An Efficient Equalization Method to Minimize Delay Spread in OFDM/DMT Systems

Transcription

1 A Efficiet Equalizatio Method to Miiize Delay Spread i OFDM/DMT Systes Roed Schur ad Joachi Speidel Istitut für achrichteübertragug Uiversität Stuttgart Pfaffewaldrig Stuttgart Geray Eail: schur@iue.ui-stuttgart.de Abstract A tie doai equalizatio techique to be used i a Orthogoal Frequecy Divisio Multiplex (OFDM or Discrete Multi Toe (DMT syste is proposed which directly iiizes the delay spread of the overall chael ipulse respose (CIR. It will be show that this techique ca provide superior perforace copared to recet developets of equalizers which are desiged to axiize the ipulse eergy i a certai widow. The perforace of the proposed ethod is deostrated for trasissio o local loop twisted pairs. The desig algorith requires low coputatioal coplexity ad is idepedet of the guard iterval legth. Siulatio ad aalytical results visualize the perforace of the ew equalizer ad copare it to those of a covetioal equalizer. I. ITRODUCTIO Multicarrier odulatio schees such as Orthogoal Frequecy Divisio Multiplex (OFDM or Discrete Multi Toe (DMT are curretly applied i ay applicatios e.g. digital audio broadcastig (DAB digital terrestrial broadcastig (DVB-T ad asyetrical digital subscriber lie (ADSL. I such systes a cyclically exteded guard iterval is iserted betwee successive ulticarrier sybols to avoid itersybol ad iterchael iterferece (ISI ICI. A chael ipulse respose (CIR exceedig the duratio of the give guard iterval will cause ISI ad ICI. To itigate sucects the idea is to feed the origial CIR ito a tie doai equalizer (TEQ such that the cascade of the coposite chael ad the equalizer yields a ipulse respose shorter tha the guard iterval. Recetly equalizers have bee developed which axiize the eergy of the overall CIR i a give widow []-[]. I the subsequet we call such a solutio a "eergy equalizer". We propose ad desig a ew equalizer which directly iiizes the delay spread of the overall CIR ("delay spread equalizer". II. SYSTEM MODEL AD DESIG PROCEDURE I this paper a discrete represetatio of a OFDM/DMT syste is assued. The followig paraeters are cosidered: The legth of the guard iterval represeted by a cyclic prefix is G saples. The size of the discrete Fourier Trasfor (DFT is saples with» G. Cosequetly there is a axiu uber of coplex odulated carriers i a DMT syste ad carriers i a OFDM syste. The tie base of the saples at the output of the trasitter is T A f A 8 MHz. Ipulse respose h ( of tie ivariat chael is of FIR type with legth K 4 saples. The TEQ is a FIR-filter with ipulse respose a ( of legth L F saples. The purpose of the TEQ is to iiize the delay spread of the overall CIR i a OFDM/DMT syste which is give by the covolutio h ( * a (. The parts of the overall CIR exceedig the guard iterval cause ISI ad ICI depedig o its distace to the guard iterval ad its eergy [3]. I cotrast to equalizatio techiques based o eergy cosideratios as desig criteria ("eergy equalizer" []-[] the "delay spread equalizer" icorporates the distace aspect. The delay spread D of a discrete tie fuctio is defied as D --- ( E ( The eergy E ad the tie ceter of are give by f A E H f eff e jπ f f A ( A df f A --- E (3 The overall CIR is give by the covolutio of h ( with a ( ad ca be writte i atrix otatio as ( ( ( K + L F Isertig ( ad (4 ito ( yields h( h( h( h( hk ( h( hk ( h( hk ( D ah H H Q H a (5 a H H H H a where Q is a diagoal atrix cotaiig the squared distaces of the saples of fro the tie ceter. H ( a( a( al ( F a (4

