in-phase and quadrature timing mismatch estimation and

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2017.2696947, IEEETransactions on Wireless Communications

1

In-phase and Quadrature Timing MismatchEstimation and Compensation in Millimeter-wave

Communication SystemsHlaing Minn, Fellow, IEEE, Qi Zhan,Student Member, IEEE,

Naofal Al-Dhahir,Fellow, IEEE, Huang Huang,Member, IEEE

Abstract—The emerging millimeter-wave MIMO systems aresubject to strong radio frequency (RF) distortions and theircompensation is crucial to realize such systems. In the existingliterature, several estimation and compensation schemes havebeen proposed for RF distortions such as carrier frequencyoffset, phase noise and in-phase and quadrature amplitude andphase imbalance (IQI). However, in-phase and quadrature timingmismatch (IQTM) is largely ignored. This paper investigatesthe effect of IQTM on millimeter-wave system performanceand reveals that IQTM causes a specific image rejection ratiocharacteristic which is substantially different from the regularfrequency-dependent IQI and it substantially prolongs the ef-fective channel length. If not compensated, IQTM can degradesystem performance significantly. As a solution to this IQTMproblem, this paper proposes novel pilot designs for transmitand receive IQTM estimation, and develops corresponding esti-mators, transmission protocol and compensation schemes. MIMOaveraging is also proposed which substantially enhances theIQTM estimation performance. Simulation results show that ourproposed pilot designs and estimators offer an efficient solutionto the IQTM problem.

Index Terms—Estimation, IQ timing mismatch, Millimeter-wave, Pilot designs, RF distortions

I. I NTRODUCTION

Millimeter-wave (mm-wave) wireless communication sys-tems are gaining increased interest [1]–[3] but they are subjectto strong RF distortions. Due to CMOS process variationsand the very high sampling frequency used in the mm-waveregime, the difference in signal propagation and processingtimes of the circuits between the in-phase and quadratureparts of the signal could be significant relative to the samplinginterval. This leads to a timing (delay) mismatch between thein-phase and quadrature paths, which we term in-phase andquadrature timing mismatch (IQTM), at both the transmitter(TX) and receiver (RX). IQTM is negligible relative to thesampling interval for systems with carrier frequency below 6GHz but it becomes significant and can substantially affectthe system performance for the emerging mm-wave systems.Not only CMOS process variations, but also device aging andharsh operating conditions could introduce/change IQTMs.

Q. Zhan, H. Minn and N. Al-Dhahir are with the Department of Elec-trical & Computer Engineering, University of Texas at Dallas, Richardson,TX, USA (Email: {qxz110230, hlaing.minn, aldhahir}@utdallas.edu). H.Huang is with Huawei Technologies Co., Ltd., Shenzhen, China (Email:[email protected]).

This work is partially supported by Huawei Technologies Co., Ltd.Parts of this work was presented at a Globecom workshop [30].

Thus, online estimation and compensation of IQTMs is animportant open problem, especially for systems operating inthe mm-wave band and above.

Related Works: RF distortions constitute important practicalproblems and most of the related works in the literature con-sider in-phase and quadrature amplitude and phase imbalance(IQI), phase noise (PN), and carrier frequency offset (CFO).Regarding IQI, [4]–[6] developed blind compensation schemesfor RX frequency-dependent IQI in single antenna systemsbut they require a large number of samples and/or iterationsto compute statistics and hence would not be suitable forpacket-based transmissions. The TX frequency-dependent IQIcalibration was addressed in [7] but its use of a feedbackcircuitry with analog-to-digital converter would not be ap-propriate for massive MIMO systems. For multiuser scenar-ios, [8] assessed performance of TX frequency-independentIQI compensation in OFDMA systems by assuming perfectknowledge of channel and IQI parameters while [9] evaluatedsymbol error rate performance of a joint compensation anddata detection approach by assuming perfect knowledge of RXsignal correlation matrices which are functions of the channeland TX and RX frequency-dependent IQI parameters. Pilot-based IQI estimation/compensation schemes were developedin [10]–[12]. Pilot designs for IQI or combined effects ofIQI and channels were presented in [13]–[17]. Some worksaddressed a few RF distortions jointly, e.g., PN plus CFO[20], PN plus IQI [18], [19] and CFO plus IQI [21], [22],[25]. Capacity or mutual information analysis was consideredfor systems with IQI in [23], PN in [24], IQI and CFOin [25], and PN and IQI in [26]. We note that the aboveIQI compensation schemes would not work well under muchstronger PN levels as experienced in mm-wave systems. Fur-thermore, in all of the above works, IQTM issue has not beenconsidered. As will be shown in the paper, IQTMs introduceadditional cascaded frequency-dependent IQIs whose imagerejection ratio (IRR) characteristics are substantially differentfrom those of the regular IQI. Furthermore, IQTMs causesubstantial prolonging of the effective channel length, thusrequiring substantially larger overhead if pilots are used toestimate the combined effects of IQTM, IQI, and channels.Such pilot overhead increase will be avoided if IQTMs can be(pre)-compensated. These facts highlight the need for separateIQTM estimation/compensation.

To the best of our knowledge, there are only a coupleof existing works [27] [28] which are related to IQTMs.



2

Dig

ital T

X B

F

Modula

tion &

Multip

lexin

g

DAC 1

(TX Filter)IDFT

+ CP

DAC UT

(TX Filter)

IDFT

+ CP

IQTM

IQTM

IQICFO

+ PNPA 1

Phase

Shifter

IQICFO

+ PN

PA

VT

Phase

Shifter

IQICFO

+ PNPA 1

Phase

Shifter

IQICFO

+ PN

PA

VT

Phase

Shifter

Fig. 1. TX-side signal model block diagram with RF impairments. (BF =beam-forming, PA = power amplifier)

Reference [27] proposed an IQTM estimation and compensa-tion method for a 4x4 MIMO-OFDM system while [28] justevaluated IQTM’s effects on the performance of an opticalcommunication system. Both [27] and [28] consider RX IQTMonly; thus, they are not applicable to systems with TX IQTMs.Furthermore, other RF distortions such as PN and CFO werenot included in their study, which limits the applicability oftheir methods/results.

Contributions: Our main contributions are i) the develop-ment of the signal model for MIMO OFDM with CFO, TXand RX PN, TX and RX frequency-dependent IQI, samplingtime offset (STO), sampling frequency offset (SFO), and TXand RX IQTM, ii) revealing the distinct IRR and effectivechannel length prolonging characteristics of IQTM, iii) pilotdesigns for IQTM estimation in the presence of other RFdistortions, iv) the corresponding IQTM estimators (withoutneeding to estimate/compensate other impairments), v) IQTM(pre)compensation schemes, and vi) protocol for uplink anddownlink IQTM compensation. The proposed pilot designsand IQTM estimators offer good estimation performance formm-wave systems with severe RF distortions.

Organization: Section II presents signal model and Sec-tion III illustrates impacts of IQTM. Section IV develops theIQTM estimation metric and Section V develops pilot designsfor IQTM estimation. Estimation and compensation of IQTMare described in Section VI. Performance evaluation resultsare discussed in Section VII. Conclusions are provided inSection VIII.

Notations: F{·} denotes the Fourier transform. For a vectorx, diag{x} represents a diagonal matrix with diagonal ele-ments given byx. ℜ{x} andℑ{x} denote the real part andimaginary part ofx while [x]RI , [ℜ{xT },ℑ{xT }]T . For asequence denoted byJ , fliplr{J } gives a left-to-right flippedversion of the sequence and|J | stands for the number of ele-ments inJ . (·)∗ and(·)T denote the complex conjugate and thetranspose operation while∗ stands for the convolution.J is anantidiagonal matrix with all-one antidiagonal elements. We usthe following Matlab notations,k : n = {k, k+1, k+2, . . . , n}and k : m : n = {k, k + m, k + 2m, . . . , k + lm} wherel = ⌊(n− k)/m⌋.

Dig

ital R

X B

F

De

mu

x, D

em

odula

tion, D

ete

ctio

n RX

Filter

-CP &

DFTIQI

CFO

+ PNIQTM

RX

Filter

-CP &

DFTIQI

CFO

+ PN

ADC UR

(SFO+STO)IQTM

ADC 1

(SFO+STO)+

LNA

1

Phase

Shifter

LNA

VR

Phase

Shifter

+

LNA

1

Phase

Shifter

LNA

VR

Phase

Shifter

Fig. 2. RX-side signal model block diagram with RF impairments. (LNA =low-noise amplifier)

II. SIGNAL MODEL

In this section, we develop the signal model under variousRF distortions such as CFO, PN, IQI, SFO, STO, and IQTM.Since the single-carrier frequency domain equalization (SC-FDE) can be viewed as discrete Fourier transform (DFT)precoded OFDM, our signal model will be presented basedon OFDM with a DFT size ofNDFT, subcarrier spacing∆fandNcp cyclic prefix (CP) samples. Fig. 1 and Fig. 2 showthe system block diagrams at the TX and the RX, respectively.The TX hasUT digital-to-analog conversion (DAC) branchesand each branch is connected toVT antenna elements. The RXhasUR analog-to-digital conversion (ADC) branches and eachbranch is connected toVR antenna elements. The DAC (ADC)branch index is denoted byu1 (u2) and the antenna indexconnected to a DAC (ADC) branch is referred to byv1 (v2).Note that IQTMs introduced by DACs/ADCs are absorbed inthe IQTM blocks in the figures.

Let {cu1,k} represent the TX IDFT output signal (after CPinsertion) for the input frequency-domain symbols{Cu1

[l]}.Denote the TX pulse shape filter impulse response asgT (t)and the TX sampling period at the output of IDFT asTs. Thefiltered OFDM signal at the DAC branchu1 without IQTM isgiven by

su1(t) =

∑

k

cu1,kgT (t− kTs) (1)

and its Fourier transformSu1(f) = F{su1

(t)} reads as

Su1(f) = GT (f)

NDFT−1∑

l=0

Cu1[l] γT (f − l

NDFTTs) (2)

whereGT (f) = F{gT (t)} and γT (f) is the TX subcarrierfilter given by

γT (f) ,NDFT +Ncp√

NDFT

e−jπTsf(NDFT−Ncp−1)

× sinc (fTs(NDFT +Ncp))

sinc (fTs). (3)

Note thatγT (f) is periodic with period1/Ts, the spectralmainlobe of γT (f) within a period has a bandwidth of2/(Ts(NDFT +Ncp) (narrow-band), andγT (f) = γ∗

T (−f).When there is IQTM at TX, such time mismatch can be

modeled by introducing a time delayτtx,u1to ℑ{su1

(t)} withrespect toℜ{su1

(t)} (or the reverse order), whereτtx can take



3

either a positive or negative value. With IQTM, the signal atthe output of the DAC branchu1 becomes

su1(t) = ℜ{su1

(t)}+ jℑ{su1(t− τtx,u1

)}. (4)

Then, Su1(f) = F{su1

(t)} is given by

Su1(f) = Su1

(f)1 + e−j2πfτtx,u1

2+S∗

u1(−f)

1− e−j2πfτtx,u1

2.

