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In wireless communication systems, correct knowledge of the correlation of a fading channel is essential for channel estimation. Both the reliability of the estimated channel impulse response (CIR) and the adjustment of an adaptive communication system need the accurate correlation information, which is difficult to identify especially when changing. By modeling the fading channel as a hybrid dynamic system, a channel estimation algorithm based on Interacting Multiple Model (IMM) is presented with the consideration of time-variant channel correlation. Applying the IMM algorithm, the proposed channel estimator can identify the channel correlation. With the accurate information of channel correlation, the proposed algorithm is capable of performing accurate estimation on the fading wireless channel with time-variant or time-invariant correlation. Our simulations demonstrate that the IMM based channel estimation algorithm has good performance in estimating CIR as well as in identifying the channel correlation.

Information transmission with high data rates and reliable performance is required in wireless communication systems. However, the performance of the communication systems suffers from the signal distortion caused by wireless channel. As a fundamental technology to ensure communication performance, a channel estimation algorithm is required to measure the channel parameters and reduce the influence on the communication system.

In the application of wireless control and wireless sensor systems, high performance communication between maneuvering objects is required. Obviously, the impulse response of the channel is time-variant because of the Doppler effect. Furthermore, the channel correlation which governs the way that the channel varies is also time-variant because of the frequent change in the speed of the moving object. This scenario can be treated as a channel that varies in different modes. In this case, the performance of channel estimation is affected not only by the variance of channel impulse response but also by the changing of channel modes.

In the adaptive communication systems, system parameters, signal modes, and transmission modes can be adjusted according to the channel quality. Therefore, the statistical information of the channel, such as channel correlation, reflecting how the channel changes, is required at each moment to adjust communication parameters. Without the channel statistical information directly provided by the channel estimator, the system parameters can only be adjusted according to the channel information calculated indirectly, for example, bit error rate (BER). Therefore, tracking the channel correlation is important for the channel estimation algorithm to achieve good performance. Channel estimation algorithms that provide CIR as well as the statistical information should be developed. This has not been systematically studied.

Among channel estimation algorithms, Linear Minimum Mean Square error (LMMSE) is widely used, since it is optimum in minimizing the Mean Square Error (MSE) of the estimated channel parameters in the presence of Additive White Gaussian Noise (AWGN). It is shown that LMMSE is very attractive for the channel estimation in [

The Kalman filter (KF), as an approximation of the Optimal Bayesian filter, is used in a wide range of engineering and econometric applications because of its high accuracy and efficiency on parameter estimating in dynamic systems [

In the case of a maneuvering receiver, channel correlation varies into different modes because of the changing Doppler shift corresponding to the relative speed between the transmitter and the receiver. Without information on channel correlation, traditional channel estimation algorithms are limited in performance in such situations, because they cannot effectively respond to the changes in channel mode [

The Multiple Model (MM) filter was developed to solve the problem of system mode changes [

The contribution of this paper is to propose a new channel estimation algorithm based on the IMM algorithm. This algorithm characterizes the fading channel using state-space models and describes the dynamic channel correlation with multiple models. The proposed method is capable of tracking the CIR and identifying the channel correlation which changes according to the maneuverability of the receiver.

This paper is organized as follows. The signal transmission of a communication system as well as the Rayleigh fading multipath wireless channel is modeled mathematically in Section

The transform and transmission of data symbols are illustrated in Figure

Diagram of a communication system.

In wireless channels, with the effect of atmospheric reflection, refraction, and reflections from object such as buildings, the signal may travel in more than one path from the transmitter to the receiver. In mobile communication systems, the attenuation and the delay of the CIR are time-variant. After being sampled with the symbol period

In mobile communication systems, the time-variance of the CIR taps is caused by the motion of the transmitter and/or receiver and is quantified by the maximum Doppler shift. When the number of multiple reflective paths is large and there is no line-of-sight signal component, the envelop of every tap of CIR can be statistically described by a Rayleigh probability density function, and the phases of each tap are uniformly distributed in

Assuming that the coherence time of the channel is

Observing (

Separating the real and image parts

The recursion of KF is given by the following equations.

The predicted mean and covariance matrix:

The predicted measurement, innovation covariance matrix, and Kalman gain:

The posterior mean (estimated value) and covariance matrix:

^{−}” of the time

From the equations of KF, it can be noticed that, to ensure the usability of KF, the covariance matrixes of process noise and measurement noise should be known. However, in real cases, they can only be calculated statistically rather than measured. In that case, noise covariance, say process noise, can be denoted as

In hybrid systems, the dynamic parameter to be estimated varies in different modes. In this case, the IMM estimator is one of the best compromises available between complexity and performance, because of its low computational requirements and the accuracy which is almost the same as that of many other algorithms with much higher complexity [

In wireless communications, CIR varies randomly. In addition, the channel features, such as channel correlation, change from time to time. As a result, channel estimation can be achieved by applying IMM estimation method.

As a result, both

In (

The IMM channel estimator is recursive.

Structure of IMM channel estimation.

The data format suitable for IMM based channel estimation algorithm on the basis of single carrier communication system is shown in Figure

Typical arrangement of training symbols.

Single carrier communication

Training mode of multiple carrier communication

Pilot mode of multiple carrier communication

From the previous analysis, although presented on the basis of single carrier communication systems, the IMM based channel estimation algorithm can also be applied in multiple carrier communication systems, such as OFDM systems in two ways, training mode and pilot mode, whose training symbols are arranged as in Figures

In this section, the performance of both KF and IMM based channel estimation algorithms is compared and analyzed based on simulation experiments in different conditions. The channel estimation algorithms are applied to a wireless system whose signal is modulated by 16-QAM with the carrier frequency 2.4 GHz and the symbol period

In the first case, the maximum Doppler frequency shift of the fading channel is a constant value

MSEs of CIR amplitude with constant maximum Doppler frequency.

MSEs of CIR phase with constant maximum Doppler frequency.

Bit error rates with constant maximum Doppler frequency.

In the second case, the maximum Doppler shift

The channel correlation

Real and estimated channel correlations.

MSEs of channel correlation with time-variant maximum Doppler frequency.

The MSE of CIR is also analyzed in this case. The MSEs of amplitude and phase are shown in Figures

MSEs of CIR amplitude with time-variant maximum Doppler frequency.

MSEs of CIR phase with time-variant maximum Doppler frequency.

Bit error rates with time-variant maximum Doppler frequency.

Correlation of a fading channel is important information for channel estimation. The performance of traditional channel estimation with the consideration of time-variance channel correlation is limited. In this paper, by modeling wireless fading channel as a hybrid dynamic system, the channel estimation algorithm based on IMM is presented. The proposed algorithm can identify the channel correlation with the help of the IMM algorithm. With accurate information of channel correlation, the proposed algorithm is capable of performing accurate estimation on the fading wireless channel with time-variant or time-invariant correlation. The results of a number of simulation experiments show that the proposed algorithm is efficient with good performance in estimating CIR as well as the channel correlation.

From (

Predicted model probability:

Mixing probability:

Mixing estimate:

Mixing covariance:

Separating the real and image parts, we can write

Predicted mean and covariance:

Residual and Kalman gain:

Update state and covariance matrix:

Model probability:

Estimated correlation matrix:

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 61240007), Natural Science Foundation of Heilongjiang Province of China (Grant no. F201337), and Research Foundation of National Police University of China. The authors would like to thank Professor Richard Yu and Professor Geoffrey G. Messier for helpful discussions and useful suggestions.