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Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Time-Frequency Based Channel Estimation for High-Mobility OFDM Systems–Part I: MIMO Case

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Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2010, Article ID 549197, 8 pages doi:10.1155/2010/549197 Research Article Time-Frequency Based Channel Estimation for High-Mobility OFDM Systems–Part I: MIMO Case ¨ Erol Onen,1 Aydın Akan (EURASIP Member),1 and Luis F. Chaparro (EURASIP Member)2 1 Department of Electrical and Electronics Engineering, Istanbul University, Avcilar, 34320 Istanbul, Turkey 2 Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA Correspondence should be addressed to Aydın Akan, akan@istanbul.edu.tr Received 1 February 2010; Accepted 14 May 2010 Academic Editor: Lutﬁye Durak ¨ Copyright © 2010 Erol Onen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Multiple-input multiple-output (MIMO) systems hold the potential to drastically improve the spectral eﬃciency and link reliability in future wireless communications systems. A particularly promising candidate for next-generation ﬁxed and mobile wireless systems is the combination of MIMO technology with Orthogonal Frequency Division Multiplexing (OFDM). OFDM has become the standard method because of its advantages over single carrier modulation schemes on multipath, frequency selective fading channels. Doppler frequency shifts are expected in fast-moving environments, causing the channel to vary in time, that degrades the performance of OFDM systems. In this paper, we present a time-varying channel modeling and estimation method based on the Discrete Evolutionary Transform to obtain a complete characterization of MIMO-OFDM channels. Performance of the proposed method is evaluated and compared on diﬀerent levels of channel noise and Doppler frequency shifts. 1. Introduction active area of research. The use of multiple antennas at both ends of a wireless link (multiple-input multiple-output (MIMO) technology) has been demonstrated to have the The major challenges in future wireless communications systems are increased spectral eﬃciency and improved link potential of achieving extraordinary data rates [5]. The corresponding technology is known as spatial multiplexing reliability. The wireless channel constitutes a hostile propaga- tion medium, which suﬀers from fading (caused by destruc- [6] or BLAST [7] and yields an impressive increase in spectral eﬃciency. Most of the previous work in the area tive addition of multipath components) and interference of MIMO wireless has been restricted to narrow-band sys- from other users. Diversity provides the receiver with several tems. Besides spatial diversity broadband MIMO channels, (ideally independent) replicas of the transmitted signal and however, oﬀer to higher capacity and frequency diversity is therefore a powerful means to combat fading and inter- due to delay spread. Orthogonal frequency division multi- ference and thereby improve link reliability. Common forms plexing (OFDM) signiﬁcantly reduces receiver complexity in of diversity are space-time diversity [1] and space-frequency wireless broadband systems. The use of MIMO technology diversity [2]. In recent years the use of spatial (or antenna) in combination with OFDM, that is, MIMO-OFDM [6], diversity has become very popular, which is mostly due to the fact that it can be provided without loss in spectral eﬃciency. therefore seems to be an attractive solution for future broadband wireless systems [8, 9]. However, intercarrier Receive diversity, that is, the use of multiple antennas on interference (ICI) due to Doppler shifts, phase oﬀset, local the receiver side of a wireless link, is a well-studied subject oscillator frequency shifts, and multi-path fading severely [3]. Driven by mobile wireless applications, where it is diﬃcult to deploy multiple antennas in the handset, the degrades the performance of OFDM systems [10]. Most of the channel estimation methods assume a linear time- use of multiple antennas on the transmitter side combined invariant model for the channel, which is not valid for the with signal processing and coding has become known under next-generation, fast-moving environments [11]. Recently the name of space-time coding [4] and is currently an
