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

Research Paper / Jan 2008

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

Research Paper / Jan 2008

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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

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

zσ

. (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)

REFERENCES

[1] W. A. Gardner and L. E. Franks, “Characterization of

cyclostationary random signal processes,” IEEE Trans. Inform.

Theory, vol. IT-21, no. 1, pp. 4-14, Jan. 1975.

[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

approach,” IEEE Trans. Commun., vol. 46, no. 3, pp. 400-412,

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

estimation in conjugate signal models with application to I/Q

imbalance compensation,” IEEE Signal Processing Lett., vol.

12, pp. 733-736, Nov. 2005.

[7] P. Rykaczewski, M. Valkama, M. Renfors, and F. Jondral,

“Non-data-aided I/Q imbalance compensation using measured

received front-end signals,” in Proc. IEEE Int. Symp. Personal,

Indoor, and Mobile Radio Communications (PIMRC’06),

Helsinki Finland, Sep 2006.

[8] I-H. Sohn, E-R. Jeong, and Y. H. Lee, “Data-aided approach to

I/Q mismatch and DC-offset compensation in communications

receivers,” IEEE Commun. Lett., vol. 6, pp. 547-549, Dec. 2002.

[9] H. Bölcskei, “Blind estimation of symbol timing and carrier

frequency offset in wireless OFDM systems,” IEEE Trans.

Commun, vol.49, no. 6, pp. 988-999, Jun 2001.

[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

Iˆ

Qˆ

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.

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