## Cross-Validation Estimation for Frequency-Dependent I/Q

## Imbalance in MIMO-OFDM Receivers

Wei-Chi Lai&Ta-Yang Juan&Terng-Yin Hsu&

Sheng-Lun Chiou

Received: 4 March 2008 / Revised: 4 August 2009 / Accepted: 6 August 2009 / Published online: 8 September 2009
*# 2009 Springer Science + Business Media, LLC. Manufactured in the United States*

Abstract The effects of variation in RF, such as I/Q imbalance and filter mismatch, are extremely important for OFDM wireless accesses. This work presents a low-computational estimation of I/Q imbalances with filter mismatches to improve performance in MIMO-OFDM receivers. For N×N MIMO-OFDM systems, the proposed cross-validation estimation is such that, only N+1 preambles are required to extract the mismatches of filters, gains and phases. With the estimated parameters, frequency-domain filters are exploited to correct frequency-dependent I/Q imbalances. Through performance evaluation of a 2 ×2 MIMO-OFDM system, with ideal channel estimations this study incurs a SNR loss of 1–1.2 dB to maintain a 10% PER at 1-dB gain error, 10°-phase error and the worst 180°-filter mismatch. In addition, this algorithm is well-matched to IEEE 802.11n and new specifications discussed in IEEE 802.11 VHT study group.

Keywords Filter mismatch . I/Q imbalance . Cross validation . Frequency-Dependent Imbalance (FDI) . MIMO-OFDM

1 Introduction

Multiple-input multiple-output orthogonal frequency divi-sion multiplexing (MIMO OFDM) is widely used for next-generation communications over frequency-selective fading [1, 2]. With the help of multiple transmitted antennas, multiple received antennas and special spatial formats, data accesses can be faster and more efficient than single-input single-output (SISO) systems. Unfortunately, RF distortions will cause serious degradations of performance in most OFDM-based designs. The key impacts are: the gain and phase mismatch between in-phase (I) and quadrature-phase (Q), namely I/Q imbalance; and a filter mismatch between I and Q denoted by frequency-dependent imbalance (FDI). In practice, the simultaneous occurrence of I/Q imbalance and filter mismatch significantly degrades the system perfor-mance. Several schemes for OFDM systems have been proposed to handle I/Q imbalance [3–7]. These schemes can be categorized as follows.

1) Data-aided (DA) methods, in which special pilots and training symbols are inserted into the transmitted data [3,4].

2) Nondata-aided (NDA) methods, in which the transmit-ted data are used without any other additional infor-mation [5,6].

3) Adaptive FIR filter based methods, in which the received data before the FFT are utilized to handle estimation, were presented to certify for an Image Rejection Ratio (IRR) of 75–97 dB after compensation. [7].

Some blind I/Q imbalance estimators that rely on signal
statistics have also been adopted to achieve estimation and
compensation [8]. Although these methods can work well
under I/Q imbalance, filter mismatches were less addressed
and a lot of cycles were required for processes. Therefore,
W.-C. Lai (**)*

### :

T.-Y. Juan### :

T.-Y. HsuDepartment of Computer Science, National Chiao Tung University, 1001, University Road,

Hsinchu 300 Taiwan, Republic of China e-mail: wclai@cs.nctu.edu.tw

T.-Y. Hsu

e-mail: tyhsu@cs.nctu.edu.tw S.-L. Chiou

Institute for Information Industry, Taipei, Taiwan, Republic of China e-mail: richard.chiou@nmi.iii.org.tw DOI 10.1007/s11265-009-0403-y

introducing the filter mismatch phenomenon into the system increases the estimation error. The I/Q imbalance with filter mismatch, namely frequency-dependent I/Q imbalance, is much critical for OFDM direct-conversion receivers to make the corresponding estimation and compensation be extremely difficult [3,9].

To ensure that MIMO-OFDM wireless accesses function at frequency-dependent I/Q imbalances, this study presents a novel method to extract the useful properties of preambles to reduce the converged cycles. As a result, the interleaving arrangement of preambles can be utilized, namely cross validation, to estimate frequency-dependent I/Q imbalances for N×N MIMO-OFDM systems within N+1 preambles. Then a frequency-domain (FD) filter is programmed by the estimated parameters to remove these I/Q distortions in each received antenna. Simulations of a 2×2 MIMO-OFDM system indicate that the proposed approach results in a 1.2-dB SNR loss with ideal channel estimations to achieve 50-dB IRR and 8% PER at frequency-dependent I/ Q imbalance with a gain error of 1 dB, a phase error of 10° and the worst FDI of 180° [3]. Although 50-dB IRR with a SNR loss of 1–1.2 dB is not excellent, it is adequate to make systems work properly. Besides, the preamble-based solution is easily compatible with IEEE 802.11n [10] and next-generation wireless LAN discussed in IEEE 802.11 Very High Throughput (VHT) study group.

The rest of this paper is structured as follows. Section II addresses I/Q model and problem statement. Section III presents the proposed algorithm and architectures of estimation and compensation for frequency-dependent I/Q imbalances. The simulations and measurements are dis-cussed in Section IV. Finally section V gives conclusions.