2 EURASIP Journal on Advances in Signal Processing frequency ωc , is caused by an object with radial velocity υ and a time-frequency varying MIMO-OFDM channel estimation approach is presented where discrete prolate spheroidal can be approximated by [15] sequences are used to obtain a robust time-varying channel υ ψ ∼ ωc , estimator that does not require any channel statistics [12]. = (3) c A time-varying model of the channel can be obtained by employing time-frequency representation methods. Here we where c is the speed of light in the transmission medium. In present a time-varying MIMO-OFDM channel estimation the new generation wireless mobile communication systems, based on the discrete evolutionary representation of the with fast moving objects and high carrier frequencies, channel output. The Discrete Evolutionary Transform (DET) Doppler frequency-shifts become signiﬁcant and have to be [13] provides a time-frequency representation of the received taken into consideration. The channel parameters cannot signal by means of which the spreading function of the multi- be easily estimated from the impulse response; however the path, fading, and frequency selective channel can be modeled estimation problem can be solved in the time-frequency and estimated. domain by means of the so called spreading function. The rest of the paper is organized as follows. In The spreading function is related to the generalized Section 2, we give a brief summary of the wireless parametric transfer function and the bifrequency function. The gener- channel model used in our approach and the MIMO- alized transfer function of the linear, time-varying channel is OFDM communication system. Section 3 presents time- obtained by taking the DFT of h(m, ) with respect to , that varying modeling and estimation of MIMO-OFDM channels is, via DET. A time-frequency receiver is also given in Section 3 L p −1 for the detection of data symbols using estimated channel α p e jψ p m e− jωk N p , H (m, ωk ) = parameters. In Section 4, we give some simulation results (4) to illustrate the performance of our algorithm for diﬀerent p=0 levels of channel noise and Doppler frequency-shifts and where ωk = (2π/K )k, k = 0, 1, . . . , K − 1. Now, the channel compare with other existing methods. Conclusions are bi-frequency function is found by computing the DFT of drawn in Section 5. H (m, ωk ) with respect to time variable, m: 2. MIMO-OFDM System Model L p −1 α p e− jωk N p δ Ωs − ψ p , B(Ωs , ωk ) = (5) In this section we give a brief introduction to the time- p=0 varying, parametric communication channel model used in where Ωs = (2π/N )s, s = 0, 1, . . . , N − 1. Calculating the our work and the MIMO-OFDM signal model. inverse DFT of B(Ωs , ωk ) with respect to ωk (or by taking the DFT of h(m, ) in (2) with respect to m), we have the 2.1. Parametric Channel Model. In wireless communications, spreading function of the channel as the multi-path, fading channel with Doppler frequency- shifts may be modeled as a linear time-varying system with L p −1 the following impulse response [14–16]: S(Ωs , k) = α p δ Ωs − ψ p δ k − N p (6) p=0 L p −1 provided that the Doppler frequency shifts are integer multi- h(t , τ ) = γ p (t )δ τ − τ p , (1) ples of the frequency sampling interval Ωs . S(Ωs , k) displays p=0 peaks located at the time-frequency positions determined by the delays and the corresponding Doppler frequencies, with where γ p (t ) are independent Gaussian processes with zero α p as their amplitudes [17]. In our approach, we extract mean and σ p variance, {τ p } are delay proﬁles describing the 2 this information from the spreading function of the received channel dispersion with τmax as the maximum delay, and L p signal and then detect the transmitted data symbol. 