Hindawi Pubishing Corporation EURASIP Journa on Wireess Counications and Networking Voue 8, Artice ID 79, pages doi:./8/79 Research Artice OFDM Link Perforance Anaysis under Various Receiver Ipairents Marco Krondorf and Gerhard Fettweis Vodafone Chair Mobie Counications Systes, Technische Universität Dresden, D-6 Dresden, Gerany Correspondence shoud be addressed to Marco Krondorf, krondorf@ifn.et.tu-dresden.de Received 8 May 7; Accepted Septeber 7 Recoended by Hiket Sari We present a ethodoogy for OFDM ink capacity and bit error rate cacuation that jointy captures the aggregate effects of various rea ife receiver iperfections such : carrier frequency offset, channe estiation error, outdated channe state inforation due to tie seective channe properties and fat receiver I/Q ibaance. Since such an anaytica anaysis is sti issing in iterature, we intend to provide a nuerica too for reaistic OFDM perforance evauation that takes into account obie channe characteristics we utipe receiver antenna branches. In our ain contribution, we derived the probabiity density function PDF) of the received frequency doain signa with respect to the entioned ipairents and use this PDF to nuericay cacuate both bit error rate and OFDM ink capacity. Finay, we iustrate which of the entioned ipairents h the ost severe ipact on OFDM syste perforance. Copyright 8 M. Krondorf and G. Fettweis. This is an open access artice distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the origina work is propery cited.. INTRODUCTION Orthogona frequency division utipexing OFDM) is a widey appied technique for wireess counications, which enabes sipe one-tap equaization by cycic prefix insertion. Conversey, the sensitivity of OFDM systes to various receiver ipairents is higher than that of singe-carrier systes. Furtherore, for OFDM syste designers, it is often desirabe to have ey to use nuerica toos to predict the syste perforance under various receiver ipairents. Within this artice the ter perforance eansbothinkcapacity and uncoded bit-error rate BER). Mosty, ink eve siuations are used to obtain reiabe perforance eures of a given syste configuration. Unfortunatey, siuations are highy tie consuptive especiay when the paraeter space of the syste under investigation is arge. Therefore, the intention of this artice is to introduce a stochtic/anaytica ethod to predict the perforance etrics of a given OFDM syste configuration. To get reaistic perforance resuts, our approach takes into account a variety of receiver characteristics and ipairents we obie channe properties such i) residua carrier frequency offset CFO) after synchronization; ii) channe estiation errors; iii) outdated channe state inforation due to tie seective obie channe properties; iv) fat receiver I/Q ibaance in ce of direct conversion receivers; v) frequency seective obie channe characteristics; vi) utipe receiver branches to reaize diversity cobining ethods such axiu ratio cobining MRC). In present OFDM standards, such IEEE 8.a/g or DVB-T, preabe or piots) are used to estiate and to copensate the CFO and channe ipuse response. Unfortunatey, after CFO estiation and copensation, the residua carrier frequency offset sti destroys the orthogonaity of the received OFDM signas and corrupts channe estiates, which worsen further the perforance of OFDM systes during the equaization process. In the iterature, the effects of carrier frequency offset on bit-error rate are osty investigated under the suption of perfect channe knowedge. The papers [, 6] consider the effects of carrier frequency offset ony without channe estiation and equaization iperfections) and give exact anaytica expressions in ters of SNR-oss and OFDM bit-error rate for the AWGN
EURASIP Journa on Wireess Counications and Networking channe. The authors of [8] extend the work of [] toward frequency-seective fading channes and derive the correspondent bit-error rate for OFDM systes in ce of CFO under the suption of perfect channe knowedge. Cheon and Hong [] tried to anayze the joint effects of CFO and channe estiation error on uncoded bit-error rate for OFDM systes, but the used Gaussian channe estiation error ode does not hod in rea OFDM systes, especiay when carrier frequency offset is arge see Section ). Additionay, receiver I/Q ibaance h been identified one of the ost serious concerns in the practica ipeentation of direct conversion receiver architectures see, e.g., []). Direct conversion receiver designs are known to enabe sa and cheap OFDM terinas, highy suitabe for consuer eectronics. The authors of [] investigated the effect of receiver I/Q ibaance on OFDM systes for frequency seective fading channes under the suption of perfect channe knowedge and perfect receiver synchronization. Additionay, in order to cope with this ipairent, the authors of [] proposed a digita I/Q ibaance copensation ethod. To our best knowedge, there is currenty no iterature avaiabe that describes a cacuation ethod for OFDM BER and ink capacity under the aggregate effect of a the entioned ipairents. Therefore, our intention is to describe the quantitative reationship between OFDM paraeters, receiver ipairents, and perforance etrics such biterror rate and ink capacity. Furtherore, we intend to provide a usefu syste engineering too for the design and diensioning of OFDM syste paraeters, piot sybos, and receiver agoriths used for frequency synchronization, channe estiation, and I/Q ibaance copensation. The structure of this artice is foows. After soe genera rearks on our proposed ink capacity evauation ethod in Section, we introduce our OFDM syste ode in section foowed by a genera probabiity density function anaysis in Section. InSection, it wi be expained how to ode the correation between channe estiates and received/ipaired signas to derive uncoded bit-error rates of OFDM systes with carrier frequency offset and I/Q ibaance in Rayeigh frequency and tie seective fading channes. It shoud be noted that the ters bit-error rate and biterror probabiity are used with equa eaning. This is due to the fact that the bit-error rate converges toward bit-error probabiity with increing observation tie in a stationary environent. Finay, we introduce our ink capacity cacuation ethod in Section 6 