Residual Synchronization Error Elimination in OFDM Baseband Receivers

Transcription

1 Resdual Synchronzaton Error Elmnaton n OFDM Baseband Recevers Xngbo Hu Yume Huang and Zhlang Hong It s well known that an OFDM recever s vulnerable to synchronzaton errors. Despte fne estmatons used n the ntal acquston there are stll resdual synchronzaton errors. hough these errors are very small they severely degrade the bt error rate (BER) performance. In ths paper we propose a resdual error elmnaton scheme for the dgtal OFDM baseband recever amng to mprove the overall BER performance. hree mprovements on exstng schemes are made: a plot-aded recursve algorthm for ont estmaton of the resdual carrer frequency and samplng tme offsets; a delay-based tmng error correcton technque whch smoothly adusts the ncomng data stream wthout resamplng dsturbance; and a decson-drected channel gan update algorthm based on recursve least-squares crteron whch offers faster convergence and smaller error than the least-mean-squares algorthms. Smulaton results show that the proposed scheme works well n the multpath channel and ts performance s close to that of an OFDM system wth perfect synchronzaton parameters. Keywords: OFDM error elmnaton offset estmaton tmng error correcton channel gan update. Manuscrpt receved Feb ; revsed May 007. hs work was supported n part by Intel Research Councl and n part by the Appled Materals Shangha Research & Development Fund under grant no Xngbo Hu (phone: emal: xbhu@fudan.edu.cn) Yume Huang (emal: yumehuang@fudan.edu.cn) and Zhlang Hong (emal: zlhong@fudan.edu.cn) are wth the State Key Laboratory of ASIC and Systems School of Mcroelectroncs Fudan Unversty Shangha P.R. Chna. I. Introducton he dgtal baseband recever one of the key parts n an OFDM system plays an mportant role n determnng the overall transmsson performance. Desgn and mplementaton of the OFDM recever have been topcs of ntense actvty for researchers and engneers snce OFDM was nvented. Numerous papers have been publshed to address ssues concernng OFDM recever desgn ncludng the ssue of synchronzaton for receved sgnals. An OFDM system s very vulnerable to synchronzaton errors and there are more strngent requrements for synchronzaton tasks n an OFDM recever than n others. In addton to the ntal estmaton for synchronzaton parameters further elmnaton of resdual synchronzaton errors (ncludng carrer-frequency and tmng errors or offsets) s absolutely necessary. he absence of an effcent error elmnaton scheme n the OFDM recever would severely degrade the overall system performance. Obvously how to desgn an effcent and robust elmnaton scheme for resdual synchronzaton errors s a crtcal ssue whle mplementng a dgtal OFDM recever. Error elmnaton s manly composed of two processng tasks error estmaton and error correcton []-[3]. Varous algorthms have been proposed to estmate both the carrer frequency offset (CFO) and the samplng tme offset (SO) usng phases or phase dfferences of sgnals at plot subcarrers [] [4]-[6]. Underlyng these algorthms s the least-squares (LS) lne-fttng method [5] or ts weghted verson [6]. In order to obtan good estmaton accuracy the number of plot subcarrers per OFDM symbol should be large enough; otherwse for many burst-type OFDM systems (such as IEEE 80.a/g WLANs and 80.6d/e BWA systems) due to very lmted number of plot subcarrers per OFDM symbol phase 596 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007

2 dfferences have to be averaged over many OFDM symbols for the sake of nose suppresson. However averagng across many symbols calls for large data storage and slows down the estmator s response to dmnshng resdual errors. Error correcton s also ndspensable for an error elmnaton scheme. A tme or frequency doman phase derotator s wdely used to correct carrer-frequency errors [] [7]. As for tmng error correcton t can be performed by adustng a resamplng numercally-controlled oscllator (NCO) [] [8]. However ths tmng error correcton desgn gnores that the drftng tmng phase caused by the SO would ump beyond sample boundares to nduce ether sample stuffng or rubbng. he large samplng dsturbance could crash the sgnal detecton. In ths paper we present an effcent resdual synchronzaton error elmnaton scheme for the dgtal OFDM baseband recever. It deals wth the CFO and SO nduced phase rotaton effects and also random phase noses amng to mprove the overall bt error rate (BER) performance va fne trackng of resdual synchronzaton errors and effcent dgtal error compensaton. It has the canoncal archtecture descrbed n [] and [6] but three mprovements on exstng schemes are proposed. We do not use an averagng-based LS lne-fttng method; rather we use a new recursve least-squares (RLS) algorthm for the ont CFO and SO estmaton. Instead of adoptng an NCO-based tmng error corrector a delay-based correcton technque s devsed to resample