A Robust Adaptive Carrier Frequency Offset Estimation Algorithm for OFDM

Transcription

1 Amercan Journal of Sgnal Processng 015, 5(A): 6-31 DOI: /s.ajsp A Robust Aaptve Carrer Frequency Offset Estmaton Algorthm for OFDM Azam Khall *, Amr Rastegarna, Bahareh Darvsh Department of Electrcal Engneerng, Malayer Unversty, Malayer, Iran Abstract The carrer frequency offset (CFO) severely egraes the bt error rate (BER) performance of orthogonal frequency vson multple (OFDM) systems. In ths paper, our am s to evelop a robust aaptve CFO estmaton algorthm for OFDM system whch s able to prove accurate estmate of CFO even n the presence of mpulsve nose. To ths en, we formulate the CFO estmaton n terms of the maxmum correntropy crteron (MCC) whch s a robust optmalty crteron for mpulsve nose. Then, we utlze the graent-ascent metho an also apply nstantaneous approxmaton to erve the robust aaptve algorthm. The propose algorthm has computatonal complexty smlar to the popular least mean-square (LMS), whle t s robust aganst the mpulsve sgnal because of usng hgher orer moments beyon just secon orer moments. The performance of the propose algorthm s evaluate uner fferent contons, nclung the Gaussan nose, mpulsve nose, an tme-varyng CFO, where smulaton results reveal the effectveness of the propose algorthm. Keywors Aaptve algorthm, CFO, maxmum correntropy crteron, OFDM 1. Introucton The orthogonal frequency vson multplexng (OFDM) enables hgh ata rate transmssons n wreless communcaton systems. It has been chosen as the stanars for the European gtal an veo broacastng, IEEE80.11a an HIPERLAN/ [1]. However, the OFDM system s very senstve to the carrer frequency offset (CFO). CFO s manly cause by two sources []. The frst one s the msmatch of carrer frequences between oscllators n transmtter an recever, an the other one s ue to the Doppler shft, whch may change from tme to tme. Unfortunately CFO estroys the orthogonalty among subcarrers, whch n turn causes nter-carrer nterference (ICI) an egraes the bt error rate (BER) performance severely []. Ths ssue motvate the evelopment of CFO estmaton algorthms. So far several schemes have been propose n the lterature to estmate the CFO of OFDM systems [3-8]. In [4] a CFO estmaton scheme has been evelope whch uses a tranng symbol wth two entcal halves. The gven algorthm n [5] utlzes a tranng symbol wth more than two entcal halves whch ncreases the estmaton range twce that of the scheme n [4]. The algorthm n [6] reles on the maxmum-lkelhoo (ML) crteron an same tranng * Corresponng author: a_rastegar@eee.org (Azam Khall) Publshe onlne at Copyrght 015 Scentfc & Acaemc Publshng. All Rghts Reserve symbol as n [5]. A peroogram-base CFO estmaton scheme has been propose n [6], whose estmaton range s as large as the banwth of the OFDM sgnal whle mantanng the same level of the estmaton performance as those of [4, 5]. Most of avalable works assume CFO as a statonary parameter an o not conser the varaton of the offset cause by the Doppler shft. Thus these types of algorthms fal when frequency offset changes from tme to tme. In [8] an aaptve LMS-base flter has been propose to estmate CFO, whch can track the varaton of the offset cause by the Doppler shft. The propose algorthm n [8] works well uner the assumpton of the Gaussan strbute nose. However, n many wreless channels, t has been observe that the nose often follows non-gaussan strbuton [9]. Thus, the conventonal estmators coul suffer from performance egraaton n the non-gaussan nose envronments. To aress ths ssue, n ths paper our am s to evelop a robust an aaptve algorthm to estmate the CFO whch s robust n presence of mpulsve nose. To ths en we nee to go beyon mean square error to explot hgher orer moments of the error. Informaton theoretc quanttes have also been propose as cost functons n aaptve flters (algorthms). For example a famly of Mnmum Error Entropy (MEE) base aaptve flters has been propose [10], [11], wheren the weghts are aapte such that the entropy or the nformaton content of the error sgnal s mnmze. Although MEE base algorthm shows robustness n presence of non-gaussan an mpulsve nose, however, t

