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Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2009, Article ID 181285, 10 pagesdoi:10.1155/2009/181285

Research Article

New Adaptive Method for IQ Imbalance Compensation ofQuadrature Modulators in Predistortion Systems

Hassan Zareian and Vahid Tabataba Vakili

Electrical Engineering Department, Iran University of Science and Technology (IUST), Tehran 16846-13114, Iran

Correspondence should be addressed to Hassan Zareian, [email protected]

Received 7 February 2009; Revised 13 May 2009; Accepted 23 June 2009

Recommended by George Tombras

Imperfections in quadrature modulators (QMs), such as inphase and quadrature (IQ) imbalance, can severely impact theperformance of power amplifier (PA) linearization systems, in particular in adaptive digital predistorters (PDs). In this paper,we first analyze the effect of IQ imbalance on the performance of a memory orthogonal polynomials predistorter (MOP PD),and then we propose a new adaptive algorithm to estimate and compensate the unknown IQ imbalance in QM. Unlike previouscompensation techniques, the proposed method was capable of online IQ imbalance compensation with faster convergence, andno special calibration or training signals were needed. The effectiveness of the proposed IQ imbalance compensator was validatedby simulations. The results clearly show the performance of the MOP PD to be enhanced significantly by adding the proposed IQimbalance compensator.

Copyright © 2009 H. Zareian and V. T. Vakili. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. Introduction

High data rate wireless communications systems use spec-trally efficient modulations such as the orthogonal frequencydivision multiplexing (OFDM). These systems are highlysensitive to the nonlinear distortion introduced by thepower amplifier (PA) due to its nonconstant envelope andhigh peak-to-average ratio (PAR) values [1]. Therefore, oneof the most challenging issues in designing PAs is thelinearity requirement. The linearization techniques improvethe linearity of PAs, and among them digital predistortionhas gained considerable attention in recent years because ofits low cost and reduced complexity [2].

A digital baseband predistortion system uses a quadra-ture modulator (QM) to transfer the predistorted basebandsignal to the RF. However, digital predistorter (PD) perfor-mance results are degraded by the IQ imbalance of QM in thedirect up-conversion transmitter chain [3]. Therefore, theeffects of IQ imbalance must be compensated, for example,by using digital signal processing.

The effect of IQ imbalance in QM on the complexgain PD with a look-up table (LUT) and the resultingperformance degradation have been previously noted in [3].

To date, several techniques have been proposed to estimateand compensate the QM imperfections in direct-conversiontransmitters [4–11]. The method presented in [4] considersa digital postcorrection at the receiver side. To estimateand compensate the IQ imbalance, some methods rely onspecial training signals [5–7]. The QM imperfections aloneare compensated with an adaptive least mean square (LMS)algorithm in [8]. In [9], a two-step compensation approachis proposed, where first the QMC parameters are adjustedby bypassing the PA and then the PD parameters are found.The method in [10], based on minimization of out-of-bandpower, jointly compensates amplifier nonlinearity and QMerrors. An adaptive technique that jointly compensates thePA nonlinearity and QM errors and does not need anyextra feedback loop for QM/QDM error compensation isproposed in [11].

To the authors’ knowledge, only a few previously pub-lished reports have considered IQ imbalance compensationin predistortion linearization systems [5, 7, 9–11]. Thedrawbacks of the existing compensation techniques arethe following: (1) offline training using special calibrationsignals is needed [5, 7], (2) interruption of the regulartransmission mode is required and generally, time-varying

2 EURASIP Journal on Advances in Signal Processing

y[n]w[n]z[n]

e[n]

x[n] Poweramplifier

Quadraturemodulator

Predistorter

Predistortertraining

Quadraturedemodulatorz[n]

+−

Figure 1: Block diagram of a direct-conversion transmitter with a predistortion linearization system.

