2d Wiener Channel Estimation Performance Analysis With Diamond-shaped

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2D Wiener channel estimation performance analysis with diamond-shaped pilot-symbol pattern in MC-CDMA systems 1 Carlos Ribeiro1,2, Atílio Gameiro1 Instituto de Telecomunicações / Universidade de Aveiro, Campo Universitário, 3810-193 Aveiro, Portugal 2 Instituto Politécnico de Leiria, Morro do Lena, Alto Vieiro, 2411-901 Leiria, Portugal Phone: +351-244820300, e-mail: [email protected] Abstract1— The influence of the pilot density and distribution on the channel estimation mean squared error of MC-CDMA systems is addressed. The adopted frame structure uses the optimum 2-dimensional diamond-shaped pilot grid and the receiver performs the estimation using a 2-dimensional Wiener interpolator filter, designed for the “Worst Case” scenario. Simulation results put in evidence that the system’s performance is strongly dependent on the pilot density and distribution and that real-time knowledge of the received signal SNR may be used to improve the performance. I. INTRODUCTION Multicarrier code division multiple access systems (MCCDMA) performance depends heavily on the ability of the receiver’s channel estimator to extract accurate channel state information. Blind estimation techniques that need to gather a large amount of information to perform the estimation exhibit a poor performance in mobile systems where the channel varies rapidly under the influence of Doppler’s effect and multipath propagation. To achieve better performance pilotaided channel estimation techniques are commonly preferred. Among these, the 2 dimension (2D) Wiener interpolator filters [1] stand out as the one that guarantees the channel estimation minimum mean squared error (MSE). The pilot-aided channel estimation is the centre of Section II, where the optimum 2D diamond-shaped pilot pattern is introduced. In Section III, the 2D Wiener interpolator is explained and the filters used in the simulations are presented. The performance analysis of channel estimation scheme is the aim of Section IV, firstly introducing the main characteristics of the simulation tool and then presenting the attained results. Finally, in Section V conclusions are drawn on this channel estimation scheme and some suggestions are given on how to improve its performance. II. THE PILOT-AIDED CHANNEL ESTIMATION Channel state information is crucial for the MC-CDMA [2] receiver’s equalizer to implement its algorithm and its performance is closely dependent on the accuracy of that information. This information is acquired by the channel estimator in the receiver. Pilot-aided techniques perform the channel estimation with the help of deterministic data Parrt of this work was developed in the WISQUAS project, supported by the European Union in the framework of CELTIC with contract no. CP2-035. symbols that are inserted in the transmitted symbols. These deterministic data symbols are known as pilots. The MC-CDMA systems use a frame structure where the pilots are multiplexed in time and frequency, Figure 1, by the transmitter. The frame is made-up of Ns symbols, and each symbol contains Nc carriers. Pilots replace data symbols in some of the carriers as depicted in Figure 1. Pilots are commonly spread in a regular structure (rectangular in Figure 1), Nf carriers apart in frequency and Nt symbols in time. Other structures that can be found in the literature [2], [3] are diagonal, random and diamond shaped. The pilots are used by the channel estimator to get the Least Squares estimates (LS), H LS , of the channel in the pilot positions ( n ', i ') Î P , H LS n ' i ' = Rn ' i ' + N n ' i ' N = H n ' i ' + n ' i ' , "(n ' i ') Î P , (1) Sn 'i ' Sn 'i ' where H ni is the complex value of the channel in position ( n, i ) of the frame, Rni , S ni and N ni are, respectively, the received symbol, the transmitted symbol and the noise component in the carrier ( n, i ) . The initial LS estimates can be interpreted as noisy samples of the channel and to be able to recover the channel state information, the pilot distances Nf and Nt should fulfill the 2D Nyquist Theorem, with cut-off frequencies given by the minimum coherence time and minimum coherence bandwidth [2], that are, respectively, functions of the mobile maximum speed and channel multipath maximum delay, Nf < Delaymax Df , (2) Nt < f Dopplermax TS where Delaymax is the maximum delay of the propagation channel, Df is the carrier separation, f Dopplermax is the maximum Doppler frequency and TS is the MC-CDMA symbol duration. The use of 2D pilot grids has been shown to outperform conventional 1D pilot patterns [1]. Among all regular patterns, diamond-shaped pilot pattern was proven to deliver Pilot Data Fig.1. MC-CDMA frame structure. Pilot Data V2 V1 å Fig. 2. 