2 System Assumptions

2.1 Packet Format

Each MIMO-OFDM symbol has 64 carriers that 56 sub-carriers are data and 8 sub-sub-carriers are nulls. The time-domain signal is preceded by the guard interval containing the last 16 samples of the OFDM symbol in every antenna. The packets are assumed to contain the common preambles (CP) and the MIMO preambles (MP) that CPs are all the same to every antenna, like the preambles of SISO-OFDM systems. Both CP and MP include two types: (1) short preambles and (2) long preambles, as illustrated in Fig. 1. The common short preambles (CSP) can be applied for packet detection and coarse synchronization, and the common long preamble (CLP), being real number, can be used for fine synchroni-zation. The MIMO short preambles (MSP) are exploited for antenna calibrations, and all MIMO long preambles (MLP) are tone interleaved across transmitted antennas for channel

estimations. The number of MLPs is equal to the number of transmitted antennas. Its valid length is equal to the number of data sub-carriers. In addition, the field of“Packet Type” records the number of antennas, a coding rate of FEC and modulation of data. Table 1 presents the tone partitions of MLPs used for 1×1, 2×2, 3×3 and 4×4 MIMO-OFDM systems as similar to IEEE 802.11n [10]. The examples of tone interleaving of MLPs across two transmitted antennas are illustrated in Figs. 2 and 3. In Fig. 2, MLPs are rearranged. (MLP3

MLP54¼ 1,

CLP3

CLP54¼ þ1) is one of the odd pairs; (MLP2

MLP55¼ 1,

CLP2

CLP55¼ þ1) is one of the even pairs. Odd pairs and even pairs all satisfy:

MLPðf Þ
MLPðfÞ¼ 1
CLPðf Þ
CLPðfÞ
) MLPk
MLP57k ¼ 1
CLPk
CLP57k*; k 2 Z and 1 k 56*
ð1Þ

where CLPk and MLPk denote the kth fraction (data sub-carrier) of CLP and MLP, respectively. Based on these pairs satisfying the property above, the proposed method can utilize these pairs to estimate the frequency-dependent I/Q imbalance. In a 2 × 2 MIMO-OFDM system, due to frequency-selective fading, different antenna will receive different ratios of the received CLPs and the received MLPs. Therefore, the frequency-dependent I/Q imbalance can be estimated in each receiver. In order to make sure the accuracy of the estimation, there are at least 1/3 fractions of MIMO long preambles distributing uniformly to satisfy Eq. (1).

2.2 I/Q Imbalance Model

Figure 4 displays the model of frequency-dependent I/Q imbalances [3]. The I/Q imbalance caused by the local oscillator (LO) can be characterized with an amplitude mismatch g=(1+ε) and a phase error θ; ideally g=1 and θ=0. In this study, I/Q signals are filtered by two mismatched LPFs with frequency responses of HI( f ) and Hg(f), respectively; different received antennas contain different frequency-dependent I/Q imbalances. In each received antenna, the LO signal of the mismatch quadrature demodulator is modeled as follows:

XLOðtÞ ¼ ILOðtÞ þ jQLOðtÞ

¼81zðtÞ þ82z *ðtÞ

ð2Þ

The coefficients81and 82in Eq. (2) are

81¼ 1 þ gejq =2 ð3aÞ 82¼ 1 gejq =2 ð3bÞ

proof
cos 2ð _{pf}ctÞ jg sin 2pfð ctþ qÞ ¼12 ej2pf
ct_{þ e}j2pfct
þ1
2g ej 2pf
ctþq
ð Þ_{ e}j 2pfð ctÞ
¼1
2ð1þ gejqÞej2pfc
t_{þ}1
2ð1 gejqÞej2pfc
t
With Fourier transformation, the baseband signal of each
received antenna is denoted as

RBBðf Þ ¼ IBBðf Þ þ jQBBðf Þ ¼ ILOðf ÞHIðf Þ þ jQLOðf ÞHQðf Þ ¼ aðf ÞZðf Þ þ bðf ÞZ* fð Þ ð4Þ where aðf Þ ¼ HIðf Þ þ HQðf Þgejq =2 ð5aÞ bðf Þ ¼ HIðf Þ HQðf Þgejq =2 ð5bÞ

In Eq. (4), Z *ðfÞ is the image aliasing effect, caused by I/Q distortions. It is also known that the baseband signal is distorted by self-mixing. The basic model [11, 12] is a special case of HIðf Þ ¼ HQðf Þ, where aðf Þ ¼

1þ gejq

ð Þ=2 and bðf Þ ¼ 1 ge_{ð} jq_{Þ=2. Figure} _{5} _{plots the}

channel frequency response (CFR) influenced by frequency-dependent I/Q imbalance with 1-dB gain error, 10°-phase error & 20°-phase error, and 180°-filter mismatch [3], in which FDI induces high-frequency distortions.

2.3 Problem Statement

For packet accesses, if one of data bits can not be decoded correctly, the entire packet will be discarded and resent. It means that I/Q distortions must be calibrated before receiving data. Most studies [6–8] can provide a high IRR without a significant loss of SNR, but special data or more than 1000 frames are needed. Besides, filter mismatches are less addressed. Further-more, CLPs and MLPs do not carry full I/Q information for all sub-carriers to balance throughput and perfor-mance. According to above issues, the objective of this study is to derive an efficient solution for frequency-dependent I/Q imbalances as well-matched to IEEE 802.11n [10] and new specifications discussed in IEEE 802.11 VHT study group.

