II. SYSTEM TRANSMISSION MODEL WITH SDT A digital communication system with the SDT is depicted in Fig. 1. A binary message sequence generated by

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1 Minimizatin f Channel Impulse Nise using Digital Smear filter Grace Oletu, Predrag Rapajic and Kwashie Anang Mbile and Wireless Cmmunicatins Research Centre, University f Greenwich, Chatham, UK {G.Oletu, P.Rapajic, K.Anang}@greenwich.ac.uk Abstract The paper describes a digital smear-desmear technique (SDT) based n plyphase multilevel sequences f unlimited length with gd autcrrelatin prperties. A design prcedure fr digital implementatin f SDT is defined and sequences with pwer efficiency higher than 5% are generated. These sequences are applied t the design f digital smear/desmear filters and cmbined with uncded and cded ITU-T V.15.1 cmmunicatin systems. The impulse nise is mdeled as a sequence f Pissn arriving delta functins with gaussian amplitudes. The impulse nise parameters are cmputed frm experimental data. Simulatin results shws that the SDT filter design methd yields a significant imprvement in bit errr rates fr bth systems subject t impulse nise, relative t systems with n SDT. The technique als cmpletely remves the errr flr caused by impulse nise. Index Terms Smear, Desmear, Pseudrandm Sequence, Impulse nise, Intersymbl Interference. I. INTRODUCTION Mst f the advances in thery and implementatin f digital transmissin ver band limited channels have been made with respect t additive white gaussian nise (AWGN), as the ultimate reliability limitatin. With imprved equalizatin, phase jitter tracking, timing recvery and trellis cded mdulatin (TCM), transmissin rates achieved ver band limited channels are clse t the theretical limit [1]. Hwever, the required errr prbabilities fr reliable data transmissin have nt been achieved even at 4.8 kb/s [2] [3]. One f the main impairment n band limited channels, causing burst errrs, is impulse nise (IN). In the present-day high speed mdems fr band limited channels are n measures against impulse nise ther than detectin [4] [5] [6]. A pssible cunter-measure t the prblem f shrt impulse nise (less than 1ms) is the smear-desmear technique (SDT) [6]. The SDT in [6] has been implemented in analg technlgy and the results were nt satisfactry due t insufficient quality f analg devices. A digital SDT technique that applies binary sequences f limited length was described in [7], [8]. The design f SDT filters in [7] is based n minimizing intersymbl interference (ISI) and maximizing filter pwer efficiency. In this design, lsses in signal-t-nise rati (SNR) can Manuscript received July 25, 212; accepted Octber 3, 212. This is an expanded versin f the wrk submitted at IEEE Vehicular Technlgy Cnference (VTC 212 -Fall), in Quebec City, Canada in September 3-6, 212. be significant since transmit and receive filters are nt matched. The purpse f this paper is t derive a mre general set f filter design criteria based n minimizing bit errr rates and als fr practical filter design. As a result, anther necessary requirement fr minimizatin f the signal-t-nise rati (SNR) lss due t mismatching filters is added t the design criteria [7]. In this paper three appraches were applied in the practical filter design. In Design 1, the smearing filters frm a pair f matched filters. The plyphase sequences used in this scheme pssess significantly better autcrrelatin prperties, measured by the merit factr than the binary sequences. Design 2 is prpsed fr systems where very lw values fr ISI variance (belw -3 db) are required. Lw ISI is achieved by designing the smearing filter t perate as equalizer. The filter sequences are required t have bth gd autcrrelatin and equalizatin prperties. It is shwn that plyphase sequences utperfrm knwn binary sequences with regard t ISI suppressin and mismatching SNR lss. Design 3 can yield ISI as lw as Design 2 with reduced system delays. The filter design is based n nncnstant amplitude sequences [9] [1], while the cmmunicatin system structure is the same as in Design 2. The required filter lengths, fr a specified level f ISI, are much smaller than in Design 2. Simulatin results shws that the SDT based n plyphase sequences, yields significant imprvement in bit errr rates cmpared t SDT based n binary sequences f the same length. The SDT is attractive n bandwidth limited channels since it des nt require bandwidth expansin. The perfrmance imprvement is btained at the cst f an additinal delay in the system which can be tlerated in applicatins f interest. The paper is rganized as fllws. In sectin II, we intrduce the mdel f a digital transmissin system with smeardesmear filters and discuss the cncept f the digital smear-desmear prcessing. Sectin III describes the digital SDT in mre detail. Sectin IV defines essential criteria and parameters fr SDT design.sectin V presents simulatin results. Finally, cnclusins are summarized in sectin VI. II. SYSTEM TRANSMISSION MODEL WITH SDT A digital cmmunicatin system with the SDT is depicted in Fig. 1. A binary message sequence generated by

