A Virtual Channel-based Approach to Compensation of I/q Imbalances in Mimo-ofdm Systems

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 communication 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 influence 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 configurations. 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.


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 multiplexing (OFDM) and multiple-input multiple-output (MI-MO) has been widely considered the most promising approach 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 transmissions [2][3][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][8][9][10] and has been standardized in a number of communication systems [11][12][13].
Although OFDM presents numerous advantages, it suffers 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 quadrature-phase branches is a major factor in performance degradation.When the received radio-frequency (RF) signal 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 ratios (SNRs) and low data rates.Although several methods were proposed to overcome ICI [18][19][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.Consequently, 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 imbalance models are reported in [16] and [17], respectively.Based on these models, the effects of I/Q imbalances are studied in OFDM systems [23][24][25][26][27][28] and in MIMO-OFDM systems [29][30][31][32][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 method 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 multipath fading channels and I/Q imbalances, this method can effectively mitigate the I/Q effects.
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 virtual 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 combining 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 expressed in closed-form.Extensive simulation results verified that the I/Q imbalances at both transmitter and receiver sides can be effectively mitigated by using the virtual channel approach.
The rest of this paper is organized as follows.In the following 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 estimators 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.
Notations: 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 represent the conjugate, the transpose, and the Hermitian transpose operations, respectively. denotes absolute value operation,  denotes Frobenius norm, and    repre- sents expectation.

 
Q  represents the Q-function. denotes convolution.

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 imbalances arise from two effects.One is the effect of the quadrature demodulator, which is frequency-independent and determined by I/Q amplitude imbalance RX g and I/Q phase imbalance RX  ; another is the effect of branch components, which is frequency-dependent and modeled as two filters with frequency response , ( ) Assuming ( )   where ( ) BB R f is the Fourier transform of ( ) BB r t , and In (3), the term * ( ) BB R f  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 signal TX R and the transmitted baseband signal TX Z with I/Q mismatch can be written as where TX g denotes the I/Q amplitude imbalance at TX,

MIMO-OFDM Model with I/Q Imbalances
A block diagram of a MIMO-OFDM wireless communication system with Alamouti diversity scheme and frequent-cy-dependent TX and RX I/Q imbalances is shown in Figure 2. In this system, there are two transmit antennas and r N (   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) Similarly, the signals to be transmitted via the second transmit antenna are The signals are then transmitted over a multi-path fading channel.The received signals at the th m receive antenna are given as The received signals at the th m receive antenna are further corrupted by I/Q imbalances in the th m receiver as follows: Assuming the noise in the system is additive white Gaussian noise (AWGN), then where ( )  , ( ) complex zero-mean Gaussian noise with variance 0 N .

( ) ( )
, and ( ) ( ) 0 m D k  .Furthermore, the received data symbols at the th k sub-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 the th k and the mirrored -th k sub-carriers.To compensate for the received signals, the joint coefficients should be estimated.The estimation methods are described in the next section.Now, assuming the accurate estimate of the joint coefficients are obtained, the received data can be combined to achieve the transmit diversity as 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 ( ) ( )  14) is reduced to the standard Alamouti diversity combining scheme as substituting (12) and ( 13) into (14), we obtain where and Finally, the received data symbols at multiple receive antennas are combined to obtain receiver diversity, and the raw data symbols are given by ( ) , intercarrier interference noise ( ) ( ) ( ) ( ) (19) where are defined as the combined coefficients, and 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.

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 knowledge of the estimated variables and is computationally intensive.An alternative is a least square (LS) estimator, which may significantly reduce the computational complexity 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.

MMSE Estimator
Assume total tr N blocks of training sequences are used in this system (That is 2 tr N OFDM symbols).Let , ( ) denote the th j training symbol in the th i block at the th k sub-carrier.According to (12), the input-output relation for one block can be written as For the total T blocks, the input-output relation is written as The MMSE estimate of v is given as [34] ( ) where

LS Estimator
Although MMSE estimator is optimal, it suffers from high computational complexity and requires the knowledge 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 s , s , s , and * s be the four consecutive training symbols within the th i and the ( 1) s .According to (12), the received data within the th i and the ( 1)th i  STC blocks at the th k sub-carrier can be represented in matrix form as Thus, the LS estimate of the joint coefficients is given as [35] 1 The estimates of the virtual channel coefficients from different training sequence groups can be averaged to obtain more accurate estimate.

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( ) Q k , 2 ˆ( ) Q k , and 3 ˆ( ) Q k , 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 the th k and the ( )th k  sub-carriers within the th i STC block can be written in matrix form as (38).Thus, the zero-forcing estimates of the transmitted data symbols are given by (39).

