Capacity Analysis for OFDM Systems with Transceiver I/Q Imbalance

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1 Capacity Aalysis for OFDM Systems with Trasceiver I/Q Imbalace Stefa Kroe ad Gerhard Fettweis Vodafoe Chair Mobile Commuicatios Systems Techische Uiversität Dresde, 6 Dresde, Germay {stefa.kroe, fettweis}@if.et.tu-dresde.de Abstract OFDM systems have gaied utmost importace for wireless commuicatios requirig ever higher data rates. The maximum data rate that ca be achieved is, however, limited by the wireless commuicatios chael as well as by hardware imperfectios that ca ot be avoided i practice. Oe of the most serious hardware imperfectios affectig the performace of OFDM systems is trasceiver I/Q imbalace. This paper studies the maximum data rate, i.e. the capacity of OFDM systems that are impaired by trasceiver I/Q imbalace of low-cost mobile termials. Both, the dowlik ad the uplik case are cosidered. Closed-form expressios are derived for the ergodic system capacity ad the outage probability cosiderig differet types of Rayleigh fadig chaels. Numerical examples are give to provide clear isight ito the system capacity degradatio due to trasceiver I/Q imbalace. I. INTRODUCTION Orthogoal frequecy divisio multiplexig OFDM) is a approved modulatio techique for high data rate wireless commuicatios systems. It allows to easily cope with the frequecy selective ature of wireless commuicatios chaels ad has bee adopted by a umber of well-kow stadards such as IEEE 8.a [] ad IEEE 8.6 []. To provide low-cost mobile termials for OFDM systems, the direct-coversio trasceiver architecture is very attractive. It gets alog with a sigle local oscillator ad does ot require ay bulky aalog image rejectio filter [3]. Thus, it eables complete moolithic itegratio [4]. However, fabricatio toleraces typically lead to a mismatch of the aalog compoets i the I- ad Q-braches of the trasceiver, which is referred to as I/Q imbalace. Ufortuately, OFDM systems appear to be quite sesitive to this kid of hardware imperfectio. A lot of research o trasceiver I/Q imbalace i OFDM systems has already bee doe. Aalytical models describig I/Q imbalace i OFDM systems eve for multiple-atea trasceivers have recetly bee detailed i [5] ad [6]. Several I/Q imbalace compesatio schemes have bee proposed, see, for istace, [7] ad [8] as well as refereces therei. The symbol error rate SER) of OFDM systems employig sigleatea trasceivers with I/Q imbalace has bee aalyzed by the authors of [6] ad [9]. It has bee show that the SER degradatio due to trasceiver I/Q imbalace strogly depeds o the type of the wireless commuicatios chael. Frequecy selective fadig chaels appear to be more critical This work has bee supported i part by the Germa Federal Miistry of Educatio ad Research BMBF) uder grat BU83. tha o-fadig additive white Gaussia oise AWGN) chaels. Furthermore, for frequecy selective fadig chaels, it has bee stated that I/Q imbalace at the receiver side is o average more destructive tha I/Q imbalace at the trasmitter side. Obviously, this implies that trasceiver I/Q imbalace of low-cost mobile termials would be more critical for the dowlik receptio tha for the uplik trasmissio, if ideal base statio hardware ca be assumed. It is, however, importat to ote that the authors of [6] ad [9] have oly cosidered the ucoded SER. As soo as chael codig is employed, which is typically the case i practice, the achievable data rate will be a much more appropriate performace measure. Hece, cosiderig the maximum data rate will lead to more sophisticated isight ito the total system degradatio due to trasceiver I/Q imbalace. The maximum data rate that a commuicatios system ca provide for reliable trasmissio is give by its capacity. Cosiderig the pure trasmissio chael without ay hardware imperfectios, the dowlik as well as the uplik system capacity equal the chael capacity. I his famous work [], C. E. Shao derived the capacity of a additive white Gaussia oise AWGN) chael ad proposed to compute the capacity of a frequecy selective chael by slicig it ito a large umber of frequecy flat chaels, which is highly related to the cocept of OFDM. Takig chael fadig ito accout, oe has to distiguish betwee ergodic capacity ad capacity with outage []. Complemetig previous results show i [] ad [3], this paper provides a comprehesive aalysis of the capacity of OFDM systems with trasceiver I/Q imbalace of lowcost mobile termials, cosiderig the dowlik as well as the uplik case. The outlie of the paper is as follows: Sectio II briefly reviews the model of a OFDM system with trasceiver I/Q imbalace. This system model is cosidered from a statistical poit of view i sectio III which builds the basis for the aalysis of the system capacity i sectio IV. Coclusios are draw i sectio V. Throughout the paper the followig otatio will be used: Radom variables are deoted by boldface letters ad their respective realizatios by cursive letters. The probability desity fuctio PDF) of a certai radom variable Z i terms of its realizatios Z will be deoted by pz). Ay subscript at pz) will be omitted for the beefit of readability.

