The Vault

Blind Estimation and Compensation of Frequency- Flat I/Q Imbalance Using Cyclostationarity
Research Paper / Jan 2008

1

Blind Estimation and Compensation of Frequency-
Flat I/Q Imbalance Using Cyclostationarity

Chia-Pang Yen, Yingming Tsai, Guodong Zhang and Robert Olesen
InterDigital Communications LLC.

2 Huntington Quadrangle
Melville, NY 11747, USA

Chia-Pang.Yen@, Yingming.Tsai@, Guodong.Zhang@, Robert.Olesen@interdigital.com


Abstract— I/Q imbalance is one of the major concerns in the
design of direct-conversion front-end receivers in high data rate
wireless networks. To address the challenge, various I/Q
imbalance estimation and compensation algorithms have been
proposed in the literature. In this paper, we propose a blind
cyclostationary method based estimation and compensation of
frequency-flat I/Q imbalance. The proposed blind estimation
algorithm uses second-order statistics to compensate I/Q
imbalance, instead of estimating the mismatch parameters
directly, and is an unbiased estimator when a DC offset exists at
the receiver. The performance of our approach is evaluated and
compared to other existing blind I/Q imbalance estimation
algorithms.

Keywords: I/Q imbalance, Cyclostationary.

I. INTRODUCTION
The next generation wireless communication networks (for

example, 3GPP LTE system) will provide high data rates such
as 100 Mbps to subscribers. In commonly used direct-
conversion front-end receiver, the received signal is I/Q
downconverted from RF to baseband signal. Due to the
imperfect oscillator in the RF front-end receiver, the I and Q
signal paths will inevitably have different amplitudes and
phases. This mismatch gives rise to image frequencies that
become interference upon down conversion to baseband. The
direct-conversion receiver is vulnerable to I/Q imbalance since
I/Q separation is performed early in the RF/analog portion. We
address the case of Direct-conversion Receiver (DCR)
topology in this study, since DCR has benefits in terms of size
and cost and is thus a preferred choice for higher levels of
integration. The impact of I/Q imbalance is more severe to the
system using high order modulations and high coding rates.
Therefore, I/Q imbalance correction is essential for the design
of higher data rates system [2].

In the past few years, extensive research has been done on
I/Q imbalance estimation and compensation in wireless
communication systems [4]-[7]. Methods such as hard
decision (HD) approach [7] and statistical approach aim to
estimate the mismatch parameters. The methods in [4]-[6] use
second-order statistics of mismatched baseband equivalent to
compensate I/Q imbalance, instead of estimating the mismatch
parameters directly. In this paper, we improve the method in

[4],[5] by using a cyclic auto-correlation based method that
exploits cyclostationarity.

The rest of the paper is organized as follows. In Section II,
the system model and I/Q imbalance model are described and
formulated. In Section III, a one-tap compensator for
frequency-flat I/Q imbalance is described. The cyclostationary
property is reviewed in Section IV. A new I/Q imbalance
estimator that uses auto-correlation method and exploits
cyclostationary property is proposed in Section IV as well. In
Section V, the numerical results are presented and discussed.
Finally, conclusions are drawn in Section VI.

II. SYSTEM AND I/Q IMBALANCE MODEL

A. System Model
The received signal )(tz is oversampled at a rate of TP / ,

where T is the symbol duration and P is an integer. Note that
the sampling rate should be greater than the Nyquist rate,
which implies that 2≥P . We let ][ns denote the transmitted
modulated symbol, ][ng = PnTttg /|)( = is the combined
transmitter and receiver pulse shaping filter, εf denote the
normalized carrier frequency offset with a uniform distribution
between [ ]2/,2/ ππ− , and θ denotes the phase offset. We
consider a multipath channel with ],[ lnh as the discrete
channel impulse response and L as the number of multipaths.
Thus, we denote the following received discrete-time signal

PnTttznz /|)(][ == as

],[][],[][
1

0

))/2(( nvlnqlnhenz
L

l

nTfPj e +−= ∑−
=

+θπ (1)

where complex additive noise )(tv is assumed to be stationary
but not necessarily white and/or Gaussian, and ][nq is
expressed as

∑ −=
u

uPngnsnq ][][][ . (2)

The analysis in the rest of the paper is based on the
following assumptions [3]:



2

Assumption 1: ][ns is a zero-mean independent identically
distributed (i.i.d) sequence which is chosen from a finite-
alphabet complex constellation with variance 2sσ , i.e.,
{ } ][][][ 21221 mmmsmsE s −= δσ , (3)
Assumption 2: The autocorrelation of a channel impulse
response is given by { }],[],[ 22*11 lnhlnhE { } ][],[],[ 2122*11 lllnhlnhE −= δ .

