IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO ...IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61,...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 9, SEPTEMBER 2013 3781

Multi-User Pre-Processing in Multi-Antenna OFDMTDD Systems with Non-Reciprocal Transceivers

Mark Petermann, Markus Stefer, Frank Ludwig, Dirk Wübben, Senior Member, IEEE,Martin Schneider, Steffen Paul, Member, IEEE, and Karl-Dirk Kammeyer, Member, IEEE

Abstract—The combination of OFDM with joint pre-processing in adaptive multi-antenna systems offers both anease of equalization in frequency-selective channels and keepingthe signal processing at the mobile stations simple. In addition,the spatial dimension can be efficiently exploited to ensurehigh system throughput. With the utilization of higher-ordermodulation the performance of the system is highly sensitive tomultiple access interference and nonorthogonal subchannels dueto hardware impairments or insufficient adaptation to the currentchannel conditions. A further source of error in TDD systemsare the non-reciprocal transceivers inhibiting the baseband-to-baseband channel reciprocity required for accurate channel stateacquisition based on the uplink channel estimate. In this paper,measurement results of a low-cost hardware-based calibrationare presented and the drawbacks are discussed leading to theutilization of a recently introduced relative calibration. Thelatter is applied to an OFDM system and achieves or at leastapproximates the baseband-to-baseband reciprocity. Thus, itenables the link adaptation using the uplink channel state in-formation. Furthermore, preliminary hardware implementationsof the relative calibration running on a real-time system showaccurate results.

Index Terms—TDD, OFDM, reciprocity, impairments, calibra-tion.

I. INTRODUCTION

THE ability to adapt to the instantaneous downlink (DL)channel is crucial to achieve ever increasing data rates.The adaptation ability relies on the downlink channel avail-ability at the base station (BS). The DL channel state informa-tion (CSI) has to be fed back by the mobile subscriber (MS) tothe base station in case of a frequency-division-duplex (FDD)system. This results in a huge overhead in the uplink (UL).To reduce the latter significantly, it is reasonable to exploit thereciprocity theorem [1]. Consequently, this means exchangingthe FDD system for a time-division-duplex (TDD) system.The theorem holds as long as the coherence time of the

Manuscript received December 21, 2012; revised April 19, 2013. The editorcoordinating the review of this paper and approving it for publication was M.Juntti.

This work was supported in part by the German Research Founda-tion (DFG) under grant KA841-21/2, PA438-3/2 and SCHN1147-2/2.

M. Petermann, D. Wübben, and K.-D. Kammeyer are with the Departmentof Communications Engineering, University of Bremen, Germany (e-mail:{petermann, wuebben, kammeyer}@ant.uni-bremen.de).

M. Stefer and M. Schneider are with the RF & Microwave Engineer-ing Laboratory, University of Bremen, Germany (e-mail: {markus.stefer,martin.schneider}@hf.uni-bremen.de).

F. Ludwig and S. Paul are with the Department of Communication Electron-ics, University of Bremen, Germany (e-mail: {ludwig, steffen.paul}@me.uni-bremen.de).

Digital Object Identifier 10.1109/TCOMM.2013.072813.120984

physical channel is large compared to the time of the duplexphase. A decisive problem arises in that the reciprocity ofthe system is lost when considering baseband-to-basebandtransmission because of the non-symmetric characteristics ofthe analog transmit (Tx) and receive (Rx) frontends. Further-more, non-linearities within the devices inhibit a reciprocalbehavior. In [2], [3], this objective is described in detailas well as the deterioration of the system performance inabsence of the reciprocity due to hardware impairments. Inaddition, receiver-side algorithms are introduced in [2], [3]to compensate for the hardware effects. Alternative solutionsexist to compensate or even avoid the effects leading to non-reciprocal communication systems. Firstly, the transceiver isdimensioned such that the identical hardware is used for thetransmit and receive path [4]. Secondly, another method aimsat calibrating the transmitter and receiver arrays separatelyusing additional hardware. The latter helps to identify thedifferences of the individual frontend characteristics and thecompensation is realized with respect to a reference an-tenna [5]. If adaptation to the DL is pursued utilizing theuplink channel state information, it is judicious to additionallyexecute a (pre-)equalization with respect to space and adaptthe modulation scheme and the power allocation per subcarrierin an Orthogonal-Frequency-Division-Multiplexing (OFDM)system. The requirements regarding the quality of the availableCSI at the BS increase especially with respect to choosing themaximum feasible modulation scheme.

So far it can be concluded that the CSI available at the BShas to be of high quality to fulfill the aforementioned demands.Conversely, the utilization of the UL-CSI at the base station ina TDD system shows a poor quality as a consequence of theviolation of the reciprocity theorem considering baseband-to-baseband communication. The hardware-based concepts arecostly and do not allow an online calibration. The signal-processing-based concepts in [2], [3] use receiver-side com-pensation algorithms.

We pursue a different concept in that a low-complexityhardware-based concept and a signal-processing-based con-cept both allowing for online calibration are investigated.

The outline of the paper is as follows. First, the systemmodel used to reflect these effects within the communicationsystem is explained in Section II and is grounded on scatteringparameters describing the different transceiver blocks andthe antennas. Based on this system model, we verify thesuggested concept of [6] in Section III that provides theopportunity of a relative calibration of the base station. In [6],

0090-6778/13$31.00 c© 2013 IEEE

3782 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 9, SEPTEMBER 2013

a low-cost hardware-based concept is proposed including therespective calibration procedure. The main advantages overthe former hardware-based concepts are the low costs andthe ability to perform an online calibration. With respect toa full calibration - this would involve the base station andthe mobile subscribers - the drawbacks are also discussed.To be able to avoid the deployment of additional hardware ithas to be accepted that the reciprocity will not be fulfilled.As a consequence, the goal has to be to compensate forthe effects conditioned by the hardware by means of signalprocessing techniques [7]. In [8]–[11], a relative calibrationprocedure is presented that is feasible for MIMO-single-carriersystems (Multiple-Input Multiple-Output). The procedure isdiscussed in detail in Section IV-A, where it is transfered to ajoint multi-user (MU) MISO-OFDM system (Multiple-InputSingle-Output). Results based on simulations are presentedand discussed comparing systems with uncoupled antennas tosystems with mutually coupled antennas. These results showthe technique to be feasible in terms of practically relevantinfluences.

Since the concept is proven to be feasible, Section IV-Bdiscusses the possibility of implementing the relative cali-bration on a system running in real-time, e.g., FPGA (FieldProgrammable Gate Array). The main challenge here is toefficiently realize the singular value decomposition (SVD) onan FPGA. To assess the performance, comparisons are drawnto existent realizations regarding the needed execution time.Section V concludes the paper by evaluating the results.

Throughout this paper, ( · )T denotes the transpose, ( · )Hthe conjugate transpose, ( · )∗ the complex conjugate, tr { · }the trace of a matrix, ( · )−1 the inverse and the Moore-Penrose pseudoinverse is denoted by ( · )†. The vec-operatorvec{ · } is defined as the operator stacking the columns ofa matrix and ⊗ denotes the Kronecker product. The diag { · }operator maps a vector onto the diagonal elements of a matrix,while diag−1 { · } maps the diagonal elements of a matrix ontoa vector. Boldface capital letters denote matrices, boldfacelower-case letters denote vectors and lower-case letters denotescalars.

II. SYSTEM MODEL

In this section, an appropriate system model is introducedthat combines the system equation in communications with thescattering parameter description established in high-frequencytechnology. Section II-A presents the considered commu-nication scenario and Section II-B describes the scatteringparameter modeling of the hardware. Section II-C delvesinto the specific effect of mutual coupling between antennaelements and Section II-D concretizes the generally introducedhardware effects and states the chosen simulation parameters.Concluding remarks about multi-user induced interference inhigh signal-to-interference plus noise regions and the resultantcalibration idea are given in Section II-E.

