UNIDIMENSIONAL PHASE DOMAIN ADAPTIVE EQUALIZER

Transcription

1 UNIDIMENSIONAL PHASE DOMAIN ADAPTIVE EQUALIZER SORIN POPESCU 1, LUCIAN IOAN 1 Key words: 1D PSK equalizer, Asymmetri trasfer futio, Dyami memory delay lie, Passbad hael model Hilbert pairs, Pseudo-gradiet. The PSK is a (2D) modulatio. So, the itersymbol iterferee (ISI) voltage error e = (r a ) depeds o the lie sigal level. But, the phase demodulatio (PD) is etirely free of the sigal level. As a oe-dimesioal (1D) solutio we here iluded both the (ISI) ad iterhael iterferee (ICI) effets ito the phase error ε = (ϕ Φ ) as uique, level free, parameter (1D) for the adaptive equalizatio. With this aim, we first geeralized the equivalet low pass filter theorem (ELPFT). O this basis we foud the phase ad the evelope formulae for the asymmetri hael respose to a retagular phase sigal. The istataeous phase demodulatio (IPD) idea works i this sese. Cosequetly we sueed i equalize the PSK sigal as a 1D oe. We here proposed the method for alulatio of the so alled pseudo gradiet. 1. INTRODUCTION Till ow, hael equalizatio for PSK was approahed as a 2D oe, usig the two ompoets of the square error ad two gradiet futios. The ompariso of the sigal with the deided data, the sigals for timig, arrier reovery ad semi duplex mode, all deped o the reeptio level. All this have bee avoided by osiderig the PSK sigal as a 1-dimesioal oe. The paper is orgaized as follows: i Se. 1 is stated a geeralizatio give by us i [1] for the equivalet lowpass filter. I Se. 2 is preseted the lowpass model of the asymmetri hael ad his oetio with the time respose. I Se. 3 we provide the respose of the passbad systems to sigals with retagular phase [2, 3]. I Se. 4 the ISI ad ICI are desribed ad the idea of the ew pass-bad equalizer (DFE) is itrodued. The ew obtaied equalizer works with zero forig ad without AGC [3]. I Se. 4 are desribed the ISI ad ICI mehaisms I Se. 5, after presetig the istataeous demodulatio ad i Se. 6 the priiples of a ew deisio feedbak pass-bad equalizer (DFE) are itrodued. It is based o a deisio feedbak 1. Politehia Uiversity Buarest, sori.popesu@omm.pub.ro Rev. Roum. Si. Teh. Életroteh. et Éerg., 56, 4, p , Buarest, 2011

2 408 Sori Popesu, Luia Ioa 2 exploitig the phase error, orrelated with the sigs of base-bad data. This fially meas that the ew equalizer works with zero forig ad without AGC [3] THE EQUIVALENT LOWPASS FILTER THEOREM Let s(t) = u T (t)osω t be a a.. retagular pulse applied to a badpass filter whose frequey respose H(ω) is etered o f. We deote H 1 (ω) ad H 2 (ω) as the brahes of H 1 (ω) for positive ad egative frequeie. If U T (ω /+ω )=U 1,2 (ω) the 1 ω t () t [ U ( ω) H ( ω) + U ( ω) H ( ω) ] e dω r = π (1) Let H 1 (ω+ω )=H 2 (ω ω)=h eq (ω) be the equivalet low pass filter of H( ω ). The r ( ) ω t = g( t) osω t eq () t = h () t u ( t) T os. (2) 1.2. THE GENERALISATION OF THE EQUIVALENT LPF THEOREM The a.. pulses are rotated whe passig through asymmetrial haels. Let the hael time domai respose be h(t)= F -1 {H(ω)}. We first fid: 1 h() t = ( ( ω) ω ( ω) ω ) ω π H re os t H im si t d. (3) 0 The BPF respose h(t) will be a pair of a.. pulses with the basebad evelopes -1 eq h ( t) = H ω. os, si F ( ( )) os, si 1 eq 1 eq h ( t) = os ω t F ( ) si F ( ) ; H ω ω os H ω si (4) h() t = hos ( t) os ωt hsi ( t) si ωt. (5) The two basebad filters are: eq eq eq eq eq eq H os( ω ) = Hre,p( ω) Him,q( ω) ; Hsi( ω ) = Him,p( ω ) + Hre,q( ω). (6) We a ow set worth the struture of the two LPFs: A H eq os (ω) has: 1. the real part the eve part of the real part of the BPF, traslated i the base bad; 2. as a imagiary part the odd part of the imagiary part of the BPF, passed ito the base bad. This is (i.e. H eq os (ω)) a hoest (preditable) filter; B H eq si (ω) has: 1. the real part the eve part of the imagiary part of the BPF, i the base bad; 2. as a imagiary part the odd part of the real part of the BPF. If we apply u T (ω)osω 0 t to H(ω) the aswer will have h os (t)*u T (t) = g os (t) ad h si (t)* u T (t) = g si (t) as evelopes. The last term disturbs all the 2D trasmissios.

