## Joint Estimation and Compensation of Transmitter and

## Receiver Radio Impairments in MIMO-OFDM Receivers

### Chen-Jiu Hsu*, and Wern-Ho Sheen**

* Department of Communication Engineering, National Chiao Tung University, Hsinchu, 300 Taiwan ** Department of Information and Communication Engineering, Chaoyang University of Technology, 413 Taiwan

E-mail : linch.ee89@nctu.edu.tw, whsheen@cyut.edu.tw

**Abstract— This paper aims to improve the performance of**

**estimation and compensation for both the transmitter and **
**re-ceiver radio impairments in the MIMO-OFDM (multiple-input,**
**multiple-output orthogonal frequency-division multiplexing) **
**sys-tems. First, a joint least-squares estimation of channel and**
**radio impairments is developed with a complete set of radio**
**impairments being taken into consideration, including **
**frequency-dependent and infrequency-dependent I-Q imbalances, dc-offsets and**
**frequency-offset. Previously, only parts of the radio impairments**
**were included and/or treated separately from the radio channel.**
**Secood, a novel two-stage compensation scheme is proposed**
**which is applicable to a general form of MIMO operations with**
**any number of transmit and receive antennas. Numerical results**
**show that the new design significantly outperforms the existing**
**ones in error-rate performance and/or the number of training**
**symbols required.**

I. INTRODUCTION

Direct-conversion radio architecture has been widely used in today’s wireless devices because it is more amenable to mono-lithic integration and thus offers a low cost, small form factor design [1]. Nevertheless, it introduces radio impairments such as I-Q imbalance and dc-offset that, along with frequency off-set, incur severe degradation in communication performance if not compensated accurately. Estimation and compensation of the radio impairments in the direct-conversion architecture has been a topic of extensive research [2]-[10]; the transmitter and receiver I-Q imbalances were investigated in [5][7][8] for the spatial-multiplexing MIMO-OFDM systems, and in [9] for the space-time block coded (STBC) systems, where a post-FFT compensation was proposed jointly with symbol detection on the extended channel that has a larger system dimension than the original system, which may largely increase the detector’s complexity. In [10], the authors proposed per-tone equalization (PTEQ) to the spatial-multiplexing systems in the presence of the transmitter and receiver I-Q imbalances and frequency offset. However, the proposed method is only applicable to a linear MIMO detection and suffers from slow convergence.

This paper aims to improve the performance of estimation and compensation for both the transmitter and receiver radio impairments in the MIMO-OFDM receivers. Taking into con-sideration both the transmitter and receiver radio impairments is crucial in a wireless peer-to-peer communication, where a less precise analog front-end is likely to be implemented at both sides of communication. This paper is unique in twofold: First, the channel and the radio impairments are

estimated jointly in an optimal way under the least-squares (LS) criterion. Second, a novel two-stage compensation is proposed which is applicable to a general form of MIMO operations with any number of transmit and receive antennas. Simulation results show that the new design significantly outperforms the existing ones in error-rate performance and/or the number of training symbols required.

II. SYSTEMMODEL

Figure 1 is the considered MIMO-OFDM system with a direct-conversion radio transceiver. Coming out of the OFDM base-band processing, the signal for transmit antenna i is

si(t) =
K−1_{X}
k=−P
N−1_{X}
n=−Ng
si,k(n) δ (t − (k (N + Ng) + n) Ts)
(1)
where
si,k(n) =
1
N
N−1_{X}
l=0
Si,k(l) exp
j2πnl
N
(2)
is the inverse fast Fourier transform (IFFT) of the transmitted
data {Si,k(l)}N−1_{l=0} in OFDM symbol k, j = √−1, Ng is
the length of cyclic prefix, N is FFT size, K + P is the
packet length (in OFDM symbols), Ts is the data (symbol)
duration, and δ(t) is the Dirac delta function. The length of
cyclic prefix is assumed larger than the maximum delay spread
of the overall channel, and therefore there is no inter-block and
inter-carrier interference. The data transmission is on a
packet-by-packet basis with the first P OFDM symbols, indexed from
K = −P to K = −1, as the training sequence for data-aided
estimation.