A. Multi-User Joint Signal Pre-Processing in OFDM Systems

In the following, a downlink scenario consisting of a basestation equipped with NB antennas and NM decentralized andnon-cooperative single-antenna mobile stations is considered.

-1

-1

-1

-1Base StationMobile Subscriber(s)

IFFT

IFFT

Fk

FFT

FFT

GI

GI

GI

GI

n1,k

nNM,k

d1,k

dNM,k

˜d1,k

˜dNM,k

βkβk

βk

βk

Fig. 1. Block diagram of the considered multi-user MISO-OFDM systemwith joint pre-processing in the frequency-domain.

As the exploitation of TDD channel reciprocity is limited bythe physical channel reciprocity depending on channel coher-ence time, an office environment with non-moving receiversis assumed throughout the paper. In addition, in such an officescenario high signal-to-noise ratios can be expected and highdata rates are usually requested (cf. Section II-E). Fig. 1 showsthe block diagram of the multi-user MISO-OFDM systemconsisting of Nsc subcarriers with joint pre-processing in thefrequency-domain. The vector of transmit symbols per antennain time-domain is obtained by pre-processing the M -QAMsymbol vector dk = [d1,k, . . . , di,k, . . . , dNM,k]

T with vari-ance INM applying the linear pre-equalization matrix Fk persubcarrier k. To satisfy a total power constraint of NB perOFDM symbol at the BS, the transmit symbols per subcarrierk are multiplied with a common scalar βk. The applicationof the Inverse Fast Fourier Transform (IFFT) yields the time-domain signals, which are extended by adding a guard inter-val (GI). In case of a sufficiently long guard time the channelimpulse response is fully contained in the GI and inter-symbolinterference (ISI) is avoided. Furthermore, a cyclic convolutionof the OFDM symbol with the mobile radio channel isachieved. As a consequence of ISI avoidance, the systemequation can be simplified to a matrix-vector multiplicationper subcarrier k. At the mobile stations, complex Gaussiani.i.d. noise samples with variance σ2n are added to describethe noise influence. At the single-antenna mobile stations, theGI is removed, followed by the transformation into frequency-domain by applying the Fast Fourier Transform (FFT). Theprocessing is completed by dividing the received signal byβk with respect to every subcarrier k. The system equationcorresponding to Fig. 1 reads

d̃k = HkFkdk + β−1k nk , (1)

where the receive signal vector d̃k contains the data of themobile subscribers and the downlink channel matrix is denotedby Hk. The filter matrix Fk is given by F

(ZF)k = G

†k in case of

Zero-Forcing (ZF) or F(MMSE)k =GHk (GkG

Hk +NM σ

2n INB)

−1

in case of minimum mean square error (MMSE) linear pre-equalization. Here, the matrix Gk denotes the transposeduplink channel matrix. The scalar

βk =

√NB

tr{FHk Fk

} (2)is chosen such that the total sum power constraint per subcar-rier k is fulfilled.

The UL- and DL-CSI available at the BS has to be estimatedand is therefore inaccurate in general. The present errors due

PETERMANN et al.: MULTI-USER PRE-PROCESSING IN MULTI-ANTENNA OFDM TDD SYSTEMS WITH NON-RECIPROCAL TRANSCEIVERS 3783

to channel estimation can be factored in by an MMSE channelpredictor model like in [12] or, as it is rendered here, bymodeling the Least Squares (LS) estimate Ĝk of the actualchannel matrix Gk in terms of the MSE with the use of acorrelation factor �e. Instead of performing channel estimation,the estimated UL channel matrix Ĝk of one subcarrier k canthen be modeled by

Ĝk =�e Gk +√�e (1− �e)Ψk , (3)

where Ψk is a Gaussian error matrix with zero mean andentry variance of one and σ2e = 1 − �2e is a normalizedestimation error power restricted to the interval σ2e ∈ [0, 1].For simulation purposes, this error power is assumed equalon all subcarriers and ensures that the power of one estimateis equal to the power of the corresponding true channel. Incontrast, correlation between subcarriers, which usually occurswhen applying LS or MMSE estimation in combination withinterpolation in pilot-based schemes, is neglected. However,this simple model is sufficient to summarize estimation errorsfor further analyses.

B. Radio-frequency-based Baseband Model

To quantify the influence of non-ideal transceiver hardwareon the effective baseband channels, an appropriate scatteringmatrix model based on microwave network theory exists inthe literature [13], [14].

This model is illustrated here in detail to give a compre-hensive overview with the focus on how to incorporate therespective hardware effects appropriately.

Fig. 2 shows a block diagram illustrating the communi-cation model by means of scattering parameter blocks at afixed frequency k. The physical downlink channel is denotedby SMB,k ∈ CNM×NB and the physical uplink channelby SBM,k ∈ CNB×NM . The mutual coupling between thedifferent antenna elements is denoted by SBB,k ∈ CNB×NB re-garding the base station and denoted by SMM,k ∈ CNM×NMregarding the mobile stations. Although the mobile subscribersare assumed to be equipped with a single antenna, the systemmodel is derived with respect to the more general case ofmobile subscribers with multiple antennas. With respect to thedownlink in the analog baseband, the transmit signal of thebase station is denoted by ãB,k and the receive signals at themobile stations by b̃M,k. Correspondingly in the uplink, thetransmit signals of the mobile stations are described by ãM,kand the receive signals at the base station by b̃B,k. The goalnow is to establish the relationship b̃M,k = Stotal,DL,kãB,kin the downlink, where Stotal,DL,k expresses the transmis-sion characteristics between the base station and mobile sta-tions. First of all, the scattering parameter description of thetransceivers is introduced. The respective transmission char-acteristics are then translated into separate diagonal matrices.Only then it is possible to determine Stotal,DL,k. The transmitand receive antenna frontends are characterized by two-portdevices defined by the matrix

TB,i,k =[

0 0αT,B,i,k γT,B,i,k

](4)

denoting the transmitter i = 1, . . . , NB of the base station andthe matrix

RM,j,k =[0 αR,M,j,k0 γR,M,j,k

](5)

denoting the receiver of mobile subscriber j = 1, . . . , NM.The matrices TB,i,k and RM,j,k consist of complex con-version gain factors αT,B,i,k and αR,M,j,k, as well as inputreflection coefficients γT,B,i,k and γR,M,j,k. The complex gainfactors αT,B,i,k and αR,M,j,k and the reflection coefficientsγT,B,i,k and γR,M,j,k are now arranged in diagonal matrices.Regarding the transmitters at the base station, the diagonalmatrices are given by

ATB,k = diag{αTB,1,k, . . . , αTB,NB,k} (6a)ΓTB,k = diag{γTB,1,k, . . . , γTB,NB,k} , (6b)

and the receivers of the mobile stations are assembled into

ARM,k = diag{αRM,1,k, . . . , αRM,NM,k} (7a)ΓRM,k = diag{γRM,1,k, . . . , γRM,NM,k} . (7b)

Hence, exploiting (6) the analog baseband-to-baseband down-link channel is derived to obtain

Stotal,DL,k = ARM,kWTRM,kSMB,kWTB,kATB,k . (8)

The coupling matrices defined by

WTB,k = (INB− ΓTB,kSBB,k)−1 (9a)WRM,k = (INM− ΓRM,kSMM,k)−1 (9b)

incorporate the influence of the reflection coefficients ΓT,B,k,ΓR,M,k and the mutual coupling between antennas SBB,k,SMM,k on the overall downlink channel Stotal,DL,k. Trans-lating (8) into the discretized domain regarding the subcarrierk finally yields

Hk = ARM,kWTRM,kSMB,kWTB,kATB,k . (10)

Replacing the index R by T and the index T by R in (10),a similar relation is obtained with respect to the matrix Gkdenoting the transposed uplink channel matrix

Gk = ATM,kWTTM,kSMB,kWRB,kARB,k . (11)

Equation (11) already features the crucial modification exploit-ing the reciprocity theorem of the physical channel

SMB,k = STBM,k . (12)

In conclusion, the total downlink scattering matrix (8)is directly linked with the downlink system equation (1)through (10). Regarding a more detailed derivation of theequations (4)-(11), the authors refer to [13] and [14].