(5)Let aIT,u1,v1

andθIT,u1,v1be the frequency-independent gain

and phase of the inphase branch at TX andaQT,u1,v1and

θQT,u1,v1be those of the quadrature branch. Let the frequency-

dependent gain and phase mismatches be represented by theinphase branch filterpIT,u1,v1

(t) and the quadrature branchfilter pQT,u1,v1

(t). Then, the (virtual) filters representing TXIQI are given by [16]

µT,u1,v1(t) =0.5(aIT,u1,v1

ejθIT,u1,v1pIT,u1,v1

(t)

+ aQT,u1,v1ejθ

Q

T,u1,v1pQT,u1,v1(t)), (6)

νT,u1,v1(t) =0.5(aIT,u1,v1

ejθIT,u1,v1pIT,u1,v1

(t)

− aQT,u1,v1ejθ

Q

T,u1,v1pQT,u1,v1(t)). (7)

Similarly, with the RX IQI parametersaIR,u2, θIR,u2

, aQR,u2,

θQR,u2, pIR,u2

(t), andpQR,u2(t), the filters representing RX IQI

are [16]

µR,u2(t) =

aIR,u2e−jθI

R,u2pIR,u2(t) + aQR,u2

e−jθQ

R,u2pQR,u2(t)

2,

(8)

νR,u2(t) =

aIR,u2ejθ

IR,u2pIR,u2

(t)− aQR,u2ejθ

Q

R,u2pQR,u2(t)

2.

(9)

If IQI is frequency-independent,pIT,u1,v1(t) = pQT,u1,v1

(t) =

δ(t) andpIR,u2(t) = pQR,u2

(t) = δ(t).The effects of TX IQI and PN can be modeled by

xu1,v1(t) = ejφT,u1

(t)[µT,u1,v1(t) ∗ su1

(t)

+ νT,u1,v1(t) ∗ s∗u1

(t)] (10)

whereejφT,u1(t) represents TX PN (+ TX CFO). From (4) and

(10), we obtain

xu1,v1(t) = ejφT,u1

(t)[µT,u1,v1(t) ∗ su1

(t)

+ νT,u1,v1(t) ∗ s∗u1

(t)] (11)

where

µT,u1,v1(t) =[µT,u1,v1

(t) + µT,u1,v1(t− τtx,u1

) + νT,u1,v1(t)

− νT,u1,v1(t− τtx,u1

)]/2, (12)

νT,u1,v1(t) =[µT,u1,v1

(t)− µT,u1,v1(t− τtx,u1

) + νT,u1,v1(t)

+ νT,u1,v1(t− τtx,u1

)]/2. (13)

Note that µT,u1,v1(t) and νT,u1,v1

(t) represent the cascadedeffects of TX IQTM and TX IQI.

The received signal at antennav2 in ADC branchu2 is

ru2,v2(t) =

UT∑

u1=1

VT∑

v1=1

L∑

l=1

xu1,v1(t− τl)e

jθT,u1,v1hu2,v2u1,v1

(t, l)

+ wu2,v2(t) (14)

where we assume that each multipath channel hasL paths,and ejθT,u1,v1 is the TX analog beamforming coefficient,hu2,v2u1,v1

(t, l) is thelth tap channel coefficient at timet betweenantenna elementv1 in TX DAC branchu1 and antenna elementv2 in RX ADC branchu2, andwu2,v2

(t) is an additive whiteGaussian noise process. Then, after analog beamforming withweights{ejθR,u2,v2 } at RX, the signal at the ADC branchu2

is given by

ru2(t) =

VR∑

v2=1

ru2,v2(t) ejθR,u2,v2 . (15)

Let αu2denote the CFO,gR(t) the RX filter impulse

response, ande−jφR,u2(t) the RX PN. After experiencing PN,

CFO and IQI (c.f. Fig. 2), the signal becomes

yu2(t) =µR,u2

(t) ∗(

ru2(t)e−jφR,u2

(t)e−j2παu2t)

+ νR,u2(t) ∗

(

ru2(t)e−jφR,u2

(t)e−j2παu2t)∗

. (16)

Next, after passing through the RX filter, the output signal isgiven by

yu2(t) = yu2

(t)∗gR(t) = µR,u2(t)∗ ru2

(t)+νR,u2(t)∗ r∗u2

(t)(17)

where

ru2(t) , (ru2

(t)e−jφR,u2(t)e−j2παu2

t) ∗ gR(t). (18)

Applying the Fourier transform, we haveYu2(f) =

F{yu2(t)}, Ru2

(f) = F{ru2(t)}, MR,u2

(f) =F{µR,u2

(t)}, NR,u2(f) = F{νR,u2

(t)} and

Yu2(f) = MR,u2

(f)Ru2(f) +NR,u2

(f)R∗u2(−f). (19)

When there is IQTM at RX, such time mismatch canbe modeled, in the same manner as in TX IQTM, and thecorresponding RX filter output signal is given by

yu2(t) = ℜ{yu2

(t)}+ jℑ{yu2(t− τrx,u2

)}. (20)

Then, Yu2(f) = F{yu2

(t)} is related toYu2(f) by

Yu2(f) = Yu2

(f)1 + e−j2πfτrx,u2

2+Y ∗

u2(−f)

1− e−j2πfτrx,u2

2.

(21)From (17) and (20), we obtain

yu2(t) = µR,u2

(t) ∗ ru2(t) + νR,u2

(t) ∗ r∗u2(t) (22)

whereµR,u2(t) andνR,u2

(t) are equivalent (virtual) IQI filtersrepresenting the effects of cascaded RX IQI and RX IQTMand given by

µR,u2(t) =[µR,u2

(t) + µR,u2(t− τrx,u2

) + ν∗R,u2(t)

− ν∗R,u2(t− τrx,u2

)]/2, (23)

νR,u2(t) =[µ∗

R,u2(t)− µ∗

R,u2(t− τrx,u2

) + νR,u2(t)

+ νR,u2(t− τrx,u2

)]/2. (24)



4

If there is a SFO between the sampling frequencies of TXIDFT and RX DFT, then the RX DFT sampling periodT ′

s

would be different from the TX IDFT sampling periodTs.Furthermore, the RX filter output sampling time instant coulddeviate from the optimal sampling instant, resulting a STOt0.The sampled version of (20) is given byyu2

[k] , yu2(kT ′

s +t0). After the CP removal, applying DFT to{yu2

[k]} of anOFDM symbol yields the frequency-domain received signalon sub-carriern (i.e., at frequencyfn , n/(NDFTT

′s)) as

Yu2[n] =

1√NDFT

NDFT−1∑

k=0

yu2[k]e

−j2πnkNDFT

=[(Yu2(f)ej2πft0) ∗ γR(f)]f=fn (25)

whereγR(f) is the RX subcarrier filter given by

γR(f) ,√

NDFT e−jπT ′

sf(NDFT−1) sinc (fT′sNDFT)

sinc (fT ′s)

. (26)

γR(f) is periodic with a period1/T ′s, has the mainlobe band-

width of 2/(T ′sNDFT) (narrow-band), andγR(f) = γ∗

R(−f).Define GR(f) , F{gR(t)}, Wu2,v2

(f) =

F{wu2,v2(t)}, Wu2

(f) ,∑VR

v2=1 ejθR,u2,v2Wu2,v2

(f),Hu2,v2

u1,v1(f) , F{hu2,v2

u1,v1(t)}, Hu2

u1,v1(f) ,

∑VR

v2=1 ejθR,u2,v2Hu2,v2

u1,v1(f), MT,u1,v1

(f) = F{µT,u1,v1(t)},

NT,u1,v1(f) = F{νT,u1,v1

(t)}, MR,u2(f) = F{µR,u2

(f)},NR,u2

(f) = F{νR,u2(f)}, ΨT,u1

(f) , F{ejφT,u1(t)rec(t)}

and ΨR,u2(f) , F{e−jφR,u2

(t)rec(t)} where rec(t) is aunit-amplitude rectangular function covering the consideredOFDM symbol.

Then, after some manipulation (See Appendix), we canexpressYu2

[n] as

Yu2[n] =

UT∑

u1=1

∑

l

{

Cu1[l]

(

Aµµu2,u1

[n, l] +Aννu2,u1

[n, l])

+Cu1[l]∗

(

Aνµu2,u1

[n, l] +Aµνu2,u1

[n, l])}

+ ηu2[n] (27)

whereAµµu2,u1

[n, l], Aννu2,u1

[n, l], Aνµu2,u1

[n, l], andAµνu2,u1

[n, l]represent the cross-coupling mechanisms from subcarrierl tosubcarriern due to the effects of TX and RX impairments, thechannels, and the analog beamforming, and they are given in(80), (81), (82), and (83), respectively. The noise termηu2

[n]is given in (84).

Denote the indexes of the used subcarriers by[k1, · · · , kN ]where (k1, kN ), (k2, kN−1), · · · , (k0.5N , k0.5N+1) are mir-

ror pairs. DefineYu2=

[

Yu2[k1], · · · , Yu2

[kN ]]T

, Cu1=

[Cu1[k1], · · · , Cu1

[kN ]]T , ηu2

= [ηu2[k1], · · · , ηu2

[kN ]]T .