QM errors, such as those caused by temperature variation,are impossible to track [9], (3) the convergence speed is veryslow and the linearization performance of the PDs is notgood [10], and (4) high computational and implementationcomplexity is involved in [11], and its application of theproposed compensation method is limited to use with thepolynomial PDs. The proposed method in this paper solvesthese drawbacks as follows: (1) no special calibration trainingsignals are needed, (2) the online calibration technique usingonly the regular transmit signal is capable of followingthe changes in the IQ imbalance parameters, (3) by usingthe recursive least squares (RLSs) adaptive algorithm theachieved convergence is faster as well as computationallymore economical, and (4) it is not dependent on thepredistortion systems and can be used in conjunction withall kinds of PDs. Moreover, the linearization performance ofPDs can be significantly improved.

In summary, the contributions of this paper are thefollowing. First, the effects of IQ imbalance in QM on theperformance parameters of a memory orthogonal polynomi-als predistorter (MOP PD), such as the convergence behaviorof PD coefficients adaptation, input amplitude to outputamplitude (AM/AM) and input amplitude to input–outputphase difference (AM/PM) characteristics of the linearizedPA, adjacent channel power ratio (ACPR), and error vectormagnitude (EVM), were investigated. Second, an adaptiveIQ imbalance compensator to enhance the PA linearizationsystem performance is proposed.

The remainder of this paper is organized as follows.Section 2 briefly describes the problem of IQ imbalance andthe predistortion compensation scheme. In this section wealso introduce a recursive algorithm based on the MOP PDas the adaptive linearization technique for the memory PA. InSection 3, we study the influence of this QM imperfection onthe MOP PD performance using simulation. To alleviate theIQ imbalance, we propose an adaptive digital algorithm fora QM compensator (QMC) in Section 4. Section 5 presentscomputer simulation results in order to evaluate the MOPPD performance in conjunction with the proposed IQimbalance compensator. Finally, conclusions are given inSection 6.

Throughout the paper, the transpose and conjugateoperations are denoted by (·)T and (·)∗, respectively.

2. SystemOverview

A simplified block diagram of a direct conversion transmitterwith a predistortion linearization system is presented in

Figure 1. Note that the digital-to-analog converter (DAC),the reconstruction filter in feedforward path, as well as theantialiasing filter and the analog-to-digital converter (ADC)in the feedback path were ignored for simplicity. Moreover,all signals were treated as complex baseband.

The baseband input first goes through the PD, theoutput distorted signal of which is converted into an analogsignal by the DAC and further up-converted in a QM tothe RF frequency. Then, the signal is amplified by PA andtransmitted. The feedback path is used for PD training. Sincewe are interested in a digital baseband PD, the passband PAoutput is translated to its equivalent digital baseband formusing the quadrature demodulator (QDM) and ADC.

Unfortunately, there are imperfections, such as the IQimbalance in the analog QM and QDM, which can affectthe PD performance greatly [12]. In this paper, we assumedan ideal QDM to form the complex baseband signal, andthe IQ imbalance in QM was assumed to be invariant withfrequency.

2.1. IQ Imbalance Compensation. Let gm and ϕm denotegain and phase imbalances between the in-phase (I) andquadrature (Q) branches in the QM, respectively. Usingmatrix notation, the distortions caused by the IQ imbalancein QM can be modeled by [13]⎡⎣wI[n]

wQ[n]

⎤⎦ =m

⎡⎣zI[n]

zQ[n]

⎤⎦, m =⎡⎣1 −gm sin

(ϕm

)0 gm cos

(ϕm

)⎤⎦,

(1)

where w[n] = wI[n] + jwQ[n] is the complex basebandrepresentation of the QM’s output distorted signal. Theterms zI[n] and zQ[n] are the inphase and quadraturecomponents of the baseband signal of z[n], respectively.

