Minimum BER Linear Transceivers for Block. Communication Systems. Lecturer: Tom Luo

Minimum BER Linear Transceivers for Block Communication Systems Lecturer: Tom Luo

Outline Block-by-block communication Abstract model Applications Current design techniques Minimum BER precoders for zero-forcing equalization Average BER and convexity Analytic solution The Message: Minimum BER precoders are MMSE precoders with a special choice of the unitary matrix degree of freedom Performance Analysis: SNR gains of several db Extensions 1

PSfrag replacements Block-by-Block Communication v(n) s(n) ŝ(n) x(n) r(n) Mod Detect H + a vector memoryless system r(n) = Hx(n) + v(n) Applications block transmission over ISI channels (OFDM, DMT) multiple antenna flat fading channels multiple antenna frequency-selective channels 2

3 ZP: achieves maximum diversity without coding; no prefix CP: channel indep. diagonalization via DFT; simple equalization Relative advantages: H is Toeplitz (and full rank) if E and C based on zero-padding H is circulant if E and C are based on cyclic-prefix extensions H 1:M Mod E h(n) + C P:1 1:P Detect M:1 s(n) v(n) ŝ(n) Stanford University EE392o Z.Q. Luo Block transmission over ISI channels

g replacements Multiple antenna flat fading channels v(1),..., v(n) s(1),..., s(n) ŝ(1),..., ŝ(n) x(1),..., x(n) r(1),..., r(n) Mod H (n) Detect + Flat channels: r(n) = H (n)x(n) + v(n) Space-time coding based on accumulating s(n) and r(n) as columns If H is constant, R = H X + V Take vec s vec(r) = (I H ) vec(x) + vec(v ) Hence H = I H 4

PSfrag replacements How should we communicate? v(n) s(n) ŝ(n) x(n) r(n) Mod Detect H + Depends on Structure of the transmitter and receiver in single-user and multiple access transmitter is usually linear, x(n) = F s(n) structure of the receiver sometimes fixed Accuracy of channel knowledge at the transmitter and receiver Design objective rate maximization; performance maximization; power minimization 5

P:1 1:P M:1 y Stanford University EE392o Z.Q. Luo Today s focus s x r F H + G v s ŝ Linear transmission Receiver: linear equalizer with elementwise detection Accurate channel knowledge at both ends Performance orientated design Uniform constellation assignment in s 6

P:1 1:P M:1 y Stanford University EE392o Z.Q. Luo Current performance orientated designs s x r F H + G v s ŝ Typically, maximize performance subject to power bound for uniform constellation Minimize MSE (arithmetic mean of elementwise MSEs) Minimize MSE subject to zero forcing equalization Maximize total signal power to total noise power subject to zero forcing Maximize arithmetic mean of elementwise SINRs Maximize geometric mean of SINRs 7

Outline Block-by-block communication Abstract model Applications Current design techniques Minimum BER precoders for zero-forcing equalization Average BER and convexity Analytic solution The Message: Minimum BER precoders are MMSE precoders with a special choice of the unitary matrix degree of freedom Simulation results: SNR gains of several db Extensions 8

P:1 1:P M:1 y Stanford University EE392o Z.Q. Luo Analysis s x r F H + G v s ŝ s = GHF s + Gv Choose length(s) rank(h) Choose G to be a zero-forcing equalizer: G = (HF ). Hence, s = s + (HF ) v Design problem: Find F that minimizes BER 9

Average bit error rate Pe: average bit error rate over all possible transmitted vectors. Pe = E{P e s } For BPSK/QPSK signals, Pe can be expressed as Pe = 1 M 2 M M Ps j j=1 m=1 mp m sj, Zero-forcing equalizer = s = s + Gv. Hence ( erfc Pe = 1 2M m ) 1 2σ 2 [GG H ]mm where E{vv H } = σ 2 I, [GG H ]ii is the ith diagonal entry of GG H, M is the block size. 10

Key Observation: Convexity If φ(x) = erfc ( ) 1, for x > 0, then 2σ 2 x d 2 φ dx = 1 ( (2σ 2 ) 1 2 2 exp 1 )( π 2σ 2 x 3 2 + 1 2σ 2 x ) x 5 2. Hence, if x < 1 3σ 2, then 2 f(x) x 2 > 0. Therefore, if 1 > 3 σ 2 [GG H ]mm then Pe is a convex function of [GG H ]mm 11