2 The impulse nise, dented by v i (t), can be written in the frm [5] Figure 1. System Transmissin mdel with SDT. the digital surce is mapped int 32-AMPM trellis cded mdulatin signals and 16-QAM fr uncded signals. The mdulatr utput symbls are then prcessed by the digital smear filter. In the smear filter the signal is expanded in the time dmain ver the filter impulse respnse. This peratin results in deliberately intrduced intersymbl interference (ISI). The channel is subject t AWGN and impulse nise. Ideal amplitude and phase channel characteristics as well as ideal phase tracking are assumed. In the receiver, the desmear filter perfrms an inverse peratin t the ne in the smear filter and thus remves the ISI intrduced in the transmitter. Bth the smear and desmear filtering are perfrmed in the baseband. After prcessing by the desmear filter the impulse nise energy is spread ut ver the filter impulse respnse length. That results in a significant reductin f the impulse nise effect n the signal. The signal is demdulated by the Viterbi decder. A. Transmitter Mdel Let b = [b(),, b(n),, b(m)] dente a cmplex symbl sequence at the utput f the mdulatr in fig. 1. The smear filter is represented by a sequence f tap cefficients, dented by s = [s(),, s(i),, s(n)] where s(i) is the ith tap cefficient and (N1) is the number f taps. The utput sequence c is btained by cnvlving the sequence b and the smear filter sequence s. We assume that the filter gain dented by A s, is nrmalized t unity. That is, A s = s(j)s (j) = 1 (1) where dentes cmplex cnjugate. The utput signal c(n) has a gaussian distributin with a zer mean and the variance P. B. Channel Mdel The input symbl t the desmear filter at time n is given by: x(n) = c(n) v(n) v i (n) (2) where v(n) is a sample f zer mean cmplex AWGN with the variance σ 2, and v i (n) is a sample f the channel impulse nise v i (t) with the variance σi 2. The impulse nise event times are represented by a Pissn prcess. v i (t) = k= z i (k)δ(t t k ) (3) where t i represent the impulse nise event times and z i is the impulse nise amplitudes. The Pissn randm prcess t i has the intensity f λ events/s and z i is a Gaussian prcess with a zer mean and the variance σi 2. The parameters λ and σi 2 are btained frm experimental data [11], [12]. Mst f the symbls received are nt crrupted by impulse nise. Due t the nnstatinary character f impulse nise we define the signal t impulse nise pwer rati ver ne symbl interval as SNR in = 1 lg( P σi 2 ) (4) where P represents the signal pwer. We assume that the average time interval between tw cnsecutive impulse nise events f v i (t) is larger than the smear/desmear filters impulse respnse length cnsisting f N symbl intervals. That means that λt s N << 1, where T s is the symbl interval. C. Receiver Mdel The desmear filter is represented by a cefficient sequence, dented by d = [d(), d(1),, d(n)]. T avid a trivial slutin in filter design fr the desmear filter cefficient set we include the fllwing pwer cnstraint A sd = s(j)d (N j) = 1 (5) where A sd is the gain f the smear-desmear filter pair. The gain A d f the desmear filter is given by A d = d(j)d (j) (6) Cmbining (1) and (5) we btain that A d 1 where equality is satisfied if and nly if s(j) = d (N j), j =, 1, 2,, N. The utput symbl f the desmear filter signal y(n) can be represented by: y(n) = b(n) b isi (n) v(n) v s (n) (7) The ttal channel distrtin at time n is given by the sum f the residual ISI, b isi (n), the additive white gaussian nise after the desmearing filter, v(n), with a zer mean and the variance σ 2 v = A dσ 2 and the impulse nise, v s, smeared ver N symbl intervals. The impulse nise v s with variance in the (n j)th symbl interval is given by: σ 2 s(n j) = E(v s (n j)v i (n j)) = σ 2 i d(j) 2 (8) The residual ISI, b isi (n), is given by the sum f N independent randm variables. Typically, N is larger than

3 1 and accrding t the central limit therem, b isi has a gaussian distributin. The sum f the three independent gaussian variables b isi, v and v i is anther gaussian variable e(n) = y(n) b(n) = b isi (n) v(n) v s (n) (9) with the variance σe 2 = E( b isi (n) 2 ) E( v(n) 2 ) E( v s (n) 2 ) (1) The variance σe 2 can be upperbunded by Merit factr F a 4 b c d Sequence length σ 2 e = σ 2 v σ 2 isi σ 2 s(n j) σ 2 v σ 2 isi max σ 2 i (11) Where σisi 2, the variance f the ISI and max σ2 i is the maximum impulse nise variance.thus the ttal channel distrtin can be cnsidered as an equivalent gaussian prcess with the variance σe. 2 The main bjective f the system design is t minimize the bit errr prbability. At high signal-t-nise rati, the bit