1. Semi-Blind (SB) Multiple-Input Multiple-Output (MIMO) Channel Estimation Aditya K. Jagannatham DSP MIMO Group, UCSD ArrayComm Presentation

2. Semi-Blind MIMO flat-fading Channel estimation. -Motivation -Scheme: Constrained Estimators. -Construction of Complex Constrained Cramer Rao Bound (CC-CRB). -Additional Applications: Time Vs. Freq. domain OFDM channel estimation. Frequency selective MIMO channel estimation. -Fisher information matrix (FIM) based analysis -Semi-blind estimation. Overview of Talk

3. A MIMO system is characterized by multiple transmit (T x ) and receive (R x ) antennas The channel between each T x -R x pair is characterized by a Complex fading Coefficient h ij denotes the channel between the i th receiver and j th transmitter. This channel is represented by the Flat-Fading Channel Matrix H MIMO System Model RxRx Transmitter Receiver = Antenna TXTX t - transmit r- receive

4. where, is the r x t complex channel matrix Estimating H is the problem of ‘Channel Estimation’ #Parameters = 2.r.t (real parameters) MIMO System Model MIMO System H

5. CSI (Channel State Information) is critical in MIMO Systems. - Detection, Precoding, Beamforming, etc. Channel estimation holds key to MIMO gains. As the number of channels increases, employing entirely training data to learn the channel would result in poorer spectral efficiency. - Calls for efficient use of blind and training information. As the diversity of the MIMO system increases, the operating SNR decreases. - Calls for more robust estimation strategies. MIMO Channel Estimation

6. One can formulate the Least-Squares cost function, The estimate of H is given as Training symbols convey no information. Training Based Estimation H(z) Training inputs Training outputs Inputs Outputs

7. Uses information in source statistics. Statistics: - Source covariance is known, E (x ( k ) x ( k ) H ) = σ s 2 I t - Noise covariance is known, E (v ( k ) v ( k ) H ) = σ n 2 I r Estimate channel entirely from blind information symbols. No training necessary. Blind Estimation ‘Blind’ data inputs ‘Blind’ data outputs H(z)

8. Channel Estimation Schemes Is there a way to trade-off BW efficiency for algorithmic simplicity and complete estimation. How much information can be obtained from blind data? –In other words, how many of the 2rt parameters can be estimated blind ? How does one quantify the performance of an SB Scheme ? Training Blind Increasing Complexity Decreasing BW Efficiency

9. Training information - X p = [x (1), x (2),…, x (L) ], Y p = [y (1), y (2),…, y (L) ] Blind information - E ( x ( k ) x ( k ) H ) = σ s 2 I t, E ( v ( k ) v ( k ) H ) = σ n 2 I r ( N-L ), the number of blind “information” symbols can be large. L, the pilot length is critical. Semi-Blind Estimation H(z) Training inputs ‘Blind’ data inputs Training outputs ‘Blind’ data outputs N symbols

10. H is decomposed as a matrix product, H= WQ H. For instance, if SVD( H ) = P Q H, W = P. Whitening-Rotation H= WQ H W is known as the “whitening” matrix W can be estimated using only ‘Blind’ data. QQ H = I Q is a ‘constrained’ matrix Q, the unitary matrix, cannot be estimated from Second Order Statistics.

11. How to estimate Q ? Solution : Estimate Q from the training sequence ! Estimating Q Unitary matrix Q parameterized by a significantly lesser number of parameters than H. r x r unitary - r 2 parameters r x r complex - 2r 2 parameters As the number of receive antennas increases, size of H increases while that of Q remains constant  size of H is r x t  size of Q is t x t Advantages

12. Output correlation : Estimate output correlation Estimate W by a matrix square root (Cholesky) factorization as, As # blind symbols grows ( i.e. N ),. Assuming W is known, it remains to estimate Q. Estimating W

13. Orthogonal Pilot Maximum Likelihood – OPML Goal - Minimize the ‘True-Likelihood’ subject to : Estimate: Properties 1. Achieves CRB asymptotically in pilot length, L. 2. Also achieves CRB asymptotically in SNR. Constrained Estimation

14. Estimator : For instance - Estimation of the mean of a Gaussian Estimator Parameter Estimation Observations  parameter p( ; )p( ; )

15. Performance of an unbiased estimator is measured by its covariance as CRB gives a lower bound on the achievable estimation error. The CRB on the covariance of an un-biased estimator is given as where Cramer-Rao Bound (CRB)

16. Most literature pertains to “unconstrained-real” parameter estimation. Results for ‘complex’ parameter estimation ? What are the corresponding results for “constrained” estimation? For instance, estimation of a unit norm constrained singular vector i.e. Constrained Estimation Complex Cons. Par. Estimator CRB

17. Builds on work by Stoica’97 and VanDenBos’93 Let  be an n - dim constrained complex parameter vector The constraints on  are given by h(  ) = 0 Define the extended constraint set f (  ) With complex derivatives, define the matrix F (  ) as Define the extended parameter vector  as p ( ,  ) be the likelihood of the observation  parameterized by  U span the Null Space of F(  ). Complex-Constrained Estimation

18. J is the complex un-constrained Fischer Information Matrix (FIM) defined as CRB Result : The CRB for the estimation of the ‘complex-constrained’ parameter  is given as Constrained Estimation(Contd.)