2 II. GENERIC SYSTEM MODEL Cosider a OFDM system with N subcarriers, where the cyclic prefix is chose to be loger tha the maximum legth L of the overall chael impulse respose. Furthermore, assume the trasmitter ad receiver to be perfectly sychroized i frequecy ad time. Let X be a complex valued symbol that is trasmitted at subcarrier. The, the trasmissio of each X over a frequecy selective chael ca be modeled i basebad by [] Y = H X + W, ) where H ad W represet the associated frequecy domai chael coefficiet ad AWGN sample, respectively. The case of a AGWN chael correspods to : H =. Note, that each X may i geeral differ from the origial trasmit symbol at subcarrier, which will be deoted by S.Olyi case of a ideal trasmitter X = S holds. Likewise, each receive symbol Y may i geeral differ from the actually received symbol R, where R = X oly holds i case of a ideal receiver. I [5] ad [6] it has bee show that I/Q imbalace of a direct-coversio trasceiver i OFDM systems iduces a mutual iterferece betwee subcarriers that are located symmetrically to the DC carrier. Sice there is at least i geeral a differece betwee the impact of I/Q imbalace at the receiver side ad I/Q imbalace at the trasmitter side, it is reasoable to model the trasceiver I/Q imbalace for the dowlik receptio ad the uplik trasmissio separately. However, for simplicity, we will use equivalet symbols for both models but distiguish betwee the I/Q imbalace characteristics of the receiver ad trasmitter frot-ed. A. Dowlik Receptio Cosiderig the dowlik receptio i a OFDM system with trasceiver I/Q imbalace of the mobile termial, I/Q imbalace at the receiver frot-ed affects the trasmitted symbols after the corruptio by the wireless chael as illustrated i Fig. a). More precisely, due to I/Q imbalace, each receive symbol Y at subcarrier will be iterfered by the complex cojugated receive symbol Y at subcarrier ad vice versa. Followig the otatio i [8], the impact of trasceiver I/Q imbalace ca thus be modeled for the dowlik receptio by R = K, Y + K, Y, ) where R deotes the actually received symbol at subcarrier. The complex valued weightig factors K, ad K, are defied through the gai ad phase imbalace at the receiver frot-ed ad may i geeral be frequecy selective. Assumig a ideal dowlik trasmitter, i.e. X = S, ad combiig ) ad ) fially yields R = K, H S + W )+K, H S + W ). 3) Based o this model, I/Q imbalace i OFDM systems is most a) b) S S wireless chael I/Q imbalace Y X I/Q imbalace R wireless chael Fig.. Mutual subcarrier iterferece due to trasceiver I/Q imbalace i wireless OFDM systems: a) dowlik receptio, b) uplik trasmissio ofte quatified i terms of a subcarrier depedet imageleakage-ratio ILR that is defied as K, ILR =. 4) K, With cotemporary semicoductor techologies a ILR of 3 db to 4 db ca be achieved, whereas the ideal case of o I/Q imbalace correspods to K, = ad K, =, i.e. : ILR = db. B. Uplik Trasmissio Lookig at the uplik trasmissio i a OFDM system with trasceiver I/Q imbalace of the mobile termial, I/Q imbalace at the trasmitter frot-ed affects the trasmitted symbols before the corruptio by the wireless chael as illustrated i Fig. b). More specifically, the origial trasmit symbol S at subcarrier will be iterfered by the complex cojugated trasmit symbol S at subcarrier ad vice versa. Similarly to ), the impact of trasceiver I/Q imbalace o the uplik trasmissio ca thus be modeled by R X = G, S + G, S, 5) where the complex valued weightig factors G, ad G, are defied through the gai ad phase imbalace at the trasmitter frot-ed ad may i geeral also be frequecy selective. Assumig a ideal