A DCT-Based Broadband Multicarrier Transceiver

Transcription

1 A DCT-Based Broadband Mulicarrier Transceier Shilpa Saish, aofal Al-Dhahir and Hlaing Minn Deparmen of Elecrical Engineering Uniersi of Teas a Dallas, Richardson, TX aldhahir@udallas.edu Absrac We inesigae he use of he Discree Cosine Transform (DCT) for mulicarrier ransmission oer frequenc selecie channels. We derie design condiions for he DCT o diagonalize he channel ino parallel, decoupled, and memorless subchannels. Furhermore, we compare he performance of DCT and DFT-based mulicarrier ransceiers in he sence of channel esimaion errors, residual frequenc offse, and narrowband inerference. Kewords: Discree Cosine Transform, Muli-Carrier Modulaion, uard Sequence, Frequenc Selecie Channel 1. Inroducion Mulicarrier Modulaion (MCM) based on he Discree Fourier Transform (DFT) has been adoped as he modulaion/demodulaion scheme in seeral digial communicaions sandards. These include wireline sandards (such as ADSL) and wireless sandards (such as DAB/DVB, IEEE a/g/n, and IEEE802.16a/e). DFT-MCM diides he frequenc response of he finieimpulse response (FIR) channel ino parallel, decoupled, and memorless subchannels using a guard sequence in he form of a cclic fi (CP). This CP is a periodic eension of he informaion smbols, conering he linear conoluion of he FIR channel ino circular conoluion. Thus, he equialen channel mari can be perfecl diagonalized b he DFT, eliminaing boh iner-bloc inerference (IBI) beween smbols and iner-carrier inerference (ICI) beween adjacen frequenc subchannels wihin each DFT-MCM smbol. Addiional aracie feaures of using he DFT ecors as a modulaion/demodulaion basis include low compuaional complei using he FFT algorihm and heir independence of he channel characerisics. In his paper, we show how o design a DCT-MCM ransceier ha enjos he aboe-menioned desirable properies of DFT-MCM in addiion o he following aracie feaures inheried from he DCT: The DCT basis is well-nown o hae ecellen specral compacion and energ concenraion properies. Thus, in he sence of ICI i ens ecessie leaage ino he adjacen subchannels [3]. The DCT is widel adoped in image/ideo coding sandards (e.g. JPE, MPE, H.261). Using i for modulaion/demodulaion on frequenc selecie channels resuls in a beer inegraed ssem design and a reduced oerall implemenaion cos. The DCT uses real arihmeic compared o he complealued DFT. This reduces he compuaional complei for he DCT-based processing and also reduces he signal power consumpion. Howeer, we show ha he main complei disadanage of DCT-MCM wih respec o DFT-MCM is he need for a filer a he receier o mae he channel impulse response (CIR) smmeric in order for i o be diagonalizable b he DCT. This paper is organized as follows. In Secion II we describe he DCT-MCM ransceier model and derie is opimali condiions. Secion III compares he performance of DFT-MCM and DCT-MCM under seeral pracical impairmens and he paper is concluded in Secion IV. 2. DCT-MCM Transceier In his secion we sar b describing he mahemaical model used for bloc ransmission oer a frequenc selecie channel. We show how he choice of he guard sequence affecs he ransceier design. This is followed b a discussion on receier signal processing including he e DCT-MCM channel diagonalizaion resul Channel Model and Assumpions We consider bloc b bloc ransmission oer a linear ime-inarian (wihin he bloc) frequenc selecie nois channel. The receied smbols are gien b m = = + hm m+ z (1) m= h where hm is he m coefficien of he oerall CIR which has memor of 2. The informaion smbols are assumed o be zero mean wih an -dimensional auo-correlaion funcion R. The addiie whie aussian noise (AW) Z and hae ariance smbols are denoed b { } 2 σ z.addiional 2 guard smbols are added before /06/$ IEEE. 175