Also, defineN×N matricesAµµu2,u1

, Aννu2,u1

, Aνµu2,u1

, Aµνu2,u1

such that their (nth row,lth column) elements are given byAµµ

u2,u1[kn, kl], Aνν

u2,u1[kn, kl], Aνµ

u2,u1[kn, kl], Aµν

u2,u1[kn, kl],

respectively. Then, we can express (27) in a matrix form as

Yu2=

UT∑

u1=1

{(Aµµu2,u1

+Aννu2,u1

)Cu1

+ (Aνµu2,u1

+Aµνu2,u1

)C∗u1}+ ηu2

. (28)

We can also express (28) in a real-valued matrix form as

[Yu2]RI =

UT∑

u1=1

Qu2,u1[Cu1

]RI + [ηu2]RI (29)

where

Qu2,u1,

[

ℜ{Uu2,u1} −ℑ{Vu2,u1

}ℑ{Uu2,u1

} ℜ{Vu2,u1}

]

, (30)

Uu2,u1, Aµµ

u2,u1+Aνν

u2,u1+Aνµ

u2,u1+Aµν

u2,u1, (31)

Vu2,u1, Aµµ

u2,u1+Aνν

u2,u1−Aνµ

u2,u1−Aµν

u2,u1. (32)

Next, to estimate TX and RX IQTMs, we decouple effectsof IQTM from those of other distortions. First, define thefollowing equations (33)-(38):

ΛD(τ) , diag{1 + e−j2πfk1

τ

2, · · · , 1 + e−j2πfkN

τ

2

}

(33)

ΛM(τ) , diag{1− e−j2πfk1

τ

2, · · · , 1− e−j2πfkN

τ

2

}

(34)

ΛµR,u2, diag {MR,u2

[k1], · · · ,MR,u2[kN ]} (35)

ΛνR,u2, diag {NR,u2

[k1], · · · ,NR,u2[kN ]} (36)

Let Dµu2,u1

andDνu2,u1

representN ×N matrices with (nthrow, lth column) elements given byDµ

u2,u1[kn, kl] in (91) and

Dνu2,u1

[kn, kl] in (92), respectively. Next, define

BDu2,u1

,ΛµR,u2Dµ

u2,u1+ΛνR,u2

JDν∗u2,u1

J, (39)

BMu2,u1

,ΛµR,u2Dν

u2,u1J+ΛνR,u2

JDµ∗u2,u1

, (40)

Qu2,u1,

[

ℜ{BDu2,u1

+BMu2,u1

} ℑ{BMu2,u1

−BDu2,u1

}ℑ{BD

u2,u1+BM

u2,u1} ℜ{BD

u2,u1−BM

u2,u1}

]

.

(41)

Then, after some manipulations (see Appendix), we have

[Yu2]RI ≈

UT∑

u1=1

Λrx,u2Qu2u1

Λtx,u1[Cu1

]RI + [ηu2]RI (42)

whereΛrx,u2represents the effect of RX IQTM,Λtx,u1

theeffect of TX IQTM, andQu2u1

the combined effects of thechannel, IQI, CFO, and PN at both TX and RX, as well as SFOand STO. The equation (42) shows the decoupled operationsof IQTM and other RF distortions. Whenτtx,u1

= τrx,u2= 0,

the IQTM matricesΛtx,u1and Λrx,u2

reduce to an identitymatrix. We will add the OFDM symbol index to the abovevariables when needed.

III. I MPACT OF IQTM

From (5) or (21), we observe that the effect of IQTM canalso be viewed as a specific type of frequency dependentIQI as it takes the same form of the signal model of afrequency-dependent IQ amplitude and phase imbalance asused in the literature (e.g., [4], [16]). But IQTM introducesan additional cascaded frequency-dependent IQI and yields afrequency-dependent IQI characteristic which is significantlydifferent from that of the regular frequency-dependent IQI inthe existing literature. To illustrate this, we evaluate the imagerejection ratio (IRR) at the RX for three cases: i) system withregular frequency-dependent IQI only (due toµR,u2

(t) and



5

Λrx,u2,

[

(ℜ[ΛD(τrx,u2)] + ℜ[ΛM(τrx,u2

)]J), (ℑ[ΛM(τrx,u2)]J−ℑ[ΛD(τrx,u2

)])(ℑ[ΛD(τrx,u2

)] + ℑ[ΛM(τrx,u2)]J), (ℜ[ΛD(τrx,u2

)]−ℜ[ΛM(τrx,u2)]J)

]

(37)

Λtx,u1,

[

(ℜ[ΛD(τtx,u1)] + ℜ[ΛM(τtx,u1

)]J), (ℑ[ΛM(τtx,u1)]J−ℑ[ΛD(τtx,u1

)])(ℑ[ΛD(τtx,u1

)] + ℑ[ΛM(τtx,u1)]J), (ℜ[ΛD(τtx,u1

)]−ℜ[ΛM(τtx,u1)]J)

]

(38)

-100 -50 0 50 10028

30

32

34

IRR

(dB

)

-100 -50 0 50 100

subcarrier index

0

20

40

60

IRR

(dB

)

dotted: IQTM onlysolid: IQTM + IQI

IQI only

τrx=0.8Ts

τrx=0.4Ts

τrx=0.1Ts

Fig. 3. Impact of IQTM on IRR of an OFDM system with frequency-dependent IQI

νR,u2(t)), ii) system with IQTM τrx,u2

only, and iii) systemwith both IQTM and regular frequency-dependent IQI. TheIRRs (in dB) at subcarriern for the three cases are given by

IRRIQI[n] = 20 log10(|MR,u2[n]/NR,u2

[n]|), (43)

IRRIQTM[n] = 20 log10(|1 + e−j2πfnτrx,u2 ||1− e−j2πfnτrx,u2 | ), (44)

IRRIQTM+IQI[n] = 20 log10(|MR,u2[n]/NR,u2

[n]|). (45)

Let us consider an OFDM system withNDFT = 256, theRX IQTM τrx,u2

, the RX IQI parametersaIR,u2= 1.02,

aQR,u2= 0.98, θIR,u2

= 1 degree,θQR,u2= −1 degree,

pIR,u2(t) = 0.01δ(t) + δ(t − Ts) + 0.01δ(t − 2Ts), and

pQR,u2(t) = 0.01δ(t) + δ(t − Ts) + 0.02δ(t − 2Ts). Fig. 3

presents the corresponding IRRs of the three cases. We observethat IQTM causes substantial IRR degradation which is moresevere at outer subcarriers. For example, at outer subcarriers,IRR is around 30 dB for the system with IQI only butIRR is less than 0 dB for the system with IQI and IQTMτrx,u2

= 0.8Ts.Another perspective of the IQTM’s effect is that it prolongs

the equivalent channel length. For a system with frequency-dependent TX and RX IQI filters ofλ taps each and theL-tap channel, the sample-spaced equivalent channel impulseresponse length (for either the direct channel or the mirrorchannel) isL + 2λ − 2 (see eq. (5) and (6) of [16]). Thus,to see the equivalent channel length prolonging effect, wecan just observe how IQTM changes the effective IQI filterlength. Considering the same system setting as in Fig. 3 with

-20 -10 0 10 20

n

0

0.5

1

µR[n

]

-20 -10 0 10 20

n

0

0.1

0.2

0.3

νR[n

]

-20 -10 0 10 20

n

0

0.5

1

bar{

µR}[

n]

-20 -10 0 10 20

n

0

0.1

0.2

0.3

bar{

νR}[

n]

Fig. 4. Equivalent channel length prolonging effect of IQTM (Top: IQI only;Bottom: IQI plus IQTM)

τrx,u2= 0.4Ts, we plot in Fig. 4 the effective IQI filter impulse

responses for the case with RX IQI only (i.e.,µR,u2[n],

νR,u2[n]) and those for the case with RX IQI and RX IQTM

(i.e., µR,u2[n], νR,u2

[n]). 1 We observe that IQTM changes theoriginal 3-taps IQI filters into equivalent IQI filters of morethan 30 taps. The consequence is that the pilot overhead wouldsubstantially increase for each transmission packet since thechannels and hence the equivalent channels (incorporating IQIand IQTM) vary over different packets. However, exploitingthe specific characteristic (namely IQ mismatch in time delay)of IQTM, we can estimate IQTM separately. The need forIQTM estimation would be infrequent since IQTM does notvary as the channel does. This provides pilot overhead saving,and serves as a motivation for separately addressing IQTM.

IV. IQTM E STIMATION METRIC

In developing the estimator and the pilot design for IQTM,we use the approximate frequency-domain signal model in(42). However, when we evaluate the system performance,the actual signals are generated according to the exact signalmodel in the time-domain. In this section, we develop pilot-based estimation metric for TX and RX IQTMs. In themetric development, we use pilot preamble(s) where non-zeropilot tones are padded with null pilot tones such that bothinter-carrier interference (ICI) and mirror tone interference(MTI) are avoided for non-zero pilot tones and their ICIs

1When the equivalent channel impulse response is obtained by IDFT ofthe frequency domain channel, the negative indexes in the plot correspondto their moduloNDFT but for presentation convenience, we plot them usingnegative indexes.



6

and MTIs are also decoupled. In addition, non-zero pilotsof different DAC branches have non-overlapping (decoupled)ICI and MTI spreads. The corresponding pilot designs willbe described later. Now, let us consider that DAC branchu1

transmits a non-zero pilot on subcarriern and a null piloton subcarrier−n, and observe the received signal (withoutnoise) on subcarriern and−n at ADC u2. From Appendix,we know thatAνµ

u2,u1[n, n] andAµν

u2,u1[n, n] are negligible and

Aµµu2,u1

[n, n] ≫ Aννu2,u1

[n, n]. Then, we have

Yu2[n] ≈ Cu1

[n]Aµµu2,u1

[n, n], (46)

Yu2[−n] ≈ C∗

u1[n]

(

Aνµu2,u1

[−n, n] +Aµνu2,u1

[−n, n])

. (47)

Next, defineξn ,Yu2

[−n]

Y ∗

u2[n]

. Then, after some simplification

(see Appendix), we obtainξn as in (48) (shown at the top ofnext page). When there are no IQTMs, i.e.,τtx,u1

= τrx,u2=

0, (48) reduces to

ξn ≈ NR,u2[−n]

M∗R,u2

[n]+

MR,u2[−n]Dν

u2,u1[−n,−n]

M∗R,u2

[n]Dµ∗u2,u1 [n, n]

. (49)

We observe thatξn without IQTM in (49) varies slowly acrossn but ξn with IQTM in (48) changes faster acrossn due to thetwo additional terms caused by IQTMs. The IQTM-inducedterms fluctuate more as the IQTM values increase. This impliesthat the sample variance ofξn, denotedσ2

ξ , is smallest whenIQTM reduces to zero. Thus, we useσ2

ξ as the metric forestimating the IQTMs(τtx,u1

, τrx,u2). As ξn can be viewed

as1/IRR[n], the same conclusion for the metric can also bereached from Fig. 3.