uplik receiver, i.e. R = Y, ad combiig ) ad 5) delivers the received symbols of the uplik trasmissio as R = H G, S + G, S )+W. 6) The amout of I/Q imbalace ca agai be quatified i terms of a subcarrier depedet image-leakage-ratio ILR that is ow give by G, ILR =, 7) G, where the ideal case of o I/Q imbalace also correspods to G, = ad G, =, i.e. : ILR = db. Typically, the receiver ad trasmitter frot-ed are itegrated o a sigle chip usig the same semicoductor techology ad are most ofte eve built of similar aalog compoets, see, for istace, [4]. Hece, the ILR of the trasmitter frot-ed will

3 usually be i the same rage as the ILR of the receiver froted, i.e. 3 db to 4 db for cotemporary semicoductor techologies. By comparig 6) to 3), it ca be see that the impact of trasceiver I/Q imbalace o the uplik trasmissio is similar to impact o the dowlik receptio, if H = H or more geerally, if the chael coefficiets H ad H i the dowlik are highly correlated, i.e. H = H. From this basic observatio, it ca already be aticipated that I/Q imbalace at the trasmitter frot-ed affects the uplik trasmissio almost like I/Q imbalace at the receiver frot-ed affects the dowlik receptio for highly correlated chael coefficiets H ad H, which is also apparet from Fig. a) ad b). This importat poit will be referred to i the ext sectio agai. III. STATISTICAL SYSTEM CHARACTERISTICS I order to aalyze the system capacity it is ecessary to cosider the geeric system model from a statistical poit of view. Therefore, the trasmit symbols S ad X as well as the AWGN samples W of the idividual subcarriers are modeled by meas of complex valued radom variables [4] which are deoted by S, X, ad W, respectively. Cosequetly, also the receive symbols Y ad R have to be modeled by meas of the complex valued radom variables Y ad R. Furthermore, each chael coefficiet H is assumed to be a realizatio of the complex valued radom variable H, where we restrict ourselves to chaels that are slowly varyig with time, i.e. each time the chael is used for data trasmissio, the chael coefficiets H will be fixed. The I/Q imbalace parameters K, ad K, as well as G, ad G, are assumed to remai costat over time, i.e. they are ot treated as radom variables. This reiterpretatio of the geeric system model ow allows to focus o the statistical system characteristics that are required for the system capacity aalysis. A. Iterferece ad Noise As ca be see from 3) ad 6) for the dowlik receptio ad uplik trasmissio, respectively, the actually received symbols R result from a superpositio of the trasmited symbols S, the iterferig trasmited symbols S ad AGWN samples. Hece, there is a corruptio due to iterferece ad oise. The average amout of this corruptio ca be stated i terms of a subcarrier depedet sigal-to-iterferece-adoise-ratio SINR. I case of a fixed chael realizatio the SINR is for the dowlik receptio give by E { K, H S } SINR = E { K, W + K, H S + W ) }, 8) ad for the uplik trasmissio by E { G, H S } SINR = E { G, H S + W }, 9) where E { } deotes expectatio [4]. The radom variables S, S, W, ad W ca i geeral be assumed to be idepedet of each other, where = ca be igored sice the DC carrier is typically ot used for data trasmissio. I additio, it is reasoable to assume the S ad W of the idividual subcarriers to be idetically distributed with zero meas ad variaces E { S } ad E { W }, respectively. Moreover, let SNR be the subcarrier depedet sigal-to-oise-ratio of the pure wireless chael which derives for a fixed chael realizatio from ) as SNR = E{ H X } E { W } = H E { X } E { W }. ) Thus, the SINR i 8) as well as i 