# Hf Channel Estimation For Mimo Systems Computer Science Essay

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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.

## Introduction

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 [1]. 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 [2]. 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[6], 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 ï¬eld of Bayesian forecasting are used for commuting CSI.

In this paper, we consider aspect of Watterson HF SISO model[7] to HF MIMO channel model , extension of the Static AR model[9] 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.

Figure 1: MIMO Communication System

## 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

(1)

Where

is the transmitted symbol in the time index k,

is the delay variable,

is the channel impulse response between jth transmitter and ith receiver of MIMO channel with correlated Rayleigh processes whose Doppler Spectrum is characterized by Gaussian Shape [7,8]. For simplicity is written as. For each time instance k, the (Mt x Nr) time-varying channel parameters have to be estimated with following auto-correlation function

(2)

And normalized spectrum for each is given as

(3)

Where superscript * denotes the complex conjugate,

is the Doppler frequency shift for path between the jth transmitter and ith receiver, and

T is the duration of each symbol.

In this paper we consider auto-regressive (AR) modeling approach for the generation of correlated Rayleigh processes whose Doppler Spectrum is characterized by Gaussian Shape [7, 8]. The analysis of [9] that treats the U shaped Doppler spectrum is extends to consider Gaussian shape of the Doppler spread of the HF channel. The implementation model for channel estimation can be approximated by following AR process of order L:

(4)

Where is lth coefficient between jth transmitter and ith receiver and ni,j,k are zero-mean identical independent distribution (i.i.d). complex Gaussian processes with variances given by

(5)

The procedure outlined in [9] has been adopted for the optimum selection of channel AR model parameters from correlation functions.

The additive noise (equation 1) can be modeled either as a complex-Gaussian distribution with argument z, mean zero, and variance, or as the Middleton Class-A noise model. This latter model has been used to model the impulsive noise commonly generated in wireless environment [10]. The probability density function of the noise model is given by

(6)

Where . The first component represents the ambient background noise with probability, while denotes the presence of an impulsive component occurring with probability.

In order to maintain a constant noise variance for a particular signal to noise ratio (SNR), the parameters, noise variance and varied such that

(7)

Finally in equation (7) if then noise model reverts to the Gaussian distribution. The nonlinear channel is modeled as (1b). This nonlinear distortion of the channel may take into account saturation effects due to transmission amplification

Equation (1a) can be written in a matrix form for flat fading as

(8)

Where is the received matrix, is the channel matrix, and is the transmitted symbol all in time index k, and is the matrix with i.i.d. AWGN elements with variance. And equal Doppler Shift between transmitter and receiver's elements in MIMO system is assumed i.e. .

With this assumption matrix coefficients of the AR model of equation (4) can be replaced by scalar coefficients. The time-varying behavior of the channel matrix can be described as

(9)

Where Nk is a matrix with i.i.d. Gaussian noise elements with variance, and is a AR coefficient modeled for HF Fading channel. Further equation (8) and (9) can be extended to frequency selective channel.

In order to parameterize (9), for time lag (Ï„) the autocorrelation of the channel fading process of equation (2) is:

(10)

Where I is the identity matrix, Ï„ is the time lag, and fD denotes the Doppler frequency resulting from relative motion between the transmitter and receiver. The Doppler shift itself is given by

(11)

where v is the mobile speed, c is the speed of light, and fc is the carrier frequency. Equating (9) to the autocorrelation of (10) for time lag Ï„ = {0, Ts}, we respectively have

(12)

(13)

where, 1/Ts is the sampling rate. For example, if the normalized desired fading rate is fDTs = 0.01, then Î± = 0.998, and = 3.94Ã-10âˆ’3.

A final comment that illustrates the suitability of the channel model is in order. By projecting (9) Ï„ time steps into the future, the expected value of a future channel state conditioned on the current value is given by

(14)

For Î± value near one, then, i.e., the best guess about a future estimate is the current estimate. Note that this is precisely what is assumed by sending periodic training codes for the wireless channel; once the channel has been estimated; it is assumed to remain approximately constant until the next set of training data is sent. Significant changes over longer periods of time are expected. Since the emphasis of the paper is on short-term prediction, one we not consider is the longer-term variation.

## 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 [3] [5], 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 [3][5]. 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 [3] [5].

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 [14]. 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 [14] [16] [18]. 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.

## Recursive Least Squares based Channel Estimation

RLS algorithm is a low complexity iterative algorithm commonly used in estimation /equalization and ï¬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 [11]. In training-based mode of the operation, this algorithm can be summarized as follows:

Initializing the parameters,

(15)

Where is a arbitrary very large number and is the Mt Ã-Mt identity matrix

and are updated for each iteration, as follows,

(16)

(17)

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.