The IQ imbalance compensation based on digital predis-tortion is more convenient, less expensive, and more preciseto implement. As shown in Figure 2, having obtained theestimates of the gain and phase imbalances, that is, gm andϕm, respectively, we use them to compensate the zI[n] andzQ[n] baseband signals before applying them to the QM.Therefore, we can obtain the compensated uI[n] and uQ[n]baseband signals by

⎡⎣uI[n]

uQ[n]

⎤⎦ = c

⎡⎣zI[n]

zQ[n]

⎤⎦, c =m−1 =⎡⎢⎣1 tan

(ϕm

)0

1gm cos

(ϕm

)⎤⎥⎦.(2)

EURASIP Journal on Advances in Signal Processing 3

w1[n]

wQ[n]

QMQMC

u1[n]

uQ[n]

z1[n]

zQ[n] 1/gm cos(ϕm)

−

gm cos(ϕm)

g msi

n(ϕ

m)

g msi

n(ϕ

m) ⎡⎣1 −gm sin(ϕm)

0 gm cos(ϕm)

⎤⎦⎡⎢⎢⎣

1 gm tan(ϕm)

01

gm cos(ϕm)

⎤⎥⎥⎦

Figure 2: QM baseband model and digital predistortion structure of QMC.

Perfect compensation (w[n] = z[n]) can be achieved ifthe parameters of the QMC are equal to the real IQ imbalanceparameters of the QM (ϕm = ϕm, gm = gm).

Since gm and ϕm are unknown in practice, they have to beestimated. In this paper a new IQ imbalance compensationscheme is presented, in which the unknown analog imbal-ance parameters are estimated digitally, without the need forany calibration or training signal.

2.2. An Adaptive MOP PD. In this section, we introducean adaptive identification algorithm for the MOP PD withan indirect learning structure, as shown in Figure 1. Thebenefit of this structure is that we can obtain directly thePD parameters instead of assuming a model for the PA,estimating the PA parameters, and then constructing itsinverse [14].

In the absence of IQ imbalance, we would have w[n] =z[n]. Ideally, the output of this system should be y[n] = x[n],when z[n] equals z[n]; that is, it makes the error term e[n] =z[n]−z[n] = 0. The output–input relation of the PD trainingblock can be described by a memory orthogonal polynomial(MOP) model [15]:

z[n] =K∑k=1Odd

Q∑q=0

αkqψkq[n], (3)

where the memory length of the PD and the polynomialorder are equal toQ andK , respectively, and αkq are complex-value coefficients. The memory orthogonal polynomial basisfunction ψkq[n] for a complex Gaussian process with zeromean and σ2

y variance, such as an OFDM signal, is defined[16] as

ψ2m+1,q[n] =m∑k=0

(−1)m−k

σ2k+1y

√m + 1

(k + 1)!

⎛⎝mk

⎞⎠φ2k+1,q[n], (4)

where φkq[n] � |y[n− q]|k−1y[n − q] is the conventionalmemory polynomial basis function. It is noted that y[n] isnon-Gaussian, before identification and predistortion, evenif x[n] is Gaussian. However, the orthogonal polynomials canalleviate the numerical instability problem associated with

the conventional polynomials even if the signal distributiondeviates from complex Gaussian [16].

We used an adaptive approach to obtain the predistortercoefficients. In the remainder of this paper the predistorterbased on this solution will be called MOP PD. MOPPD employs the RLS adaptive algorithm to track time-varying PA characteristics caused by changes in operatingtemperature, component aging, and environmental andmanual variations. Therefore, adaptation of the predistorterwas achieved by comparing z[n] with z[n] and adjusting thepredistorter parameters αkq by minimizing the criterion:

J[n] =n∑l=1

λn−l|e[l]|2 =n∑l=1

λn−l∣∣z[l]− z[l]

∣∣2, (5)

where λ is the forgetting factor. Using matrix notation we canreformulate (5) as

z[n] = αTψ[n], (6)

where

α = [α10, . . . ,αK0, . . . ,α1Q . . . ,αKQ

]T ,

ψ[n] = [ψ10[n], . . . ,ψK0[n], . . . ,ψ1Q[n], . . . ,ψKQ[n]

]T.