V. PILOT DESIGNS FORIQTMS

We can observe from (42) that the RX IQTM can becompensated by multiplying[Yu2

]RI with the inverse ofΛrx,u2

if τrx,u2is known. But we cannot apply a similar

compensation approach at the RX for the TX IQTM. Thus, weadopt a novel pilot design approach to handle the TX IQTM.In other words, our approach uses different pilot tones fordifferent TX IQTM estimation candidates but the same pilottones for all RX IQTM estimation candidates.

A. Joint TX and RX IQTMs or TX-only IQTM

Suppose{τ (i)rx,u2 : i = 1, 2, · · · , nτ,u2} and {τ (m)

tx,u1: m =

1, 2, · · · , nτ,u1} represent the sets of the trial candidate values

for the RX IQTM and the TX IQTM, respectively. LetC(m)u1

be theN×1 pilot vector to be used forτ (m)tx,u1

and its subcarrierindexes are mirror pairs.

First, we define an initial non-zero pilot tone index setJini = {JL

ini, JRini} whereJL

ini andJRini represent the index sets

of the initial non-zero pilot tones at the left side and the rightside of the DC tone, respectively. Suppose that the indexesof the used subcarriers are−NL, · · · , NR. To maintainthe mirror pair property,NL = NR. Note that other RFdistortions such as PN and IQ amplitude and phase imbalanceintroduce ICI and MTI in frequency-domain. Assume theone-side significant ICI spread isκ subcarriers and one-side significant MTI spread isι subcarriers. Recall that our

DC

X X X X X XX X XX XXX XX XXX X

Subcarriers

XX XXX XX XX X X X X

J1 J2 ICI MTIJini={J1, J2}

ICI spread offset

Spacing mirror

Fig. 5. An illustration of designingJini for a system withκ = 2, ι = 0 andtwo TX IQTM candidates.

estimation metric requires disjointness between ICI spreadsat {Jini + k : k = −κ, . . . , κ} and MTI at −Jini (or moreconservatively MTI spreads at{−Jini+k : k = −ι, . . . , ι}). Toachieve this condition, the spacing of non-zero pilot tones mustbe at least(2κ+2) (or more conservatively(2κ+2ι+2)) tonesat each side of the DC tone, and the non-zero pilot tones atthe left side and at the right side are offset by (κ+1) (or moreconservatively (κ+ι+1)) tones. The above spacing guaranteesenough spacing for decoupling ICI spreads and MTI (or MTIspread) while the offset ensures that the MTI (or MTI spreads)are disjoint from the ICI spreads. For most practical systems,we observeι = 0 and hence the two approaches are the same.Below, we present the more conservative design ofJini as

JRini = κ+ 1 : (2κ+ 2ι+ 2) : NR, (50)

JLini = fliplr(−JR

ini)− κ− ι− 1 (51)

or alternatively as

JLini = fliplr(−κ− 1 : (−2κ− 2ι− 2) : −NL), (52)

JRini = fliplr(−JL

ini) + κ+ ι+ 1. (53)

The reason the starting index in (50) isκ + 1 or in (52) is−κ − 1 is to avoid spill-over of ICI spread on the DC tone;otherwise, it may cause problems associated with the DC offsetissue2. As an example of the design in (50) and (51), considera system withNR = NL = 86, κ = 2 and ι = 0. Then,JRini = {3, 9, 15, . . . , 75, 81} and JL

ini = {−84,−78,−72,. . . , −12,−6}. Fig. 5 also illustrates this example. To seedecoupling between ICI spreads and MTI, observe that theICI spreads of tones at indexes3 and9 are at indexes{1 : 5}and{7 : 11} while the MTI of the tone at index−6 appearsat index6. Also note that the ICI spread of tone at index 3does not cause a DC offset.

We consider a single TX DAC branch first (while keepingDAC branch indexu1 in some variables for a clear connectionto the case with multiple DACs), and later we will extend thepilot design to multiple DAC branches. As we will developdifferent pilot sets for different trial candidate values{τ (m)

tx,u1},

let JLm andJR

m represent the initial non-zero pilot tone indexsets at the left and right side of the DC tone forτ (m)

tx,u1.

Furthermore, let[JLini]k and [JR

ini]k stand for thekth elementsof JL

ini andJRini, respectively. Then, we design the initial pilot

2If such issue is insignificant up toε edge tones of the ICI spread, thosestarting indexes can begin atκ− ε+ 1 andε− κ− 1.



7

ξn ≈NR,u2[−n]

M∗R,u2

[n]+

MR,u2[−n]Dν

u2,u1[−n,−n]

M∗R,u2

[n]Dµ∗u2,u1 [n, n]

+1− ej2πfnτtx,u1

1 + ej2πfnτtx,u1

[MR,u2[−n]Dµ

u2,u1[−n,−n]

M∗R,u2

[n]Dµ∗u2,u1 [n, n]

+(NR,u2

[−n]

M∗R,u2

[n]

(1− ej2πfnτrx,u2 )

(1 + ej2πfnτrx,u2 )

)Dν∗u2,u1

[n, n]

Dµ∗u2,u1 [n, n]

]

+1− ej2πfnτrx,u2

1 + ej2πfnτrx,u2

. (48)

tone index setJm = {JLm, JR

m} for τ (m)tx,u1

as

JLm ={[JL

ini]m, [JLini]m+nτ,u1

, [JLini]m+2nτ,u1

, · · · }, (54)

JRm ={[JR

ini]m, [JRini]m+nτ,u1

, [JRini]m+2nτ,u1

, · · · }. (55)

This allocatesJini among the TX IQTM candidate points in aninterleaved frequency division multiplexing manner which pro-vides frequency diversity for each candidate. As an example,consider the system mentioned above withnτ,u1

= 2 candidatepoints. Then, we haveJL

1 = {−84,−72, . . . ,−24,−12},JR1 = {3, 15, . . . , 63, 75}, JL

2 = {−78,−66, . . . ,−18,−6},and JR

2 = {9, 21, . . . , 69, 81}. Fig. 5 also illustratesJ1 andJ2 for this example.

Next, we design the pilot vectorC(m)u1 for τ

(m)tx,u1

. Let pm

denote a|Jm| × 1 vector containing constant amplitude lowpeak-to-average power ratio (PAPR) sequence forτ

(m)tx,u1

, andΠm represent theN × |Jm| matrix which assignspm to thesubcarrier indexes defined byJm. Let the TX IQTM matrixΛtx,u1

for τtx,u1= τ

(m)tx,u1

be denoted byΛtx,u1(τ

(m)tx,u1

). From

(42), we observe that we can designC(m)u1 so that the effect

of the TX IQTM for τtx,u1= τ

(m)tx,u1

is pre-compensated. Thisdesign is given by

[C(m)u1

]RI = Λtx,u1(−τ

(m)tx,u1

) [Πmpm]RI. (56)

Note thatC(m)u1 has non-zero elements only at the subcarriers

defined byJm for τ(m)tx,u1

= 0 and by{±Jm} for τ(m)tx,u1

6= 0.The TX pilot vector is

Cu1=

nτ,u1∑

m=1

C(m)u1

. (57)

The phases of{pm} can be optimized to minimize PAPR ofthe time-domain signal generated fromCu1

. A simpler low-PAPR design is to setpm = ejθmp1 for m 6= 1, and minimizethe PAPR over the parameters{θm : m = 2, 3, · · · , nτ,u1

} andthe phases ofp1.

If multiple preamble symbols with the symbol index setT are used, we denote the tone index sets by including thesymbol indext ∈ T . In this case, we have

JLini,t =JL

ini, JRini,t = JR

ini, t ∈ T , (58)

JRm,t ={[JR

ini]βt,m+m, [JRini]βt,m+m+nτ,u1

,

[JRini]βt,m+m+2nτ,u1

, · · · }, t ∈ T , (59)

JLm,t ={[JL

ini]βt,m+m, [JLini]βt,m+m+nτ,u1

,

[JLini]βt,m+m+2nτ,u1

, · · · }, t ∈ T , (60)

where{βt,m} are chosen such that non-zero pilot tone indexesof a candidate TX IQTM collected from all the preamble

symbols span the subcarriers approximately evenly, and

{JRm1,t, J

Lm1,t} ∩ {JR

m2,t, JLm2,t} = ∅, m1 6= m2, t ∈ T .

(61)

Also note that{JRm,t1 , J

Lm,t1} ∩ {JR

m,t2 , JLm,t2} = ∅, t1 6= t2.

For τ (m)tx,u1

at symbolt ∈ T , denote the constant amplitudelow-PAPR sequence vector bypm,t, the subcarrier assignmentmatrix byΠm,t, and the corresponding pilot vector byC(m)

u1,t.

Then,C(m)u1,t is designed as

[C(m)u1,t]RI = Λtx,u1

(−τ(m)tx,u1

) [Πm,tpm,t]RI, t ∈ T . (62)

The TX pilot vector at preamble symbolt is

Cu1,t =

nτ,u1∑

m=1

C(m)u1,t, t ∈ T . (63)

The same approach of PAPR-minimization can also be donefor each preamble symbol by optimizing over the phases of{pm,t : m = 1, 2, · · · , nτ,u1

} or by settingpm,t = ejθm,tp1,t

for m 6= 1, and optimizing over the parameters{θm,t : m =2, 3, · · · , nτ,u1

} and the phases ofp1,t.

B. RX-only IQTM

For the scenario with RX-only IQTM, the pilot design issimply given by the non-zero pilot tone index setJini asdefined in (50) and (51) or (52) and (53), and the constantamplitude low-PAPR sequence transmitted onJini. All of theRX IQTM estimate candidates use the same received pilotslocated at±Jini. In this case, one preamble symbol is typicallysufficient if |Jini| is reasonably large to yield a reliable samplevariance metricσ2

ξ . If multiple preamble symbols are used, thesame preamble can be repeated.

