mpacs of boh Tx and Rx Q mbalances on OFD Sysems - Analyical Approach Hassan Zareian, Vahid Tabaaba Vakili ran Universiy of Science and Technology UST, Tehran, ran Faculy of he slamic Republic of ran Broadcasing RB, Tehran, ran Email: hzareian@ee.ius.ac.ir Absrac mplemenaion of OFD sysems suffers from inphase-quadraure Q imbalance in he analog processing and can be presen a boh he ransmier and receiver. The resuling in Q disorion can severely limi he performance of OFD sysems. This paper presens a novel analyical approach o sudy he impacs of boh Tx and Rx Q imbalances on he performance of he OFD sysems and drives a closed form of bi error rae BER of hese sysems in addiive whie Gaussian noise AWGN channels. Resuls from a numerical sudy show perfec maching beween analyical and simulaion approach. Keywords - OFD, Q imbalance, BER performance, heoreical analysis. NTRODUCTON Orhogonal frequency-division muliplexing OFD has been adoped in several wireless sandards, such as digial video broadcasing-erresrial DVB-T, digial audio broadcasing DAB and EEE 8.a wireless local area nework WLAN []. This is due o is robusness agains mulipah fading and is relaively simple implemenaion. However, i is well known ha he performance of OFD sysems is severely affeced by he Q imbalances a boh up-converer modulaor in he ransmier and down-converer modulaor in he receiver []. The effecs of receiver Q imbalance on OFD sysems and he resuling performance degradaion have been invesigaed in [3] [7]. n [3] and [4], Q imbalance effecs in OFD ransmission have been invesigaed by using he simulaion. The impacs of he Q imbalance of he quadraure down-converer on he performance of a QPSK-OFD sysem are analyically sudied in [5]. Also, an analyical model of Q imbalance in mulicarrier based communicaion sysems o he heoreical sudies is presened in [6]. n [7], a novel framework for he analyical compuaion of he symbol error probabiliy in mulicarrier sysems and in he presence of boh a noisy Rayleigh fading channel and receiver Q imbalance is presened ha due o high complexiy of he formulas, heir usage for AWGN channels case migh be impracical. All previous sudies have focused on he problem of Q imbalance a he receiver. The conribuion of his paper is o exend he our previous work resuls [8] by developing an easy heoreical analysis and o derive he exac BER performances of -QA-OFD over AWGN channel in he presence of oinly ransmier and receiver Q imbalances. d d r Fig.. OFD odulaor OFD Demodulaor x w Tx Q mbalance Up-Converer odulaor Down-Converer odulaor Rx Q mbalance Baseband Equivalen y z nc AWGN Channel Simplified baseband block diagram of OFD sysem This paper is organized as follows. Secion inroduces he model of he OFD sysem wih he boh Tx and Rx Q imbalances. Evaluaion of he BER performance is carried ou in secion. The numerical resuls are shown and discussed in secion V, and finally, in secion V conclusions are given. A. OFD ransceiver SYSTE ODEL The equivalen baseband he considered OFD sysem is depiced in Fig.. Basically, he OFD signal is made up of a sum of N complex orhogonal subcarriers, each one independenly modulaed by using informaion symbols. The complex baseband samples of an OFD signal x, ransmied in he inerval [, T b ], is given by inverse discree Fourier ransform DFT of he complex daa symbols carried in he signal, x [m] = x mt s = N N k= d [k] e ω kmt s where d [k] represens he complex daa symbol for he kh subcarrier ha is generaed a rae /T s and ω k = πk f he angular frequency of he kh subcarrier. The frequency separaion beween any wo adacen subcarriers is f = /T b, where T b = NT s is he OFD symbol period. The daa
