ML Detection via SDP Relaxation

Transcription

1 ML Detection via SDP Relaxation Junxiao Song and Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) ELEC Convex Optimization Fall , HKUST, Hong Kong

2 Outline of Lecture 1 BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions 2 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 3 Dual Barrier Method for SDP J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

3 Outline of Lecture Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions 1 BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions 2 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 3 Dual Barrier Method for SDP J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

4 BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Given the linear observation model y = Hs + w, where y R m is the received signal, H R m n is the known channel matrix, s {±1} n is the vector of transmitted BPSK symbols, and w is the Gaussian noise vector w N ( 0, σ 2 I ). The maximum likelihood (ML) detection problem is minimize y Hs 2 s 2 subject to s {±1} n. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

5 How Hard the Problem is? Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions It is a nonconvex problem, due to the binary constraint set. We could solve it by evaluating all possible combinations; i.e., brute-force search. The complexity of a brute-force search is O(2 n ), not okay at all for large n! NP-hard in general no polynomial-time algorithm exists. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

6 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions What can we do? One possibility is to relax the problem. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

7 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions What can we do? One possibility is to relax the problem. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

8 Unconstrained and Box Relaxation Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Unconstrained relaxation: minimize s R n y Hs 2 2 The result is zero forcing (ZF) and the problem has a closed-form solution s = (H T H) 1 H T y. Box relaxation: minimize y Hs 2 s 2 subject to 1 s i 1, i = 1,..., n. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

9 SDP Relaxation I Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Since y Hs 2 2 = y T y + s T H T Hs 2y T Hs = x T Lx, where [ H L = T H H T y y T H y T y ] and x = [ s 1 ]. The problem can be rewritten in homogeneous form as minimize x T Lx x subject to x {±1} n+1 x n+1 = 1. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

10 SDP Relaxation II Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Note that the last constraint x n+1 = 1 is unnecessary, since if x is a solution, so is x. Thus, we can simply remove the constraint. Defining X = xx T, the problem is equivalent to minimize X,x Tr(LX) subject to diag(x) = 1 n+1 X = xx T. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

11 SDP Relaxation III Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions The constraint X = xx T is equivalent to X 0 and rank(x) = 1. The key idea in SDP relaxation is to relax the problem by removing the rank constraint, and the problem becomes which is an SDP. minimize Tr(LX) X subject to diag(x) = 1 n+1 X 0, J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

12 An Alternative Way to Derive SDR I Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Recalling the original problem (nonhomogeneous form) minimize s T H T Hs 2y T Hs + y T y s subject to s {±1} n. By letting S = ss T, we get minimize S,s subject to Tr(H T HS) 2y T Hs + y T y diag(s) = 1 n S = ss T. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

13 An Alternative Way to Derive SDR II Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Relaxing S = ss T to S ss T, we can derive an SDR minimize S,s subject to Tr(H T HS) 2y T Hs + y T y diag(s) = 1 n S ss T. This nonhomogeneous SDR is equivalent to the previously derived SDR, by Schur complement: [ ] S ss T S s X = s T 0 1 J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

14 Schur Complement Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Consider a matrix X S n partitioned as [ ] A B X = B T, C where A S k. If A is nonsingular, the matrix S = C B T A 1 B is called the Schur complement of A in X. Important characterizations: X 0 if and only if A 0 and S 0. If A 0, then X 0 if and only if S 0. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

15 SDR as the Dual of the Dual I Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Recalling the original problem in homogeneous form The Lagrangian is minimize x T Lx x subject to x {±1} n+1 n+1 L (x, λ) = x T Lx + λ i (x 2 i 1) The dual problem is i=1 = λ T 1 + x T (L + Diag(λ))x maximize λ T 1 λ subject to L + Diag(λ) 0. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

16 SDR as the Dual of the Dual II Recalling the SDP relaxation The dual function is Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions minimize Tr(LX) X subject to diag(x) = 1 n+1 X 0, g(z, λ) = inf X Tr (LX) Tr (ZX) + λt (diag(x) 1) = inf Tr ((L Z + Diag(λ))X) X λt 1 { λ T 1 L + Diag(λ) = Z 0 = otherwise. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