2 is the total number of paths. The variance σ p is a measure of the average signal power received at path p, characterized by the relative attenuation of that path, α p . Assuming that the 2.2. MIMO-OFDM Signal Model. In an OFDM communi- sampling frequency is high enough, so the delays are integer cation system, the available bandwidth Bd is divided into multiples of the sampling interval, a discrete-time channel K subchannels. The input data is also divided into K -bit model can be obtained as [15, 17] parallel bit streams and then mapped onto some transmit symbols Xn,k drawn from an arbitrary constellation points where n is the time index, and k = 0, 1, . . . , K − 1, denotes L p −1 α p e jψ p m δ h(m, ) = − Np , (2) the frequency or subcarrier index. p=0 Some pilot symbols are inserted at some preassigned positions (n , k ), known to the receiver: (n , k ) ∈ P = {(n , k ) | n ∈ Z , k = iS + (n mod (S)), i ∈ [0, P − 1]}, where m is the time index, ψ p represents the Doppler where P is the number of pilots, and the integer S = K/P frequency-shift, α p is the relative attenuation, and N p is the delay in path p. The Doppler frequency shift ψ , on the carrier is the distance between adjacent pilots in an OFDM symbol
EURASIP Journal on Advances in Signal Processing 3 (i , j ) paths are negligible, ψ p = 0, for all i, j , then the channel [10]. The nth OFDM symbol sn (m) is obtained by taking K - point inverse DFT and then adding a cyclic preﬁx (CP) of is almost time-invariant within one OFDM symbol. In that length LCP where LCP is chosen such that L ≤ LCP + 1, and L case, above equation becomes is the time-support of the channel impulse response. This is done to mitigate the eﬀects of intersymbol interference (ISI) (i, j ) L p −1 Ntx (i, j ) ( j) (i , j ) ( j) Xni,) ( α p e− jωk N p + Nn,k = Rn,k caused by the channel dispersion in time. k p=0 i=1 A MIMO-OFDM system with Ntx transmit and Nrx (10) receive antennas is depicted in Figure 1. The incoming bits Ntx are modulated to form Xni,) , where i is the transmit antenna ( (i , j ) ( j) Hn,k Xni,) + Nn,k , ( = k k index. After parallel-to-serial (P/S) conversion, the signal i=1 transmitted by the ith antenna becomes (i , j ) where Hn,k is the frequency response of the channel between K −1 the ith transmitter and j th receiver antenna, and if there are 1 X (i) e jωk m , s(i) (m) = √ (7) large Doppler frequency shifts in the channel, then the time- n K k=0 n,k invariance assumption above is no longer valid. Here we consider modeling and estimation of the channels frequency where m = −LCP , −LCP + 1, . . . , 0, . . . , K − 1, ωk = (2π/K )k, (i , j ) and each OFDM symbol is N = K + LCP sample long. The responses Hn,k and approach the problem from a time- channel output suﬀers from multi-path propagation, fading frequency point of view [17]. and Doppler frequency shifts: 3. Time-Varying Channel Estimation ⎛ ⎞ (i, j ) L p −1 Ntx for MIMO-OFDM Systems ⎜ ⎟ ( j) h(i, j ) (m, ) s(i) (m − )⎠ yn (m) = ⎝ n i=1 =0 In this section we present a time-frequency procedure to ⎛ ⎞ characterize time-varying MIMO-OFDM channels. We also (i, j ) L p −1 Ntx propose a time-frequency receiver that uses the estimated ⎜ ⎟ (i, j ) (i , j ) (i , j ) e jψ p m (i ) = m − Np ⎝ ⎠ αp sn channel fading, delay, and Doppler parameters to recover p=0 i=1 the transmitted symbols. In the following, we brieﬂy present ⎛ ⎞ the Discrete Evolutionary Transform (DET) as a tool for (i, j ) L p −1 K −1 Ntx the time-frequency representation of time-varying MIMO- ⎜1 ⎟ (i, j ) (i, j ) (i , j ) Xni,) ( jωk (m−N p ) ⎝√ jψ p m = ⎠ αp e e OFDM channels. k K k=0 p=0 i=1 ⎛ ⎞ 3.1. The Discrete Evolutionary Transform. Wold-Cramer rep- K −1 Ntx 1 ⎝√ Xni,) ⎠, (i , j ) ( jωk m = resentation [18] of a nonstationary random signal γ(n) can Hn (m, ωk ) e k K k=0 be expressed as an inﬁnite sum of sinusoids with random and i=1 (8) time-varying amplitudes and phases, or π where (·)( j ) denotes the j th receiver. (·)(i, j ) is indexing for the Γ(n, ω)e jωn dZ (ω), γ(n) = (11) −π wireless time-varying channel between the ith transmitter and the j th receiver antennas. The transmit signal is also where Z (ω) is considered a random