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Full-duplex MIMO: Spatial Processing and Characterization Alexios Balatsoukas-Stimming, Pavle Belanovic, Konstantinos Alexandris, Raffael Hochreutener, Andreas Burg Telecommunications Circuits Laboratory (TCL) cole Polytechnique Fdrale de

  1. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Cancelation results 20 MHz BW, 4 dBm transmit power 20 0 • Passive analog: -18 dB − 20 • Active analog PowerpgdBm3 − 40 Linear: -37 dB − 60 Red. phase noise: -11 dB − 80 • Total: -67 dB Txpg4dBm3 Dig.pSup.pg − 63dBm3 Rxpg − 14dBm3 − 100 RFpSup.pgsharedpref.3pg − 51dBm3 NoisepFloorpg − 85dBm3 RFpSup.pgsharedposc.3pg − 62dBm3 − 120 − 10 − 5 0 5 10 FrequencypgMHz3 • Residual power: -63 dBm 12/31

  2. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Full-Duplex MIMO • No cancellation: y = Hx + H t x t + n r ��� ��� 13/31

  3. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Full-Duplex MIMO • No cancellation: y = Hx + H t x t + n r • Cancellation signal x c : ��� ��� H c x c = − H t x t 13/31

  4. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Full-Duplex MIMO • No cancellation: y = Hx + H t x t + n r • Cancellation signal x c : ��� ��� H c x c = − H t x t • In practice, transmitted signals are affected by non-idealities : ˜ x t = x t + n t , x c = x c + n c ˜ 13/31

  5. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Full-Duplex MIMO • No cancellation: y = Hx + H t x t + n r • Cancellation signal x c : ��� ��� H c x c = − H t x t • In practice, transmitted signals are affected by non-idealities : ˜ x t = x t + n t , x c = x c + n c ˜ • Cancellation under transmit impairments: y = Hx + H t ˜ x t + H c ˜ x c + n r = Hx + H t n t + H c n c + n r 13/31

  6. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Full-Duplex MIMO • No cancellation: y = Hx + H t x t + n r • Cancellation signal x c : ��� ��� H c x c = − H t x t • In practice, transmitted signals are affected by non-idealities : ˜ x t = x t + n t , x c = x c + n c ˜ • Cancellation under transmit impairments: y = Hx + H t ˜ x t + H c ˜ x c + n r = Hx + H t n t + H c n c + n r Effective noise: n eff � H t n t + H c n c + n r 13/31

  7. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization 2 × 2 Full-Duplex MIMO Testbed hardware • We wish to characterize the effective noise n eff to: 1 Better understand transmit impairments → improved cancellation 14/31

  8. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization 2 × 2 Full-Duplex MIMO Testbed hardware • We wish to characterize the effective noise n eff to: 1 Better understand transmit impairments → improved cancellation 2 Assess whether n eff follows usual assumptions → better receivers 14/31

  9. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization 2 × 2 Full-Duplex MIMO Testbed hardware • We wish to characterize the effective noise n eff to: 1 Better understand transmit impairments → improved cancellation 2 Assess whether n eff follows usual assumptions → better receivers • National Instruments PXIe-1082 4 × NI 5791R RF transceivers Circulator-based anntena front-end 14/31

  10. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization 2 × 2 Full-Duplex MIMO Testbed hardware • We wish to characterize the effective noise n eff to: 1 Better understand transmit impairments → improved cancellation 2 Assess whether n eff follows usual assumptions → better receivers • National Instruments PXIe-1082 4 × NI 5791R RF transceivers Circulator-based anntena front-end • 1 × Desktop PC Runs Windows with LabVIEW 14/31

  11. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power 15/31

  12. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing 15/31

  13. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation 15/31

  14. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols 15/31

  15. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) 15/31

  16. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error 15/31

  17. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N 15/31

  18. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N • Statistical metrics used for effective noise characterization: 15/31

  19. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N • Statistical metrics used for effective noise characterization: 1 Autocorrelation per receiver (to assess memory ) 15/31

  20. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N • Statistical metrics used for effective noise characterization: 1 Autocorrelation per receiver (to assess memory ) 2 Pseudo-variance and correlation between real and imaginary parts (to assess circularity ) 15/31

  21. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N • Statistical metrics used for effective noise characterization: 1 Autocorrelation per receiver (to assess memory ) 2 Pseudo-variance and correlation between real and imaginary parts (to assess circularity ) 3 Histograms (to assess distribution ) 15/31