17 SDR as the Dual of the Dual III Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Thus, the dual problem of the SDR is maximize λ T 1 λ subject to L + Diag(λ) 0, which is exactly the dual problem of the original problem. Since strong duality holds for the SDP, the SDP relaxation can be considered as the dual of the dual of the original problem. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

18 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Which relaxation is better? J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

19 Comparison via Simulation Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Symbol Error Rate ZF Box Relaxation SDP Relaxation SNR, in db Figure: SER performance using quantization, n = m = 8, BPSK. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

20 Comparison of Various Relaxations Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions From the figure, we can see that the performance of the unconstrained relaxation (ZF) is the worst. This is easy to understand, since the other two relaxations are much tighter than the unconstrained relaxation. From the figure, we can also see that the SDP relaxation performs better than the box relaxation. Is the SDP relaxation tighter than the box relaxation? J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

21 Comparison of Various Relaxations Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions From the figure, we can see that the performance of the unconstrained relaxation (ZF) is the worst. This is easy to understand, since the other two relaxations are much tighter than the unconstrained relaxation. From the figure, we can also see that the SDP relaxation performs better than the box relaxation. Is the SDP relaxation tighter than the box relaxation? J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

22 Box Relaxation Vs SDP Relaxation I Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Recalling the box relaxation and the SDP relaxation minimize y Hs 2 s subject to s 2 i 1, i = 1,..., n. minimize S,s subject to Tr(H T HS) 2y T Hs + y T y diag(s) = 1 n S ss T. It seems hard to compare these two relaxations from the primal perspective. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

23 Box Relaxation Vs SDP Relaxation II Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions But we know that the optimal value of the SDP relaxation is equal to that of the dual of the original problem, thus p SDR = max λ min s y Hs 2 + For the box relaxation, by strong duality p Box = max min y Hs 2 + λ 0 s n λ i (s 2 i 1) i=1 n λ i (s 2 i 1) The feasible set of λ in SDP relaxation contains that in box relaxation, i.e., the box relaxation may be seen as further relaxation of the SDP relaxation, thus p Box p SDR p ML. i=1 J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

24 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions The original problem requires the solution to be binary. If the optimal solutions of the unconstrained relaxation and box relaxation are binary, or the optimal solution of the SDP relaxation is rank 1, then clearly we have solved the original problem. But usually the optimal solution of the relaxed problems will not be binary. How can we reconstruct binary solutions? J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

25 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions The original problem requires the solution to be binary. If the optimal solutions of the unconstrained relaxation and box relaxation are binary, or the optimal solution of the SDP relaxation is rank 1, then clearly we have solved the original problem. But usually the optimal solution of the relaxed problems will not be binary. How can we reconstruct binary solutions? J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

26 Simple Quantization Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions For unconstrained relaxation and box relaxation, a natural approach is to quantize the solution by looking at the sign, i.e., ŝ = sign(s relax ). For SDP relaxation, based on the fact that, [ when the ] solution ˆX is rank 1, it will have the structure ˆX ss T s = s T, one 1 simple approach is to quantize the last column of ˆX, i.e., ŝ = sign( ˆX 1:n,n+1 ). This is the quantization method we have used in the previous figure. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

27 Eigenvalue Decomposition Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions There are better approaches for SDP relaxation, since we have the whole matrix ˆX. Suppose λ is the largest eigenvalue of ˆX and u is the corresponding eigenvector. Like before, based on the fact that, when ˆX is rank 1, then [ ] s [ ˆX = s T 1 ] = λuu T, 1 i.e., u is a scaled version of [ s 1 ( ) u1:n ŝ = sign. u n+1 ]. Thus, we can take J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

28 Randomization BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions The idea of randomization is to generate random points using ˆX as the covariance matrix, and then quantize and keep the best of the points. A convenient way to implement this is to factorize ˆX as ˆX = VV T, multiply V by random vectors r (i) N (0, I) and then quantize ( ) ˆx (i) Vr (i) = sign V n+1,: r (i) i = 1,..., N. Then just keep the point ˆx with minimum objective value. Randomization usually can provides better performance compared with the previous two methods. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