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 successfully canceled by the proposed algorithm, the post-processing SNR at the th k sub-carrier, ( ) k  , can be calculated as 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 postprocessing SNR becomes This shows that the system with perfect I/Q over i.i.d.channel can achieve an array gain of r N .
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 -ary M QAM is given by [36] 2 Assuming that only one single bit is changed for each erroneous symbol, the equivalent bit error rate for rectangular -ary M QAM is approximated as [37] 2 1 ( ) ( ) ( )

Simulation Results
To evaluate the proposed virtual channel idea, the two estimators, and the signal compensation approach, we used MATLAB to simulate an OFDM-based 2 r N  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 independent 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 spacing and a maximum 0. Because the estimation of the joint coefficients is performed separately in each receiver, the same I/Q parameters for different I/Q models still lead to generalization 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 approach.
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 N 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 gains are differentiated by 1 dB.It should be noted that the signals after TX I/Q distortions must be normalized 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 differences of the filters are suitable to simulate the frequency-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  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 imbalances termed as "indep-IQ", and a scenario with frequency-dependent I/Q imbalances termed as "dep-IQ".The legend term of "NoComp" means that I/Q imbalances are present but no compensation scheme is applied (the CSI is estimated under I/Q imbalances), "MMSE ( N )" means that the MMSE estimator with N STC blocks of training sequences (the total 2 N  OFDM  LS estimators, the proposed compensation approach is applied to compensate the received raw data symbols.Furthermore, the BER is theoretically calculated according to (43) and is shown in the following figures with legend term "analysis".
Figure 5 and Figure 6 show the performance of a 2 1  and a 2 2  systems with frequency-independent I/Q imbalances, respectively.It is observed from the "Nocomp" curves that the I/Q imbalances significantly degrade the system performance, resulting in high error floor.These results agree with the constellation analysis mentioned above.With the proposed estimators and signal compensation approach, the I/Q distortion can be effectively mitigated and the system performance is significantly improved.By using two STC blocks of training 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    joint coefficients and compensation of the received signal, it is reasonable that the analysis results are slightly better than the ideal case.
The performance of the systems with frequencydependent I/Q imbalances are shown in Figure 7 and Figure 8 for the 2 1   and 2 2  scenarios, respectively.
The frequency dependent characteristic of the I/Q imbalances 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 reported 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 different lengths of training symbols in Figure 9 and Figure 10 for MMSE and LS estimators, respectively.It is ob- served that BER decreases with the increase of the training symbol numbers.When four blocks of training sequences are used, the performance loss compared to the ideal case is negligible.

Conclusions
In this paper, we introduce a new virtual channel concept to analyze the I/Q imbalances in a MIMO-OFDM wireless communication system over multi-path fading channels.The input-output relation is derived in frequency domain, which incorporates the effect of I/Q imbalances with multi-path fading channels.The integrated effect can be modeled by the joint coefficients of the virtual channel.By using this approach, inaccurate estimation 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 estimators, and are used to compensate the received signals.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 transmit antenna configurations by using orthogonal spacetime block coding.

Acknowledgements
This study was supported by the Advanced Telecommunications Engineering Laboratory (www.TEL.unl.edu) and was partially funded by the US Federal Railroad Administration (FRA) under the direction of Terry Tse.

Figure 1 .
Figure 1.Block diagram of frequency-dependent I/Q imbalance at a receiver.

TX
represents the I/Q phase imbalance at TX, non-linear frequency characteristics of the I and Q branches at TX.
to be transmitted over the th k sub-carrier.Based

Figure 2 .
Figure 2. Block diagram of a 2×N r MIMO-OFDM wireless communication system with transmitter and receiver I/Q imbalances.The virtual channel is illustrated in the dashed block.
. If we treat the I/Q effects as parts of the fading channels, we can model the dashed block in Figure2as a virtual channel with joint coefficients ( for data decoding, while the presence of ( ) ( ) m B k and ( ) ( ) m D k may introduce ICI.If the I/Q branches are perfect, then ( ) 1, used.If the I/Q branches are perfect, ( as the average SNR for the single-input single-output scheme.
) and (b) show the constellations of the raw data symbols under frequencyindependent and frequency-dependent I/Q imbalances, respectively.It is observed that the constellation in Figure 4 (b) becomes nearly random compared with Figure 4 (a) due to the effect of frequency-de pendent I/Q imbalances.

Figure 4 (
c) shows the constellation of signals recovered by the proposed LS algorithm, which shows that the proposed algorithm can perfectly compensate frequency-independent and frequency-dependent I/Q imbalances.

Figure 6 .
Figure 6.Performance of a 2 × 2 MIMO-OFDM system with TX and RX frequency-independent I/Q imbalances over multi-path fading channel.

Figure 7 .
Figure 7. Performance of a 2 × 1 MIMO-OFDM system with TX and RX frequency-dependent I/Q imbalances over multi-path fading channel.

Figure 8 .
Figure 8. Performance of a 2 × 2 MIMO-OFDM system with TX and RX frequency-dependent I/Q imbalances over multi-path fading channel.

Figure 9 .
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 frequencyindependent I/Q imbalances over multi-path fading channel.

Figure 10 .
Figure 10.Comparison of the performances of the LS 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 channel.