THERE has recently been considerable interest in Orthogonal

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1 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 A Robust Tmng Synchronzaton Desgn n OFDM Systems Part I: Low-Moblty Cases Yasamn Mostof, Member, I and Donald C. Cox, Fellow, I Abstract In ths paper we are nterested n desgnng a robust tmng synchronzaton algorthm for OFDM systems that utlze plot-aded channel estmaton. We frst characterze the mpact of tmng errors on the performance of a plot-aded OFDM system. We derve analytcal expressons for average channel estmaton error varance n the presence of tmng errors n hgh delay spread fadng envronments. The derved expressons show that plot-aded channel estmators are consderably senstve to tmng synchronzaton errors due to the mpact of rotatons n dfferent bases. We then show how to utlze ths senstvty to desgn a robust tmng synchronzaton algorthm, wthout tranng overhead. The proposed algorthm s a cross-block desgn that uses channel estmaton nformaton to mprove tmng synchronzaton. We confrm our analytcal results by smulatng the proposed algorthm n hgh delay spread fadng envronments. In ths paper,.e. Part I, we focus on tmng synchronzaton for low-moblty cases. The analyss and results are then extended to hgh-moblty applcatons n Part II. Index Terms Channel estmaton, orthogonal frequency dvson multplexng (OFDM), tmng synchronzaton. I. ITRODUCTIO THR has recently been consderable nterest n Orthogonal Frequency Dvson Multplexng (OFDM) systems. In OFDM systems the gven bandwdth s dvded nto narrow sub-channels. By transmttng low data rates n parallel on these sub-channels, OFDM systems can handle hgh delay spread envronments 4, 0,. By addng a guard nterval to the begnnng of each OFDM symbol, the effect of delay spread (provded that there s perfect synchronzaton) would appear as a multplcaton n the frequency doman for a tmenvarant channel. Ths feature allows for hgher data rates and has resulted n the selecton of OFDM as a standard for Dgtal Audo Broadcastng (DAB 5), Dgtal Vdeo Broadcastng (DVB 6), and wreless local area networks (80.a). The performance of OFDM systems, however, s senstve to the performance of the tmng synchronzer. Tmng synchronzaton, n ths paper, refers to the correct detecton of the start of an OFDM symbol. Tmng synchronzaton errors degrade the performance of an OFDM recever by ntroducng Inter-Carrer-Interference (ICI) and Inter-OFDM Symbol-Interference (ISI). Research n tmng synchronzaton Manuscrpt receved May, 006; revsed December 9, 006; accepted May 30, 007. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was C. Tellambura. Y. Mostof s wth the Department of lectrcal and Computer ngneerng, Unversty of ew Mexco, Albuquerque, M, 873 USA (e-mal: ymostof@ece.unm.edu). D. C. Cox s wth the Department of lectrcal ngneerng, Stanford Unversty, Stanford, CA, USA (e-mal: dcox@spark.stanford.edu). Dgtal Object Identfer 0.09/TWC.007.xxxxx /07$5.00 c 007 I of OFDM systems has manly focused on fxed wreless applcatons and can be prmarly categorzed nto two man groups: tranng-based and correlaton-based. The frst group s based on transmttng tranng nformaton for synchronzaton 3, 0,,, 7. Muller et al. has provded a good survey and comparson of such algorthms. In these algorthms, there s a waste of bandwdth n transmttng the tranng data. In the algorthm proposed by Schmdl et al. 7, half of the subcarrers are occuped by known nformaton for tmng synchronzaton. Ths nserted frequency-doman tranng nformaton results n repetton n tme doman, whch s then exploted for tmng synchronzaton n the recever. Smlarly, Mnn et al. 0 proposes a synchronzaton algorthm n the same category as Schmdl et al.,.e. based on usng tranng nformaton. ssentally, n ther case, most of the subcarrers are devoted for transmttng tranng nformaton. They further do lmted performance mprovement by estmatng the delay of the frst channel path. The second category s based on usng the redundancy of the cyclc prefx 6, 9. Then the start of the symbol s where the correlaton of the start and end data ponts s maxmzed. In the absence of delay spread, ths would work fne. However, n the presence of delay spread, the cyclc prefx would be affected by the prevous OFDM symbol resultng n performance degradaton. There are also blnd algorthms that attempt to perform tmng synchronzaton by usng cylcostatonary propertes, wthout relyng on the cyclc prefx or tranng nformaton, 4. The processng delay and computatonal complexty of these algorthms, however, can be hgh, makng t unfeasble for hgh delay and Doppler spread envronments. There are other methods that use cyclc prefx for coarse synchronzaton followed by a fne tunng based on an mpact on channel estmaton, 8. These algorthms are more n the same category as ours n the sense that they do not rely on tranng nformaton. Yang et al. uses the plots nserted for channel estmaton to perform coarse tmng synchronzaton. They fne tune ther algorthm by tryng to estmate the delay of the frst channel path. In dong so, they assume that the frst channel path s the strongest, whch may not be the case. Furthermore, they need to do averagng over a number of OFDM symbols, whch ncreases the delay. In our case we are nterested n tmng synchronzaton by usng the nformaton of only one OFDM symbol. Furthermore, we are nterested n synchronzaton n the presence of channels that do not necessarly have the frst path the strongest. Speth et al. 8 proposed usng the redundancy of the cyclc prefx for coarse synchronzaton followed by fne tunng the performance by

