Blind Adaptive I/Q Imbalance Compensation
Algorithms for Direct-Conversion Receivers Woook Nam,Member,IEEE,Heejin Roh,Jungwon Lee,Senior Member,IEEE,and Inyup Kang,Member,IEEE
Abstract—Blind adaptive I/Q imbalance compensation has be-come more popular during the last decade due to its simplicity and need for no training data.In existing work,it has been pointed out that a statistical property of a signal,which is referred to as the properness condition,can be ud for blind I/Q imbalance compen-sation.In this letter,the equi-absolute variance condition as well as the properness condition are ud to propo two blind adaptive I/Q imbalance compensation algorithms bad on the well-known LMS and RLS adaptation algorithms.The performances of the propod algorithms are evaluated through simulations,and it is shown that the propod algorithms provide nice convergence be-haviors and high image rejections.
Index Terms—Adaptivefiltering,blind estimation,direct-con-version receiver,I/Q imbalance,LMS,RLS.
I.I NTRODUCTION
A LONG with the increasing demand for low-power
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hand-held communication devices,the need for denly integrated single-chip receiver systems has been growing rapidly.As it is difficult to meet this need with the tradi-tional superheterodyne architecture,the direct-conversion architecture is regarded as an alternative solution[1].Despite its desirable characteristics and simplicity,however,the di-rect-conversion receiver suffers from imbalance between the in-pha(I)and quadrature-pha(Q)branches in practice, which results in the image signal interfering with the desired signal.Various approaches have been studied for compensating the I/Q imbalance through digital signal processing([2]–[4]and references therein).Though the I/Q imbalance is often modeled to be frequency dependent,especially for wideband systems [4],we assume a frequency-independent ca for simplicity. In this letter,we focus on blind adaptive I/Q imbalance compensation,who key feature is that no known training data is required,since only the statistical properties of the desired signal are needed.In particular,for I/Q imbalance compensation,the properness condition,which generally holds for many different communication signals in the abnce of I/Q imbalance,can be ud as the required statistical property [3],[4].In addition to the properness condition,we include the equi-absolute variance condition,which states that the vari-ances of the absolute values of the I and Q signals are the same.
Manuscript received March09,2012;revid April20,2012;accepted May 22,2012.Date of publication
June05,2012;date of current version June19, 2012.The associate editor coordinating the review of this manuscript and ap-proving it for publication was Prof.Jia-Chin Lin.
The authors are with the Mobile Solutions Laboratory,Samsung Information Systems America,San Diego,CA92121USA(e-mail:woook.;;jungwon2.; inyup.).
Digital Object Identifier10.1109/LSP.2012.2202902As with the properness condition,it can be easily shown that most practical communication signals satisfy the equi-absolute variance condition in the abnce of I/Q imbalance.Relying on the two conditions,we show that the I/Q imbalance compensation problem can be reformulated as a pair of linear minimum mean-square error(MMSE)estimation problems. Even though no prior information on the input signal statistics is available,the MMSE estimation problems can be tackled efficiently using well-known adaptivefiltering algorithms[5]. Two different blind adaptive I/Q imbalance compensation al-gorithms are propod bad on the least-mean-squares(LMS) and recursive least-squares(RLS)algorithms.Since the algorithms are built on well-founded adaptivefilter theory,the convergence to the ideal solution is almost always guaranteed. The performances of the propod algorithms are also assd through numerical simulations,and it is shown that the pro-pod algori
thms provide fast adaptation and high image signal rejection ratios.
II.S IGNAL M ODEL AND A LGORITHM D ERIV ATION
In a direct conversion receiver,the baband I/Q signals are obtained by mixing the RF signal with the sine and cosine tones of the carrier frequency,respectively,and applying low-passfil-tering.In the abnce of I/Q imbalance,the ideal baband re-ceived signal at sample time appears as
(1)
where and are the real-valued I and Q signals,re-spectively,and.We assume that the ideal baband signal is a zero-mean complex random variable satisfying the properness condition
(2)
where denotes statistical expectation.In addition to the properness condition,we impo another condition that the ideal signal should satisfy
(3)
where denotes the variance of a random variable.We refer to(3)as the equi-absolute variance condition.Though we will not explicitly prove it in this letter,it can be easily shown that the equi-absolute variance condition,as well as the properness condition,holds for a variety of baband or IF and single-or multi-carrier signals employing complex-valued al-phabets,such as quadrature-amplitude modulation(QAM)and -pha-shift keying(PSK)(for).Moreover,even in
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various channel-fading situations and in the prence of car-rier frequency offt,the properness and equi-absolute variance conditions are well established[3],[4].