))(2( 210 nnTfJ d −= πγ , (4)
where γ is a normalization constant, )(0 ⋅J is the zero-order
Bessel function of first kind, and df represents the maximum
Doppler shift.
Assumption 3: ][nv is a wide-sense stationary complex
process independent of ],[ lnh .
B. Frequency-flat I/Q Imbalance Model

The following assumptions are imposed on equation (3). In
a quadratic (I/Q) direct-conversion receiver, the received
signal )(tz is translated to baseband by mixing it with a
complex exponential generated by a local oscillator (LO) and
a low pass filter as shown in Figure 1. In this architecture, the
mixer, filters, amplifier and A/D converter are the source of
I/Q component mismatch, due to their non-equal amplitude
(gain) and phase imbalance (including imbalance at the
transmitter side). In this paper, we assume the I/Q imbalance is
frequency-flat over the entire receiver bandwidth, i.e., I/Q
imbalance parameters (gain and phase imbalance) won’t vary
in the entire receiver bandwidth. Hence, the corresponding
mismatch received baseband signal ][nx can be expressed as
the received complex baseband signal ][nz added with it’s

own complex conjugate ][* nz [4], [7], i.e., ][nx can be
expressed as

][)(][)(][ *21 nzKnzKnx Θ+Θ= , (5)
where ],[ φg=Θ incorporates the amplitude mismatch g and
phase imbalance φ , and )(1 ΘK , )(2 ΘK are complex
numbers. To evaluate I/Q imbalance distortion to the received
signal ][nz , we define the image-reject ratio (IRR) [4] as

2
2

2
1 )(/)( ΘΘ= KKIRR . (6)

If there is no I/Q imbalance at the receiver, i.e., the term
)(2 ΘK is equal to 0 then IRR ∞→ .

III. ONE-TAP COMPENSATOR FOR FREQUENCY-FLAT I/Q
IMBALANCE

The I/Q imbalance compensator is trying to eliminate the
conjugate signal (i.e., ][* nz ) from the received signal ][nx .
Since frequency-flat I/Q imbalance is assumed in this paper. A

simple one-tap compensator
)(

)(
*
1

2

Θ
Θ−=

K

K
w can be applied to

compensate the I/Q imbalance. For details of the derivation of

the one-tap compensator, please refer to the Appendix. Let
][ˆ nz denote the signal after I/Q compensation, which is given

by
][][][ˆ * nwxnxnz += . (7)

Applying the one-tap compensator
)(

)(
*
1

2

Θ
Θ−=

K

K
w to the term

][* nwx , we have

][
)(

)(
][)(][

*
1

2
2**

2
* nz

K

K
nzKnwx Θ

Θ
−Θ−= . (8)

Hence, plugging (8) into (7), we have the compensated
signal ][ˆ nz as

][)
)(

)(
)((][ˆ

*
1

2
2

1 nzK

K
Knz Θ

Θ
−Θ= . (9)

Therefore, compensation of frequency-flat I/Q imbalance does
not require estimating I/Q imbalance parameters Θ
(or )(1 ΘK and )(2 ΘK ). Instead, it only requires the estimation

of
)(

)(
*
1

2

Θ
Θ

K

K
. This motivates us to design a new estimation

approach to solve the problem of I/Q imbalance
estimation/compensation more robust with regard to any
frequency-flat I/Q imbalance parametersΘ .

IV. FREQUENCY-FLAT I/Q IMBALANCE ESTIMATOR USING
CYCLOSTATIONARY APPROACH

A. Cyclic Correlation Property
The time-varying correlation of a general non-stationary

process of ][nz is defined as ]}[][{);( * mnznzEmnRzz +=
[1], where m is an integer lag. Signal ][nz is second-order
cyclostationary with period P if and only if there exists an
integer P such that );();( mkPnRmnR zzzz += , kn,∀ [3]. To
prove that );( mnRzz is cyclostationary with period P, we
examine if );( mnRzz );( mkPnRzz += . Using (3), (4) and
definition of );( mnRzz , we have

)();()2();(
1

0
))/2(( mRmlnRTmfJemnR vqq
L

l
d

TmfQj
zz e +−= ∑−− πγ π . (10)

It can be shown that the term );( mlnRqq − is periodic with
P. That is

=+− );( mkPlnRqq ∑ −++−−+−
u

s PukmlngPuklng ])([])([
*2σ

= ∑ ++−+−
i

s iPmlngiPlng ][][
*2σ );( mlnRqq − . (11)