  22. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Measurement setup • 2.45 GHz carrier, 0 dBm transmit power • 15 cm antenna spacing • 10 MHz bandwidth, 256 OFDM carriers, QPSK modulation • N f = 100 OFDM frames consisting of 40 OFDM symbols • Remote signal x is absent (Rx at max. sensitivity) • Channel estimation is performed with a “very long” aperiodic sequence to minimize error • Residual noise recorded in 2 × N matrix N • Statistical metrics used for effective noise characterization: 1 Autocorrelation per receiver (to assess memory ) 2 Pseudo-variance and correlation between real and imaginary parts (to assess circularity ) 3 Histograms (to assess distribution ) 4 Spatial covariance matrix (to assess spatial correlation ) 15/31

  23. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Autocorrelation • The autocorrelation of each element of n eff is estimated as  � N − j − 1 N i , j + k N ∗ j ≥ 0 , i , k ,  ˆ k = 0 R i , j = ˆ R ∗ i , − j , j < 0 ,  16/31

  24. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Autocorrelation • The autocorrelation of each element of n eff is estimated as  � N − j − 1 N i , j + k N ∗ j ≥ 0 , i , k ,  ˆ k = 0 R i , j = ˆ R ∗ i , − j , j < 0 ,  1 0.9 0.8 0.7 Autocorrelation 0.6 0.5 0.4 0.3 0.2 0.1 0 −100 −50 0 50 100 Lags Time domain Frequency domain • Time domain: n eff has non-negligible memory 16/31

  25. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Autocorrelation • The autocorrelation of each element of n eff is estimated as  � N − j − 1 N i , j + k N ∗ j ≥ 0 , i , k ,  ˆ k = 0 R i , j = ˆ R ∗ i , − j , j < 0 ,  1 1 0.9 0.9 0.8 0.8 0.7 0.7 Autocorrelation Autocorrelation 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −100 −50 0 50 100 −100 −50 0 50 100 Lags Lags Time domain Frequency domain • Time domain: n eff has non-negligible memory • Frequency domain: n eff is practically memoryless 16/31

  26. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Pseudo-variance • For each chain i , the pseudo-variance is defined as: � � τ 2 i � E n 2 eff , i 17/31

  27. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Pseudo-variance • For each chain i , the pseudo-variance is defined as: � � τ 2 i � E n 2 eff , i • A smaller pseudo-variance indicates a more circular random variable 17/31

  28. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Pseudo-variance • For each chain i , the pseudo-variance is defined as: � � τ 2 i � E n 2 eff , i • A smaller pseudo-variance indicates a more circular random variable • We empirically estimate τ 2 i as N i = 1 τ 2 � N 2 ˆ i , j N j = 1 17/31

  29. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Pseudo-variance • For each chain i , the pseudo-variance is defined as: � � τ 2 i � E n 2 eff , i • A smaller pseudo-variance indicates a more circular random variable • We empirically estimate τ 2 i as N i = 1 τ 2 � N 2 ˆ i , j N j = 1 τ 2 1 | ≈ 10 − 3 • Time domain: | ˆ 1 | ≈ 10 − 5 → more circular τ 2 • Frequency domain: | ˆ 17/31

  30. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Histograms • Joint histogram of R ( N 1 , j ) and I ( N 1 , j ) Time domain Frequency domain • Time domain: R ( N 1 , j ) and I ( N 1 , j ) are strongly correlated 18/31

  31. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Histograms • Joint histogram of R ( N 1 , j ) and I ( N 1 , j ) Time domain Frequency domain • Time domain: R ( N 1 , j ) and I ( N 1 , j ) are strongly correlated • Freq. domain: R ( N 1 , j ) and I ( N 1 , j ) are practically uncorrelated 18/31

  32. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Histograms • Histogram of R ( N 1 , j ) 12 Histogram t Location−Scale Normal 10 8 Density 6 4 2 0 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Data Time domain Frequency domain • Time domain: Not Gaussian (Student’s t-distribution is good fit) 19/31

  33. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Histograms • Histogram of R ( N 1 , j ) 12 10 Histogram Histogram t Location−Scale 9 t Location−Scale Normal Normal 10 8 7 8 6 Density Density 6 5 4 4 3 2 2 1 0 0 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.2 −0.1 0 0.1 0.2 Data Data Time domain Frequency domain • Time domain: Not Gaussian (Student’s t-distribution is good fit) • Frequency domain: Gaussian (central limit theorem) 19/31