Now,in the prence of I/Q imbalance,the impaired received signal is modeled as
(4) where denotes complex conjugation,
,and,and and reprent the relative amplitude and pha imbalances, respectively.More precily,the signal model(4)corresponds to the frequency-independent I/Q imbalance ca,commonly ud in the literature[3].The cond term on the right-hand side of(4)is caud by the imbalances and results in a mirror image that interferes with the desired signal.As a measure of the impact of the interference,the image rejection ratio(IRR) for the imbalanced signal(4)is defined as
(5) For a carefully designed analog front,the IRR is in the order of 20–40dB[1],[4],which is often insufficient for advanced re-ceivers.Therefore,additional digital compensation is required. After a simple manipulation,(4)can be rewritten with respect to as
总发烧是怎么回事(6) The relationship in(6)gives us an insight into how to design an IQ imbalance compensator;we let the compensator output be
(7) where
(8)
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(9) Comparing(6)and(9),the ideal choices for the compensator parameters are and.In most cas, it is assumed that and.We can therefore assume and.
As mentioned in the introduction,tofind compensator param-eters and,we rely on the properness condition
(10) To satisfy the properness condition(10),both the real and imagi-nary parts should be zero.By the definitions of and, thefirst condition associated with the imaginary part of(10)is given by
(11)where the assumption is applied.Note that(11)is an equation of and does not depend on.Then,by the orthogo-nality principle[5],the solution to(11)is equivalent to the so-lution of a linear MMSE estimation problem
(12) For a given,the necessary and sufficient condition that sat-isfies the equi-absolute variance condition(3)and,at the same time,nulls the real part of(10)is given by
(13)
By the definitions of and,(13)is rewritten as
(14) where.As with the previous ca of,the solution to(14)is equivalent to the solution of a linear MMSE estimation problem
(15) At a glance,it ems that the MMSE formulations in(12)and (15)for the I/Q imbalance compensation parameters and do not simplify the problems at all.They still need to be solved through stochastic optimization,which requires prior knowl-edge of the input signal statistics.However,the benefit of tho MMSE formulations is that we can apply standard adaptivefil-tering techniques,such as the LMS and RLS algorithms[5],to find approximate solutions to the MMS
E estimation problems in an efficient way without any prior knowledge of the input signal statistics.The following subctions address the LMS and RLS approaches for computing and.
A.Blind LMS I/Q Imbalance Compensator
书法对联作品欣赏The LMS algorithm is the most widely ud adaptivefiltering algorithm due to its simplicity.Applying the LMS algorithm to the MMSE estimation problem(12),we have an update rule
(16) where is a properly chon step size,and is a clip function defined as
(17) for given and.In(16),the range of is limited by for practicality’s sake.Now, given,the LMS update rule for is given by
(18) where is a properly chon step size.As in the ca of, the of is confined by.As for the lec-tion of step size parameters and,we should bear in mind
NAM et al.:BLIND ADAPTIVE I/Q IMBALANCE COMPENSATION ALGORITHMS 477
TABLE I
B LIND LMS I/Q I MBALANCE C
OMPENSATOR the tradeoff between the convergence speed and the steady-state misadjustment.Though proper choices of and may de-pend on speci fic applications,the LMS algorithms im-po necessary conditions on the maximum step size,such as
(19)(20)
to guarantee stability.Note that the range of depends on through while the range of depends on the input signal statistics.To get rid of the dependency of on ,we consider an inequality
(21)
Using (21)in (20),we obtain a more stringent condition
手型画(22)
which depends only on input signal statistics.The detailed pro-cedure of the above LMS I/Q imbalance compensation algo-rithm is summarized in Table I.
