A Simple Improvement to the Viterbi and Viterbi Monomial-Based Phase Estimators

Transcription

1 Univesity of New Oleans Electical Engineeing Faculty Publications Depatment of Electical Engineeing 006 A Simple Impovement to the Vitebi and Vitebi Monomial-Based Phase Estimatos Kenneth V. Catwight College of The Bahamas Edit J. Kaminsky Univesity of New Oleans, ejbouge@uno.edu Follow this and additional woks at: Pat of the Electical and Electonics Commons Recommended Citation Catwight, K. and E. Kaminsky, A Simple Impovement to the Vitebi and Vitebi Monomial-Based Phase Estimatos", in IEEE Globecom 006 Conf. Poc., San Fancisco, CA, 7 Nov.- Dec. 006, 6 pp. This Confeence Poceeding is bought to you fo fee and open access by the Depatment of Electical Engineeing at ScholaWoks@UNO. It has been accepted fo inclusion in Electical Engineeing Faculty Publications by an authoized administato of ScholaWoks@UNO. Fo moe infomation, please contact scholawoks@uno.edu.

2 A Simple Impovement to the Vitebi and Vitebi Monomial-Based Phase Estimatos Kenneth V. Catwight School of Sciences and Technology College of The Bahamas P.O. Box N9 Nassau, N.P., Bahamas Abstact It is well known that the Vitebi and Vitebi Monomial- Based Phase Estimato, which includes the M th Powe Estimato, pefoms pooly fo coss QAM signals. Howeve, it is shown hee that by allowing the powe of the monomial to be negative, much impoved pefomance can be ealized at medium to high signalto-noise atios (SNR). Monte Calo simulations ae used to demonstate the efficacy of this novel simple extension, fo 3- and 8-QAM systems. In pinciple, this extension can also be applied to othe constellations, e.g., (,)-PSK. Keywods Synchonization, blind phase estimation, quadatue amplitude modulation, blind caie phase ecovey. T I. INTRODUCTION HE need fo blind phase ecovey in quadatue amplitude modulation (QAM) systems is well established. In ode to satisfy this need, many systems have been invented. These systems can be gouped into two aeas those that equie established gain contol and those that do not. The Fouth Powe Phase Estimato []-[3], which is a special case of the Vitebi and Vitebi (V&V) monomial-based estimatos [], the Eighth-Ode Estimato () [5], the Concentation Ellipse Oientation (CEO) [6], and moe ecently the iteative methods (DCA-a and DCA-b) of Alvaez-Diaz and Lopez-Valcace [7] ae systems in the latte categoy. Among the fome categoy ae the Reduced-Constellation Fouth Powe Estimato [], the two methods of Geoghiades [] which equie finding the mode of the pobability density of the phase, the athe complex Minimum Distance Estimato (MDE) [8], the Two- Stage Conjugate (SC) algoithm which accoding to Rice et al. [8] is simila to the Two-Pass algoithm of [9, pg. 33], the optimal method, poposed by Wang and Sepedin [0], who along with Ciblat [] have also intoduced the APP Estimato, which appoximately implements the optimal estimato, and moe ecently the Reduced Constellation Eighth-Ode Estimato (RC) fo coss QAM signals []. The pupose of this pape is to popose anothe phase estimato that does not equie established gain contol and is no moe complicated than the V&V monomial-based estimato, although with slightly inceased computational expense due to the equied ecipocal opeation. In fact, it is identical to this estimato, except negative powes of the monomial ae now allowed. Although this seems to be a tivial idea, it is not at all self-evident. Indeed, the authos could find Edit J. Kaminsky Depatment of Electical Engineeing EN 85 Lakefont Campus Univesity of New Oleans New