On joint CFO and channel estimation in single-user and multi-user OFDM systems Yik-Chung Wu The University of ong Kong Email: ycwu@eee.hku.hk, Webpage: www.eee.hku.hk/~ycwu 1
OFDM basics exp(j2pt/t) exp(j2p(2t)/t) Digital implementation S IFFT P/S exp(j2p(3t)/t) exp(j2p(4t)/t) F t F -1 f Challenges: Time offset, frequency offset, unknown channel 2
Outline Conventional OFDM [1] ML estimator under timing uncertainty Two ML estimators Relationship and comparison OFDMA [2] ML estimator Optimization theorem Importance sampling MIMO-OFDM [3] CRB and ascrb Optimal training Performances of different kinds of training 3
Conventional OFDM Packet structure Preamble section Variable number of OFDM symbols AGC & rough timing syn CFO & Channel estimation... q o 0 Observation window / FFT window Energy spill from the previous symbol ISI free region 4
Two equivalent signal models If the FFT window starts in the ISI-free region: Γ( o ) 2 p / o o N ( ) T q o F D F L h x Γ( ) T( q ) F DF h v If we treat the delay as part of the channel: x Γ( ) F DF ξ v o o : Cyclic shift matrix : N N FFT matrix : L 1 channel vector where ξ [0 h 0 ] o jn o e : Diagonal matrix with (n=0,1,...n-1) on the diagonal : Normalized frequency offset : Diagonal matrix with training data on the diagonal : First L columns of F matrix L cp T T T T q 1 ( L q L) 1 o cp o L (1) (2) 5
Two ML estimators For the first system model, with A(q)=T(q)F DF L, { ˆ, ˆ q, hˆ } arg min [ x Γ( ) A( q ) h] [ x Γ( ) A( q ) h] q,, h Since h is linear in the system model, ML for h is ˆ 1 ( ) ( ( ) ( )) ( ) ( ) h FL D DFL Γ A q x A q Γ x Put 2 nd eq. into the 1 st eq. and dropping the irrelevant terms, we have q, Eq. (3) requires a two-dimensional search { ˆ, ˆ q} arg max A ( q ) Γ ( ) x (3) 2 6
Two ML estimators (cont.) With similar procedure applied to the second system model, we have ˆ ξ FL D FΓ ( ) x cp ˆ arg max FL D FΓ ( ) x cp Eq. (4) only requires a one-dimensional search Which estimator is better? Number of unknown parameters: L+2 vs L cp +1 The first estimator will perform better But with the price of higher complexity (due to 2-D search) 2 (4) 7
Physical interpretation The cost functions in the two estimators are equivalent to J 1 (, q ) F (:, q : q L1) D FΓ ( ) x J 2 ( ) F (:,0 : L 1) D FΓ ( ) x They represent the energies of different sections of the vector It can be shown that if o is a shifted time domain channel estimate cp h( ) F D FΓ ( ) x, h( ) 2 2 h( o ) 8
Physical interpretation (cont.) J 2 (, q ) h ( ), J ( ) h ( ) 1 qq : L1 2 0: 1 L cp 2 Observation window for JCCE L cp L Shifting trial window for JTCCE First estimator (JTCCE): Locating a window of length L and finding an s.t. the energy within the window is maximized Second estimator (JCCE): Find an s.t. the energy within the window from 0 to L cp is maximized JCCE can be interpreted as an approximation to JTCCE by using a large window during frequency estimation q o L Optimal window for JTCCE N 9
Performance analysis The MSE and CRB expressions for the two estimators can be derived in closed form (eqs. not presented here) Analytical performance comparison: r F (:, L: L 1) D FMF DFLh 1 1 2 D FMF DF h MSE2 cp MSE 1 F (:, Lcp : N 1) The second term is the ratio between two projections of a common vector onto different subspaces of F When N is large, dimension of F(:,L:L cp -1) << that of F(:,L cp :N-1) Two estimators are asymptotically equivalent L 2 10
Numerical results N=64, N cp =16, L=8 Training: Chu sequence Rayleigh fading channel Exponential delay profile Normalized CFO uniformly distributed in [-0.5,0.5] q o uniformly distributed in the ISI-free region 11
