ChannelEstimationwithPilot-SymbolsoverWSSUSChannels

Transcription

1 > = ; ChannelEstimationwithPilot-SymbolsoverWSSUSChannels Ivan Periša, Jochem Egle, and Jürgen Lindner Univ of Ulm - Dept of Information Technology - Albert-Einstein-Allee Ulm - ermany Ph , Fax , ivanperisa@e-technikuni-ulmde ABSTRACT We investigate a pilot-symbol based channel estimation scheme that uses a combination of Maximum-Likelihood (ML) estimation and Wiener filtering Besides the optimum Wiener filter, we also investigate suboptimum Wiener filters which offer a better robustness against path delay time estimation errors and/or have a lower complexity The mean squared error () of the channel impulse response (CIR) is used to compare the performance of different channel estimators We also show the effect of channel estimation error on the bit error rate (BER) in computer simulations where we assume a typical short wave channel eywords channel estimation, parameter estimation ITRODUCTIO The use of pilot signals is a well known method, which can be used to obtain good and robust channel estimates A very comprehensive study of this topic for single-user communication systems is found in [] or [2] It was shown that the optimum estimator consists of a maximum likelihood (ML) estimator followed by a Wiener filter A Wiener filter approach was later also used for channel estimation in OFDM- and MC-CDMA-systems (eg see [4] and []) In [] a simplifying assumption was made for the calculation of the channel tap correlation matrices which are used in Wiener filters, though It was assumed, that the taps of the channel impulse response are mutually uncorrelated This assumption is not true, because the taps of the CIR are correlated due to the use of a band-limited basic waveform In this paper we would like to extend the work done in [], by investigating the channel tap correlation matrices which are used in the Wiener filter, in more detail In [3] optimum channel tap correlation matrices were considered, but there was no comparison to other schemes or detailed discussion about the effects of certain parameter estimation errors We propose some new schemes for the calculation of channel tap correlation coefficients, that lead to good results and are robust with respect to path delay estimation errors Further, we also consider potential errors in the estimation of the signal to noise ratio (SR) We use the mean square error () as performance criterion to compare how well the different channel estimation schemes perform under certain conditions One nice property of the as performance criterion is, that in Wiener filter theory the can be computed analytically for any filter type (see [6]) Besides the we also compare the performance of different estimators in the bit error rate (BER) sense For all comparisons we use a wide sense stationary uncorrelated scattering (WSSUS) channel model with parameters, that are typical for short wave communications 2 CHAEL ESTIMATIO WITH PILOT SEQUECES BASED O WIEER FILTER THEORY As mentioned before, the optimum channel estimator for WSSUS channels, that relies on pilot-symbols, consists of a ML channel estimation followed by a Wiener filter The WSSUS channel assumes paths, all of which have complex valued, time variant path weights In the following sections we will briefly describe ML channel estimation and then investigate the Wiener filter in more detail 2 Maximum Likelihood Channel Estimation As stated before, for ML channel estimation, we rely on a known, cyclic transmit sequence of length which we will call pilot-sequence or probe in the following Further, we assume that the channel is not varying too fast during the ML-estimation The received signal is obtained by convolution of the pilot-sequence with the channel impulse response (CIR) and the addition of white aussian noise "! # %&()+*,-/032 () where *4, is the symbol duration ote, that we see the channel as a concatenation of the basic waveform 6 and the WSSUS channel model with CIR Hence, we get where 9 <; ML > J HI?@AB CD?EF? (2) AO+ CD QP (3) At the receiver the received signal is sampled with a sufficient sampling rate, according to the yquist theorem In the following we will consider a root raised cosine with roll-off factor?r3sp TVU Therefore, it is sufficient to sample the signal at twice the symbol rate Extension to other cases, where higher sampling rates are required, is quite straightforward For channel estimation we process a sampled sequence that depends on pilot-symbols only, as illustrated in Figure, where we assume a causal CIR of finite length