(7)

The RLS adaptive algorithm to extract the predistortercoefficient vector α is initialized [17] by

P[0] = δ−1I,

α[0] = [0.1, 0, . . . , 0]T ,(8)

where δ is a small positive constant, α[0] is a K ′(Q + 1)column vector, where K ′ is (K + 1)/2. In order to updatethe coefficient vector of α at every iteration, n = 1, 2, . . . , thecalculation of the following equations is required

e[n] = z[n]− z[n] = z[n]− αT[n− 1]ψ[n],

K[n] = P[n− 1]ψ∗[n]λ + ψT[n]P[n− 1]ψ∗[n]

,

α[n] = α[n− 1] + K[n]e[n],

P[n] = 1λP[n− 1]− 1

λK[n]ψT[n]P[n− 1].

(9)


0 100 200 300 400 500

Number of iterations

0

0.05

0.1

0.15

0.2

0.25

MO

PP

Dco

effici

ents

adap

tati

on

MOP PD (K = 5,Q = 1), IBO = 9 dB

|α10|

|α30| |α50|

Solid line �� No IQ imbalanceDashed line �� IQ imbalance (gain = 3%, phase = 3◦)

Figure 3: Convergence behavior of the MOP PD coefficientsadaptation with and without IQ imbalance.

3. Effects of IQ Imbalance in QM onMOP PD Performance

The effect of IQ imbalance on the linearization performanceof the MOP PD is presented in this section. We havecarried out system simulations based on equivalent modelsto evaluate the linearization performance of the MOP PDin the presence of IQ imbalance in terms of convergencebehavior, AM/AM, and AM/PM curves and EVM and ACPRperformances.

The performance of the proposed scheme was evaluatedusing the 2K-mode OFDM signal based on the Europeandigital video broadcasting terrestrial (DVB-T) standard [18].In addition, the PA with memory effect was assumed to obeya Wiener model, implemented as a 3-tap FIR filter with thecoefficients [0.7692, 0.1538, 0.0769] [19], followed by a SalehTWTA model [20]. The operating point of the PA is usuallyidentified by the back-off. The input back-off (IBO) and theoutput back-off (OBO) are defined as

IBO = 10 logA2s

σ2x

,

OBO = 10 logA2o

σ2y

(10)

where A2s denotes the input saturation power of the PA,

and A2o presents the maximum PA output power. Moreover,

σ2x and σ2

y are the mean power of the input and outputsignals, respectively. Throughout the simulations, the IBOwas chosen as 9 dB, which is equivalent to an OBO of 4.7 dB.We have set the parameters of the MOP PD to K = 5 andQ = 1 for the order of orthogonal polynomial and lengthof memory, respectively. Moreover, we chose a forgettingfactor of λ = 0.95 and used an oversampling factor of 8 forsimulations.

Convergence properties were analyzed through RLScoefficients behavior. As shown in Figure 3, the IQ imbalanceaffects the convergence behavior of the MOP PD coefficientsadaptation and caused the coefficients of the system estima-tor not to converge to their respective optimum values.

Figure 4 gives the variation histogram of the first MOPPD coefficient (α10) during the adaptation algorithm forseveral cases. In addition to the ideal IQ balance case, thefollowing two IQ imbalance cases were considered: case A,with gm = 1.03, ϕm = 3◦ and case B, with gm = 1.1, ϕm = 10◦.From this figure, it could be concluded that the variance ofα10 in the presence of IQ imbalance has increased.

The influence of IQ imbalance on the linearization per-formance of the MOP PD can be checked out by comparingthe AM/AM and AM/PM relationship between the inputand output of the linearized PA. This will demonstratethe IQ imbalance effect on the accuracy of the MOP PDin linearizing the PA and increasing memory effects bythickening the curves.

The AM/AM and AM/PM curves obtained with predis-tortion for the three IQ imbalance cases are shown in Fig.5. We note that without IQ imbalance, the AM/AM andAM/PM curves were thinner, and the output amplitude hasbeen thoroughly linearized. We can also use the normalizedmean square error (NMSE) as a figure of merit to measurethe linearity of the power amplifier, where it is defined by

NMSE = 10 log10

(∑k

∣∣x[k]− y[k]∣∣2∑

k |x[k]|2)

, (11)

where y[n] is the linearized PA baseband output and, x[n]is its corresponding input (desired output). The measuredresults of NMSE are shown in Figure 5, where the NMSEperformance of the linearized PA was degraded by 11 and20 dB for cases A and B, respectively, in comparison with anideal IQ balance case.