process with orthogonal corrupted by additive white Gaussian noise η( j ) (m) over the increments. This is a generalization of the spectral rep- channel. The received signal for the nth frame can then be resentation of stationary processes. Priestley’s evolutionary ( j) ( j) ( j) written as rn (m) = yn (m) + ηn (m). The receiver discards spectrum [18, 19] of γ(n) is given as the magnitude square of the evolutionary kernel Γ(n, ω). Analogous to the Cyclic Preﬁx and demodulates the signal using a K -point DFT as the above Wold-Cramer representation, a discrete, time- frequency representation for a deterministic signal x(n) with K −1 1 a time-dependent spectrum is possible [13, 20]: ( j) ( j) ( j) yn (m) + ηn (m) e− jωk m Rn,k = √ K m=0 K −1 ⎛ X (n, ωk )e jωk n , x(n) = 0 ≤ n ≤ N − 1, (12) (i, j ) L p −1 K −1 Ntx ⎜1 k=0 (i, j ) (i , j ) jωs (m−N p ) ⎝√ Xni,) ( = αp e (9) s K where ωk = 2πk/K , K is the number of frequency samples, s=0 p=0 i=1 ⎞ and X (n, ωk ) is a time-frequency evolutionary kernel. A simi- K −1 lar representation can be given in terms of the corresponding (i, j ) e j (ωs −ωk )m + Nn,k ⎠, ( j) e jψ p m × bi-frequency kernel X (Ωs , ωk ): m=0 K −1 K −1 X (Ωs , ωk )e j (ωk +Ωs )n , ( j) x(n) = (13) where Nn,k is the Fourier transform of the channel noise at the j th receiver. If the Doppler eﬀects in all the channel k=0 s=0
4 EURASIP Journal on Advances in Signal Processing Cyclic Remove preﬁx cyclic preﬁx P S P Y1 S K -point / Ant #1 K -point X1 Ant #1 / / / S 1 IFFT P Time FFT S P varying wireless 1 channel Coding, Decoding, demodulation Output Input modulation Remove Cyclic bits bits preﬁx cyclic preﬁx P S Channel P S / K -point K -point / estimation / / S Ant #Ntx Ant #Nrx P 1 FFT IFFT X Ntx S Y Nrx P 1 1 Figure 1: MIMO-OFDM System Model. where ωk and Ωs are discrete frequencies. Discrete evolu- where we ignore the additive channel noise η( j ) (m) for tionary transformation (DET) is obtained by expressing the simplicity. The equation above can be rewritten in matrix kernels X (n, ωk ) or X (Ωs , ωk ) by means of the signal [13]. form as Thus, for the representation in (12) the DET that provides y(j) = H(j) x, (16) the evolutionary kernel X (n, ωk ), 0 ≤ k ≤ K − 1, is given by where N −1 T ( j) ( j) ( j) x( )wk (n, )e− jωk , y( j ) = yn (0), yn (1), . . . , yn (K − 1) X (n, ωk ) = (14) ; =0 T x = x(1) , x(2) , . . . , x(Ntx ) , where wk (n, ) is, in general, a time- and frequency- x(i) = Xni,0 , Xni,1 , . . . , Xni,) −1 ; () () ( dependent window. (17) K The DET can be seen as a generalization of the short- (i , j ) time Fourier transform, where the windows are constant. Hn (m, ωk )e jωk m √ H(i, j ) = am,k am,k = K ×K , ; The windows wk (n, ) can be obtained from either the K Gabor representation that uses nonorthogonal frames or the H( j ) = H(1, j ) , H(2, j ) , . . . , H(Ntx , j ) . Malvar wavelet representation that uses orthogonal systems. Details of how the windows can be obtained for the Gabor The input-output relation for the whole system results as and Malvar representations are given in [13]. However, for the representation of multipath wireless channel outputs, y = Hx, (18) we consider windows that are adapted to the Doppler frequencies of the channel. where ⎡ ⎤ ⎡ ⎤ H(1) y(1) ⎢ ⎥ ⎢ ⎥ 3.2. MIMO-OFDM Channel Estimation by Using DET. We H(2) y(2) ⎢ ⎥ ⎢ ⎥ H=⎢ ⎥, y=⎢ ⎥. will now consider the computation of the spreading function (19) . . ⎢ ⎥ ⎢ ⎥ . . ⎣ ⎦ ⎣ ⎦ . . by means of the evolutionary transformation of the received H(Nrx ) y(Nrx ) signal. The output of the channel, after discarding the cyclic preﬁx, for the nth OFDM symbol can be written using (8) as If H is known and then input symbols can be estimated by the following relation, ⎛ ⎞ (i, j ) L p −1 K −1 Ntx x = H−1 y. ⎜1 ⎟ (i, j ) (i, j ) (20) ( j) (i , j ) Xni,) ⎠ ( jωk (m−N p ) ⎝√ jψ p m yn (m) = αp e e k K p=0 k=0 i=1 Now calculating the discrete evolutionary representation of ( j) yn (m), we get (i, j ) L p −1 1 (i , j ) Hn (m, ωk )e jωk m Xni,) , ( =√ K −1 k K ( j) ( j) p=0 Yn (m, ωk )e jωk m . yn (m) = (21) (15) k=0