Design of the Minimum BER Precoder Our goal: minimize Pe F subject to trace(f F H ) p0 In the region that Pe is convex, applying Jensen s inequality, we have Pe = 1 erfc 2M m ( ) 1 2σ 2 [GG H ]mm 1 2 erfc ( = 1 2 erfc ( 1 2σ 2 M M m=1 [GGH ]mm ) 1 2σ 2 ( trace(ggh ) M ) ) Pe,LB equality holds if [GG H ]mm are equal, m [1, M]. 12

Design of the Minimum BER Precoder II Pe,LB defines a lower bound on Pe; minimum BER precoders can be designed in two stages. Stage 1: Minimize Pe,LB subject to the power constraint and the convex condition. minimize F Pe,LB subject to trace(f F H ) p0 [GG H ]mm < 1 3σ 2, m [1, M] Stage 2: Show that a particular solution for Stage 1 achieves the minimized lower bound. 13

Solving Stage 1 Parameterize via SVD: F = U [ Φ 0 ] V ; H = Q [ Σ 0 ] W If number columns of F rank(h), then zero forcing equalizers exist In those cases GG H = V H Φ 1 Z(U)Φ 1 V, where Z(U) = [ I M 0 ] U H W Σ 2 W H U [ I M0 ] Recall that Pe,LB = 1 2 erfc ( (2σ 2 trace(gg H )/M ) 1/2 ) Exploit: monotonicity of erfc( ); Hence, minimizing Pe,LB is equivalent to minimizing trace ( Φ 2 Z(U) ) 14

Reformulation Minimizing Pe,LB subject to power and validity constraints is equivalent to minimize U,Φ,V trace ( Φ 2 Z(U) ) subject to trace(φ 2 ) p0 [ V H Φ 1 Z(U)Φ 1 V ] mm < 1 3σ 2, m [1, M] Awkward due to the last constraint Lemma: For symmetric A 0, with A = ΨΓΨ H, min V V H =I max M m [V H AV ]mm = trace(a) and a minimizing V = ΨD, where D is the DFT matrix. 15

Remaining Problem To complete minimization of lower bound we must solve: minimize trace ( Φ 2 Z(U) ) U,Φ subject to trace(φ 2 ) p0 (2) Solution is the MMSE precoder for zero-forcing equalization p 0 U = W ; Φ = ΦMMSE-ZF = trace(σ 1 ) Σ 1/2 where, for simplicity we assumed that length(s) = rank(h). Hence, if trace ( Φ 2 MMSE-ZF Z(W )) > M/(3σ 2 ) then Fmin, LB = W ΦMMSE-ZFD otherwise lower bound is not valid 16

Minimum BER Precoder We now have that Fmin, LB = W ΦMMSE-ZFD Does the Pe for this F achieve the lower bound that it minimizes? Yes! All [GG H ]mm are equal! That is FMBER = W ΦMMSE-ZFD Interpretation: The set of all MMSE precoders for zero-forcing equalization FMMSE = W ΦMMSE-ZFV for an arbitrary unitary V. There are other choices of V which result in minimum BER All that is needed is [V ] ij = 1/ M 17

r x s s ŝ + Stanford University EE392o Z.Q. Luo Interpretation II For simplicity assume that H and F, and hence G are square Recall: G = D H ΨQ H ; H = QΣW H ; FMBER = W ΦMMSE-ZFD Hence, GHF = V H ΨΣΦV ; ṽ = Q H v ṽ1 ṽm ṽm s [ΣΦ]11 + V [ΣΦ]mm + [Ψ]mm V H [ΣΦ]M M + [Ψ]11 [Ψ]MM s Optimality of structure which synthesizes parallel sub-channels is typical However, to achieve minimum BER we must linearly combine these sub-channels 18

Interpretation III Recall: G = D H ΨQ H ; H = QΣW H ; FMBER = W ΦMMSE-ZFD ΦMMSE-ZF Σ 1/2 ; Ψ = (ΣΦMMSE-ZF) 1 Σ 1/2 Hence, equalization effort balanced between transmitter and receiver Transmitter allocates power to compensate for poor sub-channels Decision point SNR: 1 1 SNRm = 2σ 2 [GG H ]mm [DΣ 1 D H ]mm Choice of V = D makes all SNRm equal Choice of Φ = ΦMMSE-ZF makes this SNR level as small as possible 19