Oleans, LA 708, U.S.A. ejbouge@uno.edu no hint in the liteatue that negative powes would be of any inteest. In fact, all pevious authos have assumed nonnegative powes. This simple but effective extension is demonstated fo 3- QAM and 8-QAM. We show that fo these systems opeating at medium to high signal-to-noise atios (SNR), the negative powe monomial-based estimato can povide much impoved pefomance ove the conventional V&V monomialbased estimato. Howeve, in pinciple, this new extension can also be used fo othe constellations, e.g. (,)-PSK which is useful in the non-linea satellite channel [3]. The oganization of this pape is as follows: in Section II, a statement of the poblem we ae tying to solve is pesented, followed in Section III by a eview of the V&V monomialbased estimatos that ae known to solve this poblem. In Section IV, we pesent ou method that impoves the pefomance of these estimatos, and we demonstate the effectiveness of ou impovement in Section V. In Section VI, we conside some implementation issues fo these new estimatos. Finally, in Section VII, we povide some concluding emaks. II. STATEMENT OF THE PROBLEM whee { } + The eceived signal is given by jθ = e X ( n) + V ( n), n = 0,, N () X ( n) X ( n) jx ( n) is the sequence of zeomean unit vaiance, i.e., E ( X ( n ) ) i =, independently and identically distibuted (i.i.d.) QAM complex tansmitted symbols, { V ( n) = V ( n) + jvi ( n) } is a zeo-mean cicula white Gaussian noise pocess, independent of X (n) and with vaiance σ in each component, and θ is the phase angle to be detemined by obseving the eceived signal Y (n). Futhemoe, the eceived signal-to-noise atio is SNR = / ( σ ). It will also be convenient to ewite () in pola fom to get jφ( n) = ρ( n) e, n = 0,, N. () The blind estimation poblem is to find an estimate fo θ, denoted θ, without actually detecting the data X. Note that because X has quadant symmety, it is only possible to ecove

3 θ within π / ad. Without loss of geneality, we assume π / < θ < π / ad. III. REVIEW OF THE V&V PHASE ESTIMATOR In [], Vitebi and Vitebi intoduced phase estimatos suitable fo M-ay Phase-Shift Keying (M-PSK). Specifically, the V&V phase estimate is given by N ( ) = angle jmφ n θ F( ρ( n) ) e, (3) M n= 0 whee F ( ) is a eal-valued abitay nonlinea function. The monomial V&V estimatos esult fom the special case k of F( ( n) ) ρ ( n), ρ = k = 0,,,, M. Note that if k = M, (3) educes to the M th Powe Estimato [], whose phase estimate is usually stated as * M N M [ X ]. θ = angle E () M n= 0 Fo QAM signals all of which have quadant symmety M M = and E [ X ] * is negative. Hence, Wang and Sepedin [0] and Wang et al. [] investigated estimatos fo QAM that wee given specifically by N ( ) = angle ( ( )). j φ n θ F ρ n e (5) n= 0 Wang and Sepedin [0] wee able to find the optimum nonlinea function F ( ) that minimizes the vaiance of the estimato (5). (Much ealie, Paden [] had done the same fo QPSK). Howeve, this function is a complicated function of the constellation and SNR. (Please see [0] and [] fo details). Theefoe, these authos also consideed the monomial estimatos, i.e., N ( ) = angle ( ), k j φ n θ ρ n e k = 0,,. (6) n = 0 Howeve, as shown in Fig. 3 and Fig. of [], these monomial-based estimatos pefom pooly fo coss QAM constellations, a fact that is also well-known fo the Fouth Powe Estimato []-[3]. IV. SUGGESTED IMPROVEMENT TO THE V&V MONOMIAL- BASED ESTIMATOR The eason fo the poo pefomance of the V&V monomialbased estimato is that the symbols of the constellation with the highest enegy ae not on the diagonal lines X = X