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Abstract - Data transmission even at moderate data rates through ionospheric channels is subject to impairments from severe linear distortion, fast channel time variations, dynamic propagation effects and severe fading. The overall system performance strongly depends on the effective allocation of system resources. The adequacy of the effective allocation of system resources can only be derived through accurate and efficient Channel State Information (CSI). Thus, there is a clear need for accurate, efficient techniques to Estimate CSI between pairs of High Frequency links.
Performance analysis of MIMO based HF channel estimation invoking particle filtering is presented in this paper. The significant feature of this analysis is it ability to treat Non-Gaussian noise of the HF channel. The simulation results confirm the superiority of the PF techniques over the RLS in the estimation of CSI even under low SNR scenario with affordable computational complexities. Comparative performance results of various MIMO configuration such as 2x2, 4x4 relative to SISO have also been discussed. The simulation results confirm the relatively small degradation in the MIMO channel performance under non-Gaussian noise with PF incorporating. The results of the proposed analysis reaffirm the superiority of the PF technique over RLS in estimating the CSI even under non-Gaussian and low SNR scenario.
Index Terms-MIMO System, Channel Estimation, HF Communication System.
The high frequency (HF) band spanning 2-30 MHz of the spectrum has been of great interest for many years for long-distance radio communications in many military and civilian applications. The non-ideal characteristics of ionospheric channels such as severe linear distortion; fast channel time variations, dynamic propagation effects, the high interference levels and severe fading impose constraints on the achievable high-data-rate of transfer over HF channel . The increase in demand for higher capacity and reliable adaptive links over high frequency (HF) channel has motivated the researchers to explore the time, frequency and spatial dimensions of signal transmission. Thus, dynamic signal transmission with multi dimensional approach emerged as a powerful paradigm to meet these demands. Multiple-Input Multiple-Output (MIMO) communication systems offer significant capacity gain compared to conventional Single-Input Single-Output (SISO) systems by exploiting the spatial dimension . MIMO communications is an emerging technology offering significant promise for high data rates and mobility required for the next generation HF communication systems.
To achieve reliable link one has to ensure the adequate supply of real time predicted Channel State Information (CSI) for resource allocation. The impact of channel prediction can be exploited for full channel capacity during favorable channel conditions. Under channel impairment condition the Adaptive techniques based on channel condition such as modulation, channel coding, power-control and rate-control are known to improve the performance over time-varying channels on both transmitter and receiver chains of communication system.
Recent technological advances in embedded general purpose processor (GPP) , digital signal processor (DSP) technology, digital converter performance, field programmable gate array (FPGA) density and comtemplated signal processing compute devices are very prominising for high frequency , high date rate communiction with moderate to high reliablity have significantly overcome some of the inherent difficulties associated with the nature of the HF communications, which had rendered reciver strutures complicated during the past [3-5]. The focus of the next generation HF systems is likely to be directed towards the distributed networks and mobility as well as the dynamic selection of the most appropriate channel to establish and maintain communications links retaining the quality of servies intact.
In the HF environment, signals are received by the antenna array after reflection from the ionosphere, which is a dynamic and spatially inhomogeneous propagation medium. Despite the vast amount of theoretical research and simulation studies published on the subject of array signal processing, there are very few studies, which have dealt the performances analysis of the channel estimation under Non-linear channel and Non-Guassian Noisy conditions. Moreover, there is a necessity and great interest to understand how more effective adaptive algorithm should be designed and optimized for different scenarios of noisy characteristics of HF channel. Considering the futuristic state of the art networking technology for HF communication system, there is an arising necessity to evolve the next generation system with MIMO configurations to improve the link reliability and spectral efficiency that would enhance the Quality of Services (QoS). Thus, high data transmission through HF channels at a rate on the same order as or higher than the channel bandwidth is considered and generally requires powerful channel estimation techniques to avail channel state information for effective resources allocation.
In this paper, we consider the problem of providing reliable estimate of channel state information. If receiver system fails to yeild accurate estimates of fading process, receiver performance will degrade. Current wireless systems obtain the CSI through a Pilot-Assisted Transmission which is embedded periodically along with the information-bearing symbols in each frame of transmitted data. Using the training data, the receiver is then enabled to obtain an estimate of the CSI. In order to develop a dynamic state model for the time varying wireless channel, concepts from the ¬eld of Bayesian forecasting are used for commuting CSI.
In this paper, we consider aspect of Watterson HF SISO model to HF MIMO channel model , extension of the Static AR model for doppler Spectrum is characterized by Gaussian Shape. And this paper describes channel estimator for different MIMO configuation under non-linear channel and Non-Guassian Noisy environment. A comparisation with Recursive least squares (RLS) based channel estimated also considered and then presents the results of computer-simulation tests on the resulting system. In this paper we demonstrate PF has advantage with improved performance at low SNR and accurate estimation under noisy condition.