29 Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Approximation Bound of Randomization When we generate candidate solutions using the randomization approach, the expected value of the objective can be computed explicitly: E(ˆx T Lˆx) = Tr(LE(ˆxˆx T )) = 2 Tr(Larcsin( ˆX)). π We are guaranteed to reach this expected value after sampling a few points ˆx, hence we know that the optimal value p ML of the ML detection problem is bounded by Tr(L ˆX) p ML ˆx T Lˆx 2 Tr(Larcsin( ˆX)). π J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

30 SER Performance Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions Symbol Error Rate ZF Box Relaxation SDP Last Column SDP Eigenvector SDP Randomization SNR, in db Figure: SER performance, n = m = 8, BPSK. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

31 Outline of Lecture QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 1 BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions 2 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 3 Dual Barrier Method for SDP J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

32 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation So far we have considered the SDP relaxation for the case with BPSK symbols and real-valued channels. Can we handle more general complex-valued channels and constellations, such as QPSK, M-PSK and 16-QAM? Yes! There are different papers dealing with different constellations. BPSK and QPSK [Tan and Rasmussen, 2001] MPSK [Ma et al., 2004] 16-QAM [Wiesel et al., 2005], [Sidiropoulos and Luo, 2006] J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

33 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation So far we have considered the SDP relaxation for the case with BPSK symbols and real-valued channels. Can we handle more general complex-valued channels and constellations, such as QPSK, M-PSK and 16-QAM? Yes! There are different papers dealing with different constellations. BPSK and QPSK [Tan and Rasmussen, 2001] MPSK [Ma et al., 2004] 16-QAM [Wiesel et al., 2005], [Sidiropoulos and Luo, 2006] J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

34 General Channel Model QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation From now on, we will consider the following complex-valued MIMO model ỹ = H s + w, where ỹ C m is the received signal, H C m n is the channel matrix, s A n is the vector of transmitted symbols, with A being the constellation set, and w is the complex Gaussian noise vector w CN ( 0, σ 2 I ). Constellations: QPSK: A = {±1 ± j} M-PSK: A = {s = e π M (2k 1)j k = 1,..., M} 16-QAM: A = {s = s 1 + js 2 s 1, s 2 {±1, ±3}} J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

35 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation ML Detection with QPSK Constellation The signal model is where s i {±1 ± j}. ỹ = H s + w, The ML detection problem in the QPSK case: minimize ỹ H s 2 s subject to s {±1 ± j} n. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

36 SDP Relaxation for QPSK QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation The complex-valued problem can be reformulated as a real-valued one if we define [ ] [ ] Re{ỹ} Re{ H} Im{ H} y = ; H = Im{ỹ} Im{ H} Re{ H} ; s = [ Re{ s} Im{ s} ] [ Re{ w} ; w = Im{ w} The problem reduces to a BPSK one with twice the dimension: y = Hs + w, s {±1} 2n, and thus can be solved via SDP relaxation. ]. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

37 M-PSK Constellations QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation M-PSK: A = {s = e π M (2k 1)j k = 1,..., M} The complex constellation points are equally spaced on the unit-circle. For example, the 8-PSK constellation is as follows: j Im 1 Re 8-PSK J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

38 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation ML Detection with M-PSK Constellations The signal model is ỹ = H s + w, where s i {s = e π M (2k 1)j k = 1,..., M}. The ML detection problem in the M-PSK case: minimize s ỹ H s 2 subject to s {s = e π M (2k 1)j k = 1,..., M} n. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

39 SDP Relaxation for M-PSK I QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Idea: relax the constellation constraints to s i = 1, i.e., the whole unit-circle, and then apply SDP relaxation. Following the idea, we first relax the problem to be minimize ỹ H s 2 s subject to s i = 1, i = 1,..., n. By letting S = s s H, we get minimize S, s subject to Tr( H H H S) 2Re{ỹ H H s} + ỹhỹ diag( S) = 1 n S = s s H. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

40 SDP Relaxation for M-PSK II QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation As before, by further relaxing S = s s H to S s s H, we can derive a complex-valued SDR minimize S, s subject to Tr( H H H S) 2Re{ỹ H H s} + ỹhỹ diag( S) = 1 n S s s H. By Schur complement, we can rewrite the complex-valued SDR in homogeneous form: minimize Tr( LX) X C n+1 n+1 subject to diag(x) = 1 n+1 X 0, [ where L HH H = HHỹ ỹ H H ỹhỹ ]. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

