- Research Article
- Open Access
Appropriate Algorithms for Estimating Frequency-Selective Rician Fading MIMO Channels and Channel Rice Factor: Substantial Benefits of Rician Model and Estimator Tradeoffs
- Hamid Nooralizadeh^{1}Email author and
- Shahriar Shirvani Moghaddam^{2}
https://doi.org/10.1155/2010/753637
© Hamid Nooralizadeh and Shahriar Shirvani Moghaddam. 2010
Received: 8 May 2010
Accepted: 17 August 2010
Published: 19 August 2010
Abstract
The training-based channel estimation (TBCE) scheme in multiple-input multiple-output (MIMO) frequency-selective Rician fading channels is investigated. We propose the new technique of shifted scaled least squares (SSLS) and the minimum mean square error (MMSE) estimator that are suitable to estimate the above-mentioned channel model. Analytical results show that the proposed estimators achieve much better minimum possible Bayesian Cramér-Rao lower bounds (CRLBs) in the frequency-selective Rician MIMO channels compared with those of Rayleigh one. It is seen that the SSLS channel estimator requires less knowledge about the channel and/or has better performance than the conventional least squares (LS) and MMSE estimators. Simulation results confirm the superiority of the proposed channel estimators. Finally, to estimate the channel Rice factor, an algorithm is proposed, and its efficiency is verified using the result in the SSLS and MMSE channel estimators.
Keywords
- Channel Estimation
- Minimum Mean Square Error
- Channel Estimator
- Channel Estimation Error
- Linear Minimum Mean Square Error
1. Introduction
In wireless communications, multiple-input multiple-output (MIMO) systems provide substantial benefits in both increasing system capacity and improving its immunity to deep fading in the channel [1, 2]. To take advantage of these benefits, special space-time coding techniques are used [3, 4]. In most previous research on the coding approaches for MIMO systems, however, the accurate channel state information (CSI) is required at the receiver and/or transmitter. Moreover, in the coherent receivers [1], channel equalizers [5], and transmit beamformers [6], the perfect knowledge of the channel is usually needed.
In the literature, three classes of methods for channel identification are presented. They include training-based channel estimation (TBCE) [7, 8], blind channel estimation (BCE) [9, 10], and semiblind channel estimation (SBCE) [11, 12]. Due to low complexity and better performance, TBCE is widely used in practice for quasistatic or slow fading channels, for instance, indoor MIMO channels. However, in outdoor MIMO channels where channels are under fast fading, the channel tracking and estimating algorithms as the Wiener least mean squares (W-LMS) [13], the Kalman filter [14], recursive least squares (RLS) [15], generalized RLS (GRLS) [16], and generalized LMS (GLMS) [17] are used.
TBCE schemes can be optimal at high signal-to-noise ratios ( ) [18]. Moreover, it is shown in [19] that at high , training-based capacity lower bounds coincide with the actual Shannon capacity of a block fading finite impulse response (FIR) channel. Nevertheless, at low SNRs, training-based schemes are suboptimal [18].
The optimal training signals are usually obtained by minimizing the channel estimation error. For MIMO flat fading channels, the design of optimal training sequences to satisfy the required semiunitary condition in the channel estimator error, given in [7, ], is straightforward. For instance, a properly normalized submatrix of the discrete Fourier transform (DFT) matrix has been used in [7] to estimate the Rayleigh flat fading MIMO channel. In this case, a Hadamard matrix can also be applied.
On the other hand, to estimate MIMO frequency-selective or MIMO intersymbol interference (ISI) channels, training sequences are designed considering a few aspects. For MIMO ISI channel estimation, training sequences should have both good autocorrelations and cross correlations. Furthermore, to separate the transmitted data and training symbols, one of the zero-padding- (ZP-) based guard period or cyclic prefix- (CP-) based guard period is inserted. In order to estimate the Rayleigh fading MIMO ISI channels, the delta sequence has been used in [20] as optimal training signal. This sequence satisfies the semiunitary condition in the mean square error (MSE) of channel estimator. However, it may result in high peak to average power ratio (PAPR) that is important in practical communication systems.