The brief organization of the paper is as follows. Section II presents the system description with channel model. We then present channel-estimation technique in Section III. Section IV deals with performance analysis and comparison of particle filtering algorithms with RLS for channel-estimation under Gaussian/Non-Gaussian noise condition for various configurations. Finally concluding remarks are presented Section V.
II. SYSTEM MODEL
Figure 1 shows a typical MIMO communication system with Mt transmits antennas and Nr receiver antennas. The space-time (S-T) modem at the transmitter (Tx) encodes incoming bit stream using Alamouti's codes. The information bits are modulates and maps the signals to be transmitted across space and time (Mt transmit antennas). Thereafter, the S-T modem at the receiver (Rx) processes the received signal which is subjected to time-varying Ionospheric fading. The received signal also experiences Intersymbol Interference (ISI) under additive Gaussian / non-Gaussian noise. The receiver signal will be decoded on each of the Nr receiver antennas according to the transmitter's signaling strategy. The observed signal from ith receiver at the discrete time index k is
III. Channel Estimation
Channel Predictions of ionospheric propagation are typically very costly in either time or memory or both. Most of the available channel estimation schemes are based on either least mean square (LMS) algorithm or one of the Kalman based recursive least squares (RLS) algorithms as means for tracking the HF link. A Kalman filter assumes that the channel performs a degree-1 Markov process on the signal  , which is a valid assumption for both time invariant and random-walk channels. Thus, a Kalman filter is optimum for either of the two channel conditions, in the sense that it can give the minimum mean- square error in the adaptive adjustment of the receiver. Typical HF channel cannot be modeled as degree-1 Markov process, and computer-simulation tests have, in fact, confirmed that the conventional Kalman filter, together with its more recent developments is not optimum for a typical HF channel . Further the Kalman filter is limited to Gaussian stationary process but HF channel is subjected to non-stationary, time varying and Non-Gaussian noise Environment  .
In view of these considerations, an alternative approach is to develop dynamic channel estimation for the HF channel, invoking the principle of Bayesian forecasting. Bayesian forecasting deals with the optimal learning and prediction of different classes of dynamic models . Based on the concept of sequential importance sampling and the use of Bayesian theory, particle filtering (PF) is particularly useful in dealing with non-linear and non-Gaussian scenarios   . This research proposes a study that would enable the adaptive channel Prediction based on particle filters to counter the presence of non-Gaussian and non-linear Channel characteristics of HF Channel. The expected improvement in the receiver performance in lieu of the use of the particle filter in the predictor algorithm is evaluated through the system parameters like data rate and reliability. The idea of implementing the PF concept is to enable the receiver to acquire the CSI through training data and improve the receiver performance despite the presence of non-linear and non-stationary channel characteristics.
One of the main objectives of the paper is propose the adaptive HF channel estimation using particle filtering. The implementation of our adaptive algorithm starts with Preamble/Training mode that is used to acquire initial estimates, after which it reverts to a correction for data mode. In the training mode, the receiver knows the transmitted symbols the channel estimation is performed using Recursive Least Square (RLS) and Particle Filter with Extended Kalman Filter (PF-EKF) as a variant for channel estimation during training mode. The RLS and PF - EKF scheme are discussed in the following section.
A. Recursive Least Squares based Channel Estimation
RLS algorithm is a low complexity iterative algorithm commonly used in estimation /equalization and ¬ltering applications which is independent on the channel model [18-19]. The only parameter in the RLS algorithm that depends on the channel variation speed is the forgetting factor that can be empirically set to its optimum value. In this paper, the RLS algorithm is used as a channel estimator to compare with performance of channel estimation based on particle filtering.
RLS algorithm is derived for MIMO channel tracking using the analysis detailed in . In training-based mode of the operation, this algorithm can be summarized as follows:
i) Initializing the parameters,
Where is a arbitrary very large number and is the Mt -Mt identity matrix
ii) and are updated for each iteration, as follows,
Where superscript H presents the conjugate transpose operator and is the forgetting factor, which is, the optimum value of which is dependent on the Doppler frequency shift and is chosen empirically.
And Rn is Cross - correlation between received signal rn and transmitted signal sn, Qn is inverse auto correlation of transmitted signal sn.
iii) Channel matrix estimation is performed using the updated and
For next snapshot of channel matrix estimation as to proceed from step (ii),
B. Particle Filter based Channel Estimation
Particle filtering is a sequential Monte Carlo methodology where the basic idea is the recursive computation of relevant probability distributions using the concepts of importance sampling and approximation of probability distributions with discrete random measures. In this paper, PF is used for adaptive channel estimation to counter the presence of non-Gaussian and non-linear Channel characteristics of HF Channel. The following section describes in formulation channel estimation based on PF techniques.
A general state space representation of baseband communications model for a fading channel can be written as :
Where is the discrete time signal, received at the receiver, and is the state of the system composed of vectors of transmitted symbols and fading channel coefficients. The state varies in time according to a known function, which describes a Markov process driven by the noise and is additive channel noise.