2 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 mnmzng the followng metrc: l plots z l p l H(l), where z l denotes the frequency doman l th receved sample, p l s the plot at l th subcarrer and H(l) s the estmated channel at l th subcarrer. In ths paper we take a dfferent approach. We are nterested n desgnng a robust tmng synchronzer that does not rely on synchronzaton tranng nformaton, for plot-aded OFDM systems n hgh delay spread envronments. We frst analyze the mpact of tmng synchronzaton errors on the performance of a plot-aded channel estmator by dervng an analytcal expresson for channel estmaton error varance n the presence of tmng errors and for a frequency-selectve fadng channel. We fnd that for a plot-aded channel estmator, the equvalent channel and the estmated equvalent channel have rotatons n dfferent bases. Our mathematcal dervatons wll show that, due to ths effect, a plot-aded channel estmator s consderably senstve to tmng synchronzaton errors. We then show how to utlze ths senstvty to desgn a robust tmng synchronzaton algorthm, wthout tranng overhead. The proposed algorthm s a cross-block desgn that uses channel estmaton nformaton to mprove tmng synchronzaton. In ths paper,.e. Part I, we focus on tmng synchronzaton n low-moblty envronments, where channel can be taken constant over one OFDM symbol. The analyss and results are then extended to hgh-moblty applcatons n Part II. We conclude ths secton wth an overvew of the paper. In Secton II we summarze system model of an OFDM system that utlzes plot-aded channel estmaton assumng perfect synchronzaton. After brefly revewng the mpact of tmng synchronzaton errors on an OFDM system n Secton III, we derve analytcal expressons for average channel estmaton error varance of a plot-aded channel estmator n the presence of tmng errors n Secton IV. The results of Secton IV show the super-senstvty of a plot-aded channel estmator to tmng synchronzaton errors. In Secton V, we then show how to utlze ths senstvty to desgn a tmng synchronzaton algorthm that works robustly even n consderably hgh delay spread fadng envronments such as those experenced n Sngle Frequency etworks (SF). Secton VI shows the performance of the proposed synchronzer n hgh delay spread fadng envronments. It also provdes a comparson wth other exstng algorthms. Secton VII summarzes the results and brefly dscusses possble future extensons. II. IDAL CAS: PRFCT SYCHROIZATIO In ths secton we summarze the system model of an OFDM system that utlzes a plot-aded channel estmator assumng perfect synchronzaton. A. System Model Consder an OFDM system n whch the avalable bandwdth s dvded nto sub-channels and the guard nterval spans G samplng perods. represents the transmtted data pont n the th frequency sub-band and s related to the tme doman sequence, x k, as follows: = x k e jπk 0. () x pf s the cyclc prefx vector wth length G and s related to x as follows: x pf () =x G 0 G. () Let T represent the length of one OFDM symbol after addng the guard nterval. Then T s = T G represents the samplng perod. We assume that the normalzed length of the channel delay spread (normalzed to the samplng perod) s always less than or equal to G. Leth k represent the k th channel path. h k has Raylegh fadng ampltude and unformly dstrbuted phase. In ths paper, we take the channel to be constant over one OFDM symbol and leave the case of hgh moblty OFDM to Part II. Let y represent th sample of the channel output. We wll have: y = { ϑ }} { G h k x (( k)) w 0, (3) where (( )) represents a cyclc shft n the base of and w represents a sample of addtve whte Gaussan nose. Then Y, the FFT of sequence y, wll be as follows: Y = H W 0, (4) where H and W denote the FFTs of the sequences h and w. To retreve the transmtted data ponts from q. 4, the gan and phase of the channel should be estmated. In ths paper, we use frequency-doman plot tones for channel estmaton. B. Plot-Aded Channel stmaton Let ν G represent the maxmum predcted normalzed length of the channel delay spread. Therefore, only L = ν equally-spaced plot tones, plot (l ) for 0 L, are needed to estmate channel frequency response 3, where l = L. Then, Ĥ l = Y l plot (l ) = H l W l plot (l ) 0 L, (5) where Ĥl s the estmate of the channel at the l th sub-carrer. Through an IFFT of length L, the estmate of the channel n tme doman would be ĥ k = L L Ĥ l e jπk L 0 k L. (6) =0 Then through an FFT of length, the estmate of the channel at all the sub-carrers wll be L Ĥ = ĥ k e jπk 0. (7) quvalent of these operatons can also be characterzed as a frequency-doman nterpolaton. See 9 for detals.

3 MOSTOFI AD CO et al.: A ROBUST TIMIG SYCHROIZATIO DSIG I OFDM SYSTMS PART I: LOW-MOBILITY CASS 3 III. FFCT OF TIMIG SYCHROIZATIO RRORS In ths paper, we take the tmng error to be a multple of the samplng perod to ease mathematcal dervatons. The analyss and results can be extended to the sub-sample level usng the models developed n 8. Consder a case of a tmng error of m samplng perods. In ths paper, m>0 and m<0 denote tmng errors of m to the rght and left sdes respectvely. We assume that m <L(or equvalently m <Gsnce G and L are both chosen based on the maxmum predcted channel length) n ths paper. In ths secton, we wll characterze the mpact of tmng synchronzaton errors on an OFDM system. A. Case of Tmng rrors to the Rght (m >0) In ths case an error of m samplng perods to the rght sde has occurred. Then, the terms y 0,y,...,y m are mssed and nstead m data ponts of the next OFDM symbol are erroneously selected. The receved sgnal can thus be wrtten as follows: y r = ϑ ((m)) γ r s w r 0, (8) where y r s th sample of the receved sgnal for m > 0, ϑ s as defned { n q. 3, w r s a sample of 0 0 m AWG, s = ypf next and γ ( m) else r = { 0 m 0 m. ypf next () for 0 m represents the th sample of the output cyclc prefx ofthe next OFDM symbol (excludng the effect of AWG): ICI {}}{ C ypf next () = h k x k k = { ISI }} { k =0 h k x next G k, (9) where C ν represents the normalzed length of the channel delay spread and x next denotes the tme-doman transmtted data of the next OFDM symbol. Then Y r, the FFT of yr, wll be as follows for 0, Y r = Γr 0 H e jπm I r { }} { Γ r k H (( k)) (( k)) e jπm( k) S k= } {{ } ICI & ISI W r (0) Where S s the FFT of s and Γ r, the FFT of γr, s Γ r = e jπm 0 e jπ.then I r S represents the m =0 average nterference power, ncludng both ICI and ISI, where z represents average of z for an arbtrary z (we also use (z) to ndcate average of z, dependng on the formula). We next summarze an analytcal expresson derved n 8 for the average Sgnal to