The optimal training sequences of [21–25] not only satisfy the semiunitary condition but also introduce good PAPR. In [21], a set of sequences with a zero correlation zone (ZCZ) is employed as optimal training signals. In [26–28], to find these sequence sets, some algorithms are presented. In [22], different phases of a perfect polyphase sequence such as the Frank sequence or Chu sequence are proposed. Furthermore, in [23–25], uncorrelated Golay complementary sets of polyphase sequences have been used. Since both ZCZ and perfect polyphase sequences have periodic correlation properties, the CP-based guard period is employed with them. On the other hand, uncorrelated Golay complementary sets of polyphase sequences have both aperiodic and periodic types that are used with ZP- and CP-based guard periods, respectively.
Since all sequences under their conditions attain the same channel estimation error [25] and also our goal is not comparing them in this paper (this work is done in [24, 25]), we will use ZCZ sequences here.
In [25], the performance of the best linear unbiased estimator (BLUE) and linear minimum mean square error (LMMSE) estimator is studied in the frequency-selective Rayleigh fading MIMO channel. It is observed that the LMMSE estimator has better performance than the BLUE, because it can employ statistical knowledge about the channel. Nevertheless, all estimators of [23–25] are optimal since they achieve the minimum possible classical (or Bayesian) Cramér-Rao Lower Bound (CRLB) in the Rayleigh fading channels.
In most previous research on the MIMO channel estimation, the channel fading is assumed to be Rayleigh. In [29], the SLS and minimum mean square error (MMSE) estimators of [7] have been used to estimate the Rician fading MIMO channel. It is notable that these estimators are appropriate to estimate the Rayleigh fading channels, and hence the results of [29] are controversial. In [30], to estimate the channel matrix in the Rician fading MIMO systems, the MMSE estimator is analyzed. It is proved in [30] analytically that the MSE improves with the spatial correlation at both the transmitter and the receiver side. An interesting result in this paper is that the optimal training sequence length can be considerably smaller than the number of transmitter antennas in systems with strong spatial correlation.
In [31–33], the TBCE scheme is investigated in MIMO systems when the Rayleigh fading model is replaced by the more general Rician model. By the new methods of shifted scaled least squares (SSLS) and LMMSE channel estimators, it is shown that increasing the Rice factor improves the performance of channel estimation. In [31], it is assumed that the Rician fading channel has spatial correlation. It has also been shown that the error of the LMMSE channel estimator decreases when the Rice factor and/or the correlation coefficient increase.
In this paper, we extend the results of [31–33] in flat fading to the frequency-selective fading case. For channel estimator error, the new formulations are obtained so that in the special case where the channel has flat fading, the results reduce to the previous results in [31–33]. The substantial benefits of Rician fading model are investigated in the MIMO channel estimation. It is seen that Rician fading not only can increase the capacity of a MIMO system [2] but it also may be helpful for channel estimation. It is notable that the aforementioned channel model is suitable for suburban areas where a line of sight (LOS) path often exists. This may also be true for microcellular or picocellular systems with cells of less than several hundred meters in radius.
First, the traditional least squares (LS) method is probed. It is notable that for linear channel model with Gaussian noise, the maximum likelihood (ML), LS, and BLUE estimators are identical [34]. Simulation results show that the LS estimator achieves the minimum possible classical CRLB. Clearly, the performance of this estimator is independent of the Rice factor. Then, the SSLS and MMSE channel estimators are proposed. Simulation results show that these estimators attain their minimum possible Bayesian CRLBs. Furthermore, analytical and numerical results show that the performance of these estimators is improved when the Rice factor increases. It is also seen that in the frequency-selective Rician fading MIMO channels, the MMSE estimator outperforms the LS and SSLS estimators. However, it requires that both the power delay profile (PDP) of the channel and the receiver noise power as well as the Rice factor be known a priori. In general, the SSLS technique requires less knowledge about the channel statistics and/or has better performance than the LS and MMSE approaches.
Moreover, to estimate the channel Rice factor, we propose an algorithm which is important in practical usages of the proposed SSLS and MMSE estimators. In single-input single-output (SISO) channels, different methods have been proposed for estimation of the Rice factor. In [35], the ML estimate of the Rice factor is obtained. In [36], a Rice factor estimation algorithm based on the probability distribution function (PDF) of the received signal is proposed. In [37–41], the moment-based methods are used for the Rice factor estimation. Besides, to estimate the Rice factor in low SNR environments, the phase information of received signal has been used in [42]. Moreover, in [43, 44], the Rice factor along with some other parameters is estimated in MIMO systems using weighted LS (WLS) and ML criteria.