o X = X i. Indeed, elimination of all the eceived points that ae not on these lines is the basis fo the APP Estimato [], (which appoximates the optimal estimato, i.e., (5) with the optimum nonlineaity) and the sepaation of 6-QAM into two classes fo the SC estimato [8]. Howeve, in ode to do this, established gain contol is necessay. This gain contol equiement can be emoved by taking the ecipocal of the eceived symbol. In the absence of noise, this action would poduce a eceived constellation whose symbols of highest enegy will lie on the equied lines, as demonstated in Figs. (a) and (b) note the fou symbols of the inne i squae of the 3-QAM constellation of Fig. (a), shown emphasized, have been mapped to the points of the oute squae of Fig. (b). Hence, impoved pefomance can be expected at medium to lage SNR when this tansfomed constellation is applied to the V&V monomial-based estimato of (6). Howeve, the same effect can be achieved with the oiginal eceived constellation in (6) but now allowing k to be negative. As simple as this sounds, no one has appaently suggested this befoe. The next section pesents simulations that demonstate the efficacy of such an appoach. Note that fo k =, () can be used with M = instead of (6). Indeed, moe geneally, fo k = M, () can be used in place of (3) Xi Xi X Fig. (a). Constellation of 3-QAM X Fig. (b). Constellation of the tansfomed eceived 3-QAM, in the absence of noise. Each point is the ecipocal of the coesponding point in the oiginal constellation, above. V. PERFORMANCE VERIFICATION In ode to detemine the mean-squae-eo (MSE) of the phase estimate of the V&V monomial estimato with negative k values, Monte Calo (MC) simulations wee pefomed fo 3-QAM and 8-QAM. In each case, unless othewise noted,,000 MC tials wee utilized, and θ was assumed to be 0. adians, although simulations show the pefomance does not depend upon this value.

4 The esults of these simulations with,000 MC tials fo 3- QAM with N = 500 ae shown in Fig., and fo 8-QAM with N = 7500 ae shown in Fig. 3. Fom these figues, it is clea that it is possible to optimize the pefomance ove a given SNR inteval by the appopiate choice of k. Fo example, fo 3-QAM and 0 SNR 5 db, k = povides the best pefomance. This is also the case fo 8- QAM and 7 SNR 3 db. As will be shown late in this section, these SNR anges coespond to the opeating anges fo pobability of symbol eo fo pactical systems. It is also inteesting to note that the pefomance with k = 0 is bette than the pefomance with k = fo 3-QAM fo SNR above about 0 db. This is opposite to the case fo 8-QAM whee the pefomance fo k = is always bette. Also, by obsevation of Fig. and Fig. 3, it is clea that the maximum mean-squae eo is 0. ad. This is because the phase eo becomes unifomly distibuted between π / and π /, as explained by Tavaes et al. [5]. Hence, the vaiance is π / 8 = 0. ad. Note that in all the figues above, expeimental esults ae given fo the Fouth Powe Estimato ( k = ); howeve, theoetical esults could have been found with (3) of [], o moe explicitly, with (8) of [6]. Additionally, in Fig. and Fig. 3, simulation esults ae given fo the. This estimato s pefomance was chosen to epesent the pefomance of the othe estimatos which do not equie established gain contol, as [6] and [7] show that the pefomance of, CEO, DCA-a and DCA-b ae simila fo 3-QAM with N=500 and 8-QAM with N=7500. Fom Fig., it is clea that ou new extension povides impoved pefomance ove fo 3-QAM with k =, 3 o and SNR 3 db. Similaly, Fig. 3 shows that fo 8- QAM, ou method povides impoved pefomance ove with k = 3 o k = and SNR 3 db. Fo k =, howeve, SNR must be geate than 33 db. MSE (ad ) 0 0 k= Fig.. Mean Squae Eo of Phase Estimates fo 3-QAM. N = 500. MSE (ad ) 0 0 k= Fig. 3. Mean Squae Eo of Phase Estimates fo 8-QAM. N = It is also of paticula inteest to find the effects of the phase estimate on the pobability of symbol eo, P es. This was done and the esults fo 3-QAM ae shown in Fig., whee it is clealy seen that k = povides the best pefomance fo the SNR values nomally of inteest, theeby confiming the esults in Fig.. Howeve, fo SNR 5 db, k = povides the best esults. In addition, ou new extension gives impoved pefomance ove fo SNR db, again confiming the esults in Fig.. Note that the pobability of symbol eo was computed using MC simulations to find the expected value of the following: P E X Xˆ ' 0.5 efc + / 0 * SNRa + es ( X Xˆ ' + + / 0 )* SNRa ) + (7) efc( Xi Xˆ ' i + / 0 )* SNRa ) + X i Xˆ ' + i + / 0 )* SNRa )] whee ˆ ' X = X cos( θ ˆ) θ X sin( θ ˆ) θ i ˆ ' X i = X sin( θ ˆ) θ + X i cos( θ ˆ), θ and SNR a is the SNR in absolute units, i.e. not in db. (This method of simulating P es is called quasi-analytical estimation [7]. Stictly speaking, (7) is a vey tight uppe bound). Note also that 0,000 phase estimates wee used at each SNR to geneate the cuves in Fig.. As each phase estimate equies N = 500 symbols, this means that symbols wee utilized to estimate P es at each SNR. Futhemoe, the Gaussian noise only cuve in Fig. was geneated using P es efc SNR / 0. ( )

5 Pobability of Symbol Eo 0-6 Gaussian Noise Only k= Fig.. Pobability of Symbol Eo fo 3-QAM. N=500. Fo 8-QAM and N = 7500, P es is given in Fig. 5, which again shows that fo the monomial V&V estimatos, k = povides the best pefomance fo the SNR values nomally of inteest. Clealy, howeve, fo SNR 3dB, k = povides the best esults, even poviding impoved pefomance ove fo SNR 3 db, Nevetheless, the pefomance fo k = is close to optimum fo this ange, as well (at least fo SNR 33 db). Thankfully, these esults ae consistent with those in Fig. 3. Note that (7) was again used; howeve, with the 0 eplaced by 8, and with,000 phase estimates fo each SNR to geneate the cuves in Fig. 5. As each phase estimate equies N = 7500 symbols, this means that symbols wee utilized to estimate P es at each SNR. Additionally, the cuve labeled Gaussian Noise Only in Fig. 5 was geneated using P es efc( SNR / 8 ). By inspection of Fig. and Fig. 5, it is clea that substantially moe samples wee used fo 8-QAM than fo Pobability of Symbol Eo Gaussian Noise Only k= Fig. 5. Pobability of Symbol Eo fo 8-QAM, N= QAM; this is equied in ode to obtain simila pobability of symbol eo. This is also the case fo moe complex estimatos which ae gain independent, such as. It is also of inteest to detemine how these estimatos behave with espect to the numbe of samples, N. Simulations wee conducted fo 3-QAM fo low SNR and high SNR and ae shown in Fig. 6(a) and Fig. 6(b), espectively. As can be seen, in geneal, the pefomance is invesely popotional to the numbe of samples. This, howeve, is not the case fo k = 3 and k = : fo these, the pefomance is independent of the numbe of samples fo low SNR. This effect is also seen fo 8-QAM in Figs. 7(a) and 7(b). Note that 0,000 Monte Calo tials wee used fo each point in Fig. 7(a). The question impotant if coding is used then pesents itself: fo what ange of SNR, is the pefomance independent MSE (ad ) MSE (ad ) k= Fig. 6(a). Mean Suae Eo of Phase Estimates as a function of N fo 3-QAM. SNR=9 db. k= Fig. 6(b). Mean Squae Eo of Phase Estimates as a function of N fo 3- QAM. SNR=5 db.