EURASIP Journal on Advances in Signal Processing 5 ( j) ( j) (i , j ) representation of Yn (m, ωk ) only when ψr = ψ p ; in Comparing the representations of yn (m) in (21) and (15), (i, j ) we get the kernel as fact, using the window wr (m, ) = e jψ p (m− ) , the above ( j) representation of Yn (m, ωk ) becomes, Ntx 1 ( j) (i , j ) (m, ωk )Xni,) . ( √ Hn Yn (m, ωk ) = (22) k K i=1 Ntx √ ( j) (i , j ) KHn (m, ωk )Xni,) , ( Yn (m, ωk ) = (27) above relation is also valid at the preassigned pilot positions k i=1 k=k : ( j) ( j) Yn (m, ωr ) = Yn (m, ωk ) which is the expected result multiplied by K . In our estimation procedure, we use windows wu (m, ) = e jωu (m− ) (23) Ntx 1 (i , j ) where ωu is chosen in a discrete set with certain increments, (m, ωk )Xni,) , ( √ Hn = k ωu = πu/U , u = 0, 1, 2, . . . , U − 1. When ωu coincides with K i=1 one of the Doppler frequencies in the channel, the spreading (i , j ) function displays a large peak at the time-frequency position where r = 1, 2, . . . , P and Hn (m, ωk ) is a decimated (i , j ) (i , j ) (i , j ) (N p , ψ p ), corresponding to delay and Doppler frequency version of the Hn (m, ωk ). Note that P is again the number of pilots, and S = K/P is the distance between adjacent pilots. of that transmission path, with magnitude proportional to (i , j ) Finally if the pilot symbols are chosen to be orthogonal to attenuation α p . When ωu does not coincide with any of each other, the decimated frequency response of the channel the Doppler frequencies, the spreading function displays a between the ith transmitter and the j th receiver antennas random sequence of peaks spread over all possible delays. may be obtained as Then it is possible to determine a threshold that permits us to obtain the most signiﬁcant peaks of the spreading function √ K corresponding to possible delays and Doppler frequencies. In (i , j ) ( j) i Yn (m, ωr ) , Xn(,r) . (m, ωr ) = Hn (24) i Xn(,r) our experiments we observed that peaks having amplitudes larger than 65% of the maximum peak are due to an actual transmission; otherwise they are considered as noise. Thus, (i , j ) Taking the inverse DFT of Hn (m, ωr ) with respect to ωr by searching in the possible Doppler frequency range, we are and DFT with respect to m, we obtain the downsampled able to estimate all the parameters of a multi-path, fading, spreading function S (Ωs , ), and time-varying MIMO-OFDM channel via the spreading ⎛ ⎞ function of the channel. (i, j ) L p −1 (i , j ) −Np 1 According to (27), we need the input pilot symbols Xn,k δn ⎝ ⎠. (i , j ) (i , j ) (i , j ) (i , j ) (i , j ) n ( Ωs , Ωs − ψ p )= S α p δn to estimate the channel frequency response. Here we consider S S p=0 simple, uniform pilot patterns; however improved patterns (25) may be employed as well [11]. (i , j ) (i , j ) By comparing Sn (Ωs , ) and Sn (Ωs , ), we observe that (i , j ) (i , j ) (i , j ) 3.3. Time-Frequency Receiver. After estimating the spread- the channel parameters α p , N p , and ψ p calculated from ing function and the corresponding frequency response (i , j ) Sn (Ωs , ) can also be estimated from the downsampled (i , j ) Hn (m, ωk ) of the channel, data symbols Xni,) can be ( (i , j ) spreading function Sn (Ωs , ). k detected using a time-frequency receiver given in (20). On In the following, we present a method to estimate the other hand, the channel output in (9) can be rewritten as the spreading function of the MIMO channel from the received signal. The time-frequency evolutionary kernel of ⎧ ⎫ ⎛ ⎞ the channel output in the j th receiver is obtained as K −1⎨K −1 ⎬ Ntx 1 =⎝ X (i) ⎠ + Nn,k ( j) (i , j ) ( j) j (ωs −ωk )m Rn,k Hn (m, ωk ) e ⎩ ⎭ n,s K K −1 s=0 m=0 i=1 ( j) ( j) )wk (m, )e− jωk = Yn (m, ωk ) yn ( K −1 =0 1 (i , j ) = Bn (ωk − ωs , ωs )Xn,s + Nn,k ⎛ K (i, j ) L p −1 K −1 s=0 Ntx ⎜1 (i, j ) (i , j ) α p e−ωs N p ⎝√ ⎛ ⎞ Xni,) ( = (26) s K −1 K Ntx ⎝1 s=0 p=0 i=1 Xni,) ⎠ + Nn,k , (i , j ) ( j) ( = − ωs , ωs ) Bn (ωk s ⎞ K s=0 i=1 N −1 (i, j ) ⎠. (28) +ωs −ωk ) wk (m, )e j (ψ p × =0 (i , j ) where Bn (Ωs , ωk ) is the bi-frequency function of the We consider windows of the form wr (m, ) = e jψr (m− ) , for 0 ≤ ψr ≤ π presented in [17] that depends on the channel during nth OFDM symbol, and the above equation Doppler frequency ψr . This window will give us the correct indicates a circular convolution with the data symbols. Based