3 3 Uidimesioal phase domai equalizer THE LOWPASS MODEL OF THE ASYMMETRIC CHANNEL Oe a see i Fig. 1a how our equatio 6 is oeted with the wellkow model [5] of the passbad system. The pulse respose is: h ( t t ) = h ( t t ) osω ( t t ) h ( t t ) ω ( t t ). (7) 0 os 0 0 si 0 si This system ofiguratio leads us diretly to the equalizer struture i Fig. 1b [2, 3]. 0 Fig. 1a The low pass hael model: bad pass low pass bad pass (see e.g. Proakis, Dig. Comm.I MGraw Hill, 2001 [5]). Fig. 1b Our bad pass DFE: lie bad pass part itermediate bad pass part; oe a observe the two Hilbert pairs for ad d.

4 410 Sori Popesu, Luia Ioa 4 3. THE ANSWER OF THE BAND PASS FILTERS TO THE PSK SIGNALS Now we have to kow the phase ad evelope of the filtred PSK sigal. We will aalyze a sigal s 1 (t) with the phase θ(t) suddely growig from 0 o to Φ ad deayig to 0 o after a elemetary iterval T: Φ, for t T/2 s1() t = A0os ω t+θ() t, t R; θ () t =. (8) 0, elsewhere For omputig the respose of a bad pass filter to suh a sigal it is eessary to evaluate four trasiet regimes whe iterruptig ad applyig twie the a.. step futios with phases 0 o ad Φ respetively THE PHASE AND THE ENVELOPE FOR FILTERED PSK SIGNALS A simpler method [1, 3] is to osider s 1 (t) as a sum s 0 (t)+s 2 (t), where: s t) = A osω t, t R, with A 1(8b) 0 ( 0 0 = Φ π T T 2A0si os ω t+ +Φ/2, t, s2() t = , otherwise Let H (ω) be the badpass trasfer futio of the filter. The. (9) eq ( ω + ω ) U ( ω) = H ( ω) ππω/ = G( ω) H T s The respose for s 2 (t) has g(t) as evelope So the outputs are: s s 2 e 0 e () t = 2 g () t si Φ / 2 si. (10) π t + + Φ 2 [ ] () t = s () t = os ω t ; s () t = I () t os ω t + ϕ() t 0 p os ω 1e p / 2 p, (11) with I(t) ad ϕ(t) as the evelope ad phase. From the fazor diagram we have: () t () t g si Φ 2 ϕ() t = arta ; I g si Φ / 2 2 () t = 1 4 si Φ / 2 g () t [ g () t ]. (12)

5 5 Uidimesioal phase domai equalizer THE INTERSYMBOL AND INTERCHANNEL INTERFERENCE The reeived PSK sigal is: () t = os ( t kt) os ( ωt+ Φk) si ( t kt) si ( ωt+ Φk), (13) sr h h k= k= where h si (t) is resposible for the ICI ad h os (t) for the ISI PHASE ERROR STATEMENT IN VIEW OF 1D EQUALIZATION Lets the phase differees ad the system respose h be (Fig. 2): Φ = Φ Φ ; (14) k k () hos ( ) hsi ( ) r t = t os ω t t si ω t ; (15) k ( ) si Φ + ( ) os k si k 0 h k h k ( ) 1 os os Φ k si ( )si Φ k k 0 os Φ tg ε =. (16) + h k h k Fig. 2 The iterferee ISI ad ICI phasors, h os (t) ad h si (t) respetively, ad the resulted phase error ε. For eight phases ε must be up to 22.5 o. So, os x 1, ta x x. We fid: ε k 0 { ( ) si Φ ( ) os hos k - k + hsi k Φ - k }. (17)