6 of the numbe of samples? Fig. 8 shows that fo 3-QAM and k =, inceasing the numbe of samples does not impove the pefomance if SNR 0 db. Simila cuves can be detemined fo othe powes. II. SOME IMPLEMENTATION CONSIDERATIONS Having established that the monomial-based V&V estimatos with negative powes ae of benefit, especially fo 3-QAM o 8-QAM at high SNR, the question now becomes how complex is it to implement them. The answe to this question depends upon what assumptions ae made concening the available data. Fo example, it is staightfowad to ceate j (n) e φ in hadwae, simply by using a bandpass limite on the modulated signal and then tanslating to baseband. Indeed, by adding a bandpass multiplie (as used in FM tansmittes) j ( n) befoe tanslating to baseband, it is also easy to ceate e φ. Futhemoe, by amplitude demodulation of the modulated signal, ρ (n) can be made eadily available. If we wee to assume these latte two signals ae available, then each phase estimate would equie about 3N eal multiplications and N eal ecipocal opeations fo k =. (We ae ignoing the final angle detemination, as this occus once evey N samples). Howeve, we will assume a wost case scenaio, i.e., only the eal pat and the imaginay pat of () ae available, as is typical in QAM systems. The k = estimato can be implemented in two ways as N ˆ = angle ( ) θ Zi n, i =,, whee Z( n) = o 6 n= 0 ρ ( n) Z ( n) = ( n). ρ The advantage to the latte is that ρ ( n) the ode of each calculated tem is smalle than in the fome. This may be impotant to educe oveflow poblems in fixedpoint implementations. Fo the k = estimato, we can use N ˆ θ = angle. n= 0 ( ) ρ n The numbe of calculations needed fo each of these is shown in Table I, whee the computational buden of othe gain-independent estimatos as detemined by [7] is also shown. Note that DCA-a and DCA-b ae iteative estimatos which also equie some method of phase initialization which futhe adds to thei computational buden. Please see [7] fo details. As can be seen fom Table I, in ode fo the negative powe monomial-based V&V estimato to emain competitive in tems of computational buden, thee has to be a fast method to accomplish ecipocation. Fotunately, such methods exist: see, fo example, [8]-[9]. MSE (ad ) k= x 0 Fig. 7(a). Mean Squae Eo of Phase Estimates as a function of N fo 8- QAM. SNR=6 db. MSE (ad ) k= x 0 Fig. 7(b). Mean Squae Eo of Phase Estimates as a function of N fo 8- QAM. SNR=3 db. MSE (ad ) 0 0 N=500 N= Fig. 8. Mean Squae Eo of Phase Estimates fo 3-QAM and k = -.

7 TABLE I. Estimato COMPUTATIONAL BURDEN OF THE VARIOUS GAIN- INDEPENDENT PHASE ESTIMATORS No. of Real Multiplications No. of Real Recipocations No. of Real Additions N 0 8N DCA-a N/iteation 0 3N/iteation DCA-b N/iteation 0 N/iteation Fouth Powe 5N 0 N k = ( ) k =, Z k =, Z N N 5N N N 6N k = 9N N 6N III. CONCLUSION It has been demonstated using Monte Calo simulations that blind ecovey of the phase fo coss QAM signals can be geatly impoved at medium and high SNR by allowing negative powes in the V&V monomial phase estimato. Even though this is a simple idea, it appeas to be novel, as pevious authos have assumed non-negative powes, up until now. It has futhe been established that k = povides the best (o close to) pefomance fo 3-QAM and 8-QAM fo the SNR values nomally of inteest. REFERENCES [] C. N. Geoghiades, Blind caie phase acquisition fo QAM constellations, IEEE