3

Hence, we have );();( mkPnRmnR zzzz += , kn,∀ . This
means );( mnRzz is periodic with respect to n with period P
for a fixed m . Thus, );( mnRzz has discrete Fourier series
coefficients given by ≡);( mkFzz

∑−
=

−1
0

)/2();(1
P

n

knPj
zz emnRP

π which are periodic with respect

to k with period P [3]. );( mkFzz is called cyclic correlation
and }12/,,2/{ −−∈ PPk K are called cyclic frequencies or
cycles. From (12), );( mkFzz can be expressed as

∑−
=

−=
1

0

)/2();(1);(
P

n

knPj
zzzz emnRPmkF

π

∑∑ += −−
n

vv
knPj

qq
TmfPj kmRemnRe e ][)();( )/2())/2(( δγ ππ . (12)

B. Proposed I/Q Imbalance Estimator
In the following, we propose a new approach

estimate
)(

)(
*
1

2

Θ
Θ

K

K
by using cyclostationary approach. The

proposed estimation algorithm is not limited to a particular air
interface or waveform, and is therefore applicable to any
wireless application. For simplicity of denotation, we drop the
variable Θ in )(1 ΘK and )(2 ΘK , denoting them as 1K
and 2K in the rest of the paper.

First, we evaluate the autocorrelation function );( mnRxx of
][nx and using the cyclic property of signal (i.e.,

{ } 0][][ =+mnznzE ). We have
]}[][{);( * mnxnxEmnRxx +=

]}[][])([][{( *2
**

1
*

21 mnzKmnzKnzKnzKE ++++=
);();( *22

2
1 mnRKmnRK zzzz += . (13)

Secondly, by evaluating conjugate autocorrelation function
);( mnCxx of )(nx , we have { }][][);( mnxnxEmnCxx +=

]}[][])([][{( *21
*

21 mnzKmnzKnzKnzKE ++++=
);();( *2121 mnRKKmnRKK zzzz += . (14)

In the special case where 0=m , the frequency offset εf is
decoupled from )0;(nRzz . Then, we have

)0;()0;()0()0;( *2
1

0 nRlnRJnR zzvqq
L

l
zz =+−= ∑− σγ . (15)

Using (15), equations (13) and (14) are rewritten as

)0;()0;( 22
2

1 nRKKnR zzxx ⎟⎠
⎞⎜⎝

⎛ += . (16)
)0;(2)0;( 21 nRKKnC zzxx = . (17)

Finally, since );( mnRzz is periodic with P, )0;(nRxx and
)0;(nCxx are periodic with P as well. The DFT of
)0;(nRxx and )0;(nCxx are denoted as )0;(kFxx and

)0,(kFCxx respectively, and are given by

∑⎟⎠
⎞⎜⎝

⎛ += −
=

−1
0

)/2(2
2

2
1 )0;(1)0;(

P

n

knPj
xxxx enRKKPkF

π . (18)

∑= −
=

−1
0

)/2(
21 )0;(2)0;(

P

n

knPj
xxxx enRKKPkFC

π . (19)

From (18), (19), we can choose cyclic frequencies k = 0, 1
and -1 as parameters for the proposed one-tap I/Q imbalance
compensator when root-raise cosine (RRC) pulse shaping
filter is used [3]. The remaining cyclic frequencies are not
chosen because they will degrade the estimation performance
of );( mnRqq [3], [9].

Since 22
2

1 KK >> , by using the (18) and (19), the
estimate of *12 / KK can be approximated as

)0,(
)0,(

2
1

2
1

21
*
1

2
kF
kFC

K

KK

K

K

xx

xx== . (20)

for 12/ ..., ,1 ,0,2/ −−= PPk K .
Alternatively, using the method in [4], *12 / KK can be
expressed more precisely as the following

22*1

2

)0;()0;()0;(

)0;(

kFCkFkF

kFC

K

K

xxxxxx

xx

−+
= . (21)

Using (12), (18), and (19), we see that the noise
autocorrelation term ][)( kmRvv δ in the cyclic frequencies

)0;(kFxx equals zero when ≠k 0. Hence, the estimate of
*
12 / KK is unbiased for k=1 and -1. For cyclic frequency k=0,

)0;0(xxF = )0;0(xxR and )0;0(xxFC = )0;0(xxC the estimate in
(21) becomes equivalent to the one-tap I/Q imbalance
compensator proposed by Anttila [4],


22*1

2

)0;()0;()0;(

)0;(

nCnRnR

nC

K

K

xxxxxx

xx

−+
= . (22)