Interference Rato, as we wll use t n the subsequent sectons. Theorem : (from 8) The average Sgnal to Interference Rato for m>0, SIRave r = Γr 0H/, wll be as follows: I rs SIR r ave = ( m) ( m)m m σ H m C k =k σ h k () where σh = H khk = C =0 σ h wth σh representng the powerofthe th channel path, and Hk ndcatng conjugate of H k. Proof: See 8 for proof. B. Case of Tmng rrors to the Left (m <0) In ths case, due to the presence of the cyclc prefx, the number of data ponts that are mssed can be less than m. If the length of the channel delay spread spans C samplng perods, only d = max(c (G m), 0) data ponts are corrupted due to the nterference from the prevous symbol. Therefore, y l = ϑ ((m)) γ l ψ w l 0, () Where y l s { the th sample of the receved sgnal for m< ypf (G m ) 0 d 0, ψ = wth y 0 d pf () representng the th sample of the output cyclc prefx of the current { OFDM symbol (excludng the effect of AWG), 0 0 d γ l = d and w l s a sample of AWG. Then Y l, the FFT of yl, wll be, Y l = Γl 0 H e jπm I l { }} { Γ l k H (( k)) (( k)) e jπm( k) Ψ W l, k= } {{ } ICI & ISI (3) where Ψ s the FFT of ψ and Γ l, the FFT of γl, s Γ l e jπd 0 = e jπ. Then, smlar to the m > 0 d =0 case, average Sgnal to Interference Rato, SIRave, l wll be as follows: Theorem : (from 8) The average Sgnal to Interference Rato for m<0, SIRave l = Γl 0H/, wll be as follows: I lψ SIR l ave = ( d) ( d)d d σ H Proof: See 8 for proof. d Gmk k σ =0 h k IV. FFCT OF TIMIG SYCHROIZATIO RRORS O PILOT-AIDD CHAL STIMATIO (4) A. Case of m>0 In ths secton we explore the effect of tmng errors on the performance of a plot-aded channel estmator dscussed n Secton II-B. Consder the case of m > 0. Let

4 4 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 h m h 0 h h L 0 L- Fg. a Orgnal channel of normalzed length L h L h 0 h h m 0 L-m- -m - 0 -m L- L-m- Fg. b quvalent channel (m ) Fg. d quvalent channel (m ) h m h 0 h L h h m h L 0 L-m- L- 0 -m L- Fg. c stmated eq. ch. (m ) Fg. e stmated eq. ch. (m ) h 0 h h Lm h Lm h 0 h h Lm h L Fg.. ffect of rotatons n dfferent bases, the equvalent channel s rotated n base, whereas the estmated equvalent channel s rotated n base L. H eq () = k h eq(k)e jπk represent the relatonshp between the transmtted data pont at the th sub-carrer,, and the receved data pont, Y r n ths case. Usng q. 0, Heq() r = Γr 0 H e jπm and h r eq(k) = Γr 0 h ((km)). (5) rotaton n base As can be seen, a tmng synchronzaton error of m > 0 ntroduces a rotaton of m samplng perods n the base of n the equvalent channel. Ths rotaton wll result n the expanson of the channel beyond ts maxmum predcted length. To see ths, Fg. b shows the equvalent channel for the orgnal channel of length L shown n Fg. a (L s defned n Secton II). As can be seen, a rotaton has occurred and resulted n the expanson of the equvalent channel beyond ts maxmum predcted length. ven an error of one samplng perod to the rght sde results n an equvalent channel of length. Ths wll degrade the performance of the channel estmator, as t assumes an equvalent channel that spans L samplng perods at maxmum. To see the effect of tmng errors on channel estmaton analytcally, consder the case that L = ν equally-spaced frequency-doman plot tones, plot (l ) for 0 L, are nserted among the subcarrers where l = L. Then for 0 L, Ĥeq(l r Yl r )= plot (l ) = Hr eq(l ) Ir l S l Wl r. (6) plot (l ) After an IFFT of length L, the estmate of the channel n the tme doman wll be Defne U r ĥ r eq (k) = Γr 0 h ((km)) L and V r rotaton n base L as follows: u r k vk r. (7) Interference AWG L U r = I r L l α z S lz,z & V r Wl r = α z,z z=0 plot (l z ) z=0 plot (l z ). (8) Then u r and v r wll be the IFFTs of U r and V r respectvely and α,z = L L g=0 ejπg( z L ). By comparng q. 7 wth q. 5, t can be seen that there are three factors contrbutng to the channel estmaton error: rotatons n dfferent bases, nterference and nose. The frst factor occurs because the equvalent channel has a rotaton n the base of whle the estmated equvalent channel has a rotaton n the base of L. SnceL s chosen based on the maxmum predcted length of the orgnal channel, ν G, t s typcally consderably smaller than. Therefore, the msmatch between the equvalent channel and the estmated equvalent channel can be consderable, solely due to the frst factor. Fg. c shows the estmated equvalent channel for the equvalent channel of Fg. b (effect of nterference and nose s not shown on the fgure). By comparng Fg. b and c, a msmatch can be observed n the locaton of the frst m paths of the orgnal channel. Snce these samples are typcally strong, ths can result n a consderable performance degradaton of the channel estmator. To analytcally assess the contrbuton of each of the aforementoned factors, we next derve an analytcal expresson for the average channel estmaton error varance. Channel estmaton error wll be as follows for 0, ΔH r eq () = m β r,k h k rotatons n dfferent bases U r nterference V r AWG, (9) where ΔHeq r represents the frequency-doman channel estmaton error for m>0 and β,k r = m e jπ(k m) ( e jπl ). Theorem 3: Consder an OFDM system wth a tmng synchronzaton error of m samplng perods to the rght. Let Ch r error,norm () represent the normalzed average channel estmaton error varance of the plot-aded channel estmator dscussed n Secton II: Ch r error,norm () = ΔHr eq (). Then Heq r () we wll have, Ch r error,norm () = 4 m m σ k= h k kσ h k m C σ h k ϖ(m ) sn ( πl ) } {{ } factor#: rotatons n dfferent bases SIR r ave } {{ } nterference SR }{{ ave r, (0) } nose where SRave r = ( m) σ σ H and ϖ(z) =for z 0 σw and zero otherwse. σh and C are as defned n Secton III, j = σ δ j and σw s the varance of W. Proof: Usng the defnton of I l r z I r lz h k plot (l z) from q. 0, we wll have = Γ r k k = H ((l z k )) h ((lz k )) k plot (l z) e jπm(lz k ) for 0 z L and 0 k m. otng that plot (l z ) = lz wll result n ((lz k )) plot (l z) = 0 for It should be noted that the estmated equvalent channel s not a rotated verson of the equvalent channel. Therefore ther Fourer transforms,.e. H r eq and Ĥr eq, dffer both n ampltude and phase.