and concude in Section 7.. THE APPROACH We choose ink capacity, eured in bit/channe use, an iportant perforance etric for OFDM syste designs. This inforation theoretic etric aows syste designers to characterize the syste behavior subject to rea-ife receiver ipairents independenty fro any kind of channe coding and iterative detection ethods. As expained in Section 6 and iustrated in Figure, the OFDM transceiver chain incuding channe and receiver properties can be characterized effective channe between source and detector, often caed the oduation channe. The oduation channe is characterized by its conditiona PDF f Z X z x) that describes the statistica reationship between the discrete input sybos x and the continuousy distributed decision variabe z. Using any given copex M-QAM consteation aphabet X, the ink capacity can be expressed utua inforation between source and sink that ony depends on the input statistic of X and f Z X z x). Since our perforance anaysis fraework intends to describe the utua inforation and hence the ink capacity under a given input statistic), we propose the foowing work fow. ) We show how to derive f Z X z x) under receiver ipairents, given channe properties and OFDM syste paraeters. ) We use the derived f Z X z x) for uncoded BER cacuation to verify its correctness by coparing the BER prediction resuts with those obtained fro siuation. 3) We cacuate the utua inforation, that is, OFDM ink capacity, using the verified statistic f Z X z x). 3. OFDM SYSTEM MODEL We consider an OFDM syste with N-point FFT. The data is M-QAM oduated to different OFDM data subcarriers, then transfored to a tie doain signa by IFFT operation and prepended by a cycic prefix, which is chosen to be onger than the axia channe ipuse response CIR) ength L. The saped discrete copex beband signa for the th subcarrier after the receiver FFT processing can be written Y = X H + W, ) where X represents the transitted copex QAM oduated sybo on subcarrier, andw represents copex Gaussian noise. The coefficient H denotes the frequency doain channe transfer function on subcarrier, which is the discrete Fourier transfor DFT) of the CIR hτ) withaxia L taps L H = hτ)e jπτ/n. ) τ= In this paper, it is sued that the residua carrier frequency offset after frequency synchronization) is a given deterinistic vaue. Furtherore, static non-tie-seective) channe characteristics are sued during one OFDM sybo. The CFO-ipaired copex beband signa subcarrier can be written Y = X H I) + N/ k= N/, k X k H k Ik )+W. 3) The copex coefficients IK ) represent the ipact of the received signa at subcarrier k on the received signa at
M. Krondorf and G. Fettweis 3 Coding & sybo apping Discrete copex input aphabet X OFDM oduation Mobie channe, SIMO OFDM deoduation & cobining Continuous copex detector input aphabet Z Detection & decoding Moduation channe-representing PHY ipairents, oduation characteristics, and obie channe properties Figure : The oduation channe concept used for capacity evauation. subcarrier due to the residua carrier frequency offset defined in [] Ik ) = e jπk )+Δ f ) /N) sin π k )+Δ f )) Nsin π k )+Δ f ) /N ), ) where X P, and Ỹ P, denote the transitted and received preabe sybo on subcarrier. The Gaussian noise of the preabe part W h the sae variance W of the data part σ W = σ W ). The channe estiate is used for frequency doain zero-forcing equaization before data detection where Δ f is the residua carrier frequency offset noraized to the subcarrier spacing. In addition, ater in this paper, the suation N/ k= N/, k wi be abbreviated k. In3) we can see that residua CFO causes a phe rotation of the receivedsigna I)) and intercarrier interference ICI). Furtherore, there is a tie variant coon phe shift for asubcarriersduetocfogivenin[8] that is not odeed here. This is due to the fact that this tie variant coon phe ter is considered to be robusty estiated and copensated by continuous piots that are inserted aong the OFDM data sybos. I/Q ibaance of direct conversion OFDM receivers directy transates to a utua interference between each pair of subcarriers ocated syetricay with respect to the DC carrier []. Hence, the received signa Y at subcarrier is interfered by the received signa Y at subcarrier, and vice versa. Therefore, the undesirabe eakage due to I/Q ibaance can be odeed by [, ] Ỹ = Y + K Y, ) where ) represents the copex conjugation and K denotes a copex-vaued weighting factor that is deterined by the receiver phe and gain ibaance []. The iage rejection capabiities of the receiver on subcarrier can be expressed in ters of iage rejection ratio IRR) given by IRR =. 6) K In this paper, we consider fat I/Q ibaance which sipy eans IRR = IRR for a. Subsequenty, we consider preabe-bed frequency doain et-square FDLS) channe estiation to obtain the channe state inforation Ĥ ) on subcarrier : Ĥ = ỸP, = I)H + k X P,k H k Ik )+W X P, X P, X + K P,H I + )+W, X P, 7) Z = Ỹ Ĥ, 8) where Z is the decision variabe that is feed into the detector/decoder stage. The power of preabe signas and the average power of transitted data signas on a carriers are equivaent X P = σ X). In ce of utipe N Rx )receiver branches, axiu ratio cobining MRC) is used at the receiver side. Therefore, the decision variabe Z on subcarrier is given by Z = NRx κ=y,k Ĥ,κ Ĥ,κ, 9) NRx κ= where κ denotes the receiver branch index. We sue that there is the sae IRR and CFO on a branches, what is reonabe when considering one osciator used for downconversion in each branch. Furtherore, we sue uncorreated channe coefficients aong the branches, that is, E { H,κ H,κ } = ifκ κ,. ) 3.. Mobie channe characteristics To obtain precise perforance anaysis resuts in ce of subcarrier crosstak induced by CFO and I/Q ibaance, it is desirabe to use exact expressions of the subcarrier channe cross-correation properties what is shown in ore detai in Section. The cross-correation properties between frequency doain channe coefficients are ainy deterined by the power deay profie of the channe ipuse response CIR) and the CIR tap cross-correation properties. Furtherore, the discrete nature of the saped CIR is odeed tapped deay ine having L channe taps. Athough our For sake of readabiity, we ony incude the antenna branch index κ if necessary.