Before closing this subction,we should remark that the LMS I/Q imbalance compensation algorithm derived above is the same as the heuristic algorithm in [2].Nevertheless,since we rely on fi
rmly grounded adaptive filter theory instead of on an intuition-bad adaptation,we believe that the above algo-rithm now has a stronger meaning.B.Blind RLS I/Q Imbalance Compensator
Another approach to solving the MMSE estimation problems (12)and (15)in the abnce of statistics is to apply the sample average approximation method to transform the MMSE estima-tion problems into deterministic least-squares (LS)problems.This approach is called the method of LS,and has an ef ficient and recursive solution by the RLS algorithm.Let us first con-sider the MMSE
formulation for in (12),which has an LS counterpart
TABLE II
B LIND RLS I/Q I MBALANCE
C OMPENSATOR
(23)
where ,which is positive and clo to,but less than,unity,is a constant weighting factor.Then the solution to (23)is simply given by
(24)
Now,given ,the LS version of the MMSE
formulation for
in (15)is derived as
(25)
where
is a positive constant weighting factor and
.Again,the solution to (25)is
simply found to be
(26)
Note that both in (24)and in (26)can be computed recursively as new inputs and are provided,which yields the recursive procedure summarized in Table II.
III.N UMERICAL S IMULATIONS
This ction demonstrates performance results using com-puter simulations.As a test system,a 16-QAM signal with unit energy is assumed to be received through an additive white Gaussian noi (AWGN)channel.As for the I/Q imbalance pa-rameters,and are ud,which results in a 17.5dB initial IRR.The LMS and RLS algorithms in Tables I and II are applied for the I/Q imbalance compensation.In par-ticular,for the LMS algorithm,step size parameters
are ud and,for the RLS algorithm,weighting are ud.In Fig.1,the convergence
478IEEE SIGNAL PROCESSING LETTERS,VOL.19,NO.8,AUGUST
2012
Fig.1.Convergence of and with the LMS and RLS
algorithms,and dB.
behaviors of and,which are averaged over100in-dependent trials,are depicted for a12dB signal-to-noi ratio (SNR).It is en that the two algorithms show successful con-vergence within around1500iterations.In Fig.2,the improve-ment curves of IRRs,which are averaged over100independent trials,are shown for a12dB SNR.For comparison,the IRR of the one-tap blind adaptive algorithm in[4]with a step size of is also shown.Starting from the initial IRR of17.5dB, the IRRs improve as the iteration progress.It is en that the LMS algorithm and the algorithm in[4]show quite similar con-vergence and steady-state behaviors.Thefinal IRRs achieved by the LMS and RLS algorithms are around40dB and50dB,re-spectively.Thefinal IRRs are determined by the steady-state errors of the adaptive algorithms and,due to the tradeoff be-tween the convergence rate and the steady-state error,thefinal IRRs can be further improved by adjusting and,and and,at the expen of slower convergence rates.Though the are not included in this letter,both of the propod algo-rithms can easily attain over50dB IRR with a careful choice of parameters.Note that the results i
n Figs.1and2show that the RLS algorithm has faster convergence and a smaller steady-state error than the LMS algorithm,which also accords with common n on the LMS and RLS algorithms.Finally,the symbol error rate(SER)performances of the propod algorithms are shown in Fig.3as functions of the received SNR.For comparison,the performance of the one-tap blind adaptive algorithm in[4],the performance without I/Q imbalance compensation,and the ideal performance without I/Q imbalance are also shown in the same figure.It is en that SER performances clo to the ideal per-formance can be achieved by using the propod I/Q imbalance compensation algorithms.In particular,the LMS algorithm and the algorithm in[4]perform almost identically,
and the perfor-mance of the RLS algorithm coincides with the ideal perfor-mance and can hardly be distinguished.
IV.C ONCLUSION
In this letter,the blind adaptive I/Q imbalance compensa-tion problem was addresd.On the basis of the properness and equi-absolute variance conditions,it was shown that the blind I/Q imbalance compensation problem can reduce to a pair of Fig.2.Image rejection ratios of the
LMS and RLS
algorithms and the algorithm in[4]size of)
,and dB.
Fig.3.The16-QAM SER performances with the LMS
and RLS algorithms and the algorithm in (step size of)for and.
linear MMSE estimation problems.To solve the MMSE es-timation problems without any input signal statistics,the LMS and RLS algorithms were applied,which resulted in two dif-ferent blind adaptive I/Q imbalance compensation algorithms. Though the propod algorithms look similar to other existing algorithms,they arefirmly grounded on adaptivefilter theory, not heuristics,and,therefore,their stability and convergence to the ideal solution is guaranteed.Simulation results verified the nice convergence behaviors and demonstrated that the propod algorithms prent high image rejection ratios.
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