Efficient Training of Kalman Algorithm for MIMO Channel Tracking


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1 Efficient Training of Kalman Algorithm for MIMO Channel Tracking Emna Eitel and Joachim Seidel Institute of Telecommunications, University of Stuttgart Stuttgart, Germany Abstract In this aer, a Kalman algorithm is alied to track a timevarying flat fading MIMO channel. The imortance of training and aroriate initialization in combination with the Kalman tracking algorithm is shown. Adoting a eriodical training scheme with a given bandwidth efficiency, a tradeoff between investing ilots for good initialization and training the algorithm exclusively leads to the lowest. We also introduce a training on request scheme, in order to overcome the error roagation encountered by the Kalman filter after a series of detection errors. For this urose, two metrics to detect the Kalman filter divergence are develoed. We show the effectiveness of the new aeriodical training scheme in reducing the channel estimation error and saving bandwidth at the same time. I. INTRODUCTION MIMO systems with coherent detection can deliver high bit rates rovided that an accurate knowledge of the channel is available at the receiver. The erformance can even be enhanced if the channel state information (CSI) is also available at the transmitter. Algorithms to recisely estimate the CSI are therefore of aramount imortance. Often eriodical ilotassisted channel estimation (PACE) is emloyed. However, in fastvarying channels, PACE does not only decrease the bandwidth efficiency but is also incaable of detecting fast variations of the channel. Therefore, additional tracking techniques have to be alied. A method that does not require ilots is decisiondirected channel estimation. It uses reviously detected symbols and can therefore feed the channel estimation module with new measurements that ermanently reflect the current channel state. Exloiting detected data, adative filtering techniques such as Kalman filter (KF), least mean squares (LMS) or recursive least squares (RLS) filter can also be used for channel tracking. In combination with a highorder autoregressive (AR) channel model, the KF shows the best erformance among them but at the exense of higher comlexity. However, loworder AR models can cature most of the channel dynamics for small estimation lags, as it is the case for symbolwise tracking, and lead to effective tracking erformance []. A drawback of the KF is its lack of robustness with resect to wrongly detected data. To coe with this roblem, eriodical ilot atterns are often inserted to sto the filter divergence. An alternative solution exloits reliability information about detected data [2]. But this aroach requires iterative receiver structures which introduce a significant delay and a high comlexity [2], [3]. In this aer, we imrove the tracking erformance of the KF by two means. First, we show that in case of eriodical training, investing a fraction of the available training data to rovide the KF with an aroriate initialization fastens the filter convergence and imroves the tracking erformance. Second, we introduce a novel aeriodical training scheme that alies training when needed, i.e. on request. A request for ilots is initiated in case the filter diverges as a consequence of many outliers causing error roagation. Two detection methods for the error roagation are roosed and evaluated. It is clear that droing the eriodical training scheme, transmitter and receiver have the burden of a more comlicated signalling task. Nevertheless, our aroach is alicable indeendently of the detection scheme. Besides, we show that the novel aeriodical training leads to a significant tracking erformance imrovement and more than 5% reduction of required training data. II. SYSTEM MODEL We consider an M N MIMO system. The N receive signal vector at time instant n is given by: y(n) = H(n)s(n) + w(n) () where s(n) denotes the M sent signal vector, H(n) the N M MIMO flat fading channel matrix and w(n) the N additive white Gaussian noise (AWGN) vector whose comlex elements are i.i.d