusraed in Figure -. The generaor marix G is formed by he branch weighs and is given by, G T ( 0) ( 0) ( ) ( ) a b a v b v = an ( 0) b ( 0) ( ) ( ) N a N v b N v -39 where, v max ( v, v ) =. a 0 a I () T T a v c() I () T T b v b b 0 Figure -: STTC feed forward encoder srucure 3

32 If any branch has memory order less han v, hen some of is columns are allowed o be absen. For insance, if he firs branch has memory order v-, hen is column wih elemens a i (v), i=,,,n, is absen..3.. TSC Crieria.3.. A. Quasi-saic Channel The TSC-RD crieria discussed earlier requires he code disance marix o be full rank o guaranee full diversiy advanage. The geomeric uniformiy of he code rellis can be used o ensure full rank for any given code. Two heurisic design rules were described in [Taro] ha guaraneed full rank and hence maximum diversiy for a sysem wih wo ransmi anennas. Design Rule : Transiions deparing from he same sae differ in he second symbol. Design Rule : Transiions arriving a he same sae differ in he firs symbol When hese rules are followed, he code-word difference marix B( ce, ) has he form 0 * * e( L) c( L) Bce (,) =, which guaranees full rank. Hence he e() c( ) * * 0 space-ime code achieves wo-level spaial diversiy. The example codes in [Taro] are consruced using hese design rules and will be referred o as he TSC codes in his hesis. The design rules in [Taro] were generalized in [Grim] o any level of diversiy. A code ha has he propery of zeros symmery (i.e. every code-word difference marix is upper and lower riangular) is full rank. This crierion ensures full rank and reduces he search for good codes. However, his rule migh be overly resricive a imes. Example codes consruced according o his symmery propery are given in [Grim]. Space Time Trellis Codes were represened by a much more amenable Generaor form, han he rellis form in [Baro]. A sysemaic code search was hen carried ou by varying he values of he generaor marix and generaing code and error sequences specific o a generaor marix. As he rank crierion is predominan, a generaor marix was discarded as soon as i did no achieve full rank for any pair of sequences. I was found ha geomeric uniformiy could be used o limi compuer searches for good space-ime codes, bu could no be reaed as a necessary condiion for good codes. Codes were presened ha have coding gain larger han he codes presened in [Taro] for he same decoder complexiy. These codes are referred o here as he BBH (Baro, Bauch and Hansmann) codes..3.. B. Fas Fading Channels Space Time Trellis Codes which bes saisfy he TSC-DP crierion were presened in [Firm] (These codes will be referred o here as he FVY -Firmano, Vuceic and Yuan 4