errr prbability can be estimated by P b N s n b Q( d 2σ e ) (12) where N s is the average number f the nearest neighburs in the signal set, n b is the number f bits in a symbl and d is the minimum Euclidean distance in the signal set. The ultimate perfrmance limit f a cmmunicatin system is determined by gaussian nise, the nly disturbance in the channel. The perfrmance f a real cmmunicatin system with impulse nise and ISI caused by smearing filters is cmpared t an ideal system with AWGN nly. The measure f the real system perfrmance lss relative t the ideal system is defined as the rati f the ideal and the real system signal-t-nise ratis given by σe 2 L = 1 lg 1 [db] (13) σ2 Ideally, smearing filter parameters shuld be selected t btain a zer perfrmance lss. In practical filter design, the bjective is t minimize the perfrmance lss r, in ther wrds, minimize the variance f the equivalent gaussian prcess σ 2 e. D. THE SMEARING FILTER DESIGN CRITERIA The smear and desmear filters are implemented as digital filters. The initial criterin in the filter design is minimizatin f the perfrmance lss given by Eq. (13). Hence, the perfrmance lss is required t be belw a specified threshld T. L s T s (14) The perfrmance lss directly depends n the equivalent gaussian prcess variance, σe. 2 The variance σe 2 is Figure 2. Merit factr F 2 fr sequences with cnstant amplitude: a) P2 sequence, b) Frank and P1 sequence, c) P3 and P4 sequences, d) Binary sequence. given by the sum f the AWGN variance σ 2 v, the maximum smeared impulse nise variance max σs(n 2 j) j and the residual intersymbl interference variance σisi 2 as expressed in Eq. (11). The minimizatin f σe 2 can be dne by independent minimizatin f each f the three cmpnents in the sum. Hwever, minimizatin f individual cmpnents can cause significant degradatin f the thers. Fr example, minimizatin f the residual intersymbl interference variance can cause a cnsiderable increase in the AWGN variance due t mismatched smearing filters. Since the system perfrmance in the absence f impulse nise is f paramunt imprtance, the pririty is given t the minimizatin f the AWGN variance. The smearing filter design criteria can be summarized as fllws: 1) Criterin I: Minimize the AWGN variance : 2) Criterin II: Minimize the residual ISI variance : 3) Criterin III: Minimize the impulse nise variance : E. Criterin I In rder t minimize the AWGN variance the smear and the desmear filters shuld satisfy the matching cnditin in the frm f s(j) = d (N j) j =, 1, 2,, N (15) r equivalently, A d shuld be equal t 1. If the matching cnditin expressed by (15) cannt be satisfied, we define a measure f mismatching between the smear and desmear filters, called the mismatching lss in [6]. L m = 1 lg 1 σ 2 v σ 2 = 1 lg A sd [db] (16) In filter design we require that the mismatching lss L m, is less than a specified threshld T m, therefre L m T m. The mismatching lss is caused by an increase f the AWGN variance relative t the ideal system with matched filters. In practical filter design we require that the mismatching lss is smaller than.3db. But mismatched lss requirement depends n the

4 applicatin.the prpsed scheme requirement is suitable fr 64QAM applicatin but will cause a devastating effect fr 256QAM, because they are mre susceptible t nise crruptin than 64QAM. Hence mismatched lss has t be less than.3. L m.3db (17) F. Criterin II T btain the minimum ISI variance, the verall transfer functin f the smear filter, the channel and the desmear filter shuld be flat. Since we assumed that the channel des nt intrduce ISI, the abve cnditin is satisfied if the cnvlutin f sequences s and d, dented by C(k), has values z() = 1, and z(k) =, k. where cnvlutin C(k) is defined as C(k) = s(j)d(k N j) k = N 1,, 1,, 1 (18) If cnditin (15) is satisfied then Eq.(18) simplifies t C(k) = R(k) = d (j)d(k j) (19) where R(k) is the autcrrelatin functin f the sequence d. Practical difficulties make a zer ISI an unattainable bjective in filter design. Typically, a certain amunt f residual ISI after the desmearing filter is tlerated. It is measured by the variance f the residual ISI given by σ 2 isi = N 1 k=n1 C(k) 2 C() 2 (2) The signal t nise rati lss caused by the ISI is defined as Equivalently, L s = 1 lg σ2 σ 2 isi σ 2 [db] (21) L s = 1 lg ( 1 SNR ) [db] (22) F 2 where SNR is the signal t AWGN pwer rati defined as SNR = C() 2 σ and F 2 2 = C() 2 is the merit factr σisi 2 defined in [7]. The residual ISI is cnsidered as an additinal gaussian prcess intrducing a certain SNR lss at the receiver. Clearly, this lss shuld be minimized. It shuld be maintained belw a specified threshld T s, thus L s T s. The lss T s f.3db is cnsidered t be acceptable. Frm a practical pint f view it is much easier t use a quantity called nrmalized ISI level, dented by L isi, defined as L isi = 1 lg F 2 [db]. In practical