the ncomng data stream wthout samplng dsturbance. Our proposed scheme also ncorporates an effectve channel trackng loop as the further error elmnaton stage. hs channel trackng loop employs a new decson-drected effectve channel gan update algorthm based on RLS crteron other than the conventonal least- mean-square (LMS) crteron. hs paper s organzed as follows. In secton II the OFDM sgnal model wth synchronzaton mperfectons s descrbed n mathematcal expressons. Next the proposed resdual error elmnaton scheme s brefly ntroduced n secton III. hen three new algorthms or technques devsed for ths scheme are presented n detal n secton IV. Smulaton results demonstratng the performance of the proposed scheme are gven n secton V. Fnally secton VI concludes ths paper. II. OFDM Sgnal Model wth Synchronzaton Imperfectons Consder an OFDM system usng N-pont nverse fast Fourer transform (IFF) and FF for modulaton and demodulaton respectvely as shown n Fg.. Each OFDM symbol s composed of K u (< N) data symbols X k where denotes the OFDM symbol tme ndex and k denotes the subcarrer frequency ndex. he output of the IFF s dscrete tme wth samplng tme = u / N where u s the duraton of an OFDM symbol s useful part. hus accordng to [9] the baseband complex envelope for the transmtted OFDM sgnal can be expressed as Ku / π ( k/ u)( t g s) k s () = k= Ku / x() t = X e u( t ) where s s the duraton of an OFDM symbol g s the duraton of the guard nterval (or cyclc prefx CP) n an OFDM symbol and u(t) s the rectangular pulse shape functon wth unt ampltude durng 0 t s. X 0 X k X N- Y 0 Y k Y N- IFF FF P/S S/P x n CP nserton y n CP y(t) A/D removal me and frequency synchronzaton D/A Fg.. Baseband transmsson model for an OFDM system. x(t) e π fct he transmtted sgnal s corrupted n a multpath fadng channel wth addtve whte Gaussan nose (AWGN). At the recever sde the complex baseband sgnal s gven as n [9] by L l = h l (t) w(t) yt () = hl() t xt ( τ l) + wt () () where L s the total number of paths; h l (t) and τ l denote the complex gan and delay of the l-th path respectvely; and w(t) s the addtve whte Gaussan nose. he recever samples the sgnal y(t) wth the tme nterval of. Due to the exstng frequency dfference between transmtter and recever oscllators the receved n-th tme-doman sample of the -th OFDM symbol s gven by y = e y() t πδfct n t= ( n+ Ng + Nsym) ( n= N L N ) (3) where Nsym s the length of an OFDM symbol Ng s the length of the cyclc prefx and Δf C denotes the carrer frequency offset. hen synchronzaton CP strppng and FF demodulaton are performed sequentally on the receved sgnal. If the synchronzaton process s perfect we can have the receved complex frequency-doman data on the k-th subcarrer of the -th symbol Yk as g ERI Journal Volume 9 Number 5 October 007 Xngbo Hu et al. 597

3 Oversampled sgnal Decmator and tmng error corrector Baseband sgnal CORDICbased phase derotator y n FF Frequency doman sgnal One-tap FEQ Slcer Decded bts δ ε Plot-aded CFO trackng loop Plot extractor Decsondrected channel trackng loop Plot-aded tmng error trackng loop PI loop flter PI loop flter Jont CFO and SO estmator Channel gan updater Fg.. Archtecture of the proposed resdual synchronzaton error elmnaton. Y = X H + W (4) k k k k where Wk s the complex addtve nose on the correspondng subcarrer and Hk s the channel frequency response gven by H L k = hl ( s ) l= e πk ( τ l / u ). (5) However n practcal systems there s no deal synchronzaton and very small CFO wll reman even after the fnest estmaton process. Furthermore the SO nduced tmng error wll also cause dsturbances n the recever. If some resdual carrer frequency offset εδf (Δf denotes the subcarrer frequency spacng) and samplng tme offset δ exst the receved symbol Yk conssts of a sgnal term Sk an ntercarrer nterference (ICI) Ik and a nose term Wk [6] Y k S k + I k + W k =. (6) As gven n [6] the frequency offset parameter (φ qk ) the sgnal term (S k ) and the nterference term (Ι k ) can be expressed as φ = ( + δ ) ( ε + q) k (7) qk sn( πφkk ) Sk = Xk Hk Nsn( πφkk / N) e e π( N+ Ng + Ng ) φkk / N πφkk ( / N) (8) sn( πφ ) N / qk k = k k q= N /+ q k Nsn( πφqk / N) π( N+ Ng + Ng ) φqq / N πφqk ( / N) I X H e e III. Resdual Error Elmnaton Scheme As prevously mentoned both CFO and SO exst n. (9) practcal OFDM systems. For an OFDM recever good performance s acheved only by accurate offset estmaton and effcent dgtal compensaton. However there always remans more or less error for any estmaton algorthm f nose s present n receved sgnals. Resdual synchronzaton errors even f very small wll severely degrade the overall BER performance. he deleterous effects of the resdual CFO and the SO-nduced tmng offsets on recepton performance are carefully analyzed n [0]. o combat these negatve effects resdual synchronzaton offsets must be corrected va effcent fne trackng schemes. Fgure depcts the archtecture of