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5246 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007
Joint Adaptive Compensation of Transmitter and
Receiver IQ Imbalance Under Carrier Frequency
Offset in OFDM-Based Systems
Deepaknath Tandur, Student Member, IEEE, and Marc Moonen, Fellow, IEEE
Abstract—Zero-intermediate frequency (IF)-based orthogonal
frequency division multiplexing (OFDM) transmitters and re-
ceivers are gaining a lot of interest because of their potential to
enable low-cost terminals. However, such systems suffer from
front-end impairments such as in-phase/quadrature-phase (IQ)
imbalance which may have a huge impact on the performance.
Moreover, since OFDM is very sensitive to a carrier frequency
offset, this distortion needs to be taken into account in the deriva-
tion and analysis of any IQ imbalance estimation/compensation
scheme. In this paper, the effect of both transmitter and receiver
IQ imbalance under carrier frequency offset in an OFDM system
is studied and an algorithm is developed to compensate for such
distortions in the digital domain. The algorithm involved is a very
efficient post-FFT adaptive equalization which is shown to lead to
near ideal compensation.
Index Terms—Adaptive compensation algorithms, analog im-
pairments, carrier frequency offset (CFO), direct-conversion,
in-phase/quadrature-phase (IQ) imbalance, orthogonal frequency
division multiplexing (OFDM).
I. INTRODUCTION
O
RTHOGONAL frequency division multiplexing (OFDM)
[1] is a standardized technique for broadband wireless
systems. It is used for wireless LAN [2], [3], fixed broadband
wireless access [4], digital video and audio broadcasting [5],
[6], etc.
Traditionally, OFDM systems employ a superhetero-
dyne architecture to convert the baseband signal to a radio
frequency (RF) signal and vice versa. Fig. 1 shows a block
diagram of a superheterodyne receiver front-end. It converts
the RF signal down to digital baseband in several steps, passing
via intermediate frequencies (IFs). At each analog IF, filtering
by means of band-pass filters (BPF) or low-pass filters (LPF)
and amplification by means of low-noise amplifier (LNA) or
variable gain amplifier (VGA) are applied to maintain good
Manuscript received August 21, 2005; revised January 15, 2007. This work
was carried out at the ESAT Laboratory of the Katholieke Universiteit Leuven,
in the frame of Belgian Programme on Inter-University Attraction Poles, initi-
ated by the Belgian Federal Science Policy Office IUAP P5/11 (“Mobile mul-
timedia communication systems and networks”). The Scientific responsibility
is assumed by its authors. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Hongya Ge.
The authors are with the Department of Electrical Engineering, ESAT-SCD,
Katholieke Universiteit Leuven, B-3001 Leuven, Belgium (e-mail: deepaknath.
tandur@esat.kuleuven.be; marc.moonen@esat.kuleuven.be).
Digital Object Identifier 10.1109/TSP.2007.898788
selectivity and sensitivity. The drawback of this architecture,
however, is that it requires a lot of components to reach this
good signal quality. At each analog IF, the filters and the
amplifier all add to the component cost. Not only are they quite
expensive components, but as they are external, they also add
to the assembling cost.
An alternative to the superheterodyne architecture is the
zero-IF architecture (or direct-conversion architecture) shown
in Fig. 2. As the name suggests, the zero-IF architecture con-
verts the RF signal directly to baseband or vice versa without
any IFs. This clearly results in a lower component count and
consequently a lower cost. The zero-IF architecture enables an
easier integration and leads to a smaller form factor.
However, there are also drawbacks in using a zero-IF archi-
tecture. In a superheterodyne architecture, the IQ modulation
and demodulation is performed in the digital domain resulting
in perfect IQ separation. On the other hand, the zero-IF ar-
chitecture performs IQ modulation and demodulation in the
analog domain. As a result, the matching between the analog I
and Q paths and their components are imperfect. This leads to
IQ imbalance distortion which significantly degrades the signal
quality.
Rather than decreasing the IQ imbalance by increasing the
design time and the component cost, IQ imbalance can also be
tolerated and then compensated digitally. Along with IQ imbal-
ance, OFDM systems are also very sensitive to carrier frequency
offset (CFO). The performance degradation due to receiver IQ
imbalance and CFO in OFDM systems has been investigated in
[7] and [8]. Several compensation algorithms considering either
only receiver IQ imbalance or transmitter IQ imbalance indi-
vidually have been developed in [9]–[11]. Recently, joint trans-
mitter and receiver IQ imbalance compensation algorithms have
been proposed in [12] and [13].