2 ? 9? ) Transmitted Symbols x(k) Received Symbols g(k) Transmitted Probe Symbols Usable Received Symbols T Probe x(k) g(k) R Frame Datablock T2 Probe 2 T LP R2 R Frame Frame 3 Datablock 2 0 L LP Figure Frame structure T3 Probe 3 Since we deal with sequences of finite length, it is convenient to introduce a matrix notation All underlined lower case letters represent vectors, whereas matrices are represented by underlined capital letters The # -operator defines the complex conjugate transpose of a matrix or vector Scalar values within a vector at position are described by Thus, we can write the channel impulse response of length as J # # # k R3 > J (4) and the transmitted pilot-sequence of length 2 as SQ S J # # # > J S P () To increase readability we will omit absolute time indices It should be noted, that in our notation the transmit sequence is not the original probe sequence p(k) We treat it as oversampled sequence (which is described by inserting zeros into the data stream), because the received sequence is oversampled and both need to be at the same rate Then we can write the received vector as 2 (6) where is additive white aussian noise and transmit sequence matrix defined by with J # # # is the > J () being a part of the transmit-sequence with the same length as the undisturbed received sequence J # # # >4J P () It was shown in [] and [] that the ML channel estimate is then given by The estimation error covariance matrix is where # #% >4J > J is the noise variance P (9) ) "! (0) 2 Sequences with a perfect cyclic ACF If we use sequences with a perfect cyclic autocorrelation function (ACF), we obtain &(*) () where is the energy of the transmitted probe sequence Hence, the calculation of the ML estimate in Equation (9) is simplified to (see []) ( P (2) The calculation of the estimation-error covariance matrix is also simplified to # (3) which means that all taps have the same variance and are mutually uncorrelated It was shown in [] that the tap error covariance of the channel estimate is minimized, if we assume the channel estimate to be valid in the center of the undisturbed received sequence that depends on pilot-symbols, only (see Figure ) As stated before, a Wiener filter is used to improve the ML-estimate We will consider Wiener filter theory in the following section 22 Wiener Filter for Channel Estimation We assume that we have snapshots at the input of the Wiener filter We introduce a vector +-,+ that contains all snapshots that are used in the Wiener filter at a certain time instant +-,+ 0/ ) 2 46 P (4) Further, we assume a transversal filter (TF) structure, which means that Wiener filtering can be seen as a scalar product of two vectors The Wiener estimate for the E -th tap is then obtained by &9 +-,+@P () Wiener filter theory is applied to find the optimum set of filter coefficients that minimize the for a certain tap This leads us to the solution (see [6]) 9 ;;< +-,+6=*;< +-,+ +-,+>= P (6) A compact notation for a filter matrix that is used to produce the whole estimated CIR is given by 0/ 2 9 J 9 > J 46 ; < +-,+6= < +-,+ +-,+ = P () Thus, the optimum solution for estimating the entire is given +-,+ P () In the following sections we will discuss some schemes for the construction of Wiener filter matrices

3 # 2 ) / 2 ) S S 3? # + L P 22 Optimum Wiener Filter Matrix First, we will see, how to construct the optimum filter matrix that is given by () We will therefore examine the correlation of the taps of two CIRs Before doing so, we have to discuss some basic relations The noise is assumed to be white and thus we obtain, P (9) This is obviously only true, if we use sequences with perfect cyclic ACF Extension to the general case is straightforward, since equation (0) can be used to calculate the noise covariance matrix Another important relation that follows from the WSSUS model is that the path weights are uncorrelated Denoting, we can write the average path gain of path with +, P (20) In order to obtain a more compact notation we introduce the set of vectors For each path of the WSSUS model we define one vector They represent the normalized, sampled basic waveform with a shift of CIR window B S4# B # B # Then we can write the correlation matrix snapshots valid at different time instants as 4 in the P (2) of 2 L > J # (22) O+ 2 B + ( where is the noise ACF, ( is the ACF of the true CIR and is the normalized time-correlation coefficient for the -th path ( S6 ) The calculation of correlation matrices of a snapshot and the true CIR, is equivalent to the previous calculation, though we do not have to consider the noise components in this case We can use the correlation matrices described by (22) to construct the total correlation matrices For the snapshot auto-correlation matrix we obtain +-,+ / 2 SV # # #! SV # # #! % # " # # # SV 4 (23) and for the matrix holding the correlation between all snapshots and the true CIR we get >J # # # % J # # #& >J P (24) We obtain a Wiener filtered estimate for any time instant by choosing the coefficients in (24) accordingly The optimum Wiener filter matrix is then given by >4J +-,+ P (2) We see that the construction of? is quite straightforward, if the correlation of two CIRs is given Therefore, we will now discuss other schemes for the construction of the correlation matrix of two CIRs These schemes are not optimum in the -sense, but do offer advantages in certain cases 222 Wiener Filter Matrix with We will see later, that a bad estimation of a path delay time can lead to a severe degradation of performance We can therefore consider uncertain delay times in the generation of the correlation matrices used in the Wiener filter Instead of assuming that the path is at one position, we assume that it is in a window of width *)(, centered at the estimated path position Mathematically, we could express this by ML >4J,+- *6 = * ( >0/ +- +A + EF (26) where the integral is applied elementwise (integrate over each element of the matrix) In a practical implementation, this integral can be substituted by a sum The width of the window *( is adjusted, depending on the reliability of the path delay estimation 223 Wiener Filter Matrix with A simpler model, where no path estimation is required was already presented in [] In this case we assume that all taps are mutually uncorrelated We only rely on tapgain estimation and time-correlation in our Wiener 3 filter The correlation matrix we obtain is / 2 2 J 2 J > J 2 >4J 4 (2) The advantage of this scheme is obviously, that the complexity of the matrix inversion of +-,+ is greatly reduced It can be seen as a set of independent Wiener filters 224 Wiener Filter Matrix with Correlated Taps and Tap ain Estimation Another idea for a simple scheme which also considers inter-tap correlation is to assume a path at each tap position The path gain is assumed to be the tap gain As we will see later, this scheme is also robust to time shifts of a path L > P (2)

