Performance Degradation due to I/Q Imbalance in Multi-Carrier Direct Conversion Receivers: A Theoretical Analysis

Transcription

1 Performance egradation due to I/Q Imbalance in Multi-Carrier irect Converion Receiver: A Theoretical Analyi Marcu Windich, Gerhard Fettwei reden Univerity of Technology, Vodafone Chair Mobile Communication Sytem, -006 reden, Germany windich@ifn.et.tu-dreden.de Abtract I/Q imbalance ha been identified a one of the mot eriou concern in the practical implementation of the direct converion receiver architecture. In particular, at the reception of multi-carrier ignal the achievable error rate i trictly limited by the I/Q imbalance. Knowledge about the quantitative link between the hardware parameter and the reulting error rate i eential for a reaonable deign of the receiver front-end. In thi paper a novel framework for the analytical computation of the ymbol error probability in multi-carrier ytem i preented. We conider an arbitrary M-ary QAM modulated multi-carrier ignal, which i corrupted by both a noiy Rayleigh fading channel and receiver I/Q imbalance. The theoretical reult are validated exemplarily for the IEEE 80.a WLAN tandard. I. INTROUCTION The growing number of wirele communication tandar deman for highly flexible and low cot-terminal. I/Q proceing architecture, uch a the direct converion receiver, are very attractive, becaue no cotly analog image rejection filter i required []. Intead, a theoretically infinite image rejection i provided by the I/Q ignal proceing. However, the limited accuracy of the analog hardware caue mimatche between the component in the I- and the Q-branch of the receiver, known a I/Q imbalance. In the pat decade, multi-carrier ytem, uch a OFM, have gained a lot of acceptance for the deign of high data rate communication ytem. For example, the IEEE 80.a WLAN tandard [] i an OFM ytem, which i widely ued in practice. While being able to eaily cope with the frequencyelective nature of a wirele communication channel, multicarrier ytem are very enitive to I/Q imbalance. In order to cope with thee impairment, different approache for a digital compenation of the I/Q imbalance have been propoed in the literature, ee for example [3] and the reference herein. The goal of the compenation i to provide an improved image rejection, which by nature depen on the accuracy of the digital etimation and compenation approach. Knowledge about the quantitative relationhip between tranceiver parameter (uch a the image rejection ratio - Thi work wa partly upported by the German Minitry of Education and Reearch (BMBF) within the project Wirele Gigabit with Advanced Multimedia Support (WIGWAM) under grant 0BU370 i(k) (k) i Rx (k) Rx (k) ĩ Rx (k) Rx (k) a) f SC +f SC b) f SC +f SC c) f SC +f SC Fig.. Equivalent baeband repreentation of a) tranmitted RF ignal, b) received RF ignal (corrupted by the fading channel), c) received baeband ignal after direct converion with I/Q imbalance with or without digital compenation) on the one hand and ytem parameter (uch a the ymbol error probability) on the other hand i eential for the deign and the dimenioning of communication ytem. Given a targeted error probability, the hardware deigner nee to know the image rejection ratio required for reaching that goal. Conventionally, thi knowledge i gained for a pecific ytem by uing hardware meaurement or computer imulation [3], [4]. However, a comprehenive theoretical analyi i till miing. The goal of thi paper i to contribute towar cloing thi gap. The outline of thi paper i a follow: Section II define the ytem model, which i the bai of our analyi. The ymbol error probability of a ingle ubcarrier will be derived in ection III. Both exact olution and reaonable approximation are preented. A generalization of the reult to multi-carrier ytem i done in ection IV, followed by the concluion in ection V. II. SYSTEM MOEL The degradation of the ytem performance due to the I/Q imbalance i a conequence of the imperfect converion of the received radio frequency (RF) ignal down to the baeband (BB). Appropriate model for the impact of the I/Q imbalance on the received ignal have been derived in the literature. It ha been hown, that multi-carrier ignal are affected by a mutual inter-carrier interference between each pair of ymmetric ubcarrier, ee for example [3]. Becaue the effect of the I/Q imbalance i the ame for all ubcarrier, it i ufficient to conider a repreentative pair of ymmetric ubcarrier, a depicted in Fig..

