QAM CARRIER TRACKING FOR SOFTWARE DEFINED RADIO

Transcription

1 QAM CARRIER TRACKING FOR SOFTWARE DEFINED RADIO M Jame D. Scheude (Scheude Engineeing, Sydney, NSW, Autalia; jame.cheude@cheude.com.au) ABSTRACT Thi pape invetigate Quadatue Amplitude Modulation (QAM) caie tacking technue fo Softwae Defined Radio implementation. Thee technue wee developed duing a ecent invetigation poject into the baeband ignal poceing equiement of a daft TIA Public Safety Radio tandad popoal Scalable Adaptive Modulation (SAM) [1]. The popoal pecified ue of fequency multiplexed QAM channel incopoating Pilot Symbol Aited Modulation (PSAM) [2]. Thi popoal ha ince been futhe updated and now fom the bai of the cuent P34 tandad. The invetigation wa pefomed uing MATLAB baed imulation code to implement baeband adio ynchonization algoithm. The algoithm included: 4/16/64 QAM caie and ymbol clock tacking, digital AGC, and channel gain etimation technue. Thi pape will focu on explaining how a eliable QAM caie tacking technue can be developed fom the baic building block of Phae Locked Loop (PLL) implemented in oftwae a invetigated fo SAM. 1. INTRODUCTION A commonly ued method of tacking RF ignal phae i the Phae Locked Loop (PLL). In an analog cicuit implementation, a PLL i compied of an incoming inuoidal ignal which i multiplied by the inuoidal output of a Voltage Contolled Ocillato (VCO). The VCO input et the phae offet of the inuoidal output and i contolled by the DC output of the loop filte. An example block diagam of a PLL cicuit (fom [3]) i hown below: The loop filte i deigned uch that it DC output level i adjuted popotionately to the phae offet between the input inuoid and the VCO output inuoid. Auming the feedback loop i table, the input to the VCO will adjut the phae of the output inuoid until it phae offet matche that of the input inuoid. 2. IMPLEMENTING A DIGITAL PLL IN SOFTWARE In oftwae, a PLL can be implemented uing a oftwae loop to poce each ignal ample. Thi implementation eplace the analog loop filte with a digital veion and the VCO with a yntheized VCO contucted fom a imple inuoid function (e.g. co(2πf c t[i] + θ), whee t[i] i an aay of time value and θ a phae value). Howeve the continuou adjutment capability of the analog loop filte and VCO mut be eplicated in a oftwae loop uch that afte the input of each ample, the (delayed) output level of the digital loop filte can be ued to adapt the digitally yntheized VCO phae uch that it equal the phae offet of the input ignal (i.e. phae lock). Thi adaptation hould occu with each loop iteation and minimize the diffeence between the phae offet of the input ignal and digital VCO output. Thi i an example of an optimization poblem that can be olved uing Adaptive Paamete Etimation [4]. Thi method ue an iteative algoithm to etimate an unknown paamete value x by decending (o acending) the gadient of a pefomance function, denoted J(x). The deivative of the function, dj(x)/dx, i fomulated to have a zeo value when the eo between x and the cuent etimate of x[ i minimized. Thi poce can be defined by the equation: dj ( x) x[ k + 1] = x[ µ dx Whee x[ i an etimate of x at time k, x[k+1] i the etimate of x at time k+1, µ i a mall poitive numbe efeed to a the algoithm tep ize. In the PLL cae, J(x) need to be maximized when the phae of the input ignal and yntheized VCO output ae Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