our proposed resdual synchronzaton error elmnaton scheme. It s bult wth an FF processor a one-tap frequency-doman equalzer and a slcer. It s composed of three loops: a plot-aded tmng error trackng loop a plot-aded carrer frequency error trackng loop and a decson-drected channel gan trackng loop. he receved baseband sgnals wth resdual synchronzaton errors are corrected va the plot-aded error trackng loops. However tny errors (usually ε 0-3 δ 0 ppm) stll exst after the fne trackng process. In our scheme the decson-drected channel trackng loop s utlzed to further elmnate the remanng tny synchronzaton errors for BER performance mprovement because the channel estmator/updater cannot dstngush the small phase rotaton term e πφ k from the channel gan coeffcents Hk ( Ku / k Ku /). he whole error elmnaton structure conssts of several buldng blocks: the ont CFO and SO estmator the loop flters the tmng error corrector the phase derotator and the channel gan updater. he well-known CORDIC algorthm s used to mplement the phase derotaton. he proportonal-andntegral (PI) regulator s adopted here as the loop flter n both the carrer frequency and tmng trackng loops. Although our proposed archtecture s smlar to that n [] or [6] three new algorthms or technques are devsed to acheve mprovement over exstng systems: 598 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007

4 A plot-aded recursve algorthm for ont estmaton of the carrer-frequency and samplng tme offsets. Exstng systems wth a small number of plots per OFDM symbol usually adopt an averagng-based LS lne-fttng method or ts weghted verson for offset estmaton [6]. hs lne-fttng estmator has a very slow response to changng parameters as well as a large data storage requrement. he proposed estmator s based on the RLS algorthm whch can recursvely estmate resdual synchronzaton errors n an adaptve fashon. It also requres less storage than conventonal estmators. A delay-based tmng error correcton technque. Unlke the popular NCO-based tmng error correcton n [8] t can resample the ncomng data stream wthout ntroducng any samplng dsturbance. It s also easer to mplement than exstng technques. A decson-drected effectve channel gan update algorthm based on the RLS crteron. hs algorthm offers faster convergence and smaller error than the conventonal LMS algorthms. IV. Proposed Algorthms and echnques. Recursve Algorthm for Error Estmaton Error detecton or estmaton s vtal for an error trackng loop snce effcent correcton s based on a fne estmaton algorthm. An unbased estmate wth lttle varance s expected to be fed to the error correcton unt. In the desgn of OFDM error trackng loops ont estmaton of the carrer frequency and samplng tme offsets s preferred for the smple reason that the phase shfts of subcarrers caused by both offsets cannot be easly dstngushed from each other. As prevously mentoned an LS lne-fttng algorthm s popularly utlzed to estmate the CFO and SO from the extracted phase dfferences n the receved plot subcarrer sgnals between two consecutve OFDM symbols. For many OFDM systems where the number of plot subcarrers per OFDM symbol s lmted (namely four n 80.a systems and eght n 80.6d systems) the phase dfferences have to be averaged over many OFDM symbols n order to mprove estmaton. he more symbols used for averagng the more accurate the estmaton s the poorer the trackng ablty for dynamc parameters s and the more memory s requred (storage of all plot subcarrer sgnals n OFDM symbols used for averagng cannot be avoded). here s a trade-off between accuracy and complexty as well as dynamc performance for these averagng-based LS lne-fttng algorthms. In ths secton we propose a plot-aded RLS algorthm for ont estmaton of CFO and SO whch can adaptvely estmate dynamcally decreasng resdual errors n a recursve fashon. It can also acheve hardware effcency by greatly reducng data storage whle mprovng estmaton performance of the conventonal LS lne-fttng algorthms. In the appendx we show the detaled mathematcal dervaton of the proposed recursve algorthm. he complete algorthm s summarzed n Fg. 3. I) Intalzaton ( = 0). Set b0 = [0 0] and ρ I V0 = (where ρ s a small postve) then 0 N [ δ ε 0] = 0 π ( N N ) b. + g II) Iteratve computatons ( + ). ) Let b + = b and V+ = V. ) For = 0 J -repeat: a) Let ω = Yp / Xp q + = [ ω+ p ω+ ] ψ+ = ω+ ϕ+. b) Compute: K+ = ( w + q + V + q + ) V + q+ b + = b + + K+ ( ψ+ q+ b + ) V+ = ( V K q V ) /w where 0 < w <. 