5 MOSTOFI AD CO et al.: A ROBUST TIMIG SYCHROIZATIO DSIG I OFDM SYSTMS PART I: LOW-MOBILITY CASS 5 k and d s. Therefore, I r lz h k =0 plot (l z ) for 0 z L 0 k m. () Then we wll have the followng usng q. 8, L U rh k = Slz h k α,z = z=0 plot (l z ) C e jπ(k m)z L = m L h k h k k =0 k =k z=0 m C h k h k k =0 k =k z=0 α,z x k k plot (l z ) L m k = α,z e jπ(k m)z (k k )z L = k h k h k m L z=0 C k =m h k h k α,z e jπ(k m)z L. () Snce we are nterested n U rh k for 0 k m, the second term n the bracket s zero. Therefore, { U rh k = k L σ h k z=0 α,ze jπ(k m)z L 0 k =0 k (3) and = m β,k r U rh k = m m L k= kσ h k β r k= L L,k z =0 z=0 L kσh k β,k r z=0 α,z e jπ(k m)z L e jπ(z km)z L e jπz. (4) Snce z k m L m, q. 4 wll be nonzero only f z k m = L and m. Snceforany k n the gven range of q. 4, 0 L k m L for m L, there wll always exst a z that wll make z k m = L. Therefore, m,k U rh k = m k= kσ h k β,k r { jπ(lk m) e ( m) (e jπl ) βr for m. Then, m β r,k U r h k = m k= kσ h k m 0 m = (5) otng that the Gaussan nose term s ndependent of the frst two terms on the rght hand sde of q. 9 results n the followng expresson: derved next. Ilz r Ir l z plot (l z) plot (l z ) can be easly characterzed for PAM modula- m m ΔHeq() r = β,k r σh k σu r σ V rr{ β,ku r rh k }, (6) where R{.} represents the real part of the argument. We wll have β,k r =4 ( m) sn ( πl ), usng the expresson of β r from q. 9, and σv = σ r W σ. An expresson for σ U s r tons. σ U r = L L z=0 z =0,z z α (I r,zα,z L ext we calculate I r lz Ir l z plot (l z)plot (l z ) nvolves ((lz k)) ((l z k )) plot (l z)plot (l z ) z=0 α,z Ir lz S lz σ lz S lz ) (Ir l z S l ) z plot (l z)plot (l z ). (7) I r lz Ir l z plot (l z). plot (l z ) evaluatng for k, k and z z. For k, k and z z, we wll have ((l z k)) l z, ((l z k)) l z and l z l z. Therefore, If ((l z k)) = ((l z k )), ((lz k)) ((l z k )) plot (l z)plot (l z ) = ( ((lz k)) ) l z lz for d s (note that plot (g) = g ). For zero-mean constellatons (such as PSK or QAM), t can be easly verfed that lz = 0 (for nstance n QAM, f s a pont n the constellaton, e jπ s also a pont n constellaton). Therefore, ((lz k)) ((l z lz s zero resultng n k )) plot (l z)plot (l z ) = 0. Lets consder the cases where ((l z k)) ((l z k )). If ((l z k)) = l z and ((l z k )) l z or ((l z k )) = l z and ((l z k)) l z, then t can ((lz k)) ((l z be easly verfed that k )) plot (l z)plot (l z ) = 0. If ((l z k)) = l z and ((l z k )) = l z, then ((lz k)) ((l z k )) lz lz plot (l z)plot (l z ) = l z lz. It can be easly verfed that s zero for MPSK modulatons wth M > and QAM modulatons. To see ths dvde the constellaton ponts nto sub-groups wth equal ampltudes. It can be easly shown that wll be zero for each sub-group. ote that for PAM modulatons equals one. However, snce PSK and QAM modulatons are commonly used, we wll proceed wth that assumpton to facltate mathematcal dervatons ((lz k)) ((l z. Therefore, k )) plot (l z)plot (l z ) =0and I r lz Ir l z plot (l z)plot (l z ) =0. ext the followng expresson can be wrtten for g g, I r g S g g = m C g k =k k = Γ r k H ((g k )) h ((g k )) x k k k g e jπg (k m)m(g k ). g Snce k, we wll have the followng for g g, ((g k )) x k k g = g k k ((g k )) =0 g e jπk (k k ) =0. g (8) (9)

6 6 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 Therefore, I r g Sg g g =0 g g. (30) Then q. 7 can be wrtten as follows: L L σu r = α,z α,z S lz Sl z plot (l z )plot (l z ) z=0 z =0,z z L α,z Ir l z S lz σ. (3) z=0 For an arbtrary k and g where k g, the followng expresson can be wrtten, S k S g k = g. m s mke jπk (k m) m g=0 s mg e jπg (g m) k g (3) Usng the expresson of s from Secton III and notng that the ICI and ISI terms of q. 9 are ndependent, m m g=0 k g h k h g x k k x g g m m g=0 S k S g k g = C C k =k g =g e jπ(k m)k (g m)g = C k =max(k,g) σ h k e jπ(k m)(k g ) for k g. (33) Therefore, the frst term on the rght hand sde of q. 3 can be wrtten as follows: L L z=0 z =0,z z α,zα,z S lz S l z plot (l z)plot (l z ) = m C k =max(k,g) σ h k m L g=0 L L jπ(g g ) g =0 g =0 e L z=0 e jπz(g k m) L L z =0 L z=0 e jπz(g g ) L. jπz (g k m) e L (34) Snce L < C m g k m L m, then L < C m g k m L for m L. Therefore, the frst two sums nsde the bracket wll have non-zero values only for g k m = L and 0. In order to have g k m =0and g k m = L, 0 k m L and L k m should hold respectvely. Therefore, for L k m L, there wll always be a g n the range of 0 g L. We have k C n q. 34. For any k n ths range, L k m L (assumng that m L). Then for any k of q. 34, there exsts one and only one g that would result n g k m to be a multple of L (here only 0 or L). Therefore, L z=0 L z =0,z z α,zα,z S lz S l z plot (l z) plot (l z ) =0. Then, σu = L r z=0 α,z Ir lz S lz. It can be easly proved σ that, L z=0 α,z =. Then usng the defnton of SIR r ave of Theorem, we wll have, σu r =( m ) (SIRave) r σh. (35) Usng q. 5 and 6, ΔHeq r () wll then be as follows: m βr,k σ h k σ U r =4 ( m) ΔHeq() r = σv r R{ m βr sn ( πl ) m σ h k ( m ) (SIRave r ) σh σ W 4( m)sn ( πl ) m k= kσ h k σ,k U r h k } ϖ(m ), (36) where ϖ(z) =for z 0 and zero otherwse. Then the normalzed average channel estmaton error varance at the th sub-carrer, Ch r error,norm (), can be wrtten as follows: m Ch r error,norm () = ΔHr eq () m H r eq () =4 σ k= h k kσ h k m ϖ(m ) C sn ( πl σ ) h }{{ k } factor#: rotatons n dfferent bases SIR } ave r SR {{} r, }{{ ave } nterference nose (37) where SRave r = ( m) σ σ H. σw Snce, wth hgh probablty, m s consderably smaller than, q. 37 can be tghtly approxmated. For m consderably smaller than, for k m. Therefore, k m Ch r error,norm () 4Υr % sn ( πl ) factor#: rotatons n dfferent bases SIRave } r {{ SRave } r, nterference nose (38) where Υ r % = m σ h k C represents the rato of the power σ h k of the msplaced channel paths to the total