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Appropriate Algorithms for Estimating Frequency-Selective Rician Fading MIMO Channels and Channel Rice Factor: Substantial Benefits of Rician Model and Estimator Tradeoffs
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 753637 (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.
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, 
.
The discrete-time base-band model of the received training signal at symbol time 
can be described by
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.
For Rician frequency-selective fading channels, the elements of the matrix 
l,
 
are defined similar to [45, 46] in the following form:
where 
is  the  channel  Rice  factor.  The matrices 
and 
describe the LOS and scattered components, respectively. We assume that the elements of
, 
are complex as
and the elements of the matrix
,
 , are independently and identically distributed (i.i.d.) complex Gaussian random variables with the zero mean and the unit variance. The frequency-selective fading MIMO channel can be defined as the
matrix
, where 
has the following structure
Moreover, it is assumed that the elements of matrices  
 and   
, 
are independent of each other. Hence, the elements of the matrix 
are also independent of each other. Using (2), the mean value and the variance of the elements 
of 
can be computed as follows:
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.
Suppose that 
. The 
covariance matix of 
can be obtained as follows:
where
Note that the latter one is written using (5).
In order to estimate the channel matrix 
, the 
symbols are transmitted from each 
antenna. The 
first symbols are CP guard period that are used to avoid the interference from symbols before the first training symbols. At the receiver, because of their pollution by data, due to interference, these symbols are discarded. Hence, by collecting the last 
received vectors of (1) into the 
, the compact matrix form of received training symbols can be represented in a linear model as
where 
is the 
training matrix. The matrix 
is constructed by the 
of transmitted symbols in the form of 
as follows:
Note that 
is the transmitted symbol by the 
  
antenna at symbol time 
. The matrix 
in (8) is the complex 
of additive 
noise. The elements of the noise matrix are i.i.d. complex Gaussian random variables with zero-mean and 
variance, and we have
The elements of H and noise matrix are independent of each other.
The matrix 
is a complex normally distributed matrix and its 
mathematical expectation matrix can be written as 
, where the elements of the matrix 
l
are
Using (5) and (11), it is straightforward to show that the elements of the columns of 
have the following 
covariance matrix
In a particular case, when the uniform PDP is used, that is, 
, we have
When 
, (12) reduces to the Rayleigh fading channel introduced in [24, 25].
3. LS Channel Estimator
In this section, 
is assumed to be an unknown but deterministic matrix. The LS channel estimator minimizes 
and is given by
This estimator utilizes only received and transmitted signals that are given at the receiver. It has no knowledge about channel statistics. The channel estimation error is defined by 
that results in
Let us find 
which minimizes the error of (15) subject to a power constraint on 
. This is equivalent to the following optimization problem
where P is a given constant value considered as the total power of training matrix 
. To solve (16), the Lagrange multiplier method is used. The problem can be written as
where 
is the Lagrange multiplier. By differentiating this equation with respect to 
and setting the result equal to zero as well as using the constraint 
, we obtain that the optimal training matrix should satisfy
Substituting the semiunitary condition (18) back into (15), the error under optimal training is
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.
It is supposed that the transmitted power of any 
antennas at all times is 
. Then,
Substituting (20) back into (19), the minimum error can be rewritten as
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.
For optimal training which satisfies (18), the LS channel estimator (14) reduces to
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
The SSLS channel estimator of [33] is an optimally shifted type of the presented scaled LS (SLS) method of [7, 21]. The motivation of using it is the further reduction of the error in the MIMO frequency-selective Rician fading channel estimation. This estimator has been expressed in the following general form
where γ and B are the scaling factor and the shifting matrix, respectively. They are obtained so that the total mean square error (TMSE), 
, is minimized. The results are [33]
Note that in the special case, 
, the Rayleigh fading model, this estimator is identical to the SLS estimator of [7, 21]. Here, 
is given by (15). The minimum TMSE with respect to 
and 
can be given by
The minimum TMSE obtained from (25) is lower than the presented 
in [21], because always 
. Therefore, it is derived from [21] and (25) that
It means that the SSLS estimator has the lowest error among the LS, SLS, and SSLS estimators. In order to choose the optimal training sequences, let us to find 
which minimizes 
subject to a transmitted power constraint. Clearly, such an optimization problem and (16) are equivalent. Since 
, from (25) it is obvious that 
is a monotonically increasing function of 
. Note that 
is not a function of 
and so 
is the only term in (25) which depends on 
. Therefore, the optimal choice of training matrix for the SSLS channel estimator is the same as for the LS approach. Using (12), (21), and (25), we obtain that the minimum possible Bayesian CRLB (Since all of the estimators utilized in this paper attain the minimum possible CRLB, we use CRLB and TMSE interchangeably.) under the optimal training is given by
From (27), it is seen that increasing the Rice factor leads to decreasing TMSE in the introduced SSLS estimator. In other words, the SSLS channel estimator achieves lower minimum possible CRLB compared with the traditional LS estimator. The SSLS channel estimator under the optimal training can be rewritten in the following form using (20)–(24)
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.