4 J 22 Wiener Filter with Reduced umber of Correlated Taps So far we have just considered two extreme cases In the first approach we considered the correlation of all taps In the second, we assumed, that all taps are mutually uncorrelated Instead we could now consider a reduced number of mutually correlated taps For each tap we design a Wiener filter that considers a certain number of correlated taps (these could be neighbouring taps or taps with maximum cross correlation value) The advantage of this scheme is that we can adjust the complexity of our solution (from uncorrelated to fully correlated taps, both of which are basically special cases of this scheme) This approach can be applied to all previously described schemes that consider correlation between all taps D Filtering vs 2x -D Filtering All Wiener filters that exploit inter-tap correlation become very complex, if we increase the number of snapshots that are used in the filter The reason is that the size of the auto-correlation matrix +-,+ which has to be inverted is given by C# C#, and hence becomes very large for larger values of Therefore we could think of two separate Wiener filters One that exploits the intertap-correlation and one that exploits the time-correlation In the first step we calculate a filter matrix, that assumes uncorrelated taps and exploits time-correlation as described by Equation (2) In the second step we use one of the schemes, that exploits intertap correlation We have to take into account, however, that the tap- and noise-correlation is affected by the first Wiener filter The idea to use a two -D filters instead of a 2-D filter was also discussed for channel estimation for OFDM in [4] 23 Theoretical Calculation of the The aim of the Wiener filter is to minimize the for each tap Therefore, we will use the as a performance criterion, in order to compare the different channel estimators The advantage is, that we can theoretically derive the of each tap for an arbitrary filter vector 9 (see [6]) The of tap E is given by * P (which is used in most sim- The average per tap ulation results) is (29) L4 >4J P (30) 3 PERFORMACE OF DIFFERET CHAEL ES- TIMATORS For our results we have used a block transmission scheme with a pilot-sequence of length HT, a datablock of length U and a symbol duration of / * The basic wave form we used is a rootraised cosine with roll-off factor? SP T6U and we have used 3 ML estimates in each Wiener filter for the calculation For the simulation of the BER we have used ML estimate, only Further, we have focused on a typical short-wave communications channel with 2 paths and parameters as shown in the Table good Channel no 4 CCIR Poor typical/moderate bad HF path path 2 delay in ms 0 2 path gain, rms Doppler shift in Hz 0 0 Doppler spread in Hz Table Channel model used in the simulations 3 of Different Channel Estimators As seen above, the of a channel estimator can be evaluated easily by using Equation (29) 3 Average In order to compare the channel estimation schemes for different signal to noise ratios (SR), we have averaged the of each scheme over all taps The of all taps is only equal in the case of ML estimation where we use pilot sequences with a perfect cyclic ACF (see (0)) The individual of all taps for a certain SR are compared in the next section The results for a 2-path channel with Doppler spread (two-sided) of Hz are shown in Figure 2 All schemes do improve the ML estimate significantly especially for lower SR-values As we would expect, the optimum Wiener filter performs best, while all schemes that consider inter-tap correlation yield better results than the filter which assumes uncorrelated taps The performance of the Wiener filter that assumes an uncertain delay time could be improved, by choosing a smaller window size (here, we chose * ( ) In the second case we kept the same filter matrices, but shifted the second path by This means, that we have introduced an error in path delay time estimation As the results in Figure 3 show, the previously optimum filter matrix now performs worst, whereas the other simplified schemes are somewhat more robust with respect to a path delay estimation error It turns out that an error in the estimation of the SR does not have severe effects on the performance of a Wiener filter In Figure 4 we compare the performance of the optimum filter matrices to filter matrices that assume a constant SR of 20 db Over a range of about 0 db we can hardly see any performance degradation for the assumption of a constant SR Finally, we examined two schemes with reduced complexity in Figure The performance of the optimum 2D Wiener filter was also reached by two separate D filters We also see, that Wiener filters that consider a reduced