3) Let b + = b + J. 4) Compute the estmates of CFO and SO: N [ δ + ε + ] = ( ) b π N + N +. Fg. 3. Recursve algorthm for ont estmaton of CFO and SO. In addton to keepng all plot data n the last OFDM symbol for phase-dfference computatons ths recursve algorthm requres storage of only old values for the -element vector b and the -by- matrx V whle conventonal LS algorthms call for the unavodable storage of Nav J plot data (Nav s the number of OFDM symbols used for averagng and J denotes the plot number n an OFDM symbol). Generally J takes the value of 4 8 or a larger even number. o obtan good enough results Nav should be no less than 0. Obvously large storage savngs can be acheved by applyng the proposed recursve algorthm n ont estmaton of the carrer frequency and samplng clock offsets n dgtal OFDM baseband recevers. In order to evaluate the performance of the proposed recursve algorthm consder the mean behavor and the mean square behavor of the error vector Δ b() = b b (0) where b o s the optmum value for the offset vector (also the true value snce the phase nose e s zero-mean). he expectaton of the error vector can be derved from the recursve computaton formulas whch s gven by o g ERI Journal Volume 9 Number 5 October 007 Xngbo Hu et al. 599

5 ( ww ) E{ Δ b ( )} = ( ). () ρ ( ) QQ b o w Snce the forgettng factor w s a postve number less than E{ Δb ( )} 0 as. hat s to say the mean of the estmated offset sequences can always converge to ther true values. he mean-square devaton (MSD) can be expressed as D () = tr[ E{ Δb() Δb ()}] ( w) w = ρ tr[( QQ) bb( QQ) ] o o ( w ) + ( w)( w ) tr[( σ o QQ ) + ] ( + w)( w ) o () where σ s the mnmum varance of the phase nose e. From () t can be concluded that the MSD of the proposed recursve estmaton algorthm decays exponentally wth tme.. Delay-Based mng Error Correcton It s well known that tmng error correcton can be performed by means of nterpolatng among receved sgnal samples n dgtal recevers [8]. Usually the nterpolator also called an nterpolatng flter s controlled by an NCO whch offers nformaton needed to perform the nterpolatng computatons [8]. However ths conventonal error correcton technque would ntroduce resamplng dsturbance whch could crash the sgnal detecton. Here we present an effcent tmng error corrector based on a delay-varable buffer whch could overcome the weaknesses mentoned and also obvate the need for an NCO part. Fgure 4 s the block dagram of our proposed tmng error correcton structure. Snce the nterpolator can only adust fractonal tmng error a sample buffer wth adustable delay tme and a sample manpulator adoptng the skp/duplcate concept proposed n [] are mounted before and after the nterpolator respectvely to correct the ntegral tmng offset ontly. Control parameters or commands requred by these components are provded by the tmng phase controller whch derves the phase nformaton and determnes the sample Receved samples y r (m) Delay adustable buffer Integer offset z n Pecewse parabolc nterpolator Fractonal nterval µ n mng phase controller Estmated SO δ Skp/dupl manpulator Corrected samples Fg. 4. Structure of the proposed tmng error corrector. y(n) Mode select algnment modes based on the estmated value of the normalzed SO. Accordng to [8] the whole process for tmng error correcton can be mathematcally modeled as wth yn ( ) = y[( m + μ ) ] n n os I = yr mn os hi + μn os = I [( ) ] [( ) ] os (3) m = nt[ n / ] (4) n n os μ = n / m (5) where h I (t) denotes the mpulse response functon of the nterpolatng flter mn s the basepont ndex μn s the fractonal nterval s the tme nterval between two consecutve nterpolants and os s the oversamplng perod at the recever end. A four-pont pecewse-parabolc nterpolator s adopted n our proposed tmng correcton unt. For a gven desgn parameter α(0 α ) the nterpolatng flter s coeffcent formulas are C C C C 0 = αμ αμ = αμ + ( α + ) μ = αμ + ( α ) μ + = αμ αμ. n (6) Snce these coeffcents are functons of μn the nterpolator can be desgned usng the well-known Farrow structure []. he tmng phase controller does not use an NCO. It drectly updates the ntegral tmng offset and the fractonal nterval by smple computatons from ther old values as well as the estmate of normalzed SO δ and makes the current nformaton avalable to other components. he dervaton of the recursve formulas s descrbed here. wo successve nterpolatons are performed for tme nstants n and n+ and we can have the followng recurson: m n+ + n+ n os μ = m + / + μ. (7) Snce by defnton 0 μ n + < as gven n [8] the ncrement n sample count from one nterpolaton to the next s Δ m = m m = + μ. (8) n n+ n nt[ / os n] he value of / os s determned byδ. Consderng that /( = + δ ) and os = / Ros where Ros s the rate of over-samplng t can be derved that = R + δ. (9) / os os /( ) herefore (8) can be rewrtten as n 600 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007

