A Virtual Channel-Based Approach to Compensation of I/Q Imbalances in MIMO-OFDM Systems

doi:10.4236/ijcns.2010.39095

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Int. J. Communications, Network and System Sciences, 2010, 3, 711-721

doi:10.4236/ijcns.2010.39095 Published Online September 2010 (http://www.SciRP.org/journal/ijcns)

A Virtual Channel-Based Approach to Compensation of

I/Q Imbalances in MIMO-OFDM Systems

Shichuan Ma, Deborah D. Duran, Hamid Sharif, Yaoqing (Lamar) Yang

Department of C omput er and Electronics Engineering, University of Nebr aska-Lincoln, Oma ha, USA

E-mail: sma@huskers.unl.edu, dduran@mail.unomaha.edu, yyang3@unl.edu, hsharif @unl.edu

Received June 28, 2010; revised July 28, 2010; accepted September 2, 2010

Abstract

Multiple-input multiple-output (MIMO)-orthogonal frequency-division multiplexing (OFDM) scheme has

been considered as the most promising physical-layer architecture for the future wireless systems to provide

high-speed communications. However, the performance of the MIMO-OFDM system may be degraded by

in-phase/quadrature-phase (I/Q) imbalances caused by component imperfections in the analog front-ends of

the transceivers. I/Q imbalances result in inter-carrier interference (ICI) in OFDM systems and cause

inaccurate estimate of the channel state information (CSI), which is essential for diversity combining at the

MIMO receiver. In this paper, we propose a novel approach to analyzing a MIMO-OFDM wireless communi-

cation system with I/Q imbalances over multi-path fading channels. A virtual channel is proposed as the

combination of multi-path fading channel effects and I/Q imbalances at the transmitter and receiver. Based

on this new approach, the effects of the channel and I/Q imbalances can be jointly estimated, and the influ-

ence of channel estimation error due to I/Q imbalances can be greatly reduced. An optimal minimal mean

square error (MMSE) estimator and a low-complexity least square (LS) estimator are employed to estimate

the joint coefficients of the virtual channel, which are then used to equalize the distorted signals. System

performance is theoretically analyzed and verified by simulation experiments under different system con-

figurations. The results show that the proposed method can significantly improve system performance that is

close to the ideal case in which I/Q are balanced and the channel state information is known at the receiver.

Keywords: I/Q Imbalance, Multiple-Input Multiple-Output (MIMO), Orthogonal Frequency-Division

Multiplexing (OFDM), Alamouti Scheme

1. Introduction

Due to the rapid growth of broadband wireless applications,

the next generation of wireless communication systems

poses major challenges for efficient exploitation of the

available spectral resources. Among the existing techniques,

the combination of orthogonal frequency-division multi-

plexing (OFDM) and multiple-input multiple-output (MI-

MO) has been widely considered the most promising ap-

proach for building future wireless transmission systems

[1]. OFDM, as a popular modulation scheme, converts a

frequency-selective fading channel into a parallel collection

of flat-fading sub-channels, enabling high data-rate trans-

missions [2-4]. On the other hand, by deploying multiple

antennas at both ends of the transmitter (TX) and receiver

(RX), MIMO architectures are capable of combating

channel fading by taking advantage of the spatial diversity

and/or efficient in enhancing system capacity by employing

spatial multiplexing [5,6]. The MIMO-OFDM technique

has attracted significant research interest in recent years

[7-10] and has been standardized in a number of commu-

nication systems [11-13].

Although OFDM presents numerous advantages, it suf-

fers from performance degradations due to the hardware

component flaws in the analog front-ends of the transceivers,

including phase noise [14], carrier frequency offset [15],

and in-phase/quadrature-phase (I/Q) imbalance [16,17].

The imbalance between the in-phase and the quadra-

ture-phase branches is a major factor in performance

degradation. When the received radio-frequency (RF) sig-

nal is down-converted to baseband, the analog front-end

imperfections cause imbalances between the I and Q

branches, which introduces inter-carrier interference (ICI)

and frequency-dependent distortion to the received data.

This leads to a decrease in the operating signal-to-noise

S. MA ET AL.

712

ratios (SNRs) and low data rates. Although several methods

were proposed to overcome ICI [18-20], I/Q imbalances

cannot be handled using these methods. The I/Q imbalances

become more severe when a direct-conversion receiver

(DCR) [21] or lower intermediate frequency (IF) [22] is

utilized. Furthermore, the effect of the I/Q imbalances on

the system is mixed with the signal distortion from the

fading channel, making it difficult to compensate. Conse-

quently, estimation and equalization of the I/Q imbalances

from the received data are critical in OFDM-based systems.

Frequency-independent and frequency-dependent I/Q im-

balance models are reported in [16] and [17], respectively.

Based on these models, the effects of I/Q imbalances are

studied in OFDM systems [23-28] and in MIMO-OFDM

systems [29-33].

In [29], the input-output relation of a MIMO-OFDM

system with frequency-independent I/Q imbalances at the

receiver is derived. Based on this result, an adaptive

me thod is introduced to compensate for the received data.

In [31], the effect of frequency-dependent I/Q imbalances

on MIMO-OFDM system is studied, using a pilot-based

compensation scheme. In both studies, however, the

received signals are diversity combined using inaccurate

channel state information (CSI) estimated under I/Q

imbalances, leading to system performance degradation.

We have proposed a method to deal with this problem in

[33]. By jointly estimating and compensating for multi-

path fading channels and I/Q imbalances, this method can

effectively m itigate th e I/Q ef fect s.

