Impacts of both Tx and Rx IQ Imbalances on OFDM Systems - Analytical Approach

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1 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 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

2 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..

3 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

4 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 d e ΛR 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

5 Comparisons Beween Analyical and Simulaion Resuls Comparisons Beween Analyical and Simulaion Resuls Bi Error Rae BER Q imb. Case Q imb. Case Q imb. Case 3 Q imb. Case 4 Q imb. Case 5 Simulaion Signal o Noise Raio SNR per Bi, E b Bi Error Rae BER Q imb. Case Q imb. Case Q imb. Case 3 Q imb. Case 4 Q imb. Case 5 Simulaion 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 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 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 , 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 , Aug. 5. [5] C. L. Liu, mpacs of /Q imbalance on QPSK-OFD-QA deecion, EEE Trans. Consumer Elecron., vol. 44, no. 3, pp , Aug [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 , 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 ,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.