6 Δ m = nt[ R /( + δ) + μ ]. (0) n os n From (7) and (0) we may conclude that μ n+ = μn + Ros /( + δ) Δ mn. () he ntegral tmng offset can be updated as zn+ = zn +Δmn Ros. () Equatons (0) to () consttute the recursve formulas for computng the ntegral tmng offset and the fractonal nterval. he delay-adustable buffer feeds the nterpolator wth a contnuous data stream free of effects of samplng tters by automatcally alterng the delay tme for output samples. When a postve ntegral tmng offset occurs the sample delay length s decreased. On the contrary a negatve nteger offset for the receved samples leads to an ncrease n the delay length. If there s no ntegral tmng offset samples are delayed by fxed clock cycles. A sample skp/duplcate manpulator mounted after the nterpolator s ndspensable n confnng the sze of the sample buffer wthn an acceptable range. It resets the ntegral tmng offset by skppng or duplcatng a baseband sample so that the buffer sze wll reman fnte n non-stoppng transmssons. It works n one of three modes: normal skp or duplcate and ths s determned by the control command ssued by the tmng phase controller. he delay length for the receved samples at the tme nstant n s gven by Dn = D fx zn (3) where Dfx s the fxed delay length when there s no ntegral tmng offset. Assume the nteger offset takes a value n the range of [Zmn Zmax] (Zmn s negatve whereas Zmax s postve). he sample buffer s sze should be Zmax Zmn. In order to make the buffer sze as short as possble t s necessary to confne zn to a narrow range. For example consder nteger regon [ Ros Ros ]. If 0 zn Ros the skp/duplcate manpulator wll stay n the normal mode. However f zn falls outsde of the range [0 Ros ] a sample of the CP at the start of the next receved OFDM symbol wll be skpped or duplcated and at the same tme the nteger offset wll be reset to an nteger n [0 Ros ] by addng or subtractng Ros from the orgnal offset. Snce the tmng phase offset accumulated durng an OFDM symbol perod does not exceed a baseband sample tme n all OFDM systems even for as large an SO as 00 ppm the range of [ Ros Ros ] would be wde enough to ensure proper processng. 3. Decson-Drected Channel Gan Update A channel gan update scheme s manly utlzed to combat the deleterous effects of varatons n the transmsson channel. Nevertheless t can also elmnate tny synchronzaton errors remanng even after the fne offset estmaton and trackng. In the followng a new decson-drected channel gan update algorthm s descrbed n detal. At the recever end the data symbol on the k-th subcarrer of the -th OFDM symbol s gven by Yk ( ) = Xk ( ) H( + Wk ( ) (4) where X( s the transmtted data symbol W( s the addtve whte Gaussan nose and H ε ( s the effectve channel gan coeffcents ncorporatng the true channel frequency response and the modfed phase nduced by tny synchronzaton errors. A one-tap frequency-doman equalzer (FEQ) equalzes the symbol Y( wth the estmated effectve channel coeffcents ( to yeld H ε ε X ( k ) = Yk ( )/ H ( k ). (5) e he equalzed symbol Xe ( s fed to the slcer for hard or soft decson and then re-mapped to yeld Xd (. Obvously Xd ( s the constellaton pont nearest to Xe (. If the recever works properly Xd ( can be regarded as the transmtted data symbol X(. hus (4) can be rewrtten as Y ( = X d ( Hε ( + W (. (6) he estmates of the effectve channel gan coeffcents H ε ( can be computed from (6) as well as ts prevous equatons. In dgtal communcaton systems the LMS algorthm s wdely used n adaptve equalzaton due to ts smplcty. However the LMS algorthm has a slow convergence property. In our scheme the RLS crteron s adopted to adaptvely update the effectve channel gan coeffcents n real tme snce t offers faster convergence and smaller error than the LMS algorthm despte some computatonal complexty. he recursve computaton formulas are descrbed as follows: K( + = λ + X d ε P( X d ( + (7) ( + P( X ( + P( + = [ P( K ( + X d ( + P( ]/ λ (8) and H ε ( + k ) = H ε ( k ) + K ( + k ) [ Y ( + X ( ) d + k Hε ( k )] (9) where ( ) denotes conugate operaton and the weghtng coeffcent λ (0 < λ < ) s called the forgettng factor whch determnes how the algorthm treats past data nput to the d ERI Journal Volume 9 Number 5 October 007 Xngbo Hu et al. 60

7 algorthm. he recursve computatons for the effectve channel gan coeffcents can start from the results of the ntal channel estmaton. he ntal values of P(0 can de determned by d (0 X d (0 )]. (30) P(0 = [ X k It s dffcult to perform mathematcal analyss for multvarable RLS algorthms. However snce our proposed update algorthm s a sngle-varable one the performance evaluaton n analytcal forms becomes easer. Frst evaluate ts convergence property. he expectaton of the estmaton error for the effectve channel frequency response on the k-th subcarrer durng the -th symbol perod s approxmated as where λ ( λ) R (0 ΔH(0 E{ ΔH( } λ σ + x d d R(0 = X (0 X (0 σ E{ X ( X ( }. x = d d (3) Snce 0 < λ < E{ ΔH( } 0 as for any k. herefore ths update process s asymptotcally convergent. Next consder the mean square behavor for ths RLS algorthm. Its MSD can be expressed as Dk E Hk H k ( ) = { Δ ( ) Δ ( )} λ ( λ) X(0 ( λ)( + λ ) σ xσ w ( λ ) σ x ( + λ)( λ ) σ x where σ w s the varance of the addtve nose W(. V. Numercal Smulatons (3) We provde several numercal examples to demonstrate