2 ( fi smbols) and afer ( fi smbols) he informaion smbols o remoe he IBI a he epense of a hroughpu loss facor of 2. Furhermore, he guard smbols are discarded a he receier o eliminae heir inerfering effec. Equaion (1) can be resened in mari form as follows. M h L h L h L z h 1 L h L h M = + z M M M O O O M + 0 L 0 h L h0 L h z + + M + + B pariioning he inpu ecor ino fi, fi and informaion smbols, we can wrie = H = H : K+ + : + H info : + H : 1 Here, bold capial leers and small leers are used o denoe marices and ecors, respeciel. Furhermore, he subscrips of he ecor denoe he indices of is firs and las elemens separaed b a colon. The pariioning of he H mari is as follows I 0 H = H 0 ; H = H 0 ; 0 I 0 H info = H I ; (4) 0 Where I denoes he ideni mari of size and 0 denoes he all-zero mari wih rows and columns. Refer [1] for more deails. The wo lengh- guard sequences are redundan and he are relaed o he lengh- informaion sequence b some deerminisic funcions as shown below. : = (5) + : + + = Therefore, (3) can be wrien as = ( H + H + H ) = H info eq where H eq is he equialen channel mari. Equaion (6) clearl shows he effec of differen guard (6) (3) sequence designs (hrough and ) on he oerall channel mari H eq. For eample, for DFT-MCM, he guard sequences are chosen as cclic eensions of he informaion sequence; i.e. = [ 0 ( ) I ] and = [ I 0( ) ] We consider he pe-ii DCT defined b he real orhogonal mari whose (l, m) enr is gien b 2 ( l )( 2m ) π cos 2 C( l, m) = :1 l, m ; l 1 w 1 : l = 1 here C C=CC =I Fac : All marices diagonalizable b he pe-ii DCT mari can be wrien as he sum of an smmeric Toepliz mari T and an Hanel mari L, i.e. C (T+L) C T = D, where D is a diagonal mari. Moreoer, L is deermined from T from he relaions : Le1 = S Te1 (7) Le = J Le1 where S is he upper-shif mari, J is he reersal mari ande i is he i h uni ecor. For Heq o be diagonalized b DCT i mus saisf he condiions in (7). oe ha Hinfo becomes a smmeric Toepliz mari if and onl if hi = h i : 1,2. K, (8) which resuls in a smmeric linear-phase CIR. e, we need o mae he mari H + H a Hanel mari ha saisfies (7) in addiion o (8). In order o saisf he aboe consrain we choose [ J 0( ) ] [ 0 J ] = (9) = ( ) The essions for and in (9) impl ha i = i :1 i (10) + + i = + i :1 i This reeals ha he guard sequences should be smmeric eensions of he informaion smbols Receier Processing The smmer condiion in (8) can be me in pracice b implemening an FIR fron-end filer, as follows: 176

3 For channels wih long memor, denoed b L ( 2 +1), he design crierion used for he FIR channel shorening filer for DFT-based mulicarrier is modified b incorporaing he smmer consrain on he arge impulse response (TIR). In [4] i was shown ha he channel shorening mean square error (MSE) can be essed in he following quadraic form MSE = h Rh (11) where h is he smmeric and shorened CIR and R is a posiie definie mari ha depends on he original CIR and he noise ariance. The smmeric condiion on h in (8) can be imposed b defining h = I h (12) Where [ h ] I + 1 h = h0 L andi =. J 0 1 Therefore, he channel shorening MSE becomes MSE = h I R I h = h Rh (13) subjec o he consrain h h = 1. The opimum h is well nown o be he eigenecor of R = I R I corresponding o is minimum eigenalue. The opimum smmeric shorened CIR is calculaed using (12) and he opimum filer coefficiens are deermined from he Orhogonali Principle of Linear Esimaion [5] using he Wiener equaion (see [4] for deails) w = R R h (14) where R and R are he oupu-inpu crosscorrelaion and he oupu auo-correlaion marices, respeciel, which are calculaed in closed form from he original channel mari. For channels wih shor memor, we can sill use he same filer bu we need no shoren he channel. The lengh of he shorened channel is se equal o ha of he CIR. In his case he CIR is jus made smmeric wihou shorening i. For a shor and real channel, he filer could be implemened as a mached filer. Howeer his resuls in reducion of he channel hroughpu from + o Deecion Algorihms B properl designing he guard sequence and he filer, condiions (8) and (10) are saisfied. Saring from (6) we ge Y + = H eq = C DC = C = DX Z + + (15) where capial leers denoe he DCT-ransformed quaniies. D is a diagonal mari whose elemens are gien b eich eqe1 d i = (16) eice1 The elemens of X + are hus decoupled and can be indiiduall deeced b appling a simple zero forcing (or MMSE) scalar equalizer followed b a slicer. See he DCT- MCM bloc diagram in Figure Comparison of DCT-MCM wih DFT-MCM In his secion, we compare he performance of DCT- MCM wih DFT-MCM in he sence of pracical impairmens such as channel esimaion errors, residual frequenc offse, and narrowband inerference. We consider Channel B as specified in he wireless local area newor (WLA) ETSI HIPERLA2 sandard [6] which models a highl dispersie office enironmen wih a large maimum dela sad of 730nec corresponding o a CIR wih 16 smbol-spaced channel aps. We design a filer o shoren he CIR o a 5 ap impulse response for boh DFT-MCM and DCT-MCM. For he laer, he filer maes he channel smmeric as well Frequenc Offse Mulicarrier ransceiers are more sensiie o frequenc offse han single-carrier ransceiers since in he former, informaion is ransmied oer narrowband orhogonal subcarriers. An residual frequenc offse will cause ICI which degrades performance. The frequenc offse model is implemened as a diagonal mari where each enr specifies he offse a ha ime insan. The frequenc offse effec is modeled using he diagonal mari j 2πf oi S = diag e : 0 i + 2ν (17) where f o is he normalized (b subchannel widh) frequenc offse. The equialen channel mari is hen gien b H = W * S * H orig (18) where W is he filer conoluion mari and H orig is he original channel conoluion mari. All marices are of 177