EURASIP Journa on Wireess Counications and Networking anaysis is not iited to a specific type of frequency seective channe, in our nuerica exapes, we consider obie channes having an exponentia power deay profie PDP): σ τ = C e Dτ/L, τ =,,..., L, ) where σ τ = E{ hτ) } and the factor C = L τ=e Dτ/L is chosen to noraize the PDP L τ=σ τ =, what eads to σ H = E{ H }=, for a. The channe taps hτ) aresued to be copex zero-ean Gaussian RV with uncorreated rea and iaginary parts. Hence, after DFT according to ), the channe coefficients are zero-ean copex Gaussian rando variabes we. Additionay, the CIR ength L is sued to be shorter than/equa to the cycic prefix. The cross-correation coefficient of the channe transfer function on subcarriers k and in ce of frequency seective fading is defined r k, = E{ H k H }, k, ) σ H where σ H is equivaent for a subcarriers. Assuing utua uncorreated channe taps of the CIR and appying ), one gets E { L H k H } = σ τe jπk )τ/n. 3) τ= The cross-correation property of the copex Gaussian channe coefficients can be foruated to be H k = r k, H + V k,, ) where V k is a copex zero-ean Gaussian with variance σ V k, = σ H r k, )ande{v k, H } =. In current OFDM systes such 8.a/n or 8.6, there is a typica OFDM bock structure. An OFDM bock consists of a set of preabe sybos used for acquisition, synchronization, and channe estiation, foowed by a set of seriay concatenated OFDM data sybos. User obiity gives rise to a considerabe variation of the obie channe during one OFDM bock ft fading) what causes outdated channe inforation in certain OFDM sybos if there is no appropriate channe tracking. To be precise, during the tie period λ between channe estiation and OFDM sybo reception, the channe changes in a way that the estiated channe inforation used for equaization does not fit the actua channe anyore. If there is no channe tracking at the receiver side, our ai is to incorporate the effect of outdated channe inforation into the perforance anaysis fraework. Therefore, we have to define the autocorreation properties of channe coefficients H. The autocorreation coefficient of subcarrier is defined foows: r H, λ) = E{ H t)h t + λ) }. ) Appying )weget E { H t)h t + λ) } { L L = E τ= ν= σ H hτ, t)h ν, t + λ)e πτ ν)/n) }. 6) When suing uncorreated channe taps, it foows E { H t)h t + λ) } L = r h τ, λ)σ τ. 7) For sake of sipicity, it is sued that a channe taps have the sae autocorreation coefficient, that is, r h τ, λ) = r h λ), for a τ L. Substituting the reation L τ=σ τ = σ H and 6) into ), we obtain τ= r H, λ) = r h λ). 8) For the nuerica BER and ink capacity evauations done in Section. and 6., thetie seectivity ofthecopex Gaussian channe taps w odeed foows: with hτ, t + λ) = r h τ, λ)hτ, t)+v τ,λ, 9) E { hτ, t) } = E { hτ, t + λ) } = σ τ, ) where v τ,λ is a copex Gaussian RV with variance σ v τ,λ = σ τ r h τ, λ) )ande{hτ, t)vτ,λ }=. For sake of sipicity, it is sued that the channe is stationary during one OFDM sybo but changes fro sybo to sybo in the above defined anner. In our anaysis, we intentionay avoid any suptions on concrete ft-fading odes in order to obtain fundaenta resuts. Anyway, one of the coony used statistica descriptions of ft channe variations is the Jakes ode [7], where the channe autocorreation coefficient r h τ) isgivenby r h τ) = J πfd,ax ), ) and f D,ax denotes the axiu Dopper frequency that is deterined by the obie veocity and carrier frequency of the syste. It shoud be noted that r h τ) is rea due to uncorreated i.i.d. rea and iaginary parts of the CIR taps.. PROBABILITY DENSITY FUNCTION ANALYSIS The author of [9] suggested a correation ode regarding channe estiation for singe-carrier systes and derived the correspondent sybo error-rate and bit-error rate of QAModuated signas transitted in fat Rayeigh and Ricean channes. In this section, a short review of the contribution of [9] wi be given in order to further extend these resuts to OFDM systes for tie and frequency seective fading channes with CFO, I/Q ibaance, and channe estiation error. The singe-carrier transission ode without carrier frequency offset for fat Rayeigh fading channes can be written y = hx + w, ) where y, h, x, and w denote the copex beband representation of the received signa, the channe coefficient, the transitted data sybo, and the additive Gaussian noise