2 Let Rx (k) denote the received ymbol at an arbitrary ubcarrier frequency f SC at time index k, correponding to the received ignal before down converion (Fig. b). Similarly, let i Rx (k) denote the received ymbol at the image ubcarrier frequency f SC. The interference due to an imperfect direct converion with I/Q imbalance can be modelled by [3] Rx (k) = Rx (k)+ i Rx(k). () The aterik denote complex conjugation. The complex valued weighting parameter and are determined by the image rejection capabilitie of the receiver. Ideally, the image rejection ratio IRR = () approache infinity. However, the IRR which i achievable with today technologie i limited to only db []. Mot practical multi-carrier ytem are deigned uch that ubcarrier pacing i much maller than the coherency bandwidth of the wirele channel. Hence, the frequency-elective fading channel i plit into frequency-flat ubchannel in each ubcarrier. Conidering the repreentative pair of ubcarrier at f SC and f SC, the following channel model are ued: Rx (k) =h (k)(k)+n (k), (3) i Rx (k) =h i (k)i(k)+n i (k). (4) (k), i(k) denote the tranmitted ymbol. h (k), h i (k) denote the correponding time-variant channel coefficient. n (k), n i (k) denote the additive channel noie in each ubcarrier. From () and (3) one can ee, that the deired ignal (k) i rotated and caled by both the channel coefficient h (k) and the I/Q imbalance parameter. Thee effect can be removed by a proper equalization. Auming a perfect zeroforcing equalizer, the following operation will be applied: ŝ(k) = h (k) Rx(k). (5) The remaining error vector (k) =ŝ(k) (k) i a reult of both the image ignal and the additive channel noie: (k) = h i (k) h (k) i (k)+ h (k) n (k)+ h (k) n i (k). (6) The everene of the additional impairment due to the I/Q imbalance trongly depen on the propertie of the communication channel. While the performance degradation i only moderate for AWGN channel, a ignificant performance degradation can be oberved for frequency-elective fading channel [3]. Thi reult tem from the fact, that the quotient h i (k)/h (k) in (6) can take arbitrary large value in the cae of a frequency-elective channel. For fading channel, the occurrence of large value i likely, if the individual fading procee h i (k) and h (k) are independent. Independency hol, if the ditance f SC between the conidered pair of ymmetric ubcarrier i larger than the coherency bandwidth of the frequency-elective fading channel. Becaue thi cenario i mot critical from the I/Q imbalance point of view, it will be conidered in our analyi. a) d Q eciion boundarie d Symbol I Fig.. a) QAM contellation, b) ditinguihed area for the occurrence of ymbol error III. STATISTICAL SIGNAL ANALYSIS The goal of thi ection i to derive the probability of a ymbol error due to the I/Q imbalance and the channel noie. Therefore, we model the ample of the channel coefficient and the noie a complex valued random variable (RV ), which are written in bold tyle: = K i h i i + n + n i. (7) h h h The reulting diplacement i alo a RV, whoe propertie we will analyze in thi ection. The I/Q imbalance parameter and are arbitrary fixed parameter. Furthermore, the actual value of the image ignal i i conidered a an intantaneou fixed parameter. A generalization to a non-contant i will be done at the end of thi ection. The reulting error probability due to the error vector depen on both the tructure (alphabet) of the deired ignal and the ditribution of the error vector. In the following ubection we will firt conider the ymbol alphabet. Then we will analyze the ditribution of the error vector. A. Effect of the Error Vector on the Signal Contellation In our analyi we focu on the reception of M-ary quadrature amplitude modulated (QAM) ignal, which are mot frequently ued in practical multi-carrier ytem. A rectangular QAM contellation i hown exemplarily for the modulation order M =6in Fig. a. The parameter d denote the ditance between the amplitude level of adjacent ymbol in both the I and the Q dimenion. Auming equally probable ymbol, the