In the following, we evaluate the EVM performance ofthe MOP PD in the presence of IQ imbalance in QM. TheEVM criterion is of fundamental importance as it is linkedwith the symbol error rate (SER) or bit error rate (BER)performance. We used a constellation diagram to quantifythe deformation of the constellation at the output of the PAwith an EVM parameter defined by:

EVM =

√√√√√minr0,r1

∑k

∣∣∣dx[k]−((

dy[k]− r1

)/r0

)∣∣∣2

∑k |dx[k]|2 ,

(12)

where dx[k] and dy[k], respectively, denote a sequence of 16-QAM-OFDM signals and their corresponding demodulatedvalues. The parameters r0 and r1 were optimized to compen-sate for rotation and offset of the constellation.

Figure 6 shows the effects of IQ imbalance on the 16-QAM-OFDM constellation at the linearized PA output forthe differential values of gain and phase imbalances as wellas its corresponding EVM value. This result indicates thatIQ imbalances have a large impact on the EVM performanceof the MOP PD. Specifically, the EVM increased from about


0.24 0.245 0.25 0.255 0.26

|α10|

0

600

1200

1800

2400

Nu

mbe

rof

sam

ples

No IQ imbalance

(a)

0.2 0.22 0.24 0.26 0.28

|α10|

0

600

1200

1800

2400

Nu

mbe

rof

sam

ples

IQ imbalance(gain = 3%, phase = 3◦)

(b)

0.2 0.25 0.3

|α10|

0

600

1200

1800

2400

Nu

mbe

rof

sam

ples


(c)

Figure 4: The variation histogram of the first MOP PD coefficient with and without IQ imbalance.

0 0.2 0.4 0.6 0.8 1

Normalized input amplitude

−0.2

0

0.2

0.4

0.6

0.8

1

Ou

tpu

tam

plit

ude

and

phas

e(r

ad)

No IQ imbalance

AM/AM distortion

AM/PM distortion

NMSE = −40.6559 dB

(a)

0 0.2 0.4 0.6 0.8 1


−0.2

0

0.2

0.4

0.6

0.8

1

Ou

tpu

tam

plit

ude

and

phas

e(r

ad)


AM/AM distortion

AM/PM distortion

NMSE = −31.5921 dB

(b)

0 0.2 0.4 0.6 0.8 1


−0.2

0

0.2

0.4

0.6

0.8

1

Ou

tpu

tam

plit

ude

and

phas

e(r

ad)


AM/AM distortion

AM/PM distortion

NMSE = −20.4148 dB

(c)

Figure 5: The AM/AM and AM/PM curves of the linearized PA with and without IQ imbalance.

0.7% without IQ imbalance to approximately 2.2% and 8.1%for cases A and B, respectively.

Finally, to qualify the effect of IQ imbalance on the ACPRperformance of the MOP PD, we analyzed the power spectraldensities (PSDs) of the input and output signals of thelinearized PA. The ACPR characterizes the spectral regrowthand is defined as the ratio of power radiated into an adjacentchannel to the power in the main channel:

ACPR = 10 log10

⎛⎜⎝∫ fadjmax

fadjminY(f)df∫ fmainmax

fmainminY(f)df

⎞⎟⎠, (13)

where f denotes the frequency and Y( f ) the PSD of thePA output signal. The normalized PSDs of the output signaland the corresponding ACPR values computed by (13) wereshown in Figure 7. For comparison, the PSD of the originalinput signal and the output PSD of the system employingan ideal PD without IQ imbalance also are shown. As notedbefore, the IQ imbalance in QM can dramatically degrade

the ACPR performance of the MOP PD. For instance, theresulting ACPR values for case A (−41 dB) and B (−32 dB)are larger than without IQ imbalance (−52 dB).

This section illustrated the effect of IQ imbalance on theMOP PD performance and the need for a method to achieveIQ imbalance compensation, which will be presented in thenext section.