In this paper, a novel approach is proposed to analyze

MIMO-OFDM wireless communication systems with I/Q

imbalances over multi-path fading channels. A virt ual

channel is proposed to bypass the channel estimation under

I/Q influence. Based on this approach, the TX and RX I/Q

imbalances are treated as parts of the fading channels. The

effects of both the fading channels and I/Q imbalances on

the system can be jointly estimated before diversity com-

bining and are then employed for diversity combining and

signal compensation. A minimum mean square error

(MMSE) estimator and a least square (LS) estimator are

used to estimate the joint coefficients of the virtual channel.

A signal compensation approach based on a zero-forcing

algorithm is also provided. The system performance is

theoretically analyzed, and bit error rate (BER) is ex-

pressed in closed-form. Extensive simulation results veri-

fied that the I/Q imbalances at both transmitter and recei ver

sides can be effectively mitigated by using the virtual

channel approach.

The rest of this paper is organized as follows. In the fol-

lowing section, a frequency-dependent I/Q imbalances

model and an Alamouti scheme-based MIMO-OFDM

model with transmitter and receiver I/Q imbalances are

described. The input-output relation is derived in the

frequency domain. In Section 3, the MMSE and LS esti-

mators of the joint coefficients of the virtual channel are

developed. In Section 4, the BER of the system is theoreti-

cally analyzed. Simulation results are given in Section 5.

Finally, Section 6 concludes the paper.

Notatio ns: A small (capital) letter represents a variable in

time (frequency) domain, and a bold small (capital) letter

represents a vector (matrix). The superscripts *, T, H rep-

resent the conjugate, the transpose, and the Hermitian

transpose operations, respectively.denotes absolute val-

ue operation,



denotes Frobenius norm, and









repre-

sents expectation.







represents the Q-function.



denotes convolution.

2. System Model

2.1. I/Q Imbalance Model

A frequency-dependent I/Q model at the receiver side is

shown in Figure 1. As described in [17], the I/Q im-

balances arise from two effects. One is the effect of the

quadrature demodulator, which is frequency-independent

and determined by I/Q amplitude imbalance

and I/Q

phase imbalance



; another is the effect of branch

components, which is frequency-dependent and modeled

as two filters with frequency response,()

IRX

Lfand

,()

QRX

Lf. If the I/Q branches are perfect, then1







, and,,

()() 1

IRX QRX

LfLf



.

Assuming ()

rtis the baseband equivalent signal of

the received signal()

rt, the down-converted signal

()

ztcan be written as

1, 2,

()() ()() ()

RXRX BBRX BB

ztg trtgtrt (1)

where

1,, ,

2,, ,

()() ()/2

RXI RXQ RXRX

RXIRXQRXRX

gtltltge

gt ltltge

























(2)

,()

IRX

ltand ,()

QRX

ltare the time-domain representations

of ,()

IRX

and ,()

QRX

, respectively. From the point of

view of the frequency domain, the expression in (1) can

be written as

1, 2,

()()()()()

RXRX BBRX BB

fGfR fGfRf



 (3)

Figure 1. Block diagram of frequency-dependent I/Q im-

balance at a receiver.

S. MA ET AL.

713

where ()

Rfis the Fourier transform of()

rt, and

1,, ,

2,, ,

()() ()/2

RXIRXQ RXRX

GfLfL fge

Gf LfLfge

















(4)

In (3), the term*()

Rfrepresents the inter-carrier

interference (ICI) projected from the mirror frequency to

the signal frequency. This is called image projection, a

major problem caused by I/Q imbalances in signal

demodulations.

Similarly, the relation between the signalTX

Rand the

transmitted baseband signalTX

with I/Q mismatch can

be written as

1, 2,

()() ()() ()

TXTX TXTX TX

fGfRf GfRf (5)

where

1,, ,

2,, ,

()() ()/2

TXITXQ TXTX

TXITXQTXTX

GfLfL fge

Gf LfLfge

















(6)

where TX

denotes the I/Q amplitude imbalance at TX,



represents the I/Q phase imbalance at TX,

and ,()

ITX

Lfand,()

QTX

Lfindicate the non-linear fre-

quency characteristics of the I and Q branches at TX.

2.2. MIMO-OFDM Model with I/Q Imbalances

A block diagram of a MIMO-OFDM wireless communi-

cation system with Alam outi dive rsity schem e and freq uent-

cy-dependent TX and RX I/Q imbalances is shown in

Figure 2. In this system, there are two transmit anten-

nas and r

N (1) receive antennas. All signals are

represented in the form of space-time coded (STC)

blocks in the frequency domain. For example,

,1 ,2

SSdenotes an STC block data of2N



matrix,

where 1, 2,i



is the STC block index, and N is the

number of the used OFDM sub-carriers. Vector

,,, ,,

[(/2)( 1)(1)(/2)]

ij ijijijij

SNSSSN s is an

OFDM symbol to be transmitted over the system at the

th, {1,2}jj



time slot of thethiSTC block, where

,()

Sk denotes the symbol at thethk, {/2,kN





,1,1,, /2}N



 sub-carrier with average symbol en-

ergy /2

E. Vector,1i

Sis transmitted followed

by vector,2i

S. For simplicity, the block indexiis omitted

in Figure 2. ,nm

denotes the channel frequency re-

sponse between the thntransmit antenna and the

thmreceive antenna, where {1, 2}n and

{1, 2,,}



. The TX I/Q imbalance parameters are

denoted by ()

1,n

G and ()

G, and the RX I/Q imbalance

parameters are denoted by()

1,m

Gand ()

G,where again,

{1, 2}n



and {1, 2,,}



.

Assume two consecutive data symbols, ,1 ()

and

,2()

Sk, to be transmitted over thethksub-carrier. Based

Figure 2. Block diagram of a 2×Nr MIMO-OFDM wireless communication system with transmitter and receiver I/Q imbal-

ances. The virtual channel is illustrated in the dashed block.