C. Extension to Multiple DACs

For multiple DACs, the above pilot design can be directlygeneralized while maintaining the property of decoupled ICIand MTI spreads among the non-zero pilot tones. The non-zero pilot tones’ indexes are given by (50) and (51) or (53) and(52). These non-zero pilot tones are allocated among differentDAC branches in a time-division manner (i.e., using differentpreamble symbols for different DAC branches) or a distributedfrequency-division manner (similar to (55), (54)), or theircombination (e.g., time-division for different groups of DACsand frequency division for DACs within each group, or similarto (59), (60)). Then, the non-zero pilot tones of each DACbranch are allocated to different TX IQTM candidate pointsas in (55) and (54), or (59) and (60). The design principle ofthe pilot vector for each DAC branch remains the same as thatdescribed above.



8

VI. ESTIMATION AND COMPENSATION OFIQTMS

A. Joint estimation of TX and RX IQTMs

Suppose that the system has both TX and RX IQTMs andit uses the corresponding pilots (preambles) described in theprevious section. First, we apply compensation of RX IQTMbased on a candidateτ (i)rx,u2 on the received frequency-domainvector at OFDM symbolt ∈ T as

[Y(i)u2,t]RI = Λrx,u2

(−τ (i)rx,u2) [Yu2,t]RI (64)

whereΛrx,u2(−τ

(i)rx,u2) is given by (37) withτrx,u2

replacedby −τ

(i)rx,u2 . Next, for a candidate IQTM pair(τ (m)

tx,u1, τ

(i)rx,u2)

andn ∈ Jm,t with t ∈ T , we define and computeξ(m,i)n,t as

ξ(m,i)n,t =

Y(i)u2,t[−n]

(Y(i)u2,t[n])

∗(65)

and the estimation metric as a sample variance given by

σ2ξ (τ

(m)tx,u1

, τ (i)rx,u2) =

( 1∑

t∈T |Jm,t|∑

t∈T

∑

n∈Jm,t

|ξ(m,i)n,t |2

)

−∣

∣

∣

1∑

t∈T |Jm,t|∑

t∈T

∑

n∈Jm,t

ξ(m,i)n,t

∣

∣

∣

2

. (66)

Note that different TX IQTM candidate points have differentdisjoint pilot tones from which their corresponding estimationmetric values are computed separately. Among the candidateIQTM pairs {(τ (m)

tx,u1, τ

(i)rx,u2)}, the estimator chooses the one

which yields the smallest metric value as

(τtx,u1, τrx,u2

) = arg{(τ

(m)tx,u1

,τ(i)rx,u2

)}

min{σ2ξ (τ

(m)tx,u1

, τ (i)rx,u2)}.

(67)

B. Estimation of TX-only IQTM

Let us consider the scenario with TX-only IQTM and thecorresponding pilots as described in the previous section.Then, no RX side compensation is needed and we define andcompute

ξ(m)n,t =

Yu2,t[−n]

(Yu2,t[n])∗, n ∈ Jm,t, t ∈ T , (68)

σ2ξ (τ

(m)tx,u1

) =( 1∑

t∈T |Jm,t|∑

t∈T

∑

n∈Jm,t

|ξ(m)n,t |2

)

−∣

∣

∣

1∑

t∈T |Jm,t|∑

t∈T

∑

n∈Jm,t

ξ(m)n,t

∣

∣

∣

2

. (69)

Then, the corresponding TX IQTM estimator is given by

τtx,u1= arg

{τ(m)tx,u1

}min{σ2

ξ (τ(m)tx,u1

)}. (70)

C. Estimation of RX-only IQTM

Suppose that IQTM only exists at the RX and the corre-sponding pilots described in the previous section are used.

First, we apply compensation of RX IQTM based on a candi-dateτ (i)rx,u2 as in (64) to the frequency-domain received vectorat preamble symbolt ∈ T . Then, we define and compute

ξ(i)n,t =

Y(i)u2,t[−n]

(Y(i)u2,t[n])

∗, n ∈ Jini, t ∈ T , (71)

σ2ξ (τ

(i)rx,u2

) =( 1

|T ||Jini|∑

t∈T

∑

n∈Jini

|ξ(i)n,t|2)

−∣

∣

∣

1

|T ||Jini|∑

t∈T

∑

n∈Jini

ξ(i)n,t

∣

∣

∣

2

. (72)

The corresponding RX IQTM estimator is given by

τrx,u2= arg

{τ(i)rx,u2

}min{σ2

ξ (τ(i)rx,u2

)}. (73)

D. IQTM Estimation in MIMO System

In a MIMO system withUT DAC branches andUR ADCbranches, we can further improve the estimation performanceby averaging across relevant MIMO branches. Recall that thepilot tones ofUT TX DAC branches are disjoint in frequencyand/or time. TheseUT disjoint pilot sets experience the sameRX IQTM at an ADC branch. Thus, at the output of ADCbranchu2, our estimator based onUT pilot sets givesUT

estimates of the same RX IQTM for the ADC branchu2. Wepropose to average theseUT estimates to enhance the RXIQTM estimation performance. As an example of averaging,we use the majority rule to select the final estimate. In a similarmanner, TX IQTM of a DAC branch is estimated at the outputof each of theUR ADC branches and theseUR estimates ofTX IQTM of that DAC branch can be averaged to improvethe TX IQTM estimation performance.

Next, we analytically assess the performance of the majorityrule MIMO averaging. Our investigation results (as will bepresented later) show that random normalized IQTMs (i.e.,τtx,u1

/Ts and τrx,u1/T ′

s) which are uniformly distributedwithin the range[−1/8, 1/8] give almost the same bit errorrate (BER) performance as the system with no IQTMs. Thus,we consider discretized IQTM values with a normalized res-olution of 1/4, which allows us to evaluate estimation successor failure probabilities. We also observe in our investigation(as will be shown later) that the proposed estimator beforeMIMO averaging gives the correct estimate most of the timesand almost all of the errors correspond to one or two adjacentcandidate points at each side of the actual IQTM point. Thus,it is sufficient to consider a total of 5 possible points for theestimate, and denote their probabilities byPi, i = 0, 1, · · · , 4whereP0 represents the probability of correct estimation. Asthe estimates in the above averaging are independent, aftercombiningU estimates by the majority rule, the estimationsuccess probability is given by

Psuccess =

U∑

k=⌈U/5⌉

P k0 Q(U − k, k)

+

U∑

k=⌈U/5⌉

⌊(U−k)/k⌋∑

m=1

1

mP k0 Q

′(U − k, k,m) (74)



9

where ⌊X⌋ ( ⌈X⌉) is the floor (ceiling) operation whichroundsX to the nearest integer less (greater) than or equal toX andQ(X,Y ) =

∑(

Xk1

)(

X−k1

k2

)(

X−k1−k2

k3

)

P k11 P k2

2 P k33 P k4

4

with the constraint thatk1, k2, k3, k4 are non-negative integers,k1 + k2 + k3 + k4 = X andmax(k1, k2, k3, k4) < Y . Next,Q′(X,Y,m) =

∑(

Xk1

)(

X−k1

k2

)(

X−k1−k2

k3

)

P k11 P k2

2 P k33 P k4

4

with the constraint k1 + k2 + k3 + k4 = X,max(k1, k2, k3, k4) = Y andm variables fromk1, k2, k3, k4have the same maximum valueY . If there is no suchk1, k2, k3, k4 which satisfy the constraint,Q(X,Y ) orQ′(X,Y,m) equals to0. In (74), the first term correspondsto the case where the correct candidate point has more countsthan any other candidate. The second term is due to the tiedcase where the correct candidate point and one or more othercandidate points have the same majority count.

E. Protocol for IQTM Compensation

As the characteristics of the TX and RX circuit paths canbe different, the TX and RX IQTMs for uplink and downlinkcan be different. Thus, in practice, pilot transmission andestimation of TX and RX IQTMs should be done for bothuplink and downlink. The RX IQTM can be compensatedat the RX. However, the TX IQTM cannot be compensatedat the RX and hence it should be fed back to the TX andcompensation should be done at the TX. Thus, for IQTMestimation/compensation for both uplink and downlink, wepropose the transmission protocol given in Fig. 6. Note thatmajority rule MIMO averaging is applied in step 2 and itemi of step 4 in the protocol. In step 3, pilots and data tones(feedback information) of different UEs are disjoint in the timedomain (TDM), frequency domain (FDM) or a combinationof both (TDM+FDM). In items ii and iii of step 4, averagingacross UEs is performed, for example, by majority rule in thesame way as in MIMO averaging. Other parts of the protocolsteps are self-explanatory from the description, thus furtherelaboration is omitted.

We note that the UE averaging is essentially the sameas additional MIMO averaging, thus its performance can beobtained from (74) where{Pi} now represent probabilities ofthe estimates after MIMO averaging at a UE but before theUE averaging.

As IQTMs typically do not change even over a long timeinterval, it is not necessary to frequently transmit pilots andestimate IQTM. Each transceiver just stores its TX and RXIQTM estimate values and applies the corresponding compen-sation at the RX or at the TX. If desired, regular fine tuning ofthe IQTM estimate values can be done by the same approachand by appropriately averaging the estimate values.

F. IQTM Compensation

For a transceiver with stored IQTM estimates of{τtx,u1}

and{τrx,u2}, the TX IQTM compensation is performed on the

frequency-domain data vector as

[Cu1]RI = Λtx,u1

(−τtx,u1) [Cu1

]RI (75)

whereCu1becomes IDFT inputs for OFDM signal generation.

The RX IQTM compensation is performed on the DFT output

BS transmits pilots for IQTM

Each UE estimates DL TX and RX IQTM parameters

(with MIMO averaging)

UEs transmit to BS via TDM, FDM or TDM+FDM:

i) pilots for IQTM

ii) pilots for channel and other RF distortions

iii) BS’s TX IQTM

BS performs the following:

i) estimates UL TX and RX IQTMs for each UE based on

pilots for IQTM (with MIMO averaging)

ii) averages UL RX IQTM estimates across UEs, and applies

compensation of UL RX IQTM

iii) extracts feedback of DL TX IQTM based on the other set

of pilots, and averages them across UEs

BS applies compensation of its TX IQTM and transmits

i) (common) pilots for channel and other RF distortions

ii) individual TX IQTMs of UEs

Each UE applies compensation of its RX IQTM and

extracts its TX IQTM based on the pilots

All transceivers know own TX and RX IQTMs which can

be used for compensation in their TX and RX.