M. Krondorf and G. Fettweis with variance σ w,respectivey.in[9], the channe estiate ĥ is sued to be bied and used for zero forcing equaization foows: x, ) I z), ) x z = y ĥ with ĥ = αh + ν, 3) where α denotes the deterinistic utipicative bi of the channe estiates and ν represents zero-ean copex Gaussian noise with variance σ ν. The channe coefficient h and Gaussian noise ν are sued to be uncorreated. Hence, the ce of perfect channe knowedge can be eiy odeed by α = and σ ν =. In [9], the joint PDF of the decision variabe z = z r + jz i in ce of transit sybo x is derived in cartesian coordinates and can be written a f Z X z x) x) = π z bx) + a x) ). ) z=z r +jz i The PDF ainy depends on the copex paraeter bx), given by [, 9] bx) = R{b} + ji{b} =b r x)+jb i x) α r h λ)σ ) h = x α σ h + σ ν ) and the rea paraeter ax) that can be written according [, 9] a x) = x α σ h r h λ) ) + σ νσ h σ α σ h + σ + w ν) α σ h + σ. ν 6) Additionay, the cosed for integra of )withz = z r + jz i is given by [9] tobe F Z X z x) zi b i x) ) arctan zr b r x) )/ a x)+ z i b i x) ) ) = + π a x)+ z i b i x) ) zr b r x) ) arctan zi b i x) )/ a x)+ z r b r x) ) ) π a x)+ z r b r x) ). 7) In ce of N Rx receiver branches, axiu ratio cobining MRC) is used for decision variabe coputation what can be foruated z = NRx κ=y κ ĥ κ ĥ κ, 8) NRx κ= where κ represents the antenna branch index, and the κth channe estiate can be written according to the ce ĥ κ = α k h κ + ν κ. 9) x 3 x, ), ) B, R z) Figure : The QPSK consteation digra, showing the decision region for one bit position of sybo x. Since it is quite reonabe to sue that the sae channe estiation schee is used in each receive antenna branch, we have α κ = α, foraκ. The authors of [9] aso derived the PDF of z in ce of transit sybo x and N Rx receiver branches that is given by ) N Rx a f x) ) N Rx Z X,NRx z x, NRx = π z bx) + a )NRx+. 3) x) It is ey to observe that the PDF 3) for the MRC ce takes the for of ) inceofn Rx =. Additionay, the cosed for integra F Z X,NRx z x, N Rx )of f Z X,NRx z x, N Rx ) can be found in [9] that aso takes the for 7)ince of N Rx =. To enhance readabiity and to sipify our notation, we oit the receiver branch nuber N Rx in the conditiona PDF and its cosed for integra, that is, in the foowing we write f Z X z x) instead of f Z X,NRx z x, N Rx ). Finay, the resut of 7) can be used to cacuate the biterror rate of a given M-QAM consteation. In an M-QAM consteation there are Mog M) different possibe bit positions with respect to the M-QAM consteation. The probabiity of an erroneous bit with respect to the th QAM transit sybo x can be cacuated by using the cosed for integra 7) and an appropriate decision region B,ν for the νth bit position see Figure ) that takes into account the bit apping of the QAM consteation. In the paper, we aways use Gray apping in our nuerica resuts, but it is worth entioning that the described ethod can be used for arbitrary bit appings we. As aready stated, we propose to use bit-error rate prediction to verify the correctness of the derived probabiity density function f Z X z x) that is ater used to deterine the OFDM ink capacity of a given transceiver configuration. Therefore, the bit-error probabiity P b x ) takes the for P b x ) = og M) = og M) ν= [[ FZ X z x )]] B,ν, 3) where [[F Z X z x )]] B,ν denotes the -diensiona evauation of the cosed for integra F Z X z x ) subject to the
6 EURASIP Journa on Wireess Counications and Networking decision region B,ν. Finay, the bit-error probabiity can be obtained by averaging over a possibe consteation points, when suing equa probabe M-QAM sybos foows: P b = M ) P b x. 3) M =. OFDM BIT-ERROR RATE ANALYSIS In this section, the derivation of the bit-error rate of OFDM systes with carrier frequency offset, I/Q ibaance, and channe estiation error in Rayeigh frequency and tieseective fading channes wi be given. The centra idea of our BER derivation is to ap the OFDM syste ode of Section 3 to the statistics given in Section.Tobeprecise,we have to ap the OFDM syste ode to the paraeters α, a 6)andb ) expained beow... Matheatica derivation Firsty, we can rewrite the channe estiates of subcarrier in 7) with respect to the frequency seective fading characteristic given in ) tobe Ĥ = I)H + + e K k r k, X P,k Ik ) I)X