S. MA ET AL.

714

on the Alamouti scheme, ,1 ()

Skand *

,2()

Skare dis-

tributed into thethksub-carrier data stream of the first

OFDM modulator, while,2()

Skand *

,1 ()

Skare distribute d

into the thksub-carrier data stream of the second OFDM

modulator. Data streams at all sub-carriers are then proc-

essed through OFDM modulation, including operations of

padding zeros, inverse fast Fourier transform (IFFT), and

adding cyclic prefix ( CP). Because the IFFT operation only

transforms the signal from frequency domain to time do-

main, the signals (viewed in the frequ ency domain) are not

changed after OFDM modulation. Therefore, after OFDM

modulation, the two consecutive data symbols at

the thksub-carrier of the first transmitter in the frequency

domain are still ,1()

and*

,2()

Sk.

The OFDM-modulated signals are then distorted by

TX I/Q imbalances. According to (5), the signals to be

transmitted via the first transmit antenna are given by

(1)(1)(1) *

,11, ,12, ,1

(1)(1) *(1)

,21,,2 2, ,2

()()()()( )

()()()() ()

iTXiTXi

UkGkSkGkSk

UkG kSkGkSk



 (7)

Similarly, the signals to be transmitted via the second

transmit antenna are

(2)(2)(2) *

,11, ,2 2,,2

(2)(2) *(2)

,21,,1 2,,1

()() ()() ()

iTXiTXi

UkG kSkGkSk

UkG kSkGkS k



(8)

The signals are then transmitted over a multi-path

fading channel. The received signals at thethm receive

antenna are given as

() (1) (2)

,11, ,12, ,1

() (1) (2)

,21,,22, ,2

()() ()()()

imimi

VkHkUkHkUk



 (9)

The received signals at thethmreceive antenna are further

corrupted by I/Q imbalances in thethmreceiver as follows:

()()()()*()

,11, ,12, ,1

()()()()*()

,21, ,22, ,2

()()()()()

mmmmm

iRXiRXi

mmmmm

iRXiRXi

WkGkVkGkVk



(10)

Assuming the noise in the system is additive white

Gaussian noise (AWGN), then

()() ()

,1 ,1,1

()() ()

,2 ,2,2

()() ()

mmm

iii

mmm

iii

kWkN k



 (11)

where()

,()

Nkis indepe ndently ide ntically distributed (i.i.d.)

complex zero-mean Gaussian noise with variance0

Combining (7), (8), (9), (10) and (11), the received

data symbols at thethmreceive antenna before diversity

combining can be written as

()()() *

,1,1 ,1

()() *()

,2,2 ,1

()() *()

,2,1,1

() *()()

,2,2 ,2

()()()() ()

() ()() ()()

()() ()() ()

() ()() ()()

mm m

iii

mm m

iii

mm m

iii

mm m

iii

XkAkSkBkSk

CkSkDkS kNk

XkCkSkDkSk

kS kBkSkNk









(12)

where ()

()

k,()

()

Bk,()

()

Ck, and()

()

Dkare de-

fined as

()() (1)

1, 1,1,

() *(1)*

2, 2,1,

()() (1)

1, 2, 1,

() *(1)*

2, 1,1,

()() (2)

1, 1,2,

() *(2)

2, 2,

()()() ()

()( )()

()()()()

()( )( )

()()() ()

() (

RX TXm

RX TX

AkGkGkHk

GkGkHk

Bk G kG kHk

GkG kHk

Ck G kGkHk

GkG k











*

()() (2)

1, 2,2,

() *(2)*

2, 1,2,

)()

()()()()

()() ()

RX TXm

DkGkG kHk

GkGkHk







(13)

From (12) and (13), it is observed that the channel

frequency response (,nm

) and the I/Q effects

(()

1,n

G,()

1,m

Gand ()

G) are mixed together and

produce the coefficients()

()

k,()

()

Bk,()

()

Ck,

and ()

()

Dk. If we treat the I/Q effects as parts of the

fading channels, we can model the dashed block in

Figure 2 as a virtual channel with joint coefficients

()

k,()

()

Bk,()

()

Ck, and()

()

Dk. Moreover,

()

kand ()

()

Ckare critical for data decoding, while

the presence of()

()

Bkand ()

()

Dkmay introduce ICI.

If the I/Q branches are perfect, then() 1,

() ()

kHk,

()

() 0



, ()2,

() ()

CkHk, and()

() 0



. Fur-

thermore, the received data symbols at thethksub-carrier

are distorted by the mixture effects of channel and I/Q

imbalances, and interference is introduced by other data

symbols within the STC block at thethkand the mirrored

-thksub-carriers.

To compensate for the received signals, the joint coeffi-

cients should be estimated. The estimation methods are

described in the next section. Now, assuming the accurate

estimate of the joint coefficients ar e obtained, the rece ived

data can be combined to achieve the transmit diversity as

()*() ()() *()

,1,1 ,2

()*() ()() *()

,2,1,2

()() ()()()

mmmmm

iii

mmmmm

iii

YkAkXkCkX k

YkC kXkAkXk





(14)

It should be noted that the combining method is

slightly different from the Alamouti scheme, where

channel state information is employed. In our scheme,

the joint coefficients()

()

kand ()

()

Ckare used. If the

I/Q branches are perfect, (14) is reduced to the standard

Alamouti diversity combining scheme as substituting (12)

and (13) into (14), we obtain

S. MA ET AL.

715

()*()*()

,11, ,12, ,2

()*()*()

,22, ,11, ,2

()() ()()()

mm m

imimi

mmm

imimi

YkHkXkHkX k



 (15)