4 size( 2 ) ( + 2) +. The guard sequence of size 2 is sripped a he receier. The performance of DFT-MCM and DCT-MCM is compared in Fig. 2 for bloc size =64 using he achieable bi rae crierion aeraged oer channel realizaions for normalized frequenc offses ranging from 0.01 and 0.1 and inpu SR of 20dB. The number of filer coefficiens used is 64. The bi rae is calculaed using he gap approimaion [2] a a arge error rae of 6 10 assuming a coding gain of 4.5 db. The figure shows ha DCT-MCM ouperforms DFT-MCM wih increasing frequenc offse. An increase in frequenc offse increases he ICI. Since he DCT has beer specral conainmen properies han he DFT, he ICI leaage ino adjacen subcarriers is less compared o DFT and hence i achiees beer performance which also corroboraes he resuls in [3] Channel Esimaion wih Frequenc Offse The channel is esimaed in he sence of frequenc offse using raining smbols. The receied smbol a ime is modeled as j2πfo 2 () e h() i ( i) ( ) = i= 0 (19) j 2πfo where {(m)} is he raining sequence and e is he frequenc offse a ime insan. Wriing (19) in mari form, we ge = SXh + Z (20) where X is a Toepliz mari of size ( 2 + 2) 1 composed of roo-of-uni raining smbols[7], h is he CIR ecor of size and is he lengh of he raining sequence. Wihou correcion of he residual frequenc offse, he leas-squares channel esimae is gien b h es = ( X'X) X' (21) ' where (.) denoes he comple-conjugae ranspose. As in (18), he equialen channel is calculaed as H = W * S * H es (22) The performance of DFT-MCM and DCT-MCM is compared for =64 using he bi rae crierion aeraged oer channel realizaions for normalized frequenc offses ranging from 0.01 and 0.1 and inpu SR of 20dB. The lengh of he raining sequence was aen o be 64 and he number of filer coefficiens is 64. The channel was assumed consan oer 4 consecuie blocs oer which he esimae was aeraged o reduce noise effecs. Fig. 3 shows ha DCT-MCM sill ouperforms DFT- MCM wih increasing offse. Howeer, he cross-oer poin shifed o higher frequenc offses because of he increased sensiii of DCT-MCM compared o DFT- MCM o channel esimaion errors. This can be aribued o he addiional requiremen of an oerall smmeric CIR for DCT-MCM which increases he sensiii o channel esimaion errors arrowband Inerference (BI) The scarci of radio frequenc specrum necessiaes specral coeisence of seeral wireless ssems wih differen ransmission bandwidhs. One consequence of his specral oerlap is narrowband inerference o mulicarrier ssems which can affec seeral subchannels causing performance loss. In his paper, BI is modeled as aussian correlaed noise [8] wih auo correlaion sin( ωa / 2) sequence Rii () = cos( δω) ωa / 2 (23) where ω a is he bandwidh of he inerferer and δω is is cener frequenc (in radians). If J o is he oal jammer power, hen he jammer power per affeced subchannel is gien b J o * sad J = (24) where sad is he number of subcarriers oer which he inerference is sad. If fc is he cener frequenc of BI he inerference in Hz, hen 2 * π * f δω = c BI (25) BW where BW is he oal bandwidh of he channel in Hz. Knowing J o,δω, and ω a, he smmeric Toepliz inerference auo-correlaion mari can be calculaed using (23). Then, he oerall noise auocorrelaion mari is gien b 2 Rnn = J o * Rii + σ z I (27) 2 Here, σ z is he ariance of he whie aussian noise floor. The performance of DFT-MCM and DCT-MCM is compared for =64 for differen jammer o noise power raios (JR) a inpu SR=20 db using he bi rae crierion aeraged oer realizaions. The bandwidh of he channel was aen o be 20MHz and he cener frequenc of he jammer was assumed o be 3MHz. In one case, he inerference was assumed o affec onl wo subchannels and in he second case, i affeced 4 subchannels. JR was aried from -10dB o 60 db. The number of filer coefficiens was 64. Fig. 4 shows ha for moderae o high JR, DCT-MCM ouperforms DFT-MCM. Also, as he number of 178

