Tracking Time-Varying Channel in Impulse Noise

Environment Based on Kalman Filter

Linhai Li, Jinhuai Guo, Hanying Hu, Hongyi YuDept. of Commun. Engineering, Zhengzhou Information Science and Technology Institute

P.O.BOX 1001,NO.828, Zhengzhou, Henan,P.R.China, 450002cj_llhgl63.com

Abstract A new algorithm for tracking time-varying fadingchannels in impulse noise environment is proposed in thispaper, which uses the Kalman filter based on Clarke's model.However, the Kalman filter is known to be sensitive to impulsenoise. In this paper, we investigate into how to restrain theadverse effect of impulse noise. The impulsive noise is modeledas a two-term Gaussian mixture distribution. The proposedchannel tracking scheme is based on a state-spacerepresentation of the communication system and the priorinformation of the measurement noise. To reduce thecomplexity of the high-dimensional Kalman filter for channelestimation of the paths, we use a low-dimensional Kalmanfilter for the estimation of each path. Simulations show thisalgorithm is much less sensitive to impulse noise than theconventional algorithms, and is effective for the estimation ofthe fading channel when the performance of the channelestimation is presented in terms of the mean-squareerror(MSE).

I. INTRODUCTION

In wireless communications and other applications, thetime-varying channels estimation is an important subject ofresearch. Existing channel estimation techniques are mainlybased on the channel estimator itself [1-2]. In [3], higherorder statistics(HOS)-based approaches have been applied tothe discrete-time stationary signal at the receiver, which usesKalman filter to track time-varying channel. Next come thesuperimposed periodic pilot scheme for finite-impulseresponse(FIR) channel estimation [4]. Ref. [3], and [4]didn't take into account the prior information of the powerspectral density of the channel process. Generally, the time-varying impulse response of a channel can be modeled as anautoregressive moving average (ARMA) process. We focuson a second-order ARMA channel model due to itssimplicity and the true dynamics of real channels [5]. Theuncertainty in channel models is taken into consideration inthe derivation of estimation algorithm. Given the systemoutputs as the measurements and treating the state as theimpulse response coefficients, the Kalman filter can be thenemployed to estimate the time-varying channel coefficients.However, the algorithm works well in additive whiteGaussian noise channel, its performance will be significantly

degraded in impulse noise environment. It is because theKalman filter is sensitive to impulse noise, thus affecting theaccuracy of the channel estimation. In [6], A recursive leastM-estimation (RLM) algorithm is proposed to suppress theadverse effect of outliers.

In this paper,we propose a new algorithm for estimatingtime-varying channels using a robust Kalman filter based onClarke's model under impulsive noise environment. It usesan M-estimate to identify and eliminate possible outlierswhich appear in the received signal. The channel coefficientsare estimated by the Kalman filter based on Clarke's model.The simulation results show that the proposed algorithmsignificantly improves the estimation performance over theconventional approach in the presence of impulse noise.

This paper is organized as follows. Section 2 describesthe proposed fading channel model. Section 3 describes theestimation ofARMA parameters. Section 4 is devoted to theKalman filter channel tracking algorithm based on Clarke'smodel in pilot-aided wireless communication systems.Simulation results are presented in section 5 and conclusionsare drawn in section 6.

II. SYSTEM CHANNEL MODELThe time-varying fading channel is modeled as a wide-

sense-stationary complex Gaussian process with zero-mean,which makes the marginal distribution of the phase andamplitude at any given time uniform and Rayleighrespectively. The channel can be modeled by a linear time-varying filter having the complex low-pass impulse response.Assuming that there are L independent propagation pathswith different channel gain and time delay, the channelimpulse response (CIR) at time t can be represented as

L-1

h(t, ) = E a (t) exp fj (9 (t) 5(r - )1=0

(1)

where 8() is the Dirac function and y1 (t) is the phase

response of the Ith path. a, (t) and r1 are the amplitude

This work was supported by the National Nature Science Foundationof China (grants 60472064).

and time delay respectively, associated with the Ithpropagation path. The term Rayleigh fading channel refers toa multiplicative distortion h(t, r) of the transmitted signal

s(t), as in

where N(mx,c f) is the Gaussian density with mean mx2and variance c2 and £ is the contamination constant or the

probability that an impulse occurs.

I1) .h(t, r1) + n(t)

where y(t) is the received waveform and n(t) is a i.i.dcyclic Gaussian random variable with zero mean and two-sided power spectral density (PSD) No /2.