  34. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Spatial covariance matrices � ( n eff − E [ n eff ]) ( n eff − E [ n eff ]) H � • Spatial covariance matrix: K � E Measurements are specific to our setup. However, the variance of ˆ K time over time and ˆ K freq over the frequency tones is small compared to the magnitude of the entries. 20/31

  35. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Spatial covariance matrices � ( n eff − E [ n eff ]) ( n eff − E [ n eff ]) H � • Spatial covariance matrix: K � E • We empirically estimate K as ˆ N ( N − m ) ( N − m ) H K = 1 � N • m i = 1 k = 1 N i , k , i = 1 , 2 , is the ML estimate of E [ n eff ] N Measurements are specific to our setup. However, the variance of ˆ K time over time and ˆ K freq over the frequency tones is small compared to the magnitude of the entries. 20/31

  36. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Spatial covariance matrices � ( n eff − E [ n eff ]) ( n eff − E [ n eff ]) H � • Spatial covariance matrix: K � E • We empirically estimate K as ˆ N ( N − m ) ( N − m ) H K = 1 � N • m i = 1 k = 1 N i , k , i = 1 , 2 , is the ML estimate of E [ n eff ] N • Time domain: � � 0 . 0067 − 0 . 0013 − 0 . 0031 i ˆ K time = − 0 . 0013 + 0 . 0031 i 0 . 0053 Measurements are specific to our setup. However, the variance of ˆ K time over time and ˆ K freq over the frequency tones is small compared to the magnitude of the entries. 20/31

  37. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Spatial covariance matrices � ( n eff − E [ n eff ]) ( n eff − E [ n eff ]) H � • Spatial covariance matrix: K � E • We empirically estimate K as ˆ N ( N − m ) ( N − m ) H K = 1 � N • m i = 1 k = 1 N i , k , i = 1 , 2 , is the ML estimate of E [ n eff ] N • Time domain: � � 0 . 0067 − 0 . 0013 − 0 . 0031 i ˆ K time = − 0 . 0013 + 0 . 0031 i 0 . 0053 • Frequency domain: � � − 0 . 0013 − 0 . 0039 i 0 . 0070 ˆ K freq = − 0 . 0013 + 0 . 0039 i 0 . 0057 Measurements are specific to our setup. However, the variance of ˆ K time over time and ˆ K freq over the frequency tones is small compared to the magnitude of the entries. 20/31

  38. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Spatial covariance matrices � ( n eff − E [ n eff ]) ( n eff − E [ n eff ]) H � • Spatial covariance matrix: K � E • We empirically estimate K as ˆ N ( N − m ) ( N − m ) H K = 1 � N • m i = 1 k = 1 N i , k , i = 1 , 2 , is the ML estimate of E [ n eff ] N • Time domain: � � 0 . 0067 − 0 . 0013 − 0 . 0031 i ˆ K time = − 0 . 0013 + 0 . 0031 i 0 . 0053 • Frequency domain: � � − 0 . 0013 − 0 . 0039 i 0 . 0070 ˆ K freq = − 0 . 0013 + 0 . 0039 i 0 . 0057 • Spatial correlation remains in frequency domain Measurements are specific to our setup. However, the variance of ˆ K time over time and ˆ K freq over the frequency tones is small compared to the magnitude of the entries. 20/31

  39. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless 21/31

  40. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless ✗ Not Gaussian 21/31

  41. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless ✗ Not Gaussian ✗ Not circular symmetric 21/31

  42. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless ✗ Not Gaussian ✗ Not circular symmetric ✗ Spatially colored 21/31

  43. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored 21/31

  44. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored • Frequency domain: ✓ Memoryless 21/31

  45. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored • Frequency domain: ✓ Memoryless ✓ Gaussian 21/31

  46. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored • Frequency domain: ✓ Memoryless ✓ Gaussian ✓ Circular symmetric 21/31

  47. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored • Frequency domain: ✓ Memoryless ✓ Gaussian ✓ Circular symmetric ✗ Spatially colored 21/31