power of the channel (see Fg. b, c). Let factor# represent the effect of rotaton (frst term on the rght-hand sde of q. 38). As expected, t does not affect those sub-channels carryng plot tones. However, t results n a consderable ncrease of error for other sub-carrers partcularly for those at = z odd cel( L ), where z odd represents odd ntegers and cel(.) represents the celng functon. In a reasonable SRave r envronment, the overall mpact of factor# s consderably hgher than other terms. Let Ch error,rato represent the rato of the frst term to the sum of the last two terms on the rght hand sde of q. 38. xamnng Ch error,rato for dfferent values of m and Υ r % n a reasonable SR envronment shows that factor# s the domnant factor wth hgh probablty. B. Case of m<0 Smlar expressons can be derved for the case of m< 0. Usng q. 3, Heq() l = Γl 0 H e jπm and h l eq(k) = Γ l 0 h ((km)). To see the effect of rotatons n dfferent bases

7 MOSTOFI AD CO et al.: A ROBUST TIMIG SYCHROIZATIO DSIG I OFDM SYSTMS PART I: LOW-MOBILITY CASS 7 ormalzed Channel stmaton rror Varance Analyss, m=3 Smulaton, m=3 Rotaton ffect, m=3 Analyss, m= 9 Smulaton, m= 9 Rotaton ffect, m= Sub Carrer Fg.. ormalzed channel estmaton error varance vs. sub-carrer for m =3(Υ r % =4%)andm = 9 (Υl % = 50%). n ths case, Fg. d and e show the equvalent and estmated equvalent channel for the orgnal channel of Fg. a respectvely. Unlke the case of m>0, where even one error to the rght resulted n an equvalent channel of length (see Fg. b), the equvalent channel length for m<0 vares dependng on the length of the orgnal channel. For nstance, for a channel of length C ν, the equvalent channel length wll be C m for m. Therefore for C ν m, the equvalent length would stll be less than or equal to ν, whch poses no problem for the channel estmator. ote that ν s the maxmum normalzed predcted channel delay spread as defned n Secton II-B. Furthermore, the msmatch s n the locaton of the last m paths of the orgnal channel and these samples are not typcally that strong. Therefore, errors to the left sde may not degrade the performance (both n terms of effect of rotaton and nterference), dependng on the length of the channel delay spread, length of the guard nterval and number of plot tones. Followng the same procedure, an analytcal expresson can be found for the case of m<0. Let Ch l error,norm() represent the normalzed average channel estmaton error varance for m<0. Then t can be tghtly approxmated as follows, Ch l error,norm() 4sn ( πl )Υl % factor#: rotatons n dfferent bases L k=lm σ h k C σ h k SIR l ave } {{ } nterference SR }{{ ave l, (39) } nose where Υ l % = represents the rato of the power of the msplaced channel paths to the total power of the channel for the case of m<0, SRave l = ( d) σ σ H σw and d s as defned n Secton III-B. C. Smulaton Results As can be seen from the expressons for the average channel estmaton error varance, tmng errors can degrade the performance of a plot-aded channel estmator consderably. Furthermore, factor#, the effect of rotatons n dfferent bases, s the major contrbutor to channel estmaton error. Fg. shows normalzed average channel estmaton error varance, from both analyss and smulatons, as a functon of sub-carrer and for a channel wth the followng power-delay profle: System specfcatons are as follows for ths result: = 89 and G = L = 3. The channel s estmated usng the plot tones nserted at locatons 0, 4, 8,... As can be seen, for both cases of m>0 and m < 0, factor# (rotatons n dfferent bases) contrbutes almost all the channel estmaton error. Furthermore, t can be seen that channel estmaton error can be consderably hgh n the presence of tmng errors. Fnally, smulaton results verfy mathematcal dervatons of ths secton. V. TIMIG SYCHROIZATIO RROR CORRCTIO It was shown that the plot-aded channel estmator s consderably senstve to tmng synchronzaton errors due to the effect of rotatons n dfferent bases. Ths senstvty can be exploted to desgn a robust synchronzaton algorthm that requres no tranng nformaton. After a coarse tmng synchronzer has detected a start pont for the OFDM symbol (dfferent choces for ths synchronzer wll be dscussed later n ths secton), Ĥ eq (k) can be obtaned usng the plot tones. Ths channel estmate may be far from H eq (k) n the presence of tmng errors. Let ˆk = Y k and Ĥ k = Dec( ˆ k ) rep- eq(k) resent the estmated nput before and after the decson devce respectvely. Defne a decson-drected measure functon as MF = ˆ k k. In the presence of factor#, MF can be consderably hgh. Therefore, synchronzaton errors can be detected and corrected by mnmzng MF. As long as factor# s the major cause of performance loss, whch s the case wth hgh probablty, we can detect tmng errors. Due to the nature of factor#, t s possble to perform all the updates necessary for detectng and correctng tmng errors n the frequency-doman, wthout a need to go back and forth from the frequency to the tme doman. Consder correctng errors to the rght. As can be seen from Fg. c, the poston of the last m paths of the estmated equvalent channel s dfferent from that of the equvalent channel, where m s unknown. We update the estmated equvalent channel through an teratve process, correctng for one msmatched channel path at a tme. Ths means that n the frst teraton, the last channel path n ĥ r eq (k) should be transferred from poston L to poston. Followng the same procedure, the update necessary at the k th sub-channel to correct errors to the rght wll be as follow n the th teraton ( ) (a channel path s moved from poston L to ): Ĥ (),r eq jπk (k) =Ĥ(),r eq (k)ς ĥr eq (L ) e. (40) Smlarly we wll have the followng for detectng errors to the left, Ĥ (),l eq (k) =Ĥ(),l eq (k) ς ĥl eq( ) e jπ( )k, (4) Where ς = e jπlk and Ĥ(),r eq (k) = Ĥr eq(k) and Ĥ eq (),l (k) =Ĥl eq (k). In each teraton, the measure functon, MF (), wll be evaluated. Fnally the teraton that results