symbols belong o an alphabe of elemens, which depend on he modulaion forma adoped, and have he same probabiliy. The complex baseband signal x is up-convered o he desired carrier frequency before ransmission. A he receiver, he discree Fourier ransform DFT is applied o he downconvered and low-pass filered RF signal samples w [m] o recover he original daa symbols, d r [k] = N m= w [m] e ω kmt s n he absence of boh Tx and Rx Q imbalances as well as channel noise, we would have y = x, w = z and z = x, respecively. As resuls, he deeced value d r [k] equals he ransmied daa symbol d [k]. n he nex secions we will use a suiable model for Q imbalance disorions in Tx and Rx and invesigae he effecs of hem on OFD ransmission. B. Effecs of Q imbalance disorion on OFD sysem Les ε r and δ r denoe a gain imbalance and phase imbalance beween he and Q branches a he receiver, respecively. The complex baseband equaion in he ime domain for he Q imbalance effec on he received complex baseband signal z is given by [4] as w = µ r z + λ r z 3 where disorion parameers µ r and λ r express he impac of Rx Q imbalance on he received signal and are relaed o he gain and phase imbalances as follows: µ r = cos δ r / + ε r sin δ r / λ r = ε r cos δ r / sin δ r / Noe ha if he branch is balanced wih he Q branch, no disorion will exis in 3 since ε r = and δ r = which resuls µ r = and λ r =. oreover, he gain imbalance is saed in db as log + ε r / ε r. A similar approach can be used o model Q imbalance a he ransmier. Le x represens he ransmied baseband complex signal before being disored by Tx Q imbalance. Then he disored baseband signal a up-converer oupu will be given by 4 y = µ x + λ x 5 where µ and λ are defined as in 4. A derivaion of he OFD signals in he presence of boh ransmier and receiver Q imbalances are presened below. We consider he noise-free channel case, so, z = y. Applying he Q imbalance equaions of 3, 5, i is easy o driver w = µ r [µ x + λ x ] + λ r [µ x + λ x ] = µx + λx 6 where µ = µ r µ + λ r λ and λ = λ r µ + µ r λ denoe oal disorion parameers. The received OFD signal on he kh subcarrier afer he DFT processing can be expressed as [3] N N d r [k] = µ x [m] e ω kmt s + λ x [m] e ω kmt s m= = µd [k] + λd # [k] m= where d # [k] = d [ k] = d [N k] presens he ransmied OFD symbol, mirrored over he subcarriers. Therefore, i is clear ha Q imbalance causes aenuaion and roaion of he desired signal by a complex facor µ, and an inerference erm a he mirror subcarrier scaled by λ. oreover, a µ complex coefficien can be compensaed for, a he receiver, by simply inroducing a correcing facor equal o /µ. C. EV Degradaion The error vecor magniude EV measuremen is a modulaion qualiy meric widely used in digial communicaion sysems and evaluaes he effecs of imperfecions in communicaion sysems such as Q imbalance on he consellaion diagram. The EV can be measured eiher a he ransmier or receiver and is defined mosly as [8]: EV = Ns l= Ns l= 7 e [l] 8 s [l] where N s is he number of consellaion poins used o EV calculaion and e [l] = r [l] s [l] where s [l] is an ideal ransmied consellaion poin, r [l] received consellaion poin a insan l afer compensaion of he consellaion complex aenuaion. The sysem we wan o invesigae is a EEE8. WLAN sandard employs -QA modulaion. Hence he symbols ransmied over each carrier are mapped on a complex domain -QA. n his case, s [l] and e [l] belong o A and B ses, respecively. A = m λ B = m µ + λ µ Where m, n =,,..., for he noise-free channel case becomes, EV = λ µ n } n } 9 }. Hence, he resuled EV n able, he values of EV for differen Tx and Rx Q imbalances parameers are lised. A consellaion of an exemplary disored 4-QA symbol alphabe under differen cases is shown in Fig..
