Optimal Transceiver Design for Multi-Access. Communication. Lecturer: Tom Luo

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1 Optimal Transceiver Design for Multi-Access Communication Lecturer: Tom Luo

2 Main Points An important problem in the management of communication networks: resource allocation Frequency, transmitting power; Goal: high data rate, low bit error rate Transmitter receiver for multi-user high speed broadband Digital Subscribe Line application Direct formulation is nonconvex; equivalent formulation is SDP (thus convex); Further simplification to SOC, and to combinatorial polynomial time algorithm Valuable guidelines and insights for optimal practical transceiver design

3 Content Elements of data communication: OFDM, subcarriers, power loading, precoding/equalization. Linear transceiver (Transmitter Receiver) design for the two user case: SDP formulation SOC formulation O(n 3 ) strongly polynomial algorithm General multi-user case Other formulations: Sum-rate transmitter design for multi-access channel Sum-rate transmitter design broadcast channel Linear transceiver design with zero-forcing equalizer

4 Single User SISO Communication System Single Input Single Output Communication System Transmitter Filter Channel (FIR) Noise n Receiver Filter s f h x received signal g s s is input signal (assumed statistically white) h R k is a linear, time-invariant channel (assumed known) f, g are transmitter filter and equalizer filter respectively n is the additive (Gaussian) noise 3

5 An Equivalent MIMO System Multi Input Multi Output Communication System s S/P Serial Parallel Converter Transmitter Matrix Filter (precoder) F Channel (FIR) H * Noise n x received signal Receiver matrix Filter (equalizer) G P/S Parallel Serial Converter s S/P: serial-parallel converter (with cyclic prefixing); P/S: parallel-serial F: n l transmitter matrix filter (or precoder), obtained from f; data rate = l/n G: l n receiver equalizer matrix (obtained from g) H: channel matrix (obtained from h); n: noise x = H Fs n, H is circulant. 4

6 Single Input Single Output Communication System Transmitter Filter Channel (FIR) Noise n Receiver Filter s f h x received signal g s Multi Input Multi Output Communication System Transmitter Matrix Filter (precoder) Noise n Receiver matrix Filter (equalizer) s s * S/P F H G P/S x Channel Serial Parallel Converter received signal Parallel Serial Converter H * is a circulant matrix 5

7 OFDM System The circulant channel matrix H can be diagonalized via IFFT/FFT: H = D HD, where D is the standard discrete Fourier transform matrix, H is diagonal The diagonalized channel becomes a set of independent subchannels Each subchannel corresponds to a subcarrier with a particular frequency (from FFT) This diagonalization is not channel dependent Orthogonal Frequency Division Multiplexing System employs IFFT/FFT to decompose the channel H 6

8 Multi Input Multi Output Communication System Serial Parallel Converter Parallel Serial Converter Channel (FIR) received signal s S/P F IFFT H x FFT G P/S s (precoder) Transmitter Matrix Filter Noise n (equalizer) Receiver matrix Filter Equivalent to a Diagonal Channel Matrix (a set of independent parallel subchannels) H * Transmitter Filter h() h() h(3) h(4) received signal x F IFFT FFT G Subcarriers Channel (FIR) Noise n Receiver Filter Orthogonal Frequence Division Multiplexing (OFDM) System 7

9 Linearly Precoded/Power Loaded OFDM The precoder F can be a general n n matrix, subject to power constraint tr(ff ) p. The optimized design F will have a rank l n, resulting in an optimal data rate l/n. A special, and popular, linear precoder is the so called power loading precoder: F is diagonal. 8

10 Linearly Precoded/Power Loaded OFDM h() h() F IFFT FFT G h(3) h(4) received signal x Transmitter Filter Channel (FIR) Noise n Receiver Filter General Linearly Precoded OFDM System f() f() f(3) f(4) IFFT h() h() h(3) h(4) received signal x FFT g() g() g(3) g(4) Transmitter Filter Channel (FIR) Noise n Receiver Filter Power Loaded OFDM System 9

11 Two-User Multi-Access Communication Channel Transmitters Channel Matrices Receivers s F H Noise n G s x received signal s F H G s Diagram of Two user Communication System Mathematical model: x = HFs HFs ρn, ρ > 0. Linear detection: si = sign (Gix), i =,. Goal: Given the channel matrices, H, H, design transceivers F, F, G, G. 0

12 Applications and Previous Work Applications include the current and future systems of DSL, DAB, DVB. Equalizer (or receiver) design for a fixed transmitter has been researched extensively in the last decade. The joint transmitter and receiver (transceiver) design was considered recently, but only for the single user case In the single user transceiver design work, the design criteria used include: Minimum Mean Square Error, maximum information rate, channel capacity The last two require complex receiver structures.