and CN(,2σ 2 ). Without loss of generality, we assume a satially uncorrelated MIMO Rayleigh fading channel. An element h ij (n) of H(n) reresents the channel coefficient between the jth transmit and ith receive antenna and is CN(, ) distributed. The temoral autocorrelation function of h ij (n) satisfies: E {h ij (n)h ij (n ) } = J (2πf d (n n )) (2) where f d stands for the normalized Doler frequency and J is the Bessel function of first kind and order zero. In order to estimate the channel at the receiver, orthogonal ilot symbol vectors s are eriodically sent during the training eriod that takes symbol intervals T s. At the end of the training hase, a channel estimate Ĥ is comuted by means of the received ilots according to the maximum likelihood or the minimum mean squared error rincile. The training hase is followed by a data transmission hase where L d symbol vectors are sent. In the absence of tracking, the PACE estimate is used for the coherent detection of data symbols during the subsequent L d symbol eriods. We introduce the discrete time index to denote the time elased between the end of the training
2 2 training hase, we get Y = HS + W. (4) with S = [s, s,l ], Y = [y, y,l ] and W = [w, w,l ]. The PACE estimate is comuted uon the received ilots according to Ĥ ML = Y S H (S S H ) (5) Fig.. Alternating training and data hases in the hybrid training scheme hase and the end of the data transmission hase, i.e. L d, as shown in Fig.. In case of channel tracking, the PACE estimate can be used as good initial value for the tracking algorithm at the start of every training interval. A basic rerequisite is a recursive tracking algorithm, as it is the case for the KF. Doing so, the question that arises is how good the initialization has to be. In case of slow fading, increasing the number of ilots imroves the PACE estimate and we can exect a faster KF convergence. However, if the channel is varying fast, the PACE estimate has to be built uon a few ilots. Therefore, we first give a deeer insight into PACE in order to determine the otimal training length. On the other hand and to the best of our knowledge, KF training in the literature is only alied to the filter itself [4], [5], [2]. In other words, the measurements from eriodically sent ilots are used as inut to train the statistical variables of the algorithm. Instead, we adot a hybrid training scheme. This aroach was first introduced in [6] and has roved to significantly imrove the tracking erformance of the RLS algorithm. Motivated by these results and the existing corresondences between RLS and KF [7], we alied the hybrid training scheme in [6] to the KF. The simulations results in Section VII confirm the exected erformance imrovement. Additionally, we show that if the mean squared error () of the KF tracking at the end of the frame is smaller than the PACE, the initialization with PACE is disadvantageous. In this case, ilots are not needed anymore and the KF can oerate in a quasiblind decisiondirected mode. Due to the KF sensitivity to wrongly detected data, ilots have still to be requested in case of error roagation, which we introduce as a new aeriodical, ilotonrequest training scheme. III. PILOTASSISTED CHANNEL ESTIMATION During the training hase, ilot symbol vectors s,i with i are transmitted. The corresonding received y,i are imaired by AWGN vectors w,i. From () follows: y,i = Hs,i + w,i for i. (3) Assembling all ilot symbol vectors s,i, all corresonding receive symbol vectors y,i as well as w i in matrices and assuming that the channel does not change during the PACE for the ML estimate, where (.) H refers to the Hermitian of a matrix. If knowledge about the SNR and the channel satial correlation roerties is available at the receiver, a better PACE estimate can be comuted according to the M rincile. Therefore, we rewrite (4) in vector form to aly standard results from estimation theory: vec(y ) = } {{ } ŷ ( S T I ) vec(h) } {{ } } {{ } X h + vec(w ) } {{ } w where is the Kronecker roduct. The M estimate ĥ M = vec(ĥ M ) is given by: ĥ M = R hh X H ( X R hh X H ) + R w w ỹ (7) where R hh = E { hh H} { and R w w = E w w H}. When using PACE for the initialization of the tracking algorithm, we have to get a deeer insight into its estimation quality. An aroriate means to do so is to consider the channel estimation ζ(), which we define throughout this aer by: ζ() = { } MN E H() Ĥ 2 F (8) where F is the Frobenius norm and Ĥ = Ĥ in case of PACE. We now take account of the channel time variations during the training hase. With orthogonal training data, i.e. S S H = I M and the channel according to (2), we derive the ML mean squared estimation error as in (9). A similar exression was derived in [8] but only for one tx antenna. Our exression holds for an arbitrary number M of tx antennas. ζ() = M L 2 ξ(i,) + 2σ2 L i= } {{ } }{{} ζ ζ () 2 where ξ(i,) = 2( J (2πf d ( + i))). (9) shows that the PACE is comosed of two quantities: ζ () deending on the channel time variance and ζ 2 which is related to the AWGN. We see that increasing the number of ilots decreases ζ 2 but may increase ζ (). Esecially for high f d and deending on the SNR an otimal that minimizes (9) exists. This is illustrated in Fig. 2, where ζ() is lotted as a function of the SNR and for f d =.2. = is considered since ζ() is of interest when using PACE for the initialization of the tracking algorithm. For sace reasons, we only give ζ() for ML estimation. ζ() for the M estimate in (7) can be derived analogously (6) (9)
3 3 If we use the aroximation J (x) x 2 /4 for x, and set the first derivative of (9) with resect to to zero, the otimal training length which minimizes (9) can be derived to:,ot = { 2 + 3σ2 M π 2 f 2 d M 2 + 3σ2 if π 2 fd 2M > M otherwise () where refers to the floor oeration. () shows that,ot increases with increasing σ 2 and decreases with increasing f d or M. We should kee in mind that M must be satisfied which is a necessary condition for the inversion in (5). IV. THE KALMAN ALGORITHM If the fading channel can be modeled as an autoregressive rocess of order (AR()), then the KF is the otimal M estimator. However, since the first few channel correlation terms in (2) are basically imortant for symbolwise tracking, AR(2) modeling is adoted in this work as in []. The Kalman algorithm relies on a statesace formulation comosed of the observation equation () and the rocess equation(2). y(n) = X(n) z(n) + w(n) () z(n) = Fz(n ) + Bu(n) (2) SNR [db] 25 where z(n) = [h T (n) h T (n ) h T (n +)] T with h(n) = vec(h(n)) and F is the state transition matrix. X(n) contains the detected symbol vector ŝ(n) according to X(n) = [ŝ(n) I N O N NM( ) ]. u is the driving noise with E[u(n)u(n) H ] = I 2 MN. We briefly list the key equations of the Kalman tracking algorithm: Predicted channel state Predicted ẑ(n n ) = Fẑ(n n ) (3) (a) ζ () P(n n ) = FP(n n )F H + BB H (4) Kalman Gain.2. K(n) = P(n n )X H (n) (X(n)P(n n )X H (n) + R ww ) () (b) ζ 2 (c) ζ() = ζ () + ζ 2 2 SNR [db] 2 SNR [db] Fig. 2. PACE as function of SNR and for f d = Corrected channel state ẑ(n n) = ẑ(n n )+K(n)(y(n) X(n)ẑ(n n )) (6) Corrected P(n n) = (I K(n)X(n))P(n n ) (7) Some initial values for ẑ( ) and P( ) must be chosen to launch the algorithm, the socalled starting conditions. So far in the literature the starting conditions are set to arbitrary values or to the mean value of the corresonding variable if known [5]. By means of relacing actual data by training symbols (full training), we can find the amount of training needed for convergence of the filter. The full training analysis reveals that the convergence can be dramatically accelerated by choosing more aroriate starting conditions. According to the initialization of the algorithm, we differentiate between two eriodical training schemes: A scheme where only ẑ( ) is trained by means of PACE, called conventional eriodical training (CPT), and a hybrid eriodical training (HPT), where both variables ẑ( ) and P( ) are trained. Both schemes will be discussed in the next section. 2 F and B have to be comuted deending on the AR rocess order such that (2) is fulfilled. In the following they are assumed to be known at the receiver. Please refer to [] for exlicit definition.