()()() *

,11,12,1

() *()

3,2 ,1

()()*() *

,21 ,2 2,2

*( )*( )

3,1 ,2

()() ()() ()

()()()

()() ()() ()

()()()

mm m

iii

mm m

iii

YkQkSkQkS k

QkSkNk

YkQkSkQkSk

QkSkNk











(16)

where

()() 2() 2

()*() ()()*()

()*()()() *()

() |()||()|

()() ()()()

mm m

mmmmm

Qk AkCk

QkA kBkCkDk

QkA kDkCkBk





(17)

and

()*()()() *()

,1,1 ,2

()*() ()() *()

,1,1 ,2

()() ()()()

mmmmm

iii

mmmmm

iii

NkAknkCknk

NkCknkAknk







 (18)

Finally, the received data symbols at multiple receive

antennas are combined to obtain receiv er diversity, and the

raw data symbols are given by

()

,1 ,1

*()*

1,1 2,13 ,2,1

()

,2 ,2

1,2 2

signalintercarrier interferenceno i se

signal

() ()

()()()( )()( )()

() ()

()()()

ii ii

YkYk

QkSk QkSk QkSkNk

YkY k

QkS kQkS























 



**()*

,23,1,2

intercarrier interferencenoise

()()() ()

mii

kQkSkNk 





(19)

where

()

() ()

QkQ k

QkQk



































(20)

are defined as the combined coefficients, and

()

,1 ,1

()

,2 ,2

() ()

NkN k



























(21)

are the combined noise.

From (19), it can be seen that there is ICI in the signals

after receiver combining, which is caused by the image

projections from the mirrored frequency. To improve

system performance, it is necessary to compensate for

the received signals.

3. Estimators and Signal Compensation

In this section, we describe two training sequence-based

estimators to show how to estimate the joint coefficients

of the virtual channel. The first is a minimal mean square

error (MMSE) estimator. Although MMSE estimator is

an optimal linear detector, it requires some priori know-

ledge of the estimated variables and is computationally

intensive. An alternative is a least square (LS) estimator,

which may significantly reduce the computational com-

plexity at the expense of negligible BER degradation.

The estimated joint coefficients can be used to perform

diversity combining as in (14) and to compensate for the

received signals as described at the end of this section.

3.1. MMSE Estimator

Assume totaltr

Nblocks of training sequences are used in

this system (That is2tr

NOFDM symbols). Let,()

Pk,

{1, 2}j



denote thethjtraining symbol in thethiblock

at thethk sub-carrier. According to (12), the in-

put-output relation for one block can be written as

()() ()

()() ()()

mmm

ii i

kkk kuPvn

(22)

where

()() ()

,1 ,2

()() ()T

mmm

iii

kXkXk





u (23)

,1,1,2 ,2

,2,2,1 ,1

()()()( )

() ()()() ()

ii ii

iiiii

PkPk PkPk

kPk PkPkP k













 









P (24)

()()()()()

()() () ()()T

mmmmm

ikAkB kCkDk











v (25)

()()()

,1 ,2

()() ()T

mmm

iii

kNkNk











n (26)

For the totalTblocks, the input-output relation is

written as

()() ()

() () ()()

mmm

kkk kuPvn (27)

where

()()()()

()() ()()

mmTmTmT

kkk k











uuu u (28)

()() ()()

TT T

kkk k











PPPP (29)

()()()()

()() ()()

mmTmTmT

kkk k











nnn n (30)

The MMSE estimate ofvis given as [34]

() 1()

ˆ() ()

vu u



vRRu (31)

where

() ()

{()()} ()

mmH H

vu v

Ekk kRvu RP (32)

() ()

{()()}()()

mmH H

uvn

Ekkk kRuuPRP R (33)

() ()02

{()()}

mmH

EkkNRnn I (34)

() ()

{() ()}

mmH

vEkkRvv (35)

S. MA ET AL.

716

3.2. LS Estimator

Although MMSE estimator is optimal, it suffers from

h igh co mputational complex ity and requires th e knowled g e

of v

R, which must be estimated using a large amount of

transmission data. A simple but effective method is least

square estimator. To further reduce the computational

complexity, we design a special training pattern to avoid

matrix inversion operation. In order to utilize this special

pattern, training sequences must be transmitted in groups

of two blocks. Let

, and *

be the four consecutive

training symbols within the thiand the(1)thiST C

blocks at thethksub-carrier, where(1 )

pjis a com-

plex number with identical real and imaginary partsp.

The corresponding training symbols at thethksub-carrier

are also

, and*

. According to (12), the received

data within thethiand the(1)thiSTC blocks at

the thksub-carrier can be represented in matrix form as

() ()

,1 ,1

() ()

,2 ,2

()

() ()

1,1 1,1

() ()

1,2 1,2

() ()

()

() ()

kNk

ssss

kNk

ssssk

kNk

ssss

kNk

ssss



 



 





 





 



 







 



 

(36)

Thus, the LS estimate of the joint coefficients is given as [35]

1()

** ,1

()

** ,2

() ()

** 1,1

()

** 1,2

()

1,1

()

1,2

()

ˆ() ()

()

01 1()

10 1()

ssss

kXk

ssss









 





















 



















(37)

The estimates of the virtual channel coefficients from

different training sequence groups can be averaged to

obtain more accurate estimate.

3.3. Signal Compensation Approach

Based on the MMSE estimate given by (31) or LS estimate

given by (37) of the joint coefficients, the estimate of the

combined coefficients, 1

ˆ()Qk,2

ˆ()Qk, and3

ˆ()Qk, can be

calculated according to (17) and (20) , and then can be used

to equalize the raw data symbols. According to (19), the

raw data symbols at thethkand the ()thksub-carriers

within thethiSTC block can be written in matrix form as

(38). Thus, the zero-forcing estimates of the transmitted

data symbols are given by (39).