3 Joint-estimation Algorithm

To reduce the computation complexity, CFRs are estimated by the intersection of MLPs since all MLPs are tone-interleaved across different antennas, shown in Figs.2 and 3. In the case of 2 × 2 MIMO-OFDM systems, four estimated CFRs areH^11ðf Þ, ^H12ðf Þ, ^H21ðf Þ and ^H22ðf Þ

respectively, where H^ijðf Þ denotes the estimated CFR

from the ithtransmitted antenna to the jthreceived antenna. Since there are two cases of MLPk=MLP57k satisfying

MLPk=MLP57k

ð Þ= CLPð k=CLP57kÞ ¼ 1,H^11ðfÞ,H^12ðfÞ, ^

H21ðf Þ and H^22ðf Þ are first estimated from the received

Antenna number Set 0 Set 1 Set 2 Set 3

0 [−28 : 1 : −1] [1 : 1 : 28] 1 [−28 : 2 : −2] [−27 : 2 : −1] [2 : 2 : 28] [1 : 2 : 27] 2 [−28 : 3 : 1] [−27 : 3 : −3] [−26 : 3 : −2] [2 : 3 : 26] [3 : 3 : 27] [1 : 3 : 28] 3 [−28 : 4 : −4] [−27 : 4 : −3] [−26 : 4 : −2] [−25 : 4 : −2] [1 : 4 : 25] [2 : 4 : 26] [3 : 4 : 27] [4 : 4 : 28]

Table 1 Tone partitions of MLPs for 1×1, 2×2, 3×3 and 4×4 MIMO-OFDM systems.

Common Short Preambles (CSP)

Common Long Preamble (CLP)

Packet Type

MIMO Short Preambles (MSP)

MIMO Long Preamble

(1st MLP) Data Part 1 Antenna 1

MIMO Long Preamble (Nth MLP) Common Short Preambles

(CSP)

Common Long Preamble (CLP)

Packet Type

MIMO Short Preambles (MSP)

MIMO Long Preamble

(1st MLP) Data Part 2 Antenna 2

MIMO Long Preamble (Nth MLP)

Common Short Preambles (CSP)

Common Long Preamble (CLP)

Packet Type

MIMO Short Preambles (MSP)

MIMO Long Preamble

(1st MLP) Data Part N Antenna N

MIMO Long Preamble (Nth MLP) Figure 1 The preamble format of the desired N×N MIMO-OFDM system.

MLPs with MLPk=MLP57k ¼ þ1, as revealed in following equations. ^ H11ðf Þ ¼ a1ðf ÞH11ðf Þ þ b1ðf ÞH11ðfÞ ð6aÞ ^ H12ðf Þ ¼ a2ðf ÞH12ðf Þ þ b2ðf ÞH12ðfÞ ð6bÞ ^ H21ðf Þ ¼ a1ðf ÞH21ðf Þ þ b1ðf ÞH21ðfÞ ð6cÞ ^ H22ðf Þ ¼ a2ðf ÞH22ðf Þ þ b2ðf ÞH22ðfÞ ð6dÞ

where a1ðf Þ,a2ðf Þ,b1ðf Þ and b2ðf Þindicate the

frequency-dependent I/Q imbalances in the 1st and 2nd received antennas, and H11ðf Þ, H12ðf Þ, H21ðf Þ and H22ðf Þ are the

ideal CFR of frequency-selective fading in each path. The received CLP in the 1st received antenna is

R1ðf Þ ¼ a1ðf Þ Hf 11ðf ÞCLPðf Þ þ H21ðf ÞCLPðf Þg

þb1ðf Þ H11ðfÞCLPðfÞ þ H21ðfÞCLPðfÞ

ð7Þ Due to the image aliasing effect of frequency-dependent I/Q imbalance, the relation between MLPk=MLP57k and

CLPk=CLP57k can be applied to extract the mismatches of

filters, gains and phases. With MLPk=MLP57k¼ þ1 and

CLPk=CLP57k¼ 1, the mathematical derivations imply

the following ratio ofβ to α (seeAppendixfor the details).
b1ðf Þ
a
1ðfÞ
¼ R1ðf Þ CLPðf Þ ^H11ðf Þ CLPðf Þ ^H21ðf Þ
R_{1}ðfÞ CLPðf Þ ^H_{11}ðfÞ CLPðf Þ ^H_{21}ðfÞ
ð8Þ
Denote the ratio D1ðf Þ ¼ b1ðf Þ=a1ðfÞ. Since some

frac-tions of MLPs are not ðMLPk=MLP57kÞ ¼ þ1, Eq. (8) is

1( ) cos(2 )
*LO* *c*
*I* *t* = π *f t*
1( ) (1 1) sin(2 1)
*LO* *c*
*Q* *t* = − +ε *f t*+
1
θ
θ
1( )
*BB*
*I* *t*
1( )
*BB*
*Q* *t*
1
ε
1( )
*I*
*h t*
1( )
*Q*
*h* *t*
1( )
90
LPF ADC
LPF ADC
LPF ADC
*Z t*
( ) cos(2 )
*i*
*LO* *c*
*I* *t* = π *f t*
ADC
LPF
*i*
θ
( )
*i*
*BB*
*I* *t*
( )
*i*
*BB*
*Q* *t*
*i*

### ε

( )*i*

*I*

*h t*( )

*i*

*Q*

*h*

*t*( )

*i*

*Z t*

### MIMO-OFDM Modem

π o o LPF 90 ADCFigure 4 The model of frequency-dependent I/Q imbalances in MIMO-OFDM systems.