5 0 2 ML estimate Corr Taps with Tap ain Est Optimum Corr Matrix 0 2 ML estimate Optimum Corr Matrix SR [db] Figure 2 for different estimators ML Estimate "Optimum" Corr Matrix Corr Taps with Tap ain Est SR [db] Figure 4 for different estimators with wrong SR estimation (assumed 20 db) Dotted lines show the case of wrong SR estimation, whereas full lines show the case of perfect SR estimation Reduced Tap Corr (3) Reduced Tap Corr () Opt Corr Matrix Opt 2x D Filter SR [db] Figure 3 for different estimators with wrong path delay estimation number of correlated taps offer a nice solution to adjust the performance and complexity of the system It should be noted that for channels with large Doppler spread and long data blocks (as in the case we considered), time correlation does not improve performance by much 32 by Tap-Position ow that we have compared the average, we will focus on the per tap We have already discussed that for ML estimation with appropriate sequences all taps have the same error This is not true for Wiener filtered channel estimates as we can see in Figure 6 Indeed, the error greatly depends on the average path gain - or from another point of view - the SR For taps with low gain, the error is smaller than for taps with large gain This means that we could expect the Wiener filter to work better with channels that have a small number of distinct paths and many taps with low tap gain (like short wave channels for example) SR [db] Figure for different estimators with reduced complexity ML estimate Corr Taps with Tap ain Est Optimum Corr Matrix tap Figure 6 per tap for different estimators for SR=20dB

6 0 2 ML Estimate Corr Taps with Tap ain Est Optimum Corr Matrix Perfect Channel nowledge We have also discussed the impact of certain parameter estimation errors We have seen that the optimum Wiener filter is quite sensitive to path delay time estimation errors, whereas the other schemes we discussed were much more robust BER E b / 0 [db] Figure Bit error rates for different channel estimators for, filter, and snapshot used in the Wiener 32 BER for a Block Transmission Scheme It is quite comfortable to use the to compare different channel estimators evertheless, what we are really interested in, is the impact of channel estimation on the bit error performance Here, we have chosen a scheme where the regularly reinserted probe has a length of T symbols and the data-block has a length of U symbols We use Wiener filters to smoothen the ML-estimates only The channel estimates are needed for every time instant in the data block These estimates are obtained by linear interpolation between two succeeding channel estimates, which are valid for the corresponding pilot sequence As we can see from Figure, Wiener filtering can significantly reduce the BER and we get very close to the performance with perfect channel knowledge For the case we considered, we do not have much of an improvement if we consider a greater number of ML estimates, because the time-correlation coefficients are small for the Doppler spread and data block time duration we chose For slower channels or shorter data-block lenghts it could make sense to consider time correlation as well REFERECES [] SA Fechtel and H Meyr, Optimal parametric feedforward estimation of frequency-selective fading radio channels, IEEE Trans Comm,vol 42, pp ,Feb/March/April 994 [2] HMeyr, Marc Moenecklaey, and SA Fechtel, Digital Communication Receivers Synchronisation, Channel Estimation and Signal Processing, John Wiley & Sons, 99 [3] SJ rant and JCavers, Multiuser channel estimation for detection of cochannel signals, IEEE Internat Conf on Commun, Vancouver, June 999 [4] P Hoeher, S aiser and P Robertson, Two-dimensional pilotsymbol-aided channel estimation by Wiener filtering, Proc IEEE ICASSP 9, Munich, pp 4-4, April 99 [] A Bury, Butscher, J Egle, and J Lindner Pilotsymbolgestützte analschätzung für ein Erweitertes MC-CDMA-System 2 OFDM-Fachgespräch, Braunschweig, ermany, September 99 [6] S Haykin, Adaptive Filter Theory 3rd Edition, Upper Saddle River, J Prentice-Hall 996 [] S Crozier, DD Falconer, and SA Mahmoud, Least sum of sqared errors (LSSE) channel estimation, IEE Proc-F, vol3, pp 3-3, Aug 99 4 COCLUSIOS We have compared the performance of several Wiener filters and a ML estimation scheme, with respect to the of the estimated CIR and the BER in an examplary system We have seen that Wiener filters can improve the channel estimates significantly in our scenario For the system parameters we chose, the optimum Wiener filter almost reached the BER of a system with perfect channel knowledge We also introduced schemes that can be used to reduce the computational complexity of Wiener filters In the first, we separate the 2D-Wiener filter in two D filters In the second, we use a reduced number of tap correlations Thus the performance (and computational complexity) can be adjusted from worst case (uncorrelated taps, minimum complexity) to optimum solution (full correlation considered, maximum complexity)