the overall system performance of usng our proposed error elmnaton scheme. A burst-based OFDM system wth reference to the IEEE 80.6d standard [3] s used n our smulatons. In partcular we assume four prmtve parameters characterzng the OFDM symbol defned as follows. he nomnal channel bandwdth BW = 0 MHz the number of subcarrers used Nused = 00 the samplng factor n = 57/50 and the rato of CP tme to useful tme G = /8. Snce the 80.6d standard s for fxed broadband wreless access applcatons the SUI-4 channel model [4] wth the maxmum Doppler frequency of 0 Hz s adopted n our smulatons. Common smulaton condtons are the followng: the packet length (that s the number of OFDM symbols n a packet) s 00; the modulaton types are BPSK for plots and 6-QAM for data respectvely; the forgettng factor w and the value of ρ n the RLS-based resdual offset estmaton algorthm are set to and 0.0 respectvely; the forgettng factor λ n the effectve channel gan update algorthm s All statstcal smulaton results gven n ths secton are obtaned wth 0 4 Monte Carlo trals. Frst some smulaton trals are performed to ustfy the effectveness of the plot-aded error trackng loops. Fgure 5 shows a snapshot of the resdual synchronzaton error trackng convergence and steady-state performance at dfferent SNR levels. he ntal normalzed offsets for the carrer frequency and the samplng clock are 0.0 and 50 ppm respectvely. At SNR levels of greater than 0 db the convergence of both CFO and SO trackng can be acheved after approxmately 0 symbols and the curves fluctuate lttle n the steady state. Some trackng errors may reman n the steady state but they are small enough to be elmnated by subsequent channel trackng. he performance for trackng CFO and SO wll deterorate when the SNR value becomes lower. he trackng curves at a very low SNR level of 6 db are also gven for reference. he performance for CFO trackng s not poor whereas the SO trackng curve slowly fluctuates n a wde range. he performance of the RLS-based ont offsets estmaton algorthm s evaluated n comparson wth the ont weghted Normalzed CFO Normalzed SO (ppm) OFDM symbol ndex (a) 6 db db 8 db 4 db 6 db db 8 db 4 db OFDM symbol ndex (b) Fg. 5. Snapshot of the proposed resdual synchronzaton error trackng performance n multpath fadng channel at dfferent SNR levels: (a) carrer frequency offset trackng curve and (b) samplng tme offset trackng curve. 60 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007

8 least-squares (WLS) lne-fttng algorthm proposed n [6] and the lnear least-squares (LLS) lne-fttng algorthm proposed n [5]. We assume the number of OFDM symbols used for averagng Nav n each of the LLS and WLS algorthms s. he smulated estmaton RMS errors by three ont estmaton algorthms are shown n Fg. 6. In our smulatons the normalzed resdual CFO ε s fxed at the normalzed SO δ s fxed at -50 ppm and the SNR value changes from 9 db to 33 db. As can be seen n Fg. 6 the LLS algorthm s much poorer at estmatng both offsets than the other two algorthms and the proposed RLS estmaton outperforms the WLS algorthm by achevng an RMS error reducton of to 3 db. Fnally we nvestgate the overall BER performance of the proposed error elmnaton scheme n the multpath Raylegh fadng channel corrupted by AWGN. An SUI-4 model wth no Doppler effect or a Doppler frequency of 0 Hz s used n these smulatons. he whole archtecture s ncluded n a smulated RMS error (subcarrer spacng) RLS WLS LLS SNR (db) (a) RLS WLS LLS 80.6d baseband transcever system for performance evaluaton. Only 6-QAM modulaton s used and no codng s employed. A normalzed resdual CFO and a normalzed SO are assumed as 0.5% and 40 ppm respectvely. o demonstrate the benefts of the proposed scheme we consder three cases: full trackng (plot-aded trackng plus effectve channel gan update) partal trackng (only plot-aded trackng) and no trackng. Especally for the full trackng case performance comparsons are made between dfferent OFDM recevers employng LLS WLS and RLS algorthms to verfy the BER mprovement of the proposed recursve estmaton algorthms. he BER performance of an offset-free OFDM system wth perfect channel knowledge s also smulated for ntutve comparson wth that of systems employng full partal or no error trackng scheme. Note that before the decson-drected channel trackng scheme s performed channel coeffcents are ntally estmated n the followng manner. Frst the LS estmates of channel frequency response values are computed at even subcarrers by means of dvdng the receved plot symbols by the local known plots. hen lnear nterpolaton s performed between two adacent even subcarrers to get estmates of channel coeffcents at odd subcarrers. Fnally ths data s sent nto a fnte mpulse response (FIR) flter wth exponental-decay power to produce smoothed estmates of channel gans at each data subcarrers. Fgure 7 shows the uncoded BER performance curves for varous