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16QAM 5 OFDM symbols after the preamble 13
Summary Based on two different signal models, two ML estimators for joint CFO and channel estimation with timing ambiguity have been derived The first estimator needs a 2-D search The second estimator only requires 1-D search The first estimator perform slightly better than the second one, with the price of higher complexity Asymptotically, the two estimators are equivalent 14
OFDMA uplink... User 1 User 2 User 3 Each user modulates an exclusive set of subcarrier Subcarrier can be allocated blockwise, interleaved, or arbitrarily Challenges: Different user have different timing delays, CFOs and channels f 15
Timing offset problem The user s timing is roughly synchronized using the downlink synchronization channel Timing offsets in the uplink are mainly due to the propagation delay from different users => Limited to a few samples Can be treated as part of the channel Called this quasi-synchronous system As long as max(l k +q k ) < L cp, we will have an observation window free of ISI Observation window CP CP User 1 q 1 +L 1 CP CP User 2 q 2 +L 2 CP CP User 3 q 3 +L 3 16
System model Signal of user k x Γ( ) F D F h D k h k Received signal Or in matrix form k k k L k : Diagonal matrix with training data for user k : User s k channel, including the time delay q k K x Γ( ) F D F h n k1 xq( ω) h n k k L k Q( ω) [ Γ( ) F D F Γ( ) F D F... Γ( ) F D F ] 1 1 L 2 2 L K K L h [ h h... h ] T T T T 1 2 K Goal: To estimate and h, based on a single OFDM training symbol 17
ML joint CFO and channel estimator Based on the standard procedure of deriving ML estimator, we have ˆ arg max{ ( )( ( ) 1 ( )) ( ) } ω ˆ ˆ ˆ 1 ˆ ω x Q ω Q ω Q ω Q ω x h ( Q ( ω) Q( ω)) Q ( ω) x Multi-dimensional search in Computational expensive Exhaustive search impossible for K 3 Possible method: alternative projection owever, no guarantee of global maximum solution We employ an optimization theorem to solve this problem 18
Simple solution in asymptotic case If the number of subcarrier N, it can be shown that ( Q ( ω) Q( ω)) where A k =F D k F L 1 1 ( A1 A1) 0 0 ( AKAK) Therefore, the optimal CFO estimator can be decoupled as 1 ˆ argmax{ x Γ( ) A ( A A ) A Γ ( ) x} k k k k k k k k For large enough but finite N, it can be viewed as an approximate solution For small N, it suffer great performance loss 1 19
Optimization theorem Optimization theorem by Pincus [R1]: The global optimal solution (if it is unique) for a multi-dimensional optimization problem maximizing is given by 1 L'( ω)... k exp( 1L'( ω)) dω ˆ k lim k 1,..., K 1... exp( L'( ω)) dω Good news: If we are smart enough to perform the integration, we can get the optimal solution analytically!! Bad news: The integration usually is too complex to be computed analytically Physical meaning of the theorem: Taking exponential make the largest peak in L () peaker and other smaller peaks lower When 1 is large enough, exp( 1 L ()) will have only a single peak The maximum point is the mean of the resultant function (5) [R1] M. Pincus, A closed form solution for certain programming problems, Oper. Res., pp. 690-694, 1962 20
Approximation by sample mean The optimization theorem can rewritten as (for large 1 ) ˆ... L k k ( ) d k 1,..., K where L( ω) ω ω exp( L'( ω)) 1... exp( L'( ω)) dω 1 is termed pseudo-pdf This is just the statistical mean of k w.r.t. PDF L( ω) If we can generate a large number of realization of k according to the PDF L( ω), the integration can be approximated by the sample mean T 1 i ˆ k k k 1,..., K T i1 Then the question becomes how to generate samples from L( ω) 21