IQTM estimation protocol finished.

Fig. 6. Protocol sequence for TX and RX IQTM estimation/compensationfor uplink and downlink. (TDM=time-division multiplexing, FDM=frequency-division multiplexing)

frequency-domain vector at RX as

[Yu2]RI = Λrx,u2

(−τrx,u2) [Yu2

]RI (76)

whereYu2is input to the modulation slicer or data detector.

In the above, we presented baseband DSP-based compensa-tion for IQTM. If the transceiver has multi-phase filters orDAC/ADC with adjustable sampling instants for introducingdifferent delays at the TX and RX chains, IQTM compensationcan also be implemented by such hardware-based mechanism.

VII. PERFORMANCEEVALUATION

A. System Setting

We consider an OFDM system with 64 antennas at BS and4 antennas at UE. The subcarrier spacing is 1.44 MHz and thecarrier frequency is 73 GHz. We use two settings for band-width, namely, 250 MHz and 2 GHz, the corresponding DFTsizes are 256 and 2048, and the number of used subcarriers is173 and 1388, respectively. The channel model is based on the3GPP LTE channel model [29] with two clusters where eachcluster has 20 sub-paths, the second clusters delay is about 80ns (based on Huawei’s mm-wave channel measurements) andits power is−9 dB with reference to the first cluster. Analogbeamforming is applied based on the mean arrival angle of thesub-paths of the first cluster. We consider a single data streamwith 16-QAM. The PN power spectral densities (PSD) at TXand RX are independently modeled asPSD(f) = PSD(0)[1+( ffz)2]/[1 + ( f

fp)2] where PSD(0) =−60 dBc/Hz, PSD(100k)



10

= −75 dBc/Hz and PSD(∞) =−130 dBc/Hz. The CFOs atTX and RX sides are independent and uniformly distributedwithin the range of±1 ppm, the RX SFO is set at 1 ppm andthe RX STO is uniformly distributed within[−T ′

s/2, T′s/2].

Frequency-independent IQIs are independent at TX and RXsides and they are modeled with uniform distributions withthe range[−4, 4] for 20 log10[a

I/aQ], and the range[−5, 5]degrees forθI − θQ. Frequency-dependent IQI filters are setas pIR,u2

(t) = 0.01δ(t) + δ(t − Ts) + 0.01δ(t − 2Ts), andpQR,u2

(t) = 0.01δ(t)+ δ(t−T ′s)+0.02δ(t− 2T ′

s). The mobilespeed is 10 km/h. We assume no nonlinear distortion. We useκ = 2 and ι = 0 and they are set according to [17]. In thesimulation, signals are generated in the time domain with 4times oversampling of the DFT sampling frequency.

5 10 15 20

Eb/N

0 (dB)

10-3

10-2

10-1

100

Bit

Err

or R

ate

τtx

= 0.25Ts, τ

rx = 0

τtx

= 0.5Ts, τ

rx = 0

τtx

= 0.75Ts, τ

rx = 0

τtx

= Ts, τ

rx = 0

τtx

= 0, τrx

= 0.5Ts′

τtx

= 0.5Ts, τ

rx = 0.5T

s′

τtx

= 0, τrx

= 0

τtx

/Ts, τ

rx/T

s′ ∈ [-1/8, 1/8]

Compensation

Fig. 7. The impacts of IQTMs on the system’s BER performance.

B. Impact of IQTM on BER

We first evaluate the impact of IQTM on system’s BER,thus no IQTM compensation is applied here. We consider thesystem with 250 MHz bandwidth where an OFDM preamblesymbol is followed by a pilot-data multiplexed OFDM symbolcarrying 16-QAM symbols and pilot tones. Here, we assumefrequency-independent IQI3. The preamble and pilots are forestimating the channel and RF distortions (other than IQTM).These preamble and pilot designs, their corresponding estima-tors and data detection are according to [17]. In Fig. 7, wepresent the effects of IQTMs on BER for differentEb/N0

values. We evaluate BER for systems with TX-only IQTM,RX-only IQTM, and both TX and RX IQTMs. We observethat the BER performance degrades as the IQTM increasesand the BER performances for TX-only IQTM and RX-onlyIQTM are practically the same. The IQTMs at both TX andRX cause more BER degradation than the IQTM at one sideonly. We also include a random (uniform) IQTMs with thenormalized range of[−1/8, 1/8], and its BER performance is

3Pilot designs and data detection scheme for the considered system withfrequency-dependent IQI are not available in the literature and developingthem constitutes another research problem which is outside the scope of thispaper.

almost the same as that of the system with no IQTMs. Thisprovides a guideline for setting the spacing of the candidatenormalized IQTM values to be1/4 for the considered sys-tem. Furthermore, it also allows us to discretize the actualIQTM values with the spacing of1/4 (which we use inour later investigations) which in turn enables us to assessthe estimation failure probabilities of our IQTM estimators.Next, we observe that the IQTM-induced BER degradation issubstantial if the normalized IQTMs are larger than 1/8 andno IQTM compensation is applied. The above BER resultsclearly demonstrate that IQTM estimation and compensationschemes are needed to counter the effects of IQTM.

To illustrate the performance of the proposed IQTM com-pensation, we also include in Fig. 7 the BER performance(denoted “Compensation”) obtained after our proposed IQTMcompensation for the case withτtx = 0.5Ts andτrx = 0.5T ′

s.The proposed approach achieves the same BER performanceas the case without IQTMs, thus illustrating its effectiveness.

4 9 14 19

SNR (dB)

10-4

10-3

10-2

10-1

Est

imat

ion

failu

re p

roba

bilit

y

Tx-only IQTM, τtx

= 0

Tx-only IQTM, τtx

= 0.25 Ts

Tx-only IQTM, τtx

= 0.5 Ts

Rx-only IQTM, τrx

= 0

Rx-only IQTM, τrx

= 0.25 Ts′

Rx-only IQTM, τrx

= 0.5 Ts′

Proposed method

Reference method

Fig. 8. Estimation failure probabilities for systems with TX-only IQTM andRX-only IQTM

C. Estimator Performance

The estimator in [27] assumes that CFO is pre-compensatedand other RF distortions are ignored. Its estimator usestwo preambles with frequency-domain symbols{C1[k]}and {C2[k]} separately and requires thatC

∗

1 [N−k+1]C1[k]

−C∗

2 [N−k+1]C2[k]

6= 0 for k ∈ P where P is the pilot indexset. The signal is intentionally transmitted on pairs of mir-ror subcarriers to enable the key metric functionD[k] =

Y2[k]−C2[k]H[k]

Y ∗

2 [N−k+1]−Cideal[k]H[N−k+1]in the RX-only IQTM case,

whereCideal[k] = C2[k]C∗1 [N − k + 1]/C1[k] and H[k] =

Y1[k]/C1[k]. Reference [27] estimatesτrx by comparing thevariance of{D[k], k ∈ P} to heuristic thresholds.

We first evaluate the TX-only and RX-only IQTM casesfor the system with 250 MHz bandwidth and frequency-independent IQI. For the RX-only IQTM case, we use onepreamble symbol with non-zero pilot tone spacing of 6 tonesat each side of the DC tone and use the candidate set{τ (i)rx }



11

= {−T ′s,−0.75T ′

s, · · · , T ′s}. The reference estimator uses two

preamble symbols and each symbol uses the same amount ofpilots and the same pilot tone spacing as the proposed esti-mator. For the TX-only IQTM case, we use the candidate set{τ (i)tx } = {−Ts,−0.75Ts, · · · , Ts}. The number of preamblesymbols is the same as the number of TX IQTM candidatepoints. Each preamble symbol has non-zero pilot tones withtone spacing of 6 tones at each side of the DC tone and thesenon-zero pilot tones are assigned alternately among the IQTMcandidate points as described in Section V.

Fig. 8 presents the estimation failure probability of thereference estimator in [27] for the RX-only IQTM case andthe proposed IQTM estimators for the TX-only and RX-onlyIQTM cases. We see that the proposed RX IQTM estimatorhas significant advantage compared to the reference estima-tor. Note that the reference estimator is developed withoutconsidering TX IQTM, so it cannot estimate the TX IQTM.We also observe that the estimation failure probabilities of theproposed estimators for the TX-only and RX-only IQTM casesare similar.

4 9 14 19

SNR (dB)

10-3

10-2

10-1

Est

imat

ion

failu

re p

roba

bilit

y

τrx

= 0, τtx

= 0

τrx

= 0.5Ts′ , τ

tx = 0

τrx

= 0.25Ts′ , τ

tx = 0.25T

s

τrx

= 0, τtx

= 0.5Ts

τrx

= 0.5Ts′ , τ

tx = 0.5T

s

3 taps IQI, τrx

= 0, τtx

= 0.25Ts

3 taps IQI, τrx

= 0.25Ts′ , τ

tx = 0.25T

s

Fig. 9. TX IQTM estimation failure probability of the proposedmethod insystems with both TX and RX IQTMs

Next, we evaluate the estimator performance in systemswith both TX and RX IQTMs for both cases of frequency-independent IQI and frequency-dependent IQI (3 taps). Forthe proposed estimator, we use the candidate set{τ (i)tx }= {−Ts,−0.75Ts, · · · , Ts} and {τ (i)rx } = {−T ′

s, −0.75T ′s,

· · · , T ′s}. Fig. 9 and Fig. 10 show the estimation failure

probabilities of the TX IQTM and the RX IQTM, respectively,for the proposed method under several IQTM settings. InFig. 10, we also include the performance of the referenceestimator whenτtx = 0.25Ts. We observe that the referenceestimator fails in the presence of TX IQTM. The referenceestimator relies on heuristic thresholds which are calculatedbased on RX-only IQTM candidates. In the presence of TXIQTM, the metric used in [27] is no longer valid and thoseheuristic thresholds are not applicable any more. Comparingwith Fig. 8, IQTMs at both the TX and RX cause largerestimation failure probability than individual IQTM cases.

