- Research Article
- Open Access

# Low-Complexity Estimation of CFO and Frequency Independent I/Q Mismatch for OFDM Systems

- Ying Chen
^{1, 2}Email author, - Jian(Andrew) Zhang
^{1, 2}and - A.D.S. Jayalath
^{3}

**2009**:542187

https://doi.org/10.1155/2009/542187

© Ying Chen et al. 2009

**Received:**1 November 2008**Accepted:**27 April 2009**Published:**7 June 2009

## Abstract

CFO and I/Q mismatch could cause significant performance degradation to OFDM systems. Their estimation and compensation are generally difficult as they are entangled in the received signal. In this paper, we propose some low-complexity estimation and compensation schemes in the receiver, which are robust to various CFO and I/Q mismatch values although the performance is slightly degraded for very small CFO. These schemes consist of three steps: forming a cosine estimator free of I/Q mismatch interference, estimating I/Q mismatch using the estimated cosine value, and forming a sine estimator using samples after I/Q mismatch compensation. These estimators are based on the perception that an estimate of cosine serves much better as the basis for I/Q mismatch estimation than the estimate of CFO derived from the cosine function. Simulation results show that the proposed schemes can improve system performance significantly, and they are robust to CFO and I/Q mismatch.

## Keywords

- Orthogonal Frequency Division Multiplex
- Orthogonal Frequency Division Multiplex System
- Carrier Frequency Offset
- Maximal Ratio Combine
- Training Symbol

## 1. Introduction

Orthogonal Frequency Division Multiplexing (OFDM) becomes the foundation technique for broadband wireless communications because of its various advantages including high spectrum efficiency, low complexity equalization and great flexibility in resource optimization. However, one well-known disadvantage of OFDM is its high sensitivity to carrier frequency offset (CFO) [1]. CFO refers to the frequency difference between the local oscillators in the transmitter and receiver. CFO causes intercarrier interference (ICI) and could deteriorate the system performance seriously. CFO itself is not difficult to estimate and compensate, using either training-based or blind estimation schemes [2, 3]. However, when some distortions, in particular, I/Q mismatch, are entangled with CFO, the performance of conventional CFO estimator will degrade significantly [4].

I/Q mismatch is caused by the imbalance between the components of the Inphase (I-) and Quadrature (Q-) branches in I/Q modulated systems. I/Q mismatch includes gain and phase mismatches. Gain mismatch is caused by the gain difference of amplifiers or filters in I- and Q- branches. Phase mismatch is caused by the nonideal rotation in local oscillators and the phase difference between analogue filters in I- and Q- branches. In a practical receiver with analog I/Q separation, I/Q mismatch always exists and contributes as interference in general CFO estimation. On the other hand, without the knowledge of CFO, a training-based estimator cannot estimate I/Q mismatch accurately. CFO estimation in the presence of I/Q mismatch is not trivial, and has been investigated in, for example, [5–14]. Each of these schemes partially solves the CFO estimation problem in the presence of I/Q mismatch, with respective drawbacks. In [5, 6], initial CFO is estimated in the presence of errors caused by I/Q imbalance. Then, based on the CFO estimates, [5] proposes an iterative I/Q mismatch estimation approach, which requires five iterations to obtain the gain parameter. In [6], a simple time domain I/Q mismatch estimation method is proposed, but the performance degrades significantly when CFO is small. [6] also proposes a frequency domain estimator which improves performance when CFO is small, however, it is sensitive to transmitter side mismatch. In [7], an iterative scheme is proposed, requiring special training symbols which contain many zeros to suppress the I/Q mismatch effect in the receiver. In [8], a searching-based CFO estimator is developed. The high computational complexity, however, may prevent it from practical applications. In [12] iterative estimators are proposed, and they have relatively high complexity. In [13], a frequency domain adaptive I/Q mismatch compensation scheme is proposed, however, it requires perfect CFO knowledge. In [14], perfect CFO knowledge is required either in the training based RLS method or in forming the per-tone-equalizer. In [9, 10], CFO estimators based on three identical training symbols are proposed. However, [9] only uses a cosine function of the CFO to estimate the CFO parameter. The scheme is thus very sensitive to noise when CFO and/or I/Q mismatch is small, and has a phase ambiguity problem with positive and negative phases. Improvement to [9] is made in [10], using two groups of three identical training symbols. Although this estimator is robust to both transmitter and receiver I/Q mismatch, the special long training symbols designed for CFO estimation increase system overhead and are incompatible with current standards. In [11], a complete CFO and I/Q mismatch estimation and compensation scheme is proposed based on the CFO estimator in [9]. However, I/Q mismatch parameters are estimated based on the CFO estimates, which is sensitive to noise, particularly when CFO is small.