P, r, X P,I + ) I)X P, ) + ν, 33) where e denotes the ter e jφ. This coes due to the fact that the copex Gaussian channe coefficient can be written H = H e jφ.hence,wehaveh /H = e jφ = e, where φ is an equay distributed RV in the interva [ π : π]. Fro 33) weobtainan3)-ike expression foows: Ĥ = α H + ν, 3) by defining effective channe H = I)H and effective bi α k r k, X P,k Ik )+ek r, α = + X P,I + ), I)X P, 3) where α is a stochtic quantity with given subcarrier index, a set of deterinistic preabe sybos X P,k,afixedpredeterined frequency offset, a given IRR constant K and RV e = e jφ. It shoud be noted that the stochtic part of α is negigibe in ce of oderate I/Q ibaance IRR 3 db) and oderate CFO. Hence, we have that ek r, X P,I + ), 36) and α can be we odeed to be a deterinistic quantity. This is due to the fact that the piot sybos X P,k we the CFO are given deterinistic vaues and the channe crosscorreation coefficients r k, can be cacuated using ) and 3). The noise part ν of the channe estiate can be written ν = W + K W + k X P,k V k, Ik ) X P, XP,V, + K I + ). X P, 37) For σ ν, which represents the additive Gaussian noise variance of the channe estiates, we obtain σ ν = X P,k XP,nIk )I n ) r k,n r k, rn, σ ) H k n + K XP,k X P,nI k + )In + ) k n r k,n r k, r n,σ H) + σ W + K ). 38) Appying the sae ethod above for 3)and), the sae definition of effective channe H can be used to get a )- ike expression foows: Y = H X + + W = H X + W. k r k, X k Ik ) + e K I) r, X I ) + ) I) 39) Given 39), the effective sybo X can be defined that is no onger a deterinistic vaue but a stochtic quantity due to i.i.d. data sybos on subcarriers k : k r k, X k Ik )+ek r, X = X + X I + ). I) }{{} stochtic part of the effective transit sybo ) Assuing a certain transit sybo X and suing randoy transitted data sybos X k with k,wecandecopose the effective sybo X foows: X = X + J, ) which shows the stochtic nature of X due to the rando interference part J due to ICI and I/Q ibaance. Appying the centra iit theore, we sue that the interference J ter is a copex zero-ean Gaussian rando variabe J = p+ jq. The utua uncorreated rea and iaginary parts p and q have the sae variance for a consteation points σ J = k Ik ) r k, + K I + ) r, I). )
M. Krondorf and G. Fettweis 7 According to )and6), we cacuate the paraeters b = b,r + jb,i and a for M-QAM effective data sybos X on subcarrier in frequency and tie seective fading channes: ) α r h λ)σ H ) b X = X α σ H + σ, ν a ) X = X α σ H r h λ) ) + σ ν σ H σ W 3) ) σ + σ, Ĥ Ĥ where σ H = I) σ H and σ = α Ĥ I) σ H + σ ν.fro 3) one can observe that the paraeter σ W h to be cacuated exacty to obtain reiabe resuts. The ter W rep- resents the effective noise of the received signa that consists of AWGN parts W, W, and ICI parts, respectivey. If we substitute 3)and) into ), we get W = W + K W + k V k, Ik ) k X +K XV, I + ). ) For an exact expression of σ W,wetake), σ V k, = σ H r k, ) together with the suptions of utuay uncorreated data sybos and obtain σ W = σ W + K ) + σ H Ik ) r k, ) k + K σ H I + ) r, ). ) As an exape, for one QPSK consteation point with index = on subcarrier, X, = / ), ) = / )+ j), we need to recacuate b X, ) and paraeter a X, ) separatey for each effective sybo reaization X, = X, + p + jq = + j)+p + jq 6) to use the cosed for integra and 3) for BER cacuation. Subsequenty, the bit-error rate on subcarrier for the th consteation point can be expressed using 3) by the foowing doube integra invoving the Gaussian PDFs of p and q: P b X, ) = P b X, + p + jq ) π σ J e p +q )/ σ J dpdq. 7) Finay, to obtain the genera bit-error rate, we have to average 7) overan C data subcarriers with index and M- QAM consteation points with index foows: P b = MN C N C/ = N C/ = M ) P b X,. 8) BER 3 Siuation Δ f = %, cacuation SNR db) 3 MRC N Rx = 3 Δ f = %, cacuation Δ f = 7%, cacuation Figure 3: The coparison of siuated and cacuated uncoded BER versus SNR for 6-QAM OFDM under residua CFO in nontie-seective channe environent and IRR = 3 db... Bit-Error rate perforance: nuerica resuts In this section, the derived anaytica expressions for bit-error rate are copared with appropriate siuation resuts for both singe-input singe-output) OFDM transission we SIMO singe-input utipe-output) OFDM using MRC and two receiver antenna branches. Furtherore, we consider an IEEE 8.a-ike OFDM syste [3] with 6- point FFT. The data is 6-QAM oduated to the data subcarriers, then transfored to the tie doain by IFFT operation and finay prepended by a 6-tap ong cycic prefix. The data is randoy generated and one OFDM piot sybo w used for channe estiation. The used BPSK piot data in the frequency doain is given by X P, = ) for subcarrier index = [ 6 : : 6],. 