13 Mean Square Error The detection with receiver (equalizer) Gi: ŝi = sign (Gix). Let ei denote the error vector (before making the hard decision) for user i, i =,. Then e = Gx s = G(HFs HFs ρn) s = (GHF I) s GHFs ρgn. This further implies E(ee ) = (G HF I) (GHF I) (GHF) (GHF) ρ GG Similarly, we have E(ee ) = (G HF I) (GHF I) (GHF) (GHF) ρ GG.

14 Formulation: MMSE Equalizer Case Our goal is to design a set of transmitting matrix filters Fi and a set of matrix equalizers Gi such that the total mean squared error MSE = tr(e(ee )) tr(e(e e )) is minimized. As is always the case in practice, there are power constraints on the transmitting matrix filters: tr(ff ) p, tr(ff ) p The above is nonconvex. We first eliminate the variables G, G: the MMSE equalizers. 3

15 Formulation: MMSE Equalizer Case By minimizing E(ee ) with respect to G, we obtain the following MMSE equalizer for user : G = F H W, where W = ( HFF H H FF H ρ I). Substituting this into E(ee ) gives: E(ee ) = F H WH F I. Similarly, the MMSE equalizer G for user is given by G = F H W and resulting minimized (with respect to G) mean square error for user is given by: E(ee ) = F H WH F I. 4

16 Total MSE Substituting into the above expression gives rise to MSE = tr(e(ee )) tr(e(e e )) ( ) = tr F H WH F tr ( F H WH F ) ( ) = tr WHFF H tr ) = tr (W(HFF H H FF H ) n ( WHFF H ) n n = ρ tr (W) n, where the last step follows from the definition of W. 5

17 Formulation: MMSE Equalizer Case Introduce matrix variables: U = FF, U = FF. Then the MMSE transceiver design problem becomes minimizeu,u tr ((HUH H UH ρ I) ) subject to tr(u) p, tr(u) p, U 0, U 0. Reformulate using the auxiliary matrix variable W: minimizew,u,u tr (W) subject to tr(u) p, tr(u) p, W (HUH H UH ρ I) U 0, U 0. 6

18 SDP Formulation The constraint W (HUH H UH ρ I) is equivalent to LMI: [ W I I HUH H UH ρ I ] 0. (3) We obtain an SDP formulation: minimizew,u,u tr (W) subject to tr(u) p, tr(u) p, W satisfies (3), U 0, U 0. Interior point method with arithmetic complexity O(n 6.5 log(/ɛ)), ɛ > 0 is the solution accuracy. 7

19 OFDM: Diagonal Designs are Optimal! Result If H and H are diagonal, as in the OFDM systems, then the optimal transmitters are also diagonal. Implication The MMSE transceivers for an multi-user OFDM system can be implemented by optimally setting the data rates and allocating power to each subcarrier for all the users. 8

20 Linearly Precoded/Power Loaded OFDM h() h() F IFFT FFT G h(3) h(4) received signal x Transmitter Filter Channel (FIR) Noise n Receiver Filter General Linearly Precoded OFDM System f() f() f(3) f(4) IFFT h() h() h(3) h(4) received signal x FFT g() g() g(3) g(4) Transmitter Filter Channel (FIR) Noise n Receiver Filter Power Loaded OFDM System 9

21 From SDP to SOC Formulation Restricting to diagonal designs, the SDP becomes SOC: minimizew,u,u n subject to i= wi n u(i) p, i= n u(i) p, i= ( wi h (i) u(i) h(i) u(i) ρ ), u(i) 0, u(i) 0, i =,,..., n. There exist highly efficient (general purpose) interior point methods to solve the above second order cone program. Arithmetic complexity O(n 3.5 log(/ɛ)), ɛ > 0 is the accuracy. 0

22 Properties of Optimal MMSE Transceiver Let u 0, u 0 be the optimal transceivers. Define: { I = {i u (i) > 0, u (i) = 0}, I = {i u (i) = 0, u (i) > 0}, Is = {i u (i) > 0, u (i) > 0}, I u = {i u (i) = 0, u (i) = 0}. I, I: subcarriers allocated to user and user ; Is and Iu: subcarriers shared and unused; data rates: ( I Is )/n, ( I Is )/n For each i I and j I, we have h (i) h (i) h (j) h (j). For all i, j Is, we have h (i) h (i) = h (j) h (j). For any i Iu and any j I Is, we have h(i) < h(j). Similarly, for any i Iu and any j I Is, we have h(i) < h(j).