C. Mutual Coupling Between Antennas

Since the antenna coupling in combination with the reflec-tion coefficients of the transceivers influences the downlinkand uplink channel, it justifies to take the antenna couplinginto account. However, if antenna coupling is considered,heterogeneous interpretations exist [15]–[17]. In [16] and [17],the mutual coupling between isotropic radiators is introduced.


Reciprocal Physical MIMO Channel

Base Station Mobile Station(s)

SBB,k SMM,k

SBM,k

SMB,k

ãB,1,k ãM,1,k

ãB,NB,k

ãB,i,k ãM,j,k

ãM,NM,k

b̃B,1,k b̃M,1,k

b̃B,NB,k

b̃B,i,k b̃M,j,k

b̃M,NM,k

TB,1,k

TB,i,k

TB,NB,k

TM,1,k

TM,j,k

TM,NM,k

RB,1,k

RB,i,k

RB,NB,k

RM,1,k

RM,j,k

RM,NM,k

Fig. 2. Extended channel model using S-parameter description with BS and MS in downlink mode.

Iν

Vg,ν

Vind,νZ0

ZA

antenna

Fig. 3. Circuit theory motivated antenna model for one antenna element ν.

This might make sense in terms of a pure circuit theory pointof view but from an antenna engineering point of view thiswould be related to the array factor [15]. Therefore, in additionto stating the mutual coupling in terms of the scattering matrix,the underlying physical context is also presented here.

If at least two antennas are placed closely to each other withrespect to the wavelength and one of these antennas is drivenby, e.g., a voltage source, the second antenna is irrevocably inthe near-field of the driven and therefore transmitting antenna.Consequently, the transmitting antenna induces a voltage at theports of the non-transmitting antenna, which is called mutualcoupling. To obtain a measure for the mutual coupling, theinduced voltage can be related to the impressed current onthe transmitting antenna. The ratio between induced voltageand impressed current is called the mutual impedance. Here,the mutual impedance is examined to describe the mutualcoupling between two antennas with indices ν and ξ. Themutual impedance Zξν according to circuit theory is definedas [15]

Zξν =VξνIν

, (13)

where Vξν is the induced voltage at port ξ due to a current Iνimpressed at port ν [18]. Translating this to antenna theory, acircuit model for an antenna ν within an array of NB elementscan be established as is depicted in Fig. 3. The antenna ismodeled by a series connection of its input impedance ZAand an additional voltage source Vind,ν . According to Fig. 3,

the ideal generator voltage Vg,ν with generator impedance Z0can be expressed as

Vg,ν = (Z0 + ZA) · Iν + Vind,ν . (14)The voltage Vind,ν is the induced voltage on antenna ν stem-ming from the current excitations of the remaining antennaelements ξ = 1, . . . , NB ∀ ξ �= ν, which is expressed by

Vind,ν =

NB∑ξ=1,ξ �=ν

Zξν Iξ . (15)

Additionally, (15) makes use of the superposition princi-ple for assembling the total induced voltage [18]. Rewrit-ing the current and voltage relations indicated by (14)in matrix form, the impedance matrices ZBB,k ∈ CNB×NBand ZMM,k ∈ CNM×NM are obtained. With the calculatedimpedance matrices ZBB,k and ZMM,k, the according scat-tering matrices SBB,k and SMM,k are obtained by employ-ing [19]

SBB,k =(ZBB,kZ0

+ INB

)−1 (ZBB,kZ0

− INB)

(16)

exemplary for the scattering matrix of base station. Withrespect to the antenna coupling between the mobile stations thescattering matrix SMM,k is obtained by replacing the indexB in (16) with M. Here, Z0 = 50 Ω denotes the referenceimpedance of the antenna ports.

D. Modeling of the Individual System and Simulation Param-eters

This section states the actual modeling and assumptionsmade regarding the different hardware components apparentin the downlink and the uplink.

• Ideally, the baseband-to-baseband transfer function re-sembles an allpass function. In reality however, thisassumption cannot be preserved. Although the allpasscharacteristic of the baseband-to-baseband transfer func-tion is not present, it is physically reasonable to modelthe transfer function to possess an allpass-like transferfunction. The allpass-like behavior is implemented byadding a complex perturbation term to unity. This is


TABLE ISIMULATION PARAMETERS USED THROUGHOUT THE PAPER.

Parameter Symbol Valuenumber of subcarriers Nsc 256guard interval length Ng 6coded modulation scheme M 16-QAMnumber of channel taps L 6punctured 3GPP Turbo Code Rc 0.5transmission scenario NM ×NB 4× 4

explained and applied in more detail in the followingsub-item.

• The downlink involves the conversion gains of thetransmitters at the base station αTB,i,k and the conver-sion gains of the mobile stations αRM,i,k. Accordingto the preceding sub-item, the conversion gains at thebase station are modeled as αTB,i,k = 1 + δTB,i,k andαRB,i,k = 1 + δRB,i,k while setting the conversion gainsof the mobile stations equal to one. The additional errorterms δTB,i,k and δRB,i,k denote zero mean complexGaussian random variables on each antenna i and sub-carrier k. The variance of the error terms is assumed tobe subcarrier-independent and identical for each antennai, i.e., σ2TBδ,i,k and σ

2RBδ,i,k simplify to σ

2δ . It has to be

pointed out that the behavior of the transceivers regardingthis allpass-like transfer function behavior is feasiblewhen considering relative bandwidths of ≈ 1 %. Thelatter is maintained throughout this paper and is in accor-dance with the long-term-evolution (LTE) standardization[20].

• From an engineering point of view, it is reasonable toassume the mean value of the input reflection coefficientsγTB,i,k and γRB,i,k to be equal to a typical value, in thefollowing set to −20 dB ∧= 0.1 [21]. This results in theinput reflection coefficient model γTB,i,k = 0.1+ κTB,i,kand γRB,i,k = 0.1 + κRB,i,k, where κTB,i,k and κRB,i,kdenote zero mean complex Gaussian random variableson each antenna i and subcarrier k. The error varianceof the modeled input reflection coefficients is subcarrier-independent and identical for each transceiver i, i.e.,σ2TBκ,i,k and σ

2RBκ,i,k simplify to σ

2κ.

• With respect to the base station, the antenna elements areconsidered to be “infinitesimally thin” λ/2 dipoles [22]and the input as well as the mutual impedance can becomputed by using the results presented in [23]. Since thebandwidth of the baseband signal can be considered to berelatively small compared to the carrier frequency f , theimpedances are presumed to be frequency-independentand are therefore only evaluated at the carrier frequencyf . The impedance matrix ZBB,k as well as the accord-ing scattering matrix SBB,k simplify to ZBB and SBB ,respectively.

• From a physical point of view, it is legitimate to presumethat no coupling is present between the single-antennaMS as the spacing between the users is at least of severalwavelengths such that SMM reduces to a scaled identitymatrix SMM = ZAZA+2Z0 INM .