6 412 Sori Popesu, Luia Ioa 6 5. ALL UNIDIMENSIONAL DIGITAL PSK RECEIVER 5.1. INSTANTANEOUS PSK DEMODULATION (IPSKD) The basi idea [4] of the Istataeous DPSK Demodulatio (IPSKD) is to ompare the phases of the sigal with the states of a Demodulator Otal Couter. The sigal diagram from Fig. 3a is the heart of the reeiver. Here we a see how the risig edges (RE) of the lie sigal will fid the DOC i oe of the states, = 0, 1, 2,...7. Just ow the outer state umber is to be memorized. It is ust the trasmitted data tribit N3, N2, N1 i the BCD ode. After a Gray deodig, the tribits G3, G2, G1 are set up ito a parallel/serial shift register R3, R2 (Fig. 3b). G3 ad G2 will be used as deided data for the DFE. 6. THE 1D EQUALIZER The omplexity of a equalizer depeds o the sigal dyamis, the SNR ad o the eletrial legth of the hael. Turbo equalizatio [6, 7, 8, 9, 10] requires huge omplexity, a drawbak for real time work. I [6] are itrodued redued omplexity estimatio algorithms ad i [7] is performed a oied equalizatio ad deodig of ompoet odes. There, the hael ad the equalizer play respetively the roles of iteral eoder ad deoder. I [8] ad [9], for turbo equalizatio with DFE is proposed the a posteriori probability (APP) estimatio uder delay ostraits. Fig. 3a 1) left-had diagram [4]: DOC sigal diagram related with the lie PSK sigal the outer states ad the eight phases are bietive groups; 2) right-had diagram [6]: the risig edges RE of T +/ (i.e. the first betwee T + or T from the PSK sigal) pulses are phase samples seleted by R B symbol lok. The ϕ(t) has a aperture error ε due to the T+/ eetri positio.

7 7 Uidimesioal phase domai equalizer 413 Fig. 3b The sig of the phase osie is the G2 data bit; the G3 bit is for sie sig;they are the data dibits for a QPSK with their better (tha 8-PSK) oise protetio DYNAMIC MEMORY DELAY LINE (DMDL) After the data trasfer from the demodulator otal outer (DOC) i the data register, this outer is deleted i the 0 state (000) by the same positive risig edge (PRE) of the T +/ trasitio. So, the DOC is opyig the lie sigal phases ad saves it idefiitely, util a ew trasfer. But the lie sigal phase is umpig with Φ k uder the emissio data otrol. I the ext iterval the phase differee Φ k betwee the lie sigal ad the DOC state will represet the data tribit i the BCD ode. The DOC saves the sigal s phase ump Φ k = lπ/4 by its dyami outig proess. The DOC is thus a ew phase modulator. It provides us with deided replia of the trasmitted phases. O the ew iterval (k+1)t the Φ k phase from DOC (OC1) is loaded i OC2 (it is a loadig outer with 8f lok too). OC2 memorizes this old phase o the ew iterval (k+1)t. The OC2 is outig ad dividig from 8f till 1f, with this iitial phase. It geerates a PSK sigal with 1f frequey ad with the old phase Φ k. So we ustify the ame of dyami memory delay lie (DMDL) for all the ext OC s. We otiue the operatio for Φ k + 2 ad so o. It is easily possible to exted its delay idefiitely. The DMDL ells are memorizig the ideal, deided. Moreover, very well time-limited, this sigal is oiseless too THE EQUALIZATION CRITERIA If we sueed i fidig the evelopes h os, si (k) the their iipiet values will be used to adust the taps ad d of the DFE equalizer. The phase error ε is obtaied from eq. (17) replaig the hael resposes h os/si (t) by g os/si (t), the equalizer s resposes to it: ε gsi ( ) os Φ- +gos ( ) si Φ- 0, (18)