2 Q ( (( K + L F (6 D is take as cost fuctio ad has to be iiized. A good choice for the tie ceter of the effective CIR is orig where orig deotes the tie ceter of the origial CIR h (. The expressio i (5 leads to a quadratic optiizatio proble where the deoiator cotais the eergy of. Therefore the atrix product H H H results i a heritia positive defiite atrix ad a Cholesky decopositio ca be perfored. Chol{ H H H} L H L (7 The atrix L i (7 is a upper triagular atrix. Furtherore the substitutios v L a ad a L v (8 are itroduced. The isertio of (7 ad (8 ito (5 results i D vh W v v H v with W ( L H H H Q H L ( I atheatics the ter o the right had side of (9 is kow as Rayleigh-quotiet. It is idepedet of the absolute value of v ad rages betwee the sallest ad the largest eigevalue µ i ad µ ax of the heritia atrix W [4]. Cosequetly the iiu delay spread is give by D i µ i. ( To obtai D i v has to be chose as the eigevector correspodig to µ i. The coefficiet vector a of the "delay spread equalizer" ca the be calculated by usig (8. (9 III. SIMULATIO AD COMPUTATIO RESULTS We have aalyzed the perforace of the described delay spread equalizatio ethod i a DMT trasissio syste over a twisted pair subscriber loop. Hece the CIR h ( is assued to be real-valued. As a cosequece real-valued equalizer coefficiets are sufficiet. The results of the proposed equalizer are copared with those of the "eergy equalizer" [] []. I Fig. the CIR of a subscriber loop with 3 k legth ad 4 wire diaeter as well as the correspodig copressed ipulse resposes for both types of equalizatio are show. A cyclic prefix with G 6 saples is assued. The effective CIR obtaied by the "delay spread equalizer" is deoted by. I cotrast to that h deotes the effec- eff tive CIR obtaied by the "eergy equalizer". The copressig effect of the TEQ is clearly visible for both equalizatio techiques. The axiu legth G + of the CIR to avoid ISI/ICI i the DMT syste is arked with grey color i the subplots of Fig.. The delay spread of the overall CIR is about 4µs without tie doai equalizatio. It is reduced to µs with the "eergy equalizer" ad eve to 5µs with the proposed "delay spread equalizer" h ( Delay-Spread Equalizatio discrete tie Fig. : oralized CIR h ( of a subscriber loop (legth 3 k wire diaeter 4 without equalizatio. is the CIR with iiized delay spread ad is the copared CIR copressed with ethod []. A. Iterferece Power I the followig we give a approxiatio siilar to [3] to estiate the ISI/ICI power of the subchaels at the output of the DFT filter bak at the receiver. For this purpose we chose a widow of size G + that covers as uch eergy of the overall CIR as possible. The saples outside of the widow are deoted pre- ad postcursors. They cotribute to the ISI ad ICI power. Accordig to Fig. we defie the ipulse resposes w pre ϑ ( ϑ P ad w post ξ ( ξ P of the pre- ad postcursor which reai outside the widow of G + cosecutive saples. The ubers P ad P of such ipulse resposes are deteried by the legth of the cyclic prefix ad the overall CIR. Furtherore the axiu legth of w pre ϑ ad w post ξ is give by the size of the DFT. The trasitter sigal of the DMT syste ca be assued as zero ea ad white Gaussia process if early all carriers are odulated with ucorrelated sybols of equal power. The a approxiatio for the ISI/ICI power σ iter of the th subchael ( is give by (. Eergy Equalizatio Origial h (