2 Amercan Journal of Sgnal Processng 015, 5(A): has hgh computatonal complexty. The use of Correntropy as a cost functon n orer to tran the flter weghts has been propose n [1]. The gven algorthm n [1] whch uses maxmum correntropy as the cost functon has lower computatonal complexty than MEE, whle ts robustness s smlar to the MEE base algorthm. Thus, n ths paper we evelop a MCC base aaptve algorthm for CFO estmaton. To erve the propose algorthm, we frstly formulate the CFO estmaton accorng to the MCC, an then we utlze the graent-ascent metho wth sutable nstantaneous approxmatons to solve t. We compare the performance of the propose algorthm wth the LMS-base algorthm n [8] n fferent contons nclung the Gaussan nose, mpulsve nose, an tme varyng CFO, where the results show the superor performance of the propose algorthm. Notaton: Through the paper we use lower case bol letters to enote vectors, captal bol letters for matrces. We also use () T to enote transposton, an () H to enote Hermtan transpose. Moreover Im{} enotes the magnary part of ts argument an E stans for the statstcal expectaton. The rest of ths paper s organze as follows: In secton, we present some backgroun on CFO estmaton problem an also ntrouce the MCC. The propose algorthm s presente n secton 3. Smulaton results are presente n secton 4. Secton 5 conclues the paper.. Backgroun.1. CFO Estmaton Problem s( n) [ s ( n), s ( n),..., s ( n)] T Let us efne 0 1 N 1 as the nth OFDM ata block to be transmtte, where N s the number of subcarrers n the OFDM system. The ata are use to moulate orthogonal subcarrers an ths moulaton can be mplemente by nverse screte Fourer transform (IDFT). Usng the matrx representaton, the nth block of the moulate sgnal s [14] 0 1 N 1 T N IDFT matrx whch s gven by ( n ) [ ( n ), ( n ),..., ( n )] Ws ( n ) (1) where W s the N j j( N 1) 1 1 e e W N j( N 1) j( N 1)( N 1) 1 e e where n () / N. Then, a cyclc prefx (CP) s nserte where t s assume that the length of CP s longer than the maxmum elay sprea of the channel to avo ntersymbol nterference (ISI). Fnally, the resultant () baseban sgnal s up-converte to the rao frequency (RF) before transmsson. At the recever se, the sgnal s frstly own-converte an emoulate usng screte Fourer transform (DFT) to recover the esre sgnal. Wthout CFO, the receve sgnal can be expresse by r( n) WCs( n) q ( n) (3) where q ( n) enotes the atve whte Gaussan nose (AWGN) an C ag{ C0( n), C1( n),..., CN 1( n)} (4) represents the channel characterstcs n the frequency oman. In ths paper, we assume AWGN channel whch means we have C ag{1, 1,..., 1}. In the presence of CFO the receve sgnal, after the removal of the CP, s gven as where r( n) WCs( n) q( n) (5) jv j ( N 1) v ag 1, e N,..., e N (6) represents the CFO matrx an s the frequency offset normalze to subcarrer spacng. To guarantee the system performance, the CFO must be estmate an compensate before the DFT emoulaton. In the presence of the CFO, the receve sgnal, after DFT emoulaton, s gven by H x( n) W Cr( n) W WCs( n) q( n) (7) where H H W represents the DFT emoulaton matrx... Maxmum Correntropy Crteron Gven two ranom varable X an Y, the correntropy s efne as follows where (, ) V( X, Y ) ( X, Y ) V( X, Y ) ( x, y) FXY, ( x, y) (8) s a shft-nvarant Mercer kernel, an FXY, ( x, y) enotes the jont strbuton functon of ( XY, ). The Gaussan kernel s the most wely use kernel n correntropy 1 ( xy, ) exp Where x y, an 0 s the kernel wth. The maxmum correntropy cost functon s gven by J MCC () exp (9) (10)