4 9 14 19

SNR (dB)

10-3

10-2

10-1

100

Est

imat

ion

failu

re p

roba

bilit

y

τrx

= 0, τtx

= 0

τrx

= 0.5Ts′ , τ

tx = 0

τrx

= 0.25Ts′ , τ

tx = 0.25T

s

τrx

= 0, τtx

= 0.5Ts

τrx

= 0.5Ts′ , τ

tx = 0.5T

s

3taps IQI, τrx

= 0.25Ts′ , τ

tx = 0

3 taps IQI, τrx

= 0.25Ts′ , τ

tx = 0.25T

s

Ref., τrx

=0.25Ts′ , τ

tx=0.25T

sProposedmethod

Fig. 10. RX IQTM estimation failure probability of the proposed method insystems with both TX and RX IQTMs

But the proposed method still gives good performance. Theestimation failure probabilities of differentτtx andτrx valuesare similar.

4 9 14 19SNR (dB)

10-3

10-2

10-1

Est

imat

ion

failu

re p

roba

bilit

y

τTX

= 0.25Ts, τ

RX = 0, one preamble

τTX

= 0.25Ts, τ

RX = 0.25T

s′ , one preamble

τTX

= 0.25Ts, τ

RX = 0.5T

s′ , one preamble

τTX

= 0.25Ts, τ

RX = 0, two preambles

τTX

= 0.25Ts, τ

RX = 0.25T

s′ , two preambles

τTX

= 0.25Ts, τ

RX = 0.5T


Fig. 11. TX IQTM estimation failure probability for systems with both TXand RX IQTMs (2 GHz bandwidth)

Fig. 11 and Fig. 12 present the TX and RX IQTM estimationperformance of the proposed method in the system with 2GHz bandwidth in the presence of both the TX and RXIQTMs. We use{τ (i)rx } = {−T ′

s,−0.75T ′s, · · · , T ′

s} and{τ (i)tx }= {−0.5Ts,−0.25Ts, · · · , 0.5Ts}. Here, we test with oneor two preamble symbols only. We observe that using twopreamble symbols gives a reasonably good estimation failurerate of about1% for both TX and RX IQTMs.

Next, to illustrate performance improvement of MIMOaveraging, we consider a MIMO system withUT = UR = 4as an example. Suppose that the error probability mass func-tion before MIMO averaging is given by[P0, P1, P2, P3, P4]which correspond to probabilities of selecting5 candi-



12

Psuccess =P 40 +

(

4

1

)

P 30 (1− P0) + 3!P0(P1P2P3 + P1P2P4 + P1P3P4 + P2P3P4)

+

(

4

2

)

P 20

[

2(P1P2 + P1P3 + P1P4 + P2P3 + P2P4 + P3P4) +1

2

4∑

i=1

P 2i

]

. (77)

4 9 14 19SNR (dB)

10-3

10-2

10-1

Est

imat

ion

failu

re p

roba

bilit

y

τTX

= 0.25Ts, τ

RX = 0, one preamble

τTX

= 0.25Ts, τ

RX = 0.25T

s′ , one preamble

τTX

= 0.25Ts, τ

RX = 0.5T

s′ , one preamble

τTX

= 0.25Ts, τ

RX = 0, two preambles

τTX

= 0.25Ts, τ

RX = 0.25T


τTX

= 0.25Ts, τ

RX = 0.5T


Fig. 12. RX IQTM estimation failure probability for systems with both TXand RX IQTMs (2 GHz bandwidth)

date IQTM values with the normalized estimation error[0,−0.25, 0.25,−0.5, 0.5], respectively. From (74), the esti-mation success probability after the majority rule MIMOaveraging can be obtained as in (77). Taking the results inFig 9 as an example, the estimation failure probability of acertain DAC branch TX IQTM is about5.6 × 10−3 at 4dBSNR for each single estimator (before MIMO averaging). Thecorresponding probabilities obtained by simulation are givenby P0 = 0.9944, P1 ≈ P2 = 0.0024, andP3 ≈ P4 = 0.0004.Substituting these values into (77) gives the estimation failureprobability of the MIMO averaging as3.579 × 10−5 and thesimulation gives a matched result of3.6 × 10−5. Similarly,in Fig 10, the estimation failure probability of a certain ADCbranch RX IQTM is about6.1 × 10−3 at 4dB SNR for eachsingle estimator (before MIMO averaging). The correspondingprobabilities obtained by simulation are given byP0 = 0.9939,P1 ≈ P2 = 0.0026, and P3 ≈ P4 = 0.0004. With thesevalues, (77) gives the estimation failure probability of theMIMO averaging as4.188 × 10−5 and the simulation givesa matched result4.165 × 10−5. Thus, we observe that theestimator performance can be significantly improved by meansof MIMO averaging. As UE averaging plays the same role,its additional performance improvement can also be directlyunderstood.

D. Pilot Overhead and Estimator Complexity

As IQTMs typically do not change much, their estimationneed not be performed frequently. The pilot overhead dependson the number of TX IQTM candidate points, the phase

noise one-side ICI spread (as it determines the non-zero pilottone spacing), and the number of used subcarriers (as theICI level on a subcarrier depends on it). For example, forτtx,u1

/Ts ∈ [−0.5, 0.5] with 5 TX IQTM candidate pointsand 250 MHz bandwidth, we can use 5 pilot symbols insteadof 9 symbols and obtain similar estimation performance (theplots of which are omitted due to space limitation). For reliableIQTM estimation in the systems with the phase noise one-sideICI spread of two subcarriers, and the subcarrier spacing of1.44 MHz, the pilot overhead we use is one OFDM symbolof duration0.844µs (about 28 non-zero pilot tones) per TXIQTM candidate point for 250 MHz bandwidth (173 usedsubcarriers) and two OFDM symbols (with OFDM symbolduration of0.787µs) per 5 TX IQTM candidate points (about92 non-zero pilot tones per TX IQTM candidate point) for 2GHz bandwidth (1388 used subcarriers). The pilot overheadcould be less for milder phase noises.

Next, we provide the computational complexity of the jointTX and RX IQTM estimator after the RX DFT operation.Suppose fornτ,u1

TX IQTM candidate points,nτ,u2RX

IQTM candidate points, andUT TX DAC branches, thereare M OFDM symbols, each withK non-zero pilot tones.Then, from (64), (65), (66), and (67), the estimator requiresnτ,u2[8KM + (2KM + nτ,u1

)UT ] real multiplications ordivisions,nτ,u2[5KM +(4KM −nτ,u1

)UT ] real additions orsubtractions, andUT (nτ,u1

nτ,u2−1) real comparisons. For ex-

ample, for the considered system withNDFT = 256, UT = 1,τtx,u1

∈ [−0.5Ts, 0.5Ts], and τrx,u2∈ [−0.5T ′

s, 0.5T′s], we

can useK = 28, nτ,u1= 5, nτ,u2

= 5, M = 5 andthe estimator requires 7,025 real multiplications, 6,275 realadditions, and 24 real comparisons.

VIII. C ONCLUSIONS

We have studied the effects of IQTMs on mm-wave OFDMsystems with RF distortions and developed correspondingsignal model, novel pilot designs, IQTM estimators, and corre-sponding protocols for IQTM compensation. Our investigationresults are summarized below. IQTMs cause a cascaded IQIscenario with substantially different characteristics from theregular (commonly considered) IQI. These IQTMs can prolongthe effective channel length and significantly degrade the IRR(more pronounced at outer subcarriers) and the BER perfor-mance. An IQTM value larger than1/8 of the normal samplingperiod causes substantial BER degradation to the consideredsystem and the larger the IQTM value, the worse the BERis. Our estimators apply RX-side IQTM compensation foreach RX IQTM candidate and a disjoint set of specificallydesigned pilot tones for each TX IQTM candidate. Resolutionof IQTM candidate points should be at least 4 points per thenormal sampling period. Our estimation metric exploits the



13

fact that a larger IQTM causes more fluctuation of IRR acrosssubcarriers and it proves to be an effective metric. The pro-posed IQTM estimators and compensators can be implementedwithout requiring compensation of other RF distortions. Thesimulation results show that the proposed pilot designs andIQTM estimators substantially outperform the existing methodin terms of estimation performance and general applicability.Furthermore, the proposed MIMO averaging and UE averagingsubstantially enhance IQTM estimation performance. Overall,IQTM is an important issue for mm-wave systems and theproposed approach offers an effective solution.

APPENDIX

This appendix presents detailed steps in developing thefrequency domain signal models. First, substitutingF{Eq.(18)} into F{Eq. (22)}gives

Yu2(f) = MR,u2

(f){Ru2(f) ∗ΨR,u2

(f + αu2)}GR(f)

+ NR,u2(f){R∗

u2(−f) ∗Ψ∗

R,u2(−f + αu2

)}G∗R(−f).

(78)

Next, substitutingF{Eq. (14)} into F{Eq. (15)}gives

Ru2(f) =

VR∑

v2=1

{

UT∑

u1=1

VT∑

v1=1

ejθT,u1,v1Xu1,v1(f)Hu2,v2

u1,v1(f)

+Wu2,v2(f)

}

ejθR,u2,v2 . (79)

Similarly, substitute (2) intoF{Eq. (11)}and then into (79),(78), and (25), successively. Then, some lengthy but straight-forward manipulation gives (27) with its variables defined in(80)-(84) (shown at the top of next page).