filter design parameter L isi is mre cnvenient. Fr systems emplying multilevel mdulatin schemes we assumed that the SNR is at least as high as 2dB, t meet the requirement (14), we require L isi 3dB (23) G. Criterin III Criterin III cnsists f minimizatin f the maximum smeared impulse nise variance. T be cnsistent with the already established design criteria [7], we intrduce the merit factr defined as the rati f the maximum impulse nise variance and the maximum smeared impulse nise variance in a single symbl interval F 2 = σ2 i (n) σs(n 2 j) = 1 max j d(j) 2 (24) in db L F2 = 1 lgf 2 [db]. Where σi 2 is the maximum impulse nise variance and σs(n 2 j) is the maximum smeared impulse nise. In practical filter design we required that L F2 T F2 = 2[dB] (25) The impulse nise variance befre spreading can be as large as the signal variance [11]. That is, ccasinal impulse hits can reach well abve the signal level. Impulse nise spreading shuld reduce the impulse nise variance t the level f the AWGN variance. A further spreading is nt effective [13] [14]. Since multilevel mdulatin systems shwn in figure 5.18 in [15] and figure 9.17 in [16] required the minimum SNR f 2 db. Clearly the merit factr F 2, as defined by Eq. (24), shuld be as large as pssible. It shws hw much the impulse nise variance in a single symbl interval has been reduced by smearing. A filter with a larger length can prduce a larger merit factr F 2. It is cnvenient t intrduce a measure fr filter smearing efficiency which des nt depend n filter length in [17], [18]. The pwer efficiency f a sequence d, dented by η, is defined by η = N d j 2 (N 1) max j d j 2 (26) The pwer efficiency has its maximum value f 1, fr cnstant amplitude sequences, while it is less than 1 fr nncnstant amplitude sequences. Cmbining Eqs. (6), (24) and (26) we btain the fllwing expressin fr the merit factr F 2 F 2 = η(n 1) A d (27) In rder t maximize the merit factr F 2, sequences shuld have the pwer efficiency as large as pssible. Fr a finite desmear filter length N, subject t cnstraints (1) and (5), it has been shwn in [7] that the maximum value fr F 2 is bunded by N 1, i.e. F 2 N1. The equality is satisfied if and nly if d j 2 = 1 N 1 and s(j) = d (N j) (28) That is, the ptimum merit factr F 2 is achieved nly when the smearing filters frm a matched filter pair and bth f them are represented by sequences with cnstant amplitude.

5 III. PRACTICAL FILTER DESIGN The ptimum values fr the three filter design criteria cannt be achieved simultaneusly. There are three appraches in practical filter design. In Design 1 we search fr sequences with cnstant amplitude and gd autcrrelatin prperties. The smear/desmear filter pair cnsists f matched filters. This design satisfies the requirements fr ptimum values f Criteria I and III, as defined by Equatin. (28).The value f Criterin II clearly depends n the autcrrelatin prperties f the filter sequences.the cnstant amplitude sequences with the best knwn autcrrelatin prperties are plyphase sequences [14]. The smearing filter design based n these sequences achieves an imprvement f 7.78 db in merit factr F 2 relative t the design based n binary sequences fr the sequence length f 2. This type f filter design based n plyphase sequences can prduce filters with ISI level, L isi, f -15 db with sequence lengths f 2, while filters designed in [7] cannt achieve σisi 2 less than -8 db. Design 2 is suitable fr systems where very lw values fr ISI level ( 3dB) are required. The imprvement in suppresin f intersymbl interference is achieved by sequence equalizatin. That is, the smearing filter is designed as an equalizer and therefre the smear/desmear filter pair is nt matched. While the smearing filters in this design will prduce lw ISI (Criterin II), the mismatched filters will result in an increased AWGN variance (Criterin I) and smeared impulse nise variance (Criterin III). Therefre, the three design criteria shuld be mnitred and adjusted simultaneusly. It shuld be nted that the smear/desmear filters in this design might intrduce a significant system delay f several sequence lengths. Design 3 relies n sequences with nncnstant amplitude. This design apprach can achieve ISI as lw as in Design 2, with an additinal advantage f a lwer system delay. In this methd Criteria I and II are satisfied at the expense f the filter pwer efficiency η. Sequences with nncnstant amplitude will result in a nn-ptimum value fr the impulse nise variance (Eq. (28)) fr the given desmear filter length. A pssible chice is Huffman sequences which have gd autcrelatin prperties [17] r sequences presented in [19] which have bth gd autcrrelatin prperties and pwer efficiency. A. Design 1 In Design 1 we fcus n cnstant amplitude plyphase sequences with gd autcrrelatin prperties. Plyphase sequences with cnstant amplitude, knwn as Frank and P1-P4 sequences [14], have better autcrrelatin prperties than M-sequences. It is imprtant t nte that plyphase sequences als are resilient t carrier phase and timing instabilities. In the sequel, we will discuss the prperties f these sequences with respect t SDT applicatins. 