Figure 1 (a) shows the direct-conversion radio transmitter for transmit antenna i, where f0,i= f0,iI +jf

Q

0,iis the dc-offset, and hI

t,i(t) + jh Q

t,i(t) is the (unit-energy) base-band transmit filter. If hI

t,i(t) 6= h Q

t,i(t), it is said that there is a frequency-dependent I-Q imbalance. The frequency-infrequency-dependent I-Q imbalance, on the other hand, is characterized by the param-eters αt,i and θt,i, which are the gain and phase imbalance respectively due to imperfect analog circuits of the mixer. fcis the carrier frequency which is same for all transmit antennas; in other words, only one oscillator is used for all antennas at the transmitter for implementation simplicity. Figure 1(b) is the MIMO channel, where ˜hj,i(t) = Re{hj,i(t)ej2πfct} is the channel response from transmit antenna i to receive

π α π θ − + π α π θ − +

Fig. 1. Direct-conversion radio transceiver with I-Q imbalances, dc-offsets and frequency-offset (a) transmitter, (b) MIMO channel, and (c) receiver.

antenna j, and hj,i(t) is the equivalent base-band. ˜wj(t) = Rew0,j(t)ej2πf0t

is the pass-band additive white Gaussian noise, and w0,j(t) is its base-band equivalent.

Figure 1(c) is the direct-conversion radio receiver for antenna j. Let αr,j and θr,j be the frequency-independent gain and phase imbalance, respectively, hI

r,j(t) + jh Q r,j(t) be the (unit-energy) base-band filter, and d0,j = dI0,j + jd

Q 0,j be the dc-offset. Again, it is said to have frequency-dependent I-Q imbalance if hI

r,j(t) 6= h Q

r,j(t). f0 = fc − ∆f is the local oscillator frequency of the receiver, where ∆f is the frequency-offset between transmitter and receiver which is same for all receiver branches. After some manipulations, the received signal after sampling can be expressed as [6]

rj(n) = rjI(n) + jr Q j (n) = hr+,j(n) ⊗yj(n)ej2πνn+ w0,j(n) + hr−,j(n) ⊗ yj(n)ej2πνn+ w0,j(n) ∗ + d0,j (3) where yj(n) = nt X i=1 [si(n) ⊗ ht+,i(n)

+s∗i (n) ⊗ ht−,i(n) + f1,i] ⊗ hj,i(n) , (4) hr±,j(n) = 1/2 · [hIr,j(n) ± αr,je∓jθr,jhQr,j(n)], ht±,i(n) = 1/2 · [hI

t,i(n) ± αt,iejθt,ihQt,i(n)], f1,i = f0,i⊗ ht+,i(n) + f∗

0,i⊗ht−,i(n), ν = ∆fTSis the normalized frequency-offset, and w0,j(n)is a zero mean additive white Gaussian noise. [x]∗ and ⊗ denote the complex conjugate of x and the operation of linear convolution, respectively. In addition, hr−,j(n) ⊗

yj(n) ej2πνn+ w0,j(n)∗ in (3), and s∗i(n) ⊗ ht−,i(n) in (4) are the mirror interferences induced by the receiver and transmitter I-Q imbalances, respectively.

III. JOINTESTIMATIONOFRADIOIMPAIRMENTSAND

CHANNEL

From (3) and (4), it can be seen that the mirror interferences due to the transmitter and receiver I-Q imbalances appear at different mirror frequencies in the presence of frequency offset, and thus they cannot be combined as one and compensated by applying the technique developed in [6], where only the receiver radio impairments were considered. (The same argument applies to the dc-offset). This motivates us to propose a two-stage compensation scheme (detailed in

this and next sections). With the two-stage compensation in mind, a time-domain filter is introduced as follows for each receive branch, say branch j, aiming to cancel out the mirror interference due to the receiver I-Q imbalance

rj(n) − ρj(n) ⊗ r∗j(n)
= ˜hr+,j(n) ⊗ yj(n)ej2πνn+ w0,j(n)
+ ˜hr−,j(n) ⊗ yj(n)ej2πνn+ w0,j(n)
∗
+ dj (5)
where ˜hr±,j(n) = hr±,j(n) − ρj(n)⊗h∗_{r∓,j}(n), and
dj = d0,j − ρj(n) ⊗ d∗_{0,j}. For a perfect cancellation,
˜