9) The data and piot sybos are oduated on data carriers. The DC carrier we the carriers at the spectra edges are not oduated and are often caed virtua carriers. For siuation and nuerica BER anaysis, we use an 8 taps exponentia PDP frequency seective Rayeigh fading channe with D = 7 see Section 3). Furtherore, we choose statistica independent channe reaizations for the two antenna branches in ce of SIMO OFDM transission. Thedoubeintegraof7) isevauatednuericayusing Matab buit-in integration functions having a nuerica toerance of 8 and upper/ower integration bounds of ±. Figure 3 iustrates the cacuated and siuated 6-QAM BER versus SNR σ X/σ W) with given carrier frequency offset Δ f in % subcarrier spacing) and IRR = 3 db under nontie variant obie channe conditions. Figure iustrates the cacuated and siuated 6-QAM BER versus SNR σ X/σ W) with given carrierfrequency offset
8 EURASIP Journa on Wireess Counications and Networking BER 3 6 IRR = 3 db, siuation IRR = db, siuation SNR db) MRC N Rx = 3 3 IRR = 3 db, cacuation IRR = db, cacuation Figure : The coparison of siuated and cacuated uncoded BER versus SNR for 6-QAM OFDM with residua CFO of 3% under non-tie seective channe conditions under IRR = 3 db/ db. BER 3 Siuation r h λ) =.99, cacuation SNR db) 3 MRC N Rx = 3 r h λ) =.99, cacuation r h λ) =.998, cacuation Figure : The coparison of siuated and cacuated uncoded BER versus SNR for 6-QAM OFDM with residua CFO of 3% and IRR = 3 db under tie seective channe conditions. Δ f in % subcarrier spacing) and IRR = 3 db under nontie variant obie channe conditions. In Figure, we use a fixed Δ f of 3% to investigate 6- QAM BER versus SNR for tie variant obie channe properties, characterized by the channe tap autocorreation coefficients r h λ). The resuts iustrate that our anaysis can approxiate the siuative perforance very accuratey if the channe power deay profie, the iage rejection ratio of the direct conversion receiver, and carrier frequency offset are known. 6. CAPACITY ANALYSIS OF IMPAIRED OFDM LINKS To perfor OFDM ink capacity anaysis, it sees andatory to review the ain principes and bic equations of how to cacuate average utua inforation between source and sink of a oduation channe. An exceent overview of this topic can be found in [7] that is suarized in the foowing. In an OFDM syste, we have a nuber of parae channes, that is, data subcarriers. Hence we propose to cacuate the utua inforation for each of the parae data carriers independenty and to finay average the ink capacity aong the data carriers. Let us consider rea input and output aphabets X and Z. Both aphabets can be characterized in ters of inforation content carried by the eeents of each aphabet what eads to the concept of inforation entropy HX) and HZ). The entropy of the discrete aphabet X having eeents X with appropriate probabiity PX )isgivenby HX) = P ) )) X og P X. ) Conversey, Z is sued to be a rea continuousy distributed RV having reaizations z. As a resut, Z can be characterized by its differentia entropy HZ) = f Z z)og fz z) ) dz, ) Z where f Z z) denotes the PDF of Z. Finay, the utua inforation IX; Z)ofX and Z can be foruated [7] IX; Z) = P ) ) X f Z X z X Z ) ) f Z X z X og ) ) dz. z Xn P Xn n f Z X ) Itcanbeseenfro) that IX; Z) requires knowedge of a- priory probabiities PX ) and conditiona PDFs f Z X z X ) ony. Mosty we have that PX ) = /M in ce of M-ary consteations. Since the above defined utua inforation cacuation schee sues one-diensiona output variabes and z is a two-diensiona copex RV of rea part z r and iaginary part z i,wehavetosoveadoubeintegratoobtain the corresponding utua inforation foows: IX; Z) = P ) ) X f Z X zr + jz i X Z r Z i ) ) f Z X zr + jz i X og ) ) dz r d Zi. zr + jz i X n P Xn n f Z X 6.. Mutua inforation under carrier crosstak 3) Recaing the two-diensiona conditiona PDF f Z X z r + jz i X ) on subcarrier given in Section, we have that ) a ) f Z X zr + jz i X = X π ) z r + jz i b X )), + a X )