Figure 3 Tone interleaving of MLPs across two transmitted antennas.

7 _{1}
50
*MLP*
*MLP* = +
7 _{1}
50
*MLP*
*MLP* = +
8 _{1}
49
*MLP*
*MLP* = +
8 _{1}
49
*MLP*
*MLP* = +
8 _{1}
49
*CLP*
*CLP* = −
1st MIMO Preamble
Null
3 _{1}
54
*MLP*
*MLP* =−

Null Null Null Null Null Null Null Null Null Null Null Null Null Null

2 _{1}
55
*MLP*
*MLP* = −
Null
2 _{1}
55
*CLP*
*CLP* = +
3 _{1}
54
*CLP*
*CLP* = +
Null

Null Null Null Null Null Null

Null
2 _{1}
55
*MLP*
*MLP* = −
Null
3 _{1}
54
*MLP*
*MLP* =−

Null Null Null Null Null Null Null

2nd MIMO Preamble
8 _{1}
49
*CLP*
*CLP* = −
3 _{1}
54
*CLP*
*CLP* = +
7 _{1}
50
*CLP*
*CLP* = −
2 _{1}
55
*CLP*
*CLP* = +
Common Preamble
7 _{1}
50
*CLP*
*CLP* = −
1 2 3 4 5 6 7 8 49 50 51 52 53 54 55 56
1 2 3 4 5 6 7 8 49 50 51 52 53 54 55 56
1 2 3 4 5 6 7 8 49

**represents the odd pairs which satisfy** **represents the even pairs which satisfy**

represents a matching pair between CLPs and MLPs

50 51 52 53 54 55 56 1 2 3 4 5 6 7 8 49 50 51 52 53 54 55 56 1 2 3 4 5 6 7 8 49 50 51 52 53 54 55 56 1 2 3 4 5 6 7 8 49 50 51 52 53 54 55 56 Set 1 Set 0 Set 0 Set 1 Tx 1 Tx 2 … … … … … …

deficient in such subcarriers. Via Eqs. (6a–6d)–(8), the mathematical derivations with ðMLPm=MLP57mÞjm6¼k ¼

1 and ðCLPm=CLP57mÞjm6¼k ¼ þ1 are conducted to fulfill

the ratio D1(f), expressed as

b1ðf Þ
a
1ðfÞ
¼ R1ðf Þ CLPðf Þ ^H11ðf Þ CLPðf Þ ^H21ðf Þ
R_{1}ðfÞ þ CLP fð Þ ^H_{11}ðfÞ þ CLP fð Þ ^H_{21}ðfÞ
ð9Þ
Then the complete ratio D1(f) is to join Eqs. (8) and
(9) together. In the 2nd received antenna, the same

process are employed to measure the
frequency-dependent I/Q imbalance. Only one CLP and two MLPs
are used in operations. For the case of an N × N
MIMO-OFDM system, the ratio D1(f) of the jthreceived antenna
becomes
bjðf Þ
a
jðfÞ
¼
Rjðf Þ
PN
k¼1CLPðf Þ ^Hkjðf Þ
R_{j}ðfÞ P
N
k¼1CLPðf Þ ^H
kjðfÞ
ð10aÞ
**0** **10** **20** **30** **40** **50**
**0.5**
**0.6**
**0.7**
**0.8**
**0.9**
**1** **1**
**1.1**
**1.2**
**1.3**
**1.4**
**1.5**
**0.5**
**0.6**
**0.7**
**0.8**
**0.9**
**1.1**
**1.2**
**1.3**
**1.4**
**1.5**
**Subcarrier Index**
**Amplitude**
**0** **10** **20** **30** **40** **50**

**Channel Frequency Response**
**1dB 20o, H _{I}=[0.1, 1], H_{Q}=[1, 0.1]**

**Channel Frequency Response**

**1dB 10o, H _{I}=[0.1, 1], H_{Q}=[1, 0.1]**

**Channel Frequency Response**
**1dB 20o, H _{I}=[0.1, 1], H_{Q}=[1, 0.1]**

**Channel Frequency Response**

**1dB 10o, H _{I}=[0.1, 1], H_{Q}=[1, 0.1]**

**Subcarrier Index**

**Subcarrier Index**

**Amplitude**

### (a)

### (b)

**Ideal CFR**

**0**

**10**

**20**

**30**

**40**

**50**

**Subcarrier Index**

**0**

**10**

**20**

**30**

**40**

**50**

**-3**

**-2**

**-1**

**0**

**1**

**2**

**3**

**-3**

**-2**

**-1**

**0**

**1**

**2**

**3**

**Phase**

**Phase**

**1dB 10o, HI=[0.1, 1], HQ=[1, 0.1]**

**Ideal CFR**

**1dB 20o, HI=[0.1, 1], HQ=[1, 0.1]**

**Ideal CFR**

**1dB 20o, HI=[0.1, 1], HQ=[1, 0.1]**

**Ideal CFR**

**1dB 10o, HI=[0.1, 1], HQ=[1, 0.1]**

Figure 5 Amplitude and phase of CFR with hI(t): [1 0.1], hg(t):

