Estimation and Equalization Based on a Parametric Channel Representation Presenter: Kostas Berberidis University of Patras Computer Engineering & Informatics Department Signal Processing & Communications Lab 2 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Presentation Outline Part 1: Parametric channel estimation • Formulation / A new cost function • Algorithmic technique (Decoupled Estimation of Channel Parameters – DECP) • Theoretical and experimental results Contributors: Athanasios A. Rontogiannis 1 Kostas Berberidis 2 Alexandra Marava 2 Jacques Palicot 3 1 Institute of Space Applications and Remote Sensing, N.O.A, Athens, Greece 2 Dept. of Computer Eng. & Informatics and RACTI, Univ. of Patras, Greece 3 SUPELEC / ETSN, Rennes, France
2: Efficient equalization of sparse channels • From the CIR to DFE filters • Adaptive equalization • Experimental results Contributors: Athanasios A. Rontogiannis and Kostas Berberidis 4 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Part 1: Problem Definition Ø Motivation: The Semi-blind concept Ø Desired properties § Saving in bandwidth § Reasonable computational cost Ø Assumptions § The channel has a specular form § The pulse shaping filters are known § A limited number of information symbols are known at the receiver (of the channel order)
G. Xu, H. Liu, L. Tong, T. Kailath “A least-squares approach to blind channel identification”, IEEE Trans. on Signal Proc., Dec. 1995. Ø S.V. Schell, J.L. Bapat, ‘Partially Blind Identification of FIR channels for QAM Signals ’, Proc. of MILCOM 95, pp.592-596, San Diego, 1995. Ø G. Li, Z. Ding, ‘Semi-blind Channel Identification for Individual Data Bursts in GSM Wireless Systems ’, Signal Processing, vol. 80, pp. 2017- 2031, 2000. Ø L. Perros-Meilhac, E. Moulines, K. Abed-Meraim, P. Chevalier, P. Duhamel, “Blind Identification of Multipath Channels: A Parametric Subspace Approach”, IEEE Trans. on Signal Proc., July 2001. 6 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Channel Estimation: Formulation (1/3) Ø We assume that the overall impulse response of the propagation channel is given as: p: number of multipath components a i : complex attenuation factor of the i-th component t i : delay of the i-th component g(t): pulse shaping function Ø The main objective of the channel estimation algorithm is the estimation of a i , t i , i=0,1,…p-1, using a combination of output information and a limited number of input symbols. 1 0 ( ) ( ) p i i i h t a g t τ − = = − ∑
(2/3) Ø Starting point of new method: The SRM concept, which exploits the multichannel structure. w 2 (n) w 1 (n) h 1 h 2 s(n) + + y 2 (n) y 1 (n) 8 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Channel Estimation: Formulation (3/3) Ø The subchannels’ output samples are with − g(t) is a symmetric, truncated raised cosine pulse shaping function which is non-zero at the interval [0, 2Lg T]. ( ) ( ) ( ), 1,2 T i i L i y n n w n i = + = h s ( ) , 1,2 i i G i = = h t a 0 1 0 1 0 1 ( 1) ( 1) ( ) . . . ( ) 2 2 ( 1) ( 1) ( ) . . . ( ) 2 2 ( ) . . . . . . ( 1) ( 1) ( ) . . . ( ) 2 2 p p i p i i g T g T i i g T T g T T G i i g LT T g LT T τ τ τ τ τ τ − − − − − − − − − + − + − = − − + − + − t O
Method (1/7) Ø Application of the SRM concept: Ø Inclusion of the known information symbols: where SML contains the known symbols and y1M , y2M the subchannels’ outputs. [ ] 1 2 1 2 Y Y − = h 0 h 2 1 1 1 2 2 ML M ML M Y Y S S − = 0 h 0 y h 0 y 10 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis The Channel Estimation Method (2/7) Ø In parametric form: where Ø In blind case: We apply the constraints: § t 0 is known, for example t 0 = 0. § a0 =1 or aTe1 =1 which gives: ( ) S Y G = t a z 1 2 ( ) ( ) ( ) G G G = t t t ( ) YG = t a 0 bl z Yg YG = = ) ( - a ˆ ) ˆ ( 0 τ τ
Method (3/7) Ø The unknown parameters are estimated from the LS problem: Ø The LS objective function is § linear with respect to a § non-linear with respect to t § separable with respect to a and t. Ø Golub-Pereyra approach to obtain the delay parameters: Ø The vector of attenuation parameters is the LS solution: , min ( ) , ( ) ( ) S Y G −Φ Φ = a t z t a t t ( ) 2 † argmin ( ) ( ) opt I = −Φ Φ t t t t z † ( ) opt opt = Φ a t z 12 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis The Channel Estimation Method (4/7) Ø The non-linear problem concerning t can be treated by § a multidimensional search ⇒ high complexity § a non-linear optimization search method ⇒ risk of local minimum (e.g. Newton type) Ø Proposed solution Investigation of the form of the objective function, in order to restrict the area of search for the global minimum. Ø It is proven that the non-linear optimization function is decoupled with respect to the delay parameters t i , i=0, 1, …, p-1 Ø The optimization search can be performed separately for each t i and independently of the other delay parameters.