2 equal which povide the highet DC output value fom the digital loop filte. J(x) i defined by [4] a: J ( φ ) = LPF [ kt ].co(2π. f0kt + φ[ ) PLL { } 0.14 econd. Thi ovehoot can be minimized by educing the tep ize, µ. Howeve a tade-off exit in that malle µ value inceae the numbe of loop iteation equied. Auming a mall µ, the deivative of J PLL (φ) with epect to φ at time k can be appoximated a: dj PLL ( φ) = LPF{ [ kt ]in(2πf 0kT + φ[ ) } dφ The coeponding adaptive update equation i then: φ [ k + 1] = φ[ µ LPF [ k. T ]in(2πf 0kT + φ[ )} { } The figue below how the digital PLL containing a digital VCO, a ignal multiplication tep, a digital loop filte and an adaptive element update: φ[k+1] µ z -1 φ[ LPF in(2.π.f 0. k.t + φ[) The MATLAB code ued to implement thi PLL i a follow: T=1/2000; time=1; t=0:t:time; % time vecto f0=200; phoff=pi/2; % caie feq. and phae fc=200; % aumed feq. at eceive p=co(2*pi*fc*t+phoff); % implified eceived ignal caie = p; fl=100; ff=[ ]; fa=[ ]; h=fipm(fl,ff,fa); % LPF deign mu=.01; % algoithm tepize theta=zeo(1,length(t)); theta(1)=0; % initialize vecto fo etimate z=zeo(1,fl+1); % initialize buffe fo LPF fo k=1:length(t)-1 % z contain pat fl+1 input VCO(k) = in(2*pi*f0*t(k)+theta(k)); z=[z(2:fl+1), p(k)*vco(k)]; update=flipl(h)*z'; % new output of LPF theta(k+1)=theta(k)-mu*update; % algoithm update end The plot below how the oftwae PLL ynchonizing with input 200Hz coine ignal with phae offet of π/2 adian. The VCO initially ha a zeo phae offet. In the fit plot, the input ignal i hown in ed and the VCO ignal in blue. Iteation of the loop adjut the phae of the VCO to match the phae of the input ignal. At appoximately 0.2 econd, the two ignal ae aligned and the PLL i phae locked. 3. QAM PHASE RECOVERY USING PLLS A common method ued to tack a digital adio caie i to extact a inuoidal caie ignal fom the eceived ignal and then tack the ignal caie phae uing a PLL. Thi method i ecommended fo digital modulation uch a Pule Amplitude Modulation (PAM) whee a naowband filte can extact a eplica of the caie ignal due to it lage fequency component magnitude. Howeve, QAM modulation epeent a uppeed caie modulation cheme. The QAM pectum ha no ditinguihable caie fequency component fo a PLL to tack. Fo example the plot below how the pectum of a 4-QAM (QPSK) ignal a ued by SAM (note that the meage ignal ha been Raied Coine filteed) with a uppeed caie at 2500Hz. The econd plot how the VCO phae offet a it adapt to the input ignal phae offet. The plot confim that phae lock i achieved at appoximately 0.2 econd and the phae locked value i the expected value of π/2 adian. The plot alo how that the algoithm initially ovehoot the coect phae etimate and peak at 0.52 adian at appoximately Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

3 Depite thi, [5] decibe a method fo QAM phae ecovey by tacking the caie of the ignal, pemanipulated with a 4 th -powe opeation, followed by a naowband filte centeed on the quadupled caie fequency component. The pefomance function uitable fo a 4 th powe PLL i then detemined to be: 1 J = co(4( θ )) 4PLL φ 4 whee θ i the phae of the eceived ignal and φ i the phae of the VCO in the PLL. Thi will be maximized when θ = φ. Uing a gadient decent tategy, the update equation i then defined a: dj4pll φ[ k + 1] = φ[ + µ dφ giving an update equation of: φ[ k + 1] = φ[ + µ in(4( θ[ φ[ )) A uitable PLL deign i hown in the figue below: Both [6] and [7] tate that the method wok fo 4-QAM, but [6] tate that it only wok paably well fo 16-QAM due to the difficulty in emoving the non caie fequency component intoduced. And [7] tate that 16-QAM contellation patten jitte will be appaent but that the oute point dominate uch that the fouth-powe ignal ha a component at 4f c that i uable if a vey naow band PLL i ued to efine it. Howeve, a naow band filte will equie a high filte ode that will lead to inceaed DSP computation cot and hould be conideed in any SDR deign. 