hr−,j(n) = 0, that is, ρj(n) = (h∗r+,j(n))−1⊗ hr−,j(n), where (h∗

r+,j(n))−1 is the inverse filter of h∗r+,j(n). In practice, however, the cancellation may not be perfect. In this case, rj(n) − ρj(n) ⊗ rj∗(n) = e j2πνn " Xnt i=1 si(n) ⊗ h+,j,i(n) + s∗ i(n) ⊗ h−,j,i(n) ! + fj # + dj+ wj(n) (6) where wj(n) =˜hr+,j(n) ⊗ w0,j(n) + ˜hr−,j(n) ⊗ yj(n)ej2πνn+ w0,j(n) ∗ , (7) h±,j,i(n) = h. t±,i(n) ⊗ hj,i(n) ⊗ (˜hr+,j(n) e−j2πνn), and fj =

nt

P i=1

f1,i⊗ hj,i(n) ⊗ (˜hr+,j(n) e−j2πνn). Here, h+,j,i(n)is the overall impulse response from transmit branch i to receive branch j involving transmit filter, channel, and receive filter after canceling out the receiver mirror interfer-ence while h−,j,i(n) is the overall impulse response due to the transmitter I-Q imbalance. fj and dj are the equivalent transmitter and receiver dc-offset to be compensated, and wj(n) is the composite effects of additive white Gaussian noise and residual error after receiver I-Q cancellation. In (6), {ρj(n)}, {h±,j,i(n)}, {fj}, {dj}, and ν are the parameters needed to be estimated and compensated from the received signal before it can be passed on to later signal processing. The estimation will be based on the least-squares principle; that is, the optimum parameters are sought to minimize the square error of {wj(n)}.

Similar to [6], ρj(n) and h±,j,i(n) are approximated by the FIR filters ρj, h±,j,i, with the length Lρ and Lh±, respectively.1 Recall that the first P OFDM symbols,

indexed from k = −P, · · · , −1, serve as the training sequence for the estimation of radio impairments and channel. Let rj(k) = [rj,k(0), · · · , rj,k(N − 1)]T with rj,k(n) = r. j(k(Ng+ N ) + n) be the useful part of OFDM symbol k, wj(k) = [wj,k(0), wj,k(1), · · · , wj,k(N − 1)]T with wj,k(n) = wj(k(Ng + N ) + n), Rj(k) be the N × Lρ received signal matrix with the (p, q)-th entry [Rj(k)]p,q = rj,k(p − q), 0 ≤ p ≤ N − 1, 0 ≤ q ≤ Lρ− 1, and T±,i(k) be the N × Lh± signal matrix with

[T±,i(k)]p,q = si,k(p − q), 0 ≤ p ≤ N − 1, 0 ≤ q ≤ Lh±− 1.
Furthermore, define rj = [rTj(−P ), · · · , rTj(−1)]T,
Rj = [RTj(−P ), · · · , RTj(−1)]T, T(k) =
[T+,1(k), · · · , T+,nt(k), T
∗
−,1(k), · · · , T∗−,nt(k), 1N],
hj = [hT+,j,1, · · · , hT+,j,nt, h
T
−,j,1, · · · , hT−,j,nt, fj]
T_{, and}
wj = [wTj(−P ), · · · , wTj(−1)]T. From (6) and using these
notations, we have

rj− R∗jρj = Γ (ν) Thj+ dj1+ wj (8)

where T = [TT

(−P ), · · · , TT

(−1)]T_{, 1 is the}
all 1 vector with dimension NP , and Γ (ν) =
diag{Γ−P(ν) , · · · , Γ−1(ν)} with Γk(ν) = ej2πk(Ng+N )ν ·
diag{1, ej2πν_{, · · · , e}j2πν(N −1)_{}. Since the optimization}
problem formulation in (8) carries the same form as that in
[6], the recursive optimization procedure developed there can
be used here to obtain the joint estimates.