Method (5/7) The form of the non-linear cost function – An example § Binary alphabet § 2-ray multipath channel § t = [1.2 7.4]T § a = [0.5 0.5]T § Two 1-dimensional searches are adequate for the location of the global minimum 14 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis The Channel Estimation Method (6/7) Ø The basic steps of the DECP algorithm 1. Set values for unknown L, p, respectively (overestimated). 2. Initialize ti , i=0, 1, …, p-1 with distinct random values in the interval [0, LT]. 3. Choose a linear step size d and set i=1. 4. Minimize with respect to ti . Find ti,opt by evaluating the function at ti =jd, j=0, 1, …, 5. Set ti =ti,opt , i=i+1 and repeat from step 3 until i = 6. Obtain the attenuation parameters from 7. Run a Gauss-Newton search in the neighborhood of topt to improve the estimation accuracy for t and a ( ) 2 † ( ) ( ) ( ) f I = −Φ Φ t t t z † ( ) opt opt = Φ a t z p L ˆ , ˆ p ˆ δ T L ^
Method (7/7) Alternative version of DECP algorithm (DECPa) By exploiting the form of cost function f(t ) an alternative version of DECP can be derived. Computational complexity of DECP: O(L2 + p3 ) Computational complexity of DECPa: O(L2 + p2 ) 16 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis DECP: Theoretical Issues (1/2 ) Ø The form of the cost function has been studied theoretically to justify the inherent decoupling property. Ø To this end we have stated (and proved) a proposition concerning the “diagonal dominance” of a matrix. AQ Q B H = A : nxn, Q : nxm Unitary Ø Matrix B has a larger degree of dominance w.r.t. matrix A provided that m2 ≤ n Ø If A is strongly diagonally dominant then B tends to a diagonal matrix | | | | , , 1 , i i n i j j j i A i a a r ∑ ≠ = =
(2/2 ) Ø Case of closely spaced time delays (the estimation capabilities of the algorithm have been quantitatively studied w.r.t. the involved parameters, namely, L, p, degree of dominance of the associated autocorrelation matrix, and autocorrelation properties of the pulse shaping function) Ø Overmodeling with respect to channel order L Ø Overestimation of the number of CIR dominant components p 18 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis DECP: Simulations (1/3) Performance of the new method Comparison with SRM § 16-QAM alphabet § 4-ray multipath channel § t = [0 6.11 18.17 23.34 ]T § a = [0.9+0.1j -0.31-0.26j -0.4-0.3j 0.3-0.15j ]T § Lg =3, L=29, rolloff=0.3, d=0.2 § 50 training symbols, 200 output symbols, 100 independent runs § New method significantly outperforms SemiblindSRM § Gauss-Newton method can be used for improvement 2 2 ( ) 1 1 (1/ ) ( ()) Q L k a c t i est k RMSE Q h h i = = − ∑∑
Performance of the new method and SRM for closely spaced delays § 16-QAM alphabet § 4-ray multipath channel § t = [0 6.11 22.77 23.34 ]T § a = [0.9+0.1j -0.31-0.26j -0.4-0.3j 0.3-0.15j ]T § Lg =3, L=29, rolloff=0.3, d=0.2 § 49 training symbols, 200 output symbols, 100 independent runs § Much better performance than Semiblind SRM, especially at high SNRs. § The performance of the new method is not affected by closely spaced delays. 20 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis DECP: Simulations (3/3) Performance of the new method and SRM for overestimated L § 16-QAM alphabet § 4-ray multipath channel § t = [0 6.11 18.17 23.34 ]T § a = [0.9+0.1j -0.31-0.26j -0.4- 0.3j 0.3-0.15j ]T § Lg =3, L=29, rolloff=0.3, d=0.2 § 56 training symbols, 200 output symbols, 100 independent runs § SNR=25dB § The new method is tolerant in channel order over- estimation, in contrast with SemiblindSRM. § The slight increase in the new algorithm's RMSE is due to constant training length
of sparse channels Ø Channels with sparse impulse response appear in several wireless communication applications § HDTV terrestrial transmission § High Speed Communications over DS loops § Underwater digital communications § Outdoor high data rate mobile communications Ø In most cases the involved sparse channels exhibit also large delay spreads. Ø Important issues: § Low complexity § Fast convergence ⇒ Reduction of training sequence § Good tracking capabilities 22 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Typical IR of sparse channels ∑ − = − = 1 0 ) ( ) ( p k k k t g a t h τ ak : complex attenuations tk : time delays
of CIR parameters Ø Aim: To estimate ak and tk ( h= G(t )a ) Ø This can be achieved by using DECP or DECPa Ø In the following we use a simplified version of DECP Ø Specifically, only the non-blind part of the combined cost function is employed (smaller processing delay / less complexity) 24 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis 2nd Step: DFE initialization (1/5) Ø Problem: Given an estimate of the channel IR, compute the optimum DFE coefficients Ø Optimum MMSE filter(Wiener Solution): Ruu c = rus c = [aT bT]T is the (? +? )×1 vector of equalizer coefficients Ruu = the autocorrelation matrix of the input vector u=[yT dT]T rus = crosscorrelation of input vector and the desired response Ø Leads to system of equations: Ryy a + Ryd b = rys (Ryd )Ha + Rdd b = 0 *M, N : lengths of FF and FB