123 1,1

****

2131 ,

31 2*

321

()

ˆˆ ˆ

()()0()() ()

ˆˆˆ

()()()0()

ˆˆˆ ()

0()()()

()

ˆˆˆ

()0() ()

Qk QkQkSk

QkQkQkSk N

Qk QkQkSk

Qk QkQk













































1

()



































(38)

12 31,1

****,1

213

31 22*

** ,2

321

ˆˆ ˆˆ()()0()

() ()

ˆˆˆ

ˆ()

()() ()0()

ˆˆˆˆ()

0()()()

() ()

ˆˆˆ

ˆ()0() ()()

Qk QkQk

Sk Yk

QkQkQkSk

Qk QkQk

Sk Yk

Qk QkQkSk











































 













(39)

4. Performance Analysis

According to (19), the received raw data symbols are

contaminated by noise and interference from the signals

at the mirrored sub-carriers. If the ICI can be success-

fully canceled by the proposed algorithm, the post- proc-

essing SNR at thethksub-carrier, ()k



, can be calcu-

lated as







1,1 1,2

,1 ,2

() 2() 2

|()()||() ()|

() |()||( )|

|( )||( )|

1|()||()|

Nmm

QkS kQkSk

knk nk

AkCk N

Ak Ck



























(40)

where /2

Eis the average transmit energy per symbol

period per antenna and0





can be interpreted as

the average SNR for the single-input single-output scheme.

This post-processing SNR is determined by the joint

effects of channels and I/Q imbalances. For classical i.i.d.

channels [5] and perfect I/Q characteristics, the post-

processing SNR becomes

() r







(41)

This shows that the system with perfect I/Q over i.i.d.

channel can achieve an array gain ofr

In OFDM systems, each sub-carrier can be treated as a

frequency-flat channel. Assuming optimum detection at

the receiver, the corresponding symbol error rate for

rectangular -a ryMQAM is given by [36]

S. MA ET AL.

717

13()

()11 211

PkQ M





 







(42)

Assuming that only one single bit is changed for each

erroneous symbol, the equivalent bit error rate for rec-

tangular -aryMQAM is approximated as [37]

() ()

()

Pk Pk

log M

 (43)

5. Simulation Results

To evaluate the proposed virtual channel idea, the two

estimators, and the signal compensation approach, we

used MATLAB to simulate an OFDM-based2r



MI-

MO system with frequency-dependent TX and RX I/Q

imbalances. The size of fast Fourier transform (FFT) is

128, the number of used sub-carriers is 96, and the length

of CP is 32. The multi-path channel is modeled by six in-

dependent complex taps with a power delay profile of a 3

dB decay per tap. It should be noted that the actual channel

length can be estimated [38]. The simulation bandwidth is

set to 20 MHz, leading to a 156.25 KHz sub-carrier spac-

ing and a maximum 0.3



excess delay. Sixty-four qua-

drature amplitude modulations (64QAM) are used.

The non-linear frequency characteristics of the I and Q

branches, ()

Lfand( )

Lf, are modeled as two first-

order finite impulse response (FIR) filters. A form of

parameters



,,[ ,],[,]

II QQ

ab ab



is used to describe

the I/Q imbalance, where[,]

aband[ ,]

abare the co-

efficients of the FIR filters for the I and Q branches.

I/Q parameters of



1.03,3,[0.01,0.9],[0.8,0.02] and



[0.8,0.02],[0.01,0.9] are used for the two transmitters,

respectively. For simplification of simulation, all receivers

use I/Q parameters of





1.05,3,[0.8,0.02],[0.9,0.01].

Because the estimation of the joint coefficients is per-

formed separately in each receiver, the same I/Q pa-

rameters for different I/Q models still lead to generaliza-

tion of the simulation results. It should be noted that the

I/Q imbalance parameters are chosen to be worst case in

order to evaluate the robustness of the proposed app ro ach.

A frame-by-frame transmission scheme is employed in

the simulation. The multi-path channel is independently

generated for each frame. One frame consists of tr

STC blocks of training symbols followed by 50 STC

blocks of data symbols. A total of 5,000 frames (288

Mbits) are simulated for each scenario at a given SNR.

Figure 3 shows the frequency response of two of the

FIR filters used for simulating the frequency-dependent

characteristics of I/Q imbalances. For both filters, the

variations of the amplitude are within 0.5 dB, but the

average g ains are d ifferentiated b y 1 dB. It shou ld be noted

that the signals after TX I/Q distortions must be nor malized

in the simulations to compensate for the energy lost due to

attenuations of the filters. While the variation of the phase

response for the first filter is small, the variation is very

large for the second filter. The amplitude and phase differ-

ences of the filters are suitable to simulate the fre-

quency-dependent characteristics of I/Q imbalances.

Figure 4 shows the typical constellations of one frame of

symbols generated in an ideal channel environment (without

multi-path fading and noise). Figures 4 (a) and (b) show the

constellations of the raw data symbols under frequency-

independent and frequency-dependent I/Q imbalances, re-

spectively. It is observed that the constellation in Figure 4 (b)

beco mes nearly random co mpared with Figure 4 (a) due to

the effect of frequency-de pendent I/Q imbalances. Figure 4

posed LS algorithm, which shows that the proposed algo-

rithm can perfectly compensate frequency-independent and

frequency -depen dent I/Q im balances.