IV. TWO-STAGECOMPENSATIONOFRADIOIMPAIRMENTS

We consider the general MIMO structure of
linear-dispersion (LD) codes, which subsumes spatial multiplexing
and STBC as special cases and is applicable to any number of
transmit and receive antennas [11]. Without loss of generality,
the first code block that starts from the zero-th OFDM symbol
is considered for notation simplicity. From [11], a set of ns
data symbols {Dm(l)}n_{m=1}s , which is to be transmitted on
sub-carrier l of κ consecutive OFDM symbols, is encoded as
a κ × nt LD code matrix S(l) as follows

S(l) =
S1,0(l) . . . Snt,0(l)
... ... ...
S1,κ−1(l) · · · Snt,κ−1(l)
=
ns
X
m=1
(Re {Dm(l)} Am+ jIm {Dm(l)} Bm) (9)
where Am and Bm are κ × nt complex-valued dispersion
matrices that are designed to reap diversity and/or degree of
freedom gains of the MIMO channel. As in [11], with a proper
selection of the dispersion matrices, spatial-multiplexing and
STBC can be viewed as special cases of (9). Recall that
{Si,k(l)}N−1_{l=0} is the data symbols input to the IFFT in (2).

1_{Throughout this paper, bold uppercase letters denote matrices and bold}

lowercase letters denote vectors. (·)T _{and (·)}H _{represent the operations of}

transpose and conjugate transpose of a matrix or vector, respectively.

( ) − ( ) πνε − ( ) − − ( ) ρ ( ) ! "" { }+ { }− ( )− ( ) ( )− ( )

Fig. 2. The two-stage compensation scheme.

The proposed two-stage compensation is shown in Figure 2. From (6), the time-domain compensation is straightfor-ward, where the receiver I-Q imbalance is compensated first, followed by the receiver dc-offset, frequency-offset and the transmitter dc-offset. With perfect compensation, the received signal becomes zj,k(n) = e−j2π ˆνε(n,k)[rj,k(n) − ˆρj(n) ⊗ r∗j,k(n) − ˆdj] − ˆfj = nt X i=1

si,k(n) ⊗ h+,j,i(n) + s∗i,k(n) ⊗ h−,j,i(n)

+ ωj,k(n) (10) n = 0, · · ·, N − 1, where ε (n, k) = k (N + Ng) + n, and ωj,k(n) = e−j2π ˆνε(n,k)wj,k(n). (10) says that at this point the mirror interference due to the transmitter I-Q imbalance is the only impairment left to be compensated. Here, we propose a new method, called frequency-domain mirror-interference cancellation, for the compensation. Starting from (10), taking FFT on zj,k(n),

Zj,k(l) = nt

X i=1

[H+,j,i(l)Si,k(l) + H−,j,i(l)Si,k∗ (−l)] + Ωj,k(l) (11) where {X (l)}N−1 l=0 . = FFTh{x(n)}N−1n=0 i , and {Ω (l)}N−1 l=0 . = FFTh{ω(n)}N−1n=0 i

. In addition, using a matrix form for those Zj,k(l) corresponding to the LD code matrix S(l), one has

Z(l) = S (l) G+(l) + S∗(−l) G−(l) + Ω (l) (12) where Z(l) = Z1,0(l) · · · Znr,0(l) ... ... ... Z1,κ−1(l) · · · Znr,κ−1(l) (13) Ω(l) = Ω1,0(l) · · · Ωnr,0(l) ... ... ... Ω1,κ−1(l) · · · Ωnr,κ−1(l) (14) and G±(l) = H±,1,1(l) · · · H±,nr1(l) ... · · · ... H±,1,nt(l) · · · H±,nr,nt(l) (15)

Furthermore, let zj(l), ωj(l), and g±,j(l) denote the j-th column of Z(l), Ω(l), and G±(l), respectively. Define ˜d(l) =

[Re{D1(l)}Im{D1(l)}, . . . , Re{Dns(l)}Im{Dns(l)}]
T_{,}
˜

z_{(l) = [Re{z}T

1(l)}Im{zT1(l)}, . . . , Re{zTnr(l)}Im{z T nr(l)}]

T_{,}
˜

ω_{(l) = [Re{ω}T

1(l)}Im{ωT1(l)}, . . . , Re{ωTnr(l)}Im{ω
T
nr(l)}]
T_{,}
e
A±,m=
Re {Am} ∓Im {Am}
±Im {Am} Re {Am}
,
and
e
B±,m=
−Im {Bm} ∓Re {Bm}
±Re {Bm} −Im {Bm}
.