In our early work [15], we independently developed a CFO estimation scheme partially similar to the approach in [10]. Different to [10], our scheme only requires one group of three identical training symbols by forming an approximated estimator for the CFO. The scheme works well for various I/Q mismatch values when the CFO is not too small (say, of the normalized CFO), and the performance otherwise degrades. In this paper, we propose some novel estimation schemes which are robust to any values of both transmitter and receiver I/Q mismatch, and have better accuracy of the I/Q mismatch estimation for small CFO. The schemes use a group of at least three identical training symbols, which are generally present in the preamble of current systems, for example, WLAN and WiMAX systems. They serially estimate I/Q mismatch and CFO with low complexity, without incurring iterative process. The schemes mainly consist of three steps. Firstly, a cosine function of the CFO, which is free of I/Q mismatch interference, is formed using a group of three identical training symbols. Secondly, based on the estimated value of the cosine function instead of the CFO estimate, the I/Q mismatch parameters are estimated. Thirdly, the I/Q mismatch is compensated using the estimates, and a sine function of the CFO is formed based on the compensated signal. Combining the results of cosine and sine functions, CFO can then be estimated accurately. The use of cosine value instead of the CFO estimate for I/Q mismatch estimation is from the insight that the cosine value is much more robust to noise than the CFO estimate. The rest of the paper is organized as follows. Section 2 formulates the problem of CFO and I/Q mismatch estimation in OFDM systems. In Section 3, the proposed CFO and I/Q mismatch estimation schemes are developed. Simulation results are presented in Section 4. Section 5 concludes the paper.

## 2. Problem Formulation and System Structure

Equation (6) shows that the transmitter side and the receiver side I/Q mismatch impacts can be decoupled and the transmitter side I/Q mismatch is only contained in and . If the channel is static during CFO estimation, periodically transmitted training symbols lead to periodical and at the receiver. In the CFO and I/Q mismatch estimation algorithms to be presented, only the periodicity of the baseband signal is required and exploited, and the detailed information of and is not required. After the CFO and receiver side I/Q mismatch are compensated, the transmitter-side I/Q mismatch can be estimated via joint estimation of channel and I/Q mismatch proposed in [6] or by a least square estimator. In the following, we propose some CFO and I/Q mismatch joint estimators, which only require the periodicity of training sequences instead of the actual signal values.

and the superscript "*" denotes the conjugate.

According to (8), the received signal becomes the sum of the scaled original signal and the interference from its own conjugation. It is clear that CFO is always entangled with I/Q mismatch. Even when CFO is known, without the information of I/Q mismatch, the second part in (8) cannot be eliminated, so CFO cannot be compensated correctly. Thus it is a natural task to estimate CFO and I/Q mismatch jointly.

## 3. CFO and I/Q Mismatch Estimation

Referring to Figure 1, the proposed scheme consists of three steps, including forming a cosine estimator for CFO which is free of I/Q mismatch interference, estimating I/Q mismatch using the estimated cosine value, and forming a sine estimator for CFO by removing I/Q mismatch in the received signal using the estimated I/Q mismatch parameters. The CFO is then estimated by combining the sine and cosine estimator. In the process, both CFO and I/Q mismatch are estimated in the presence of minimum interference from each other, introduced by the residual estimation error due to the noise.

### 3.1. Cosine Estimator Free of I/Q Mismatch Interference

where the sum and difference formulas of sine and cosine functions are used.

where for real and . The combiner is similar to an equal gain combiner (EGC), with the function ensuring samples to be combined in a constructive way. This combiner, which will be called as EGC hereafter, only requires one division, plus additions.

The EGC estimator even promises better performance than MRC when the number of training symbols is large and the CFO is small. The reason is that the MRC is the best one only when (1) signal and noise are independent and (2) noise samples are uncorrelated. However, when more than three training symbols are used in averaging, each noise samples could appear several times in combining. These repeated noise samples are scaled by , and in EGC, some of the items have opposite phases and a noise cancellation effect can be achieved when approaches . Thus the total noise can be partially cancelled due to the noise correlation in the EGC estimator when is approaching .

For Q-branch, we can form a similar estimator. By combining I- and Q- branches, the final cosine estimator using EGC is given by(14)

To eliminate the phase ambiguity and reduce the estimation error for small , a complementary sine estimator is generally needed. Such a sine estimator free of I/Q mismatch cannot be constructed directly. In [10], a sine estimator is proposed based on special training symbols, which are created by taking the original training sequences and superimposing an artificial CFO to generate point-wise -degree phase rotation. In [15], we introduce an approximated sine estimator, which can work without changing the training symbols for the cosine estimator. However the estimator in [15] sees interference from I/Q mismatch, particularly when the mismatch is large. It is thus natural to consider the approach of forming a sine estimator free of I/Q mismatch after estimating and compensating it.