Next, the Fourier transforms of (12), (13), (23), and (24)give

MT,u1,v1(f) =MT,u1,v1

(f)1 + e−j2πfτtx,u1

2

+NT,u1,v1(f)

1− e−j2πfτtx,u1

2, (85)

NT,u1,v1(f) =MT,u1,v1

(f)1− e−j2πfτtx,u1

2

+NT,u1,v1(f)

1 + e−j2πfτtx,u1

2, (86)

MR,u2(f) =MR,u2

(f)1 + e−j2πfτrx,u2

2

+N ∗R,u2

(−f)1− e−j2πfτrx,u2

2, (87)

NR,u2(f) =M∗

R,u2(−f)

1− e−j2πfτrx,u2

2

+NR,u2(f)

1 + e−j2πfτrx,u2

2. (88)

We observe thatMT,u1,v1(f), NT,u1,v1

(f), MR,u2(f), and

NR,u2(f) do not change much within a subcarrier bandwidth

(or within the main bandwidth ofγT (f) andγR(f)) since theunderlying frequency-dependent IQI filter frequency responsesdo not vary much within a subcarrier bandwidth. Using this

fact in (80), we obtain (89) and (90) shown at the next pageand (91)-(92) shown below:

Dµu2,u1

[n, l] ,

VT∑

v1=1

MT,u1,v1[l]ejθT,u1,v1Du2,u1,v1

[n, l] (91)

Dνu2,u1

[n, l] ,

VT∑

v1=1

NT,u1,v1[l]ejθT,u1,v1Du2,u1,v1

[n, l] (92)

Similarly, we obtain (93)-(95) shown at the next page. For theused subcarriers, we can express (89), (93), (94), and (95) inthe matrix form as

Aµµu2,u1

≈ΛMR,u2

(

Dµu2,u1

ΛD(τtx,u1) +Dν

u2,u1ΛM(τtx,u1

))

,

(96)

Aννu2,u1

≈ΛNR,u2

(

JDµ∗u2,u1

Λ∗M(τtx,u1

)J

+ JDν∗u2,u1

Λ∗D(τtx,u1

)J)

, (97)

Aνµu2,u1

≈ΛMR,u2

(

Dµu2,u1

ΛM(τtx,u1)J

+Dνu2,u1

ΛD(τtx,u1)J)

, (98)

Aµνu2,u1

≈ΛNR,u2

(

JDµ∗u2,u1

Λ∗D(τtx,u1

)

+ JDν∗u2,u1

Λ∗M(τtx,u1

)J)

. (99)

From (90), we know thatDu2,u1,v1[n, l] is negligible if n

is not aroundl. Consequently, this characteristic applies toDµ

u2,u1[n, l] andDν

u2,u1[n, l]. This renders that dominant terms

of Aµµu2,u1

and Aνµu2,u1

[n, l] are around their diagonal whilethose ofAνµ

u2,u1and Aµν

u2,u1are around their anti-diagonal;

other terms are insignificant. Define

Cu1, ΛD(τtx,u1

)Cu1+ΛM(τtx,u1

)JC∗u1, (100)

Ru2,

UT∑

u1=1

Dµu2,u1

Cu1+Dν

u2,u1JC∗

u1, (101)

Ru2, ΛµR,u2

Ru2+ΛνR,u2

JR∗u2, (102)

and successively substitute them. Next, substitute (96), (97),(98), and (99) into (28). Then, after appropriate re-arrangingthe terms in the resulting lengthy equation, we obtain

Yu2≈ ΛD(τrx,u2

)Ru2+ΛM(τrx,u2

)JR∗u2

+ ηu2. (103)

From (101), (102), (39), (39), we can also express

Ru2= BD

u2,u1Cu1

+BMu2,u1

C∗u1. (104)

Next, we convert (100), (104), and (103) into the real-valuedmatrix form and obtain

[Cu1]RI = Λtx,u1

[Cu1]RI, (105)

[Ru2]RI =

UT∑

u1=1

Qu2,u1[Cu1

]RI, (106)

[Yu2]RI = Λrx,u2

[Ru2]RI + [ηu2

]RI. (107)

Substituting the above equations successively gives (42).



14

Aµµu2,u1

[n, l] =

VT∑

v1=1

ejθT,u1,v1

[({[

Hu2u1,v1

(f){

[GT (f)γT (f − l∆f)MT,u1,v1(f)] ∗ΨT,u1

(f)}]

∗ΨR,u2(f + αu2

)}GR(f)ej2πft0MR,u2

(f))

∗ γR(f)]

f=fn, (80)

Aννu2,u1

[n, l] =

VT∑

v1=1

e−jθT,u1,v1

[({[

Hu2∗u1,v1

(−f){

[GT (f)γT (f − l∆f)N ∗T,u1,v1

(−f)] ∗Ψ∗T,u1

(−f)}]

∗Ψ∗R,u2

(−f + αu2)}

G∗R(−f)ej2πft0NR,u2

(f))

∗ γR(f)]

f=fn, (81)

Aνµu2,u1

[n, l] =

VT∑

v1=1

ejθT,u1,v1

[({[

Hu2u1,v1

(f){

[G∗T (−f)γ∗

T (−f − l∆f)NT,u1,v1(f)] ∗ΨT,u1

(f)}]

∗ΨR,u2(f + αu2

)}G∗R(−f)ej2πft0MR,u2

(f))

∗ γR(f)]

f=fn, (82)

Aµνu2,u1

[n, l] =

VT∑

v1=1

e−jθT,u1,v1

[({[

Hu2∗u1,v1

(−f){

[G∗T (−f)γ∗

T (−f − l∆f)M∗T,u1,v1

(−f)]

∗Ψ∗T,u1

(−f)}]

∗Ψ∗R,u2

(−f + αu2)}

G∗R(−f)ej2πft0NR,u2

(f))

∗ γR(f)]

f=fn, (83)

ηu2[n] =

[{

[Wu2(f) ∗ΨR,u2

(f + αu2)]GR(f)µR,u2

(f)ej2πft0

+[W ∗u2(−f) ∗Ψ∗

R,u2(−f + αu2

)]GR(f)ej2πft0 νR,u2

(f)}

∗ γR(f)]

f=fn. (84)

Aµµu2,u1

[n, l] ≈VT∑

v1=1

ejθT,u1,v1

[({[

Hu2u1,v1

(f){

[GT (f)γT (f − l∆f)MT,u1,v1[l]] ∗ΨT,u1

(f)}]

∗ΨR,u2(f + αu2

)}GR(f)ej2πft0MR,u2

[n])

∗ γR(f)]

f=fn

=MR,u2

[n]

2

[

(1 + e−j2πflτtx,u1 )Dµu2,u1

[n, l] + (1− e−j2πflτtx,u1 )Dνu2,u1

[n, l]]

(89)

Du2,u1,v1[n, l] ,

[{[(

{[γT (f − l∆f)GT (f)] ∗ΨT,u1(f)}Hu2

u1,v1(f)

)

∗ΨR,u2(f + αu2

)]

GR(f)ej2πft0

}

∗ γR(f)]

f=fn(90)

Aννu2,u1

[n, l] ≈ NR,u2[n]

2

[

(1− e−j2πflτtx,u1 )Dµ∗u2,u1

[−n,−l] + (1 + e−j2πflτtx,u1 )Dν∗u2,u1

[−n,−l]]

(93)

Aνµu2,u1

[n, l] ≈ MR,u2[n]

2

[

(1− ej2πflτtx,u1 )Dµu2,u1

[n,−l] + (1 + ej2πflτtx,u1 )Dνu2,u1

[n,−l]]

(94)

Aµνu2,u1

[n, l] ≈ NR,u2[n]

2

[

(1 + ej2πflτtx,u1 )Dµ∗u2,u1

[−n, l] + (1− ej2πflτtx,u1 )Dν∗u2,u1

[−n, l]]

(95)

Next, we present development for the expression ofξn in(48). From (46) and (47), we have

ξn ≈ Aνµu2,u1

[−n, n] +Aµνu2,u1

[−n, n]

Aµµu2,u1 [n, n]

. (108)

SinceMT,u1,v1[n] ≫ NT,u1,v1

[n], from (89), we have

Aµµu2,u1

[n, n] ≈ 0.5MR,u2[n](1 + e−j2πfnτtx,u1 )Dµ

u2,u1[n, n].(109)

Similarly, since MR,u2[n] ≫ N ∗

R,u2[−n], we have

MR,u2[n] ≈ 0.5MR,u2

[n](1 + e−j2πfnτrx,u2 ). Then, weobtain MR,u2

[−n]/M∗R,u2

[n] ≈ MR,u2[−n]/M∗

R,u2[n]

and NR,u2[−n]/M∗

R,u2[n] ≈ NR,u2

[−n]/M∗R,u2

[n] +1−ej2πfnτrx,u2

1+ej2πfnτrx,u2. By substituting (94), (95), and (109) into (108)

and applying the above approximations, we obtain (48).

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Hlaing Minn (S’99-M’01-SM’07-F’16) receivedB.E. degree in Electrical Engineering (Electronics)from the Yangon Institute of Technology, Yangon,Myanmar, in 1995; M.Eng. degree in Telecommu-nications from the Asian Institute of Technology,Pathumthani, Thailand, in 1997; and Ph.D. degreein Electrical Engineering from the University ofVictoria, Victoria, BC, Canada, in 2001. DuringJanuaryAugust 2002, he was a Postdoctoral Fellowwith the University of Victoria. He has been withthe University of Texas at Dallas since 2002 and

is currently a Full Professor. His research interests include wireless com-munications, statistical signal processing, signal design, cognitive radios,dynamic spectrum access and sharing, next-generation wireless technologies,and wireless health-care technologies.

Dr. Minn serves as an Editor-at-Large since 2016 and served as an Editorfrom 2005 to 2016 for the IEEE Transactions on Communications. He wasalso an Editor for International Journal of Communications and Networksfrom 2008 to 2015. He served as a Technical Program Co-Chair for theWireless Communications Symposium of the IEEE GLOBECOM 2014 andthe Wireless Access Track of the IEEE VTC, Fall 2009, as well as a TechnicalProgram Committee Member for several IEEE conferences.

Qi Zhan received the B.E. and M.S. degree inElectrical Engineering from Huazhong Universityof Science and Technology in 2008 and 2011; andthe Ph.D. degree in Electrical Engineering from theUniversity of Texas at Dallas, TX, USA, in 2016. Heis currently a senior system engineer with SamsungSemiconductor Inc. at San Diego, CA, USA. Hisresearch interests are in the broad areas of wirelesscommunications, signal processing, channel stateinformation and advanced transmission schemes.

Naofal Al-Dhahir (F’08) is Erik Jonsson Distin-guished Professor at UT-Dallas. He earned his PhDdegree in Electrical Engineering from Stanford Uni-versity. From 1994 to 2003, he was a principalmember of the technical staff at GE Research andAT&T Shannon Laboratory.

He is co-inventor of 41 issued US patents, co-author of over 350 papers and co-recipient of 4 IEEEbest paper awards. He is the Editor-in-Chief of IEEETransactions on Communications.

Huang Huang (S’09-M’11) received the B.Eng.and M.Eng. (Gold medal) degrees from the HarbinInstitute of Technology (HIT), Harbin, China, in2005 and 2007, respectively, and the Ph.D. degreefrom The Hong Kong University of Science andTechnology (HKUST) in 2011. He is currently aResearch Engineer with Huawei Technologies Com-pany, Ltd.

in-phase and quadrature timing mismatch estimation and

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