2D diamond-shaped pilot pattern. the best performance [3], [4]. Figure 2 shows an example of a MC-CDMA frame with this pilot pattern. The diamond-shaped pattern, like any regular pilot pattern, can be represented using 2 basis vectors V1 and V2 [5], V 1 = [i1 n1] T V 2 = [i 2 n 2] , T (3) The optimum pilot pattern for a given pilot density is achieved [3], [4] when n1 = 0 and i 2 = i1 2 . Such a pattern is exemplified in Figure 2, where i1 = Nt and n 2 = Nf , resulting in, é Nt ù é Nt ù (4) V1 = ê ú V 2 = ê 2ú . êë Nf úû ë0û Defining the matrix V, é Nt Nt ù 2ú , (5) V = [V 1: V 2 ] = ê ëê 0 Nf ûú the pilot density D is inversely proportional to the pilot spacing and is defined [5] for any regular 2D pattern as, D = det (V ) The estimation error is inversely proportional to the size of the set of LS estimates used [7]. The LS estimates used to estimate each position should be the ones closer to estimated position, according to the rules defined in [7]. The calculation of the optimum filter coefficients for each point to estimate is based on the Wiener-Hopf equations [2], θn - n '',i -i '' = wn ',i ',n,ifn '-n '',i '-i '' , " {n '', i ''} Î t ni , (8) {n ',i '}Ît ni where the filter’s input-output cross-correlation θn - n '',i - i '' is -1 -1 = ( NtNf ) . (6) III. THE “WORST CASE” 2D WIENER INTERPOLATOR This estimation filter assures the minimization of the channel estimation MSE, when the channel samples are corrupted by noise [2], [3], [6]. The use of this type of static filter assumes that the propagation channel is wide-sense stationary for the duration of the MC-CDMA frame. The optimum filter coefficients vary from estimated point to estimated point (time and frequency), by what the filter is usually described as shiftvariant. The filter input is made up of the LS estimates of the channel in the pilot positions. The filter output Hˆ ni is channel’s discrete transfer function estimation and it’s expressed as, Hˆ ni = wWienn ',i ',n ,i H LS n ' i ' , "t ni Î R , (7) n ', i ' Î t { } ni where wWienn ',i ',n ,i represents the 2D shift-variant Wiener filter, å com NTap coefficients per estimated position and t ni is the set of LS estimates used to get the estimate of the ( n, i ) position of the frame. t ni is a sub-set of set of pilots present in each frame P . The size of t ni is necessarily coincident with the length of the filter, t ni = NTap . defined by, { } * θn - n '',i -i '' = E H ni H LS = Rf (n - n '') Rt (i - i '') , n '' i '' (9) considering that the noise and the transmitted symbols are statistically independent. Rf and Rt are, respectively, the frequency and time channel correlations. The filter’s input correlation fn '- n '',i '- i '' is defined as, { * fn '- n '',i '-i '' = E H LSn ' i ' H LS n '' i '' = q n '- n '',i '-i '' + } 1 d ( n '- n '', i '- i '' ) gC = Rf (n '- n '') Rt (i '- i '') + , (10) 1 d ( n '- n '', i '- i '' ) gP where g P is the average signal-to-noise ratio per path. Considering that each element of the MC-CDMA frame is estimated using every pilot in it, given a frame with Ns symbols and Nc carriers per symbol, using a 2D shift-variant Wiener filter with NTap coefficients, the total number of coefficients needed for the frame will be, NTotal = NTap .Ns.Nc . (11) In the analysis performed in this document, channel estimation is performed using 2 orthogonal 1D Wiener interpolator filters. First filter is applied in the frequency domain and the second in time domain, using the estimates obtained from the first. These filters perform similarly to the 2D Wiener interpolator filter [1], with a considerably lower complexity. The filters are designed assuming uniform delay and Doppler power spectrums (worst case scenario: maximum expected channel delay and mobile terminal (MT) speed) and the calculation of its coefficients is based on (8), using all the pilots present in the frame. The average signalto-noise ratio per path is known and used in the filter project. IV. PERFORMANCE ANALYSIS The performance analysis of the effect of the pilot density and distribution in the mean squared error of the channel estimation was carried out using a MC-CDMA system simulation tool with single antenna receiver and transmitter. The main characteristics of the simulation tool are presented in Table I. The adopted MC-CDMA frame always uses the optimum 2D diamond-shaped pilot pattern, with 128 symbols per frame. The pilot distances adopted for simulations where Nt Î {2, 4, 6,K, 20} and Nf Î {1, 2,K, 20} . Fig. 3. Examples of frequency responses of simulated BRAN E channel. The simulated wireless channel followed the time related parameters from ETSI BRAN-E channel model [8], with an impulse response given by, h ( i ,t ) = L -1 åa d (t - t ) , i l l (12) l =0 where L is the number of paths, ai l is the complex value of path l in instant i and t l is the delay of path l. The paths are assumed to be statistically independent, with normalized L -1 average power, ås l 2 = 1 , where s l 2 is the average power l =0 of path i. Table I - Simulation tool main characteristics. Carrier frequency Bandwidth OFDM symbol duration Carrier spacing Cyclic prefix Number of carriers