average power of a QAM ignal i [5] b) + d d Q d A A 0 + d A = 6 (M )d. (8) In order to ditinguih between the parameter of the deired ignal and the image ignal, we will ue the ubcript and i, repectively. Both the deired ignal (k) and the image ignal i(k) are aumed to be rectangular QAM ignal of the order M and M i, repectively. Neverthele, the analyi framework preented in thi paper can be eaily adapted to alternative contellation. With the deciion threhold of the detector placed at the the midpoint of adjacent amplitude level, 3 different area for the location of the error vector can be ditinguihed (ee Fig. b): I

3 If the error vector i located within area A 0, no ymbol error occur. A location within area A (coniting of 4 partition, which are haded in light gray) may reult in a ymbol error. No ymbol error occur, if the deired ymbol i located at the correponding boarder of the contellation ( M out of M ymbol). Similarly, a location within area A (coniting of 4 partition, which are haded in dark gray) may reult in a ymbol error. No ymbol error occur, if the deired ymbol i located at the correponding corner of the contellation ( out of M ymbol). Hence the ymbol error probability of a QAM ignal corrupted by an additive random error vector i determined by: P (i) = M M P ( A )+ M P ( A ), (9) M M where P ( A i ) denote the probability, that i located within area A i. The argument in P (i) tree the fact, that the error probability i baed on an intantaneou value of the image ignal i. In order to calculate the probabilitie of being located in a certain area, the probability denity function (pdf) of the error vector will be derived in the next ection. B. itribution of the Error Vector In our analyi we conider a Rayleigh fading channel [5], i.e. the RV h and h i repreenting the channel coefficient are complex Gauian ditributed with zero mean and variance h and h i, repectively. Furthermore, we aume an independent additive Gauian noie. Therefore, the RV n and n i are complex Gauian ditributed with zero mean and variance n and n i, repectively. Baed on (7), the reulting RV can be rewritten a a quotient of two RV : = N K = h i i + n + Ki n i. (0) h The numerator N i a linear combination of Gauian ditributed RV. Hence N i alo Gauian ditributed, if the channel coefficient h and the noie term n, n i are mutually independent.with thi aumption, N ha zero mean and a variance of N = h i i + n + n i. () Similarly, the denominator i found to have zero mean and a variance of = h. Hence i the quotient of two zero-mean Gauian ditributed RV. While the derivation of it ditribution function i quite extenive in general, the derivation become much impler in the cae of N and being independent. Thi aumption hol, if h i independent from both the noie term n, n i and the channel coefficient h i. In order to calculate the ditribution function of the diplacement vector, we repreent = re jϕ in polar coordinate. The real RV r and ϕ denote the magnitude and the phae of the the complex RV, repectively. Similarly, the complex Gauian ditributed RV of the numerator and the denominator can be repreented a N = r N e jϕn and = r e jϕ. The magnitude r N and r are Rayleigh ditributed with the variance N and, repectively. It can be hown (ee Appendix), that the pdf of the magnitude r = r N /r yiel { ra (r +a ) r 0 () 0 r<0. The ditribution function i parameterized by the parameter a, which i defined a a = N = h i h i + n h + n i h. (3) The phae ϕ N and ϕ are uniformly ditributed. Hence the phae ϕ = ϕ N ϕ i alo uniformly ditributed, i.e. it pdf i f ϕ (ϕ) = π. (4) It ha been hown, that the ditribution of the complex RV can be eaily decribed in polar coordinate uing the pdf of the magnitude r and the phae ϕ. However, for the following calculation it i reaonable to conider carteian coordinate = x + jy intead. In general, the joint pdf of the polar coordinate r, ϕ and the carteian coordinate x, y are linked by [6] f xy (x, y) = r f rϕ(r, ϕ). (5) In our cae, the joint pdf of the polar coordinate i eparable, i.e. f rϕ (r, ϕ) = f