In the above-mentioned references, the channel Rice factor is estimated using the received signals. However, in this paper, we suggest an algorithm based on training signal and LS technique. Simulation results corroborate the good performance of this algorithm in channel estimation. In practice, such algorithms are required to identify the type of environment (Rayleigh or Rician) in several applications, for instance, adaptive modulation for MIMO antenna systems.
The next section describes the MIMO channel model underlying our framework and some assumptions on the fading process. The performance of the LS, SSLS, and MMSE estimators in the frequency-selective Rician fading MIMO channel estimation and optimal choice of training sequences are investigated in Sections 3, 4, and 5, respectively. Numerical examples and simulation results are presented in Section 6. Finally, concluding remarks are presented in Section 7.
Notation: is reserved for the matrix Hermitian, for the matrix inverse, for the matrix (vector) transpose, for the complex conjugate, for the Kronecker product, for the trace of a matrix, for the mean value of the elements in a matrix, for the mode value of the elements in a vector and for the absolute value of the complex number. stacks all the columns of its matrix argument into one tall column vector. is the mathematical expectation, denotes the identity matrix, and denotes the Frobenius norm.
2. Signal and Channel Models
We assume block transmission over block fading Rician MIMO channel with transmit and receive antennas. The frequency-selective fading subchannels between each pair of antenna elements are modeled by taps as , and . We suppose identical PDP as for all subchannels. Then, the taps of all the subchannels have the same power , that is, . It is also assumed unit power for each sub-channel, that is, .
where and are the complex vector of received symbols on the antennas and the vector of transmitted training symbols on the antennas at symbol time , respectively. The vector in (1) is the complex additive noise at symbol time . The matrices constitute the taps of the multipath MIMO channel.
where . According to (4) and (5), the channel Rice factor can vary the mean value and the variance of the channel in the defined model.
Note that the latter one is written using (5).
The elements of H and noise matrix are independent of each other.
When , (12) reduces to the Rayleigh fading channel introduced in [24, 25].
3. LS Channel Estimator
For flat fading, , (19) is similar to that of [7]. In order to achieve the minimum error of (19), the training sequences should satisfy the semiunitary condition (18). Due to the structure of in (9), it means that the optimal training sequence in each antenna has to be orthogonal not only to its shifts within taps, but also to the training sequences in other antennas and their shifts within taps. Here, we consider the ZCZ sequences as optimal training signals without loss of generality.
From (21), holding constant, the minimum error of the LS estimator decreases when increases. On the other hand, holding constant, the minimum error of this estimator increases when increases.
This estimator obtains the minimum possible classical CRLB (21). However, the error of (21) is independent of the Rice factor. Clearly, the LS estimator cannot exploit any statistical knowledge about the frequency-selective Rayleigh or Rician fading MIMO channels. In the next sections, we derive new results in the frequency-selective Rician channel model by the proposed SSLS and MMSE estimators.
4. Shifted Scaled Least Squares Channel Estimator
This estimator offers a more significant improvement than the LS and SLS methods. However, from (28), it requires that and or equivalently the Rice factor as well as be known a priori. The required knowledge of the channel statistics can be estimated by some methods. For instance, the problem of estimating the MIMO channel covariance, based on limited amounts of training sequences, is treated in [47]. Moreover, in [48], the channel autocorrelation matrix estimation is performed by an instantaneous autocorrelation estimator that only one channel estimate (obtained by a very low complexity channel estimator) has been used as input.
From (30), it is seen that increasing SNR leads to increasing which is restricted by 1. Then, the SSLS estimator in (24) reduces to the LS estimator when . Moreover, decreasing the Rice factor to zero (which implies that and hence ) leads to increasing which is restricted by . Hence, the SSLS estimator in (24) reduces to the SLS estimator of [21] when . On the other hand, at or for (which implies that ), the SSLS estimator in (24) reduces to .
Generally speaking, the scaling factor in (24) is between 0 and 1. When the channel fading is weak ( or AWGN) or the transmitted power is small, that is, , the scaling factor . Also, when the channel fading is strong ( or Rayleigh) or the transmitted power is large, that is, , the scaling factor . Finally, in the Rician fading channel , we have .
5. MMSE Channel Estimator
Proof.
See the appendix.
Then, the SSLS and MMSE channel estimators are identical within the uniform PDP.