9) obviously results i SNR SINR = + ILR SNR + ), ) where for the uplik trasmissio, i.e. for ) resultig from 9), the SNR i the deomiator part has to be replaced by SNR. Hece, the SINR of the uplik trasmissio ca be expressed by the SINR of a equivalet dowlik receptio with SNR = SNR, i.e. : H = H. This cofirms the observatio discussed i the previous sectio. Moreover, this fact motivates to focus o the impact of I/Q imbalace at the receiver frot-ed, takig ito accout that the impact of I/Q imbalace at the trasmitter frot-ed ca be expressed by meas of a special case as stated above. From ) it is easy to see that the SINR strogly depeds o the ILR as well as o the relatio of SNR ad SNR. Furthermore, it is obvious that SINR SNR, ) which shows that trasceiver I/Q imbalace will always cause a more or less sigificat performace degradatio. B. Wireless Chael So far, fixed chael coefficiets have bee cosidered, eablig already the derivatio of the system capacity for fixed chael realizatios. However, it is expediet to have a look at the chael statistics, too, i order to aalyze the system capacity with respect to all possible chael realizatios. Give the L taps h l of the time domai chael impulse respose i basebad, the frequecy domai chael coefficiets H are obtaied by meas of the discrete Fourier trasform L H = h l e jπl/n. 3) l= Each tap of the chael impulse respose is treated as a realizatio of the complex valued radom variable h l with zero mea ad variace σh l =E{ h l }. Assumig the h l of the idividual taps to be mutually idepedet, which is a commo assumptio for wireless commuicatios chaels, the variace of the frequecy domai chael coefficiets yields L σh =E{ H } = σh l 4) l=

4 for all subcarriers. Thus, the variace of the frequecy domai chael coefficiets simply derives from the chael power delay profile []. Furthermore, the cross correlatio of two chael coefficiets yields L E{H H k} = σh l e jπ k)l/n. 5) l= From 3) it ca already be aticipated that the system capacity of the dowlik will deped o the cross correlatio of the chael coefficiets of subcarriers located symmetrically to the DC carrier. Therefore, it is coveiet to defie the cross correlatio coefficiet ρ give by E{H ρ = H } E{ H } E{ H } = E{H H } σh 6) for each subcarrier. As already discussed, the impact of trasceiver I/Q imbalace o the uplik trasmissio ca be expressed by meas of the impact o a equivalet dowlik receptio, where : ρ =. Cosiderig Rayleigh fadig chaels, which are of special iterest for wireless commuicatios systems, the joit PDF of the chael coefficiets H ad H is give by the multivariate Gaussia distributio [5] ph,h )= π σ 4 H ρ ) exp H + H Re{ρ H H } σh ρ ) ), 7) where σh ad ρ have bee defied i 4) ad 6), respectively, ad Re{ } deotes the real part. The system capacity aalysis requires the joit characteristics of the absolute values of the chael coefficiets H ad H. Hece, they are expressed i terms of the ormalized absolute values Ψ, ) ad phase values φ π, π) with H σ =Ψ e jφ, 8) H where Ψ ad φ are treated as realizatios of the real-valued radom variables Ψ ad φ, respectively. By cartesiato-polar-trasformatio ad margial PDF computatio, 7) delivers the joit PDF of the radom variables Ψ ad Ψ pψ,ψ )= 9) ) ) 4 Ψ Ψ ρ exp Ψ +Ψ ρ Ψ ρ I Ψ ρ, where I ) deotes the modified Bessel fuctio of first kid ad zeroth order [4]. For the two special cases of fully correlated ad ucorrelated chael coefficiets, i.e. for ρ = ad ρ =, 9) reduces to pψ, Ψ ) ρ = = Ψ exp Ψ ) δψ Ψ ), ) where δ ) deotes the Dirac delta fuctio ad pψ, Ψ ) ρ = = 4 Ψ Ψ exp Ψ Ψ ). ) IV. SYSTEM CAPACITY ANALYSIS From [6] it is kow that