23 Intuitive Interpretation x = HFs HFs ρn, with Hi, Fi diagonal; x(i) = h(i)f(i)s(i) h(i)f(i)s(i) ρ n(i). In a fading environment, the path gains h(i), h(i) are random, the probability of having two equal path gains is zero. Is is singleton: at most one subcarrier should be shared by the two users. The remaining subcarriers are allocated to the two users according to the path gain ratios: subcarrier i to user and subcarrier j to user only if h(i) h(i) h (j) h(j). The subcarriers in Iu have small path gains for both users (i.e., both h(i) and h(i) are small), and they should not be used by either user, i.e., they are useless subcarriers!

24 A Strongly Polynomial Time Algorithm The properties of optimal MMSE transceivers can be used to design a combinatorial algorithm. Assume h() h() > h () h() > > h (n ) h(n ) > h (n) h(n). Then I {,..., i} and I {i,..., n} for some i. Leads to an O(n 3 ) strongly polynomial time (combinatorial) algorithm (vs. O(n 3.5 log /ɛ) interior point algorithm for SOC). 3

25 4 g (4) f (4) h (4) g (3) f (3) h (3) g () f () h () s f () h () g () Subcarrier Allocation and Power Loading s received signal x f (4) h (4) g (4) f (3) h (3) g (3) f () Noise n g () s h () s f () h () g () h (4) h (3) F G h () s h () s received signal h (4) x h (3) F G s h () Noise n h () s Transmitters Channel Matrices Receivers Stanford University EE39o Z.Q. Luo Practical Implications

26 General m-user Case Mathematical model: x = HFs HFs HmFmsm ρn. Let Gi be the linear MMSE matrix equalizer at the i-th receiver. Then the total MSE is given by ρ tr ( (HFF H H ifif i H i ρ I) ) (m )n. Let Ui = FiF i. Then the power constrained optimal MMSE transmitter design problem can be described as: minimizeu,...,um tr ((HUH H mumh m ρ I) ) subject to tr(ui) pi, Ui 0, i =,..., m. 5

27 SDP/SOC Formulation SDP formulation minimize tr (W) subject to tr(ui) pi, Ui 0, i =,,..., m, [ W I I HUH H mumh m ρ I ] 0. SOC formulation minimizew,u,...,um subject to n i= wi n uj(i) pj, j =,,..., m, i= ( wi h (i) u(i) hm(i) um(i) ρ ), uj(i) 0, i =,,..., n, j =,,..., m. 6

28 Simulation Scenario Uplink with 6 active users and 60 available subcarriers Each user sees its own Rayleigh channel (complex-valued) Three schemes:. AMOUR No channel knowledge; Each user uses 0 subcarriers, spreads 8 bits over these carriers using a DFT-type spreading.. Individually MMSE power-loaded OFDM Same subcarrier allocation as AMOUR. Each user sends bit per subcarrier, i.e 0 bits per block; knows its allocated channels and does MMSE power loading for these bits. 3. Multi-user MMSE power loaded OFDM Using the SOCP formulation. In this case the subcarrier allocations and the number of bits per block vary from block to block, but the average number of bits per block remains 0. 7

29 8 BlockSNR, db Multi usr MSE power loading via SOCP, 0 bits/blk on ave AMOUR, no channel knowledge, 8 bits/blk Single usr MSE power loaded OFDM with AMOUR subcarrier alloc, 0 bits/blk 0 3 average BER user OFDM in a length 3 Rayleigh channel, 60 subcarriers, Coding gain smaller here Stanford University EE39o Z.Q. Luo Simulation Results

30 Efficiency of the Design Approach On a PIII 600Mhz PC, Two users, symbols per block, length 3 channel; SDP 0.65 secs SOCP 0.3 secs 6 users, 0 symbols per block, length 3 channel SOCP 0.65 secs 9

31 Formulation: Max Sum Rate Capacity for MAC Let Σk 0, pk denote the covariance matrix and the transmit power of the k-th user signal. The total sum rate of multi-access channel is K log det(i HkΣkH H k ) k= which is achievable by successive nulling and cancellation at BS. The multi-user transmitter design is then maximize log det(i K k= H kσkh H k ) subject to tr(σk) pk, Σk 0, k =,,..., K. A convex problem; can be solved by interior point methods, iterative water-filling. 30

32 Formulation: Max Sum Rate Capacity for BC Let Σk 0 denote the covariance matrix for the k-th user signal, and let p denote the total transmit power. By duality, total sum rate of a broadcast channel channel is K log det(i H H ΣkHk) k k= which is achievable by dirty paper coding technique, where Σk (new variables) depends on Σk linearly. The multi-user transmitter design is then maximize log det(i K k= H k ΣkH H k ) K subject to k= tr( Σk) p, Σk 0, k =,,..., K. A convex problem; can be solved by interior point methods (and iterative water-filling?). 3