As in [11], it is convenient to rewrite (12) in the form of real matrices and vectors as follows

˜

z(l) = eG+(l) ˜d(l) + eG−(l) ˜d(−l) + ˜ω(l) (16) where eG±(l) shown at the bottom of the page is a tall or square matrix with dimension 2κnr× 2ns. From (16), the mirror-interference cancellation is proposed, by introducing the filter Φ (l), as follows

˜
z_{(l) − Φ (l) ˜z (−l) =}hGe+(l) − Φ (l) eG−(−l)
i
˜
d(l)
+hGe−(l) − Φ (l) eG+(−l)
i
| {z }
=0

to cancel mirror interference ˜

d_{(−l) + [ ˜}ω_{(l) − Φ (l) ˜}ω_{(−l)]}

= H (l) ˜d(l) + ω (l) (17)

where H (l) = eG+(l) − Φ (l) eG−(−l) and ω (l) = ˜ω(l) − Φ(l) ˜ω(−l) is the effective channel and noise after compensa-tion. Clearly, to cancel out the mirror interference completely, we need to have

e

G−(l) − Φ (l) eG+(−l) = 0 (18) For the case κnr= ns, (18) has a unique solution of

b

Φ(l) = eG−Ge−1+ (−l) (19)
for the case κnr > ns, however, there are infinite solutions.
Naturally, the one with minimum noise power E{kω(l)k2_{} is}
sought in this case. That is, the optimal bΦ(l) is obtained by
solving the following constrained optimization problem.

b

Φ(l) = arg Φ(l)

min E{kω(l)k2}, s.t. eG−(l) − eΦ(l) eG+(−l)=0 (20) Moreover, it can be shown that

Enkω (l)k2o≈ 2κnrσ2+ σ2tr

ΦT(l)Φ(l) (21) Here we have used the approximations E{ ˜ω(l) ˜ωT

(l)} ≈
σ2_{I}
2κnrand E
˜
ω(l) ˜ωT(−l) ≈ 0. Therefore, (20) becomes
b
Φ(l) = arg
Φ(l)
min tr{ eΦT(l) eΦ_{(l)}, s.t. e}G−(l)− eΦ(l) eG+(−l)=0
TABLE I
RF PARAMETERVALUE
(αt,i, θt,i) (1.05, 5o),(0.94, −6o)
(αr,j, θr,j) (1.08, 5o),(0.91, 6o),(0.92, −5o),(1.09, −6o)
{hI
t,i(n), h
Q
t,i(n)}, Ipart : [1 0.3 0.2 0.1]
{hI
r,j(n), h
Q
r,j(n)} Qpart : [0.9 0.4 0.15 0.15]

∆f uniform over -0.5 and 0.5 subcarrier spacing |f0,i|, |d0,j| 0.15, −0.1, 0.08, −0.12

The problem amounts to solve the minimum norm solution of
the linear equations of eGT_{+}_{(−l) e}ΦT(l) = eGT_{−}(l) for eΦT(l)
which is given by
b
Φ(l) =
e
G_{+}_{(−l)}GeT_{+}_{(−l) e}G_{+}_{(−l)}
−1
e
GT_{−}(l)
T
(22)
After the mirror-interference cancellation, any type of MIMO
detectors can be used for detecting ˜d(l) from (17). Note that
the second-stage compensation deals the same problem as that
tackled in [5] and [7]-[9], where only the impairment of I-Q
imbalance was considered with the detection done on extended
channel. This will largely increase the detector’s complexity
especially if MAP (Maximum a posteriori) or ML (Maximum
Likelihood) type of detection is employed. In our approach, on
the contrary, the system dimension is kept the same as the one
with no I-Q imbalance, which may result in a lower detection
complexity.