initialization (2/5) Ø Taking into account that: y(n) = H⋅s(n) + w(n) Ø And substituting in the system of equations we get the solution: where eM+L1 = [0 … 0 1]T, S0 =sw 2/ss 2 * L1 , L2 : lengths of anticausal, causal parts of CIR 2 2 ( ) 1 H N L − − = × H a b 0 1 1 1 1 0 1 ( ) H M L S − + = + ⋅ ⋅ ⋅ a H H I H e 26 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis 2nd Step: DFE initialization (3/5) 1 1 1 1 ( 1) ( ) ( 1) ( 2) ( ) 1 (0) ( ) p p p M L L p p M L p p L − − − + − − − = h h h 0 h h H 0 0 h h K K M O O M K 2 2 2 2 2 ( ) ( ) ( 1) ( ) ( 1) ( ) 2 (1) ( 1) ( ) p p M L p p p p M M L L p p p L M L − − − + = h 0 h 0 0 h h h h 0 H h h h K K K K M O M O K K
initialization (4/5) Ø FF filter : It has been shown recently* that for sparse channels (and under reasonable assumptions) the FF filter can be approximated as: M H e = a H 12 where eM = [0 . . . 0 1]T and H12 is the right MxM block of matrix H1 . Due to the Toeplitz structure of H12 the above system can be solved efficiently with O(M2) complexity. *A.A. Rontogiannis, K. Berberidis, “Efficient DFE for Sparse Wireless Channels”, IEEE Trans. on Wireless Communications, vol.2, no.3, pp.570-581, May 2003. 28 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis 2nd Step: DFE initialization (5/5) Ø FB filter : § Direct solution by (Toeplitz matrix)×(vector) multiplication (Complexity O(L2 ·M)) § Efficient solution with the help of FFT (Complexity O(L2 ·logL2 )) 2 (1: ) 2 H L = − b H a
Adaptive DFE Ø Having obtained reliable estimates of the FF and FB filters we use them as initial values in the conventional LMS-based Decision Feedback Equalizer Ø The DFE runs directly in the decision directed mode. 30 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis The whole scheme Remarks: The 1st part can be blind, semi-blind or non-blind depending on the application conditions In a burst transmission mode (with short bursts and channel invariant within the burst) a fixed DFE may be adequate and therefore part 3 can be omitted.
method (non-blind version) vs LS channel estimation for a sparse channel § 16-QAM alphabet § 6-ray multipath channel § t = [-23.89T, –11.61T, 0, 32.71T, 52.95T, 53.71T ]T § a = [0.1+0.1j, 0.2, 1, 0.08-0.17j, 0.15-0.2j, -0.16-0.19j ]T § rolloff=0.125, d=0.2T, § 260 and 140, 260 known symbols, 100 independent runs § The new method outperforms LS estimation even with almost 50% of the training symbols . 32 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Simulations (2/3) New DFE vs constantly trained conventional adaptive DFE § 16-QAM alphabet § 6-ray multipath channel § t = [-23.89T, –11.61T, 0, 32.71T, 52.95T, 53.71T ]T § a = [0.1+0.1j, 0.2, 1, 0.08-0.17j, 0.15-0.2j, -0.16-0.19j ]T § rolloff=0.125, d=0.2T § 140 known symbols for the new DFE, 200 independent runs § SNR=25dB, (FF, FB) = (60,100) § The convergence time of DFE is improved 5 -10 times
Performance of the new method vs DFE with known channel (or LS estimated) § 16-QAM alphabet § 6-ray multipath channel § t = [-23.89T, –11.61T, 0, 32.71T, 52.95T, 53.71T ]T § a = [0.1+0.1j, 0.2, 1, 0.08-0.17j, 0.15-0.2j, -0.16-0.19j ]T § rolloff=0.125, d=0.2T § (FF, FB) = (60,100) § Training=140,260 § Frame length=2000 data symbols § The performance of the new DFE with short training is very close to the performance of the MMSE DFE with known channel 34 SUPELEC, Feb. 2004 Presenter: Kostas Berberidis Conclusions (1/2) • A new semi-blind technique was developed for the estimation of multipath channel parameters • The cost function under consideration is separable with respect to time delays and attenuation factors • The time delays’ non-linear cost function has a special structure which allows for a decoupled estimation of the unknown parameters • The proposed method: − is computationally efficient − achieves a lower estimation error compared to other related methods − is robust against the overmodelingproblem
on the new parametric channel estimation technique a new equalization scheme has been developed • The technique is appropriate for sparse channels • The technique achieves considerable BER improvement w.r.t. the conventional adaptive DFE • Alternatively the same BER performance is attained with much less training symbols (thus offering savings in bandwidth) • The technique has affordable computational cost, due to efficient schemes employed in both parts of the technique