The autocorrelation properties of the random processh(t) are governed by the maximal Doppler frequency fD:

R(r) = E{h(t)h*(t r)} JO(2;TfDr) (3)where J0 () is the zero-order Bessel function of the firstkind. There is a non-rational power spectral density andDoppler spectrum of the channel process for each channeltap.

1 1

)2D

0,

fIfl<fD

otherwiseEach channel tap is a zero-mean complex Gaussian

random process like h(t) described above, which isuncorrelated with and thus also independent from any othertap process, but having time-autocorrelation as described by(3). If there is a significant line-of-sight component in thechannel, the only change in above model of multiplicativedistortion is that each channel tap now has non-zero mean.

But its correlation properties are still described by (3), and itsPSD is given by (4) to within an additive constant.

In practice, n(t) might consist of impulse noise and itcan be modeled as the contaminated Gaussian noise. Theimpulsive noise b(k) can be modeled as [7]

b(k) =-(k)g(k) (5)

where {/B(k)} stands for Bernoulli process, i.e., an i.i.d.

sequence of zeros and ones with P(,8(k) = 1) = 8, and

g(k) is a complex white Gaussian noise with zero mean

and variance 7b . We consider cb= W72 with K >> 1.

Under this model, the probability density of the channelnoise n(k) = w(k) + b(k) can be expressed as a Gaussianmixture

p(n(k)) = (1-)N(O, u72) ±+ N(O, (K + 1)c0 ) (6)

III. ARMA PARAMETER ESTIMATION

The filter with frequency response equal to the squareroot of (4) performs a linear operation. By filtering of theindependent Gaussian variables, the resulting sequenceremains Gaussian, with a spectrum

Sout(f) - Sin(f)|H(f)I , where |H(f)2 is the squaredmagnitude response of the filter, chosen equal to (4). If theinput sequence is independent, the spectral shape of theoutput Gaussian sequence will follow (4). Obviously, thisstructure can be replicated as many path as the receiverdesires, each path with possibly the same Doppler rates.

An IIR filter of order 2K, synthesized as a cascade ofK second-order canonic sections(biquads),is designed toapproximate the square root of the spectrum of (4) as it istranslated into discrete frequency. We follow an approachproposed by K.Steiglitz [9] in 1970 and use the methodadopted by Christos [8] to search for the optimum realcoefficients akt bk, ck, dk , k = 1, ,K and the scalingfactor A, such that the magnitude response of the filter isgiven by (7).For z = eJW approaches the desired magnituderesponse.

K 1±akz'± bkz 2

H(Z) = Afn +kz +b (7)k=1l±ckZ±1 dkZ2

Based on the channel transport function H(z), a state-space model for the fading channel can be built. By defining

x[n] := [h(n),.., h(n + p 1)]T , we get

x[n] = Ov[n -1] + Gwfn] (8)

which is the state equation with w[n] being the white

Gaussian process noise. The matrices 0 and G are definedaccording to (7). The observation equation of the state-spacemodel is

y(n) = A[n]x[n] + v(n) (9)

where A:= [a, ..., ap ]. The state-space model of (8) and(9) allows us to use Kalman filter to adaptively track thetime-varying fading channel.

IV. CHANNEL ESTIMATION UNDER IMPULSIVENOISE ENVIRONMENT

We focus on a second-order channel model by (7) forsimplicity and its true dynamics of real channels. Following

L-1

y(t) = Zs(t1=0

Shh(f)-

the similar procedure of the previous section, we obtain thestate-space model for the Ith path as

x[k] = b(k, k - I)x[k -1] + B(k - I)w(k) (10)

where x[k] = [h(k) h(k +)] is the fading channeldiscrete-time random variable, k is the channel observationtime index. The terms w(k) is the process noise with

covariance Q(k), and the measurement noise v(k) is

v(k) = z(k) -C(k)x(k|k-1) (11)They are assumed to be additive white Gaussian noise

with different PSD. In practice, v(k) might consist ofimpulse noise and it can be modeled as the contaminatedGaussian noise.

The Kalman filter is known to be sensitive to impulsenoise. The accuracy of the channel estimation can deteriorateby impulsive noise coming from the communicationschannel, which leads to significant degradation of systemperformance. In this section, we introduce a robust Kalmanfilter as shown in TABLE I that can be used to improve therobustness of the state estimation by means of the state-spacemodel given by section 3. The robust Kalman filter inTABLE I differs from the conventional Kalman filter inupdating the Kalman gain

K(k) = q(e(t)) P(k|k _1)CT (k)[C(k)P(k|k -1)T 1 ~~~~~~~~~~(12)CT(k) + R(k)]-1

where a weight function q(*) as defined in following thatprevent the Kalman gain from being corrupted by impulseswith abnormally large amplitude.