4. Proposed IQ Imbalance Compensation

A complete block diagram of the system model for theproposed IQ compensator in conjunction with the MOP PDis shown in Figure 8.

Based on the baseband input u[n] and the passbandoutput of the QM, we developed a method to estimate theQM model parameters and construct the associated QMC bythe digital predistortion technique (see Figure 2). An inversememory PA model (the MOP PD) is also estimated using theRLS algorithm to achieve PA linearization.


−5 −3 −1 1 3 5

In-phase

−5

−3

−1

1

3

5

Qu

adra

ture

MOP PD (K = 5,Q = 1), IBO = 9 dB

w/PD (no IQ imbalance), EVM = 0.69612%w/PD (case A), EVM = 2.2082%w/PD (case B), EVM = 8.1264%Ideal PD + PA, EVM = 0.2855%

Figure 6: 16-QAM-OFDM constellation and the correspondingEVM value of simulated output of the linearized PA with andwithout IQ imbalance.

Two parameters, gm and ϕm, are to be estimated usingany algorithm, depending on the desired performance inthe asymmetric model of Figure 2, with the LMS and RLSalgorithms being the most obvious ones resulting in differentconvergence speeds and computational complexities. TheRLS algorithm was used in this paper to provide fastconvergence [17].

The actual signal applied to the PA was a bandpass signalwith a complex envelope produced by the QM, which is itselfpreceded by the QMC.

The ideal envelope detector (ED) in Figure 8 can becharacterized using an ideal square-law envelope detectorfollowed by the ideal ADC as shown in Figure 9. Thediscrete output baseband signal s[n] is related to the complexenvelope of the RF input signal by

s[n] = |w[n]|2. (14)

In the following, we derived an RLS algorithm forestimation and compensation of the IQ imbalance based onthe envelope of the output signal at QM. The adaptation ofthe RLS algorithm is to minimize the criterion J on the erroreqm[n] = s[n]− s[n],

J[n] =n∑l=1

λn−l∣∣∣eqm[l]

∣∣∣2, (15)

where λ is the forgetting factor. We first rewrite (1) in linearform:

w(t) = [zI(t)− gm sin

(ϕm

)zQ(t)

]+ jgm cos

(ϕm

)zQ(t).

(16)

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency normalized by symbol frequency

−70

−60

−50

−40

−30

−20

−10

0

Pow

ersp

ectr

alde

nsi

ty(P

SD)

(dB

)

MOP PD (K = 5,Q = 1), IBO = 9 dB

Original, ACPR = −56.1243 dBw/PD (no IQ imbalance), ACPR = −52.1366 dBw/PD (case A), ACPR = −41.2066 dBw/PD (case B), ACPR = −32.1631 dBIdeal PD + PA, ACPR = −53.3925 dB

Figure 7: Normalized PSD and the corresponding ACPR valueof the simulated input and output of the linearized PA with andwithout IQ imbalance.

z[n]x[n]

e[n]

y[n]

u[n]

s[n]

QMmodeling

PDtraining

PAQM

QDM

QMCPD

ED

s[n]−

−

eqm[n]

y(t)w(t)

z[n]

Figure 8: The proposed adaptive compensator structure in con-junction with the MOP PD.

s(t)Square-law

detector

Ideal envelope detector (ED)

s[n]ADCw(t)

Figure 9: Block diagram of internal ideal ED structure.

By substituting (16) into (14) through some manipula-tions, (14) can be rewritten as

s[n]− z2I [n] = ζTχ[n], (17)


0 100 200 300 400 500


0

0.2

0.4

0.6

0.8

1

1.2

QM

mod

elco

effici

ents

adap

tati

on


ζ1

ζ2

(a)

deg

rad

0 100 200 300 400 500


0

1

2

3

4

5

6

Para

met

ers

esti

mat

ion

ofIQ

imba

lan

ce


gmϕm

(b)

Figure 10: Convergence behavior of (a) QM model coefficients adaptation (b) the corresponding parameters estimation of IQ imbalance.