41 16-QAM Constellation QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 16-QAM: A = {s = s 1 + js 2 s 1, s 2 {±1, ±3}} J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

42 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation ML Detection with 16-QAM Constellation The signal model is ỹ = H s + w, where s i {s = s 1 + js 2 s 1, s 2 {±1, ±3}}. The ML detection problem in the 16-QAM case: minimize ỹ H s 2 s subject to s {s = s 1 + js 2 s 1, s 2 {±1, ±3}} n. As in the QPSK case, we can rewrite the problem as a real-valued one minimize y Hs 2 s subject to s {±1, ±3} 2n. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

43 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation SDP Relaxation with Polynomial Constraints I The constellation constraints can be written in a polynomial form (s 2 i 1)(s2 i 9) = 0 (i = 1,, 2n) to obtain the following equivalent formulation: minimize y Hs 2 s,t subject to s 2 i = t i, i = 1,, 2n t 2 i 10t i + 9 = 0, i = 1,, 2n. Since there are quadratic equality constraints, the problem is nonconvex. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

44 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation SDP Relaxation with Polynomial Constraints II Defining a new variable s X = t [ s T t T 1 ] = 1 ss T st T s ts T tt T t s T t T 1 = then the quadratic equality constraints become diag(x 1,1 ) X 2,3 = 0 diag(x 2,2 ) 10X 2,3 + 9 = 0 X 1,1 X 1,2 X 1,3 X 2,1 X 2,2 X 2,3 X 3,1 X 3,2 X 3,3 and the objective function can be written as H T H 0 H T y y Hs 2 = Tr y T H 0 y T y X. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

45 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation SDP Relaxation with Polynomial Constraints III The problem can be rewritten as minimize X subject to Tr H T H 0 H T y y T H 0 y T y X diag(x 1,1 ) X 2,3 = 0 diag(x 2,2 ) 10X 2,3 + 9 = 0 X 0 X 3,3 = 1 rank(x) = 1 Finally, the problem can be relaxed to an SDP by dropping the rank-one constraint. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

46 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation SDP Relaxation with Bound Constraints I Still consider the equivalent real-valued problem By letting S = ss T, we get minimize y Hs 2 s subject to s {±1, ±3} 2n. minimize S,s Tr(H T HS) 2y T Hs + y T y subject to S ii {1, 9}, i = 1,..., 2n S = ss T. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

47 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation SDP Relaxation with Bound Constraints II Relaxing S = ss T to S ss T is not enough to yield a convex relaxation. We also relaxes {1, 9} to the interval [1, 9], leading to minimize S,s Tr(H T HS) 2y T Hs + y T y subject to 1 S ii 9, i = 1,..., 2n S ss T. Rather unexpectedly, it is shown in [Ma et al., 2009] that the two SDP relaxations are equivalent. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

48 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation So far, we have seen different specialized SDP relaxations for different constellations. Can we come up with a universal SDP relaxation to deal with all the constellations? Yes! Next we are going to talk about a universal binary SDP relaxation [Fan et al., 2013], which can handle any constellations. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

49 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation So far, we have seen different specialized SDP relaxations for different constellations. Can we come up with a universal SDP relaxation to deal with all the constellations? Yes! Next we are going to talk about a universal binary SDP relaxation [Fan et al., 2013], which can handle any constellations. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

50 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Binarization of Arbitrary Constellations I The starting point of the universal binary SDR is the existence of a binary space covering the original non-binary signal space. For any constellation A C of size A = M, there always exists a covering binary space B defined by a vector α C q, such that A B = {s s = α H b, b {±1} q }, with q satisfying log 2 (M) q M + 1. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

51 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Binarization of Arbitrary Constellations II For example, if A = {s 1, s 2,..., s M }, we can construct a covering binary space B by setting [ ] H 1 α = 2 s 1, 1 2 s 2,, 1 2 s M, 1 s i, 2 which is not very desirable as it achieves the most expensive binary expansion with q = M + 1. But the binary mapping (i.e., the choice of α) is not unique for a given constellation A, and for most constellations we can have tighter binary mappings with smaller q. For example, the 16-QAM constellation, we can construct B by using α = [2, 2j, 1, j] H and thus q = log 2 16 = 4. i J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