Sub-channel dropping scheme We define block SNR ρ p0/(p σ 2 ). Our analytic MBER precoder is valid for moderate-to-high SNR, ρ 3( trace(σ 1 ) ) 2 P M ρc, This condition can be tested before the minimum BER precoder is assembled This condition can be ensured by increasing transmitting power dropping the lower-gain sub-channels, reallocating transmission power on the surviving channels. 20

Minimum BER precoders for CP Systems Applying our optimum design FMBER to the CP system, we have FCP-MBER = D H MMSE-ZFD where MMSE-ZF is the MMSE power loading matrix for ZF equalization. The CP-MBER precoder is related to standard DMT in that Water-filling power loading is replaced by MMSE power loading the sub-channels are linearly combined via a second DFT matrix 21

22 r x s s ŝ + 5 10 15 20 25 30 Index of the subchannel m 0 H(e j2π(m 1)/M ) 0.5 1 P:1 1:P M:1 y 1.5 Index of the channel taps k 0 0 0.5 1 1.5 2 2.5 3 0.1 0.2 h(k) 0.3 0.4 0.5 0.6 0.7 (M, L, P ) = (32, 3, 35); ρ = p0/(p σ 2 ) F G H v 1:M PSfrag replacements Stanford University EE392o Z.Q. Luo Performance Analysis: One channel

23 FZP-OFDM D; SNR gain > 5 db @ BER=10-4 P:1 1:P M:1 y r x s s ŝ + 0 5 10 15 20 25 Block SNR, db 10 4 10 3 BER P:1 1:P M:1 y r x s s ŝ + 0 5 10 15 20 25 Block SNR, db 10 4 10 3 BER 10 2 10 2 H v 1:M 10 1 MBER ZF ZP OFDM MMSE ZF I Max SNR ZF H v 1:M 10 1 MBER ZF ZP OFDM MMSE ZF I Max SNR ZF ZP I F ŝ G + F ŝ G 0 5 10 15 20 25 + 10 0 Block SNR, db 0 5 10 15 20 25 10 0 Block SNR, db PSfrag replacements P:1 1:P M:1 y r x s s PSfrag replacements 10 4 10 4 10 3 10 3 BER BER 10 2 P:1 1:P M:1 y r x s s 10 2 F G H v 1:M 10 1 10 1 MBER ZF ZP OFDM MBER ZF ZP OFDM MMSE ZF I 10 0 F G H v 1:M 10 0 PSfrag replacements PSfrag replacements ZP: BER, one channel Stanford University EE392o Z.Q. Luo

24 0 5 10 15 20 25 Block SNR, db P:1 1:P M:1 y r x s s ŝ + CP OFDM MMSE ZF power loading CP MBER 10 4 BER (CP) 10 3 10 2 10 1 10 0 CP: BER, one channel Stanford University EE392o Z.Q. Luo F G H v 1:M PSfrag replacements

25 True SNR gain of ZP is larger, as no power wasted sending cyclic prefix r x s s ŝ + SNR in db 10 5 0 5 10 15 20 25 ZP MBER ZP MBER DROP CP MBER CP MBER DROP Waterfilling DMT 10 4 P:1 1:P M:1 y 10 3 BER 10 2 F G H v 1:M 10 1 10 0 ZP and CP BER M=32 L=3 P=35 PSfrag replacements ZP vs CP, one channel Stanford University EE392o Z.Q. Luo

BER analysis: average over channels Randomly generated length 5 FIR channels i.i.d. zero-mean circular complex Gaussian taps impulse response normalized M = 16, P = 20 Average BER over 2000 realizations 26

27 r x s s ŝ + 0 5 10 15 20 25 Block SNR, db ZP MBER ZP MMSE I ZP OFDM ZP MSNR 10 4 P:1 1:P M:1 y 10 3 BER 10 2 F G H v 1:M 10 1 10 0 ZP: BER, average over channels PSfrag replacements Stanford University EE392o Z.Q. Luo

28 r x s s ŝ + 0 5 10 15 20 25 Block SNR, db CP OFDM MMSE ZF power loading CP MBER 10 4 P:1 1:P M:1 y 10 3 BER 10 2 F G H v 1:M 10 1 10 0 CP: BER, average over channels PSfrag replacements Stanford University EE392o Z.Q. Luo