6 EURASIP Journal on Advances in Signal Processing Spreading function |S(Ωs , ℓ )| on aforementioned equality it is possible to write the MIMO- OFDM system consisting of Ntx transmitter and Nrx receiver antennas in a matrix form as ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ Relative attenuation B(1,1) · · · B(Ntx ,1) r(1) x(1) z(1) 1 ⎢.⎥ ⎢ . ⎥⎢ . ⎥ ⎢ . ⎥ . ⎢ . ⎥=⎢ . ⎥⎢ . ⎥ + ⎢ . ⎥ . ⎣.⎦ ⎣ . ⎦⎣ . ⎦ ⎣ . ⎦ ··· . 0.5 (Nrx ,1) · · · B(Ntx ,Nrx ) (Nrx ) x(Ntx ) z(Nrx ) r B 0 (29) 1 0.75 0. 5 or 128 0.25 Fre 96 qu enc 1(0) .75 64 32 r = Bx + z, y ( 0 0.5 (30) 96 128(0) ple) ×π rad 0.25 (sam 64 Time 32 0 .) 0 (i , j ) where B(i, j ) = [bs,k ]K ×K = Bn (ωk − ωs , ωs ) is a K × K matrix; r, x, and z are K × 1 vectors deﬁned by Figure 2: An example of estimated spreading function for the 2 × 2 ( j) T ( j) T MIMO-OFDM system. ( j) ( j) ( j) ( j) = [Rn,1 , Rn,2 , . . . , Rn,K ] , x( j ) = [Xn,1 , Xn,2 , . . . , Xn,K ] , r( j ) ( j) T ( j) ( j) and z( j ) = [Nn,1 , Nn,2 , . . . , Nn,K ] , respectively. Finally, data ( j) 101 symbols Xn,k can be estimated by using a simple time- frequency receiver: x = B−1 r. The exhaustive search for the channel Doppler fre- 100 quencies may seem to increase the computational cost of the proposed method. However, considering the carrier 10−1 frequencies and maximum possible velocities in the envi- ronment, Doppler frequencies lie in a certain band which MSE 10−2 can be easily covered by the algorithm. Furthermore, our channel estimation approach does not require any a priori information on the statistics of the channel as in the case of 10−3 many other channel estimation methods [11]. In the following, we demonstrate time-varying MIMO- 10−4 OFDM channel estimation performance of our time- frequency-based approach by means of examples. 5 10 15 20 25 30 35 4. Simulations SNR (dB) fD = 1, 2D Slepian-based fD = 1, TF-based In the experiments, a 2-input, 2-output MIMO-OFDM fD = 0.2, 2D Slepian-based fD = 0.2, TF-based system is considered, and the wireless channels are simulated randomly; that is, the number of paths, 1 ≤ L p ≤ 5, the Figure 3: MSE for proposed channel estimation and the method delays, 0 ≤ N p ≤ LCP − 1, and the Doppler frequency presented in [12] for 2 × 2 MIMO-OFDM system for fD = 0.2 and shift 0 ≤ ψ p ≤ ψmax ( p = 0, 1, . . . , L p − 1) of each fD = 1. path are picked randomly. Input data is QPSK coded and modulated onto K = 128 subcarriers, 16 of which are assigned to pilot symbols. The Signal-to-Noise Ratio (SNR) of the channel noise is changed between 5 and 35 dB, for two We show the mean square error (MSE) of our chan- diﬀerent values of the maximum Doppler frequency. Figure 2 nel estimation approach and the 2D Slepian-based two- depicts an example of the estimated spreading function for dimensional channel estimation method [12] for diﬀerent a 2 × 2 MIMO system, during one OFDM symbol. In our channel noise levels and for normalized Doppler frequencies simulations, we use Doppler frequencies normalized by the fD = 0.2 and fD = 1 in Figure 3. As seen from the graphs, our subcarrier spacing [21]: method outperforms the Slepian-based approach in terms of both estimation error and robustness against increased υ fc fD = (31) Ts N. Doppler frequencies. Note that our method is capable c of estimating and compensating for large Doppler shifts yielding a similar MSE for both fD = 0.2 and fD = 1. The performance of our channel estimation method is investigated and compared with that of a recently proposed We then investigate the bit error rate (BER) versus time-frequency varying MIMO-OFDM channel estimation SNR of the channel noise performance of both channel estimation approaches for diﬀerent numbers of the pilot approach [12]. This method applies the discrete prolate symbols in one OFDM block, P = {8, 16, 32} and show the spheroidal sequences to obtain a robust time-varying chan- nel estimator that does not require any channel statistics, results in Figure 4. Notice that for a ﬁxed number of pilots, similar to our proposed method. proposed TF-based approach achieves the same BER with