3 w pre w pre w pre 3 precursor ISI ICI postcursor Fig. : ISI ad ICI due to pre- ad postcursor of the CIR at the DMT-receiver. G+ B. Ipact of Chael oise Tie doai equalizatio chages oise properties at the receiver. Parts of the DMT receiver are show i Fig. 4. We exaied the oise power σ oise at the th subchael output of the DFT filter bak i presece of zero ea additive white Gaussia oise (AWG with variace σ AWG o the chael as show i Fig. 4. eglectig the reoval of the guard iterval ca be calculated by (4 ad (5 [5]. ISI ICI w post w post σ iter ( j π zv ( e ( P z( --- w pre ϑ ( *w pre ϑ ( ϑ P (3 w post ξ ( *w post ξ ( ξ The covolutio i (3 is deoted by *. Fig. 3 shows the ISI/ICI power of the th subchael of a DMT syste for both equalizatio ethods. Trasissio over a twisted pair subscriber loop with 3 k legth ad 4 wire diaeter is cosidered. As a result the ISI/ICI power of all subchaels decreases by ore tha 3 db for both equalizatio techiques. A copariso with the siulated results i Fig. 3 shows that ( provides quite acceptable results. σ oise σ oise σ AWG cv ( e ( c( a( * a( ak ( ak ( k j π (4 (5 log( σ iter without TEQ (Calculatio without TEQ (Siulatio delay spread TEQ (Calculatio delay spread TEQ (Siulatio eergy TEQ (Calculatio eergy TEQ (Siulatio subchael idex Fig. 3: ISI/ICI power of subchael of a DMT receiver with G 6 ad 8. Subscriber loop with 3 k legth 4 wire diaeter TEQ with L F 6 coefficiets. th Accordig to (4 ad (5 σ oise depeds o the equalizer coefficiets a (. Siulatio ad aalytical results for the oise power σ oise are show i Fig. 5. As ca be see siulatio results cofir (4 quite well. Further Fig. 5 illustrates that the white chael oise becoes colored after tie doai equalizatio. As a cosequece the variace σ oise of the oise chages fro subchael to subchael. Chael σ AWG TEQ a( Fig. 4: DMT receiver with TEQ ad oisy chael. C. Sigal-to-distortio ratio As log as the guard iterval is of adequate legth the sigal power at the th output chael of the DFT is give by (6. It is assued that the trasitter odulates all carriers with equal power. σ sigal reove guard iterval λ DFT - σ oise FEQ (6

4 log( σ oise σ AWG without TEQ (Calculatio without TEQ (Siulatio delay spread TEQ (Calculatio delay spread TEQ (Siulatio eergy TEQ (Calculatio eergy TEQ (Siulatio subchael idex 35 3 log( γ without TEQ delay spread TEQ eergy TEQ subchael idex Fig. 5: Output oise power of subchael of a DMT receiver with G 6 ad 8 i presece of AWG at the iput of the receiver. TEQ with L F 6 coefficiets. The paraeter λ i (6 deotes the agitude of the overall trasfer fuctio of the th subchael specified by the chael ad the equalizer trasfer fuctio. The ISI/ICI power is assued to be sall copared to the sigal power at the receiver. Cosiderig the ISI/ICI power ( the oise power (4 ad the sigal power (6 the sigal-todistortio ratio γ at the output of the th subchael of the DMT receiver is give by (7 The sigal-to-distortio ratio γ with ad without tie doai equalizatio is show i Fig. 6 for trasissio over a twisted pair subscriber loop of 3 k legth. Apart fro the otches preset with the "eergy equalizer" γ is icreased i ost subchaels for both equalizatio ethods due to the reductio of ISI/ICI. However otches are preset i the γ fuctio whe the eergy equalizatio ethod is applied. I cotrast to that there occur o otches by the "delay spread equalizer". The effect of such otches is a perforace degradatio of the trasissio syste. As a easure for iproveet we use the geoetric ea of γ averaged over all subchaels. This value iproves fro 36 db without TEQ to 5 db with eergy equalizatio ad eve to 6 db with delay spread equalizatio. The ipact of o the achievable data rate is for further study. γ th λ γ zv σ j π AWG ( cv ( e ( Fig. 6: Sigal-to-distortio ratio γ at the output of the subchael of a DMT receiver with 8 G 6 ad σ AWG 6. Subscriber loop with 3 k legth 4 wire diaeter TEQ with L F 6 coefficiets. D. Ipact of sychroizatio error A critical proble of OFDM/DMT systes is give by sybol tiig at the receiver [6]. For this purpose the sesitivity agaist sychroizatio errors at the receiver is exaied for both equalizatio ethods. Fig. 7 visualizes the ISI/ICI power of the th subchael as a fuctio of sychroizatio offset by trasissio over a subscriber loop of 3 k legth whe equalizatio is doe with the covetioal eergy equalizer. log( σ iter subchael idex 63 Fig. 7: ISI/ICI power at the receiver due to sychroizatio offset by usig the eergy equalizatio ethod. DMT odulatio with 8 ad G 6. Subscriber loop with 3 k legth 4 wire diaeter TEQ with L F 6 coefficiets. th sychroizatio offset i saples