However, all the previously mentioned algorithms ignore the
problem of joint IQ imbalance at both transmitter and receiver
along with CFO estimation and compensation. In [12], Tsui and
Lin analyze the performance of a joint transmitter and receiver
IQ compensation technique under a small residual CFO. How-
ever, the compensation scheme does not eliminate the effect of
the CFO. The combination of CFO with receiver IQ imbalance
has been studied in [14] and [15]. To the best of our knowledge,
no general solution has been proposed so far for the problem
combining CFO with both transmitter and receiver IQ imbal-
ance.
This paper is organized as follows. In Section II, we introduce
an IQ imbalance model and describe its impact in OFDM based
1053-587X/$25.00 © 2007 IEEE
TANDUR AND MOONEN: JOINT ADAPTIVE COMPENSATION OF TRANSMITTER AND RECEIVER IQ IMBALANCE 5247
Fig. 1. Superheterodyne receiver architecture.
Fig. 2. Zero-IF receiver architecture.
systems. Section III describes the mutual impact of IQ imbal-
ance and CFO. In Section IV, the basics of a suitable transmitter
and receiver IQ imbalance and CFO compensation scheme are
explained. Section V presents the adaptive compensation algo-
rithm. Our simulation results are shown in Section VI and, fi-
nally, the conclusion is given in Section VII.
II. IQ I
MBALANCE
MODEL AND IMPACT
In the up-conversion of a zero-IF architecture, the incoming
signal in the I path is up-converted by the local oscillator (LO)
signal at the carrier frequency, while the Q path is up-converted
by the LO signal with a 90
phase shift. The combination of
these two signals results in the RF signal. As the IQ modulation
is performed in the analog domain, the matching of the I and Q
paths is not perfect: the filtering at baseband may not be 100%
matched, the 90
phase rotation of the LO signal will be slightly
off and the I and Q paths may not be matched with perfectly
equal power, especially if cheap components or architectures are
used. All these contributions result in a mismatch between the I
and Q path, the so-called transmitter IQ mismatch or transmitter
IQ imbalance. A similar mismatch and imbalance occurs in the
down-conversion stage at the receiver and is referred to as re-
ceiver IQ mismatch or receiver IQ imbalance.
We analyze the effect of IQ imbalance in the time and
frequency domain. Frequency domain signals are underscored,
while time domain signals are not. Vectors are indicated in bold
and scalar parameters in normal font. Superscripts
, , and
represent conjugate, transpose, and Hermitian, respectively.
Let
represent the transmitted baseband complex signal
vector of length
before being distorted by the IQ imbalance
at the transmitter. The distorted signal in the time domain can
then be modeled as [16]
(1)
where
is the time domain signal vector with transmitter IQ
imbalance.
denotes the real part and the imaginary
part. The distortion parameters
and are the amplitude and
phase orthogonality mismatch between the I and Q branches in
the RF modulation process at the transmitter.
For analytical derivations, (1) can be rewritten as
(2)
with
(3)
(4)
Equation (2) shows that the effect of transmitter IQ imbalance
on the time domain signal is twofold. First, the signal is scaled
by a complex factor
. Second, a small scaled version
of its complex conjugate is added. If no IQ imbalance
is present, then
and, thus, and
and then (2) reduces to .
Parameters
are used for analytical derivations, as they
form a more tractable mathematical description of the IQ imbal-
ance. For simulations, the more physically relevant parameters
will be used.
5248 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007
The IQ imbalance impact on the frequency-domain signal
vector
is
(5)
Here,
denotes the mirroring operation in which the vector
indices are reversed, such that
,where
for
for
is the mirror carrier of and is the number of subcarriers
in an OFDM symbol.
We now consider OFDM transmission over a time invariant
frequency selective channel. Let
be the frequency response
of the multipath channel of length
. The channel then adds a
filtering that should be included in formula (2). If the channel
impulse response is shorter than the OFDM cyclic prefix, which
is a standard assumption, formula (2) is changed to
(6)
where “
” denotes component-wise vector multiplication and
is assumed to be a circular (proper) complex additive white
Gaussian noise (AWGN) and
is its frequency response.