smulaton cases mentoned above n the same multpath fadng channel where there s no Doppler effect. Smlarly raw BER performances correspondng to a slowfadng channel wth a 0 Hz Doppler frequency are shown n Fg. 8. As Fgs. 7 and 8 demonstrate BER performance RMS error (sample) SNR (db) (b) Fg. 6. Estmaton RMS errors versus SNR by dfferent algorthms n multpath channels: (a) carrer frequency offset and (b) samplng tme offset. BER No trackng Partal trackng Full trackng (LLS) Full trackng (WLS) Full trackng (RLS) Error free SNR (db) Fg. 7. Raw BER performance versus SNR of OFDM systems wth and wthout error trackng compared wth that of an offset-free system wth perfect channel knowledge when the Doppler frequency s 0 Hz. ERI Journal Volume 9 Number 5 October 007 Xngbo Hu et al. 603

9 mprovement on conventonal schemes by adoptng LLS or WLS lne-fttng estmaton algorthms s acheved when the proposed RLS algorthm s employed. he recever also performs much better n the full trackng mode than n the partal trackng mode. From Fg. 7 as the Doppler frequency s set to zero we conclude that decson-drected channel trackng can work to further mprove the BER performance by effectvely elmnatng the tny CFO and SO remanng after the plotaded trackng. As the SNR gets hgher the mprovement become more obvous and t can reach 7 to 8 db n BER reducton at most. As shown n Fgs. 7 and 8 the overall BER performance of the proposed error trackng scheme s closest to that of an OFDM system wth perfect synchronzaton parameters especally at hgh SNR levels. Fnally f there s not any trackng scheme the performance wll become terrbly poor. herefore the effectveness of the proposed synchronzaton error elmnaton scheme has been well ustfed. BER No trackng Partal trackng Full trackng (LLS) Full trackng (WLS) Full trackng (RLS) Error free SNR (db) Fg. 8. Raw BER performance versus SNR of OFDM systems wth and wthout error trackng compared wth that of an offset-free system wth perfect channel knowledge when the Doppler frequency s 0 Hz. VI. Concluson In ths paper we proposed an effcent synchronzaton error elmnaton scheme to trace and remove the resdual carrer frequency and samplng tme offsets n the dgtal OFDM baseband recever. hree new algorthms or technques are devsed for ths scheme to acheve mprovement on exstng systems. A plot-aded recursve algorthm s used for ont estmaton of CFO and SO. It can acheve hardware effcency by greatly reducng data storage and also mprove estmaton performance compared wth conventonal LS algorthms. he delay-based tmng error correcton drectly updates the control parameters by teratve computatons. It can effectvely obvate the possble large samplng dsturbance whch could crash the sgnal detecton. he channel trackng loop employs a new decson-drected channel gan update algorthm based on the RLS crteron whch can acheve faster convergence and smaller error than the conventonal LMS algorthms. Smulaton results demonstrate that the new channel updatng technque can mprove BER performance by elmnatng tny synchronzaton errors remanng even after the fne offset trackng. he effectveness of the whole scheme has been ustfed by varous numercal smulatons and the overall BER smulaton has shown that the performance of the proposed error elmnaton scheme s close to that of a system wth perfect synchronzaton parameters. Appendx. Mathematcal Dervaton of the Proposed Recursve Algorthm As seen n (8) when both the carrer frequency and samplng clock offsets are present S k /H k wll be shfted n phase as gven n [6] by π ( ε + δk + δε)[( N + N + N ) / N + / N]. (A) Snce δ ε has a much smaller order of magntude than ε or δ ths term wll be gnored n the followng dervaton. Assume that J plots are nserted nto N subcarrers and these plot subcarrer ndexes are denoted by p = 0 J-. Plot data s dfferentally encoded wth a PN-sequence whch s known at the recever so we can have g X C X C { + }. (A) p = p p p For any and {0 J-} compute the phase dfference of the receved plot data between two consecutve OFDM symbols as gven n [6] as ϕ = arg( C Y Y ) p p p = arg( C ps ) ps p + e N + Ng = π [ p ] [ δ ε] + e N g (A3) where arg( ) s the phase of ts argument and e s the phase rotaton due to ICI and addtve nose. Pror to further dervaton we weght (A3) wth a coeffcent ω : ψ = ω ϕ N + N g = π [ ω p ω ][ δ ε ] + ω e.(a4) N Consderng that the phases of the subcarrers wth lttle fadng are more relable than those of the deeply faded subcarrers n transmsson the coeffcent ω can be set to the LS estmate of 604 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007