8 8 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 Relatve power of the paths Fg μ s Delay as a multple of samplng perod Power-delay profle of channel#. n the mnmum MF,orMF below a predefned threshold, s selected 3. Call ths teraton corr,z,wherez =denotes that a tmng error to the rght has been detected. Smlarly z = ndcates detecton of an error to the left. Then t s possble to go back to the tme doman and correct the start of the symbol. For z =, the ntal start pont of the symbol should be moved to the left by corr,z. The opposte wll be the case for z =. In practce, a coarse synchronzer could be used for the ntal tmng synchronzaton. A tradtonal choce would be a sldng correlator that correlates the data ponts at the begnnng and end of the symbol, lookng for a peak 9. Ths wll be then followed by the proposed fne synchronzaton algorthm. Snce the proposed algorthm works robustly, there s no requrement on the qualty of the ntal coarse synchronzaton (as long as m <L) and s used solely to have a start pont for the proposed algorthm. ote that the senstvty of the plot-aded channel estmator to factor# makes t possble to desgn a tmng synchronzer wth a consderably better performance than the tradtonal one, wthout relyng on synchronzaton tranng nformaton. Furthermore, t allows for tmng error detecton solely n the frequency doman whch reduces the complexty consderably, compared to gong back and forth from frequency to tme doman. If a sub-optmum channel estmator were to be used nstead, the same mprovement would not be acheved for two reasons. Frst, the hgh senstvty of the plot-aded channel estmator to tmng synchronzaton errors allows for robust detecton and correcton of synchronzaton errors n the presence of nose and Doppler. However, such senstvty does not exst for other sub-optmum estmators, resultng n a rather hgh probablty of false detecton n the fne synchronzaton stage. Second, due to the lack of factor# for other estmators, tmng synchronzaton errors do not affect the estmated channel sgnfcantly. Therefore detectng the tmng error solely n the frequency doman, as was done n the proposed algorthm, would not result n tmng error detecton. Then to perform a decson-drected correcton, there would be a need to go back and forth between tme and frequency 3 Unless t s known apror that m s postve or negatve, both drectons should be tred. domans, whch would ncrease the complexty consderably wthout resultng n an acceptable performance. VI. SIMULATIO RSULTS An OFDM system s smulated n a hgh delay spread fadng envronment of a Sngle Frequency etwork 4 (SF). SF refers to the networks such as DAB and DVB that broadcast on the same frequency n adjacent cells and therefore can experence consderably hgh delay spreads. The followng parameters are chosen for smulatons 5. Input modulaton s 8PSK. Useful bt rate s 7.3Mbps. =89, L = G=3 and T s =.6μs. The power-delay profle of the smulated channels have two man clusters each wth 9 non-zero paths, to represent an SF channel. Two power-delay profles are consdered. Power-delay profle of Channel#, shown n Fg. 3, has delay spread of 36.5μs, spannng 64% of the guard nterval. Channel# has a smlar profle, but the delay between ts two clusters has ncreased such that the total length spans 00% of the guard nterval (worst case channel) n order to make the mpact of errors to the left sde more pronounced. In both channels two clusters have equal power as can be seen from Fg. 3. ach channel path s generated as a random varable wth Raylegh dstrbuted ampltude and unformly dstrbuted phase 7, 5. Three methods are smulated. The frst two methods are based on the tradtonal correlatonbased tmng synchronzaton, usng the redundancy of the cyclc prefx. Method I s method of Van de Beek et al., 9, whch pcks the maxmum correlaton pont. Method II s bascally method I wth a smple varaton to mprove the performance. It pcks the pont from whch the sum of the next L correlaton ponts s maxmzed, hence reducng the chance of an error to the rght sde (whch s more costly) at the prce of ncreasng the probablty of errors to the left sde. Method III utlzes method I for ntal coarse synchronzaton followed by our proposed synchronzaton error correcton method of the prevous secton. The ntal uncertanty range for the start of the OFDM symbol s taken to be from the mddle of the prevous symbol to the mddle of the next symbol. To evaluate the performance of these methods, P error, the average probablty of makng a tmng error of m samplng perods, s measured. To evaluate the cost of makng an erroneous offset, we use P b, the average BR (Bt rror Rate) n case of an offset of m samplng perods (ths s the pre-decodng BR). The frst and thrd curves of Fg. 4 show P error of the three methods, for channel# n the absence of nose. The second and fourth curves of Fg. 4 show P b for the three methods 6. The delay spread of channel# spans 64% of the guard nterval. Ths allows for tmng offsets of up to 36% of the guard nterval (whch becomes 8 samplng ponts) to the left to occur wthout any loss of performance. 4 Smlar (or more) mprovement s also obtaned wth non-sf channels. 5 System parameters are based on the Srus Rado second generaton system specfcaton. 6 ote that P b of method III s dfferent from that of method I and II. To obtan P b at a specfc tmng synchronzaton offset, bt error rate should be averaged over dfferent nput, nose and channel realzatons at that synchronzaton offset. For a gven offset of m, channel realzatons that would lead to that offset are dfferent for method III than those for methods I and II. Ths s due to the fact that method III uses channel estmaton to correct tmng errors.