If not otherwise stated, the simulation parameters summa-rized in Tab. I are used throughout the remaining sections.

E. Transceiver Impact and Calibration Objectives

The utilization of the UL channel matrix for pre-equalization in (1) leads to interference terms caused bythe differences between (10) and (11). Thorough analyticaltreatments of the interference terms and the influence onthe receive signal-to-interference plus noise ratios (SINR) fordifferent types of pre-equalization are given in [14] and [24],[25]. The authors also provide some receive SINR analysesin [26]. All references commonly show that multi-user inducedinterference dominates the receive SINR deterioration in non-reciprocal systems especially if the users are in high transmitSNR regions, e.g., close to the base station. When beingclose to the base station high data rate processing is possible.In contrast, higher order modulations are more sensitive tointerference effects.

These SINR considerations emphasize the contemplatedindoor scenarios because of the influence of non-reciprocaltransceivers at high transmit SNR regions. In the LTE context,these situations may occur more often when thinking aboutthe femto-cell concept. To come full-circle, cheap low-powerfemto-cell MIMO base stations usually suffer even morefrom non-reciprocal transceiver characteristics substantiatingthe need for calibration. All calibration approaches pursue thesuppression of multi-user or multi-antenna interference as theprimal objective, which is the reason why similar methods donot apply in established single-antenna systems.

III. HARDWARE-BASED DOWNLINK CHANNELCALIBRATION

The motivation for the hardware-based relative calibrationis the compensation of the effects of different transmit andreceive gains at the base station. Mathematically, the compen-sation can be accomplished by an additional diagonal calibra-tion matrix Ck that is incorporated into the pre-equalizationfilter matrix, i.e., Fk = CkG

†k in case of Zero-Forcing [6],

[27]. Consequently, the compensation of the in general non-diagonal coupling matrices WTB,k and WRM,k cannot beaccomplished and the approach is actually only reasonablewhen being able to neglect the mutual coupling. This isjustified if the distance between antenna elements is aroundthe half of a free-space wavelength in terms of the carrierfrequency. Regarding this specific distance, the coupling canbe assumed to be small and hence allows the assumptionof WTB,k ≈ INB and W−1RM,k ≈ INM . In this case, thecalibration matrix has to be designed such that

WTB,k︸︷︷︸≈INB

·AT B,k ·Ck ·A−1R B,k · W−1RM,k︸︷︷︸≈INM

=αTB,1,kαRB,1,k

· I (17)

is fulfilled. The remaining factor on the right hand side of (17)specifies transceiver 1 of the base station as the referencetransceiver. Consequently, the described calibration approachillustrates a relative calibration. It has to be pointed outthat this approach relies on an additional equalization at themobile stations to at least compensate for αTB,1,k/αRB,1,k.The design regulation in (17) leads to the following values of


attenuator

calibrationsetup

ãB,1,k

ãB,2,k

b̃B,1,k

b̃B,2,k

TB,1,k

TB,2,k

RB,1,k

RB,2,k

S12,k S21,k

S11,k

S22,k

Fig. 4. Calibration setup attached to base station transceivers.

the calibration matrix Ck = diag {c1,k, . . . , cNB,k}c1,k = 1

ci,k =αTB,1,kαRB,1,k

αRB,i,kαTB,i,k

. (18)

The ci,k can be calculated using a calibration setup proposedby [6] resulting in an improvement in terms of a lower bit errorrate (BER) without providing measurement results. In [27],simulation results are reported and discussed that prove thisconcept to be feasible. The additional hardware needed to re-alize the calibration consists of two single-pole-double-throw(SPDT) switches and an attenuator in case of a base stationequipped with two transceivers. The MIMO demonstratorintroduced in [28], [29] implements a base station with twotransceivers and can be used to verify the concept with respectto measurement results. The attenuator is necessary to avoidoverdriving the receive chains of the transceivers [6], [27] andis realized by a commercially available digital-step-attenuatorto adjust for a good signal-to-noise ratio with respect to thecalibration procedure.

1) Calibration Procedure: Fig. 4 schematically shows thebase station with the attached calibration setup.

b̃B,2,k = S̃21,k αTB,1,k αRB,2,k ãB,1,k + n21,k (19a)

S̃21,k =S21,k

1+ γRB,2,k(S21,k S12,k−S22,k+ γTB,1,k S11,k S22,k)(19b)

b̃B,1,k = S̃12,k αTB,2,k αRB,1,k ãB,2,k + n12,k (19c)

S̃12,k =S12,k

1+ γRB,1,k(S21,k S12,k−S11,k+ γTB,2,k S11,k S22,k)(19d)

By transmitting a pilot signal from the transmitter oftransceiver 1 to the receiver of transceiver 2, (19a) contains thegain factors αTB,1,k and αRB,2,k, the overall forward transmis-sion coefficient S̃21,k of the calibration setup and an additionalnoise term n21,k regarding each subcarrier k. Changing thedirection of transmission, i.e., transmitting from transceiver 2to transceiver 1 yields (19c) containing the gain factors αTB,2,kand αRB,1,k, the overall reverse transmission coefficient S̃12,kof the calibration setup and again an additional noise termn12,k. The overall forward transmission coefficient S̃21,kin (19b) and the overall backward transmission coefficient

0 125 250

−3

−2

−1

0

1

k

μk

Re{b̃B,2,k

/ãB,1,k

}Re

{b̃B,1,k

/ãB,2,k

}Im

{b̃B,2,k

/ãB,1,k

}Im

{b̃B,1,k

/ãB,2,k

}

Fig. 5. Mean value μk of the base station’s measured transfer functions oftransmitter 1 to receiver 2 (̃bB,2,k

/ãB,1,k ) and transmitter 2 to receiver 1

(̃bB,1,k/ãB,2,k ).

S̃12,k in (19d) also contain the influences due to finite match-ing of the calibration setup, i.e., S11,k �= 0 and S22,k �= 0.Furthermore, the influence of imperfect transceiver matchingis present in (19b) and (19d). Both effects, imperfect matchingof the calibration setup as well as the imperfect matching ofthe transceivers deteriorate the calibration performance.

By taking several measurements according to (19a)and (19c), the noise can be averaged out to obtainthe calibration parameter c2,k

c2,k =b̃B,2,kãB,1,k

ãB,2,kb̃B,1,k

=αTB,1,kαRB,1,k

αRB,2,kαTB,2,k

S̃21,k

S̃12,k. (20)

The only difference between (20) and (18) is the ratioS̃21,k/S̃12,k of the forward and reverse transmission coeffi-cients of the calibration setup. The desired value of this ratiois one and its deviation influences the achievable calibrationaccuracy.