6. Simulation Results
Here, we denote a ZCZ set with length size and ZCZ length by In the following subsections, we present several numerical examples to illustrate both the superiority and reasonability of the proposed SSLS and MMSE channel estimators in the frequency-selective Rician fading models.
6.1. The Shorter Training Length to Estimate the Rician Fading Model
The sequences under test in Figures 1 through 3 are and sets [26]. It is notable that these results are obtained based on both the channel model and the channel Rice factor which are defined in Section 2.
6.2. Comparing the LS-Based and MMSE Channel Estimators
6.3. The Rician Fading Model with a Higher Number of Antennas
6.4. Increasing Rice Factor
6.5. Substantial Benefits of the Rician Fading MIMO Channels
N _{ T } | N _{ P } | SNR (dB) | κ | Normalized TMSE |
---|---|---|---|---|
2 | 72 | 5 | 0 | 8.17 × 10^{-2} |
2 | 72 | −4.56 | 10 | 8.17 × 10^{-2} |
2 | 40 | 5 | 10 | 6.02 × 10^{-2} |
4 | 72 | 5 | 10 | 6.02 × 10^{-2} |
It is generally true that the less the channel estimation error, the better the bit error rate (BER) performance for a fixed data detection scheme. The proposed methods can also guarantee the best BER performance for a given detection method.
6.6. A New Algorithm to Estimate the Rice Factor
The difference of the proposed estimators with the other estimators such as SLS of [7, 21] or LMMSE of [25] is that the performance of our proposed estimators can be improved because of exploiting the Rice factor, while the other methods cannot use this factor. In order to perform the proposed SSLS and MMSE channel estimators in the Rician fading MIMO channels, it is required that the channel Rice factor be known at the receiver. In this subsection, we propose an algorithm to estimate . This algorithm has the following steps.
Step 1.
Step 2.
Step 3.
Step 4.
Step 5.
Step 6.
In simulation processes, it is seen that for some restricted values of , the estimated Rice factors in Step 5 deviate from the actual values of the Rice factor randomly (not shown). This event especially occurs at low SNRs and high values of . Step 6 is used to remove this deficiency. In this step, we use MATLAB FUNCTION (HIST and MAX) to calculate the mode value of the elements in vector . Hence, the accurate Rice factor can be obtained. It is assumed that the channel Rice factor is stable during the received consecutive blocks. It should be noted that the channel Rice factor estimator can be updated using a sliding window comprising N blocks, which would be useful in real-time estimation of .
7. Conclusion
In this paper, the performance of training-based channel estimators in the frequency-selective Rician fading MIMO channels is investigated. The conventional LS technique and proposed SSLS and MMSE approaches have been probed. The MMSE channel estimator has better performance among the tested estimators, but it requires more knowledge about the channel. For channels with uniform PDP or a lower number of taps, the SSLS estimator is acceptable. However, for nonuniform PDP with a higher number of taps, the MMSE channel estimator is required to attain a lower TMSE. In general, the SSLS technique provides a good tradeoff between the TMSE performance and the required knowledge about the channel. Moreover, the computational complexity of this estimator is lower than that of MMSE and near to that of LS estimator. Finally, we proposed an algorithm to estimate the channel Rice factor. Numerical results validate the good performance of this algorithm in Rician fading MIMO channel estimation.
The estimators suggested in this paper can be practically used in the design of MIMO systems. For instance, in order to obtain a given value of TMSE in the Rician channel model, either the required SNR may be decreased or the training length can be reduced. Then, resources will be saved. Besides, for the given values of the SNR, training length, and TMSE in the aforementioned channel model, the number of antennas can be increased. It is worthwhile to note that the excess of antenna numbers in MIMO systems leads to a higher capacity. It is also remarkable that the Rician fading is known as a more appropriate model for wireless environments with a dominant direct LOS path. This type of the fading model, especially in the microcellular mobile systems and LOS mode of WiMAX, is more suitable than the Rayleigh one.
Appendix
Declarations
Acknowledgments
This work has been supported by the Islamshahr Branch, Islamic Azad University, in Islamshahr, Tehran, Iran. We would like to thank Dr. Masoud Esmaili, Faculty Member of Islamic Azad University for the selfless help he provided. Also, the authors would like to thank the reviewers for their very helpful comments and suggestions which have improved the presentation of the paper.
Authors’ Affiliations
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