deal Case Case 4 Case Case 3 Case 5 for recangular -QA, is easily deermined from SER = P C where P C = P P Q is he probabiliy of correc decision for he -QA modulaion. P and P Q are he probabiliy of error of PA wih signal poins for each and Q signal componens, respecively. To calculae of he SER for he ransmied symbol in he presence of Tx and Rx Q imbalances in AWGN channel, we can carry ou he following procedure. Firs, a saisical model for he decision variables is developed by aking ino accoun he effecs of boh Tx and Rx Q imbalances, and hen assuming he ih ransmied symbol and inerference erm are caused by he h symbol, we calculae he corresponding P i, P Q i and P C i. Finally, we average he above P C i expression over all of complex daa symbols and he inerference erms belong o A and B ses in 9, respecively. Applying Tx Q imbalance equaion 5, he signal received hrough he AWGN channel represened in Fig. is expressed by z = y + n c = µ x + λ x + n c Where n c denoes AWGN wih one-sided power specral densiy PSD N inroduced a he receiver. The symbols received on he kh carrier are expressed, afer he DFT processing and considering Rx Q imbalance equaion 3, by Fig.. The effecs of Tx and Rx Q imbalances on 4-QA-OFD consellaion. BER PERFORANCE ANALYSS n his secion, we evaluae he BER performance for he OFD sysems in presence of Q imbalance assuming ha he aenuaion and roaion will be compensaed for before he decision process. Since every subcarrier has he same -QA modulaion and he channel is assumed AWGN, he calculaed BER performance is also equal o he BER performance of - QA-OFD sysem. According o [9], he symbol error rae SER performance TABLE LST OF THE EV VALUES FOR DFFERENT CASES OF TX AND RX Q BALANCES Q case Tx Q Parameers Rx Q Parameers EV Case deal UP-Converer ε r =.5dB, δ r = 5 o 5.3 Case ε =.5dB, δ = 5 o deal Down-Converer 5.3 Case 4 ε = db, δ = 5 o ε r = db, δ r = o 7. Case 3 ε =.5dB, δ = 5 o ε r =.5dB, δ r = 5 o.43 Case 5 ε =.5dB, δ = 5 o ε r = db, δ r = o 5.6 d r [k] = N m= N w mt s e ω kmt s = µ r z [m] + λ r z [m] e ω kmt s m= = µd [k] + λd # [k] + µ r N c [k] + λ r N # c [k] 3 Where µd [k] and λd # [k] erms are he useful one and inerference, respecively. The Gaussian noise erm N c [k] obained by DFT of he hermal Gaussian noise of he receiver has zero mean and variance σn = N /. As menioned before, he µ complex coefficien will be assumed o be compensaed for before he decision process, herefore, he scaled decision variable can be expressed by r [k] = d r [k] µ = s [k] + e [k] + n [k] 4 where s [k] = d [k] is ransmied symbol over he kh subcarrier and e [k] = λ/µ d # [k] presens he scaled version of inerference erm; and n [k] = µ r /µ N c [k] + λ r /µ N c # [k] denoes he oal noise samples ha have zero mean and
variance σt. Due o he uncorrelaion beween N c [k] and N c # [k], i is easy o drive σt µ r = µ + λ r µ σn 5 We assume a signal consellaion wih disance wo d = beween adacen symbols, hen he average bi energy of ransmied -QA-OFD signal is equal o E b = log 3 µ + λ 6 By using 6, he hermal Gaussian noise variance σ n in 5 can be replaced by signal o noise raio per bi γ = E b : a 4 ΛL 8 3 7 6 5 9 d e ΛR 6 5 4 3 Λ U Λ D Decision Boundaries for i h Symbol b d e Q i e σ n = E b γ 7 From now onwards we will consider only kh subcarrier. Therefore, for simpliciy and convenience, we can drop k from he above noaions so ha he received decision variables for he and Q branches, can be wrien as c d+ e d e d d+ e Q e Q d r = s + e + n r Q = s Q + e Q + n Q 8 where and Q subscribes presens real and imaginary pars, respecively. To illusrae proposed mehod ha has been used in his paper, he signal consellaion for 6-QA is depiced in Fig. 3a. The decision hresholds for a prooype symbol affeced by inerference due o Q imbalance are also shown in Fig. 3b. Wih he aid of Fig. 3c., he symbol error probabiliy condiioned o he ih symbol and inerference erm caused by he h symbol for componen, i.e. P i, can be deermined by where, Q P i = Q Q e +e, i Λ L, i Λ R e + Q +e, else 9 Λ L =,,..., } Λ R = +, } +,..., Using Fig. 4d., similar resuls are obained for P Q i wih hese differences ha e, Λ L and Λ R are subsiued by e Q, Λ U and Λ D, respecively. Where, } Λ U =,,..., Λ D =, + },..., + Fig. 3. a 6-QA consellaion wih Q imbalance b Decision boundaries for ih symbol c componen d Q componen The probabiliy of correc decisions should be evaluaed by averaging he condiional