r (r)f ϕ (ϕ). Merging (), (4), (5) and uing the dependency r = x + y, the joint pdf of the carteian coordinate reult in: f xy (x, y) = a π (x + y + a ). (6) C. Symbol Error Probability Given the joint pdf in carteian coordinate, the deired probabilitie P ( A ) and P ( A ) of equation (9) can be calculated by a two-dimenional integration within the appropriate integration boundarie: P ( A )=4 P ( A )=4 + f xy (x, y) dx dy (7) f xy (x, y) dx dy (8) Here we ued the circular ymmetry of the joint pdf f xy (x, y), reulting in 4 equiprobable partition of the area A and A, repectively. The equation (7) and (8) are olvable in cloed form. By merging the reult into (9), the ymbol error probability can be repreented a P (i) = M M (9) M M +γ (i) ( M ) 4 M π +γ (i) arctan +γ (i),

4 where γ (i) =[ hi h i + n h + n i h ] 4. (0) Recall, that (9) repreent the ymbol error probability for an intantaneou value of the complex-valued image ignal i. In order to calculate the total ymbol error probability, the ditribution function of the image ignal mut be conidered. With the realitic aumption, that the ample of the image ignal are generated baed on an alphabet with dicrete equally probable ymbol, the total ymbol error probability yiel P = M i P (i m ), () M i m= where M i denote the modulation order of the image ignal and i m denote the mth complex-valued ymbol of the modulation alphabet.. Second order approximation Baed on the et of equation (9)-() the exact ymbol error probability can be calculated for arbitrary etting of the individual variance. However, due to the high complexity of the formula, their uage might be impractical. Therefore we aim for a implification of the formula, while preerving the accuracy of the predicted error probability in region of interet. Our approach i to approximate the intantaneou ymbol error probability in (9) by a econd order Taylor erie, which reult in: [ P [] M (i) = + ( M ) ] γ (i). () M π M By merging (0),() and (), the econd order approximation of the total ymbol error probability yiel: P [] M i = M i [ = hi P [] (i m ) (3) m= M i h M i m= 4 d [ M + M π i m + n h ( M ) M d + n i h ]. (4) ] Interetingly, the pecific tructure of the image ignal (modulation order M i, ymbol alphabet) i irrelevant in the econd order approximation of the ymbol error probability. The only relevant parameter i the average power of the image ignal i = M i i m. (5) M i m= By uing (8), the ymbol ditance d can be replaced by the average power of the deired ignal : 4 d = 3 (M ). (6) Correction factor alpha Modulation Order M Fig. 3. Shape of the correction function α(m ) In order to derive a concie repreentation of (4), the impact of the modulation order M i eparated from the remaining parameter. We introduce the correction parameter α(m )= M 3 M [ M + M π ( M ) M ], (7) which i approximated by α(m ) / for higher order modulation (ee Fig. 3). Uing thi correction factor, (4) can be rewritten a [ P [] = hi i h + n h + n i ] h M α(m ). (8) The 3 term of the um inide the bracket of (8) deerve a more detailed dicuion. i i the average power of the image ubcarrier at the tranmitter ide, while h i i the variance of it aociated channel coefficient. Hence h i i i the average power of the image ubcarrier meaured at the receiver ide. Similarly, the term h repreent the average power of the deired ubcarrier meaured at the receiver ide. Therefore, the term SIR SC = h h i i (9) denote the ignal-to-image power ratio of the conidered pair of ubcarrier, meaured at the receiver. Similarly, the term SNR SC = h n (30) denote the ubcarrier-baed ignal-to-noie power ratio, meaured at the receiver. Finally, the third term decribe the reidual impact of the additive noie in the image ignal. Thi term i much maller than the econd term for imilar noie power n n i, becaue IRR for practical I/Q imbalance parameter. Hence, the third term of the um will be neglected. With thee definition, (8) implifie to an eay-to-ue approximation of the ymbol error probability for a ingle ubcarrier: P [] [ SIR SC IRR + SNR SC ] M α(m ). (3) Equation (3) indicate an interchangeability between ditortion due to the channel noie (SNR SC ) on the one hand and ditortion due to the I/Q imbalance (SIR SC IRR) on