the capacity of a OFDM system for a fixed chael realizatio derives as the sum capacity of the idividual subcarriers used for data trasmissio. Hece, cosiderig the system capacity C at a sigle subcarrier will be sufficiet for drawig coclusios with respect to the overall system capacity. The computatio of each C ca be based o [], where it has already bee show that i case of a AWGN chael the capacity of a commuicatios system is achieved by usig Gaussia distributed trasmit symbols. OFDM trasforms a frequecy selective chael ito a set of frequecy flat chaels, which i terms of capacity boils dow to a AWGN chael for each subcarrier []. Thus, without hardware imperfectios, the system capacity C at a sigle subcarrier is simply give by the AWGN chael capacity derived i [], which yields for a fixed chael realizatio C = log + SNR ) ) i bits per chael use. Note that complex valued trasmit symbols are cosidered here. For the derivatio of the system capacity C with trasceiver I/Q imbalace, complex Gaussia distributed trasmit symbols are presumed as well, focusig o the geeral case where the mutual iterferece betwee the subcarriers is ukow ad therefore ca ot be exploited. Assumig complex Gaussia distributed trasmit symbols, the iterferece raised at each subcarrier due to I/Q imbalace will be complex Gaussia distributed, too. Thus, it ca be iterpreted as part of the AWGN, which meas that the origial SNR becomes the SINR give i ). Accordigly, the system capacity C with trasceiver I/Q imbalace derives from ) by replacig the SNR with the SINR, which yields C = log + SNR + ILR SNR + ) ). 3) Itroducig the average sigal-to-oise-ratio SNR of the pure wireless chael such that SNR =Ψ SNR 4) for each subcarrier, the system capacity for a fixed chael realizatio give i 3) ca be rewritte as Ψ C = log ) SNR + + ILR Ψ, 5) SNR + ) where : Ψ =Ψ has to be cosidered the uplik trasmissio. From ) directly follows that the system capacity C with trasceiver I/Q imbalace will always be less tha without trasceiver I/Q imbalace. Moreover, for SNR approachig ifiity, C is upper bouded by C SNR= = log + Ψ Ψ ILR ), 6) where the upper boud depeds o the actual ILR as well as o the ratio of Ψ ad Ψ.

5 Chael fadig will cause the istataeous system capacity C to vary each time the wireless commuicatios chael is used for data trasmissio. Hece, it is reasoable to ivestigate the ergodic system capacity ad the system capacity with outage takig all possible chael realizatios, i.e. the statistical characteristics of the wireless chael, ito accout. A. Ergodic System Capacity Assume the istataeous system capacity C give by 5) to be realized at each data trasmissio. The ergodic system capacity C is the defied as the average system capacity give by the double itegral C = C pψ, Ψ ) dψ dψ, 7) where pψ, Ψ ) is the joit PDF of the radom variables Ψ ad Ψ, represetig the ormalized absolute values of the chael coefficiets at subcarrier ad. Itiseasyto show that the overall ergodic system capacity agai derives as the sum of the C of the idividual subcarriers that are used for data trasmissio. Hece, it is sufficiet to cocetrate o C of a sigle subcarrier. Cosiderig Rayleigh fadig chaels, pψ, Ψ ) is give by 9) ad more specifically for ρ = ad ρ = by ) ad ), respectively. The computatio of C requires umerical itegratio i each case. Fig. shows the obtaied results. It ca be observed that the ergodic system capacity C may be sigificatly degraded due to trasceiver I/Q imbalace depedig o the actual ILR.ForafixedILR, the degradatio becomes more ad more severe with icreasig SNR. However, the degradatio does ot cosiderably