8 414 Sori Popesu, Luia Ioa 8 where: os/ si( t) = ( / d ) hos/si( t T) ; gos/si( ) = ( / d ) hos/ si( ). g (19) The phase error ad the gradiet of the m.s.e. beomes, with 2 E = ε : ε ( ) os Φ - ihsi i + si Φ - dihsi ( i) ; (20) 0 i i ε E / i di = 2 ε. /d i The gradiet has two terms: ε ε = os Φ- hsi ( i ) ; si Φ- hos ( i) i 0 =. (22) di 0 The gradiet algorithm desribes adustmet of the oeffiiet vetors ad d: + 1 ( d ) = ( / d ) i / i i i α E i / di (21), (23) ad α is the equalizatio step. It must to kow the derivative of ε. Now we suggest to use aother kid of m.s.e. derivative, deomiated pseudo-gradiet, defied by: [ os Φ /si - Φ - ] ε ψ E gos / gos = ε = ε. (24) g ( ) / g ( ) 0 os si <xx> is the expetatio. We eed to multiply both sides of ε formula from eq. (20) first with os Φ -k, the with si Φ -k ad take the mea values. First oe observe: 0 for k os Φ - os Φ -k =, (25) 0 for k = - -k The we a write the fial, desired result: os o si i = 0, k, Z. (26) ψ E g = os = g ( ) ; E g = si = g ( ) si ε Φ - si ψ ε Φ-. (27, 28) The ad d taps will be adusted aordig to eq. (27) ad (28). Their effet exerts oly o the phase error. So g os () is erased by adustig the values util this g os () disappears ad aquires its fial value. The g si () is assoiated with d i a similar way. So the phase error beomes miimum. os os

9 9 Uidimesioal phase domai equalizer ANTERIOR COEFFICIENTS ADJUSTMENT For the adustmet of the aterior equalizer taps: -1, -2, d -1, d -2, the (27) ad (28) formulae do ot apply, beause they imply (for < 0) to kow Φ -. This a however happe e.g. at +3, before the deisio momet <. The solutio is to hage i the previous formula the time origi: '=, = '+. That meas: ( T ), 0 ε os Φ - = g si >, (30) ' + { Φ Φ + } g ( ) ε + os = Φ =, < 0. (31) ' ' si So, oe delay the phase error sig ad orrelate it with deided phase umps. The 90 o rotatio for the aalogial orretio of both g os (t)os(ω t) ad g si (t)si(ω t) is doe by traslatig the lie sigal o a higher frequey with two itermediary quadrature arriers f it ad a BPF. After filterig the lateral upper bad, oe add the deided sigal from dyami memory delay lies (DMDL ) adusted by the posterior oeffiiets d POSTERIOR TAPS ADJUSTMENT BY DFE EQUALIZATION For posterior taps we eed: the phase error ε ad the sie ad osie sigs of Φ ' '- i.e. the MSB G3, G2 (Fig. 2b) of the data tribit. We detet the phases at T usig as arriers the deided ad delayed lie sigals from DMDL (Fig. 3, 4). I Fig. 4 oe a see the posterior lobs orretio usig the deided sigals from DMDL. This deided sigals r^p, q (t T) exist o a sigle iterval eah, with ostat phase. They retur through the feedbak loop, by the BPF. Here they aquire their ow evelope, whose maximum is at (+1)T. This omes exatly whe we have to erase the iterferee sigal. This sigals are weighted with the tap values, d. After summig up, these orretios pass a BPF. The eessary phase errors ε appear, prie free by their sig from the arrier reovery blok. Some ew ideas i the field appear i [10] ad [11]. I [10] is itrodued the oept of DFE equalizatio with bidiretioal arbitrary DFE (BAD), havig a 1-2 db gai ompared to DFE ad oly 1dB less tha the MAP oe. However, the BAD omplexity is oly liearly ireasig with the hael legth, while the MAP oe is ireasig expoetially. I [11] is itrodued the idea of softfeedbak equalizer, SFE, for highly dispersive haels, whih overpasses the lassial BCRJ-based turbo equalizer i this situatio.