Outline Block-by-block communication Abstract model Applications Current design techniques Minimum BER precoders for zero-forcing equalization Average BER and convexity Analytic solution The Message: Minimum BER precoders are MMSE precoders with a special choice of the unitary matrix degree of freedom Performance Analysis: SNR gains of several db Extensions 29

Simple extensions Higher-order QAM constellations When Gray labelled, BER dominated by nearest and next to nearest neighbour errors Our minimum BER precoder minimizes both simultaneously Coloured noise: Current design implicitly depends on eigen structure of HH H Coloured noise case depends on eigen structure of HR 1 vv HH 30

Linear MMSE equalization Linear MMSE equalizer: G = F H H H ( Rvv + HF F H H H) 1 Now residual ISI in s: s = Diag(GHF )s + OffDiag(GHF )s + Gv However, as M gets large, ISI is almost surely Gaussian Algebra more difficult, but theme is the same: Minimum BER precoder is an MMSE precoder with the unitary degree of freedom V chosen so that [V ] ij = 1/ M. Performance gains of the same order 31

32 r x s s ŝ + Block SNR (db) 10 6 15 10 5 0 5 10 15 20 25 10 5 10 4 P:1 1:P M:1 y BER 10 3 10 2 10 1 ZP MBER ZP MMSE I ZP OFDM ZP I theoretical 10 0 BER (ave d): MMSE equalization, ZP Stanford University EE392o Z.Q. Luo F G H v 1:M PSfrag replacements

ZF/MMSE Decision Feedback Equalizers We have considered schemes with fixed detection order We have found a computationally-efficient algorithm for an optimal precoder However, F I is often close to being optimal, hence performance gains lower 33

Channel unknown at transmitter with perfect channel knowledge at receiver H is unknown, but structured F I is optimal for CP schemes with linear equalization, in sense of minimum BER (Lin and Phoong) for ZP schemes with ML detection, in sense of minimizing Chernoff bound on pairwise error probability (Giannakis + students) for ZP schemes with ML detection, in sense of minimizing worst-case averaged pairwise error probability (Zhang, Davidson, Wong) if H unknown and unstructured (space-time case) any unitary F is optimal in the PEP sense 34

Rate Adaptive Design Today s focus has been on performance objectives for uniform constellations Precoder tends to compensate for low-gain sub-channels In practice, channel knowledge at the transmitter allows joint constellation assignment and precoding (power loading) via water-filling Precoder tends to allocate power and bits to high-gain sub-channels However, achieving reliable performance at rates promised by water-filling requires ideal Gaussian codes In practice, constellations are assigned by rounding to a small set Once that is done one ought to optimize precoder for performance Several ad-hoc methods available. Reasonable performance Can also apply minimum BER technique to groups of sub-channels assigned the same constellation: Significant SNR gains 35

36 r x s s ŝ + 5 10 15 20 25 30 Block SNR, db P:1 1:P M:1 y r x s s ŝ + Block SNR, db 0 5 10 15 20 25 30 50 bits per block 100 P:1 1:P M:1 y 150 10 3 BER F G H v 1:M 200 250 F G H v 1:M PSfrag replacements WF Design A Design B 10 2 g replacements SNR Gap 8 db; square QAM; (M, L, P ) = (32, 4, 36); iid Gaussian channel taps with normalization; 50 realizations; Stanford University EE392o Z.Q. Luo Rate Adaptive Design II

Speculation: Implications for coded systems ṽ1 ṽm ṽm s [ΣΦ]11 + V [ΣΦ]mm + [Ψ]mm V H [ΣΦ]M M + [Ψ]11 [Ψ]MM s Our design minimizes BER of uncoded system What are implications for coded systems? Choice of V = D makes all sub-channel SNRs equal. Hence single code systems should perform well, at least for hard decision schemes Choice of V = D correlates outputs Hence elementwise detection and decoding is suboptimal; Complexity of optimal? For belief propagation decoders, correlation between sub-channels may create short loops in graphs, and may lead to inferior performance 37 1

Conclusion Analytic expression for minimum BER precoder for block transmission with uniform constellation and zero-forcing equalization. Valid at moderate to high SNRs Is a special MMSE precoder Possible extensions abound. Some completed, some being considered In particular, an adaption of the idea to rate-adaptive design provides considerable SNR gain 38