3 8 Azam Khall et al.: A Robust Aaptve Carrer Frequency Offset Estmaton Algorthm for OFDM In the sequel, we wll use ths cost functon to erve the propose algorthm. 3. Propose CFO Estmaton Algorthm In ths secton, we present our aaptve CFO estmaton algorthm. To ths en, let us enote th sample of recovere sgnal on subcarrer k as x (we gnore the subcarrer nex for the sake of smplcty). We also use the tranng symbols n where every s ( n ) n () s a quarature ampltue moulaton (QAM) or phase-shft-keyng (PSK) symbol. Then, by comparng the estmaton wth the tranng symbols, we efne the error sgnal as j w / N e x (11) where w s a real-value. We now efne the cost functon n terms of the error sgnal as we have J( ) E exp j w j w N * N * e x e x x (1) (13) To fn the optmum weght that maxmzes the cost functon n (1), we can use teratve graent ascent approach as w 1 w J() (14) w where s the step-sze parameter an w enotes the graent wth respect tow. By computng the graent J () wth respect w to we obtan 4 j w / * ( ) Im{ N w J e x } N (15) E exp( ) Substtutng w J () from (15) n (14) gves 4 j w / * 1 Im{ N w w e x } N (16) e E exp( ) Then, the propose algorthm can be obtane by replacng the statstcal moment n (16) by nstantaneous approxmaton as 4 j w / * 1 Im{ N w w e x } N (17) exp( ) The estmate of CFO s gven n terms of w as v w, where the number of teratons equals the number of plot symbols. 4. Smulaton Results In ths secton, we evaluate the performance of the propose algorthm an compare t wth the gven LMS-base algorthm n [8]. We conser the number of subcarrers n the OFDM system as N 104. We conser the mean-square-error (MSE) as the performance metrc whch s efne as MSE P 1 v vˆ (18) P t 1 where P s the number of Monte Carlo runs, whch n our smulatons was P The CFO s assume as v 0. n the followng set of smulatons. In the frst set of smulaton we conser Gaussan nose conton where Eb / N0 0 B. Fg. 1 shows the learnng curves of the propose algorthm an the LMS-base algorthm for. We can see that the propose 0.005, algorthm exhbts a better performance than the algorthm LMS algorthm snce t has smaller estmaton varance. Fg. presents the MSE versus Eb/N0 for varous step-sze parameters. It s observe that for both algorthms, the MSE ecreases as Eb / N0 ncreases or when the step sze parameter ecreases.

4 Amercan Journal of Sgnal Processng 015, 5(A): Fgure 1. The learnng curves of the propose algorthm an the LMS-base algorthm for E b /N 0 =0 B Fgure 3. The MSE performance of both algorthms n terms of the Eb/N0 for fferent step sze parameters Fgure. The MSE performance of both algorthms n terms of the E b /N 0 for fferent step sze parameters For small Eb / N 0 values, the propose algorthm performs better than the LMS- base algorthm for all step-sze values. As Eb / N0 ncreases, the performance of both algorthms becomes smlar, specally for small step-sze values. In the next smulaton setup, we assume agan an OFDM system escrbe above but wth mpulsve nose. To moel the mpulsve nose we a a zero-mean complex-value oubly whte Gaussan nose at 0.B SNR wth probablty Pr = 0.03, where Pr enotes the probablty that the mpulsve nose occurs. The learnng curves of the propose algorthm an the LMS-base algorthm s shown n Fg. 3. The MSE versus Eb/N0 for varous step-sze parameters nfferent ranges of Eb/N0 are shown n Fg. 4 an Fg. 5. We can see that n the presence of mpulsve nose, the performance of LMS-base algorthm ecreases, whle the propose algorthm works well n ths conton. In Fg. 6, we compare the estmaton range of both the propose LMS-base algorthm an the propose algorthm uner the Gaussan nose an mpulsve nose contons. It s clear that the propose algorthm has better performance for the whole range of CFO than the LMS-base algorthm. Fgure 4. The MSE versus Eb/N0 for varous step sze parameters uner mpulsve nose conton Fgure 5. The MSE versus Eb/N0 for varous step sze parameters uner mpulsve nose conton In Fg. 7, we compare the trackng performance of both algorthms n the presence of mpulsve nose. In ths case we assume that the normalze frequency offset changes n tme accorng to the followng moel v v z (19) 1