Importance sampling In general, generating samples directly from L( ω) is difficult since it is a multi-dimensional PDF Generating realization from an arbitrary but fixed PDF is a well-studied problem in statistics ere, we use a technique called importance sampling It is based on the observation that L( ω) ˆ k... k L( ω) dω... k g( ω) dω g( ω) g( ω) where g( ω) is called normalized importance function... g( ω) dω If g( ω) is chosen s.t. realization of k can be easily generated T i 1 i L( ω ) ˆ k k i T g( ω ) i1 22
Further simplification In our problem, k is the CFO, so it is a circular R.V. The estimator can be further rewritten as T i T i 1 i L( ω ) 1 1 i L( ω ) ˆ k k exp( j2 k ) i p i T g( ω ) 2 p T g( ω ) where L()=exp( 1 x Q(Q Q) -1 Q x) is the non-normalized version of L( ω) and g() is the non-normalized version of g( ω) Advantages: i1 i1 Eliminates potential bias Computation of normalization constants for L() and g() can be avoided 23
Choosing g() By the strong laws of large number, the estimate ˆk will converge to optimal value, regardless of choice of g() Choice of g() only affects computational complexity ow fast the estimate converge to the true value General guidelines for choosing g() : Easy sample generation Close to L() in order to reduce variance of the estimate From the discussion in asymptotic case, propose to choose g() as (with 2 < 1 ) g K 1 ( ω) exp( 2 k1x Γ( k ) Ak ( Ak Ak ) Ak Γ ( k ) x) K 1 k1exp( 2x Γ( k ) Ak ( Ak Ak ) Ak Γ ( k ) x) g k ( ) k 24
Numerical results N=64, L cp =16, L=8, no time delay, random subcarrier allocation, k uniformly distributed [-N/2,N/2] 1 =2/K, 2 =1/K, T=2000 Training: constant modulus white seq in frequency domain hˆ h K 2 2 MSE ˆ CFO k1( k k ), MSEch 25
Asymptotic performance 26
Complexity comparison Complexity order has been derived for the proposed estimator, the decoupled estimator and the alternative projection method (APFE) 27
Summary Direct implementation of ML joint CFO and channel estimation is impractical due to the multi-dimensional search An optimization theorem, together with the importance sampling technique is used to solve this problem The proposed method can guarantee global optimality without the need of providing a good initial estimate 28
Multi-user MIMO-OFDM system Assume quasi-synchronous s.t. the small time delays can be lumped into the channels With spatial multiplexing (e.g., BLAST), each user is using all the subcarriers at the same time The receive antennas at BS share the same oscillator All users are driven by different oscillators Can be easily generalized to the case where user equips with more than one antenna 29
System model Received signal (one OFDM symbol) at the jth receive antenna K x 1 Γ( ) A h n Stack all the received vector from different receive antenna where j i i i ij j xq( ω) h n x [ x x... x ], n [ n n... n ] T T T T T T T T 1 2 M 1 2 M h [ h h... h ] with h [ h h... h ] T T T T T T T T 1 2 M j 1 j 2 j Kj Q( ω) I [ Γ( ) A... Γ( ) A ] with A F D F M 1 1 K K i i L r This linear model is in the same form as OFDMA case For estimation of and h, we can use a similar procedure as in OFDMA case What is the optimal training sequences? r r r 30
CRB It can be shown that the CRB for the joint CFO and channel estimation problem is 2 1 CRB( ω) ( R{ Z ΠQZ}) 2 2 1 1 1 CRB( h) 2( ) ( ) ( { Q }) ( ) 2 Q Q Q Q Q Z R Z Π Z Z Q Q Q where 1 ΠQ IM N Q( Q Q) Q r Z11 Z21 ZK1 Z12 Z22 ZK 2 Z, with Z diag(0,1,..., N 1) Γ( ) A h M Z1M Z r 1M Z r KMr ij k i ij 31