[0.1 1], (1-dB gain error, 10° phase error) and (1-dB gain error, 20° phase error): (a) am-plitude; and (b) phase.

bjðf Þ a jðfÞ ¼ Rjðf Þ PN k¼1CLPðf Þ ^Hkjðf Þ RjðfÞ þ PN k¼1CLPðfÞ ^H kjðfÞ ð10bÞ The converged cycles are “N+1” (one CLP and N MLPs). Due to insufficient information in the CLP and MLPs, the complete D1(f) for all sub-carriers can not be accomplished by preambles directly. For example, only 24 sub-carriers of IEEE 802.11n enable to measure D(f) in each received antenna; however, the ratios of the remainder sub-carriers can be discovered using interpolation.

After interpolation and smoothing, the compensation factor Z1(f) for the 1st received antenna can be given by

Z1ðf Þ ¼ a

1ðfÞR1ðf Þ b1ðf ÞR1ðfÞ

a1ðf Þa1ðfÞ b1ðf Þb1ðfÞ

ð11Þ Assuming thata1ðfÞ ¼ 1 and b1ðf Þ ¼ D1ðf Þ, Eq. (11)

becomes

Z1ðf Þ ¼

R1ðf Þ D1ðf ÞR1ðfÞ

1 D1ðf ÞD1ðfÞ

ð12Þ
Based on Eq. (12), a FD filter with the coefficients of
D1( f ) is acquired to compensate for frequency-dependent I/
Q imbalance in the 1st received antenna. For the jth
received antenna, the same method can be applied to obtain
the complete Djðf Þ ¼_{a}bjðf Þ

jðfÞ and Zj(f). According to Eqs. (8), (9) and (12), Fig. 6 is the architecture of the cross-validation estimator, and Fig. 7 is the architecture of the FD-filter compensator. In Figs. 6 and 7, total 56 sub-carriers (without nulls) must share all dividers, multipliers and adders of Dj(f) and Zj(f) to balance clock rate, hardware cost and designed complexity. The function of−f is easily implemented by the inversed buffer access with [57−k] for the kthdata sub-carrier. According to MLPk=MLP57k and

CLPk=CLP57k, two kinds of buffers, namely first-input

first-output (FIFO) and first-input last-output (Stack), are adopted to measure Dj(f) using one complex divider as displayed in Fig.7. As interpolation is not essential for each sub-carrier, only one interpolator is needed in designs. The output of the cross-validation estimator must be serial to match the shared structure of the FD-filter compensator. All modules can be realized by hardware-description language (HDL) and synthesized for an in-house CMOS library by Design Analyzer (Synopsys).

4 Simulation and Discussion

A typical model [13] of frequency-selective fading was utilized to evaluate the proposed approach. The worst case of frequency-dependent I/Q is 2-dB gain error, 20° phase error and 180°-filter mismatch. This severe condition is applied to evaluate the proposed method for its robustness and effectiveness. The packet-error rate (PER) is the performance index and the required PER is 10% in the following conditions.

(1) 2×2 MIMO OFDM with a bandwidth of 20MHhz. (2) Data length of 1024 bytes.

(3) MCS 13: 64-QAM modulation and 2/3-coding rate. (4) FEC: 6-bit Soft Viterbi decoder.

(5) Alamouti STBC.

(6) Frequency-selective fading: TGnD (rms:50 ns, 8 taps), and TGnE (rms:100 ns, 15 taps)

Three settings of frequency-dependent I/Q imbalances are:

a) Typical I/Q:

1-dB gain error, 10° phase error in the 1st antenna; 0.7-dB gain error, 8° phase error in the 2nd antenna. Complex Divider MUX 56-to-1 MUX Remove Nulls Moving Average Two-Port Buffer (56) Buffer Stack Control Signal Complex Adder Complex Adder Complex Adder Complex Adder FIFO CFR (H21) One-Shot Channel Estimator (H=Y/X)

CFR (H11) (56) (56) ( *) ( *) ( *) Inter-56x Symbol Rate Symbol Rate (2) Control polator Signal (serial output) D(-f) D(f)

b) High I/Q:

1-dB gain error, 20° phase error in the 1st antenna; 2-dB gain error, 10° phase error in the 2nd antenna.

c) The worst FDI: 180°-filter mismatch [3]

i): The LPF of I: hIðtÞ ¼ ½1 0:1 (time domain)

ii): The LPF of Q: hQðtÞ ¼ ½0:1 1 (time domain)