4. QAM PHASE RECOVERY USING DECISION- DIRECTED CARRIER TRACKING The plot below how the pectum fo a fouth powe 4- QAM ignal. A tong fequency component can be een to exit at 4 time the caie fequency (4 * 2500Hz = 10,000Hz). Unfotunately the technue i not a ucceful at extacting highe ode M-QAM baed caie ignal. The plot below how the fouth-powe ignal pectum fo 16-QAM and 64- QAM. The magnitude of the caie component i hown to diminih elative to the meage fequency component a the QAM ode inceae. The final QAM caie tacking method invetigated take a diffeent appoach to caie tacking. The method geneate a phae eo ignal by exploiting the phae and amplitude diffeence between each eceived ymbol value and the neaet ideal QAM contellation ymbol value. Thi method i called Deciion-Diected Caie Tacking (DDCT) becaue the deciion (the choice of the neaet allowable ymbol in the contellation) diect the adaptation of the eceive caie phae to match the tanmitte phae by minimizing the mean-quae of an eo function. Thi eo function (popoed in [8]) i defined a the phae diffeence between each eceived baeband ymbol [kt ] and it neaet ideal contellation point c on the complex plane: e kt ] = c [ kt ] [ Theefoe the pefomance function to be optimized in thi algoithm i the mean quaed eo (MSE) of the baeband ymbol demodulation eo: 2 J MSE = E{ e[ kt ] } The adaptive update algoithm can then be defined a: µ J φ[ k + 1] = φ[ + MSE φ The analyi in [8] doe not diectly deive the expeion fo J MSE. Intead the following elationhip between the demodulated ymbol phae α and neaet ideal contellation point α c i detemined: in( α [ e[ kt ] [ kt ] Im ] αc ) = c Thi ha the ame ign a the phae eo between the ideal contellation point c and the eceived demodulated baeband ymbol [kt ], povided the phae eo i not too Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

4 lage. Since in(φ) i appoximately equal to φ fo mall φ, thi tem i alo nealy a linea function of the eo fo mall phae φ value. In addition, the plot below how that the algoithm uccefully tack a vaying elative caie phae at the eceive: Thi phae eo ignal can be ued in a phae-locked loop to iteatively adjut φ o that the demodulated baeband ymbol value ae aligned in phae angle with the ideal contellation point. Changing φ by ome angle ha the effect of otating the baeband ymbol value by the negative of the phae angle. Thi eo ignal deivation give the final adaptive update equation a: Im[ e[ kt ] [ kt ]] φ[ k + 1] = φ[ + µ c The MATLAB code ued to implement thi PLL i a follow: CARRIER = 1000; k = 1; mu = 0.1; M = 16; T = 1 / 4800; phaenow = 0; phaeet = phaenow; phaeinc = 2*pi*CARRIER * T; fo = pbsymbol(1:end) % An aay of paband QAM ymbol % Demodulate the paband ymbol and toe in aay Symbol[ =.* exp(-j * phaenow); % Find the neaet QAM contellation point to ymbol deciionsymbol = qammatch(, M); % Calculate the phae eo deciioneo = deciionsymbol - ; % Calculate the new phae etimate theta[ = phaeet; phaeet = phaeet + mu * (imag(conj(deciioneo)*) / (ab(deciionsymbol)*ab())); % Calculate the next demodulation phae value phaenow = phaenow + phaeinc + phaeet; k = k + 1; end The following plot how the eult of the above eceive PLL achieving caie phae lock with a fixed tanmitte caie phae offet of 0.2π adian and fequency 1000Hz. Reult fo 4, 16, and 64-QAM ignal ae hown fo the SAM ymbol clock ate of 4800 baud. The ed line how the known phae offet intoduced in the tanmitte. The blue line how the value of φ[ a the PLL convege to the fixed tanmitte phae. Each plot wa geneated uing a µ value of QAM CARRIER FREQUENCY OFFSETS Peviouly it wa aumed that the caie fequency ued by the tanmitte wa exactly known by the eceive. In pactice, RF and IF demodulato will nomally exhibit mall fequency toleance leading to a fequency offet between tanmitte and