10 416 Sori Popesu, Luia Ioa 10 Fig. 4 Deisio feedbak equalizer priiple for the bak lobes: a uequalized sigal r (t T) ; b deided sigal r^p(t T) with argumet Φ -1 for 1 ; BPF respose for r^p(t T); d BPF respose for r^q(t T). By ombiig liear equalizatio ad soft itersymbol-iterferee aellatio, the SFE oeffiiets are hose to miimize the mea-squared error (MSE) betwee the equalizer output ad the trasmitted sequee, uder a Gaussia approximatio to the a priori iformatio ad the SFE output. The resultig omplexity grows oly liearly with the umber of oeffiiets. It is opposed to the quadrati omplexity of reported miimum-mse strutures. Uexpeted appliatios have bee proposed i [12] ad [13]. I [12] a zeroforig equalizer is used for iter-proessor oetios ad i [13] is used a blid equalizatio. The simplifyig sig algorithm is similar to the here preseted oe, but our approah is ot a blid equalizatio. The Tomliso [14] equalizer must obtai a 2D reeiver error o a miimum four level, so equalized, haell. Our 1D solutio allows to use a biary, possibly uequalized, hael. 7. CONCLUSIONS We foud the ext theoreti ad prati results: a. The oetio betwee the evelopes h si, os (t) of the time respose of the asymmetri hael ad the odd ed eve parts of the real ad imagiary parts of the frequey respose H(ω). b. The ilusio of the ISI ad ICI effets ito the phase error ε as uique parameter for adaptive equalizatio.

11 11 Uidimesioal phase domai equalizer 417. The phase ad the evelope for the respose of the hael to a retagular phase exitatio, usig oly the time resposes of the equivalet basebad filter. d. The simplest delay lie DMDL for the multilevel arrier PSK sigal. e. The oe-dimesioal (ε ) 2 parameter as uique square error. f. The equalizer is a zero-forig oe, iludig the aterior taps, ulike the DFE. g. The isesibility to quik level hages is a huge advatage for atual wireless etworks. Reeived o April 29, 2010 REFERENCES 1. S. Popesu, I. Băiă, Adelaida Mateesu, A pass bad Equalizer with Digital Delay Lie, Proeedigs of sixth Summer Shool Symposium o Ciruit Theory, pp , Prague, July Adelaida Mateesu, I. Băiă, S. Popesu, A Digital Method of Filterig for DPSK Data Sigals, Rev. Roum. des Si. Teh. Életroteh. et Éerg., 27, pp (1983). 3. Ad. Mateesu, I. Băiă, S. Popesu, Equalizers for Data Trasmissio over Telephoe Chaels, Eur. Coferee o Ciruit Theory ad Desig, ECCTD'80, Warsaw, Sept. 1980, pp A. Tahauser, High Data Trasmissio with Differetial Phase Modulatio NTZ Haft 7, J. Proakis, Digital Commuiatios, MGraw-Hill, 2001, pp Ad. Pău, Algoritmi u omplexitate aritmetiă redusă utilizaţi î turbo-egalizare şi estimarea iterativă a aalului, Politehia Uiversity of Buarest, D.Raphaeli, Combied Turbo Equalizer ad Deodig, IEEE Comm.Lett. 2, 4, pp , J. Moo; F.R. Rad, Turbo equalizatio via ostraied-delay APP estimatio with deisio feedbak, IEEE Tras.Comm., 53, 12, pp , L. Jug-Tao, S. B. Gelfad, Optimized deisio-feedbak equalizatio for ovolutioal odig with redued delay, IEEE Tras.Comm., 53, 11, pp , J.K. Nelso, A.C. Siger, U. Madhow, C.S. MGahey, BAD: bidiretioal arbitrated deisiofeedbak equalizatio, IEEE Tras.Comm., 53, 2, pp , R. Lopes, J.R. Barry, The soft-feedbak equalizer for turbo equalizatio of highly dispersive haels, IEEE Tras.Comm., 54, 5, pp , T. Toifl et all., Low-omplexity adaptive equalizatio for high-speed hip-to-hip ommuiatio paths by zero-forig of itter ompoets, IEEE Tras.Comm., 54, 9, pp , Gi-Hog Im, Cheol-Ji Park, Hui-Chul Wo, A blid equalizatio with the sig algorithm for broadbad aess, IEEE Commuiatios Letters, 5, 2, 2001, pp M. Tomliso, A ew Automati Equalizer Employig modulo Arithmeti Elletr. Lett., 7, pp , 1971.