  48. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Summary of residual noise properties • Time domain: ✗ Not memoryless Traditional receiver ✗ Not Gaussian ✗ Not circular symmetric assumptions do not hold ✗ Spatially colored • Frequency domain: ✓ Memoryless ✓ Gaussian ✓ Circular symmetric ✗ Spatially colored OFDM: Need to study and undo effects of colored noise 21/31

  49. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Impact of colored noise on ZF and ML receivers x ZF = D � H − 1 y • Zero-forcing (ZF) receiver: ˆ � 22/31

  50. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Impact of colored noise on ZF and ML receivers x ZF = D � H − 1 y • Zero-forcing (ZF) receiver: ˆ � x ML = arg min x ∈O M � y − Hx � • Maximum-likelihood (ML) receiver: ˆ 22/31

  51. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Impact of colored noise on ZF and ML receivers x ZF = D � H − 1 y • Zero-forcing (ZF) receiver: ˆ � x ML = arg min x ∈O M � y − Hx � • Maximum-likelihood (ML) receiver: ˆ 0 10 −1 10 −2 10 FER −3 10 ZF −4 ML 10 ZF (w. self−interference) ML (w. self−interference) −5 10 0 5 10 15 20 25 30 SNR (dB) 22/31

  52. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Impact of colored noise on ZF and ML receivers x ZF = D � H − 1 y • Zero-forcing (ZF) receiver: ˆ � x ML = arg min x ∈O M � y − Hx � • Maximum-likelihood (ML) receiver: ˆ 0 10 −1 10 −2 10 FER −3 10 ZF −4 ML 10 ZF (w. self−interference) ML (w. self−interference) −5 10 0 5 10 15 20 25 30 SNR (dB) Colored noise → ∼ 3 dB worse performance 22/31

  53. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Noise whitening • Whitening filter: W = K − 1 / 2 23/31

  54. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Noise whitening • Whitening filter: W = K − 1 / 2 x ZF = D ( H − 1 W − 1 Wy ) = D ( H − 1 y ) ZF receiver: ˆ 23/31

  55. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Noise whitening • Whitening filter: W = K − 1 / 2 x ZF = D ( H − 1 W − 1 Wy ) = D ( H − 1 y ) ZF receiver: ˆ x ML = arg min x ∈O M � Wy − WHx � ML receiver: ˆ 23/31

  56. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Noise whitening • Whitening filter: W = K − 1 / 2 x ZF = D ( H − 1 W − 1 Wy ) = D ( H − 1 y ) ZF receiver: ˆ x ML = arg min x ∈O M � Wy − WHx � ML receiver: ˆ 0 10 −1 10 −2 10 FER −3 ZF 10 ML ZF (w. self−interference) ML (w. self−interference) −4 10 ZF (w. self−interference whitening) ML (w. self−interference whitening) −5 10 0 5 10 15 20 25 30 SNR (dB) 23/31

  57. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Noise whitening • Whitening filter: W = K − 1 / 2 x ZF = D ( H − 1 W − 1 Wy ) = D ( H − 1 y ) ZF receiver: ˆ x ML = arg min x ∈O M � Wy − WHx � ML receiver: ˆ 0 10 −1 10 −2 10 FER −3 ZF 10 ML ZF (w. self−interference) ML (w. self−interference) −4 10 ZF (w. self−interference whitening) ML (w. self−interference whitening) −5 10 0 5 10 15 20 25 30 SNR (dB) ML: Noise whitening → ∼ 1 dB reclaimed 23/31

  58. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Estimation of covariance matrix • Whitening filter requires knowledge of covariance matrix K 24/31

  59. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Estimation of covariance matrix • Whitening filter requires knowledge of covariance matrix K • K can be estimated in training phase We have observed that K does not vary significantly with low mobility 24/31

  60. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Estimation of covariance matrix • Whitening filter requires knowledge of covariance matrix K • K can be estimated in training phase We have observed that K does not vary significantly with low mobility • Since the setup is highly static, we can attempt to build a model to predict K No need to estimate K Possibility of optimizing the setup to reduce coloring 24/31

  61. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Setup • Two RF chains, antenna distance d 25/31

  62. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Setup • Two RF chains, antenna distance d • Cancellation channel: � � h CX 1 , RX 1 0 H c = h CX 2 , RX 2 0 25/31

  63. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Setup • Two RF chains, antenna distance d • Cancellation channel: � � h CX 1 , RX 1 0 H c = h CX 2 , RX 2 0 • Self-interference channel � � h TX 1 , RX 1 h TX 1 , RX 2 H t = h TX 1 , RX 2 h TX 2 , RX 2 25/31