deped o the correlatio of the chael coefficiets, i.e. o ρ. Oly i case of exceptioally high ILR, correlated chael coefficiets cause a aggravated capacity degradatio. Sice the impact of trasceiver I/Q imbalace o the uplik trasmissio ca be expressed by meas of the impact o a equivalet dowlik receptio, where : ρ =, this observatio leads to the coclusio that, regardig the ergodic system capacity, trasceiver I/Q imbalace affects the uplik trasmissio almost like the dowlik receptio. More aalytically, the results depicted i Fig. also show C ρ = C C ρ =, 8) which cofirms, that ρ = ad ρ = defie the lower ad upper asymptotes of C, respectively. For SNR approachig ifiity, closed-form solutios of 7) ca be foud for ρ = ad ρ = by usig 6) together with ) ad ), which leads to C ρ =, = log ILR ) 9) SNR= ILR ad C ρ =, = log SNR= + ). 3) ILR Accordig to Fig., 9) delivers a etire upper boud of the ergodic system capacity C for the dowlik receptio C i bits per chael use SNR = 3 db SNR = db SNR = db SNR = db 4 SNR = 3 ILR i db ρ data data data3 = ρ =.8 ρ = Fig.. Degradatio of the ergodic system capacity C due to trasceiver I/Q imbalace for differet sigal-to-oise ratios SNR ad Rayleigh fadig chael correlatio coefficiets ρ with trasceiver I/Q imbalace i case of a Rayleigh fadig chael, whereas 3) provides the correspodig etire upper boud for the uplik trasmissio. B. System Capacity with Outage Assume that each data trasmissio requires oe ad the same data rate, where the data rate is defied via a miimum SINR deoted by SINR mi, that has to be provided to trasmit reliably. Aytime the istataeous SINR is less tha SINR mi,, the data trasmissio will ot be reliable, i.e. there will be a outage. Give a certai SINR mi,,the maximum data rate that ca be achieved i case of reliable trasmissio is defied as the outage capacity C out, give by C out, = log + SINR mi, ) 3) for each subcarrier. The associated outage probability P out, derives from the joit PDF pψ, Ψ ) as SINR mi, + ILR +Ψ SNR )) / SNR P out, = pψ, Ψ ) dψ dψ, 3) ad cosequetly depeds o the ILR as well as o SNR. Cosiderig Rayleigh fadig chaels, pψ, Ψ ) is agai give by 9), where closed-form solutios of 3) ca be foud for ρ = ad ρ = with ) ad ), respectively. For ρ =, P out, yields P ρ = out, = 33) exp SINR ) mi, + ILR ), +SINR mi, ILR SNR whereas ρ = leads to the expressio give i 34). Numerical results for P out, cosiderig ILR = 3 db are show i Fig. 3, where umerical itegratio has bee used to compute P out, for ρ =.8. It ca be observed that trasceiver I/Q imbalace may severely raise the outage probability at each subcarrier, especially for moderate ad high SNR. Furthermore, for practically relevat P out, <.5,

6 P ρ = out, = exp SINR ) mi, + ILR ), for SINR mi, SNR SINR mi, ILR ) ILR, else 34) P out,.8.6 SNR = db SNR = db SNR = db SNR = 3 db SNR =.4 ρ =. ρ =.8. ρ = o I/Q imbalace C out, i bits per chael use Fig. 3. Impact of trasceiver I/Q imbalace o the outage probability P out, for ILR = 3 db, differet sigal-to-oise ratios SNR ad Rayleigh fadig chael correlatio coefficiets ρ highly correlated chael coefficiets cause the outage probability to be affected eve worse tha for ucorrelated chael coefficiets, i.e. for ρ =. This observatio ca also be substatiated by the results show i Fig. 4, where the average system capacity with outage C out,, which is give by C out, = P out, ) log + SINR mi, ) 35) for each subcarrier, has bee cosidered for P out, =. ad ILR = 3 db. Ulike the ergodic capacity C,the average capacity with outage C out, appears to strogly deped o the chael correlatio coefficiet ρ with C out, ρ = C out, C out, ρ =. 