10 the channel gan of the relevant plot subcarrer: Y p ω =. (A5) X p By stackng (A4) for = 0 J - and expressng them n vector form we have where = [ ψ 0 ψ L ψ J ] ψ ψ = Qb + e (A6) ω0 p0 ω p L ω J pj Q = ω0 ω L ω J N + N g b = π [ δ ε ] N e [ 0 0 ] = ω e ω e L ω J ej. he prevous vector equaton s based on the plot data of the (-)th and -th OFDM symbols. hus all of the plots n the OFDM symbols pror to the (+)th symbol can be utlzed to form a larger vector equaton where Ψ = Rb+ E (A7) [ ψ ψ ψ ] Ψ = L [ Q Q Q ] R = L [ e e e ] E = L. hus accordng to (A7) the LS estmate of b s gven by b = ( R R ) R Ψ. (A8) On recevng the (+)th OFDM symbol we get a new vector equaton or where ψ (A9) + = Q+ b + e+ ψ + = q + b+ e + ( = 0 J ) ψ + = ω+ ϕ+ q + = [ ω+ p ω+ ] e + e+ + = ω. L (A0) Defne Ψ = Ψ R = R E = E and [ Ψ+ L ] Ψ + = ψ + [ R ] + R + = L q + [ E+ ] E + = L. e + Evaluate (A0) for = 0 J - respectvely and add them nto (A7) n turn to form new vector equatons. For any {0 J - } we have Ψ = R b E. (A) Expand the above equaton to yeld Ψ+ R = ψ + + q whch can also be denoted as E b + e = R + + b + E+ + (A) Ψ. (A3) Accordng to the RLS algorthm for a forgettng factor w (0 < w < ) the recursve formulas for estmatng b are gven by and where References = b + K + + ( ψ + + q+ + b + ) (A4) b+ + + V = / w ( V K q V ) (A5) K = ( w + q V q ) V q.(a6) [] M. Speth S. Fechtel G. Fock and H. Meyr Optmum Recever Desgn for OFDM-Based Broadband ransmsson Part II IEEE rans. Commun. vol. 49 no. 4 Apr. 00 pp [] S.P. Jmenez M.J. Garca F.J. Serrano and A.G. Armada Desgn and Implementaton of Synchronzaton and AGC for OFDM-Based WLAN Recevers IEEE rans. Consum. Electron. vol. 50 no. 4 Nov. 004 pp [3] S.J. Yang Y.C. Le and.d. Chueh Desgn and Smulaton of a Baseband ranscever for IEEE 80.6a OFDM-Mode Subscrber Statons Proc. 004 IEEE Asa-Pacfc Conf. on Crc. and Syst. vol. Dec. 004 pp [4] J. Lu and J. L Parameter Estmaton and Error Reducton for OFDM-Based WLANs IEEE rans. Moble Computng vol. 3 no. Apr.-June 004 pp [5] I.H. Hwang H.S. Lee and K.W. Kang Frequency and mng Perod Offset Estmaton echnque for OFDM Systems Electron. Lett. vol. 34 no. 6 Mar. 998 pp ERI Journal Volume 9 Number 5 October 007 Xngbo Hu et al. 605

11 [6] P.Y. sa H.Y. Kang and.d. Chueh Jont Weghted Least- Squares Estmaton of Carrer-Frequency Offset and mng Offset for OFDM Systems over Multpath Fadng Channels IEEE rans. Vehcular echnol. vol. 54 no. Jan. 005 pp. -3. [7] L. Kuang Z. N J. Lu and J. Zheng A me-frequency Decson-Feedback Loop for Carrer Frequency Offset rackng n OFDM Systems IEEE rans. Commun. vol. 4 no. Mar. 005 pp [8] F.M. Gardner Interpolaton n Dgtal Modems Part I: Fundamentals IEEE rans. Commun. vol. 4 no. 3 Mar. 993 pp [9] M. Speth S.A. Fechtel G. Fock and H. Meyr Optmum Recever Desgn for Wreless Broadband Systems Usng OFDM Part I IEEE rans. Commun. vol. 47 no. Nov. 999 pp [0] X. Wang.. hung Y. Wu and B. Caron SER Performance Evaluaton and Optmzaton of OFDM System wth Resdual Frequency and mng Offsets from Imperfect Synchronzaton IEEE rans. Broadcast vol. 49 no. June 003 pp []. Pollet and M. Peeters Synchronzaton wth DM Modulaton IEEE Comm. Mag vol. 37 no. 4 Apr. 999 pp [] L. Erup F.M. Gardner and R.A. Harrs Interpolaton n Dgtal Modem Part II: Implementaton and Performance IEEE rans. Commun. vol. 4 no. 6 June 993 pp [3] IEEE 80.6a-03/0 IEEE Standard for Local and Metropoltan Area Networks Part 6: Ar Interface for Fxed Broadband Wreless Access Systems June [4] IEEE Std Channel Models for Fxed Wreless Applcatons IEEE 80.6 Broadband Wreless Access Workng Group echncal Reports 003. Xngbo Hu receved the BS degree n electrcal engneerng from Wuhan Unversty Hube Chna n 998 and the MS degree n electrcal engneerng from snghua Unversty Beng Chna n 00. He s currently workng towards a PhD degree n mcroelectroncs at Fudan Unversty Shangha Chna. Hs research nterests nclude desgn and VLSI mplementaton for dgtal wreless transcevers sgnal processng algorthms nvolved n wreless communcatons and RF-baseband co-desgn. Yume Huang receved the BS and MS degrees from Huazhong Unversty of Scence and echnology Hube Chna n 993 and 996 respectvely and the PhD degree from Fudan Unversty Shangha Chna n 004. She s currently an assstant professor at the Department of Mcroelectroncs Fudan Unversty. Snce 00 she has partcpated n the development of CMOS RF transcever and baseband crcuts for Bluetooth and WLAN 80.b/g. Her research nterests nclude CMOS RF/analog IC and RF system desgn for wreless communcaton and hgh frequency nose optmzaton. Zhlang Hong receved the BS degree n physcs from the Chnese Unversty of Scence and echnology Beng Chna n 970 and the PhD degree from EH Zurch Swtzerland n 985. Snce 996 he has been wth the Department of Electronc Engneerng Fudan Unversty Shangha Chna where he s currently a professor and vce drector of the State Key Laboratory of ASIC and Systems. From 99 to 994 as a vstng scholar he vsted the Department of Informaton Processng of Hannover Unversty Germany. Hs research nterests nclude hgh-performance analog and mxed ntegrated crcuts desgn and RF system desgn for dgtal wreless communcaton. 606 Xngbo Hu et al. ERI Journal Volume 9 Number 5 October 007