2) Measurement Results: A measurement setup was pro-posed in [28] to evaluate the influence of non-reciprocaltransceivers in a realistic environment. By attaching thehardware calibration setup to the MIMO demonstrator, therelative hardware-based calibration (HC) of the base stationtransceivers can be executed and evaluated. The calibrationsetup was verified using a network analyzer and the meanvalue of the ratio S̃21,k/S̃12,k was computed to be equalto S̃21,k/S̃12,k ≈ 1.0433 − j 0.0466 within the ISM-bandat 2.4 GHz. The value of this ratio results in an errormagnitude of approximately −24 dB. To determine thecalibration parameter c2,k, 1000 measurements were takenregarding (19) and the variance of c2,k versus subcarrier kstays below −22 dB. Fig. 5 shows the mean value of themeasured transfer functions with respect to each subcarrierk of the demonstrator transceivers. Fig. 6 depicts the mea-surement results determined exploiting the multiple antennademonstrator operating in a multiuser scenario with applied


0 3 6 9 12 1510−4

10−3

10−2

10−1

10 0

Rx SNR / dB

BE

R

UL-CSIUL-CSI / HCDL-CSI

Fig. 6. Measured bit error rates (BER) vs. received signal-to-noise ratio (RxSNR) of an adaptive multi-user MISO-OFDM system using the MIMOdemonstrator for a quasi-static 2 × 2 multi-user laboratory scenario anddifferent degrees of CSI with applied channel coding.

channel coding. At a BER of 10−3, the hardware-basedcalibration outperforms the non-calibrated base station (UL-CSI) by approximately 2.7 dB but it is inferior to the DL-CSI-based transmission by approximately 4 dB. This gap motivatesa further investigation in terms of a signal-processing-basedcalibration approach proposed in the following section. Onereason for the remaining gap is the temperature dependencyobserved with respect to the calibration parameters. Althoughthe calibration setup enables an online calibration, this couldnot be accomplished in our setup due to the inability ofsimultaneously running one transceiver in transmit mode andthe other transceiver in receive mode. Furthermore, it ispossible that the assumption of negligible antenna couplingdoes not hold. In contrast to the presented approach, theadditional hardware is avoided in [30] making use of a low-power signal exchange between the different antennas. Theadvantage of this method is that the load impedance seen bythe transceivers remains the same regarding calibration anddata transmission phase. This can in general not be ensuredwhen using additional hardware components. Although thismay lead to an improved performance in terms of the BER,both hardware-based calibration setups are not able to copewith the coupling effects which underlines the pursuit of asignal-processing-based calibration approach.

IV. SOFTWARE-BASED DOWNLINK CHANNELCALIBRATION

The system model is established in Section II-B and thecorresponding system equations with respect to the downlinkchannel matrix Hk as well as the transposed uplink chan-nel matrix Gk are derived. The hardware-based calibrationpresented in Section III is only capable of compensatingthe effect of different gains of the base station transceivers.To be able to compensate for the coupling between thetransceiver branches at the base station and to also includethe hardware imperfections of the mobile stations into the

Calibration Phase Transmission Phase (τT � τP)

DLDLDL ULULCalibration

τCτP

UL-CSI

DL-CSI

c

τT

Fig. 7. General workflow of relative calibration procedure.

calibration process, a relation between Hk and Gk needsto be established that can be solved with the help of signalprocessing techniques.

Once this relation is set up, both Hk and Gk have tobe made accessible to the base station. In other words, acalibration phase according to Fig. 7 needs to be insertedbetween regular transmission phases. During the calibrationphase, the instantaneous channel state information of thedownlink Hk has to be fed back by the mobile stations tothe base station. To build the aforementioned relation betweenHk and Gk, the expression for the transposed uplink matrixGk established in (11) is solved for SMB,k and the result isinserted into the expression for the downlink matrix Hk givenby (10) leading to the following equations

Hk = CM,k ·Gk ·CB,k (21)CM,k = ARM,kWRMW−TTMA

−1TM,k (22)

CB,k = A−1RB,kW−TRB WTBATB,k . (23)

The matrices CM,k and CB,k are in general full matricesbecause they include the mutual coupling effects due to closelyspaced antennas. Once these matrices are known it is feasibleto exploit the uplink channel state information initially basedon the physical reciprocity theorem to adapt to the downlinkchannel. The calibration vector c in Fig. 7 consists of thematrix entries of CM,k and CB,k and is applied to thetransmitted data of the base station to mitigate the hard-ware effects. The following section introduces the rearrangedrelation of (21) to make the so-called total least squares(TLS) principles applicable and discusses different methodsto solve this specific TLS problem. Section IV-B covers theimplementation aspects when running the calibration processon a fixed-point system, e.g., an FPGA.

A. Total Least Squares Principles for Frequency-Domain Cal-ibration

The goal is to obtain a set of linear equations that can beutilized to obtain a solution to the calibration matrices CM,kand CB,k. Here, the total least squares principle is exploited.Starting with the rearrangement of (21), the following equationis obtained

HkC−1B,k −CM,kGk = 0NM×NB . (24)

Applying the vec-operator to (24) and making use of theidentity vec{M ·X ·N} = (NT ⊗M)·vec{X} leads to(INB⊗Hk)vec

{C−1B,k

}−(GTk ⊗INM)vec{CM,k}=0NMNB×1

(25a)


and augmenting the matrix INB ⊗ Hk with the matrix−GTk ⊗ INM yields

[INB ⊗Hk , −GTk ⊗ INM

]︸︷︷︸�Ek

·[

vec{C−1B,k

}vec{CM,k}

]︸︷︷︸

�ck

= 0NMNB×1 ,

(25b)where the matrix Ek ∈ CNMNB×(N

2M+N

2B) and the vector

ck ∈ C(N2M+N

2B)×1 have been defined. Literally, a calibration

vector ck has to be determined for every single subcarrier.Since it is reasonable to assume the baseband-to-basebandbehavior of the transceivers by allpass-like transfer func-tions, the determination of one calibration vector c regard-ing the Nsc subcarriers is justified enduring only a smallerror (cf. Fig. 5). After acquiring the channel state informationof several subcarriers K , it is possible to include the additionalinformation by forming the matrix E =

[ET1 , . . . ,E

TK

]Twith E ∈ CKNMNB×(N2M+N2B). With the consideration ofK ≤ Nsc subcarriers and using (25b), the linear system ofequations can be written as

E · c = 0KNMNB×1 . (26)Obviously, matrix E depends on estimates of Gk andHk (cf. [8]). Here, K defines the number of subcarriers usedfor calibration, where the subcarriers can be arbitrarily chosendue to the frequency-flat transfer functions of the transceivers.To be able to obtain a solution for (26), the number ofsubcarriers K has to be larger than K > NM/NB+NB/NM.

Due to the inherent estimation errors in the observationmatrix E, the solution for the overdetermined set of equa-tions (26) can be obtained by solving a total least squaresoptimization problem [8] given by (27).

minimizeΔE

‖ΔE‖F (27a)such that (E+ΔE) c ≈ 0KNBNM×1 . (27b)

The goal is to find a perturbation matrix ΔE depending onE with minimum Frobenius norm that lowers the rank ofE + ΔE such that a solution to (27b) can be computed.The matrix ΔE is called TLS correction of the optimizationproblem. Consequently, the solution to (27), the calibrationvector c, lies in the right null space of E and can becomputed with the singular value decomposition of E asshown in [31], [32]. Furthermore, it was proven in [31], [32]that the SVD gives the maximum-likelihood (ML) solutionto problem (27). Then, if E = UΣVH depicts the SVD andmatrix V =

[v1, . . . ,vN2B+N2M

]denotes the right singular

vector space, the estimated solution depends on the rightmostsingular vector corresponding to the smallest singular value inΣ such that

c = vN2B+N2M . (28)

Thus, the vector c in (26) can be fully determined (up toa scalar coefficient, which vanishes due to the reciprocalmultiplication in (21)) [8], [33]. With (28) the UL matricesGk can be adjusted according to (21) and are used afterwardsfor the calculation of the pre-equalization filters.

0 10 20 30 4010−5

10−4

10−3

10−2

10−1

10 0

Eb/N0 / dB

BE

R

MMSE, σ2e =0, reciprocal

robust MMSE

MMSE calibrated (K=1)



σ2e =10−4, reciprocal

σ2e =10−4,σ2δ=−30dB

σ2e =10−4,σ2δ=−20dB

Fig. 8. BER versus Eb/N0 for an uncoupled system with MMSE pre-equalization and different reciprocity conditions (with (gray-colored) andwithout (black-colored) applied channel coding).