C. Impact of DC offset
If a DC offset in the receiver [8], then the right-hand side of

(12) has to be increased by a term 2d , where d is the DC
offset amplitude. If cyclic frequency k =0 is used in the
proposed algorithm, the proposed algorithm in (21) becomes
equivalent to Anttila’s algorithm as mentioned in (22). It was
established in [5] that for Anttila’s algorithm there is a bias in
I/Q imbalance estimation when a DC offset exists. Therefore,
we choose not to use cyclic frequency k =0 for the proposed
algorithm in the presence of DC offset. For k=1 and -1, the
impact of DC offset will be removed because DC offset’s
discrete Fourier series coefficient is a pulse only at k =0 and is



4

zero elsewhere. Hence, the DC offset doesn’t impact the
proposed algorithm in (20) or (21) with k=1 and -1.

V. PERFORMANCE ANALYSIS AND NUMERICAL RESULTS
The performance of the proposed I/Q imbalance

compensation algorithm was simulated and compared to the
moment-based blind estimation algorithm proposed by Anttila
[4]. The simulation parameters are summarized in table 1. The
simulation results are averaged by 1000 runs. Note that the
performance of I/Q imbalance estimation depends not on the
sampling rate but on the roll-off factor of the pulse shaping
filter [10]. In the simulation, we adopt the frequency-flat I/Q
imbalance model

where
2

1)(1
φjgeK

−+=Θ ,
2

1)(2
φjgeK −=Θ , and ],[ φg=Θ .

TABLE I. SIMULATION PARAMETERS

Transmission BW 1.25 MHz
Modulation QPSK, 16QAM

Mobility 3 km/hr
Channel model Rayleigh

Channel delay profile [0, 0.5, 2.3] µs
Channel power profile [-3, 0, -6] dB

Amplitude mismatch, g 1.07 dB
Phase imbalance, φ 2 degree

Image-reject ratio (IRR) at analog
front end

24 dB

DC offset amplitude 0 and 0.1
Pulse shaping filter RRC with roll-off factor 0.22
Up-sampling rate Nyquist rate

Number of observed modulation
symbols

10,000 symbols per
simulation run



The IRR performance of I/Q imbalance estimation
algorithms are plotted and compared in Figures 2-5 with
different modulations, DC offset and SNR ranges. As shown in
the QPSK modulation scenario in Figure 2, the proposed
algorithm with cyclic frequency k=1 outperforms the Anttila’s
algorithm at low SNR (4~9.4 dB) and underperforms at SNR
above 9.4 dB. It is because Anttila’s estimate is unbiased at
high SNR, however biased at low SNR. On the other hand, the
proposed algorithm with cyclic frequency k=0 provides about
the same IRR performance as Anttila’s algorithm in the entire
range of SNR of interest. As shown in the 16QAM modulation
scenario in Figure 3, the proposed algorithm with cyclic
frequency k=1 outperforms the Anttila’s algorithm with SNR
below 9.2 dB and underperforms with SNR above 9.2 dB.
Similar to the trend observed in Figure 2, the proposed
algorithm with cyclic frequency k=0 provides about the same
performance as Anttila’s algorithm.

With the presence of DC-offset (as shown in Figures 4 and
5), the proposed algorithm with cyclic frequency k=1 yields
better IRR performance than the Anttila’s algorithm at both
low and high SNR regardless of modulation. This is because
proposed algorithm is unbiased when DC-offset exists, while
Anttila’s algorithm is biased. In the meanwhile, the proposed

algorithm with cyclic frequency k=1 yields about the same IRR
performance as the Anttila’s algorithm.

VI. CONCLUSIONS
In the paper, we proposed a new blind I/Q imbalance

estimation algorithm that exploits cyclostationarity. The
proposed blind estimation algorithm uses second-order
statistics to compensate I/Q imbalance instead of estimating
the mismatch parameters directly, and is an unbiased estimator
when DC offset exists. The performance results show that the
proposed algorithm is a very promising solution to I/Q
imbalance estimation and compensation.