1) Cnstant Amplitude Plyphase Sequences: : Fr a plyphase Frank sequence f length N = L 2, the phase f a sequence element is ϕ(k, l) = 2π/L(k 1)(l 1) and sequence elements are d[k L(l 1)] = exp(jϕ(k, l)), where k = 1,, L, l = 1,, L. The phases f P1 and P2 sequence elements are given by ϕ(k, l) = π/l[l (2k 1)][(k 1)L (l 1)] and ϕ(k, l) = π/2l(l 1 2k)(l 1), respectively. It is imprtant t bserve that bth P1 and P2 sequences are available nly fr square integer lengths, i.e N =...36, 49, 64,... P2 sequences are further restricted t even lengths nly. Odd length P2 sequences pssess rather bad autcrrelatin prperties. Sequences P3 and P4 are defined fr any integer length. Phases f their elements are ϕ(k) = π/n(k 1) 2 and ϕ(k) = π/4n(2k 1) 2 π/4(2k 1)f r1 k N. The mst imprtant prperty f cnstant amplitude plyphase sequences, relevant t SDT applicatins, is that the mainlbe t sidelbe pwer rati is a mntnically increasing functin f the sequence length. This prperty makes them much mre effective in suppressing ISI then binary sequences [7], [8]. In additin, these sequences have cnstant amplitude and cnsequently, the ptimum Criterin III (28). The methd fr generating binary sequences with high F 2 invlves cmputer search and sequence eliminatin, which fr large values f sequence lengths becme prhibitively time cnsuming. On the ther hand, plyphase sequences are generated analytically. B. Design 2 A distinguishing prperty f this design methd is a very lw ISI level (L isi 3dB) achieved by sequence equalizatin. On the ther hand, mismatched filters inevitably intrduce a certain level f SNR lss [19]. This SNR lss can be maintained belw a specified value by chsing a prper sequence fr the desmearing filter. The sequence shuld have bth, gd autcrrelatin prperties measured by F 2 and gd equalizatin prperties. In the sequel a simple criterin t estimate the equalizatin prperties f a sequence is discussed. 1) Zer Frcing Sequence Equalizatin : Fr a sequence s j, j =, 1, N, we define zer frcing equalizer r inverse filter [4] as a digital filter with a Krnecker delta sequence respnse t the sequence s. The Z - transfrm f the zer frcing equalizer is given by D(z) = 1 N s jz j (29) A necessary requirement fr the filter existence is that the Z-transfrm, S(z), des nt have zers n the unit circle [4]. Where S(z) is the Z-transfrm f the sequence s j. If the sequence pwer is nrmalized t unity (Eq. (1)), the SNR lss f the inverse filter, dented by L ZF, is given by the rati L ZF = 1 lg 1 2π 1 π π D(ejω ) 2 dω (3) The quantity D(e jω ) can be evaluated at a clsely spaced set f pints by the use f fast Furier transfrm,

6 and the integral can thus be calculated t a clse apprximatin. The L ZF lss shws the difference in SNR between a zer frcing equalizer and a matched filter. This quantity is an upper bund n the SNR lss in all ther equalizatin methds [4]. Therefre, the L ZF can be used in sequence search as an indicatr f their perfrmance with respect t the equalizatin SNR lss. It is wrth nting that the equalizatin perfrmance shuld be evaluated by bth L ZF and F 2 parameters, since a gd merit factr F 2 des nt guarantee a lw L ZF lss. 2) Sequences with Gd Equalizatin Prperties : Fig. 2 shws the merit factr F 2 binary sequences and the plyphase sequence fr length 2 and 4 respectively. The main lbe t side lbe pwer rati has a flr fr binary sequences, while it increases mntnically with the sequence length fr plyphase sequences. It has been bserved that this rati is prprtinal t the square rt f the sequence length. Fig. 3 and 4 illustrate the smearing filter lengths needed t btain a specified level f ISI suppressin. The results are presented fr shrt and lng Frank and binary sequences f cmparative lengths. As a rule f thumb, binary sequences require cnsiderably larger filter lengths relative t crrespnding Frank sequences. Bth figures indicate that filter lengths f binary sequences are almst dubled relative t the Frank sequences. Fig. 3 shws the required filter lengths fr the P2 (36) sequence. Althugh this sequence has very gd autcrrelatin prperties (F 2 = 15.22) it requires very lng filters, relative t the binary and Frank sequences, in rder t achieve a lw level f ISI suppressin. This prperty f P2 sequences is a cnsequence f their infinite zer frcing equalizatin lss. Fig. 5 shws the L ZF lss fr binary sequences btained by limited search [9] based n the merit factr F 2 nly. The L ZF lss f selected plyphase sequences are indicated in fig. 5 fr cmparisn. It can be bserved frm Fig. 5 that many binary sequences (abut 6%) have a small L ZF lss. Hwever, mst sequences with lengths abve 1 