After compensations, the amplitude and phase of ^

H11ðf Þwith 1-dB gain error, 20°-phase error and 180°-filter

mismatch of hIðtÞ ¼ ½1 0:1 and hQðtÞ ¼ ½0:1 1 in

TGnE (15 taps and 100-ns RMS delay spread) are shown in Fig. 8. It reveals that the high-frequency distortion of

FDI can be eliminated to make the CFR be close to that of frequency-selective fading. Even though there is a shift error, this solution performs well enough to make Alamouti equalization work. In Figs. 9 and 10, the conditions of frequency-selective fading are TGnD (8 taps and 50-ns RMS delay spread) and TGnE (15 taps and 100-ns RMS delay spread); I/Q imbalances are typical I/Q and high I/Q; and FDI is the worst 180°-filter mismatch. When channel estimation is not perfect, the system degradation is within 2.9–3.1 dB under typical I/Q with the worst FDI. With ideal channel estimations, the SNR losses of the proposed algorithm are within 1 dB and 1.2 dB. In order to be compatible with IEEE 802.11n, only 3 preambles are available to estimate FDI I/Q imbalance via the proposed Cross-Validation Estimator Data(f) D(f) Data(-f) ( *) Control Signal 56-to-2 MUX Data*(-f) D(-f) ( *) D*(-f)

One-bit 1 Divider Complex

Symbol Rate 56x Symbol Rate

Corrected output Z(f) S/P FFT Remove Nulls

Figure 7 Architecture of the FD-filter compensator in each received antenna.

**0** **10** **20** **30** **40** **50** **60**
**0**
**0.5**
**1**
**1.5**
**2**
**2.5**
**Subcarrier**
**Amplitude**
**Amplitude of TGnE**
**H11 Ideal Comp.**
**H11 w/o Comp.**
**H _{11} w. Comp.**

### (a)

**0**

**10**

**20**

**30**

**40**

**50**

**60**

**-2**

**-1**

**0**

**1**

**2**

**3**

**4**

**Subcarrier**

**Phase**

**Phase of TGnE**

**H11 Ideal Comp.**

**H11 w/o Comp.**

**H11 w. Comp.**

### (b)

Figure 8 Amplitude and phase of CFRH11with h^ I(t):[1, 0.1], hg(t):[0.1,1] , 1-dB gain error and 20°-phase error in TGnE: (a) amplitude; and (b)

method. It results in an additional 1.2-dB SNR loss. It is also known that the SNR losses are more sensitive to dependent I/Q imbalances than frequency-selective fading. Since 50-dB IRR is not excellent, OFDM

pilots with the adaptive scheme [7] as a fine estimation can be exploited to enhance performance.

Since only 3 preambles with interleaving property are valid to estimate the frequency-dependent I/Q imbalance,

**14** **15** **16** **17** **18** **19** **20**

**SNR**

**PER**

**PER vs SNR, MCS13, TGnD, Typical IQ **

**TGnD, TypicalIQ, w. Ideal Comp.**

**TGnD, TypicalIQ, WorstFDI, w. Comp., w. Ideal Channel Est.**
**TGnD, TypicalIQ, w. Comp.**

**TGnD, TypicalIQ, WorstFDI, w. Comp.**
**TGnD, TypicalIQ, w/o Comp.**
**TGnD, TypicalIQ, WorstFDI, w/o Comp.**
**SNR Loss is 2.9 dB**

**SNR Loss is 1 dB**

**Channel Estimation Loss**
**1.9 dB**

### (a)

**14**

**15**

**16**

**17**

**18**

**19**

**20**

**21**

**10-2**

**10-1**

**100**

**10-2**

**10-1**

**100**

**SNR**

**PER**

**PER vs SNR, MCS13, TGnD, HighIQ**

**TGnD, HighIQ, w. Ideal Comp.**

**TGnD, HighIQ, WorstFDI, w. Comp., w. Ideal Channel Est.**
**TGnD, HighIQ, w. Comp.**

**TGnD, HighIQ, WorstFDI, w. Comp.**
**TGnD, HighIQ, w/o Comp.**
**TGnD, HighIQ, WorstFDI, w/o Comp.**
**SNR Loss is 3.5 dB**

**SNR Loss is 1.8 dB**

**Channel Estimation Loss**
**1.7 dB**

### (b)

Figure 9 PER versus SNR, MCS 13 and TGnD with/without FDI: (a) Typical I/Q and (b) High I/Q.**14** **15** **16** **17** **18** **19** **20** **21**

**SNR**

**PER vs SNR, TGnE, Typical IQ **

**TGnE, TypicalIQ, w. Ideal Comp.**

**TGnE, TypicalIQ, WorstFDI, w. Comp., w. Ideal Channel Est.**
**TGnE, TypicalIQ, w. Comp.**

**TGnE, TypicalIQ, WorstFDI, w. Comp.**
**TGnE, TypicalIQ, w/o Comp.**
**TGnE, TypicalIQ, WorstFDI, w/o Comp.**

**SNR Loss is 3.1 dB**

**SNR Loss is 1.2 dB**

**Channel Estimation Loss**
**1.9 dB**
(a)
**14** **15** **16** **17** **18** **19** **20** **21** **22**
**10-2**
**10-1**
**100**
**SNR**
**PER**
**10-2**
**10-1**
**100**
**PER**

