Channel Tracking and Equalization using Kalman

Estimation for MIMO Systems in Non-Isotropic

Ricean Fading Environment

Abbas H. Kazmi Department of Electronic Engineering

Mohammad Ali Jinnah University Islamabad, Pakistan

Email: s.haiderkazmi@gmail.com

Abstract- The problem of channel tracking and equaliza­tion for multi-input mUlti-output (MIMO) time-varying non isotropic Ricean fading channels is investigated. A Kalman filter based on high order autoregressive (AR) model with statistics closely matching those of the directional Ricean fading process is proposed to track channels variations. A finite­length MIMO minimum-mean-squared-error decision-feedback equalizer (MMSE-DFE) is used to perform equalization task. The proposed algorithm works much better in tracking time varying inter-symbol interference (lSI) channels at the expense of high computational cost.

Index Terms-MIMO, Non isotropic Ricean fading, Autore­gressive model, Kalman filter.

I. INTRODUCTION

The use of mUltiple antennas at the receiver and/or trans­mitter in wireless systems, commonly known as space-time (ST) wireless communications is an emerging tecnology that promises substantial improvements in system performance. One of the main problems in this area that still needs to be solved is the channel tracking and equalization of such multiantenna systems in highly time-varying fading environ­ments. A solution that is proposed in literature to address this problem is to use state space approach [1-4]. The state space model developes a channel estimator based on the valid assumption that time-varying fading channel is markovian in nature. In [2], the problem of channel tracking for MIMO time­varying frequency-selective channels is addressed by using low order AR model. In [3], the authors have used higher order AR model to estimate MC-CDMA fading channels based on Kalman filtering. In [4], authors consider the estimation of rapidly time varying DS-CDMA channels in more realistic non isotropic Rayleigh fading based on high order AR model with Kalman filtering.

In this paper, we consider the estimation of rapidly time varying non isotropic Ricean fading channels based on Kalman filtering and MMSE DFE to equalize the channel effect. More­over, we also use higher order AR model channel estimator with kalman filtering based approach.

978-1-4244-8003-6/1 0/$26.00® 20 1 0 IEEE

Noor M. Khan Department of Electronic Engineering

Mohammad Ali Jinnah University Islamabad, Pakistan

Email: noor@ieee.org

:( ----/ ...... .,,-.::---)0----- 0--->

Fig. I. Block Diagram of typical MIMO Channel model

The remainder of the paper is organized as follows. In section II, we present the proposed system model and the directional approach for ricean fading channel environment. In the same section, we introduce Kalman filter based Chan­nel estimation algorithm. Simulation results are presented in section III, while section IV concludes the paper.

II. SYSTEM MODEL

A typical MIMO communication system model is depicted in Fig. 1. Due to multipath propagation in the radio channel, a linear combination of all transmitted data sequences, each distorted by lSI, is observed at each receiver antenna of the nT-input nwoutput MIMO channel under white Gaussian noise. Thus, the received baseband signal y! at receiver j at time t can be expressed as [5],

nT Z(i,j) y! = L L h�j)(t)xLm + v! ,

i=l m=O j = 1, ..... , nR (1)

where h�j) represents the mth tap of the impulse response of order Z(i,j) between the ith input x(i) and the jth output y(j) of the MIMO channel. xLm is the baseband constellation

point transmitted by the ith input at time t -m, and v{ is the complex noise sample at the jth receiver. The over all channel impulse response between the ith input and the jth output with memory denoted by l(i,j) can be represented by vector h(i,j) as,

(2)

where l = maxi,j l(i,j) and h�j)(t) = 0 for m >

l(i,j). By collecting the outputs y{ from all receiver an­tennas at time t into an nwdimensional column vector Yt we can write the MIMD channel input output relationship of (1) in vector form as,

(3)

where Xt is the nR x nRnT(l + 1) dimensional matrix with the transmitted symbols repeated diagonally, according to the Kronecker product

X - [ (1) (nT) (1) (nT) (1) (nT) ] t - Xt ... Xt Xt-1···Xt-1 .... Xt_I·.·Xt_1 ® InR (4)

and ht is a long vector of length nRnT(l + 1) containing all the channel taps at time t ,

hP,l) (t) ... hp,nR) (t) ... h}nT,l) (t) ... h}nT,nR) (t)J

A. Channel Model

(5)

According to Bello's WSSUS model [6], all channel taps are zero-mean, wide-sense-stationary complex Gaussian processes, uncorrelated with each other and have discrete time­autocorrelation function (ACF),