5 It ca be see that ISI/ICI power rises eorously whe sybol tiig is ot ideal. So we ca recogize that the syste with the "eergy equalizer" is rather sesitive agaist sychroizatio error. I cotrast to that the syste with the delay spread equalizatio ethod is relatively isesitive as it is show i Fig. 8. For various sychroizatio errors a direct copariso of the ISI/ICI power betwee both equalizatio ethods is illustrated i Fig. 9 as a cotour plot. log( σ iter subchael idex sychroizatio offset i saples Fig. 8: ISI/ICI power at the receiver due to sychroizatio offset by usig the delay spread equalizatio ethod. DMT odulatio with 8 ad G 6. Subscriber loop with 3 k legth 4 wire diaeter TEQ with L F 6 coefficiets. subchael idex Cotour lie : log( σ iter D-Spread σ iter Eergy db -db -db -db -db -db db -db db db db db db -db -db -db -db -db -db Fig. 9: Copariso of ISI/ICI power betwee "delay spread equalizer" ad "eergy equalizer" i the subchaels of a DMT receiver with 8 ad G 6 as a fuctio of sychroizatio offset. Subscriber loop with 3 k legth 4 wire diaeter TEQ with L F 6 coefficiets. db db db db db sychroizatio offset i saples Assuig ideal sybol tiig ISI/ICI power is lower i ost subchaels by usig the "eergy equalizer". However i presece of sychroizatio errors the ISI/ICI power with the "delay spread equalizer" rises uch less with icreasig sychroizatio offset tha with the "eergy equalizer". Cosequetly the pealty i syste perforace due to sychroizatio error is uch lower by usig the delay spread equalizatio ethod. IV. COCLUSIO We have desiged a equalizer which iiizes the delay spread i a OFDM/DMT syste. The solutio was applied o a twisted pair subscriber lie with 3 k legth ad 4 wire diaeter. As a result the equalizer ca reduce the ISI/ICI power i the DMT subchaels sigificatly. The solutio was copared to a "eergy equalizer" which axiizes the eergy of the overall CIR i a certai widow. It was deostrated that tie doai equalizatio chages oise properties at the receiver. Cosiderig oise ad ISI/ICI together as distortio the proposed "delay spread equalizer" ca achieve a higher sigal-to-distortio ratio tha the "eergy equalizer". Furtherore we have exaied the perforace of the ew equalizer i the presece of sybol sychroizatio offsets at the receiver. It was deostrated that the perforace degradatio due to sychroizatio errors is lower with the delay spread equalizatio ethod tha with the eergy equalizatio ethod. Already for a tiig offset of about saples the differece i ISI/ICI power betwee both equalizatio ethods is ore tha db i ost subchaels. REFERECES [] J. W. Melsa R. C. Youce C. E. Rohrs "Ipulse Respose Shorteig for discrete Mutlitoe Trasceivers" IEEE Tras. o Co. o. Dec. 996 [] R. Schur J. Speidel R. Agerbauer "Reductio of Guard Iterval by Ipulse Copressio for DMT Modulatio o Twisted Pair Cables" Proceedigs of IEEE GLOBECOM Sa Fracisco Vol. 3 pp ov. [3] J. L. Seoae S. K. Wilso S. Gelfad "Aalysis of Itertoe ad Iterblock Iterfereces i OFDM whe the legth of the cyclic Prefix is shorter tha the legth of the Ipulse Respose of the chael" Proceedigs of IEEE GLOBECOM 97 Phoeix Vol. pp ov. 997 [4] H. R. Schwarz H. Rutishauser E. Stiefel "uerik syetrischer Matrize" (i gera Teuber-Verl. 968 [5] S. Kapoor S. edic "Iterferece Suppressio i DMT Receivers usig Widowig" Proceedigs of IEEE ICC ew Orleas Vol. pp Jue [6] T. Pollet M. Peeters "Sychroizatio with DMT Modulatio" IEEE Couicatios Magazie pp o. 4 Apr. 999