**PER vs SNR, TGnE, HighIQ **

**TGnE, HighIQ, w. Ideal Comp.**

**TGnE, HighIQ, WorstFDI, w. Comp., w. Ideal Channel Est.**
**TGnE, HighIQ, w. Comp.**

**TGnE, HighIQ, WorstFDI, w. Comp.**
**TGnE, HighIQ, w/o Comp.**
**TGnE, HighIQ, WorstFDI, w/o Comp.**
**SNR Loss is 4 dB**

**SNR Loss is 2 dB**

**Channel Estimation Loss**
**2 dB**

(b) Figure 10 PER versus SNR, MCS 13 and TGnE with/without FDI: (a) Typical I/Q ; and (b) High I/Q.

fewer converged cycles are needed than other methods. The designed features are summarized in Table2. Considering computational effort, 5 multiplications and 4 additions are required for acquiring one ratio ofβ to α in Eq. (8). Since there are 24 ratios of β to α needed to calculate, it costs 120 multiplications and 96 additions. Furthermore, taking interpolation and moving average into account, the total multiplications and additions are 206 and 358 respectively, which is more efficient than other references. Two major contributions of the proposed cross-validation scheme are summarized as following: 1) Backward compatible with IEEE 802.11n; 2) Low-computational complexity.

For algorithm verifications, a 2×2 software-defined radio (SDR) was constructed, as displayed in Fig. 11. The field programmable gate array (FPGA) chips (Xilinx Virtex-II) with on-board 14-bit digital-to-analog (D/A) and analog-to-digital (A/D) converters are the interface between in-house RF modules and software. The packets are first generated by MATLAB and then transmitted to RF front-ends through 14-bit D/As. In order to make MIMO transmissions coherent, there is an additional D/A module, as a hardware trigger, to control all D/As coherently at TX. After down-converting the RF signals into baseband at the receiver, analog signals are fed into four 14-bit A/Ds for quantization. The proposed Table 2 Summary of the designed features.

Ref [3] Ref [4] Ref [5] Ref [6] Ref [7] Ref [8] Proposed work

System type OFDM+ QPSK OFDM MIMO OFDM MIMO OFDM+64 QAM Single Carrier+16 QAM OFDM+64 QAM MIMO OFDM+64 QAM Method NLS estimation & FIR filter

LS estimation MMSE estimation & RLS filter LS estimation LMS based adaptive FIR filter Blind estimation Cross-validation estimation & FD filter Required format Special Pilot Symbol Special pilot Long preamble Alamouti-coded data

Data Data Long preamble

I/Q Estimation

One-shot One-shot Adaptive One-shot Adaptive Adaptive One-shot

Converged cycles

10 10 30–35 2 (2×1 MIMO) 4000 Frames 1000 3 (2×2 MIMO )

N+1( N×N MIMO) # of Multiplication >1600 616 5376 594 8000 19000 206 # of Addition >1500 448 1312 216 8000 6000 358 Filter mismatch Yes No No No No No Yes

Compatibility No No Yes Yes Yes Yes Yes

Tolerant range 1dB, 5° 2dB, 8° 0.45dB, 2.81° 2dB, 5° 3dB, 30° 1.05dB, 5° RX1: 1dB, 10°

RX2: 0.7dB, 8°

Fading channel Exponential decay (3 taps) Frequency selective (4 taps) N/A Frequency selective (4 taps)

Rayleigh ETSI-A IEEE TGnE

(rms:100 ns, 15 taps)

SNR loss < 1 dB About 1 dB 3 dB < 1 dB N/A 1–2dB 1.2dB

Figure 11 SDR platform for a 2×2 MIMO-OFDM system.

algorithm then processes the quantized signals by using software (MATLAB). The measured 16-QAM constellations in the 1st received antenna are plotted in Fig. 12, where carrier synchronization [2] and equalization [14] have been included in the platform. The measured EVM without compensation and with compensation are –1.9 dB and – 21.35 dB, respectively, which indicate that the proposed scheme can also work well for practice. Both simulations and measurements hint that this study ensures a MIMO-OFDM receiver overcoming the problems of frequency-dependent I/Q imbalances.

5 Conclusion

This work presents a low-computational estimation, using interleaving arrangement of preambles, to extract the mis-matches of filters, gains and phases for MIMO-OFDM wireless accesses. Simulations and measurements indicate that the proposed scheme can solve frequency-dependent I/Q imbal-ances with 2-dB gain errors, 20°-phase errors and the worst 180°-filter mismatches to achieve 10% PER in frequency-selective fading. With the help of cross validation, for N×N MIMO-OFDM systems, just N+1 preambles are used in actions. This study does not only derive an efficient solution for frequency-dependent I/Q imbalances in OFDM-MIMO receivers, but also is well-suited to next-generation wireless LAN discussed in IEEE 802.11 VHT study group. Consequent-ly, the system performance can be improved via the proposed method, which can loose the specification of RF design.

Acknowledgements This work was conducted under “A plan to actively participate in international standard organizations for wireless communications” of the Institute for Information Industry, MOEA, and supported by the National Science Council of Taiwan, ROC under Grant and NSC-97-2220-E-009-016.