9 MOSTOFI AD CO et al.: A ROBUST TIMIG SYCHROIZATIO DSIG I OFDM SYSTMS PART I: LOW-MOBILITY CASS 9 P error P b P error P b method II method I Safe Zone method I, II method III (Proposed) Safe Zone Safe Zone method III (Proposed) m (offset from the start of the symbol n samplng perod) Fg. 4. Performance of the proposed tmng synchronzer for channel# n the absence of nose. Average Pb Fg method I method II method III (Proposed) Perfect Synchronzaton σ σh/σw (n db) Average P b for channel# n the presence of nose. P error P b P error P b method II method I method I, II method III (Proposed) method III (Proposed) m (offset from the start of the symbol n samplng perod) Fg. 5. Performance of the proposed tmng synchronzer for channel# n the absence of nose. P error Fg Safe Zone Schmdl et al. proposed algorthm m (offset from start of the symbol) Comparson wth the algorthm proposed by Schmdl et al. We refer to ths regon as the safe zone whch can be seen n Fg. 4. The proposed method has a consderably lower probablty of synchronzaton error n the places that BR s hgh and manly has tmng offsets n the safe zone. From P error and P b, the average BR due to tmng synchronzaton errors, Pb, can be found: Pb = m P bp error. Pb would be as follows for channel#: P b,method I =.398, P b,method II =.0079, P b,proposed method = Fg. 5 shows smlar curves for channel#. In ths case, there wll be more senstvty to m, snce the delay spread spans 00% of the guard nterval. Ths can be seen from P b curves of Fg. 5, where there s no safe zone. Ths affects the shape of P error of method III. It can be seen that the proposed synchronzer reduces the probablty of tmng errors at offsets that have hgh costs. However, method II stll makes a consderable number of tmng errors to the left whch has a hgh cost for ths channel. To see the effect of nose, Fg. 6 shows average bt error rate as a functon of average receved SR for channel #. The dashed lne shows the BR of a perfect synchronzer for comparson. As can be seen, the performance of the proposed method s very close to that of the perfect synchronzer. The results confrm that the proposed algorthm s a robust tmng synchronzaton method that works effectvely even n consderably hgh delay spread fadng envronments, wthout requrng synchronzaton tranng nformaton. ext, we compare our proposed algorthm wth some of the already exstng ones n the lterature. The method proposed n 7 s one of the most cted algorthms that use tranng nformaton for synchronzaton. In ther case, half of the subcarrers are occuped by known nformaton for tmng synchronzaton. Fg. 7 shows the performance of ther algorthm for channel# and the aforementoned system parameters at SR =0dB. It can be seen that our proposed algorthm outperforms thers consderably. ote that ths s the case even though they transmt tranng nformaton (and therefore known nformaton) on half of the subcarrers, whereas we only use 5% of the subcarrers to send the plot nformaton. Therefore, wth less waste of bandwdth, we acheved a consderably better performance. Yang et al. uses the plots nserted for channel estmaton to perform tmng synchronzaton. However, they try to estmate the power-delay profle of the channel by averagng

10 0 I TRASACTIOS O WIRLSS COMMUICATIOS, VOL. 6, O., OVMBR 007 One Sample of the rror Correcton Metrc Robustness Margn Safe Zone Tmng Offset Fg. 8. rror correcton metrc as a functon of the tmng offset showng the robustness of the proposed method, our proposed method seeks to fnd the mnmum of the metrc. over a number of OFDM symbols (around 0). Also they rely on the frst path of the channel beng the strongest, whch may not be the case. In our case, we can correct for large tmng offsets for channels that do not have the frst powerdelay profle path the strongest (see SF channel of Fg. 3). Furthermore, we do the synchronzaton usng only one OFDM symbol and wthout a need for averagng over a number of symbols. Fnally, Speth et al. 8 proposed usng the redundancy of the cyclc prefx for coarse synchronzaton followed by fne tunng the performance by mnmzng the followng metrc: l plots z l p l H(l), as descrbed n Secton I. They provde a fgure whch represents ther metrc values as a functon of the tmng error (Fg. 7 of 8). A more drastc change out of the safe zone s an ndcaton of a better and more robust performance. In all ther cases, the metrc gets around 75% of ts safe zone value even after 50 errors out of the safe zone. Fg. 8 shows a sample (note that ths s one sample and not the average) of our metrc functon (MF of Secton V) for comparson. In can be seen that the metrc changes orders of magntude mmedately out of the safe zone, resultng n a consderably more robust tmng synchronzaton, for a fadng channel wth even longer delay spread and more paths. VII. SUMMARY AD FURTHR TSIOS In ths paper we proposed a robust tmng synchronzaton algorthm for plot-aded OFDM systems n low-moblty envronments. We analytcally characterzed the mpact of tmng errors on the performance of plot-aded channel estmaton by dervng an expresson for channel estmaton error varance n the presence of tmng errors and for a frequency-selectve fadng channel. The analyss showed that a plot-aded channel estmator s consderably senstve to tmng synchronzaton errors, due to the mpact of rotatons n dfferent bases. We then showed how to utlze ths senstvty to desgn a robust tmng synchronzer that requres no tranng nformaton. The algorthm s a cross-block desgn that uses the senstvty of the channel estmator to correct for tmng synchronzaton Robustness Margn errors. Smulaton results confrmed mathematcal dervatons and showed the robust performance of the proposed algorthm n hgh delay spread fadng envronments. In part II of ths paper, we show that the mathematcal framework and the proposed algorthm can be extended for robust synchronzaton n hgh moblty cases and n the presence of a frequency offset. RFRCS J. Bngham, Multcarrer modulaton for data transmsson: An dea whose tme has come, I Commun. Mag., vol. 8, no. 5, pp. 5-4, May 990. H. Bolcske, Blnd estmaton of symbol tmng and carrer frequency offset n wreless OFDM systems, I Trans. Commun., vol. 49, pp , June B. Park, H. Cheon, C. Kang, and D. Hong, A novel tmng estmaton method for OFDM systems, I Commun. Lett., vol. 7, no. 5, pp , May L. Cmn, Analyss and smulaton of a dgtal moble channel usng orthogonal frequency