1) Simulation Results for Uncoupled Systems: In case of asystem with negligible mutual coupling, the coupling matricesWTB,k and WRM,k in (9) are strictly weighted diagonalmatrices, resulting in diagonal matrices CB and CM in (21).As a consequence, the optimization problem simplifies asthe number of unknowns in (26) reduces to NM +NB. Thisconsequently changes the structure and reduces the size of thematrix E and the calibration vector c. Multiple measurementsare not necessary to obtain a solution if NM > 1 and NB > 1.Nevertheless, still further overdetermining this set of equationsby incorporating more measurements (K > 1) achieves moreaccurate results [34].

Fig. 8 presents BER results versus Eb/N0 for linearMMSE pre-equalization. The channel estimation error is set toσ2e = 10

−4. For completeness, it has to be mentioned that theSNR loss due to the GI is also considered in the results. Toensure a minor complexity for the applied calibration at theBS, only up to twelve subcarriers (K ∈ {1, 5, 12}) are usedin the calibration process. It can be seen that with occurringreciprocity mismatches the calibrated ordinary MMSE solu-tion clearly outperforms a robust pre-equalizer (see [35], [36])in terms of uncoded transmission (solid lines). The increasingerror rates at high signal-to-noise ratios originate from a filtermismatch term due to the imperfect CSI with constant σ2e .Furthermore, this interference term is inversely proportionalto the noise σ2n [26]. The error rates can be slightly improvedby increasing the number of subcarriers K used for thecalibration. The coded results instead (dashed lines) show thatwith either using a robust approach or applying calibrationexcellent results can be obtained as long as the reciprocitymismatch remains smaller or equal to σ2δ = −30 dB. Inthis case, almost the performance of a MMSE pre-equalizerwith perfect reciprocity and without estimation errors can beachieved. If the reciprocity mismatch is increased to −20 dB,the robust MMSE pre-equalizer shows severe degradationsand only calibration can deal with such a high reciprocitymismatch. This result underlines the need for calibration


0 10 20 30 4010−4

10−3

10−2

10−1

10 0

Eb/N0 / dB

BE

Ra) σ2δ = −30 dB

recipro.,σ2e =0

no calib.

SVD

CG �=50

K=32

K=64

K=128

0 10 20 30 4010−4

10−3

10−2

10−1

10 0

Eb/N0 / dB

b) σ2δ = −20 dB

Fig. 9. BER versus Eb/N0 with BS antenna coupling and variance ofthe transceiver reflection coefficient of σ2κ = −30 dB and without appliedchannel coding. The channel estimation error variance is σ2e = 10

−4.

procedures.2) Simulation Results for Systems with BS Coupling:

Increasing the number of subcarriers K or the number ofantennas (or both) makes the exploitation of the SVD to solvethe TLS problem (27) computationally prohibitive. In [21],a low-complexity implementation for coupled systems due tothe necessary large-scale problem for an increased numberof parameters was presented to approximate the SVD. It wasshown that the constrained minimization problem in (27) isequivalent to minimizing the so-called Rayleigh quotient

f(c) =cHEHEc

cHc. (29)

The minimization of the Rayleigh quotient in turn is equivalentin finding the eigenvector c associated with the smallest eigen-value of matrix EHE such that min

{‖ΔE‖2F

}= min{f(c)}

equals the minimum singular value and the TLS solution isobtained. An advantage of the Rayleigh quotient is the factthat (29) can be minimized iteratively. One possibility is touse an inverse power method to find the corresponding eigen-vector [31]. To avoid the matrix inverse in the inverse powermethod, this procedure can be efficiently solved via a con-jugate gradient (CG) method [21]. Accordingly, Fig. 9 com-pares the BER results for uncoded systems applying MMSEpre-equalization for different numbers of subcarriers K ∈{32, 64, 128} exploited during the calibration process and dif-ferent reciprocity mismatches σ2δ ∈ {−30 dB,−20 dB}. Thevariance of the reflection coefficients at the BS is fixed toσ2κ = −30 dB for these simulations. The calibration withthe considered mutually coupled BS array is carried out withthe CG method applying = 50 iterations. If the stoppingcriterion given by a marginally small update of the Rayleighquotient is reached, the algorithm stops earlier. The resultsin Fig. 9 a) and b) show a significantly decreased errorperformance if no calibration is applied. The calibration resultsfor K = 32 worsen the performance in terms of the BERcompared to the uncalibrated case. Additionally, the resultsin case of applied calibration are almost independent of the

0 50 100 150 200 25010−4

10−3

10−2

10−1

10 0

measurement

MS

E

UL-CSI

ideal FB

4 Bit

5 Bit

Fig. 10. MSE measurements with 2×2 MIMO demonstrator system in Line-of-Sight scenario assuming uncoupled antenna elements in the calibrationalgorithm [28].

reciprocity mismatch. It can be concluded that only with alarge number of calibration carriers, sufficient linear equationsare available to ensure a good estimate of the calibration vectorc. While the SVD solution achieves a good performance forK = 64, the CG method needs more carriers to significantlyimprove the average BER at the mobile stations. In case ofK = 128 the CG method has the same performance gaincompared to the direct SVD but with considerably reducedcomplexity. This substantiates the fact that the CG method isespecially applicable for large-scale matrix problems.

Overall, the presented frequency-domain-based calibrationis able to match the downlink channel with the uplink suffi-ciently well so that the adaptation to the downlink becomesfeasible when exploiting the reciprocity theorem. In contrast,a time-domain-based calibration presented in [37] is notcapable of achieving similarly good results. The former canbe represented by a structured total least squares problem thatis not well suited for OFDM systems due to its complexityand numerical instabilities in certain scenarios.

On the other hand, the high amount of DL channel feed-back in UL direction in the calibration phase still rendersthe application in high-rate adaptive communication systemsdifficult. Nevertheless, in [38], a recursive implementation ofthe TLS principles containing an additional QR decomposition(QRD) was presented in combination with directly quantizedfeedback that is distributed over the OFDM time-frequencygrid. Interpolation between calibration carriers results in thereduction of feedback necessary in frequency direction [39]while efficiently updating the R matrix of the QRD can helptracking the varying parameters in time [38].

Regarding an evaluation of the calibration scheme in a prac-tical system, the MIMO demonstrator is used and uncoupledantenna elements are assumed throughout the measurements.This assumption leads to the reduced TLS problem withNM + NB unknown coefficients. Hence, Fig. 10 shows themean square error (MSE) results between the true measuredDL channel, the measured UL and the calibrated UL channel,


respectively. In case of the uncalibrated MIMO demonstrator,the MSE remains at around 10% in each duplex phase,whereas the possible MSE reduction is from 10% to 0.2%on average for ideal (analog) feedback. With quantized feed-back, assuming five bits per real coefficient per DL channel,the MSE performance shows comparably good results. Theimplementation of only four bit feedback per real coefficientleads to MSE results around one order of magnitude largerthan in case of perfect feedback. Consequently, the amount offeedback must be large enough. To reduce the feedback, thedata to be fed back can be split up into multiple packets or(in terms of multicarrier systems) into several OFDM symbolsand distributed over multiple subcarriers.