eceive. To undetand the impact of thi offet, a eceived paband ignal pb can be witten in tem of the oiginal QAM ymbol meage team m[: (2π +θ ) = [ ] j f t kt pb m k e The tanmitte caie fequency i identified by f t and phae offet θ. Following demodulation by the eceive, the eceived QAM ymbol baeband value ae then defined a: j(2π ftkt + θ ) j(2πf kt + φ ) = m[ e e j[2π ( ft f ) kt + ( θ φ )] = m [ e whee φ i the phae of the eceive. Now auming that f = f t, thi equation then become: j( θ φ ) = m[ e Theefoe the demodulation tep eult in phae otation of the demodulated ignal equal to the phae diffeence between the tanmitte phae θ and eceive VCO phae φ, (θ - φ). Demodulation of a 16-QAM ignal with 0.2π adian phae offet then eult in the following 16-QAM contellation: Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

5 The contellation plot how the eceive QAM ymbol ae otated by 0.2π adian a pedicted. Auming that f f t and that (f f t ) = 10Hz then the eceived baeband ymbol team can be defined a: [ j(2π 10 kt + 0.2π ) = m[ e The fequency offet will theefoe intoduce a phae otation in the eceived QAM ymbol contellation popotional to the ize of the fequency offet (the ymbol contellation i otating with time). The plot below how the contellation eult of demodulating a 16-QAM ignal with 0.2π adian phae offet and 10Hz fequency offet: The eceived ymbol contellation i pinning a pedicted. And the phae otation can be defined a: θ[ 2π ( f f ) kt + ( θ φ) = t Thi indicate that the input to the PLL VCO, φ[, will exhibit a phae offet of (θ φ) and a gadient equal to the fequency diffeence (f f t ). Thi i confimed by the following plot uing the QAM baed PLL with a fequency offet of 10 Hz and phae offet of 0.2π adian. The blue plot how the φ[ value geneated by the PLL loop, the ed plot the contant phae offet, and the geen plot the pedicted φ[ value calculated fom: φ[ 2π 10 kt π = Inteetingly, the 4-QAM plot convege exactly to the pedicted plot, but the 16 and 64-QAM plot convege to the ame gadient but with a diffeent phae offet. Thi i due to the inheent π/2 phae ambiguity in the Deciion- Diected algoithm. Thi ambiguity eult fom the fact that the eceive cannot ditinguih between phae offet that ae multiple of π/2 due to the quae ymmety of M-QAM contellation (a quae contellation look exactly the ame when it i otated by π/2 adian a it doe otated by π adian etc). It i ecommended in [5] that thi ambiguity can be eolved uing one of the following method: diffeentially encoding the meage ouce o that the change in ymbol value between each ymbol i known letting a tained equalize automatically add a otational phae to achieve a match to the taining ymbol coelating the down-ample output with a known/taining ignal Howeve, [9] alo ugget by inetion of known data ymbol into the ymbol team. Since SAM wa pecified to ue ineted Pilot and Synchonization ymbol in the data team (a PSAM cheme), the pilot and ync ymbol wee ued to ovecome the π/2 phae ambiguity in the poject. 6. TRACKING FREQUENCY OFFSETS IN DECISION-DIRECTED QAM CARRIER TRACKING When the cuent DDCT PLL i opeating with a caie fequency offet, the contellation ceae to otate with time but emain fixed with an angula offet that i popotional to the fequency offet. Howeve, ince the fequency offet ha been hown to be imply the gadient (deivative φ/ T) of the changing phae, the contellation angula offet can be coected uing an additional accumulato in the PLL (a 2nd-ode loop). In [8], the exiting QAM PLL i extended to tack the fequency offet. The QAM phae eo ignal that wa popotional to ine of the angle between the demodulated ymbol and the neaet ideal contellation point wa deived to be: Im [ e[ kt ] [ kt ]] in( α αc ) = c Theefoe, the phae eo to be tacked on each PLL loop iteation i: φ = [ e[ kt ] [ kt ] Im ] c In ode to tack the additional phae change due to a caie fequency offet, the following additional phae accumulation tep i included in the PLL: ψ [ k + 1] = ψ [ + µ 2 φ[ Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