4 4 V. PERIODICAL TRAINING OF THE KALMAN TRACKING ALGORITHM As can be seen in Fig., the eriodically sent ilots are divided into two sequences. One sequence of length that is attributed to the PACE block rovides the tracking algorithm with a good initial estimate. The second sequence trains the algorithm and takes L t T s time. For a fair comarison of the new training scheme with the reviously established ones, the otimal and L t are chosen such that ( +L t )/L d is ket constant. In order to study the convergence of the KF, Fig. 3 and Fig. 4 lot the ζ() in case of full training. The erformance of the tracking algorithm deends highly on the quality of the initialization. In case of initialization with zero, the filter might even not converge within a frame. The convergence is however drastically accelerated if an amount of the training data is sent on more aroriate initialization with PACE. = L = (a) SNR = db = L (a) SNR = db L = (b) SNR = 42dB Fig. 4. Channel estimation ζ() for full training with different at f d =. and the error covariance P rovide us with an efficient tool to do so. If the theoretical is much smaller than the trace of P, reinitialization makes sense. Otherwise, the received ilots are not needed. Because of the eriodical training, this means that ilots are transmitted at the beginning of each frame but are not used which is a real waste of bandwidth. This leads us to the aeriodical training scheme where ilots are only sent when needed, i.e on request. The ilot on request training scheme (PRQT) is discussed in the next section (b) SNR = 42dB Fig. 3. Channel estimation ζ() for full training with different at f d =. The full training analysis leads to a further result. In the steady state at the end of a frame, the can be smaller than the PACE at the beginning of a frame, i.e. ζ(l d ) < ζ(o). This haens for examle in the desiciondirected mode at high SNR when the detected data is mostly correct. In this case, reinitializing the algorithm is not advantageous. Therefore, we have to think about a mechanism to decide whether to reinitialize with PACE or not. The theoretical PACE (9) VI. APERIODICAL TRAINING: PILOTS ON REQUEST In this training scheme, ilots are transmitted on request 3. The necessity for ilots arises when the Kalman filter diverges as a consequence of a series of detection errors. KF works robustly as long as the detected symbols are almost correct. In case of misdetections the model in () is not matched anymore and the channel estimation quality deteriorates which may result in more misdetections in the following stes and to error roagation. Accurate detection of error roagation is a key issue for the novel PRQT in order not to imair the sectral efficiency. 3 We assume that the transmission of the ilots is delayfree. Consideration of a stochastically delayed time of arrival of the receive ilot signal is subject to future work.
5 5 2 true real h (n) estimated real h (n) x 3..5 tr(p(n n )) tr(p(n n)) tr(r ee (n)) Fig. 5. time n e(n) Analysis of different variables in the KF as function of the discrete In [2], [9] reliability information about the detected data is used to detect wrongly detected symbols and exclude them from channel tracking. This aroach is feasible as long as the channel is almost invariant on a received block. Besides, it requires an iterative receiver that can rovide statistical reliability information. Instead, if symbolwise tracking with a noniterative receiver is required due to fast fading, this scheme is not alicable anymore and we have to think of other indicators for the error roagation. Closer analysis of different statistical quantities involved in the KF algorithm suggests that a filter divergence occurs in most of the cases just after a stee eak has aeared in their rogress. This is for examle the case for the magnitude of the innovation rocess e(n) = y(n) ˆX(n)ẑ(n n ). However, other variables such as P(n n ), P(n n), and R ee (n) remain unchanged. R ee (n) is the covariance of the innovation defined by R ee (n) = E[e(n)e(n) H ]. Further mathematical maniulations on R ee yield (8). Comutation of (8) is erformed within the Kalman gain in () at each iteration and therefore does not require any further comutational resources. R ee (n) = X(n)P(n n )X H (n) + R ww (8) Fig. 5 shows some variables involved in the KF tracking rocess for f d =.4, L t = 2 and L d = 2. We can see that a large e(n) due to an instantaneous high noise value gives birth to a filter divergence. The error roagates until the beginning of the next frame where the divergence is interruted by setting the estimate to the PACE value. Armed with these observations, we develo a first metric m to detect a filter divergence. If m = e(n) exceeds a threshold Q th which is related to