Using (12) and (21), the scaling factor in (24) can be rewritten as
The SNR is defined as 
Then, we have
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
For the linear model described in Section 2, the MMSE, LMMSE, and maximum a posteriori (MAP) estimators are identical [34]. Hence, we obtain a general form of the linear estimator, appropriate for Rician fading channels, that minimizes the estimation error of channel matrix 
. It can be expressed in the following form
where 
has to be obtained so that the following TMSE is minimized
The optimal 
can be found from 
and it is given by
Proof.
See the appendix.
Substituting 
back into (31), the linear MMSE estimator of 
can be rewritten as
It is notable that in the frequency-selective Rayleigh fading MIMO channel,
. The performance of MMSE channel estimator is measured by the error matrix 
whose pdf is Gaussian with zero mean and
The MMSE estimation error can also be computed as
Let us find 
which minimizes the channel estimation error subject to a transmitted power constraint. This is equivalent to the following optimization problem
By using ZCZ training sequences that satisfy(18), 
will be a diagonal matrix. Note that 
in (12) is a diagonal matrix. Therefore, according to the lemma 1 in [7] (see also the proposition 2 in [24]) and by using (12) and (20), we obtain that the TMSE (37) will be minimized as
When 
(39) is analogous to the acquired result in [24, 25] for LMMSE estimator. For 
, the minimum CRLB (39) is lower than the minimum CRLB of this channel estimator. Equation (39) will be equal to (27) when the channel has uniform PDP. In this case, using (13), (18), and (20) the MMSE channel estimator (34) reduces to
where
Then, the SSLS and MMSE channel estimators are identical within the uniform PDP.
6. Simulation Results
In this section, the performance of the LS, SLS, SSLS, and MMSE channel estimators is numerically examined in the frequency-selective Rayleigh and Rician fading channels. It is assumed that each sub-channel has the exponential PDP as
As a performance measure, we consider the channel TMSE, normalized by the average channel energy as
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
Figure 1 shows the normalized TMSE 
of the LS channel estimator versus SNR in the Rayleigh 
and Rician 
fading channels. As it is expected, the performance of the LS estimator is independent of the fading model. In order to improve the performance of this estimator, the training length may be increased. It is notable that the bandwidth is wasted when the training length is increased.
Normalized TMSE of the LS estimator in the Rayleigh and Rician fading channels (
).
Figures 2 and 3 show the normalized TMSE of SSLS and MMSE channel estimators, respectively, versus SNR in the Rayleigh 
and Rician 
fading channels. It is observed that for the given length of training sequences, the performance of SSLS and MMSE estimators in the Rician fading channel is significantly better than the Rayleigh one. In the Rayleigh fading model, increasing the training length improves the normalized TMSE of the estimators. However, in the Rician fading channels, the performance of both SSLS and MMSE estimators with a shorter training length is better than the Rayleigh fading model with a longer training length particularly at low SNRs and high Rice factors. Then, the training length can be reduced in the presence of the Rician channel model. At higher SNRs, the normalized TMSEs of each estimator with various Rice factors are nearly identical. In practice, for the given values of TMSE, SNR, and 
, the optimum training length can be calculated from (27), (39), or these figures.
Normalized TMSE of the SSLS estimator in the Rayleigh and Rician fading channels (
).
Normalized TMSE of the MMSE estimator in the Rayleigh and Rician fading channels (
).
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
All estimators are optimal because they achieve their minimum possible CRLB. However, the performance of the estimators is different. This subsection compares the computational complexity and performance of the LS, SLS, SSLS, and MMSE estimators. As illustrated in Table 1 and Figures 4 and 5 due to lower number of multiplications and additions, the LS-based (LS, SLS, and SSLS) estimators have lower computational complexity than MMSE estimator. Moreover, LS-based algorithms do not include the matrix inverse operation. However, the LMMSE channel estimator of [25, 29] cannot fundamentally benefit from the Rice factor of the Rician fading channels. The general form of this estimator has a complexity near to it, while it can fully exploit a priori knowledge of the 
and 
.
.
Computational complexity of the LS-based and MMSE channel estimators (Real multiplications for 
and 
).
Computational complexity of the LS-based and MMSE channel estimators (Real additions for 
and 
).
In Figures 6 and 7, the performances of LS-based and MMSE estimators are compared in the cases of 
and 
respectively. The 
and 
sets are used in these figures, respectively. We obtained the 
set using the algorithm of [28] and the 
code of [26]. Table 2 shows the generated 
set. As depicted, the MMSE channel estimator has the best performance among all the methods tested. However, it requires that the channel PDP and 
as well as 
be known a priori. For the large values of 
, the MMSE channel estimator outperforms the SSLS channel estimator. However, for the small values of 
, the performances of both estimators are similar. Practically, even small values of 
lead to enough accuracy for the channel order approximation if there is a good synchronization. Hence, the SSLS channel estimator that requires less knowledge about the channel statistics and has lower complexity than the MMSE estimator can be used. Furthermore, the normalized TMSEs of the SSLS and MMSE estimators coincide at low SNRs when the Rice factor increases. It is noteworthy that the performances of the two above-mentioned estimators are always identical in uniform PDP.
set.