The BER performances of the proposed estimators and

compensation approach are shown in Figures 5-10. To

make better comparisons, a series of simulations under

different scenarios was conducted, including a scenario of

perfect I/Q and CSI known at receivers termed as “ideal

case”, a scenario with frequency-independent I/Q imbal-

ances termed as “indep-IQ”, and a scenario with fre-

quency-dependent I/Q imbalances termed as “dep-IQ”.

The legend term of “NoComp” means that I/Q imbal-

ances are presen t but no compensation scheme is applied

(the CSI is estimated under I/Q imbalances), “MMSE

(N)” means that the MMSE estimator withN STC

blocks of training sequences (the total2N OFDM

Figure 3. Frequency response of the FIR filters used for

simulating the frequency-dependent characteristics of I/Q

imbalances; coefficients of FIR 1 are [0.8, 0.02], and coeffi-

cients for FIR 2 are [0.01,0.9].

S. MA ET AL.

718

(a) (b) (c)

Figure 4. Constellations of signals in a frame generated by a 2 × 1 MIMO-OFDM system under ideal channel environment. (a)

raw data constellations under freque ncy -independe nt I/Q imbalanc es; (b) r aw data constellations unde r fre quenc y- depe ndent

I/Q imbalances; (c) constellations of the compensated data under frequency-dependent I/Q imbalances.

symbols) is used, and “LS(N)” means that the LS esti-

mator withNSTC blocks of training sequence is used to

estimate the joint coefficients. For both MMSE and

10 1520 2530 35 40

−6

−5

−4

−3

−2

−1

SNR (dB)

BER

2x1 analysis

2x1 ideal case

2x1 indep−IQ NoComp

2x1 indep−IQ MMSE(2)

2x1 indep−IQ LS(2)

Figure 5. Performance of a 2 × 1 MIMO-OFDM system

with TX and RX frequency-independent I/Q imbalances

over multi-path fading channel.

10 15 20 25 30 35 4

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BER

2x2 analysis

2x2 ideal case

2x2 indep−IQ NoComp

2x2 indep−IQ MMSE(2)

2x2 indep−IQ LS(2)

Figure 6. Performance of a 2 × 2 MIMO-OFDM system

with TX and RX frequency-independent I/Q imbalances

over multi-path fading channel.

LS estimators, the proposed compensation approach is

applied to compensate the received raw data symbols.

Furthermore, the BER is theoretically calculated accord-

ing to (43) and is shown in the following figures with

legend term “analysis”.

Figure 5 and Figure 6 show the performance of a



and a22



systems with frequency-independent I/Q

imbalances, respectively. It is observed from the “No-

comp” curves that the I/Q imbalances significantly de-

grade the system performance, resulting in high error

floor. These results agree with the constellation analysis

mentioned above. With the proposed estimators and sig-

nal compensation approach, the I/Q distortion can be

effectively mitigated and the system performance is sig-

nificantly improved. By using two STC blocks of train-

ing sequences, the system performance resulting from

both MMSE and LS estimators is close to the ideal case.

The performance degradation is less than 1 dB. Although

LS estimator performs a little worse than MMSE estimator,

the low computational complexity makes the LS estimator

more competitive. By assuming perfect estimation of the

1015 2025 303540

−6

−5

−4

−3

−2

−1

SNR (dB)

BER

2x1 analysis

2x1 ideal case

2x1 dep−IQ NoComp

2x1 dep−IQ MMSE(2)

2x1 dep−IQ LS(2)

Figure 7. Performance of a 2 × 1 MIMO-OFDM system

with TX and RX frequency-dependent I/Q imbalances over

multi-path fading channel.

S. MA ET AL.

719

10 15 20 25 30 35 40

−6

−5

−4

−3

−2

−1

SNR (dB)

BER

2x2 analysis

2x2 ideal case

2x2 dep−IQ NoComp

2x2 dep−IQ MMSE(2)

2x2 dep−IQ LS(2)

Figure 8. Performance of a 2 × 2 MIMO-OFDM system

with TX and RX frequency-dependent I/Q imbalances

over multi-path fading channel.

10 15 20 2530 35 40

−6

−5

−4

−3

−2

−1

SNR (dB)

BER

2x1 ideal case

2x1 indep−IQ NoComp

2x1 indep−IQ MMSE1

2x1 indep−IQ MMSE2

2x1 indep−IQ MMSE4

Figure 9. Comparison of the performances of the MMSE

estimator with different lengths of training symbols for the

2 × 1 MIMO-OFDM system with TX and RX frequency-

independent I/Q imbalances over multi-path fading chan-

nel.

10 15 20 25 30 35 40

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BER

2x1 ideal case

2x1 indep−IQ NoComp

2x1 indep−IQ LS(1)

2x1 indep−IQ LS(2)

2x1 indep−IQ LS(4)

Figure 10. Comparison of the performances of the LS esti-

mator with di fferent lengths of t raining sym bols for the 2 × 1

MIMO-OFDM system with TX and RX frequency-indepen-

dent I/Q imbalances over multi-path fading channel.

joint coefficients and compensation of the received sig-

nal, it is reasonable that the analysis results are slightly

better than the ideal case.

The performance of the systems with frequency-

dependent I/Q imbalances are shown in Figure 7 and

Figure 8 for the21



and 22scenarios, respectively.

The frequency dependent characteristic of the I/Q im-

balances causes fatal influence on the MIMO-OFDM

systems. Without compensation, the BER remains one

half regardless of the SNR. Our proposed approach can

also successfully combat the worst fading effect caused

by frequency-dependent I/Q imbalances, resulting in

good performance that is close to the ideal case. This

significant performance improvement has not been re-

ported by other literatures.