the exectation in (8), an error roagation is occuring and ilots are requested to sto it. Intuitively, Q th is exected to deend on the SNR γ db. If Q th is small, we would be requesting and transmitting ilots all the time instead of data, reducing the sectral efficiency. On the other hand, a large Q th can fail in detecting many α 2 3 CPT a=. b= a=. b a=. b=5 a=. b=3 a=. b= a=.5 b= a=.5 b a=.5 b=5 a=.5 b=3 a=.5 b= a=.25 b= a=.25 b a=.25 b=5 a=.25 b=3 a=.25 b= a= b= a= b a= b=5 a= b= CPT a=. b= a=. b a=. b=5 a=. b=3 a=. b= a=.5 b= a=.5 b a=.5 b=5 a=.5 b=3 a=.5 b= a=.25 b= a=.25 b a=.25 b=5 a=.25 b=3 a=.25 b= a= b= a= b a= b=5 a= b= Fig. 6. as function of the SNR for various (a, b) (to). Ratio ilots over data α as function of the SNR for various (a, b) (bottom) error roagations. Thus, finding the otimal threshold is a constrained otimization roblem. We have to search for the threshold that minimizes the under the constraint that the sectral efficiency remains beyond a certain value. Out of lack of mathematical tractability for this roblem and for the sake of simlicity, Q th is defined as an affine function of γ db by means of the coefficients a and b as follows: Q th = (a γ db + b) tr {R ee } (9) By intensive simulations, we determine the coefficients a and b which minimize the keeing the sectral efficiency beyond a certain value. To evaluate the quality of sectral efficiency, we introduce the arameter α which denotes the ratio of number of ilots over data. Some results of this otimization rocess are illustrated in Fig. 6. Therein, the tradeoff between small and large sectral efficiency is lain to see. For instance, (a,b) = (,3) leads to the lowest but the required number of ilots is very high. On the other hand, (a,b) = (,) requires the smallest number of ilots but at the exense of large. As a second aroach, we suggest to consider the normalized innovation squared (NIS), in order to rovide a metric m 2 indeendent of the SNR. The NIS m 2 is defined as: m 2 = e(n) H R ee (n)e(n) (2)
6 6 2 3 HPT = =8 3 =3 =5 = =5 = results suggest that training the algorithm exclusively erforms worse than allocating an amount of the training to suly the algorithm with a PACE initialization for both f d. Furthermore, we notice that HPT with = 2,L t = 6 erforms best for f d =.. Indeed, at f d =., = 2 leads to the smallest PACE. At smaller f d, investing all ilots for PACE initialization leads to the lowest on the whole considered SNR range. α HPT = =8 3 =3 =5 = =5 = = L t L t L t L t L t = (a) f d =. Fig. 7. as function of the SNR for various M d (to). Ratio ilots over data α as function of the SNR for various M d (bottom) An error roagation occurs if m 2 exceeds a threshold M d. This aroach is known as validation gating and is widely known in the field of target tracking to exclude very unlikely measurementtotrack associations []. The NIS follows a chisquare robability density function. Thus e(n) H R ee e(n) < M d means that for a robability that % of true associations are acceted, M d can be comuted from = P(N 2, M d 2 ) = Γ(N/2) Md /2 e t t N/2 dt (2) where Γ is the Gamma function. For instance, for = 99,99% and N = 2, M d = The for different M d values and the corresonding α are illustrated in Fig. 7. The tradeoff between low and high sectral efficiency is again lain to see. VII. SIMULATION RESULTS A 2 2 MIMO system with BPSK modulation and zeroforcing receiver is considered. We assume that a constant sectral efficiency is given. This means that for eriodical training, we kee the ratio of training and data transmission hase lengths α = ( +L t )/L d constant. For our simulations, we take L d = and + L t = 8. The corresonding results for HPT with different f d are illustrated in Fig. 8. These 2 3 = L t L t L t L t L t = (b) f d =. Fig. 8. as function of the SNR for hybrid training scheme with + L t = 8 The results for PRQT, otimized under the constraint that α 8% for comarison fairness, are lotted in Fig. 9 for f d =.. PRQT with both suggested metrics outerforms CPT and HPT significantly. The ratio of required ilots is even reduced to less than 2% as can be seen in Fig.. For PRQT with m 2, M d = 3 is chosen for an SNR < 24dB and M d = 5 beyond since this yields to a good tradeoff between and sectral efficiency, resecting the constraint α 8% on the whole considered SNR range. PRQT with m 2 outerforms PRQT with m for SNR < 2dB. For low SNR however, it leads to similar as for CPT and HPT. The discontinuities in Fig. arise from adoting different arameters ((a,b) for m, and M d for m 2 ) deending on the SNR. Considering that it is less comlicated to otimize the threshold M d in comarison to Q th (2 degrees of freedom with the coefficients a and b), the NIS metric m 2 is referred over m if the oerating SNR is large enough.