EURASIP Journal on Advances in Signal Processing 7 101 101 100 100 10−1 10−1 BER 10−2 BER 10−2 10−3 10−3 10−4 10−4 10−5 4 6 8 10 12 14 16 18 20 22 24 10−6 Number of paths (SNR = 15 dB) TF-based, 8 pilot 2D Slepian-based, 8 pilot 0 5 10 15 20 25 30 35 TF-based, 16 pilot 2D Slepian-based, 16 pilot SNR (dB) Figure 5: BER versus the number of channel paths for the TF-based 2D Slepian-based, 8 pilot TF-based, 8 pilot channel estimation and the method presented in [12] for 8 and 16 TF-based, 16 pilot 2D Slepian-based, 16 pilot pilots and 15 dB SNR. 2D Slepian-based, 32 pilot TF-based, 32 pilot Figure 4: BER versus SNR performance of the TF-based and 2D Slepian-based method presented in [12] for 2 × 2 MIMO-OFDM Acknowledgments system for 8, 16, and 32 pilots. This work was partially supported by The Research Fund of The University of Istanbul, Project nos. 6904, 3898, and 4382. about 15 dB less SNR than the Slepian-based method. Note also that increasing the number of pilots improves the BER performance in both methods. References Finally the eﬀect of the number of channel propagation paths is investigated. BER is calculated for both TF and [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time Slepian-based channel estimation approaches by changing codes for high data rate wireless communication: performance the number of paths from 3 to 25 and for P = 8 and criterion and code construction,” IEEE Transactions on Infor- 16 pilots. Results given in Figure 5 show that for the same mation Theory, vol. 44, no. 2, pp. 744–765, 1998. number of paths, proposed TF-based approach achieves [2] H. A. Cirpan, E. Panayirci, and H. Doˇ an, “Nondata-aided g approximately 100 times less BER than the 2D Slepian-based channel estimation for OFDM systems with space-frequency channel estimation method. Also notice that the BER per- transmit diversity,” IEEE Transactions on Vehicular Technology, vol. 55, no. 2, pp. 449–457, 2006. formance goes down rapidly for the number of paths larger than 12. [3] A. Shah and A. M. Haimovich, “Performance analysis of opti- mum combining in wireless communications with Rayleigh fading and cochannel interference,” IEEE Transactions on 5. Conclusions Communications, vol. 46, no. 4, pp. 473–479, 1998. [4] S. M. Alamouti, “A simple transmit diversity technique for In this work, we present a time-varying estimation of wireless communications,” IEEE Journal on Selected Areas in MIMO-OFDM channels for high-mobility communication Communications, vol. 16, no. 8, pp. 1451–1458, 1998. systems by means of discrete evolutionary transform. The [5] A. J. Paulraj and T. Kailath, “Increasing capacity in wireless main advantage of the proposed method is that it does not broadcast systems using distributed transmission/directional assume any statistics on the communication channel. The reception,” US patent no. 5,345,599, 1994. parametric channel model used in this approach allows us [6] H. B¨ lcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of o to obtain a two-dimensional representation for the channel OFDM-based spatial multiplexing systems,” IEEE Transactions and estimate its parameters from the spreading function. We on Communications, vol. 50, no. 2, pp. 225–234, 2002. observe that the method is robust against large variations [7] G. J. Foschini and M. J. Gans, “On limits of wireless com- on the channel frequency response, that is, fast fading. munications in a fading environment when using multiple Simulations show that our time-frequency-based method antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. has considerably better channel estimation and bit error 311–335, 1998. performance compared to a similar time-frequency varying [8] A. Goldsmith, Wireless Communications, Cambridge Univer- channel estimation approach [12]. sity Press, New York, NY, USA, 2005.
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