52 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Binarization of Arbitrary Constellations III In the case of the 16-QAM constellation, it just so happens that B = A. In other cases, however, like the 12-QAM constellation shown below, B will contain more points than A, i.e., B A and we need to add some constraints to exclude all the undesired points. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

53 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Binarization of Arbitrary Constellations IV Lemma In the following lemma, we derive linear constraints to remove all the undesired points. For any constellation A C of size A = M, there always exists a vector α C q (with q satisfying log 2 (M) q M + 1) and a matrix D such that A = {s s = α H b, b T D (q 2)1 T, b {±1} q }. One possible choice for D is D = [d 1, d 2,..., d l ], where d 1, d 2,..., d l are the binary vectors that generate the difference B A, i.e., α H d i B A, i = 1,..., l. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

54 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Table: Binarization for different constellations Constellation A α D BPSK A = { 1, 1} 1 QPSK A = {±1 ± j} [1, j] H 16-QAM A = {s s = s 1 + js 2, s 1 S, s 2 S} [2, 2j, 1, j] H S = { 3, 1, 1, 3} 64-QAM A = {s s = s 1 + js 2, s 1 S, s 2 S} [4, 4j, 2, 2j, 1, j] H S = { 7, 5, 3, 1, 1, 3, 5, 7} 256-QAM A = {s s = s 1 + js 2, s 1 S, s 2 S} [8, 8j, 4, 4j, 2, 2j, 1, j] H S = {±(2i + 1), i = 0, 1,..., 7} 12-QAM A = {s s A 16 QAM, s 2 < 18} [2, 2j, 1, j] H 32-QAM A = {s s A 64 QAM, s 2 < 50} [4, 4j, 2, 2j, 1, j] H 8-PSK A = {s s = e j (2i+1)π 8, i = 0, 1,..., 7} [a, b, aj, bj] H (a = 2 2 cos( π 8 ) b = 2 2 sin( π 8 )) J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

55 Universal Binary SDP Relaxation I QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation With the binary representation of arbitrary constellations in the previous lemma, the signal space A n can be expressed as: A n = { s s = α H b, bt D (q 2)1 T, b {±1} qn }, where α H = α H I n n, b = [b T 1,..., bt q ] T (with the n-dimensional vector b q containing the q-th bit of the n transmitted symbols) and D = D I n n. The implication is that any n-dimensional signal space A n can be expressed by all qn-dimensional binary vectors b, that satisfy some linear inequalities constraints b T D (q 2)1 T. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

56 Universal Binary SDP Relaxation II QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation Recalling the general ML detection problem minimize ỹ H s 2 s subject to s A n. By applying the binary representation of A n, the problem becomes minimize ỹ H α H b 2 b subject to b T D (q 2)1 T b {±1} qn, which is now a qn-dimensional binary ML detection problem. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

57 Universal Binary SDP Relaxation III QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation The problem reduces to the BPSK case, but with additional linear inequality constraints. By letting x = [ b T 1] T and X = xx T, the problem becomes minimize X,x subject to Tr(LX) diag(x) = 1 qn+1 X = xx T X qn+1,1:qn D (q 2)1 T where [ H L = H H H H ỹ ỹ H H ỹ H ỹ ], with H = H α H. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

58 Universal Binary SDP Relaxation IV QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation By relaxing X = xx T to X 0, finally we get the SDP relaxation minimize X subject to Tr(LX) diag(x) = 1 qn+1 X 0 X qn+1,1:qn D (q 2)1 T. To sum up, given a constellation, we first find α and D, then let α H = α H I n n, D = D I n n and construct L as in the previous page. With D and L, we then solve the above SDP relaxation. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

59 Outline of Lecture Dual Barrier Method for SDP 1 BPSK Signal Detection Problem Statement Convex Relaxations Comparison of Various Relaxations Reconstruct Binary Solutions 2 QPSK Constellation M-PSK Constellations 16-QAM Constellation Universal Binary SDP Relaxation 3 Dual Barrier Method for SDP J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