have an excessively high zer frcing equalizatin lss which makes them unsuitable fr equalizatin. Finally, a simple example is singled ut t illustrate that the merit factr F 2 and the L ZF lss are nt clsely related and that bth f them have t be used in evaluating sequence equalizatin prperties. Plyphase sequence P2 (36) has the merit factr F 2 f which is superir t the best knwn binary Barker(13) sequence with F 2 f Hwever, the L ZF lss fr P2 (36) is infinite, while Barker (13) sequence has the lwest knwn value fr L ZF f.21 db. 3) Evaluatin f Design 2 in Cmmunicatin Systems : T evaluate the perfrmance f filters btained by Design 2, a number f sequences have been selected. They are used t design smearing mismatched filters in a real system where the receive filter perates as an MMS equalizer. The MMS equalizatin methd has been chsen because it prvides the best trade-ff in reducing the effects f residual ISI and gaussian nise. The principle f MMS equalizatin can be summarized as fllws. Let u filter pair defined by sequence s = (s(),, s(k)) at the transmitter and d = (d(),, d(n)) at the receiver. Nte that the lengths f the smear and desmear filters are in general different due t varius equalizatin requirements fr the ISI level. If the utput f the smearing filter is sequence s, the utput c f the desmearing filter d can be expressed in matrix frm as c = A.s where A is a [(N K 1) (K 1)] Teplitz matrix defined by the first rw (d(), 1, K ) and first clumn (d(), d(n), N1,, NK ) T. If a desired desmearing filter respnse t sequence s is a sequence z = ( z(), z(1),...z (N K)) then the mean squared errr between the actual filter respnse dented by c, and the desired filter respnse dented by z, is given by ε = NK 1 c(j) z(j) 2 (31) The filter sequence s which minimizes the mean squared errr (31) is given by [3] s = (A H.A) 1.A H.z (32) where () H dentes a transpsed and cnjugated matrix. Matrix A H.A is a (K 1) (K 1) crrelatin matrix f sequence s. Nte that A H.A is a Teplitz matrix whse inverse can be calculated by the Levinsn-Durbin algrithm [3]. We define the desired desmearing filter respnse z as a sequence with n ISI, with elements z(l) = 1, and L is the largest integer (N K)/2 and z(j) =, j L. It is imprtant t nte that thugh fr sme sequences the zer frcing equalizer (Eq. (29)) might nt exist, the minimum mean square apprximatin, defined by Eq. (32), always exists. Nte that the selected binary sequences were chsen t ptimize the merit factr F 2 as prpsed in [9]. In general, it has been bserved that binary sequences have the SNR lss in MMS equalizers, L m, clse t the SNR lss fr zer frcing equalizers, L ZF. C. Design 3 In system with sequence equalizatin the requirements fr Criteria I and II, expressed by equatin (17) and (23), respectively, cannt be met simultaneusly due t prhibitively large filter lengths. T imprve the filter perfrmance fr practical sequence lengths, we prpse Design 3 based n equalizatin and nn-cnstant amplitude sequences. In this design apprach we slightly sacrifice the filter pwer efficiency, η, (Eq. 26) relative t its maximum value btained in Design 2 in rder t satisfy Criteria I and II. Huffman sequences [17] are nncnstant amplitude sequences with best knwn ISI and pwer efficiency prperties. A thrugh examinatin f results presented in [17] reveals that the pwer efficiency f Huffman sequences des nt exceed.43. A drawback f Huffman sequences is that gd pwer efficiency is nt guaranteed and it usually ranges between.3 and.4 [18]. We prpse t use nncnstant amplitude sequences,

7 Filter length ISI level SNR lss Pwer efficiency N1 L ISI [db] L m[db] η TABLE I. THE SMEARING FILTER PARAMETERS FOR THE UNCODED AND CODED SYSTEMS Design Seq. ISI SNR IN merit System type length level lss factr delay N1 L ISI [db] L m[db] F[dB] NK1 Design Design TABLE II. THE SMEARING FILTER DESIGN METHOD COMPARISON Filter length a b Desired level f ISI [db] Figure 3. The smearing filter length, K 1 fr systems with shrt sequence equalizatin: a) Frank sequence f length 36, b) Binary sequence f length 33, c) P2 sequence f length 36. c generated by a methd presented in [2] [21] [22] [23] which are superir t Huffman sequences with regard t pwer efficiency. The design methd can be summarized as fllws. Step 1 Chse a Frank sequence with gd zer frcing equalizatin lss (L ZF 1dB) as an input d sequence in equatin (32). Step 2 Calculate a filter sequence s by the MMS algrithm (Eq.