Special case: CFO-free If there is no CFO, the CRB reduces to It is shown [R2] that the condition of minimizing Tr{ CRB( h) } is It is further shown in [2] that two types of training satisfy the condition L1 1 () l FDM pilot: d ( n) b [ n ln / L k] n N CDM(F) pilot: CRB( h) I ( A A) CFO free k 2 1 CFO free M A A l0 k [R2]. Minn and N. Al-Dhahir, Optimal training signals for MIMO OFDM channel estimation, IEEE Trans. Commun., May 2006 32 I KL () l { b k } are constant-modulus symbols r 1 1 1 1 A [ A1 A2... A K ] L is any integer s.t. N / L is an integer while L L N / K d ( n) exp( j )exp( j )exp( j2 p( k 1) n/ N) k n k n and k are R.V.s in [0,2 p] w.r.t. n and k 0,1,..., 1 k 0,1,..., K 1
Asymptotic CRB If the CFOs are not zero, the conditions for the optimal training cannot be obtained in closed-form We turn to asymptotic CRB: 2 6 1 ascrb( ω) ( R{ R}) 3 N 2 1 3 1 ascrb( h) ( { }) N R R R 2 Much Simpler!!! where [... ] with diag([ h h... h ]) T T T T 1 2 M j 1 j 2 j Kj r R R R 11 12 1K R21 R22 R2K Ai A j M ij i j R I, with R ( ) r N R K1 R K 2 R KK 33
Minimizing the ascrb Using the fact that with equality holds iff It can be shown that Tr( ascrb( ω)) with equality hold iff 1 (( R{ R}) ) kk, 2 K 6 1 N N 2 1 Tr( ascrb( h)) Tr( R ) 3 R is diagonal M r k 1 i1 ki k, k ki R h R 3 2 1 ( R{ R}) It can be further shown that this is equivalent to I A A M KL r I h h h K Mr i1 ki ki M r k 1 i1 hki Rk, khki KL kk, 34
Optimal training The optimal training for CFO-free case (FDM and CDM(F) sequences) are also asymptotically optimal for joint CFO and channel estimation Question: for finite number of subcarriers, will they perform differently? Fact 1: CRB ascrb Fact 2: Training with correlation in time domain would have the CRB depart from the ascrb FDM sequence is repetitive in time domain correlated CDM(F) sequence has relatively a long correlation Prediction: CDM(F) performs better than FDM sequence 35
Numerical results N=64, L cp =16, L=8, M r =2, K=2 Exponential delay profile 36
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Summary Condition for optimal training has been derived by minimizing the ascrb Both CDM(F) and FDM sequences are asymptotically the best For finite number of subcarriers, CDM(F) seq perform better than FDM seq 38
References [1] Jianwu Chen, Yik-Chung Wu, Shaodan Ma and Tung-Sang Ng, ML Joint CFO and Channel Estimation in OFDM systems with Timing Ambiguity, IEEE Trans. on Wireless Communications, vol. 7, no. 7, pp. 2436-2440, Jul 08. [2] Jianwu Chen, Yik-Chung Wu, S. C. Chan and Tung-Sang Ng, Joint maximum-likelihood CFO and channel estimation for OFDMA uplink using importance sampling, IEEE Trans. on Vehicular Technology, vol. 57, no. 6, pp.3462-3470, Nov. 2008. [3] Jianwu Chen, Yik-Chung Wu, Shaodan Ma and Tung-Sang Ng, Joint CFO and channel estimation for multiuser MIMO-OFDM systems with optimal training sequences, IEEE Trans. on Signal Processing, vol. 56, no. 8, pp. 4008-4019, Aug 08. Further related readings: Kun Cai, Xiao Li, Jian Du, Yik-Chung Wu and Feifei Gao, CFO Estimation in OFDM Systems under Timing and Channel Length Uncertainties with Model Averaging, IEEE Trans. on Wireless Communications, Vol. 9, no. 3, pp. 970-974, Mar 2010. Xun Cai, Yik-Chung Wu, ai Lin and Katsumi Yamashita, Estimation and Compensation of CFO and I/Q Imbalance in OFDM Systems under Timing Ambiguity, IEEE Trans. on Vehicular Technology, Vol.60, no.3, pp.1200-1205, Mar 2011. Gongpu Wang, Feifei Gao, Yik-Chung Wu and Chintha Tellambura, Joint CFO and Channel Estimation for OFDM-Based Two-Way Relay Network, IEEE Trans. on Wireless Communications, Vol.10, no.2, pp.456-465, Feb 2011. 39