M. Krondorf and G. Fettweis 9 where a X) andb X) contain the entire OFDM ink ipairent inforation channe estiation error, I/Q ibaance, CFO, outdated channe inforation, and channe power deay profie). According to Section, the copexvaued transit sybo is stochtic by nature due to CFO and I/Q ibaance carrier crosstak and can be expressed X + J = X + p + jq,where represents the consteation point index whie p and q represent the effects of I/Q ibaance and residua CFO. Both, p and q can be odeed i.i.d. zero-ean Gaussian RV done in Section. Additionay, both paraeters a X) andb X) are subcarrier-dependent. As a resut ) h to be reforuated for subcarrier f Z X,P,Q zr + jz i X + p + jq ) a X + p + jq ) = π z r + jz i b X + p + jq ) + a X + p + jq )). ) Hence, the cacuation of p/q-independent conditiona argina PDFs can be done via nuerica doube integration ) f Z X zr + jz i X = f Z X,P,Q zr + jz i X + p + jq ) f Q q) f P p)dpdq. 6) According to Section, we have the Gaussian distribution for each p and q: f P p) = f Q q) = πσ J e p,q) /σ J, 7) where σ J isgivenin). In ce of MRC utiantenna reception, we have to proceed in the sae anner. 6.. OFDM ink capacity: nuerica exapes The quantitative reationship between receiver ipairents, OFDM syste paraeters and ink capacity is an essentia piece of inforation for the diensioning of I/Q ibaance copensation agoriths we frequency synchronization ethods. Moreover, the effects of tie-seective obie channes on ink capacity can be used to design scattered piot structures for channe estiation and tracking done in []. Generay, ink capacity indicates the axiu data rate that can be achieved with strong channe coding under a given input consteation and a specified receiver architecture. The nuerica exapes of average utua inforation are chosen such that we iustrate the effects of channe estiation error, outdated channe state inforation CSI), residua CFO, and fat receiver I/Q ibaance on the ink capacity of and SIMO OFDM inks. Therefore, we choose the sae IEEE 8.a-ike OFDM syste paraeters introduced in Section., sue an 8 taps exponentia PDP obie channe and the use of 6-QAM oduation on each data carrier. Again, statistica independent channe reaizations for the N RX antenna branches in ce of SIMO OFDM Mutua inforation in bit/channe use 3. 3... 6-QAM upper bound SNR db), perfect CSI MRC N Rx =, perfect CSI 3, FDLS MRC N Rx =, FDLS Figure 6: The utua inforation, averaged over a data carriers, coparison between perfect channe-state inforation and rea FDLS channe estiation for and SIMO OFDM, CFO = %, no I/Q ibaance, static Rayeigh fading channe. transission are sued. The utua inforation eured in Bit/Channe Use) is averaged aong the data carriers and potted over SNR σ X/σ W). In Figure 6, we iustrate the effect of rea-ife frequency doain et-square FDLS) channe estiation on the ink capacity of and SIMO OFDM, respectivey, suing no I/Q ibaance, a perfect frequency synchronization CFO = %) and static non-tie-seective) channe properties. As reference, we potted the ce of perfect channe state inforation that can eiy be odeed by α = andσ ν =. In Figure 7, we show the aggregate effect of I/Q ibaance and FDLS channe estiation under static-channe conditions and perfect frequency synchronization. It is ey to see that I/Q ibaance h ony itte effect on the averaged utua inforation perforance, what is especiay the ce at reaistic iage rejection ratios above 3 db. Interestingy, a worst ce IRR of db heaviy ipacts the perforance but causes ony a sa perforance oss in ce of receiver diversity cobining. Figure 8 depicts the effect of CFO on averaged ink capacity under rea FDLS channe estiation and no I/Q ibaance under static-channe conditions. It can be shown that a oderate CFO of 3% causes ony a negigabe degradation of and SIMO OFDM ink capacity. The worst ce perforance in ce of CFO = % is potted to iustrate the ower sensitivity of the SIMO ink copared to the ink. Nevertheess, we have to state that in ce of reaistic frequency synchronization techniques, it is highy iprobabe to have a residua CFO arger than 3% at oderate SNR > db). This fact is aso entioned in [] where the authors derived the PDF of the residua CFO in ce of rea frequency synchronization under Rayeigh fading channes and given SNR.