5 30 Azam Khall et al.: A Robust Aaptve Carrer Frequency Offset Estmaton Algorthm for OFDM where z s a zero-mean Gaussan nose wth varance As we can see the propose algorthm can track the varaton of the CFO an the estmaton accuracy s not affecte by the mpulsve nose 5. Conclusons The carrer frequency offset egraes the performance of OFDM systems. In ths paper, we have propose a correntropy-base aaptve algorthm to estmate the CFO smultaneously n OFDM systems. The propose algorthm acheves comparable performance an a wer estmaton range n comparson wth the LMS-base algorthm. The tme-varyng nature of Doppler shft causes the frequency offset to change from tme to tme, whch affects the estmaton accuracy of many exstng estmators. The propose algorthm can track the varaton of the Doppler shft well an the estmaton accuracy s not affecte n the presence of mpulsve nose. REFERENCES [1] M. Morell, C.-C. J. Kuo, an M.-O. Pun, Synchronzaton technques for orthogonal frequency vson multple access (OFDMA): a tutoral revew, Proc. IEEE, 95(7), , July 007. [] T. Pollet, M. Van Blael, an M. Moeneclaey, BER senstvty of OFDM systems to carrer frequency offset an Wener phase nose, IEEE Trans. Commun., 43(), , Feb [3] W. Xu an X. Fu, Bln Frequency Ambguty Resoluton n OFDM Systems Usng a Lnear Precoer, Wreless Pers Commu, , 01. [4] T. M. Schml an D. C. Cox, Robust frequency an tmng synchronzaton for OFDM, IEEE Trans. Commun., 45(1), , Dec Fgure 6. The estmaton range of the propose LMS-base algorthm an the propose algorthm uner the Gaussan nose (top) an mpulsve nose (bottom) contons [5] S. Chang an E. J. Powers, Effcent frequency-offset estmaton n OFDM-base WLAN systems, Electron. Lett., 39(1), , Oct [6] M.-H. Cheng an C.-C. Chou, Maxmum-lkelhoo estmaton of frequency an tme offsets n OFDM systems wth multple sets of entcal ata, IEEE Trans. Sg. Process., 54 (7), , July 006. [7] G. Ren, Y. Chang, H. Zhang, an H. Zhang, An effcent frequency offset estmaton metho wth a large range for wreless OFDM systems, IEEE Trans. Vehc. Technol., 56(4), , July 007. [8] F. Yang, K. H. L an K. C. Teh, Aaptve LMS-ase algorthm for Carrer Frequency Offset Estmaton n OFDM Systems, Proceengs of the 8th Internatonal Symposum on Sgnal Processng an Its Applcatons, , 005. [9] T. K. Blankenshp an T. S. Rappaport, Characterstcs of mpulsve nose n the 450-MHz ban n hosptals an clncs, IEEE Trans. Antennas, Propag., 46(), , Feb Fgure 7. Trackng performance of both algorthms n the presence of mpulsve nose [10] D. Erogmus an J. Prncpe, Generalze nformaton potental crteron for aaptve system tranng, IEEE Transactons on Neural Networks, 13(5), , September 00.

6 Amercan Journal of Sgnal Processng 015, 5(A): [11] D. Erogmus, Informaton theoretc learnng: Reny's entropy an ts applcatons to aaptve system tranng, Ph.D. ssertaton, Unversty of Flora, 00. [1] A. Sngh an J. C. Prncpe, Usng correntropy as a cost functon n lnear aaptve flters, n Proc. Int. Jont Conf. Neural Networks (IJCNN), 009, [13] B. Chen, L. Xng, J. Lang, N. Zheng, an J. Prncpe, Steay-state mean-square error analyss for aaptve flterng uner the maxmum correntropy crteron, IEEE Sgnal Processng Letters, 1(7), , July 014. [14] F. Yang, K. H. L, an K. C. Teh, A carrer frequency offset estmator wth mnmum output varance for OFDM systems, IEEE Commun. Lett., 8, , Nov. 004.