An intuitive sense is that a longer training sequence

could result in better performance. To demonstrate this

point, we compare the system performances with differ-

ent lengths of training symbols in Figure 9 and Figure

10 for MMSE and LS estimators, respectively. It is ob-

served that BER d ecreases with the increase of the train-

ing symbol numbers. When four blocks of training se-

quences are used, the performance loss compared to the

ideal case is negligible.

6. Conclusions

In this paper, we introduce a new virtual channel concept

to analyze the I/Q imbalances in a MIMO-OFDM wire-

less communication system over multi-path fading

channels. The input-output relation is derived in fre-

quency domain, which incorporates the effect of I/Q im-

balances with multi-path fading chann els. The integrated

effect can be modeled by the joint coefficients of the

virtual channel. By using this approach, inaccurate esti-

mation of the channel state information can be avoided

at the diversity combining stage. The joint coefficients

are estimated by using the proposed MMSE and LS es-

timato rs, an d are us ed to co mpen sate th e receiv ed sign als .

Simulation results show that the proposed approach can

effectively mitigate the TX and RX I/Q imbalances in

MIMO-OFDM systems. Although only a two transmit

antenna scheme is illustrated in this paper, our proposed

approach can be easily extended to any number of trans-

mit antenna configurations by using orthogonal space-

time block coding.

7. Acknowledgements

This study was supported by the Advanced Telecommu-

nications Engineering Laboratory (www.TEL.unl.edu)

and was partially funded by the US Federal Railroad

Administration (FRA) under the direction of Terry T se.

S. MA ET AL.

720

8. References

[1] M. Jang and L. Hanzo, “Multiuser MIMO-OFDM for

Next-Generation Wireless Systems,” Proceedings of the

IEEE, Vol. 95, No. 7, July 2007, pp. 1430-1469.

[2] R. Chang and R. Gibby, “A Theoretical Study of Per-

formance of an Orthogonal Multiplexing Data Transmis-

sion Scheme,” IEEE Transactions on Communications

Technology, Vol. 16, No. 4, August 1968, pp. 529-540.

[3] Y. Wu and W. Y. Zou, “Orthogonal Frequency Division

Multiplexing: A Multi-Carrier Modulation Scheme,”

IEEE Transactions on Consumer Electronics, Vol. 41,

No. 3, August 1995, pp. 392-399.

[4] L. Hanzo, M. Muinster, B. Choi and T. Keller, “OFDM

and MC-CDMA for Broadband Multi-User Communica-

tions,” Wlans and Broadcasting, Wiley, 2003.

[5] A. Paulraj, R. Nabar and D. Gore, “Introduction to Space-

Time Wireless Communications,” Cambridge University

Press, Cambridge, 2003.

[6] Y. Yang, G. Xu and H. Ling, “An Experimental Investi-

gation of Wideband MIMO Channel Characteristics

Based on Outdoor Non-Los Measurements at 1.8 GHz,”

IEEE Transactions on Antennas Propagation, Vol. 54,

November, 2006, pp. 3274-3284.

[7] H. Sampath, S. Talwar, J. Tellado, V. Erceg and A. J.

Paulraj, “A Fourth-Generation MIMO-OFDM Broadband

Wireless System: Design, Performance and Field Trial

Results,” IEEE Communications Magazine, Vol. 40, No.

9, September 2002, pp. 143-149.

[8] G. L. St¨uber, J. R. Barry, S. W. McLaughlin, Y. G. Li,

M. A. Ingram and T. G. Pratt, “Broadband MIMO-

OFDM Wireless Communications,” Proceedings of the

IEEE, Vol. 92, No. 2, February 2004, pp. 271-294.

[9] Y. Li, J. H. Winters and N. R. Sollenberger, “MIMO-

OFDM for Wireless Communications: Signal Detection

with Enhanced Channel Estimation,” IEEE Transactions

on Communications, Vol. 50, No. 9, September, 2002, pp.

1471-1477.

[10] M. Shin, H. Lee and C. Lee, “Enhanced Channel-

Estimation Technique for MIMO-OFDM Systems,”

IEEE Communications Letters, Vol. 53, No. 1, January

2004, pp. 262-265.

[11] IEEE Standard for Local and Metropolitan Area Net-

works, Part 16: Air Interface for Fixed Broadband Wire-

less Access System, Octobe r 2004.

[12] IEEE Candidate Standard 802.11n: Wireless LAN Me-

dium Access Control (MAC) and Physical Layer (PHY)

Specifications, 2004.

[13] 3GPP Technical Specification Group Radio Access Net-

work, Overall Description of E-UTRA/E-UTRAN, 2007.

[14] S. R. Herlekar, C. Zhang, H.-C. Wu, A. Srivastava and Y.

Wu, “OFDM Performance Analysis in the Phase Noise

Arising from the Hot-Carrier Effect,” IEEE Transactions

on Consumer Electronics, Vol. 52, No. 3, August 2006,

pp. 757-765.

[15] J. Lei and T.-S. Ng, “A Consistent OFDM Carrier Fre-

quency Offset Estimator Based on Distinctively Spaced

Pilot Tones,” IEEE Transactions on Wireless Communi-

cations, Vol. 3, No. 2, March 2004, pp. 588-599.

[16] C. L. Liu, “Impacts of I/Q Imbalance on QPSK-OFDM-

QAM Detection,” IEEE Transactions on Consumer Elec-

tronics, Vol. 44, No. 3, August 1998, pp. 984-989.

[17] M. Valkama, M. Renfors and V. Koivunen, “Compensa-

tion of Frequency-Selective I/Q Imbalances in Wideband

Receivers: Models and Algorithms,” Proceedings of the

3rd IEEE Workshop on Signal Processing Advances in

Wireless Communications, Taiwan, March 2001, pp.

42-45.