5 0.0 Symbol Error Probability Fig. 4. noie Signal to Noie Ratio [db] M = 56 M = 64 M = 6 M = Theory: Approximation Theory: Exact olution Simulation Image Rejection Ratio [db] Single ubcarrier ymbol error probability: SIR =, no channel Image Rejection Ratio [db] Fig. 5. Contour plot of the ingle ubcarrier ymbol error probability: SIR =, modulation order: M =64(olid line), M = 56 (dahed line) the other hand. In the preence of a dominating channel noie (SNR SC SIR SC IRR), the et of equation (9)- () implify to the pecial cae of an M-ary QAM ignal tranmitted over a noiy Rayleigh fading channel, which ha been analyzed in variou textbook [5], [7]. Equivalently, for SNR SC SIR SC IRR, the ytem performance i mainly determined by the I/Q imbalance. Figure 4 how the ymbol error probability a a function of the image rejection ratio for different modulation order M = M i = M. The theoretical reult are confirmed by computer imulation. Approximation (3) matche ufficiently with the exact olution (9)-() for ymbol error probabilitie of P 0.. Finally, Fig. 5 addree the performance degradation due to both channel noie and I/Q imbalance for two exemplary modulation order. IV. MULTI-CARRIER SYSTEMS The derived equation for the ymbol error probability of a ingle repreentative ubcarrier contitute a bai for the calculation of the total ymbol error probability in arbitrarily deigned multi-carrier ytem. The total error probability can be obtained by merging the individual error probabilitie of all data ubcarrier. Note, that thi approach i capable to deal with frequency-elective I/Q imbalance a well. In thi cae, the parameter IRR become a function of the ubcarrier frequency. For example, we conider a imple multi-carrier ytem with the following propertie: all data ubcarrier are tranmitted with the ame power, i.e. = i = for all SC, the average power of the channel coefficient i frequency-flat, i.e. h = h i = h for all SC, the additive channel noie i white, i.e. n = n i = n for all SC, the I/Q imbalance i frequency-flat, i.e. the IRR i contant for all SC. In thi cae, the ubcarrier-baed ymbol error probability P will be identical for all data ubcarrier. Conequently, the total ymbol error probability of the multi-carrier ytem P equal P a well. When conidering the total ytem performance, it i more meaningful to determine P a a function of the ignal-to-noie power ratio SNR MC, which i defined baed on the entire multi-carrier ignal. The caling factor between SNR MC and SNR SC depen on the actual tructure of the multi-carrier ignal. Let N total denote the total number of ubcarrier. Auming N ued ued ubcarrier with equal tranmit power, we get: SNR MC = N ued h N total n = N ued N total SNR SC. (3) The validity of the derived theoretical reult will be demontrated exemplarily for the IEEE 80.a WLAN tandard []. Figure 6 how the imulation reult for the ETSI Hiperlan/ channel model A and B [8]. For high SNR, the error floor due the I/Q imbalance become obviou. A careful comparion between the theoretical and the imulative reult reveal a mall mimatch, if the I/Q imbalance i the dominating impairment. Thi effect can be undertood by conidering the individual ymbol error rate in each ubcarrier (ee Fig. 7). In the theoretical analyi, the channel coefficient of the deired ubcarrier and the image ubcarrier are aumed to be independent. However, in practical ytem a reidual dependency between the channel coefficient may exit, in particular around the C ubcarrier. Obviouly, the degree of reidual dependency i trongly related to the coherency bandwidth of the channel. An increaed coherency bandwidth correpon to a reduced ymbol error probability (here: channel ETSI-A). The gap between the theoretical and the imulative reult vanihe a the coherency bandwidth get maller. Therefore, the theoretical reult preented in thi paper can be conidered a a wort cae analyi.