( 32)). It is assumed that the filter and input sequences have the same length. Step 3 Nrmalize the sequence s t satisfy ss = 1. Step 4 Cmpute the ISI level, L isi, and SNR lss, L m. Test whether they satisfy cnditins (17) and (23). If the answer is psitive, stp the prcedure, therwise use sequence s as the input vectr and repeat Step 2. This methd was used t generate a series f sequences that satisfy Criteria I-III. The sequence prperties are illustrated in Fig. (6-8). The results are shwn fr tw sequence lengths f 256 and 484. These lengths were selected t achieve the specified impulse nise suppressin in (Eq. (25))fr the uncded 16QAM and cded 32AMPM fr ITU.T V.15.1 data transmissin systems.the results shws that increasing the number f iteratins in the sequence design prcedure generally results in a lwer SNR lss (Fig.6) and ISI level (Fig.7). Hwever, the pwer efficiency, η, is als reduced cmpared t the cnstant amplitude sequences, and its variatins as a functin f the number f iteratins are shwn in Fig. 8. Criteria I-III was met after 16 and 2 iteratins fr sequence lengths f 484 and 256, respectively. The sequence parameters are listed in Table 1. Table 2 cmpares Design 2 and Design 3 techniques. The tw design methds are cmpared n the basis f values fr Criteria I-III. Clearly, design 3 ffers lwer values fr the SNR lss and fr the same values f ISI and smeared impulse nise variances intrduces a lwer system delay. IV. SIMULATION RESULTS The simulatin results in are fr cded and uncded ITU-T V.15.1 data cmmunicatin systems. Fig. [9-1] shws simulatin results fr 32-AMPM trellis cded Filter length b a Desired level f ISI [db] Figure 4. The smearing filter length, K 1 fr systems with lng sequence equalizatin: a) Frank sequence f length 196, b) Binary sequence f length 195. mdulatin signals and 16 QAM fr uncded systems in the presence f impulse nise (IN). Parameters f IN are: λ = 1 3 events/s, SNR in = [db]. The average time interval between tw cnsecutive impulse nise is.8s. In a cmmunicatin system subject t impulse nise, it is highly likely that all affected symbls will be incrrect, resulting in errr bursts with the bit errr rates clse t.5. Typical errr rate curves exhibit an errr flr which cannt be eliminated by increasing SNR. The results indicate that the SDT ffers a significant reductin in the SNR required t achieve the same bit errr rate as in a system with n SDT fr bth cded and uncded systems. The cding gain f the cded system relative t the reference uncded system is 2.5 db at the BER at 1 5, which is almst the same as the cding gain n gaussian channels. Als, the SDT cmpletely remves the errr flr in bth systems. In the abve example fr the filter impulse respnse f length N=256 and the pwer efficiency η =.54, the theretical SD gain is F=22 db. In mst cases a gain f this rder is sufficient t suppress the influence f IN n the bit errr rate. The real SD gain is reduced due t errr clustering caused by impulse nise spreading.

8 Zer frcing equalizatin lss [db] d c b a lg(F2) [db] Sequence length Number f iteratins Figure 5. The zer frcing equalizatin lss fr sequence with cnstant amplitude: a) Frank sequence, b) P1 sequence c) P3 and P4 sequence d) Binary sequence. Figure 7. Desired level f residual ISI, L ISI in terms f the number f iteratins in design 3.Slid line: Sequence f length 256, Dash line: Sequence f length SNRlss in [db].15.1 Pwer efficiency Number f iteratins Number f iteratins Figure 6. The SNR lss, Lm fr the system emplying sequences with nncnstant amplitude btained by design 3. Slid line: Sequence series f length 256, Dash line: Sequence series f length 484. Figure 8. Sequence efficiency, η in design 3.Slid line: Sequence f length 256, Dash line: Sequence f length 484. V. CONCLUSION The paper presents a digital smear-desmear technique (SDT) applied t data transmissin ver band limited channels. A generalized set f filter design criteria based n minimizing the average bit errr prbability is intrduced. The design criteria were applied t practical filter design and used in digital implementatin f the SDT. Plyphase sequences that meet the design requirements were generated. The SDT is simulated and cmbined with uncded and cded cmmunicatin systems fr high data transmissin. Simulatin results shw that the SDT yields a significant imprvement in bit errr rates fr bth systems, subject t impulse nise, relative t the systems with n SDT and the systems with filters based n binary sequences f crrespnding length. The technique als cmpletely remves the errr flr caused by impulse nise. REFERENCES [1] G. Frney Jr, R. Gallager, G. Lang, F. Lngstaff, and S. Qureshi, Efficient mdulatin fr band-limited channels, Selected Areas in Cmmunicatins, IEEE Jurnal n, vl. 2, n. 5, pp , [2] J. Bingham, Multicarrier mdulatin fr data transmissin: An idea whse time has cme, Cmmunicatins Magazine, IEEE, vl. 28, n. 5, pp. 5 14, 199. [3] S. Marple Jr, Digital spectral analysis with applicatins, Englewd Cliffs, NJ, Prentice-Hall, Inc., 1987, 512 p., vl. 1, [4] J. Prakis, Digital cmmunicatins. McGraw-hill, 1987, vl [5] D. Sargrad and J. Mdestin, Errrs-and-erasures cding t cmbat impulse nise n digital subscriber lps, Cmmunicatins, IEEE Transactins n, vl. 38, n. 8, pp , 199. [6] R. Wainwright, On the ptential advantage f a smearingdesmearing filter technique in vercming impulse-nise prblems in data systems, Cmmunicatins Systems, IRE Transactins n, vl. 9, n. 4, pp , [7] G. Beenker, T. Claasen, and P. Hermens, Binary sequences with a maximally flat amplitude spectrum, Philips J. Res, vl. 4, n. 5, pp , [8] J. Fennick, Amplitude distributins f telephne channel nise and a mdel fr impulse nise, Bell Syst. Tech. J, vl. 48, n. 1, pp , [9] G. Beenker, T. Claasen, and P. van Gerwen, Design f smearing filters fr data transmissin systems, Cmmunicatins, IEEE Transactins n, vl. 33, n. 9, pp , [1] G. Oletu, P. Rapajic, K. Anang, and R. Wu, The smearing filter design techniques fr data transmissin, in Vehicu-