33 Formulation: Zero-Forcing Equalizer Case Recall ei denotes the error vector (before making the hard decision) for user i, i =, and e = Gx s = G(HFs HFs ρn) s = (GHF I) s GHFs ρgn. Moreover, E(ee ) = (G HF I) (GHF I) (GHF) (GHF) ρ GG E(ee ) = (G HF I) (GHF I) (GHF) (GHF) ρ GG. Consider the zero-forcing equalizers: G = (HF), G = (HF). 3

34 Formulation: Zero-Forcing Equalizer Case Substituting the ZF conditions into the MSE expressions gives ( MSE = tr (GHF) (GHF) ) ( ) ρ tr GG ( tr (GHF) (GHF) ) ( ) ρ tr GG. Introduce new matrix variables U = FF, U = FF, V = G G, V = G G. Then the MSE can be rewritten as MSE = tr(vhuh ) ρ tr(v) tr(vhuh ) ρ tr(v) The power constraint becomes tr(u) p, tr(u) p. The ZF condition reduces to V = HUH, V = HUH. 33

35 Formulation: Zero-Forcing Case The Minimum MSE transceiver design problem can be cast as minimize MSE = tr(vhuh ) ρ tr(v) tr(vhuh ) ρ tr(v) subject to tr(u) p, tr(u) p, V = HUH, V = HUH, Vi 0, Ui 0, i =,. Note the constraints are nonlinear (due to the matrix inverse) The objective function is nonconvex quadratic, due to the cross terms tr(vhuh ) and tr(vhuh ). Reformulation is necessary. 34

36 Reformulation: ZF Case Use monotonicity and Schur complement technique, we obtain the following equivalent formulation: minimize MSE = tr(vhuh ) ρ tr(v) tr(vhuh ) ρ tr(v) subject to tr(u) p, tr(u) p, Vi 0, Ui 0, i =, [ ] [ ] HUH I I V 0, HUH I I V 0. () Note that the constraints are all linear matrix inequalities (LMIs), and in particular convex. But the objective function is nonconvex. 35

37 Alternating Direction Method Fixing the designs for user (namely, U, V), the objective function MSE is linear in U, V, resulting in a SDP. Similarly, fixing U and V yields a semidefinite program in U and V. Alternating Direction Method: At iteration 0, let U (0) i = V (0) i = I. At iteration k, Solve () with U, V fixed to the values of U (k ), V (k ). Update U (k) and V (k) to the resulting optimized values of U and V. Solve () with U, V fixed to the values of U (k ), V (k ). Update U (k) and V (k) to the resulting optimized values of U and V. Repeat with k := k. Convergence: bounded iterates the minimum principle necessary optimality condition. 36

38 Power-loaded OFDM Optimal? Let channel matrices H, H be diagonal. If we fix U, V at some positive definite diagonal matrices in () and optimize with respect to U, V, then the resulting optimized matrices U, V can also be taken to be positive definite and diagonal. The proof uses reduction and a property of bipartite matching polytope. Conjecture: the optimal solutions of () are always diagonal. Imply the power-loaded OFDM is optimal. 37

39 Diagonal Designs Restrict to diagonal designs (Ui, Vi diagonal) The formulation () reduces to a geometric program: minimize subject to n ( v (i)v (i) v(i)v i= n i= h (i)v (i) p, (i) ) n ρ (v(i) v(i)) i= n h (i)v (i) p, i= vj(i) 0, j =,, i =,..., n. () () can be turned into a convex program by using the standard logarithmic transformation. The dual of () is a linearly constrained entropy maximization problem. 38

40 The Dual Program maximize 6n i= δi log δi n i= ciδ4ni λ log λ λ log λ subject to δi δi ρ δni δ4ni = 0, i n, δi δi ρ δni δ5ni = 0, i n, λ = 4n i= 5n i=4n δi, λ = 6n i=5n δi, δi =, δi 0, i 6n, where the coefficients ci are defined as { log ( ph ci = (i)), i n, log ( ph (i)), n i n. 39

41 Summary So far we have Presented various SDP/SOC formulations and algorithms for the optimal transceiver design problems Studied the properties of the optimal transceiver designs. Demonstrated the potential of SDP/SOC/interior point methods in digital communication. Results provide valuable guidelines and insights for the practical system design. Future work Incorporating QoS and other receiver structures in the formulation. Extension to the multi-user downlink case. 40

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