(6)

where Jo(.) denotes the zero-order Bessel function and !m =

!dT is the maximum Doppler frequency normalized by sam­pling rate liT. In Bello's channel model, one of the main assumption is that the propagation path consists of two di­mensional isotropic scattering so that the probability of angle of arrival (ADA) at the receiver becomes uniformly distributed on (-71", 71"). However in [4], the effect of directional scattering on Rhh(t) is captured by using Von Mises distribution for the probability density function (pdt) of ADA for scatter component as,

PBcat(B) = 1 eJ<cos(IJ-p,) 27l"Io(J<) BE (-71", 71") (7)

where 10(.) denotes the zero-order modified bessel function of first kind, p, E (-71", 71") represents the mean direction of the ADA, and K, � 0 controls the beamwidth.

For specular component , the pdf of ADA can be written as,

(8)

The pdf of ADA of multipath components (scatter and specu­lar) at the receiver can be expressed as [7]:

pBcat (B) + K pBpec(B) Pe (B) = =----'-''--------=--....:.......:.. K+l

(9)

Where K is the Rice factor. For the ADA given in (9), the corresponding autocorrelation function is,

where

and

RBcat(t) + K RBPeC (t) R (t) =

hh hh hh K + 1 (10)

Rs:,.ec (t) = exp(j271"cos(P,)!m It!) (12)

For the case of uncorrelated scattering (US), Rhihq (t) becomes zero for i =I- q. Autoregressive models can be used for the computer simulation of correlated Ricean fading processes. The complex AR process of order p , AR(P), can be generated via the pth-order autoregression [8] as,

p ht = -LA kht-k + Wt

k=l (13)

where Wt = [W1(t) W2(t) ... wnRnT(l+l) (t) J is a zero mean, complex white Gaussian sequence vector with covariance matrix Q and

l all(k) a21(k)

aM1(k)

a1M(k) 1 a2M(k)

aMM(k)

is M x M matrix (where M = nRnT(l + 1» containg the multichannel AR model cofficients. The relationship between the AR parameters and the ACF of the fading process is given by multichannel Yule-Walker equations as [8],

where

l Rhh[OJ Rhh[IJ

Rhh� -IJ

(14)

Rhh[-p+ IJ Rhh[-p+2J

Y'-A-I� I_� 11:=:=

x:=:=�_

Channel h'-l> Channel [l�>-l> +', .,l�,l a) Kalman Filter Based Tracking and Channel Predic--

:, Estimator

KF Predictor I

[hi-Nt' .. ,h t-.o.-I· r----------------------------------------- , I

� I I

I I I I

'+- Decision 'N,·P' ,B;P' � I ,

A=

I I+--[Y'-Nt' · · ,Y, l Design � ,

I , � I l .u.1SE DFE , �------------------------------------------�

Fig. 2. Block Diagram of the proposed Receiver

AH[P]

and Y=

l Rhh[l] 1 Rhh[2]

Rhh[P]

The submatrices in model covariance block matrix Rhh are nRnT(l + 1) x nRnT(l + 1) toeplitz matrices Rhh[j] = E{h[ t + j]h[ t]H}. Once the A[k] coefficient matrices have been determined by using ( 14), the nRnT(l+l) xnRnT(l+l) covariance matrix Q of driving noise process can be computed as [8],

p Q = Rhh[O] + L Rhh[-k]AH[k] (15)

k=l After obtaining Q, the driving noise process can be realized

by first computing the factorization GG H = Q and then taking the product w [ n] = Gi, where i is an nRnT(l +l) x 1 vector of independent zero mean complex Gaussian variates with unit variance.

B. Receiver Model

The proposed receiver shown in Fig.2 consists of a Kalman filter (KF) based estimator to track the time-varying channels, a decision feedback equalizer (DFE) to combat lSI and a channel prediction module to bridge the time gap between the channel estimates made by Kalman filter and those needed for the DFE adaptation realizing the fact that an optimum DFE produces decisions with delay � > O. First at time t, the Kalman filter assumes that the DFE hard decisions Xt-�-l' .... Xt-�-l-l are correct and uses them along with received vector Yt-�-l and p previously estimated channel vectors to estimate the next channel value ht_�. In the next step, -predictor co�putes the sequence of � predicted chan­nels ht-�:±:l' .... , ht. by �xploiting the most recent Kalman estimates ht-�-p+l' .... , ht_�. Finally DFE uses these � predicted channels, along with the Nt - � (where Nt is the order of feedforword filter (FFF) co-efficients wopt of DFE) most recent channel estimates from the kalman filter to decode one more set of nT-dimensional symbols Xt-� which is then used by Kalman filter to estimate new channel at the next iteration.