A similar expression as formulas (1) and (2) can be used to
model IQ imbalance at the receiver. Let
represent the received
baseband complex signal of length
before being distorted by
IQ imbalance. Then, the distorted baseband signal in the time
domain will be given as
(7)
where the distortion parameters
and are defined as in (3)
and (4) and
is the time domain signal vector with receiver
IQ imbalance.
When the impact of transmitter and receiver IQ imbalance
along with channel distortions are considered, then (7) is
changed as follows:
(8)
Equation (8) shows that due to transmitter and receiver IQ im-
balance power leaks from the signal on the mirror carrier
to the carrier under consideration and thus causes inter-car-
rier-interference (ICI). As OFDM is very sensitive to ICI, IQ im-
balance results in severe performance degradation. This is later
illustrated in Section VI.
III. CFO M
ODEL AND
IMPACT
Since OFDM is very sensitive to a CFO [7], this distortion
also needs to be taken into account. In this section, a model is
proposed that incorporates both IQ imbalance and CFO.
During the up and down conversion of a zero-IF architecture,
ideally the LO should produce a pure sine wave at the standard-
ized RF carrier frequency to translate the baseband signal to RF
and vice versa. However, in practice, the produced carrier fre-
quency may vary from one realization to another, resulting in a
possible CFO between the transmitter and the receiver.
For a receiver with no IQ imbalance, the time domain effect
of a CFO
on an incoming signal is a phase rotation pro-
portional with time. This results in a rotated signal vector
given as
(9)
with
the element-wise exponential function on the vector
and where is a time vector.
When CFO is present together with receiver IQ imbalance,
the resulting baseband signal can be written as [14]
(10)
which effectively means that the signal is first distorted by CFO
and then by the receiver IQ imbalance.
For the case involving CFO as well as transmitter and re-
ceiver IQ imbalance and channel distortions, the previous equa-
tion changes to
(11)
The joint effect of both transmitter and receiver IQ imbalance
along with CFO results in a severe performance degradation,
as will be shown in Section VI, and so a digital compensation
scheme is required.
IV. IQ I
MBALANCE AND CFO COMPENSATION
Following the joint model from Section III, formula (11), we
know that the received OFDM symbol in the time domain is
given as
(12)
where
. Equation (12) ex-
plicitly shows only the CFO, the receiver IQ imbalance and the
noise contribution. The channel filtering and the distortion due
to transmitter IQ imbalance are hidden in the definition of
and
so not considered for the time being.
We now assume that the CFO
can be estimated accurately
in the system. In practice, several CFO estimation schemes in-
deed exist that are found to be sufficiently robust against the IQ
TANDUR AND MOONEN: JOINT ADAPTIVE COMPENSATION OF TRANSMITTER AND RECEIVER IQ IMBALANCE 5249
imbalance [14] and [15]. Given a good estimate of , obtained
using one of these schemes, we first perform an element-wise
multiplication of the received distorted symbol with the esti-
mated negative frequency offset
. This results in a
vector
as follows:
(13)
where
and
is now a zero mean improper complex noise vector [17] mainly
due to the presence of receiver IQ imbalance.
Similarly, we also perform an element-wise multiplication of
the complex conjugate of the received signal with the negative
frequency offset
resulting in a vector as follows:
(14)
where now
is also a zero mean
improper complex noise vector.
Both
and consist of two contributions “ ” and “ ”
scaled by different weighing factors. Here “
” is called the de-
sired signal and “
” the undesired signal. This is because in the
frequency domain, the former gives rise to the desired signal,
while the latter yields a mirror image and causes ICI (because
of the complex conjugate), subject to leakage caused by the ex-
ponential term
.
Transforming (13) and (14) to the frequency domain, we ob-
tain
(15)
The resulting matrix
is of dimension . Equation
(15) can be written more explicitly for each component (fre-
quency bin) “l” as
(16)
From the definition of
it follows that
, hence
(17)
Substituting this in (16) results in
(18)
By replacing
by , we then obtain
(19)
Merging (18) with the complex conjugate of (19), finally re-
sults in
(20)
From this it follows that in the noiseless case,
can be
obtained by taking an appropriate linear combination (corre-
sponding to the first column of
) of the left-hand side quan-
tities, i.e.,
(21)
where the weights
, , , and are of length . This formula
demonstrates that a receiver structure can be designed that ex-
actly compensates for the transmitter and receiver IQ imbalance,
the CFO and the channel effect. The coefficients
, , , and
can be computed from , , , , , and , if these are
available. In the noisy case a suitable set of coefficients can be
estimated based on an MSE minimization
where is the expectation operator.