V. NUMERICALRESULTS

The performance of the proposed receiver is evaluated for an uncoded MIMO-OFDM system with 64-QAM modulation order and MMSE detection. The system parameters are set as FFT length N = 64, cyclic prefix length Ng = 16, and symbol time Ts = 50ns. The dc-offset is given by d0,j = |d0,j| · (1 + j)/

√

2and f0,i= |f0,i| · (1 + j)/ √

2with signal
power normalized to 1. Table 1 gives the impairments
param-eters. The transmission is done on a packet-by-packet basis
beginning with the training sequence similar to 802.11a spec
[12]. An exponential decay multipath channel is considered
with root-mean square delay spread TRM S = 50ns. The length
of channel is 10 taps, and each tap is zero mean independently
complex Gaussian random variable. The parameters are set as
Lh_{+}= 13and Lh−= 9.

Figure 3 shows the BER performance over fading channels with different Lρ’s that characterize the effect of FIR approxi-mation of the filters {ρj(n)}

nr

j=1. Lρ≥ 5 is usually enough to obtain good performance in all our numerical results. The BER performance with receiver radio impairments compensation only proposed in [6] is also shown for comparison purpose. Clearly, the transmitter radio impairments incurs error floor if left not compensated, as one can expect. Figure 4 compares

e
G±(l) =
e
A±,1ge±,1(l) Be±,1eg±,1(l) · · · Ae±,nsge±,1(l) Be±,nseg±,1(l)
... ... ... ... ...
e
A±,1ge±,nr(l) eB±,1eg±,nr(l) · · · Ae±,nseg±,nr(l) Be±,nseg±,nr(l)
, ˜g_{±,j}(l) =
Reg_{±,j}(l)
Img_{±,j}(l)

5 10 15 20 25 30 10−5 10−4 10−3 10−2 10−1 100

64 QAM, n_{t}=2, n_{r}=4, P=4, Fading Channel

SNR(dB)
Uncoded BER
Rx Comp [6]
*L*_{ρ} =5
Proposed
*L*_{ρ} =1
*L*_{ρ} =3
*L*_{ρ} =5
*L*_{ρ} =7
*L*_{ρ} =9
*L*_{ρ} =15
Ideal Receiver

Fig. 3. The effect of FIR approximation on the time-domain filter ρj(n).

5 10 15 20 25 30 10−5 10−4 10−3 10−2 10−1 100

64 QAM, n_{t}=2, n_{r}=4, Fading Channel

SNR(dB)
Uncoded BER PTEQ [10]_{P=48}

P=60
P=80
P=120
*Proposed, L*_{ρ} = 5
P=4
P=8
Ideal Receiver

Fig. 4. Performance comparisons between the proposed method and per-tone equalization [10] (spatial-multiplexing MIMO).

the BER performance of the proposed receiver with the PTEQ in [10], as an example of the spatial-multiplexing MIMO systems. Since there is no method for frequency and dc-offset estimation in [10], ideal frequency compensation with no dc-offset (both sides) is assumed for comparison purpose. As can be seen, the new receiver significantly outperforms PTEQ in terms of BER and the required training symbols. Figure 5 is the comparison of the proposed method with the one in [9] as an example of STBC-MIMO systems. Again, ideal frequency compensation with no dc-offsets is assumed. Clearly, the method [9] does not work properly in the presence of frequency-offset which ranges from -0.5 to 0.5 subcarrier spacing in our simulations.

VI. CONCLUSION

In this paper, a new estimation and compensation method is proposed for the transmitter and receiver radio impairments in the liner-dispersion coded MIMO-OFDM systems. The radio

5 10 15 20 25 30 35 10−5 10−4 10−3 10−2 10−1 100

64 QAM Alamouti Scheme, n_{t}=2, n_{r}=1, Fading Channel

SNR(dB) Uncoded BER Zou [9]

P=20
P=40
P=80
*Proposed, L*_{ρ} = 5
P=4
P=8
Ideal Receiver

Fig. 5. Performance comparisons between the proposed method and the one in [9] (STBC MIMO).

impairments and channel are jointly estimated under the least-squares criterion. A novel two-stage method consisting of time and frequency-domain compensation is proposed that is applicable to a general form of MIMO operations with any number of transmit and receive antennas. Numerical results show that significant performance improvement is observed for both the spatial-multiplexing and STBC MIMO systems.

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