5 subchannels affeced b BI increases, he reducion in bi rae is more graceful for DCT-MCM. This demonsraes he improed robusness of DCT-MCM compared o DFT_MCM in he sence of BI due o he superior specral compacion and energ concenraion properies of he DCT which reduce ecess leaage of he ICI energ ino adjacen subchannels and hence resul in beer performance. 4. Conclusion The DCT is an opimal modulaion/demodulaion basis when he oerall CIR is smmeric and he fi and fi guard sequences are a smmeric eensions of he informaion smbols in each ransmied bloc. The firs condiion can be me b implemening an FIR filer and he second condiion can be me b placing smmer condiions onl on he guard sequence. DCT-MCM has comparable complei o DFT-MCM for long channels where boh ransceiers require a filer o limi he guard sequence hroughpu oerhead. Howeer, for shor channels, DCT-MCM sill requires a filer (unlie DFT-MCM) which can be implemened as a ime reersed mached filer. Boh DFT-MCM and DCT- MCM can perfecl diagonalize frequenc selecie channels wihou channel nowledge a he ransmier. Our simulaions show ha DCT-MCM is more robus o frequenc offse and narrowband inerference han DFT- MCM. Boh impairmens resul in inercarrier inerference which leas ino adjacen subcarriers. Howeer, he enhanced specral conainmen proper of he DCT (compared o DFT) resuls in beer performance. In summar, our resuls in his paper show ha DCT- MCM is a iable mulicarrier ransceier which can be compeiie wih DFT-MCM in some pracical scenarios. Our ongoing research includes comparison of DFT-MCM and DCT-MCM in a wireless enironmen in he sence of I/Q imbalance, iming offse, and Doppler (mobili) condiions. In addiion, we are comparing he performance of boh ransceiers in a realisic ADSL enironmen. [3] P. Tan and. Beaulieu, Precise Bi Error Rae Analsis of DCT OFDM in he sence of Carrier Frequenc Offse on AW Channels. In IEEE lobecom Conference, oember [4]. Al-Dhahir and J.M.Cioffi. Efficienl-Compued Reduced- Parameer Inpu-Aided MMSE Equalizers for ML Deecion: A Unified Approach. IEEE Trans. Informaion Theor, Page(s): , Ma 1996 [5] T. Kailah, A. Saed and B. Hassibi. Linear Esimaion. Prenice Hall, [6] ETSI ormalizaion Commiee. Channels for HIPERLA/2 in Differen Indoor Scenarios. Documen o. 3ERI085B. Sophia- Anipolis, France, [7] C. Fragouli,. Al-Dhahir and W. Turin. Training-Based Channel Esimaion for Muliple-Anenna Broadband Wireless Communicaions, IEEE Transacions, Page(s): , March [8] L. B. Milsein and R. A. Ilis. Signal Processing for Inerference Rejecion in Sad Specrum Communicaions. IEEE ASSP Magazine, Page(s):18 31, April Informaion Bloc IDCT Deeced Informaion Bloc Slicer Slicer Add Smmeric uard P/S 1-ap equalizer 1-ap equalizer Channel DCT noise Prefiler Remoe Smmeric uard Figure 1. DCT-MCM bloc diagram for baseband signaling S/P 5. Acnowledgemens The wor of. Al-Dhahir is suppored in par b ATP conrac and b SF conracs CCF and DMS References Fig 2. Achieable bi raes a differen frequenc offses [1]. Al-Dhahir, H. Minn, S. Saish, Opimum DCT-Based Mulicarrier Transceiers for Frequenc-Selecie Channels. To appear in IEEE Transacions on Communicaions, [2] J. Cioffi, A Mulicarrier Primer.In ASI T1E1.4 Commiee conribuion no Aailable online hp:// 179

6 Fig 3. Achieable bi raes wih esimaed channel a differen frequenc offses Fig 4. Achieable bi raes a differen jammer o noise power raios 180