### 3.2. Estimation of I/Q Mismatch Parameters

As can be seen from Figure 2, when is small, the estimate of is much more robust to noise than . Next we develop an algorithm to estimate the I/Q mismatch parameters based on the estimate of instead of . This approach can estimate mismatch parameters more accurately, particularly when is small.

Since I/Q mismatch is generally fixed during one transmission, is a fixed constant, and it will not contribute to the CFO estimation and can be absorbed in channel coefficients for I/Q mismatch compensation. Thus we only need to know to compensate the I/Q mismatch for the moment. The value of can be computed via , which can be estimated from . The formulation of estimating from is shown in the appendix, and the result is given by

where denotes the real part of .

As pointed out in the appendix, the estimation accuracy of becomes low when CFO is small and is approaching zero. This is the common drawback of general I/Q mismatch estimation schemes based on the periodicity of the training sequence. To improve the performance of the proposed schemes, further processing can be applied. For example, a threshold can be set to initiate a frequency domain least square estimator or a joint estimator for I/Q mismatch and channel response [6] when the estimated CFO from is smaller than the threshold. This threshold can be set as according to our simulation results. The detailed discussion is beyond the scope of this paper.

### 3.3. CFO Estimation after I/Q Mismatch Compensation

#### 3.3.1. Autocorrelation-Based CFO Estimation

The performance of this estimator depends on the accuracy of the estimated I/Q mismatch parameters.

#### 3.3.2. Sine Estimator

The estimator given by (24) depends on the estimation of I/Q mismatch, and estimation error of I/Q mismatch affects both the and parts of the CFO estimate. Alternatively, we can form a complementary sine estimator to exploit the cosine estimator developed in Section 3.1 which is free of I/Q mismatch. With estimated I/Q mismatch parameters, a sine estimator can be formed as follows.

The complexity of this scheme is approximately half of the autocorrelation-based one as only needs to be estimated. Its performance, however, could be better than the latter because of the use of the interference-free cosine estimator.

### 3.4. CFO and I/Q Mismatch Compensation

where we note that the term will be absorbed in the channel coefficients and does not need to be known and compensated here.

### 3.5. Implementation Issues

Although the proposed schemes are divided into three steps, they can be implemented in a parallel manner. Thus very little memory is required in the hardware and the processing delay is very small. As can be seen from (12), (13), (17), (24), and (28), all the sums can be implemented in parallel because the CFO and I/Q imbalance parameters in these equations are fixed and independent of the received signal samples. Parameters can be estimated based on the final sums.

## 4. Simulation Results

The proposed schemes can be used in any OFDM systems with more than three periodical training symbols in the preamble. In our simulation, the Mbps option in the IEEE802.11a standard is followed, and QAM modulation and convolutional coding are used. We use ETSI Multipath A [16] in the simulation. Both mean square error (MSE) of estimates and bit error rate (BER) are used to evaluate the performance of the proposed systems. Each result presented was averaged over packets, each with bytes. Basically, at least 3 periodical training sequence are required for the proposed estimator, however, we assume short training symbols are available for CFO and I/Q mismatch estimation which is the minimum requirement of the methods in [10].

- (i)
*MRC+*sin: MRC-based cosine and sine estimators using MRC cosine estimator and (28); - (ii)
*EGC+Phase*: EGC-based cosine estimator (14) to generate I/Q mismatch estimate, and autocorrelation estimator (24) for CFO estimation; - (iii)
*Tubbax(Li)*: Joint CFO and time domain I/Q mismatch estimator in [6] which applies conventional autocorrelation-based CFO estimator in [3]; - (iv)
*Fan/Fan+Tubbax*: CFO estimator in [11] plus the time domain I/Q mismatch estimator in [6] (combination of the two schemes generates much better performance than the single one in [11].); - (v)
*Rore*: CFO estimator in [10] using special training sequence.

## 5. Conclusions

In this paper, some low-complexity joint CFO and I/Q mismatch estimators are proposed. The estimators are formed based on the observation that a cosine estimator of the CFO, which is free of I/Q mismatch, serves much better as the basis for I/Q mismatch estimation than an initial estimate of CFO. The proposed schemes are robust to any values of CFO and I/Q mismatch, and can improve the accuracy of CFO and I/Q mismatch estimates significantly. The proposed schemes are applicable to systems with conventional training symbols and have low complexity, and they are very promising for broadband systems where I/Q mismatch could deteriorate system performance significantly.

## Appendix

We derive the estimation of from here.

which gives us the denominator of . An averaging is performed on (A.6).

which gives the numerator of .

Equation (A.9) shows that the noise effect in the estimates of is inversely proportional to . So the smaller the is, the larger the noise effects in the are. When is small and is approaching zero, the estimation accuracy of degrades. Averaging over a group of samples and replacing with its estimate establishes the final estimate as shown in (17).

## Declarations

### Acknowledgment

NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.