6 Symbol Error Probability Theory: Approximation Theory: Exact olution Simulation: ETSI A channel Simulation: ETSI B channel IRR = 0dB IRR = 30dB IRR = 40dB IRR = 50dB IRR Signal to Noie Ratio SNR [db] MC Fig. 6. Performance degradation in a 64-QAM mode IEEE 80.a WLAN ytem due to I/Q imbalance for different image rejection ratio Symbol Error Probability 0 0 Theory Simulation: ETSI A channel Simulation: ETSI B channel Subcarrier index Fig. 7. Comparion between theoretical and meaured ymbol error probability of individual ubcarrier: 64-QAM, SNR MC =40dB, IRR=30dB V. CONCLUSION AN FUTURE WORK A novel framework for the theoretical computation of the ymbol error probability in multi-carrier ytem ha been derived in thi paper. We conidered a M-ary QAM modulated multi-carrier ignal, which i corrupted by both a noiy Rayleigh fading channel and receiver I/Q imbalance. The theoretical reult have been confirmed by computer imulation. In addition to analyzing the ymbol error probability, the preented framework can be ued a a bai for calculating the bit error probability of the multi-carrier ytem. In many practical application, the bit error probability i even more expreive than the ymbol error probability alone. Furthermore, an extenion of the ytem model towar correlated channel coefficient i deirable. A olution of thee iue i under active reearch. APPENIX Given two independent Rayleigh ditributed RV r N, r with the variance N and, repectively. The pdf of thee RV i r x f rx (r x )= e rx rx rx r x 0, (33) 0 r x < 0 where the index x i replaced by N and, repectively. We want to calculate the pdf f r (r) of the random variable r = r N /r, which i a function of the random variable r N and r. According to [6], the pdf of the quotient of two RV i r f(rr,r ) dr, (34) where f(r N,r ) denote the joint pdf of r N and r. Becaue r N and r are independent, the joint pdf can be eparated a r f rn (rr ) f r (r ) dr. (35) In order to calculate f r (r), the cae r 0 and r<0mut be ditinguihed. Firt we conider the cae r<0. According to (33), f rn (rr ) = 0 for r 0. On the other hand, f r (r )=0for r < 0. Hence (35) become r f rn (rr ) f r (r ) dr =0 (36) }{{} 0 for r<0. Next we conider the cae r 0. According to (33), f r (r )=0for r < 0. Hence the integral in (35) become 0 = 4 r N =r N for r 0. r rr N e r r N r e r dr (37) ( ) r 3 e r N + r dr (38) 0 }{{ ( ) } ( r + N r N + REFERENCES ). (39) [] B. Razavi, eign Conideration for irect-converion Receiver, IEEE Tranaction on Circuit and Sytem Part II: Analog and igital Signal Proceing, vol. 44, pp , June 997. [] IEEE, Part: Wirele LAN Medium Acce Control (MAC) and Phyical Layer (PHY) pecification, IEEE Std 80.a-999, 999. [3] M. Windich and G. Fettwei, Standard-Independent I/Q Imbalance Compenation in OFM irect-converion Receiver, in Proc. 9th Intl. OFM Workhop (InOWo), (reden, Germany), pp. 57 6, 5-6 Sept [4] C.-L. Liu, Impact of I/Q imbalance on QPSK-OFM-QAM detection, IEEE Tranaction on Conumer Electronic, vol. 44, pp , Aug [5] J. G. Proaki, igital Communication. McGraw-Hill, Inc., 3rd ed., 995. [6] A. Papouli, Probability, Random Variable and Stochatic Procee. McGraw-Hill, Inc., 3rd ed., 99. [7] K.. Kammeyer and V. Kühn, MATLAB in der Nachrichtentechnik. J. Schlembach Fachverlag, 00. [8] ETSI EP BRAN, Channel model for HIPERLAN/ in different indoor cenario, Mar. 998.