Channel matrix estimation is performed using the updated and

(18)

For next snapshot of channel matrix estimation as to proceed from step (ii),

## 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 [12]:

(19)

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

(20)

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

(21)

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

(22)

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:

(23)

Where L is the number of particles, Î´(Â·) 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.

. (23a)

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.

(24)

In order to increase the sampling efficiency, we analysis extended Kalman Filter as the proposal distribution [11].

Following the selection of the particles from (24), the weights for at time t are sequentially updated as follows [12]:

(25)

To monitor the degeneracy of weight or sample impoverishment, a suggested measure called the effective sample size is defined as,

(26)

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.

PF Channel Estimation algorithm can be summarized as follows:

For time steps t, t + 1, t + 2â€¦

Starting from posterior estimate for time t - 1:

and

For some mean mt-1 and variance Pt-1.

Update the prior distribution and perform prediction.

(27)

Where, estimated means (28)

(29)

(30)

is process noise.

is measurement noise

Posterior estimate for time t:

Where,

(31)

(32)

Using convergence results for the limiting behavior of the recurrence relations equation 27 through 30 can be modified and it can be shown that

(33)

(34)

(35)

is referred as the rate of adaptation and takes the value .

Once and are found out, the channel estimation is perform using the method of importance sampling, predict the state density by propagating particles â„“ = 1, ..., L, from time t âˆ’1 to t using (9),

(36)

Where and noise variance.

## 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.

The STBC matrix for 2x2 is

(36)

Where and is QPSK symbol.

Symbol [] is transmitted over two antenna at time slot, and Symbol [] is transmitted over two antenna at time slot.

And the STBC matrix for 4x4 is

(37)

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.

Figure 2 shows comparison MSE graph of Channel Estimation between RLS and PF-EKF (Particle Filtering - Extend Kalman Filter) for various antenna configurations such as SISO, 2x2 and 4x4 under normalized Doppler rate of 0.04 for both Gaussians noise HF Channel.

Figure 3 illustrates the comparison MSE of channel estimation between RLS and PF-EKF for non-Gaussian noise environment under various antenna configurations.

From the Figure 2 and 3, shows MSE Metric shows that EKF-PF performs better than RLS. For higher configuration MSE shows much improved performance due to diversity factor in STBC. Comparing MSE results of PF-EKF verse RLS for 4x4 antenna configurations gain improvement of 5-6dB is intended for Gaussian Noise condition. Similarly for non-Gaussian scenario, the corresponding gain improvement is about 4-5 dB. It is pertinent to point that the above data on gain improvement of PF-EKF refers to low SNR.

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.

Figure 4 and 5 shows SER (Symbol Error Rate) comparison between RLS and EKF-PF for various antenna configuration such as 1x1,2x2 and 4x4 under normalized Doppler rate of 0.04, in both Gaussian and non-Gaussian noise HF Channel respectively

Figure 2 : MSE vs SNR for Gaussian Noise HF Channel

Figure 3 : MSE vs SNR for Non Gaussian Noise HF Channel

Figure 4 : SER vs SNR for Gaussian Noise HF Channel

Figure 5 : SER vs SNR for Non Gaussian Noise HF Channel

From the Figure 4 and 5, it is seen that PF-EKF performs better than RLS for both Gaussian and Non-Gaussian. There is an improved in SER with higher MIMO configuration. This is to be expected due to diversity factor in STBC. Comparing EKF-PF verse RLS for 4x4 antenna configurations under Gaussian Noise Channel 0.8-1dB gain as we move toward lower configuration gain is around 0.2 to 0.5 dB gain and for Non-Gaussian Noise around 0.5 to 0.8dB gain is achieved at lower SNR.

Figure 6 shows the combine graph of figure 4 and 5 for 4x4 MIMO SER Comparison between RLS and PF-EKF under Gaussian and Non-Gaussian Noise HF Channel to have better clarity individually.

Figure 6: SER vs SNR for 4x4 under Gaussian and Non-Gaussian HF Channel

Figure 7 shows the Estimated Channel response comparison made between the RLS & PF-EKF for normalized Doppler Spread 4 Hz, order of AR model 3 and SNR 8 dB under non-Gaussian noise. It is found that PF-EKF tracks the channel states more accurately compared to RLS and variance for RLS and PF is 0.0197 and 0.00259 respectively.

Figure 7: Comparison of channel tracking based RLS and EKF-Particle Filter (PF)

## Conclusion

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.