B. Hardware Implementation Aspects

The investigation of the presented algorithms based onTLS principles indicated that the accuracy of the calibra-tion is of great importance at the expense of meeting anylatency constraints. Therefore, this section introduces efficientaccurate low-complexity calculations suitable for real-timeimplementations. Finally, a calibration module is implementedon an FPGA and first measurement results show excellentperformance in terms of accuracy and FPGA area consump-tion. As FPGAs already execute many digital signal processingfunctions/algorithms in the BS, the calibration can potentiallybe embedded in one of the existing FPGAs. An advantage ofFPGAs towards very-large-scale integration circuits (VLSI) isthe possibility of an easy reconfiguration and lower costs withrespect to small-scale production. In addition, FPGAs havebetter timing behavior compared to Digital Signal Processors(DSPs).

1) Low-Complexity TLS Calibration: The implementationof the low-complexity TLS calibration on an FPGA isachieved regarding an uncoupled system, i.e., CB and CM arediagonal matrices. According to Section IV-A the optimizationproblem of (27) can be solved by the application of the SVD.In order to achieve a reduction in complexity an efficientapproximation to the SVD is obtained. The latter can beaccomplished by several existing algorithms [31], [32], inparticular by the rank-revealing ULV and URV decomposi-tion [40], where U and V are acronyms for a unitary, R fora right and L for a left matrix. Here, the ULV decomposition(ULVD) represents the matrix E by

E = U′LV′H (30)

where the unitary matrices U′ and V′ are approximations ofU and V of a SVD on E. L is a lower triangular matrix,which has the same singular values as E. This ULVD will beachieved by two QR decompositions. The first QRD obtainsQR = E, then the second QRD is applied on the conjugatetranspose of R such that

E = Q

(R00

)= QR̃HQ̃H (31a)

with RH0 = Q̃R̃ (31b)

where Q � U′, Q̃ � V′ and R̃H � L. Finally, the rightmostcolumn of Q̃ contains an approximation of the TLS solutionvector c as in (28). The computational complexity is reduced

by the fact that the second QRD in (31b) is only performedon RH0 ∈ CNB+NM×NB+NM , so that the dimension of R0depends solely on the number of BS antennas NB and thenumber of users NM. Furthermore, the computational com-plexity is significantly reduced by omitting the calculation ofthe orthogonal matrix Q in the first QRD.

In order to compute the QRDs in (31a) some considerationsfor a suitable hardware implementation have to be made.Especially three methods have been established to computethe QRD: the Householder transformations, the Gram-Schmidtprocess and Givens rotations (sometimes also called Jacobirotations). Concerning an FPGA implementation, the House-holder transformation has the disadvantage that it cannotbe parallelized. The Gram-Schmidt algorithm is numericallyunstable. Furthermore, the Gram-Schmidt process requiressquare-root operations sharing this necessity with the nu-merically more stable modified Gram-Schmidt [31]. On theother hand, the Givens rotation allows a parallel computationalstructure and with the well-known CORDIC (COordinateRotation DIgital Computer) algorithm [41] costly divisionsor square root computations can be avoided in the Givensrotation. Another advantage of the CORDIC based Givensrotation used in this work is that the QRD can be adaptivelyimplemented in hardware. If this is considered in the design,the QRD can be used for several numbers of calibrationcarriers K . In this way, an adaptive calibration is possible.Because the number of columns of E is independent of K ,the Givens rotations can be parallelized for a fixed numberof columns. If K is incremented, only the computation timeis rising slightly because further rotations are executed on theadditional rows of matrix E in the first QRD. The computationtime of the second QRD is not affected by changing K .

2) FPGA Implementation: As mentioned before, theCORDIC algorithm is a well-known iterative method to cal-culate trigonometric and algebraic functions like sine, cosine,square root or division [41], [42]. The principle of CORDICis based on m serial micro-rotations. As CORDIC onlyuses bitshifting, addition and subtraction operations, it isparticularly suitable for hardware implementation of complexalgorithms. The disadvantage of an increasing latency foriterative algorithms is reduced by a pipelined implementation.As a consequence of the iterative topology of the CORDICalgorithm, the numerical accuracy depends on the numberof micro-rotations. Furthermore, the accuracy is reduced bythe conversion from floating-point to fixed-point, which isnecessary for efficient hardware realization. A quantizationscheme that ensures a solution very close to that obtained byfloating-point operations is acquired by simulations. Regardingfloating-point results, an optimal quantization is achieved byusing 16 bit word-length with 12 fractional bits. Fig. 11 showsthe dependency of the BER on the number of micro-rotationsm with linear MMSE pre-equalization. The channel estimationerror variance is again set to σ2e = 10

−4 and the reciprocityerror variance is σ2δ = −20 dB. The calibration is simulatedwith an equivalent fixed-point FPGA-model. For more thanm= 11 micro-rotations, no improvement can be achieved inthe uncoded case, while for the coded transmission m = 8micro-rotations are sufficient to obtain the best performance.

An interesting effect can be observed by the error prop-


3 4 5 6 7 8 10 1210−4

10−3

10−2

10−1

10 0

m

BE

Ra) Uncoded

K = 1

K = 3

K = 6

16 Bit

14 Bit

3 4 5 6 7 810−4

10−3

10−2

10−1

10 0

m

BE

R

b) Coded

Fig. 11. BER at Eb/N0 = 40 dB for an uncoupled MU-MISO-OFDMsystem versus different numbers of micro-rotations m in the CORDIC;channel estimation error variance set to σ2e = 10

−4, σ2δ = −20 dB,a) without, b) with applied channel coding.

agation in the ULVD in case of a smaller number of bitsused for quantization and larger K . As Fig. 11 shows the bestperformance for 14 bit quantization is achieved by K = 1, anincrease of K does not induce a better performance such asfor 16 bit quantization. Instead, the performance is gettingworse. The increasing BER for the 14 bit quantization atm > 10 in Fig. 11 a) originates from the circumstance thatfor this scenario 10 fractional bits are used and for m > 10the fixed-point data will be shifted for m > 10 positions tothe right within the CORDIC. In this case the rounding erroris increasing because m is greater than the fraction length ofthe fixed-point data. This behavior has to be considered if thecalibration is realized for coupled antennas, where K 1calibration carriers are needed. In this case, a longer wordlength or a more precise rounding is required.

As shown, the number of micro-rotations has a significantimpact on the accuracy of the calibrated UL channel. Fora sufficient calibration performance m = 8 micro-rotationsare required in the coded case. In order to enhance thethroughput the CORDIC and the QRD can be parallelizedand pipelined. To get a trade-off of precision, computing timeand hardware usage a 2nd-order unrolled CORDIC-unit isapplied [43]. The order of the CORDIC architecture denotesthe number of micro-rotations in one clock cycle. Concerningthe 2nd-order architecture, four 2nd-order CORDIC unitsperform one Givens-rotation in four clock cycles with m = 8micro-rotations. If a higher order of the CORDIC is used,the combinatorial path is getting longer and the maximumfrequency fclk of the circuit is reduced.