0 100 200 300 400 500


0

0.1

0.2

MO

PP

Dco

effici

ents

adap

tati

on

MOP PD (K = 5,Q = 1), IBO = 9 dB

|α10||α30||α50|

(a)

0 100 200 300 400 500


0

0.02

0.04

0.06

MO

PP

Dco

effici

ents

adap

tati

on

MOP PD (K = 5,Q = 1), IBO = 9 dB

|α11||α31||α51|

(b)

Figure 11: Convergence behavior of the MOP PD coefficients adaptation.

where ζ and χ[n] are complex vectors defined by

ζ = [ζ1, ζ2]T ,

χ[n] =[z2Q[n],−2zI[n]zQ[n]

]T.

(18)

The QM model coefficients of ζ1 and ζ2 are related to thegain and phase imbalances as follows:

ζ1 = g2m,

ζ2 = gm sin(ϕm

).

(19)

Note that in order to obtain the parameters of the IQimbalance in the asymmetric QM model, we have applieda two-step technique for each instant of n. During the firststep, ζ1 and ζ2 were estimated, and for the second step, thegm and ϕm were derived and reserved as the parameters ofcompensator for the next instant, n + 1.

By defining s[l] � s[l] − z2I [l] and s[l] � s[l] − z2

I [l], wecan modify the criterion J in (15) as

J[n] =n∑l=1

λn−l∣∣∣s[l]− s [l]

∣∣∣2. (20)


0 500 1000 1500 2000


10−6

10−5

10−4

10−3

10−2

10−1

100

Rel

ativ

eam

plit

ude

erro

r

IQ imbalance (gain = 3%, phase = 3◦), IBO = 9 dB

QM + PAMethod in [11]w/ PD, w/ QMC

(a)

0 500 1000 1500 2000


10−6

10−5

10−4

10−3

10−2

10−1

100

Ph

ase

erro

r


QM + PAMethod in [11]w/ PD, w/ QMC

(b)

Figure 12: Instantaneous errors between the input and output of the linearized PA with the proposed IQ imbalance compensator (a) relativeamplitude error (b) phase error.

Therefore, we have the following adaptive algorithm forupdating ζ[n] for every n = 1, 2, . . ., directly computed from(21)–(24):

eqm[n] = s[n]−s[l] = s[n]−z2I [n]−ζT[n−1]χ[n], (21)

K[n] = P[n− 1]χ∗[n]λ + χT[n]P[n− 1]χ∗[n]

, (22)

ζ[n] = ζ[n− 1] + K[n]eqm[n], (23)

P[n] = 1λP[n− 1]− 1

λK[n]χT[n]P[n− 1]. (24)

The RLS adaptive algorithm was initialized by setting

P[0] = δ−1I,

ζ[0] = [0.1, 0]T ,(25)

where δ is a small positive constant. The IQ imbalanceparameters calculated from (2) at instant n were applied asthe new values of proposed QMC for instant (n + 1):

gm[n + 1] =√ζ1[n],

ϕm[n + 1] = arcsin

⎛⎝ ζ2[n]√ζ1[n]

⎞⎠. (26)

5. Simulation Results

In this section, we evaluate the linearization performance ofthe MOP PD in conjunction with the proposed IQ imbalancecompensator using simulation with parameters similar tothose of Section 3. For both RLS algorithms, the forgettingfactor we used was 0.95. For comparison, we also consideredanother IQ imbalance compensation method introduced in[11], which jointly estimates the PD and QMC parameters.

−5 −3 −1 1 3 5

In-phase

−5

−3

−1

1

3

5

Qu

adra

ture


QM + PA, EVM = 14.8339%Method in [11], EVM = 0.61325%w/ PD, w/ QMC, EVM = 0.55266%Ideal PD + PA, EVM = 0.2855%

Figure 13: EVM performance of the MOP PD without and with theproposed QMC.

First, the convergence behavior of the proposed QMCand MOP PD adaptive algorithms was studied for an IQimbalance case with gm = 1.03, and ϕm = 3◦.