As the observation signal , the channel is estimated or the transmitted symbols are detected sequentially.
This implies obtaining estimates of where. The signal of equation (19) can be rewritten as
Where forms state sequence, which consists of transmitted symbol st and transition vector F and form observations Sequence. Each state is represented by the M previous channel information.
An important objective of the recursive estimation is to infuse a level of confidence in accepting the validity channel coefficient at time t, taking different values, given the data up to time t. Thus recursive estimation demands the probability density function (pdf) .
It is assumed that the initial pdf (prior) .The pdf may be obtained recursively in two stages, namely prediction and update.
The prediction stage involves using the system equation 19 and 20 with assumption that pdf at time is available; state will evolve over time t via the Chapman-Kolmogorov equation
Where describes how the state density evolves with time k, and is defined by the state equation. When the current observation becomes available, prior pdf of equation 21 gets updated via Bayes' rule, resulting
Where is the likelihood of receiving the observation, given the state. The likelihood is determined by the observation equation (20). The denominator term in (22) is necessary in order to keep the new estimate of the posterior properly normalized such that for all t. From the distribution can be obtain channel estimate.
In order to recursively evaluate the updates, the method of Importance Sampling, is utilized, which is a common Monte Carlo (MC) method for sequential MC filters [12-14].
The idea of importance sampling is to represent the required posterior density function by a set of weighted particles:
Where L is the number of particles, δ(·) is the Dirac delta function, and is the state of particle at time t. The weights themselves are normalized such that at each time t.
As the number of particles increase to larger value, the approximation in (23a) converges to the true posterior pdf.
New particles are drawn from a known distribution referred to as the proposal distribution.
In order to increase the sampling efficiency, we analysis extended Kalman Filter as the proposal distribution .
Following the selection of the particles from (24), the weights for at time t are sequentially updated as follows :
To monitor the degeneracy of weight or sample impoverishment, a suggested measure called the effective sample size is defined as,
Whenever is below a predefined threshold NT (typically NT = 2/3 L), a resampling procedure is performed. Specifically, particles with low weights are discarded, forming a subset of particles. New particles are generated by resampling with replacement particles from the subset with probability to keep L constant. The weights must now be normalized by resetting them to . In a sequential filtering framework, the resampling procedure is almost inevitable; however, it also introduces increased random variation into the estimation procedure.
IV. Performance Analysis
Analysis of simulation result of MIMO channel estimation for both Gaussian and non-Gaussian Noise scenario is presented. Extensive comparative studies have been carried out on MIMO HF channel estimation employing RLS and PF with EKF as proposal distribution. The MIMO Channel estimation simulations have been compared with corresponding SISO HF channel also. Simulation parameter consider are flat fading HF channels model for both Gaussian and non- Gaussian noise with parameter, QPSK modulation for symbol mapping ,order of AR model is 3 and length of particle L is 50. In addition the effect of choice of order of MIMO on channel estimation has also addressed.
Where and are QPSK symbols. The Symbol  is transmitted over four antenna at time slot, similarly other column symbol is transmitted in respective time slot and.
In addition, the simulation studies are including the analysis of SER (Symbol Error Rate) under Gaussian and non-Gaussian noise of MIMO and SISO HF channel.
This paper presents analysis of MIMO based HF channel invoking particle filtering. The proposed PF based analysis has been demonstrated to show an improved performance in comparison to that obtained with RLS algorithm. The noteworthy feature of the paper is the treatment of even a non-Gaussian noise in estimating the HF channel estimation. In addition, the influences of MIMO configurations on the performance of HF channel have also been investigated. The performance of various MIMO configurations has been compared with that of SISO also. This paper provides convincing evidence for the benefit of incorporating dynamic Bayesian modelling technique for use in estimating a rapidly changing MIMO HF wireless channel.
It is inferred from the simulations, that the performance of the channel estimation with the PF technique is superior to the RLS technique and other techniques with affordable computational complexity even in low SNR. The results presented in this paper indicate that the PF techniques can be handled with a better trade off between computational complexities and desirable performance suitable for HF communication system. The results of the simulation studies indicate that the PF based HF channel estimation algorithm out perform the other algorithm like RLS in Gaussian noise conditions. While the prior algorithms have been unable to treat the non-Gaussian noise condition, the proposed PF technique is demonstrated to deal with these scenarios. The simulation results of the MIMO based HF channel with PF technique confirm that there is a degradation in the channel performance under non-Gaussian noise conditions and the degradation is relatively small in comparison with Gaussian noise considerations.
The particle filter technique is having the better trade-off between computational complexities and performance which most suitable for HF Environment. It can be seen that the proposed algorithm outperforms the existing methods in Gaussian noise and the degradation of performance in the non-Gaussian noise case is really small whereas other standard methods are not actually designed to treat this case. Moreover, increasing the number of particles does not modify the results significantly.
The analysis of this paper has a strong potential to treat the non linear time dispersive HF channel which is the current subject of research investigation of the author.
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