19. Let Q = [q 1, q 2,…., q t ]. q i is thus a column of Q. The constraints on q i s are given as: Unit norm constraints : q i H q i = ||q i || 2 = 1 Orthogonality Constraints : q i H q j = 0 for i  j Constraint Matrix : Let SVD ( H ) be given as P Q H. CRB on the variance of the (k,l) th element is Semi-Blind CC-CRB

20. has only ‘n’ un-constrained parameters, which can vary freely. has only (n = ) 1 un-constrained parameter. t x t complex unitary matrix Q has only t 2 un-constrained parameters. Hence, if W is known, H = WQ H has t 2 un-constrained parameters. Unconstrained Parameters

21. Let N  be the number of un-constrained parameters in H. Also, X p be an orthogonal pilot. i.e. X p X p H  I Estimation is directly proportional to the number of un- constrained parameters. E.g. For an 8 X 4 complex matrix H, N  = 64. The unitary matrix Q is 4 X 4 and has N  = 16 parameters. Hence, the ratio of semi-blind to training based MSE of estimation is given as Semi-Blind CRB

22. Simulation Results Perfect W, MSE vs. L. r = 8, t = 4.

23. Time Vs. Freq. Domain channel estimation for OFDM systems. Consider a multicarrier system with # channel taps = L (10), # sub-carriers = K (32,64) h is the channel vector. g = F s h, where F s is the left K x L submatrix of F (Fourier Matrix). Total # constrained parameters = K (i.e. dim. of H ). # un-constrained parameters = L (i.e. dim. of h ). OFDM Channel Estimation

24. FIR-MIMO System H(0),H(1),…,H(L-1) to be estimated. r = #receive antennas, t = #transmit antennas (r > t). #Parameters = 2.r.t.L (L complex r X t matrices) D + DD x(k) H(1)H(2)H(L-1) + y(k) + H(0)

25. Fisher Information Matrix (FIM) Let p(ω;θ) be the p.d.f. of the observation vector ω. The FIM (Fisher Information Matrix) of the parameter θ is given as Result: Rank of the matrix J θ equal to the number of identifiable parameters. –In other words, the dimension of its null space is precisely the number of un-identifiable parameters.

26. SB Estimation for MIMO-FIR FIM based analysis yields insights in to SB estimation. Let the channel be parameterized as θ 2rtL. Application to MIMO Estimation: J θ = J B + J t, where J B, J t are the blind and training CRBs respectively. It can then be demonstrated that for irreducible MIMO- FIR channels with (r >t), rank(J B ) is given as

27. Implications for Estimation Total number of parameters in a MIMO-FIR system is 2.r.t.L. However, the number of un- identifiable parameters is t 2. For instance, r = 8, t = 2, L = 4. –Total #parameters = 128. –# blindly unidentifiable parameters = 4. This implies that a large part of the channel, can be identified blind, without any training. How does one estimate the t 2 parameters ?

28. Semi-Blind (SB) FIM The t 2 indeterminate parameters are estimated from pilot symbols. How many pilot symbols are needed for identifiability? Again, answer is found from rank(J θ ). J θ is full rank for identifiability. If L t is the number of pilot symbols, L t = t for full rank, i.e. rank(J θ ) = 2rtL.

29. SB Estimation Scheme The t 2 parameters correspond to a unitary matrix Q. H(z) can be decomposed as H(z) = W(z) Q H. W(z) can be estimated from blind data [Tugnait’00] The unitary matrix Q can be estimated from the pilot symbols through a ‘Constrained’ Maximum-Likelihood (ML) estimate. Let x(1), x(2),…,x(L t ) be the L t transmitted pilot symbols.

30. Semi-Blind CRB Asymptotically, as the number of data symbols increases, semi-blind MSE is given as Denote MSE t = Training MSE, MSE SB = SB MSE. –MSE SB α t 2 (indeterminate parameters) –MSE t α 2.r.t.L (total parameters). Hence the ratio of the limiting MSEs is given as

31. Simulation SB estimation is 32/4 i.e. 9dB lower in MSE r = 4, t = 2 (i.e. 4 X 2 MIMO system). L = 2 Taps. Fig. is a plot of MSE Vs. SNR.

32. Talk Summary Complex channel matrix H has 2rt parameters. –Training based scheme estimates 2rt parameters. –SB scheme estimates t 2 parameters. –From CC-CRB theory, MSE α #Parameters. –Hence, FIR channel matrix H(z) has 2rtL parameters. –Training scheme estimates 2rtL parameters. –From FIM analysis, only t 2 parameters are unknown. –Hence, SB scheme can potentially be very efficient.

33. References Journal Aditya K. Jagannatham and Bhaskar D. Rao, "Cramer-Rao Lower Bound for Constrained Complex Parameters", IEEE Signal Processing Letters, Vol. 11, no. 11, Nov. 2004. Aditya K. Jagannatham and Bhaskar D. Rao, "Whitening-Rotation Based Semi-Blind MIMO Channel Estimation" - IEEE Transactions on Signal Processing, Accepted for publication. Chandra R. Murthy, Aditya K. Jagannatham and Bhaskar D. Rao, "Semi-Blind MIMO Channel Estimation for Maximum Ratio Transmission" - IEEE Transactions on Signal Processing, Accepted for publication. Aditya K. Jagannatham and Bhaskar D. Rao, “Semi-Blind MIMO FIR Channel Estimation: Regularity and Algorithms”, Submitted to IEEE Transactions on Signal Processing.