9 Bit errr rate SNR [db] Figure 9. Bit errr rate fr uncded system in the presence f impulse nise: Slid line: with SDT, Dash line: withut SDT n. 16, pp , [2] S. Al-Araji, M. Al-Qutayri, K. Belhaj, and N. Al- Shwawreh, Impulsive nise reductin techniques based n rate f ccurrence estimatin, in Signal Prcessing and Its Applicatins, 27. ISSPA 27. 9th Internatinal Sympsium n. IEEE, 27, pp [21] D. Kai and Y. Ruliang, New methd f chatic plyphase sequences design, Aerspace and Electrnic Systems, IEEE Transactins n, vl. 43, n. 3, pp , 27. [22] A. Azari, S. Esfahani, and M. Rzbahani, A new methd fr timing synchrnizatin in fdm systems based n plyphase sequences, in Vehicular Technlgy Cnference (VTC 21-Spring), 21 IEEE 71st. IEEE, 21, pp [23] J. Pereira and H. Da Silva, Generalized chu plyphase sequences, in Telecmmunicatins, 29. ICT 9. Internatinal Cnference n. IEEE, 29, pp Bit errr rate Miss Grace Oletu received he BSc degree in Physics in 1998 frm Ed State University Ekpma, Nigeria and MSc degree in Persnal and Radi Mbile Cmmunicatin frm Lancaster University, UK. She is currently a PhD student f University f Greenwich with the mbile and wireless research centre at the schl f Engineering, University f Greenwich, United Kingdm. Her research interest includes the design and perfrmance evaluatin errr cntrl cding and Equalizatin SNR [db] Figure 1. Bit errr rate fr cded system in the presence f impulse nise: Slid line: with SDT, Dash line: withut SDT lar Technlgy Cnference (VTC Fall), 212 IEEE 76th. IEEE, 212, pp [11] M. Carey, H. Chen, A. Desclux, J. Ingle, and K. Park, 1982/83 end ffice cnnectin study: Analg vice and viceband data transmissin perfrmance characterizatin f the public switched netwrk, AT&T Bell Labratries technical jurnal, vl. 63, n. 9, pp , [12] J. Fennick, Understanding impulse-nise measurements, Cmmunicatins, IEEE Transactins n, vl. 2, n. 2, pp , [13] F. Kretschmer and B. Lewis, Dppler prperties f plyphase cded pulse cmpressin wavefrms, Aerspace and Electrnic Systems, IEEE Transactins n, n. 4, pp , [14] M. Ackryd, Synthesis f efficient huffman sequences, Aerspace and Electrnic Systems, IEEE Transactins n, n. 1, pp. 2 8, [15] J. Wzencraft and I. Jacbs, Principles f cmmunicatin engineering, New Yrk, p. 319, [16] S. Haykin, Cmmunicatin systems. Jhn Wiley & Sns, 28. [17] F. Kretschmer Jr and F. Lin, Huffman-cded pulse cmpressin wavefrms, Naval Research Lab. Reprt, vl. 1, [18] E. Msca, Sidelbe reductin in phase-cded pulse cmpressin radars (crresp.), Infrmatin Thery, IEEE Transactins n, vl. 13, n. 1, pp , [19] P. Rapajic and A. Zejak, Lw sidelbe multilevel sequences by minimax filter, Electrnics letters, vl. 25, Prf. Predrag Rapajic received his BE in 1982, University f Banjaluka, Bsnia and Herzegvina, ME in 1988, University f Belgrade and his PhD in 1994 frm the University f Sydney, Australia. Befre and after his PhD, he held several full time industrial psitins, the latest ne in 1996 with Mtrla, Sydney. He held academic psitins at the University f Sydney , the Australian Natinal University, Canberra, Australia and the University f New Suth Wales, Sydney, Australia He is currently hlding chair as Prfessr f Cmmunicatin Systems and he is directr f research prgram in Cmputer and Cmmunicatins Engineering, University f Greenwich, UK. Prf. Rapajic s research interests include adaptive multi-user detectin, equalizatin, errr cntrl cding, mbile cmmunicatin systems, multi-user infrmatin thery, infrmatin capacity and mdelling f time variable cmmunicatin channels. Prf Rapajic has published mre than 18 highly influential research papers in IEEE Transactins, bks, peer-refereed internatinal jurnals and cnferences in the area f mbile cmmunicatins. Kwashie A Anang received his Diplma in Electrical and Electrnic Engineering frm the Kwame Nkrumah University f Science and Technlgy, Kumasi, Ghana in 1997 and his BEng degree in Electrical and Electrnic Engineering and MSc degree in Wireless Cmmunicatin frm University f Greenwich in 25 and 27. He is currently ding his PhD with the University f Greenwich. His research interest includes next generatin cellular wireless systems, mbile ad hc wireless netwrks ruting prtcls and antenna design.