36) Thus, it also turs out that, regardig the average system capacity with outage, the impact of trasceiver I/Q imbalace o the uplik trasmissio, correspodig to ρ =, will be at least as much severe as o a equivalet dowlik receptio. This is i some respects cotradictory to the results give i [5] ad [6] but ca be traced back to the fact that the actual system capacity has bee cosidered istead of the SER. V. CONCLUSIONS I this paper the capacity of OFDM systems with trasceiver I/Q imbalace of low-cost mobile termials has bee ivestigated. It has bee show, that the impact of trasceiver I/Q imbalace o the uplik trasmissio ca be expressed by meas of the impact o a equivalet dowlik receptio. It was foud that trasceiver I/Q imbalace limits the ergodic system capacity as well as the system capacity with outage to upper bouds that deped o the iterferece raised at each subcarrier. For Rayleigh fadig chaels, closed-form expressios for the ergodic system capacity ad the outage probability have bee derived. Thus, OFDM system desigers are eabled to verify which amout of trasceiver I/Q imba- C out, i bits per chael use ρ. = ρ =.8 ρ = o I/Q imbalace SNR i db Fig. 4. Impact of trasceiver I/Q imbalace o the average outage capacity C out, for P out, =., ILR = 3 db ad differet Rayleigh fadig chael correlatio coefficiets ρ lace ca be tolerated to esure reliable trasmissio i the dowlik ad uplik at a give maximum data rate. REFERENCES [] IEEE Std 8.a-999, Part: Wireless LAN Medium Access Cotrol MAC) ad Physical Layer PHY) Specificatios, 999. [] IEEE Std 8.6-4, Part6: Air Iterface for Fixed Broadbad Wireless Access Systems, 4. [3] A. A. Abidi, Direct-Coversio Radio Trasceivers for Digital Commuicatios, IEEE Joural of Solid-State Circuits, December 995. [4] P. Zhag et al., A 5-GHz Direct-Coversio CMOS Trasceiver, IEEE Joural of Solid-State Circuits, December 3. [5] Y. Zou ad M. Valkama ad M. Refors, Aalysis ad Compesatio of Trasmitter ad Receiver I/Q Imbalaces i Space-Time Coded Multiatea OFDM Systems, EURASIP Joural o Wireless Commuicatios ad Networkig, Jauary 8. [6] T. C. W. Schek, E. R. Fledderus ad P. F. M. Smudlers, Performace Aalysis of Zero-IF MIMO OFDM Trasceivers with IQ Imbalace, Joural of Commuicatios, December 7. [7] T. C. W. Schek, E. R. Fledderus ad P. F. M. Smudlers, Estimatio ad Compesatio of Frequecy Selective TX/RX IQ Imbalace i MIMO OFDM systems, Proc. of the IEEE Iteratioal Coferece o Commuicatios, Jue 6. [8] M. Widisch ad G. Fettweis, Blid Estimatio ad Compesatio of I/Q Imbalace i OFDM Receivers with Ehacemets Through Kalma Filterig, Proc. of the IEEE/SP Workshop o Statistical Sigal Processig, August 7. [9] M. Widisch ad G. Fettweis, O the Impact of I/Q Imbalace i Multi-Carrier Systems for Differet Chael Scearios, Proc. of the IEEE Iteratioal Symposium o Circuits ad Systems, May 7. [] C. E. Shao, Commuicatio i the Presece of Noise, Proc. of the I.R.E, Jauary 949. [] A. Goldsmith, Wireless Commuicatios. Cambridge Uiv. Press, 5. [] M. Krodorf ad G. Fettweis, OFDM Lik Performace Aalysis uder Various Receiver Impairmets, EURASIP Joural o Wireless Commuicatios ad Networkig, Jauary 8. [3] S. Kroe ad G. Fettweis, O the Capacity of OFDM Systems with Receiver I/Q Imbalace, Proc. of the IEEE Iteratioal Coferece o Commuicatios, May 8. [4] A. Papoulis ad S. U. Pillai, Probability, Radom Variables ad Stochastic Processes, 4th ed. McGraw-Hill, Ic.,. [5] J. G. Proakis, Digital Commuicatios. McGraw-Hill, Ic.,. [6] D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatio. Cambridge Uiv. Press, 5.

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