APPENDIX
The detailed mathematical derivation of the one-tap

compensator in Section III is described here.
First, we have

⎥⎥⎦


⎢⎢⎣

⎥⎥⎦


⎢⎢⎣


ΘΘ
ΘΘ=⎥⎥⎦


⎢⎢⎣


][

][

)()(
)()(

][

][
**

1
*
2

21
* nz

nz

KK
KK

nx

nx
. (23)

Then, we can write (23) in the matrix form as Kzx = .
Due to the cyclostationary property of received signal z ,

we have { } 0=zzE . However, we have { } 0≠xxE because of I/Q
imbalance. Hence, we seek a compensation filter

[ ]TT ww 21,=w which restores the output xwTz =ˆ , which
yields { } 0ˆ =zzE . We have { }zzE ˆ expressed as

{ }rrE ˆ = { }rET xw = { }zzETΚw , (24)
where { }zE z is given by

{ }zE z = ⎪⎭
⎪⎬


⎪⎩
⎪⎨


⎥⎥⎦


⎢⎢⎣


][
][

][
* nznz

nz
E =

{ }{ }⎥⎥⎦⎤⎢⎢⎣⎡ ][][
][][

* nznzE

nznzE
=

⎥⎥⎦


⎢⎢⎣


2

0


. (25)

Then, we have

{ }zzE ˆ =
⎥⎥⎦


⎢⎢⎣


2

0

z

T

σΚw = )()(
*
1221 Θ+Θ KwKw = 0 (26)

Hence, we have the one-tap compensator given by

w =
)(
)(

2

*
1

2

1

Θ
Θ−=

K
K

w
w . (27)



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cyclostationary random signal processes,” IEEE Trans. Inform.
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[2] B. Razavi, “Design considerations for direct-conversion
receivers,” IEEE Trans. Circuits Syst. II, vol. 44, pp. 428-435,
Aug. 1987.

[3] F. Gini and G. B. Giannakis, “Frequency offset and symbol
timing recovery in flat-fading channels: A cyclostationary
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Mar. 1998.

[4] L. Anttila, M. Valkama, and M. Renfors, “Blind moment
estimation techniques for I/Q imbalance in quadrature
receivers,” in Proc. IEEE Int. Symp. Personal, Indoor, and



5

Mobile Radio Communications (PIMRC’06), Helsinki Finland,
Sep 2006.

[5] L. Anttila, M. Valkama, and M. Renfors, “Circularity-based I/Q
imbalance compensation in wideband direct-conversion
receivers ” Proc. IEEE,Trans. Vehicular Technology, vol 99,
issue 99 pp 1-15, Sep 2007.

[6] M. Valkama, M. Renfores, and V. Koivunen, “Blind signal
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[7] P. Rykaczewski, M. Valkama, M. Renfors, and F. Jondral,
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[8] I-H. Sohn, E-R. Jeong, and Y. H. Lee, “Data-aided approach to
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[9] H. Bölcskei, “Blind estimation of symbol timing and carrier
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[10] Y. Wang, P. Ciblat, E. Serpedin, and P. Loubaton, “Performance
Analysis of a Class of Non data-Aided Frequency Offset and
Symbol Timing Estimators for Flat-Fading Channels,” IEEE
Trans. Signal Processing vol.50, no. 9, pp. 2295-2305, Sep 2002.





I/Q
Im

ba
la

nc
e

Es
tim

at
or

&


C
om

pe
ns

at
or






Figure 1: Direct-conversion receiver structure.



4 5 6 7 8 9 10
20

25

30

35

40

45

50

55

SNR (dB)

Im
ag

e
R

ej
ec

t R
at

io
(I

R
R

)

QPSK IQ imbalance parameters g=1.07 phi=2, roll off factor=0.22

Anttila’s
cyclo 0
cyclo 1
analog




Figure 2: IRR performance of the compensated signal with QPSK
modulation at low SNR.

8 10 12 14 16 18 20
20

25

30

35

40

45

50

55

SNR (dB)

Im
ag

e
R

ej
ec

t R
at

io
(I

R
R

)

16-QAM IQ imbalance parameters g=1.07 phi=2, roll off factor=0.22

Anttila’s
cyclo 0
cyclo 1
analog


Figure 3: IRR performance of the compensated signal with 16 QAM

modulation at high SNR.


4 5 6 7 8 9 10
15

20

25

30

35

40

45

50

SNR (dB)

Im
ag

e
R

ej
ec

t R
at

io
(I

R
R

)

QPSK IQ imbalance parameters g=1.07 phi=2, DC offset = 0.1, roll off factor=0.22

Anttila’s
cyclo 0
cyclo 1
analog


Figure 4: IRR performance of the compensated signal with QPSK

modulation at low SNR with DC offset.


12 13 14 15 16 17 18 19 20
20

25

30

35

40

45

50

SNR (dB)

Im
ag

e
R

ej
ec

t R
at

io
(I

R
R

)

16-QAM IQ imbalance parameters g=1.07 phi=2, DC offset=0.1, roll off factor=0.22

Anttila’s
cyclo 0
cyclo 1
analog




Figure 5: IRR performance of the compensated signal with 16 QAM
modulation at high SNR with DC offset.