The weight function q(*) is given by

II le(t)<<A (13)

If |e(t)| is great than A, the Kalman gain vector K(t)is equal to zero and the abnormal estimation error isprevented from entering the channel estimates. To estimatethe threshold A/, for simplicity, e(t) is assumed to consistof a Gaussian process that corresponds to the 'impulse-free'error signal and an impulsive component. Suppose thevariance of the 'impulse-free' error signal e(t) is ^e the

probability of |e(t)| > A is [6]

)A(t)=Pr >e(t)A5>A=I- erfI A (14)LFt~

where Pr(.) is the probability operator, erf(.) is the error

function and Qe is the variance of e(t). )yA is chosen as

0.01 and the corresponding threshold is A = 2.57 e .

The matrix 0(k, k -1) in the difference equation (10)relates the state at the previous time step k Ito the state atthe current step k. The coefficients of the measurementmatrix C(k) and process matrix 0(k, k -1) aredetermined by the channel transfer function H(z) .

0(kik-1) = ° B(k -1) [°

According to the process and measurement model, a newestimate state x[k] is a linear combination of both the

previous state x[k-1] and the process noise w(k) . Therecursive estimated state value x[k] is the components ofthe impulse response of the channel. We initialize theKalman filter with x[O] = 0 and P(O) = po, where po is

the stationary covariance of x(k-1) and can be computedanalytically from (3). For each k , do the Kalman filterupdate according to TABLE I .

TABLE I. ROBUST KALMAN FILTER CHANNEL ESTIMATION ALGORITHM

Nk|k -1] = 0(k, k - 1)x[k - 1 k -1]

x[kk] = 0(k, k - l)x[k - 1|k -1] + K(k)v(k)P(k|k - 1) = 0(k,k - I)P(k - 1|k _ I)ST (k,k -1)

+ B(k - I)Q(k -l)BT (k - 1)K(k)= q(e(t)) P(kk _l)CT (k)[C(k)P(k|k- 1)

CT (k) + R(k)]-1

P(k|k) = [I - K(k)C(k)]P(k k -1)where [k|k -1] is the prior state estimate at step k, and

i[k k] is the posterior state estimate at step k given

measurement z(k) . P(k|k -1) and P(k|k) are prioriestimate error covariance and posteriori estimate errorcovariance respectively. K(k) is the Kalman gain. Channelestimate at instance k is

h(k) = [1,O]x[k k] (15)

Note e, as the MSE of the Ith path channel estimator

22

e= E{h1(k)-h1(k) } (16)

where E{x} denotes the ensemble average of x

V. SIMULATION RESULTS

Matlab simulations are used to examine the performanceof the proposed approach in a time-varying channel. In our

simulation, a two-order ARMA model is used to simulate thechannel. The transmitted sequence of QPSK modulation isemployed. The parameter of Bernoulli sequence is£ = 0.05 and the impulsive noise parameter K = 100

Simulation results of the method proposed in [3] is alsoincluded as a comparison. The channel estimation error isassessed by the normalized MSE.

10

LUcm

1o62

1 0-2

0 5 1 0 1 5 20 25 30EbINO(dB)

Figure I Channel estimate MSE of the Kalman filter Estimators indifferent Doppler

1o-6

10

LU

[o-4 5 10Eb/NO

Figure 2 Bit-error rate of the Kalman-filter and conv

Fig. 1 shows the mean square error(MSEfilter channel estimation versus the receive(ratio(SNR) with different JDT and DoppleiMSE is the average of the channel estimatioFrom Fig. 1 we can see that the algorithm ]

paper yields excellent tracking capability in presence ofimpulsive noise.

Fig. 2 shows Bit-error rate (BER) of the robust Kalmanfilter and conventional Kalman filter channel estimatorversus the received SNR. The receiver BER performance isimproved significantly by using the proposed algorithm ofthis paper. Especially in impulse noise environment, theBER performance is relatively better compared withconventional Kalman filterchannel estimation algorithm.

VI. CONCLUSION

A new algorithm for tracking time-varying channels inpulsive noise environment using the Kalman filter ispresented. It employs a simple dynamical model of thechannel to reduce the arithmetric complexity, while offeringreasonable good performance. The robust Kalman filter isemployed to restrain the adverse effect of impulsive noise. Itmakes the most of the acquired information in the channelestimator. We give some simulations of different cutofffrequency JDT and different Doppler frequency. The noisychannel estimate from the Kalman filter can be used toestimate the parameters of the channel coefficients whenthey are governed by an ARMA model. Simulations showthat the proposed algorithm gives more stable performancethan the conventional methods under impulsive noiseenvironment.

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