Appendix

Derivation of Equation (8)

According to CLP *ðf Þ ¼ CLPðf Þ, Eqs. (6a–6d) and (7), the baseband signal in the 1st received antenna is

R1ðf Þ ¼ a1ðf ÞH11ðf ÞCLPðf Þ þ b1ðf ÞH11ðfÞCLP fð Þ

þa1ðf ÞH21ðf ÞCLPðf Þ þ b1ðf ÞH21ðfÞCLP fð Þ

ðA:1Þ Since CLPð k=CLP57kÞ¼1 or CLPðf Þ=CLP fð Þ¼1,

Eq. (A.1) can be rewritten as

R1ðf Þ ¼ a1ðf ÞH11ðf ÞCLPðf Þ þ b1ðf ÞH

11ðfÞ CLPðf Þ 2CLPðf Þ½ þa1ðf ÞH21ðf ÞCLPðf Þ þ b1ðf ÞH

21ðfÞ CLPðf Þ 2CLPðf Þ½ ðA:2Þ Substituting Eqs. (6a) and (6c) into Eq. (A.2), it yields

R1ðf Þ ¼ CLPðf Þ ^H11ðf Þ þ CLPðf Þ ^H21ðf Þ

2b1ðf ÞCLPðf Þ H11ðfÞ þ H21ðfÞ

ðA:3Þ

With CLPk=CLP57k¼1 or CLPðf Þ=CLPð f Þ¼1,

Eq. (A.1) can be also expressed as

R1ðf Þ¼a1ðf ÞH11ðf Þ CLP f½ ð Þ2CLP fð Þþb1ðf ÞH

11ð ÞCLP ff ð Þ þa1ðf ÞH21ðf Þ CLP f½ ð Þ2CLP fð Þþb1ðf ÞH

21ð ÞCLP ff ð Þ ðA:4Þ Substituting Eqs. (6a) and (6c) into Eq. (A.4), it yields

R1ðf Þ ¼ CLP fð Þ ^H11ðf Þ þ CLP fð Þ ^H21ðf Þ 2a1ðf ÞCLP fð Þ H½ 11ðf Þ þ H21ðf Þ ðA:5Þ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

### (a)

### (b)

After replacing f with *−f ,and making conjugate, Eq.*
(A.5) becomes
R*_{1}ðfÞ ¼ CLP*ðf Þ ^H *_{11}ðfÞ þ CLP*ðf Þ ^H *_{21}ðfÞ
2a*1ðfÞCLP*ðf Þ H*11ðfÞ þ H*21ðfÞ
h i
ðA:6Þ
According to CLP*ðf Þ ¼ CLPðf Þ, the ratio of β to α can
be derived by Eqs. (A.3) and (A.6).

b1ðf Þ
a*1ðfÞ
¼ R1ðf Þ CLPðf Þ ^H11ðf Þ CLPðf Þ ^H21ðf Þ
R*_{1}ðfÞ CLPðf Þ ^H *_{11}ðfÞ CLPðf Þ ^H *_{21}ðfÞ
ðA:7Þ
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Wei-Chi Lai received the B.S. degree from the Department of Computer Science Information Engineering, National Cheng-Kung University, Tainan, Taiwan, in 2004, and the M.S. degree from the Institute of Computer Science, National Chiao-Tung University, Hsinchu, Taiwan, in 2006. He is currently pursuing the Ph.D. degree in computer science from the National Chiao-Tung University, Hsinchu, Taiwan. In 2006, he joined the Institute of Computer Science, National Chiao-Tung University. His current research interests include signal processing for wireless communications, multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems, and associated very large-scale integration (VLSI) architectures.

Ta-Yang Juan received the B.S. and M.S. degrees in computer science and information engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 2004 and 2006, respectively, where he is currently working toward the Ph.D. degree in computer science at the Institute of Computer Science. His current research interests include signal processing and channel estimation for MIMO-OFDM communications, and associated very large-scale integration (VLSI) architectures.

Terng-Yin Hsu (M’07) received the B.S. and M.S. degrees from Feng Chia University, Taichung, Taiwan, in 1993 and 1995, respectively, and the Ph.D. degree from National Chiao-Tung University, Hsinchu, Taiwan, in 1999, all in electronic engineering. In 2003, he joined the Department of Computer Science, National Chiao-Tung University, where he is currently an Assistant Professor. His current research interests include VLSI architectures, wireless communications, multi-spec transmissions, high-speed networking, analog-like digital cir-cuits, system-on-chip (SoC) design technology, and related application-specific ICs (ASIC) designs.

Sheng-Lun Chiou received the B.S. degree in electronic engineering from National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, in 2002 and the M.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 2004. He is currently working towards the Ph.D. degree in the Department of Electrical at the National Cheng Kung University, Tainan, Taiwan. His research interests include wireless communication signal processing, estimation and detection theory, infor-mation theory, multiple antenna techniques. Since 2008, he worked in Institute for Information Industry (III), Taipei, Taiwan, in the 3GPP LTE area of developing Technical Specifications (TS) and Technical Reports (TR).