In Section V, a training-based initialization of
, , , and
is described, based on such an MMSE criterion.
V. A
DAPTIVE COMPENSATION ALGORITHM
In a standard OFDM receiver with no front-end distortion,
single tap equalizers are sufficient for channel equalization.
However, for an OFDM system with transmitter and receiver
IQ imbalance along with CFO and channel distortions, more
complex equalization schemes are needed. To mitigate these
distortions we propose an OFDM system with four-tap adaptive
equalizer for each subcarrier in the OFDM symbol. The four
taps of the equalizer can be represented as
, , , and
each of length corresponding to the length of the OFDM
5250 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007
Fig. 3. Four-tap adaptive equalizer to compensate front-end nonidealities along with channel distortion.
symbol. The taps are trained to the desired coefficient vectors ,
, , and using the training symbols available in the OFDM
system.
The four-tap adaptive equalizer is as shown in Fig. 3. The
four inputs of the adaptive equalizer are
, , , and
obtained from the received signal as
discussed in Section IV. The equalizer coefficients
, ,
, and are updated here based on the well-known
LMS scheme but any other adaptive filtering algorithm (mini-
mizing on MMSE criterion) can be used. To better illustrate the
update equations, we introduce the time (or iteration) index
,
i.e.,
, , , and represent the equalization coef-
ficients at time instant
. Similarly, represents the output of
the equalizer at time instant
which is given as
(22)
The equalization coefficients are then updated according to
the LMS rule
where is the error signal generated at
iteration
for the tone index using a training symbol . is
the LMS step-size parameter. Once the equalizer coefficients are
trained with a suitable number of training symbols, the obtained
coefficients are used to equalize the received signal according to
(21).
Although LMS is the simplest adaptive filtering algorithm,
it suffers from a slow convergence. This problem could be se-
vere for the application at hand, since current OFDM systems
usually deploy only short training sequences in order to reduce
the training overhead in packet-based data transmission. A short
training length is acceptable in an OFDM system with ideal I
and Q branches as a good channel estimation can be achieved
only with a few training symbols. However, due to IQ imbal-
ance, there is a cross-coupling between every tone and its mir-
rored tone, resulting in ICI. The ICI is more severe when CFO
is also considered and thus resulting in a very slow convergence
rate. Better adaptive algorithms like RLS could then be used
which have a faster convergence rate, if the added complexity
can be afforded [18].
An end-to-end OFDM system with the compensation scheme
is shown in Fig. 4. The shaded blocks are the front-ends of the
communication system where IQ imbalance and CFO distortion
is introduced.
It is noted that the receiver structure shown in Fig. 4 general-
izes earlier structures for specific subproblems. References [11]
and [14], for instance, apply to the compensation of CFO with
either transmitter or with receiver IQ imbalance but not both. In
this case, the output of any one FFT branch can be taken (in-
stead of both outputs) as the compensation can then be obtained
with a two tap adaptive equalizer.
VI. S
IMULATION RESULTS
A typical OFDM system (similar to IEEE 802.11a) is simu-
lated to evaluate the performance of the compensation scheme
for transmitter and receiver imbalance under CFO. The per-
formance comparison is made with an ideal system with no
TANDUR AND MOONEN: JOINT ADAPTIVE COMPENSATION OF TRANSMITTER AND RECEIVER IQ IMBALANCE 5251
Fig. 4. OFDM system with post-FFT compensation of transmitter and receiver IQ imbalance with CFO.
Fig. 5. BER versus SNR simulated for 64 QAM constellation with LMS adaptive equalization. Transmitter phase imbalance of , transmitter amplitude
imbalance of
, receiver phase imbalance of , receiver amplitude imbalance of , and CFO of . (a) AWGN
flat channel (nonfading). (b) Four-tap complex Gaussian channel (fading).
front-end distortion and with a system with no compensation
algorithm included.
The parameters used in the simulation are: OFDM symbol
length of
, cyclic prefixof . There are two
different channel profiles: 1) an additive white Gaussian noise
(AWGN) channel with a single tap unity gain and 2) a multipath
channel with
taps where the taps are chosen inde-
pendently with complex Gaussian distribution. Every channel
realization is independent of the previous one and the BER re-
sults depicted are from averaging the BER curves over several
independent channels. The step size
of the adaptive equalizer
is kept at 0.2.