tion: To estimate the sequence ht of the time-varying fading channels modeled by a pth order AR process in (13), we write its p( l + 1 )nTn R x 1 state vector as,

(16)

Thus the resulting matrix state space representation of the fading channels system is given by

(17)

Yt = [Xt OnRx(p-l)nrnR(l+1)] Zt + Vt (18)

Where transition matrix F and the input matrix G are respectively defined as,

F= [ -A(l) -A(2) I(p-l)nrnR(l+l)

-A(p)

J O(p-l)nrnR(l+l)xnrnR(I+l (1 )

G - [ InrnR(l+1)xnrnR(l+l) (20) - O(p-l)nrnR(l+l)xnrnR(l+1) Given the above state space representation of the system

and assuming F, Q, Pt-�-l and Zt-�-lare known from a preceding training phase, Kalman filtering at time t with delay � can be carried out by the following series of equations [10],

1) First nR x nR dimensional covariance matrix Re of innovation process e can be obtain as,

Re,t-�-l = Rvv + Xt-�-lPt-�-l:XL�_1t (21)

2) Next Kalman gain matrix KpnrnR(l+l)xnR can be ob­tain as,

Kt-�-l = (FPt-�-lX;_�_l)R;'�_�_l 3) Innovation process enRxl is then obtain as,

(22)

et-�-l = Yt-�-l - Xt-�-lZt-�-l (23)

4) State vector Z of length pnTnR(l + 1) is then updated as,

Zt-� = FZt-�-l + Kt-�-let-�-l (24)

5) In the end, error covariance matrix

ppnrnR(l+l)xpnrnR(l+l) is then updated as,

Pt-� FPt-�-lF* + GQG* - Kt-�-lRe,t-�-lK;_�_l (25)

Since at time t, the last channel estimate from kalman !liter is ht_� and DFE needs chan}lel es!imation upto

ht , the last � channels estimate ht, ... ht-�+l have to be predicted. In [2], optimum linear prediction, given that the channel follows the model of (17) is suggested as

(26)

b) Decision-Feedback Equalizer (DFE) : Different design methodologies are proposed in [9] for obtaining the coeffi­cients of optimum MMSE feedforward (woPt) and feedback (Bopt) filters with lengths Nf and Nb + 1 matrix taps, respectively, as well as for the optimum selection of the delay .6. for any system. Here we use causal feedback filtering scenario where only past decisions for all inputs are available for any input at present time. The most necessary part of the DFE design is the formulation of nRNf x nT(Nf + l) dimensional block channel Matrix H

iIt 0 iIt I 0 0 0 iIt-1 iIt-1 0

H= 0 I

0 0 iIt-NJ+1 0 iIt-NJ+1

I 27)

where iI�, m = 0,1, ... , l are the channel estimate of the nR x nT channel matrices Hm(k), k = t, t -1, ... , t -Nf + 1

which contains the taps h�,j) (k) of (1) in its (j, i)th position. Once H has been formed, we can easily design MIMO DFE by using procedure describe in [9]. Here we only replicate the important equations from [9].

The FIR MIMO MMSE-DFE shown in Fig 3 consists of feedforward filter matrix

w* = [wo W� ... W�J-l] (28)

and a feedback filter matrix equal to,

-B� ... - B* ] Nb (29)

Where B* is defined as,

B* = [Bo B� ... B�J (30)

The enteries of the matrices Wand B are in the following form,

(l,nT) 1 w·

(:l,nT) Wi

(31)

b(l,nT) 1 bl:�,nT)

(32)

The optimum matrices of the feedforward and feedback filters are related as,

(33)

where B* can be defined on the basis of B* as,

B *] (34)

l1Rxl ----l

y, FFF l1Txl Decision \VtOpt

l1Tx 1 FBF BOP' ,

Fig. 3. Block Diagram of MMSE DFE

l1Txl X'_6

�

with decision delay 0 :::; .6. :::; N f +l-l satifying the condition (.6. + Nb + 1 = N f + l). The input-output crosscorrelation and the output auto-correlation matrices given in (33) can be taken respectively as,

(35)

Ryy = HRxxH* + Rvv (36)

where Rxx is nT(Nf +l) x nT(Nf +l) input auto-correlation matrix and Rvv is nRNf x nRNf noise auto-correlation matrix. In the special case (which we will use in simulations) when noise and input processes are uncorrelated between different time samples and from one input-output channel to the other, the matrices Rxx and Rvv become block diagonal and take the following simple form,

diag(0';1'0';2''' ,0';n )) , , , T (37)