7 7 Fig. 9. α CPT L t = CPT = L t HPT L t 4 CPT L t = erfect decoding PRQT m PRQT m 2 erfect CSI as function of the SNR for CPT, HPT, PRQT and erfect CSI CPT, HPT +L t REFERENCES [] C. Komninakis, C. Fragouli, A. Sayed, and R. Wesel, Multiinut multioutut fading channel tracking and equalization using Kalman estimation, IEEE Transactions on Signal Processing, vol. 5, no. 5, , May. 22. [2] I. Nevat and J. Yuan, Joint channel tracking and decoding for BICM OFDM systems using consistency tests and adative detection selection, IEEE Transactions on Vehicular Technology, vol. 58, no. 8, , Oct. 29. [3] J. Choi, M. Bouchard, and T. H. Yea, Adative filteringbased iterative channel estimation for MIMO wireless communications, IEEE International Symosium on Circuits and Systems, Vol. 5, May. 25. [4] E. Karami and M. Shiva, Decisiondirected Recursive Least Squares MIMO Channel Tracking, EURASIP Journal on Wireless Communications and Networking, Dec. 25. [5] S. Haykin, Adative Filter Theory. PrenticeHall, Inc., 996, ISBN X. [6] E. Eitel, R. A. Salem, and J. Seidel, Imroved decisiondirected recursive least squares MIMO channel tracking, IEEE International Conference on Communications,. 5, Jun. 29. [7] A. Sayed and T. Kailath, A statesace aroach to adative RLS filtering, IEEE Signal Processing Magazine, vol., no. 3,. 8 6, Jul [8] Q. Sun, D. Cox, H. Huang, and A. Lozano, Estimation of continuous flat fading MIMO channels, IEEE Transactions on Wireless Communications, vol., no. 4, , Oct. 22. [9] I. Nevat and J. Yuan, Channel tracking using runing for MIMOOFDM systems over GaussMarkov channels, IEEE International Conference on Acoustics, Seech and Signal Processing, vol. 3,. III 93 III 96, Ar. 27. [] T. Bailey, B. Ucroft, and H. DurrantWhyte, Validation gating for nonlinear nongaussian target tracking, 9th International Conference on Information Fusion,. 6, Jul. 26. PRQT m PRQT m Fig.. Ratio ilots over data α for PRQT and CPT with 8% training A further very remarkable result is that the curves for KF with PRQT and KF oerating with CPT and erfect detection are overlaing between 2dB and 36dB. For SNR<2dB, PRQT generally erforms worse due to the constraint of sectral efficiency. In the high SNR however, it even outerforms the tracking with erfect detection. The erformance ga between PRQT and erfect CSI is basically due to the model mismatch with AR(2). We exect this ga to be smaller with higher AR order. VIII. CONCLUSION In this aer, we deal with tracking fastvarying MIMO channels by alying the Kalman algorithm. Different training schemes for this algorithm are suggested such as the hybrid eriodical scheme and the ilot on request scheme. They are comared to the conventional eriodical training. The hybrid scheme can decrease the floor by an order of magnitude in comarison to the conventional eriodical training while maintaining the same sectral efficiency. For the aeriodical ilot on request training, we develo two different metrics for detecting the error roagation. Finally, we show that with the novel training on request scheme the can be significantly decreased while the number of required ilots is remarkably reduced.
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