We consider IQ amplitude imbalance of
, , and
phase imbalance of
, at both transmitter and
receiver. For CFO, the standard [2] specifies a maximum toler-
able frequency deviation of 20 ppm. For a transmitter and re-
ceiver LO operating between 5 and 6 GHz, this translates in
a maximum
240 kHz. For the simulation, we consider
100 kHz. The CFO is usually considered as the ratio of
the actual carrier frequency offset
to the subcarrier spacing
, i.e., , where is the sampling period. In
our case, the subcarrier spacing is 312.5 kHz, thus
is 0.32.
Fig. 5 shows the performance curves obtained for BER versus
SNR for an uncoded 64QAM OFDM system. With no compen-
sation scheme in place, the OFDM system is completely unus-
able. Even for the case when there is only transmitter and re-
ceiver IQ imbalance and no CFO, the BER is very high. For
the case with the compensation scheme employed, the curves
are very close to the ideal situation with no front-end distortion.
The compensation performance depends on how accurately the
adaptive equalizer coefficients can converge to the ideal values.
The design of zero-IF receivers typically yields an IQ imbal-
ance on the order of
[19]. The
performance curves clearly demonstrate that for such IQ imbal-
5252 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007
ance values compensation is absolutely necessary to enable a
high data rate communication. When CFO is also present, the
system certainly cannot work without a suitable compensation
technique. Moreover, very large IQ imbalance values can be cor-
rected just as easily. Thus, the presented IQ-CFO mitigation al-
lows to greatly relax the zero-IF design specifications.
VII. C
ONCLUSION
In this paper, the joint effect of both transmitter and receiver
IQ imbalance under carrier frequency offsets in an OFDM
system is studied and an algorithm has been developed to
compensate for such distortions in the digital domain. The
algorithm involved is a very efficient post-FFT adaptive equal-
ization which leads to near ideal compensation.
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Deepaknath Tandur (S’05) received the B.S. degree
in electronics and communications engineering from
RV College of Engineering, Bangalore University,
Bangalore, India, in 2001 and the M.S. degree in em-
bedded systems design engineering from University
of Lugano, ALaRI, Lugano, Switzerland, in 2004.
He is currently pursuing the Ph.D. degree in elec-
trical engineering from the Katholieke Universiteit
Leuven, Leuven, Belgium.
His research interests include communication sys-
tems and signal processing, including MIMO OFDM
systems, algorithms for front-end impairments compensations, channel identi-
fication and equalization, and experimental and practical communication sys-
tems.
Marc Moonen (M’94–SM’06–F’07) received the
electrical engineering and Ph.D. degrees in applied
sciences from Katholieke Universiteit Leuven,
Leuven, Belgium, in 1986 and 1990, respectively.
Since 2004, he has been a Full Professor with
the Electrical Engineering Department, Katholieke
Universiteit Leuven, where he is heading a research
team working in the area of numerical algorithms
and signal processing for digital communications,
wireless communications, DSL, and audio signal
processing.
Prof. Moonen was a recipient of the 1994 K.U.Leuven Research Council
Award, the 1997 Alcatel Bell (Belgium) Award (with P. Vandaele), the 2004
Alcatel Bell (Belgium) Award (with R. Cendrillon), a Journal Best Paper Award
from the IEEE T
RANSACTIONS ON SIGNAL
PROCESSING (with G. Leus) and
from Elsevier Signal Processing (with S. Doclo), and was a 1997 “Laureate
of the Belgium Royal Academy of Science.” He was chairman of the IEEE
Benelux Signal Processing Chapter (1998–2002), and is currently President of
the European Association for Signal, Speech and Image Processing (EURASIP)
and a member of the IEEE Signal Processing Society Technical Committee on
Signal Processing for Communications. He has served as Editor-in-Chief for
the EURASIP Journal on Applied Signal Processing (2003–2005) and has been
a member of the editorial board of the IEEE T
RANSACTIONS ON CIRCUITS AND
SYSTEMS—II: REGULAR PAPERS (2002–2003) and the IEEE Signal Processing
Magazine (2003–2005). He is currently a member of the editorial board of In-
tegration, the VLSI Journal, EURASIP Journal on Applied Signal Processing,
EURASIP Journal on Wireless Communications and Networking, and Signal
Processing.