Tans. Commun., vol. 5, no., pp , Nov [] E. Sepedin, P. Ciblat, G. B. Giannakis and P. Loubaton, Pefomance analysis of blind caie phase estimatos fo geneal QAM constellations, IEEE Tans. Signal Poc., vol. 9, no. 8, pp , Aug. 00. [3] K. V. Catwight, Blind phase ecovey in geneal QAM communication systems using altenative highe ode statistics, IEEE Signal Pocessing Lettes, vol. 6, no., pp , Dec [] A. J. Vitebi and A. M. Vitebi, Nonlinea estimation of PSKmodulated caie phase with application to bust digital tansmission, IEEE Tans. Infom. Theoy, vol. IT-9, pp , July 983. [5] K. V. Catwight, Blind phase ecovey in coss QAM communication systems with eighth-ode statistics, IEEE Signal Pocessing Lettes, vol. 8, no., pp , Dec. 00. [6] P. Campisi, G. Panci, S. Colonnese and G. Scaano, Blind phase ecovey fo QAM communication systems, IEEE Tans. Signal Poc., vol. 53, no., pp , Apil 005. [7] M. Alvaez-Diaz and R. Lopez-Valcace, Diamond contou-based phase ecovey fo coss QAM constellations," 005 IEEE/SP 3 th Wokshop on Statistical Signal Pocessing, Bodeaux, Fance, July 7-0, 005, pp [8] F. Rice, B. Cowley, B. Moan, and M. Rice, Came-Rao lowe bounds fo QAM phase and fequency estimation, IEEE Tans. Commun., vol. 9, pp , Sept. 00. [9] M. Luise and S. Pupolin, Boadband Wieless Communications, New Yok: IEEE Pess and Pentech, 99. [0] Y. Wang and E. Sepedin, A class of blind phase ecovey techniques fo highe ode QAM modulations: estimatos and bounds, IEEE Signal Pocessing Lettes, vol. 9, no. 0, pp , Oct. 00. [] Y. Wang, E. Sepedin and P. Ciblat, Optimal blind nonlinea leastsquaes caie phase and fequency offset estimation fo geneal QAM modulations, IEEE Tans. Wieless Commun., vol., pp , Sept [] K. V. Catwight and E. J. Kaminsky, Blind phase ecovey in coss QAM communication systems with the educed-constellation eighth-ode estimato (RC), Global Telecommunications Confeence, GLOBECOM 05, IEEE,, vol., St. Louis, MO, 8 Nov- Dec. 005, pp [3] R. De Gaudenzi, A. Vicente, B. Ponticelli and A. Guillen i Fabegas, APSK coded digital modulation schemes fo nonlinea satellite channels with high powe and spectal efficiency, in Poc. 0 th Intenational Communications Satellite Systems Confeence of the Ameican Institute of Aeonautics and Astonautics (AIAA-ICSSC), Monteal, Canada, pape no. 86, May 00. [] B. E. Paden, A matched nonlineaity fo phase estimation of a PSKmodulated caie, IEEE Tans. Infom. Theoy, vol. 3, pp. 9-, May 986. [5] G. N. Tavaes, L. M. Tavaes, and M. S. Piedade, Impoved Came-Rao bounds fo phase and fequency estimation with M-PSK signals, IEEE Tans. Commun., vol. 9, pp , Dec. 00. [6] K. V. Catwight and E. J. Kaminsky, Asymptotic pefomance of the P th powe law phase estimato, Global Telecommunications Confeence, GLOBECOM 05, IEEE, vol., St. Louis, MO, 8 Nov- Dec. 005, pp [7] M. C. Jeuchim, Techniques fo estimating the bit eo ate in the simulation of digital communications systems, IEEE J. Select. Aeas Commun., vol., pp , Jan. 98. [8] P. H. Hung, H. Fahmy, O. Mence and M. J. Flynn, Fast division algoithm with a small lookup table, Confeence Recod of the Thity-Thid Asiloma Confeence on Signals, Systems, and Computes, Pacific Gove, CA, -7 Oct. 999, pp [9] M. D. Ecegovac, T. Lang, J-M. Mulle, and A. Tisseand, Recipocation, squae oot, invese squae oot, and some elementay functions using small multiplies, IEEE Tans. Computes, vol. 9, no. 7, pp , July 000.