Normalized TMSEs of LS-based and MMSE estimators for various Rice factors in the case of 
Normalized TMSEs of LS-based and MMSE estimators for various Rice factors in the case of 
6.3. The Rician Fading Model with a Higher Number of Antennas
In Figures 8 and 9, the effect of both the channel fading type and the number of 
antennas is considered in a joint state. The two sets of 
, that is, 
and 
of Table 2, and 
are employed in 
and 
MIMO systems, respectively, the former system has the Rayleigh fading and the latter one has the Rician model. At low SNRs, it is seen that the performance of the SSLS and MMSE estimators in the Rician fading model with a higher number of antennas is still better than the Rayleigh fading model with lower number of antennas especially at high Rice factors. At higher SNRs, the performances of the above mentioned estimators in both models are analogous. It is noteworthy that the capacity of MIMO system increases almost linearly with the number of antennas. It should also be noted that Rician fading can improve capacity, particularly when the value of 
is known at the transmitter [2].
Normalized TMSEs of the SSLS estimator versus SNR in Rayleigh and Rician fading MIMO systems with 
Normalized TMSEs of the MMSE estimator versus SNR in Rayleigh and Rician fading MIMO systems with 
6.4. Increasing Rice Factor
Figure 10 indicates the channel estimation normalized TMSE of the LS, SSLS, and MMSE estimators versus 
  for 
 dB. From this figure, it is observed that increasing the Rice factor leads to decreasing the normalized TMSE of the SSLS and MMSE channel estimators. At high Rice factors, the performances of the proposed estimators are analogous particularly at low SNRs and for the small values of 
(see also Figures. 6 and 7). It is noteworthy that the TMSE of LS and SLS estimators is independent of 
. The channel will be no fading or AWGN when 
.
Normalized TMSE of the LS, SSLS, and MMSE estimators versus Rice factor for SNR = 10 dB, 
6.5. Substantial Benefits of the Rician Fading MIMO Channels
In Tables 3 and 4, substantial benefits of the frequency-selective Rician fading MIMO channels are shown using the SSLS and MMSE estimators. According to these tables, a lower SNR or shorter training length can be used to estimate the channel in the presence of the Rician model. In practice, the Rice factor can be measured at the receiver and fed back to the transmitter to adjust the SNR or training length. Hence, resources can be saved in the interested channel model. As illustrated in these tables, a higher number of antennas may be used in the mentioned channel without increasing TMSE. This means that the capacity of MIMO systems is increased.
.
.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.
Calculate the mathematical expectation matrix of the channel by using the LS estimates of 
during the observed 
previous blocks as follows:
Step 2.
Partition 
to 
where 
.
Step 3.
Estimate the 
parameter (based on (11)) for all paths of the multipath channel as
Step 4.
Calculate the Rice factor for all paths of the multipath channel as
Step 5.
Calculate the channel Rice factor by calculating the mean value of the several paths' Rice factors in the following form:
Step 6.
Estimate the final Rice factor by calculating the mode value of the several estimated Rice factors during the observed 
consecutive blocks as
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 
.
For example, the performance of the SSLS and MMSE channel estimators using the aforementioned algorithm is probed in Figures 11 and 12. First, the channel Rice factor is estimated using the proposed algorithm. Then, the result is applied to the channel estimator. In order to compare the results with other works, normalized TMSE of the SLS estimator of [21] and the MMSE estimator in the case of 
is plotted in Figures 11 and 12, respectively. Also, the CRLB of the channel estimators is displayed as a reference. As depicted, the normalized TMSE of the channel estimators using the proposed algorithm is very close to the CRLB, especially for low values of 
. However, for high values of 
, the results diverge from CRLB, particularly at low 
. Nevertheless, it is observed that increasing the number of received blocks, 
leads to a better result for normalized TMSE of the channel estimators.
Normalized TMSE of the SSLS estimator by using the Rice factor estimation algorithm (
).
Normalized TMSE of the MMSE estimator by using the Rice factor estimation algorithm (
).
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
Proof of (33)
Using (31), the TMSE (32) yields
With some calculations, the TMSE (A.1) is given by
The optimal 
can be found from
Finally, we have
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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.
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Nooralizadeh, H., Shirvani Moghaddam, S. Appropriate Algorithms for Estimating Frequency-Selective Rician Fading MIMO Channels and Channel Rice Factor: Substantial Benefits of Rician Model and Estimator Tradeoffs. J Wireless Com Network 2010, 753637 (2010). https://doi.org/10.1155/2010/753637
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DOI: https://doi.org/10.1155/2010/753637