Figure 10 shows the convergence behavior of QM modelcoefficients adaptation, and the corresponding parametersestimation of IQ imbalance as a function of the number ofiterations. According to these figures, it was clear that themerit of the proposed compensator was rapid convergenceof parameters within fewer than just 100 samples. As shownin Figure 10(a), the estimated values of gain and phaseimbalance were extremely near to the true values of the given


−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency normalized by symbol frequency

−70

−60

−50

−40

−30

−20

−10

0

Pow

ersp

ectr

alde

nsi

ty(P

SD)

(dB

)


Original, ACPR = −56.1243 dBQM + PA, ACPR = −34.8659 dBMethod in [11], ACPR = −51.9977 dBw/ PD, w/ QMC, ACPR = −52.0022 dBIdeal PD + PA, ACPR = −53.3925 dB

Figure 14: ACPR performance of the MOP PD with and withoutthe proposed IQ imbalance compensator.

IQ imbalance parameters (gm = 1.03, ϕm = 3◦). The rapidand reliable convergence of the MOP PD coefficients towardtheir optimal values by adding the proposed adaptive IQimbalance compensator is shown in Figure 11.

Figure 12 shows the instantaneous errors (amplitude andphase) as a function of the number of iterations, with andwithout a joint compensator. After convergence, both theproposed method and the method of [11] reduced the errorsignificantly and showed fast convergence, but the proposedmethod was still somewhat more accurate and reliable thanthe method in [11].

To evaluate EVM performance of the MOP PD, the trans-mitted signal constellation with the proposed correctionmethod is shown in Figure 13. For comparison, the EVMperformance of the method in [11] and an ideal PD withoutIQ imbalance (i.e., soft limiter) are presented as well. It isnoted that the EVM performance improves from about 2.2%(see Figure 6) to approximately 0.55% (slightly less than themethod in [11]) after the proposed QMC method is applied,obtaining a residual vector error of less than 0.3% of the idealcase of linear amplification without IQ imbalance.

Figure 14 compares the normalized PSD of the outputsignal and the corresponding ACPR value for a linearizedPA with and without the proposed adaptive correctionmethod. From Figure 14, it is evident that with the proposedcompensation techniques or method of [11], the ACPRperformance improved from −41 (see Figure 7) to −52 dB,an improvement of about 11 dB.

Based on the above discussion, the digital IQ imbalancecompensator similar to the method in [11] was an effectivetechnique for improving the linearization performance ofthe MOP PD. In the following, the complexity of both

the proposed method and the method in [11] is evaluated.Although the proposed method used an extra loop forQM compensation that has only an envelope detector foradaptation, unlike the method in [11], it estimated the QMCparameters and the MOP PD coefficients directly, and thelength of coefficients vector was deduced from 2K ′(Q +1) = 12 (see [11]) to K ′(Q + 1) + 2 = 8 by usingthe proposed method. Consequently, the computationalcomplexity and implementation of the proposed algorithmwere quite modest.

The major advantage of the proposed method over thatin [11] is its independence from the predistortion systemsthat can be used in conjunction with any variety of PDs.

6. Conclusions

In this paper, the effects of QM imperfections, such as gainand phase imbalances on the MOP PD performance, wereexamined through computer simulations. The results showthat the MOP PD does not achieve sufficient compensationof the PA effects in the presence of IQ imbalance in QM.Therefore, we have proposed an adaptive algorithm toestimate and correct the IQ imbalance of QM and to enhancethe MOP PD performance.

The proposed compensation technique was used inconjunction with the MOP PD performance, and usingcomputer simulations we showed that the MOP PD achievessignificant improvements over the case without any IQimbalance compensation. For instance, with a gain imbal-ance of approximately 3% and a phase imbalance of around3◦, the EVM and the ACPR performances of the MOP PDimproved by 3% and 8 dB, respectively, after applying theproposed correction method.

Unlike previously published techniques, the introducedadaptive method is capable of online IQ imbalance compen-sation without using special training signals. It presents fasteradaptation and more reliable convergence than previouslypublished methods, which enables it to track the possiblevariations of IQ imbalance parameters of QM.

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