3) Implementation Results: The ULVD has been imple-mented in a calibration module for a 2×2 MU-MISO-Systemon a Xilinx Virtex 5 FPGA (XC5VSX50T) [44]. To benefitfrom the fact that more channel measurements improve thecalibration performance, we decided to design an adaptivecalibration where the number of calibration carriers can bechanged from K = 1 to K = 6 in single steps. Therefore,also the latency caused by the calibration can be adjusted from

TABLE IISYNTHESES RESULTS FOR THE XILINX VIRTEX 5 FPGA (XC5VSX50T).

adaptive fix K = 1

total % total %

Slices 24 739 97 16 842 66LUT 40 417 79 31 965 63FF 17 076 33 7 785 15Multiplier 34 26 34 26

maximum fclk 108.5 MHz 129.2 MHztTLS/K 2.61 µs 2.09 µs

15.66 µs to 2.61 µs. In addition, this short calculation timeallows quick updates in a recursive implementation of the TLSwith iterative feedback using a QR-based algorithm describedin [38]. This reduces the amount of reserved bits for calibrationin the UL transmission. The adaptive ULVD implementationcan decompose matrices up to a size of C24×4. In order tocompare the device utilization and the calculation time, asecond design is synthesized, where K is fixed to one, whichimplies that only the matrix E ∈ C4×4 can be decomposed.According to Fig. 11, a quantization of 16 bit word lengthwith 12 fractional bits is chosen. Table II shows the usageof slices, lookup-tables (LUT), flip-flops (FF) and multipliers.Furthermore, the percental utilization of the available hardwareof the FPGA after syntheses is given. Additionally, the max-imum clock frequency fclk and the computation time of theTLS solution per calibration carrier tTLS/K at maximum fclkare listed. In consequence of the increased control complexityfor the adaptive module the calibration time per subcarrierraises. In order to evaluate the computational reduction byusing the ULVD instead of a SVD the calculation time iscompared to the work given in [45]. There, a complex-valued4 × 4 SVD unit for 180 nm CMOS technology is given andwith 8 CORDIC micro-rotations the SVD takes 10.75 µs at aclock frequency of 272 MHz. In this work, the TLS solutionis calculated out of a 4× 4 calibration-matrix in 2.09 µs at aclock frequency of 129 MHz. However, both implementationsuse completely different technologies so that a fair comparisonis not possible, especially for the power consumption and areaor device utilization, respectively. But it shows that the ULVDis a promising approach to obtain the TLS solution with lesseffort compared to the SVD.

V. CONCLUSION AND FUTURE WORK

The application of calibration methods has been studiedfor multi-user MISO-OFDM systems utilizing time-divisionduplex. The latter additionally exploit joint pre-processing atthe base station. The successfully applied calibration methodsenable the utilization of the uplink channel estimate due tothe reciprocity theorem. With low-cost transceiver solutionsthe direct utilization of the baseband uplink channel esti-mate for downlink adaptation is impossible due to the non-reciprocal properties of the transceivers. Additional hardware-based circuitries ensuring an online calibration capability showimprovements in terms of BER performance in measurementswhile significant receive SNR losses still remain due to theinability of accounting for antenna coupling. The signal-processing-based relative calibration is able to compensate for


these effects almost perfectly when using the channel estimateson every other subcarrier during the calibration process. Unfor-tunately, the approach of a software-based relative calibrationrequires a special calibration phase degrading the total systemthroughput. Exploiting the suggested recursive implementa-tion of the calibration algorithm mitigates this degradationefficiently. To bridge the gap between signal processing andhardware implementation, the effect of a quantized feedbackwas also investigated based on a MIMO demonstrator. Inthis case, 5 bits per real-valued channel coefficient lead toalmost identical results when comparing to the ideally fed backchannel state information. Furthermore, initial aspects of thecomputation of the needed total least squares solution on anFPGA is studied. The theoretical background provides insightin low-complexity solutions based on the CORDIC algorithm,which can be applied to solve the required matrix decom-positions. In the future, the considerable improvements usingsignal processing calibration will be further investigated onoff-the-shelf hardware MIMO solutions [28], [46], especiallythe implementation of the recursive algorithm on an FPGAwill be pursued.

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Mark Petermann received the Dipl.-Ing. degree inelectrical engineering from the University of Bre-men, Germany, in 2005. From 2005 to 2012 he waswith the Department of Communications Engineer-ing at the University of Bremen, where he receivedthe Dr.-Ing. degree in 2012. Currently, he is withthe ATLAS ELEKTRONIK GmbH. His main fieldsof interest are multi-user MIMO communications,image and correlation processing.

Markus Stefer received the Dipl.-Ing. degreein electrical engineering from the University ofAachen, Germany, in 2007. Currently, he is withthe RF & Microwave Engineering Laboratory at theUniversity of Bremen, Germany, where he is alsoworking towards his Ph.D. degree. His main fieldsof interest are the modeling of RF components forcommunication systems, signal processing in multi-ple antenna systems and localization and tracking.

Frank Ludwig received the Dipl.-Ing. degree inelectrical engineering from the University of Bre-men, Germany, in 2009. Currently, he is withthe Institute of Electrodynamics and Microelectron-ics, University of Bremen, Germany, where he iscurrently working towards his Ph.D. degree. Hisfocus is on the implementation of digital signalprocessing algorithms for wireless communicationson FPGAs and prototype development for wirelessmultiantenna LTE systems.

Dirk Wübben (S’02-M’06-SM’12) received theDipl.-Ing. (FH) degree in electrical engineering fromthe University of Applied Science Münster, Ger-many, in 1998, and the Dipl.-Ing. (Uni) degree andthe Dr.-Ing. degree in electrical engineering from theUniversity of Bremen, Germany, in 2000 and 2005,respectively. He was a visiting student at the DaimlerBenz Research Departments in Palo Alto, California,in 1997, and in Stuttgart, Germany, in 1998. From1998 to 1999 he was with the Research and Devel-opment Center of Nokia Networks, DÃ 1

4sseldorf,

Germany. In 2001, he joined the Department of Communications Engineering,University of Bremen, Germany, where he is currently a senior researcher andlecturer.

Martin Schneider received his Diploma and Doc-torate degree in Electrical Engineering and Infor-mation Technology from the University of Han-nover, Germany, in 1992 and 1997, respectively.From 1997 to 1999, he was with Bosch TelecomGmbH in Backnang, Germany, where he developedmicrowave components for point-to-point and point-to-multipoint radio link systems. In November 1999,he joined the Corporate Research and AdvancedDevelopment division of Robert Bosch GmbH inHildesheim, Germany. As a project and section man-

ager of the “Wireless Systems” group he focused on research and developmentof phased array and smart antenna concepts for automotive radar sensors at24 GHz and 77 GHz. From 2005 to 2006, he was with the business unitAutomotive Electronics of Robert Bosch GmbH in Leonberg, Germany. As asection manager he was responsible for the “RF electronics” for automotiveradar sensors. Since March 2006, he has been a full professor and head ofthe RF & Microwave Engineering Laboratory at the University of Bremen,Germany.

Steffen Paul (S’86-M’93) studied electrical engi-neering at Technical University Dresden and Tech-nical University Munich, Germany, from 1984-1989,where he received his Diploma (Dipl.-Ing.) degreein 1989. From 1989 to 1997 he was working at theInstitute of Network Theory and Circuit Design ofthe same institution. In 1993 he received the Dr.-Ing. degree. He held a postdoctoral position at theEECS department of the University of California,Berkeley in 1994 to 1995. From 1997 to 2007 he wasworking at Infineon Technologies (former Siemens

Semiconductor group) in the areas of memory design, xDSL and UMTSconcept engineering. He joined the University Bremen in 2007 as a fullprofessor for electromagnetic theory and microelectronic systems.

Karl-Dirk Kammeyer (M’95) received the Dipl.-Ing. in electrical engineering from Berlin Universityof Technology, Berlin, Germany, in 1972, and thePh.D. degree (Dr.-Ing.) from Erlangen University,Erlangen, Germany, in 1977. From 1972 to 1979,he worked in the field of data transmission, digitalsignal processing, and digital filters at the Univer-sities of Berlin, Saarbrucken, and Erlangen, all inGermany. From 1979 to 1984, he was with Pader-born University, Paderborn, Germany, where he wasengaged in the development of digital broadcasting

systems. During the following decade, he was a Professor for digital signalprocessing in communications at Hamburg University of Technology, Ham-burg, Germany. In 1995, he was appointed as Professor for telecommunica-tions at the University of Bremen, Bremen, Germany.

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