60 Dual Barrier Method for SDP Solving the SDP Relaxation Having the SDR formulation, a general purpose software, such as CVX, can be used to solve the SDP conveniently. But to improve the computational efficiency, we can write our own interior point method (IPM) to explicitly exploit the problem structure. For ease of illustration, we take the SDP relaxation in the BPSK case as an example, to show the derivation of a specialized (and fast) IPM. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

61 A First Look at the Problem Dual Barrier Method for SDP Recalling the problem formulation minimize Tr(LX) X subject to diag(x) = 1 n+1 X 0. We know that in the barrier method we need to compute the gradient and Hessian. But in the above problem, the variable is an (n + 1) (n + 1) matrix, and the Hessian will be of size (n + 1) 2 (n + 1) 2, which is hard to handle. Thus, instead of directly dealing with the primal problem, we resort to the dual problem. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

62 Dual Barrier Method for SDP I Dual Barrier Method for SDP The dual problem is maximize 1 T v v subject to L + Diag(v) 0. The logarithmic barrier of L + Diag(v) 0 is logdet (L + Diag(v)). The dual barrier method will solve the following problem based on Newton s method: minimize v f (v) = t ( 1 T v ) logdet (L + Diag(v)), where t is the barrier parameter. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

63 Dual Barrier Method for SDP II Dual Barrier Method for SDP Denoting v (t) the optimal solution of the problem for a given t, the gradient will be zero ( f(v (t)) = t1 diag (L + Diag (v (t))) 1) = 0. Defining X (t) = 1 t (L + Diag (v (t))) 1, then diag (X (t)) = 1 n+1, so it s a feasible point for the primal. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

64 Dual Barrier Method for SDP Dual Barrier Method for SDP III The duality gap is Tr(LX (t)) ( v (t) T 1) = Tr(LX (t)) + v (t) T diag (X (t)) = Tr(LX (t)) + Tr(Diag (v (t)) X (t)) = Tr((L + Diag (v (t)))x (t)) = 1 t Tr(I n+1) = n + 1 t which means Tr(LX (t)) is not far away from the desired optimal value p of the original problem by more than n+1 t. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

65 Dual Barrier Method for SDP IV Dual Barrier Method for SDP The algorithm for solving the SDP is summarized as follows: Algorithm: dual barrier method for SDP Input: L, µ > 1, t > 0, tolerance ɛ > 0 Repeat 1. Centering step. Compute v (t) by solving minimize t ( 1 T v ) logdet (L + Diag(v)) v using Newton s method. 2. Stopping criterion. Quit if (n + 1) /t < ɛ 3. Increase t. t = µt. Output: X = (1/t) (L + diag (v (t))) 1 J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

66 Computational Complexity Dual Barrier Method for SDP 10 0 SDPT3 SeDuMi Tailored dual barrier method Average running time (s) Problem size, n (n = m) Figure: Average running time vs problem size. J. Song and D. Palomar (HKUST) SDP Relaxation ELEC / 63

67 References Fan, X., Song, J., Palomar, D. P., and Au, O. C. (Nov. 2013). Universal Binary Semidefinite Relaxation for ML Signal Detection. IEEE Trans. on Communications, 61(11): Ma, W. K., Ching, P. C., and Ding, Z. (Oct. 2004). Semidefinite relaxation based multiuser detection for M-ary PSK multiuser systems. IEEE Transactions on Signal Processing, 52(10): Ma, W. K., Su, C. C., Jaldén, J., Chang, T. H., and Chi, C. Y. (Jun. 2009). The equivalence of semidefinite relaxation MIMO detectors for higher-order QAM. IEEE Journal of Selected Topics in Signal Processing, 3(6): Sidiropoulos, N. D. and Luo, Z. Q. (Sept. 2006). A semidefinite relaxation approach to MIMO detection for high-order QAM constellations. IEEE Signal Processing Letters, 13(9): Tan, P. H. and Rasmussen, L. K. (Aug. 2001). The application of semidefinite programming for detection in CDMA. IEEE Journal on Selected Areas in Communications, 19(8): Wiesel, A., Eldar, Y. C., and Shamai, S. (May 2005). Semidefinite Relaxation for Detection of 16-QAM Signaling in MIMO Channels. IEEE Signal Processing Letters, 12(9):

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