[18] H.-C. Wu and Y. Wu, “Distributive Pilot Arrangement

Based on Modified M-sequences for OFDM Inter-Carrier

Interference Estimation,” IEEE Transactions on Wireless

Communications, Vol. 6, No. 5, May 2007, pp. 1605-

1609.

[19] X. Huang and H.-C. Wu, “Robust and Efficient In-

ter-Carrier Interference Mitigation for OFDM Systems in

Time-Varying Fading Channels,” IEEE Transactions on

Vehicular Technology, Vol. 56, No. 5, September 2007,

pp. 2517-2528.

[20] H.-C. Wu, X. Huang, Y. Wu and X. Wang, “Theoretical

Studies and Efficient Algorithm of Semi-Blind ICI Equa-

lization for OFDM,” IEEE Transactions on Wireless

Communications, Vol. 7, No. 10, October 2008, pp. 3791-

3798.

[21] A. A. Abidi, “Direct-Conversion Radio Transceivers for

Digital Communications,” IEEE Journal of Solid-State

Circuits, Vol. 30, No. 12, December 1995, pp. 1399-

1410.

[22] J. Crols and M. Steyaert, “Low-IF Topologies for High-

Performance Analog Front Ends of Fully Integrated Re-

ceivers,” IEEE Transactions on Circuits System, Vol. 45,

No. 3, March 1998, pp. 269-282.

[23] A. Tarighat, R. Bagheri and A. H. Sayed, “Compensation

Schemes and Performance Analysis of IQ Imbalances in

OFDM Receivers,” IEEE Transactions on Signal Process-

ing, Vol. 53, No. 8, August 2005, pp. 3257-3268.

[24] J. K. Hwang, W. M. Chen, Y. L. Chiu, S. J. Lee and J. L.

Lin, “Adaptive Baseband Compensation for I/Q Imbalance

and Time-Varying Channel in OFDM System,” Intelli-

gent Signal Processing and Communications, Japan, De-

cember 2006, pp. 638 – 641.

[25] J. Tubbax, B. Come, L. van der Perre, S. Donnay, M.

Engles and C. Desse t, “Joi nt Co mpe nsa ti on of I Q Imbal an ce

and Phase Noise,” IEEE Semiannual on Vehicular Tech-

nology Conference, Vol. 3, April 2003, pp. 1605-1609.

[26] D. Tandur and M. Moonen, “Joint Compensation of

OFDM Frequency-Selective Transmitter and Receiver IQ

Imbalance,” EURASIP Journal on Wireless Communica-

tions and Networking, Vol. 2007, No. 68563, May 2007,

pp. 1-10.

[27] L. Anttila, M. Valkama and M. Renfors, “Frequency-

Selective I/Q Mismatch Calibration of Wideband Di-

rect-Conversion Transmitters,” IEEE Transactions on

Circuits System, Vol. 55, No. 4, April 2008, pp. 359-363.

[28] S. Ma, D. Duran and Y. Yang, “Estimation and Compen-

S. MA ET AL.

721

sation of Frequency-Dependent I/Q Imbalances in OFDM

Systems over Fading Channels,” Proceedings of the 5th

International Conference on Wireless Communications,

Networking and Mobile Computing, Beijing, September

2009.

[29] A. Tarighat, W. M. Younis and A. H. Sayed, “Adaptive

MIMO OFDM Receivers: Implementation Impairments

and Complexity Issues,” IFAC Workshop on Adaptation

and Learning in Control and Signal Processing, Yoko-

hama, August 2004.

[30] Y. Zou, M. Valkama and M. Renfors, “Digital Compen-

sation of I/Q Imbalance Effects in Space-Time Coded

Transmit Diversity Systems,” IEEE Transactions on

Signal Process, Vol. 56, No. 6, June 2008, pp. 2496-

2508.

[31] Y. Zou, M. Valkama and M. Renfors, “Analysis and

Compensation of Transmitter and Receiver I/Q Imbal-

ances in Space-Time Coded Multi-Antenna OFDM Sys-

tems,” EURASIP Journal on Wireless Communications

and Networking, No. 391025, January 2008, pp. 1-10.

[32] D. Tandur and M. Moonen, “Compensation of RF Im-

pairments in MIMO OFDM Systems,” IEEE Interna-

tional Conference on Acoustics, Speech and Signal Proc-

essing, Las Vegas, April 2008, pp. 3097-3100.

[33] S. Ma, D. Duran, H. Sharif and Y. Yang, “An Adaptive

Approach to Estimation and Compensation of Fre-

quency-Dependent I/Q Imbalances in MIMO-OFDM

Systems,” Proceedings of IEEE Global Communications

Conference, Honolulu, 2009.

[34] A. H. Sayed, Adaptive Filters, Wiley-IEEE Press, New

Jersey, 2008.

[35] D. Manolakis, V. K. Ingle and S. M. Kogon, “Statistical

and Adaptive Signal Processing: Spectral Estimation,

Signal Modeling, Adaptive Filtering and Array Processing,”

MA: Artech House Publishers, Boston, 2005.

[36] J. G. Proakis, “Digital Communications,” 4th edition,

Mc-Graw-Hill, New York, 2000.

[37] R. L. Peterson, R. E. Ziemer and D. E. Borth, “Introduc-

tion to Spread-Spectrum Communications,” 3rd edition,

Prentice Hall, New Jersey, 1995.

[38] X. Wang, H. C. Wu, S. Y. Chang, Y. Wu and J. Y.

Chouinard, “Efficient Non-Pilot-Aided Channel Length

Estimation for Digital Broadcasting Receivers,” IEEE

Transactions on Broadcasting, Vol. 55, No. 3, September

2009, pp. 633-641.