Spreading factor 5GHz 39,424MHz » 26 m s 38,5kHz 20% OFDM symbol duration 1024 16 Figure 3 shows examples of the simulated channel, respectively (from left to right), for MT speed of 30km/h, 100km/h and 300km/h. Figure 4 shows the channel estimation MSE performance surfaces as function of the pilot distances Nt and Nf,. Pilot density ranges 50% (Nf=1, Nt=2) to 0,25% (Nf=20, Nt=20). Columns (from left to right) represent the MT speeds of 30km/h, 100km/h and 300km/h, respectively. Rows (from top to bottom) represent the power efficiency, Eb N 0 , of 10dB, 20dB and 30dB, respectively. Each surface has a larger dot that points out the optimum value in each surface and a set of smaller dots that show the pairs (Nt,Nf) over which the achieved channel estimation MSE is not higher than 3dB from the optimum. Table II summarizes the achieved channel estimation minimum MSE in the scenarios depicted in Figure 4. Table III summarizes the optimum pilot densities D and associated optimum pilot distances pairs (Nt,Nf) in the scenarios depicted in Figure 4. Observing the surfaces in Figure 4 it is obvious that the achieved channel estimation MSE is strongly dependent on the pilot distribution and density. Looking at each row, it’s clear that the performance of the channel estimator remains fairly constant with the MT speed. Both the optimum value of density and the set of points that will result in a near optimum performance are nearly the same for the 3 simulated speeds. In addition, the optimum density is always surrounded by a set of near optimum points. The achieved MSE varies little with the pilot distance in time Nt, and a value in the range [2,…,8] will most likely result in a near optimum performance of the channel estimator. The main issue is that the MSE surface does not evolve smoothly with Nf and looking at surfaces in columns is easy to see that there‘s no value of Nf that will achieve a near optimum performance for the various simulated values of power efficiency. A frame structure with a static pilot grid will not achieve a near optimum channel estimation. If better performance is mandatory then real-time knowledge of the received signal SNR is needed and pilot distance Nf must dynamically adapt. The pilot distance Nf should vary inversely with the estimated received signal SNR, but always fulfilling the Nyquist Theorem. Simulations carried out with BRAN-C and BRAN-A channel models showed a similar behavior pointing out that, for this channel estimation scheme, the rules adopted for the choosing of Nt and Nf will also be valid. Table II - Achieved channel estimation minimum MSE. MT Speed Eb/N0 10dB 20dB 30dB 30km/h -6,4 -15,4 -24,0 100km/h -6,4 -15,7 -24,1 300km/h -6,3 -15,7 -24,0 Table III - Optimum pilot density D and pilot distances in time and frequency (Nt,Nf). MT Speed Eb/N0 10dB 20dB 30dB 30km/h 7,1% (2,7) 10% (2,5) 16,7% (6,1) 100km/h 300km/h 7,1% (2,7) 10% (2,5) 16,7% (6,1) 7,1% (2,7) 10% (2,5) 16,7% (6,1) Fig. 4. Channel estimation performance surfaces. Some care should be taken when finding the pilot distances Nt and Nf using the method proposed in [1] and [2], as it not always provides the best results. V. CONCLUSIONS The simulation results presented in this paper showed that the performance of the 2D Wiener interpolator used for channel estimation is strongly dependent on the pilot symbols density and distribution. Some rules are given that most likely will result in a near optimum channel estimation for a variety of MT speeds and wireless channels. This work puts in evidence the need for a dynamic frame structure and the realtime estimate of system parameters like received signal SNR. REFERENCES [1] P. Hoeher, S. Kaiser, P. Robertson, ”Two-dimensional pilot-symbolaided channel estimation by Wiener filtering,” in Proc. ICASSP97, Munich, Germany, pp. 1845-1848, April 1997. [2] S. Kaiser, P. Hoeher, “Performance of multi-carrier CDMA systems with channel estimation in two dimensions,” in Proc. IEEE PIMRC’97, Helsinki, Finland, pp. 115-119, September 1997. [3] F. Bader, R. Gonzalez, “Pilot time-frequency location adjustment in OFDM systems based on the channel variability parameters,” in IEEE International Workshop on Multi-Carrier Spread Spectrum (MCSS'2005), Munich, Germany, September 2005. [4] J.-W. Choi, Y.-H. Lee, “Optimum pilot pattern for channel estimation in OFDM systems,” IEEE Transactions on Wireless Communications, vol. 3, no. 5, pp. 2083-2088, September 2005. [5] D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, New Jersey, 1984. [6] M. Ekstrom, “Realizable Wiener filtering in two dimensions,” IEEE Transactions on Acoustics, Speech and Signal Processing, pp. 31-40, Vol. 30, Feb. 1982. [7] S. Kaiser, “Multi-carrier CDMA mobile radio systems – analysis and optimization of detection, decoding, and channel estimation”, PhD. Dissertation, Munich, Germany, Jan. 1998. [8] ETSI Project Broadband Radio Access Networks (BRAN); HIPERLAN Type 2, Technical specification; Physical layer, Oct. 1999.