6 The final completed econd-ode adaptive update equation fo DDCT i then: ϕ [ k + 1] = ϕ[ + 2πf T + µ 1 φ[ + ψ [ The peviou MATLAB code example can then be extended a follow: CARRIER = 1000; k = 1; mu = 0.1; M = 16; T = 1 / 4800; phaenow = 0; pi = 0; phi = 0; phaeinc = 2 * pi * CARRIER * T; fo = pbsymbol(1:end) % An aay of paband QAM ymbol % Demodulate the paband ymbol and toe in aay Symbol[ =.* exp(-j * phaenow); % Find the neaet QAM contellation point to ymbol deciionsymbol = qammatch(, M); % Calculate the phae eo deciioneo = deciionsymbol - ; % Calculate the new phae etimate theta[ = phi; phaeeo = (imag(conj(deciioneo)*)) / (ab(deciionsymbol)*ab()); pi = pi + mu2 * phaeeo; phi = mu1 * phaeeo + pi; % Calculate the next demodulation phae value phaenow = phaenow + phaeinc + phi; k = k + 1; end The two plot below how the eult of PLL code with a caie fequency of 2700Hz caie and fequency offet of 210Hz. The left plot how the coectly demodulated contellation depite the lage caie fequency offet. The ight plot how the contellation of a demodulated 16-QAM ignal with channel noie intoduced (SNR = 20dB). The algoithm i effective even with noie intoduced. Thi i becaue the Adaptive Paamete Etimation technue inheently povide a low-pa filteing (aveaging) function on the PLL VCO output. ignal. Thi method can then be extended to tack the phae and fequency of QAM ignal uing the appopiate pefomance function. Deciion Diected Caie Tacking povide a eliable method fo tacking caie phae and fequency offet fo M-QAM ymbol. It can be implemented with low computation cot a it only opeate on the eceived paband M-QAM ymbol value athe than on evey ignal ample. The method can alo opeate uccefully in channel noie. Howeve the method ha the diadvantage of the inheent π/2 phae ambiguity at the demodulato output and the equiement fo the ymbol clock timing to be known pio to demodulation. In the TIA SAM popoal, both diadvantage wee eolved by the ue of Synchonization and Pilot ymbol ineted into the potocol data team. 8. REFERENCES [1] TIA Wideband Ai Inteface Scalable Adaptive Modulation (SAM) Phyical Laye Specification, TIA- 902-BAAB, [2] J.M. TORRANCE, L. HANZO, Compaative Study of Pilot Symbol Aited Modem Scheme. Sixth Intenational Confeence on Radio Receive and Aociated Sytem, pp 36 41, Septembe [3] R. E. BEST, Phae-Locked Loop: Deign, Simulation and Application, McGaw-Hill [4] R. JOHNSON, W. A. SETHARES, W. A., Telecommunication Beakdown: Concept of Communication Tanmitted by Softwae-Defined Radio, Peaon Pentice Hall, [5] R. JOHNSON, A Digital Quadatue Amplitude Modulation (QAM) Radio, Peaon Pentice Hall, 2003 [6] J.B.ANDERSON, Digital Tanmiion Engineeing, Pentice Hall, [7] J. A. C. BINGHAM, The Theoy and Pactice of Modem Deign, Wiley Pe, CONCLUSION It ha been hown that a imple PLL loop can be implemented in oftwae uing Adaptive Paamete Etimation. The etimation method equie the deivation of a pefomance function that the algoithm decend (o acend) to lock onto the unknown phae of a eceived [8] S. A. TRETTER, Communication ytem deign uing DSP algoithm: with laboatoy expeiment fo the TMS320C6701 and TMS320C6711, Kluwe Academic/Plenum Publihe, New Yok, [9] L. E. FRANKS, Caie and Bit Synchonization in Data Communication - A Tutoial Review. IEEE Tanaction on Communication, COM-28, 1980 Poceeding of the SDR 08 Technical Confeence and poduct Expoition, Copyight 2008 SDR Foum, Inc. All Right Reeved

7