  64. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Setup • Two RF chains, antenna distance d • Cancellation channel: � � h CX 1 , RX 1 0 H c = h CX 2 , RX 2 0 • Self-interference channel � � h TX 1 , RX 1 h TX 1 , RX 2 H t = h TX 1 , RX 2 h TX 2 , RX 2 25/31

  65. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Assumption: Single frequency f c 26/31

  66. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Assumption: Single frequency f c • Cancellation channel H c is constant → modeled as constant gain α and constant phase φ α : � � α e j φ α 0 H c = α e j φ α 0 26/31

  67. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Assumption: Single frequency f c • Cancellation channel H c is constant → modeled as constant gain α and constant phase φ α : � � α e j φ α 0 H c = α e j φ α 0 • Self-interference channel from transmitter i to receiver i is constant → modeled as constant gain β and constant phase φ β : � � β e j φ β ? H t = β e j φ β ? 26/31

  68. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Self-interference channel from transmitter i to receiver j → wireless channel of distance d 27/31

  69. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Self-interference channel from transmitter i to receiver j → wireless channel of distance d • Can be modeled as gain γ ( d ) and phase φ γ ( d ) : � λ � 2 φ γ ( d ) = 2 π d γ ( d ) = 4 π d λ 27/31

  70. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Self-interference channel from transmitter i to receiver j → wireless channel of distance d • Can be modeled as gain γ ( d ) and phase φ γ ( d ) : � λ � 2 φ γ ( d ) = 2 π d γ ( d ) = 4 π d λ • Model for self-interference channel: � � β e j φ β γ ( d ) e j φ γ ( d ) H t = γ ( d ) e j φ γ ( d ) β e j φ β 27/31

  71. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Recall that: n eff � H t n t + H c n c + n r 28/31

  72. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Recall that: n eff � H t n t + H c n c + n r • Assume that n t , n c , n r are independent and K n t = K n c = I and K n r = σ 2 I 28/31

  73. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model • Recall that: n eff � H t n t + H c n c + n r • Assume that n t , n c , n r are independent and K n t = K n c = I and K n r = σ 2 I • Then, we get � � A ( d ) B ( d ) K y ( d ) = , B ( d ) A ( d ) where A ( d ) = α 2 + β 2 + γ ( d ) 2 + σ 2 and e j ( φ γ ( d ) − φ β ) + e − j ( φ γ ( d ) − φ β ) � � B ( d ) = βγ ( d ) 28/31

  74. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Avoiding colored noise • Optimal distance d ∗ to minimize off-diagonal elements (i.e., minimize spatial correlation): d ∗ = arg min e j ( φ γ ( d ) − φ β ) + e − j ( φ γ ( d ) − φ β ) � � γ ( d ) d 29/31

  75. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Avoiding colored noise • Optimal distance d ∗ to minimize off-diagonal elements (i.e., minimize spatial correlation): d ∗ = arg min e j ( φ γ ( d ) − φ β ) + e − j ( φ γ ( d ) − φ β ) � � γ ( d ) d • Using Euler’s formula, we get: � 2 k + 1 − φ β � d ∗ = k ∈ Z , λ, 4 2 π which gives B ( d ∗ ) = 0 ! 29/31

  76. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Avoiding colored noise • Optimal distance d ∗ to minimize off-diagonal elements (i.e., minimize spatial correlation): d ∗ = arg min e j ( φ γ ( d ) − φ β ) + e − j ( φ γ ( d ) − φ β ) � � γ ( d ) d • Using Euler’s formula, we get: � 2 k + 1 − φ β � d ∗ = k ∈ Z , λ, 4 2 π which gives B ( d ∗ ) = 0 ! • Suitably chosen antenna spacing eliminates coloring 29/31

  77. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model verification • Carrier frequency: 2.40 GHz , signal bandwidth: 10 KHz 30/31

  78. Introduction Full-Duplex MIMO Full-Duplex MIMO Testbed Self-interference Characterization Covariance matrix model verification • Carrier frequency: 2.40 GHz , signal bandwidth: 10 KHz −4 2 x 10 Covariance between y(1) and y(2) 1 0 6 8 10 12 14 16 18 Antenna distance (cm) Measurements Model 30/31

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