probabiliies P C i over all of he daa symbols and he inerference erms. Assuming all he symbols are equally likely, i can be easily shown all inerference erms have he same probabiliy equal o /, yields P C = i= = ] [ P i P Q i Finally, according o and assuming a Gray-coded signal se, BER is can expressed by BER = log V. NUERCAL RESULTS P C 3 The validiy of he presened analyical mehod resuls is confirmed by compuer simulaion in his secion. The OFD signal for simulaions was similar o he EEE8. WLAN sandard [], which has 5 acive subcarriers. Figs. 4-6 compares he analyical and he simulaion BER performances for he -QA-OFD sysem wih differen Tx and Rx Q mismaches cases as well as he ideal case no Q imbalance. The values of each cases are illusraed in able. We find perfec agreemen wih he analyical and simulaion resuls. However, analyical approach can be used
Comparisons Beween Analyical and Simulaion Resuls Comparisons Beween Analyical and Simulaion Resuls Bi Error Rae BER 3 4 5 6 Q imb. Case Q imb. Case Q imb. Case 3 Q imb. Case 4 Q imb. Case 5 Simulaion 4 6 8 4 Signal o Noise Raio SNR per Bi, E b Bi Error Rae BER 3 4 5 6 Q imb. Case Q imb. Case Q imb. Case 3 Q imb. Case 4 Q imb. Case 5 Simulaion 5 5 5 3 Signal o Noise Raio SNR per Bi, E b Fig. 4. Comparisons beween analyical and simulaions BER resuls for 4-QA-OFD sysem wih differen Tx and Rx Q mismaches values Fig. 6. Comparisons beween analyical and simulaions BER resuls for 64-QA-OFD sysem wih differen Tx and Rx Q mismaches values Bi Error Rae BER 3 4 5 6 Comparisons Beween Analyical and Simulaion Resuls Q imb. Case Q imb. Case Q imb. Case 3 Q imb. Case 4 Q imb. Case 5 Simulaion 5 5 5 Signal o Noise Raio SNR per Bi, E b Fig. 5. Comparisons beween analyical and simulaions BER resuls for 6-QA-OFD sysem wih differen Tx and Rx Q mismaches values o derive he OFD sysem performance, wihou he need o run exensive simulaions. Resuls show he Q imbalance caused by imperfecions of he Tx and Rx analog processing degrades BER performance of an OFD sysem. This is due o he presence of inerference erms in he received signal. Also, sensiiviy of -QA- OFD signals o Q imbalance increases wih he alphabe size. Finally, by comparison of differen cases in Figs. 4-6, we can see he influence of Tx and Rx Q imbalances on sysem performance us depends on EV value in no individually Tx and Rx Q imbalances values. For case only Rx Q imbalance and case only Tx Q imbalance which have he same EV value, BER degradaion compared o an ideal case wihou Tx and Rx Q imbalances are he same. V. CONCLUSONS The impacs of ransmier and receiver induced Q imbalance on he OFD sysems BER performance is sudied in his paper. We proposed a new heoreical analysis of he effecs of boh Tx and Rx Q imbalances on he performance of -QA-OFD sysems. Theoreical resuls showed perfec agreemen wih hose obained by simulaion and hey can be used o derive he OFD sysem performance, wihou he need o run exensive simulaions. REFERENCES [] R. Van Nee, and R. Prasad, OFD for Wireless ulimedia Communicaions, Arech House,. [] B. Razavi, RF icroelecronics. Englewood Cliffs, Prenice-Hall, 998. [3] J. Tubbax, B. Come, L. Van der Perre, S. Donnay, and. Engels. Q imbalance compensaion for OFD sysems, EEE nernaional Conference on Communicaions CC, pp.343347, ay 3. [4] A. Tarigha, R. Bagheri, and A. H. Sayed, Compensaion schemes and performance analysis of Q inhabians in OFD receivers, EEE Trans. Signal Process., vol. 53, no. 8, pp. 357368, Aug. 5. [5] C. L. Liu, mpacs of /Q imbalance on QPSK-OFD-QA deecion, EEE Trans. Consumer Elecron., vol. 44, no. 3, pp. 984989, Aug. 998. [6]. Windisch and G. Feweis, Performance Degradaion due o /Q mbalance in uli-carrier Direc Conversion Receivers: A Theoreical Analysis, in Proc. EEE nl. Conference on Communicaions CC, pp. 57-6, June 6. [7] P. Rykaczewski,. Valkama,. Renfors, Analyical approach o Q imbalance in OFD, CDA and C-CDA based sysems, EEE Radio and Wireless Symposium, pp. 555-558, Jan. 6. [8] H. Zareian and V. Tabaabvakili, Analyical BER Performance of - QA-OFD Sysems in he Presence of Q mbalance, Fourh nernaional Conference on Wireless and Opical Communicaions Neworks WOCN7, vol. 48, pp. 43-44,Jul. 7. [9] P. B. Kenningon, High-lineariy RF amplifier design, Arech House, [] J. G. Proakis, Digial Communicaions, cgraw-hill,. [] EEE, Par: Wireless LAN edium Access Conrol AC and Physical Layer PHY specificaions, EEE Sd 8.a-999, 999.