dvson multplexng, I Trans. Commun., vol. 33, pp , July B. Le Floch, R. Hallbert-Lasalle, and D. Castellan, Dgtal audo broadcastng to moble recevers, I Trans. Consumer lectron., vol. 35, no. 3, pp , Aug M. Hseh and C. We, A low-complexty frame synchronzaton and frequency offset compensaton scheme for OFDM systems over fadng channels, I Trans. Veh. Technol., vol. 48, no. 5, Sep. 999, pp Wllam C. Jakes, Mcrowave Moble Communcatons. ew York: I Press, Y. Mostof and D. Cox, Mathematcal analyss of the mpact of tmng synchronzaton errors on the performance of an OFDM system, I Trans. Commun., Feb Y. Mostof, ICI mtgaton and tmng synchronzaton for plot-aded OFDM moble systems, Ph.D. thess, Stanford Unversty, ov H. Mnn, V. K. Bhargava, and K. B. Letaef, A robust tmng and frequency synchronzaton for OFDM systems, I Trans. Wreless Commun., vol., no. 4, July 003, pp H. Mnn, V. K. Bhargava, and K. B. Letaef, A combned tmng and frequency synchronzaton and channel estmaton for OFDM, I Trans. Commun., vol. 54, no. 3, Mar. 006, pp S. Muller, On the optmalty of metrcs for coarse frame synchronzaton n OFDM: A comparson, n 9th I Internatonal Symposum PIMRC, 998, pp R. eg and J. Coff, Plot tone selecton for channel estmaton n a moble OFDM system, I Trans. Consumer lectron., vol. 44, no. 3, pp. -8, Aug B. Park, H. Cheon;. Ko, C. Kang, and D. Hong, A blnd OFDM synchronzaton algorthm based on cyclc correlaton, I Sgnal Processng Lett., vol., no., pt., pp , Feb Theodore S. Rappaport, Wreless Communcatons: Prncples and Practce. Upper Saddle Rver, J: Prentce Hall, H. Sar, G. Karam, and J. Janclaude, Transmsson technques for dgtal TV broadcastng, I Commun. Mag., vol. 36, pp , Feb T. M. Schmdl, Synchronzaton algorthms for wreless data transmsson usng orthogonal frequency dvson multplexng (OFDM), Ph.D. dssertaton, Stanford Unversty, June M. Speth, F. Classen, and H. Meyr, Frame synchronzaton of OFDM systems n frequency selectve fadng channels, n Proc. I 47th VTC, 997, pp J. Van de Beek, M. Sandell, and M. Isaksson, Low-complex frame synchronzaton n OFDM systems, n 4th I Internatonal Conf. Unversal Pers. Commun., 995, pp R. D.J. van ee and R. Prasad, OFDM for Wreless Multmeda Communcatons. Boston, MA: Artech House, 998. S. Wensten and P. bert, Data transmsson by frequency-dvson multplexng usng the dscrete Fourer transform, I Trans. 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11 MOSTOFI AD CO et al.: A ROBUST TIMIG SYCHROIZATIO DSIG I OFDM SYSTMS PART I: LOW-MOBILITY CASS Yasamn Mostof (S 98-M 04) receved the B.S. degree n electrcal engneerng from Sharf Unversty of Technology, Tehran, Iran, n 997, and the M.S. and Ph.D. degrees from Stanford Unversty, Stanford, CA, n 999 and 004, respectvely. She s currently an assstant professor at the Department of lectrcal and Computer ngneerng at the Unversty of ew Mexco. Pror to that, she was a postdoctoral scholar at the Calforna Insttute of Technology from 004 to 006. Her research nterests nclude cooperatve sensor networks, moble communcatons, control and dynamcal systems and sgnal processng. Donald C. Cox (S 58-M 6-SM 7-F 79) receved the B.S. and M.S. degrees n lectrcal ngneerng from the Unversty of ebraska n 959 and 960, respectvely, and the Ph.D. degree n lectrcal ngneerng from Stanford Unversty n 968. He receved an Honorary Doctor of Scence from the Unversty of ebraska n 983. From 960 to 963, Mr. Cox dd mcrowave communcatons system desgn at Wrght-Patterson AFB, Oho. From 963 to 968 he was at Stanford Unversty dong tunnel dode amplfer desgn and research on mcrowave propagaton n the troposphere. From 968 to 973 hs research at Bell Laboratores, Holmdel, ew Jersey n moble rado propagaton and on hgh-capacty moble rado systems provded mportant nput to early cellular moble rado system development, and s contnung to contrbute to the evoluton of dgtal cellular rado, wreless personal communcatons systems and cordless telephones. From 973 to 983 he was Supervsor of a group at Bell Laboratores that dd nnovatve propagaton and system research for mllmeter-wave satellte communcatons. In 978 he poneered rado system and propagaton research for low-power wreless personal communcatons systems. At Bell Laboratores n 983 he organzed and became Head of the Rado and Satellte Systems Research Department that became a Dvson n Bell Communcatons Research (Bellcore) wth the breakup of the Bell System on January, 984. He was Dvson Manager of that Rado Research Dvson untl t agan became a department n 99. He contnued as xecutve Drector of the Rado Research Department where he champoned, led and contrbuted to research on all aspects of lowpower wreless personal communcatons enttled Unversal Dgtal Portable Communcatons (UDPC). He was nstrumental n evolvng the extensve research results nto specfcatons that became the U.S. Standard for the Wreless or Personal Access Communcatons System (WACS or PACS). In September 993 he became a Professor of lectrcal ngneerng and Drector of the Center for Telecommuncatons at Stanford Unversty where he contnues to pursue research and teachng of wreless moble and personal communcatons. He was apponted Harald Trap Frs Professor of ngneerng n 994. Dr. Cox was a member of the Admnstratve Commttee of the I Antennas and Propagaton Socety (986-88), was an Assocate dtor of the I TRASACTIOS O ATAS AD PROPAGATIO (983-86), s a member of the atonal Academy of ngneerng, s a member of Commssons B, C and F of USC/URSI, and was a member of the URSI Intercommsson Group on Tme Doman Waveform Measurements (98-84). He was awarded the I 993 Alexander Grapham Bell Medal For poneerng and leadershp n personal portable communcatons; was a corecpent of the 983 Internatonal Marcon Prze n lectromagnetc Wave Propagaton (Italy); receved the I 985 Morrs. Leeds award and the I Thrd Mllennum Medal n 000; and receved the 983 I Vehcular Technology Socety paper of the year award, and the I Communcatons Socety 99 L. G. Abraham Prze Paper Award and 990 Communcatons Magazne Prze Paper Award. He receved the Bellcore Fellow Award n 99. Dr. Cox s a fellow of AAAS and the Rado Club of Amerca. He s author or coauthor of many papers and conference presentatons, ncludng many nvted and several keynote addresses, and books. He has been granted 5 patents. Dr. Cox s a member of Sgma, Sgma Tau, ta Kappa u and Ph Mu pslon, and s a Regstered Professonal ngneer n Oho and ebraska.