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Wideband Communications

Wideband Communications . Lecture 18-19: Multi-user detection Aliazam Abbasfar. Outline. Multi-user detection (MUD) Optimum detection De-correlator MMSE detector Nonlinear detector. Multi-user detection. Single user detection Require single signature waveform + timing

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Wideband Communications

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  1. Wideband Communications Lecture 18-19: Multi-user detection Aliazam Abbasfar

  2. Outline • Multi-user detection (MUD) • Optimum detection • De-correlator • MMSE detector • Nonlinear detector

  3. Multi-user detection • Single user detection • Require single signature waveform + timing • Single user matched filter • Optimum receiver when one correlation is available • Was believed to be close to optimum • The interference can be approx. as Gaussian RV • Multi-user detection • Multi-user matched filters • Improves the estimation when other correlations are available • y1 is not the sufficient statistics for b1 • y2, y3, …, yKalso have information about b1

  4. Two-user optimum MUD • y(t) = A1b1 s1(t) + A2b2 s2(t) + n(t) • y1 = A1b1 + A2b2r21+ n1 • y2 = A1b1r12+ A2b2 + n2 • y = R A b + n • Noises are correlated • E[n nT] = s2 R • MAP estimation : • Jointly optimum : max P{ (b1, b2)| y1, y2} • Individually optimum : max P{ b1 | y1, y2} • USUALLY gives the same answer • Not necessarily (counter example : P++=0.26, P+-=0.27, P-+=0.26, P--=0.21) • Usually there is only one pair with high posteriori probability • Which one to compute • Individually optimum : best BER for one user • Jointly optimum : Low complexity • ML estimation : • Jointly optimum : : max P{y1, y2|(b1, b2)} • Individually optimum : : max P{y1, y2| b1} • ML = MAP estimation when bits are equiprobable

  5. Two-user ML estimation • Cross-correlation matrix decomposition • Cholesky decomposition : R = LLH • y’ = L-1y = LH A b + L-1n = LH A b + n’ • y'1 = A1b1 + r A2b2+ n’1 • y'2 = lA2b2 + n’2 • New noises are uncorrelated • E[n’ n’T] = s2 I • Jointly optimum ML estimation : • The nearest point to one of 4 hypotheses • min [(y'1 – (A1b1 + A2b2r))2+(y'2 - lA2b2)2)] • min [ |L-1y - LH A b|2] • No near-far effect

  6. Signal space analysis • y = X1 s1 + X2 s2 + n • Orthonormal bases that span the signal • p1 = s1 • p2 = (s2 – s2Hs1 s1)/|s2 – s2Hs1 s1| • p2 = (s2 – r s1)/ l • projections on orthogonal bases that span the signal space • y'1 = p1H y = y1 • y'2 = p2H y = (y2 - r y1)/ l • Noises are uncorrelated • ML solution : nearest point to hypotheses • Any orthonormal set works

  7. Two-user ML estimation (2) • y’ = LH A b + n’ • y'1 = A1b1 + r A2b2+ n’1 • y'2 = lA2b2 + n’2 • E[n’ n’T] = s2 I • Individually optimum ML estimation : • Min[ P(y’|b1,+)+P(y’|b1,-) ] • min { exp[((y'1 – (A1b1 + A2r))2+(y'2 - lA2)2))/s2] + exp[((y'1 – (A1b1 - A2r))2+(y'2 + lA2)2)) /s2] } • More complex than jointly estimation • No near-far effect

  8. K-user • y’ = LH A b + n’ • E[n’ n’T] = s2 I • Jointly optimum ML estimation : • The nearest point to one of hypotheses • min { |L-1y - LH A b|2 } • max { 2bTAy - bTARA b } • Combinatorial optimization • Complexity : O(2K) • Really complex for long codes • Complexity is more for asynchronous CDMA

  9. Power trade-off regions • Optimum MUD Single user • BER = 3x10-5 • SNR = 12 dB • No penalty if r < 0.5

  10. De-correlator • y = R A b + n • y’ = R-1 y = Ab + R-1 n • Data bits are de-correlated • bk = sgn( y’k) • New noises are correlated • E[n’ n’T] = s2 R-1 • Advantages: • Ak ’s not needed • Can be de-centralized • y’k = pkH y • The best estimate when Ak’s not known • BER = Q( Ak/(sRkk) ) • No near-far problem • Error free with no noise

  11. Signal space analysis • y = X1 s1 + X2 s2 + … + XKsK + n • For each user find the a basis vector orthogonal to other users vector • Two user case : • p1 = s2 • p2 = s1 • Decision regions • Independent of A • Passes through origin • Noises are correlated • Noise enhancement

  12. Power trade-off regions • vs. Single user Optimum MUD • BER = 3x10-5 • SNR = 12 dB • Minimum total power when users have the same power

  13. MMSE detector • y = R A b + n • y’ = G y = G( RAb + n) • bk = sgn( y’k) • Minimize MSE : E[|y’-b|2] • G = (R- s2 A-2)-1 • Notes: • Ak ’s are needed • Can be de-centralized • y’k = pkH y • Performance • De-correlator in high SNR • Optimum in low SNR • No near-far problem • Error free with no noise

  14. Successive cancellation • y = R A b + n • y’ = G y = G( RAb + n) • bk = sgn( y’k) • Minimize MSE : E[|y’-b|2] • G = (R- s2 A-2)-1 • Notes: • Ak ’s are needed • Can be de-centralized • y’k = pkH y • Performance • De-correlator in high SNR • Optimum in low SNR • No near-far problem • Error free with no noise

  15. Successive cancellation • Detect one user at a time and cancel its interference • Popular order : decreasing received power • Best order : decreasing correlator output power • For two-user case • we detect user 2 first • Subtract from Rx signal • Near-far effect can happen

  16. Power trade-off regions • vs. Optimum MUD • User 2 is detected first • BER1,2 = 3x10-5 • r= 0.5  SNR1 = 12 dB, SNR2 = 15 dB • Same power : SNR1 = 18 dB, SNR2 = 18 dB

  17. Decision feedback detector • y’ = L-1y = LH A b + L-1n = LH A b + n’ • y'1 = A1b1 + r A2b2+ n’1 • y'2 = lA2b2 + n’2 • Detect b2 first, • Cancel b2 in y’1 , then detect b1 • BER  Q( AkLkk/s) • Lower signal amplitude • No near-far effect

  18. Reading • Verdu 4.1, 5.1, 5.5, 6.1, 6.2, 6.3, 7.1,7.5

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