EURASIP Journa on Wireess Counications and Networking Mutua inforation in bit/channe use 3. 3... 6-QAM upper bound MRC N Rx = No I/Q ibaance IRR = 3 db IRR = db SNR db) Figure 7: The utua inforation averaged over a data carriers under the aggregate effect of I/Q ibaance and FDLS channe estiation for OFDM, 6-QAM, CFO = %, static 8 taps exponentia PDP Rayeigh fading channe. 3 3 Mutua inforation in bit/channe use 3. 3... 6-QAM upper bound MRC N Rx = r h λ) = r h λ) =.99 SNR db) r h λ) =.99 r h λ) =.98 Figure 9: The utua inforation averaged over a data carriers under tie-seective channe properties and FDLS channe estiation for OFDM, 6-QAM, CFO = %, IRR = 3 db, tie variant 8 taps exponentia PDP Rayeigh fading channe. 3 Mutua inforation in bit/channe use 3. 3... 6-QAM upper bound MRC N Rx = Mutua inforation in bit/channe use 3. 3... 6-QAM upper bound MRC N Rx = SNR db) 3 3 SNR db) 3 No CFO CFO = 3% CFO = % Figure 8: The utua inforation averaged over a data carriers under CFO, FDLS channe estiation is sued, 6-QAM oduation on a subcarriers, no I/Q ibaance, tie variant 8 taps exponentia PDP Rayeigh fading channe. CFO = 3%, IRR = 3 db and rea FDLS CFO = 3%, IRR = 3 db and perfect CSI No ipaients, perfect CSI Figure : The utua inforation averaged over a data carriers, coparing the effect of receiver ipairents in ce of perfect CSI and rea FDLS channe estiation, 6-QAM oduation on a subcarriers, static 8 taps exponentia PDP Rayeigh fading channe. Figure 9 depicts the effect of outdated channe-state inforation quantified by appropriate channe autocorreation coefficients r h λ), FDLS channe estiation and I/Q ibaance under 8 taps exponentia PDP Rayeigh fading channe conditions and perfect frequency synchronization. Again, the perforance oss in ce of diversity cobining is saer than the oss that we have in ce of conventiona receiver designs. Moreover, we have to state that even in ce of very sa deviations of r h λ) fro the idea static ce r h λ) =, the effect of outdated channe-state inforation causes uch arger perforance osses than reaistic CFO and I/Q ibaance. Finay, we want to highight the fact that in ce of oderate receiver ipairents the perforance oss ainy coes due to channe-estiation errors. This iportant observation is iustrated in Figure where we potted
M. Krondorf and G. Fettweis averaged utua inforation versus SNR under CFO = 3% and IRR = 3 db suing static channe properties. As reference we use a pot without any I/Q ibaance, CFO, or channe estiation error. Interestingy the ipairent pots in ce of perfect CSI are aost equivaent to the reference curves but we observe a severe perforance degradation in ce of rea FDLS channe estiation. 7. CONCLUSIONS In this paper, we show how to anayticay evauate the uncoded bit-error rate we ink capacity of OFDM systes subject to carrier frequency offset, channe estiation error, outdated channe state inforation, and fat receiver I/Q ibaance in Rayeigh frequency and tie-seective obie fading channes. The probabiity density function of the frequency doain received signa subject to the entioned ipairents is derived. Furtherore, this PDF is verified by eans of bit-error rate cacuation. We show that our approach can be used to exacty evauate uncoded bit-error rates when a priori knowedge of the obie channe power deay profie, the iage rejection ratio and receiver CFO is used. Furtherore, we show how to use the derived PDF to cacuate OFDM ink capacity under the aggregate effects of receiver ipairents and obie channe characteristics. Finay, we highight the fact that channe uncertainty induced by channe estiation errors we outdated channe state inforation have uch severer ipact on OFDM capacity than CFO or I/Q ibaance. [9] S. K. Wison and J. M. Cioffi, Probabiity density functions for anayzing uti-apitude consteations in Rayeigh and Ricean channes, IEEE Transactions on Counications, vo. 7, no. 3, pp. 38 386, 999. [] M. Windisch and G. Fettweis, Standard-independent I/Q ibaance copensation in OFDM direct-conversion receivers, in Proceedings of the 9th Internationa OFDM-Workshop In- OWo ), pp. 7 6, Dresden, Gerany, Septeber. [] M. Windisch and G. Fettweis, Error probabiity anaysis of uti-carrier systes ipaired by receiver I/Q ibaance, in Proceedings of the Internationa Syposiu on Wireess Persona Mutiedia Counications WPMC 6), SanDiego, Caif, USA, Septeber 6. [] M. Windisch and G. Fettweis, Perforance degradation due to I/Q ibaance in uti-carrier direct conversion receivers: a theoretica anaysis, in Proceedings of the IEEE Internationa Conference on Counications ICC 6), vo., pp. 7 6, Istanbu, Turkey, June 6. REFERENCES [] H. Cheon and D. Hong, Effect of channe estiation error in OFDM-bed WLAN, IEEE Counications Letters, vo. 6, no., pp. 9 9,. [] I. Cosovic and G. Auer, Capacity achieving piot design for MIMO-OFDM over tie-varying frequency-seective channes, in Proceedings of the IEEE Internationa Conference on Counications ICC 7), pp. 779 78, Ggow, Scotand, UK, June 7. [3] IEEE. Part, Wireess LAN ediu access contro MAC) and physica ayer PHY) specifications, in IEEE Std 8.a- 999, 999. [] M. Krondorf and G. Fettweis, Bit error rate cacuation for OFDM with synchronization errors in tie and frequency seective fading channes, in Proceedings of 3th European Wireess Conference EW 7), Paris, France, Apri 7. [] K. Sathananthan and C. Teabura, Probabiity of error cacuation of OFDM systes with frequency offset, IEEE Transactions on Counications, vo. 9, no., pp. 88 888,. [6] T. Poet, M. van Bade, and M. Moenecaey, BER sensitivity of OFDM systes to carrier frequency offset and Wiener phe noise, IEEE Transactions on Counications, vo. 3, no., pp. 9 93, 99. [7] J. G. Proakis, Digita Counications, McGraw-Hi, New York, NY, USA, th edition,. [8] L. Rugini and P. Banei, BER of OFDM systes ipaired by carrier frequency offset in utipath fading channes, IEEE Transactions on Wireess Counications, vo.,no.,pp. 79 88,.