(38)

As we previously mentioned that we only consider the case when only past decisions for all inputs are available for any input at present time (i.e., Bo = InT). In this case the solution for the optimum matrix of the feedback filter cofficients is given by,

(39)

where Rll(nT(.6. + 1) x nT(.6. + 1)) and R12 are taken from the matrix R defined as, R = R;; + H*R;;-,; Hand

partitioned as R = [ �i: ��� ], and matrix C is defined

as, C* = fOnT xnT� InT ] ' Using (39) along with (35) and (36), the coefficients of the feedforward matrix given in (33) can easily be found where as using (39) along with (34) and (29), one easily found the coefficients of the feedback matrix.

1�r. -.. -..

-..

�.

-..

-...

-.

� .. - .. - .. - .. �, .. - .. -.. -.��-.�-�._���:._�_�1_ .. �._ .. _ .. _ .. � ________ --, ... ____ _______ ________ , _______ _______ , _______ _______ -- Actual Channel . ---- -. - -- - --- -- ---- - -,-- - -- - ­- _ .. _ _ . " .. _--- -------- ,-------------- ------- -------- ,-------

---- ---,- -- ---- ----. - . ----- _ .'-- - ---- - -----

, , ,

-- AR(1O)

-------,-- - ----,- - -----.,--- --_______ , _______ � _______ J. __ _ _ , , , ------_ . . _---_ ... . _---_ .... _-- -· . . , -------' ------_ .. - - - - - - -'" - - - - - - - - - - - � - ------· . . , ------- ; - - - - - - - � --------:- - - - - - - - - - - - r - ------

· . . , ------_ . ------_ .. . -----_ ... - - - - - - - - - - - � - ----- -· . . , · . . , ----- - - ; - - - - - - - � --------:- - - - - - - - - - - - - r -------· . . , , " , · . . , ------- ,------- ,------- ,----- - ------- r-------· . . . .

, " " ' " - -- - --. ---- -- - .. -- - _ .. .. ,.. ... . . . ... ... .. . . ... . , .. ..... ., . .. . .. ..... .... . � ..... . .

1 D·' 0:---::5':- 0 ---"'1 0:::0---:-150:::-----:::200:::-----:::2�50,---::3�00,---::3�50,---,4�00,---,4�50,----:!500 Samples

Fig. 4. Envelope of the estimated fading processes. SNR=15 dB

Ill. SIMULATION RESULTS

In this section, we implement the Kalman-aided MIMO OFE algorithm outlined in previous section and compare the BER performance of the system using kalman channel estimator with high order AR model with that of low order AR model, with standard LMS and with perfect channel state information (CSI) scenarios, under realistic directional Ricean channel model. We considered (2, 2) system where each input generates BPSK constellation points. The doppler rate, JdT , is considered to be 0.05 and specular-to-diffuse power ratio is taken as K = lOdE for all taps. We considered first order channel ( i.e, channel with memory l = 1). The OFE has Nf = 3 and Nb = 1 matrix taps and decision delay is kept as D. = 2. Fig. 4 shows the estimated envelopes of the fading processes (only one out of eight taps is shown here) using the kalman filter based estimaters with AR(l) model (blue solid line) and with AR(10) model (red solid line), where as Fig. 5 shows BER performance of system versus SNR in highly non isotropic environment (i.e, we considered K, = 2). The results shows that kalman filter based tracking gives generally better performance than model-independent, less complex LMS based traking when using either low or high order AR approxmation of channel. From Fig. 4 and 5, one can also notice that increasing the order of AR model significantly increases the BER performance of the system. The high order channel modeling helps to lower the possibilty of error propagation.

IV. CONCLUDING REMARKS

In this paper we have presented a receiver structure for tracking and equalizing directional MIMO Ricean fading channels. A kalman filter based on high order AR model with statistics matching those of the directional Ricean fading process has been used to track channels and OFE has been

---e--- perfect CSI ---t--LMS

�?V-�ri'--"", ... ... " ...... ........... ........... � KF with AR1 --A- KF with AR10

10.3 -----------� - --- --. -- - -� ----------

10.4 '-------'-------'-------'----&--'-------'-=.<'r------' o 5 10 15 SNR(dB)

20 25 30

Fig. 5. Performance of (2,2) system in nonistropic fading environment with 11:=2

used to combat lSI. Simulation results shows that high order AR model based kalman filter offer improved performance when compared with simple LMS and low order AR model based kalman filter. However complexity increases with in­crease in order of AR based kalman filtering but this can be compromise over better system performance.

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