sparse bayesian learning: a beamforming and toeplitz...
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![Page 1: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/1.jpg)
Sparse Bayesian Learning: A Beamforming andToeplitz Approximation Perspective
Bhaskar D RaoUniversity of California, San Diego
Email: [email protected]
![Page 2: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/2.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 3: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/3.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 4: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/4.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 5: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/5.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 6: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/6.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 7: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/7.jpg)
Some References
1. M. Tipping, Sparse Bayesian learning and the relevance vectormachine, J. Machine Learning Research, vol. 1, pp. 211-244, 2001.
2. David Wipf and Bhaskar D. Rao, Sparse Bayesian learning for basisselection. IEEE Transactions on Signal processing, 2004.
3. David Wipf and Bhaskar D. Rao,(2007), An empirical Bayesianstrategy for solving the simultaneous sparse approximation problem.IEEE Transactions on Signal Processing, 55(7), 3704-3716.
4. D. Wipf, B. Rao, S. Nagarajan, Latent Variable Bayesian Models forPromoting Sparsity, IEEE Trans. Info Theory, 2011.
5. D. Wipf and S. Nagarajan, Iterative Reweighted `1 and `2 Methodsfor Finding Sparse Solutions. IEEE Journal of Selected Topics inSignal Processing, 2010.
6. D. Wipf and S. Nagarajan,(2007, June), Beamforming using therelevance vector machine. In Proceedings of the 24th internationalconference on Machine learning. ACM.
7. M. Al-Shoukairi, M. and B. Rao, Sparse Signal Recovery UsingMPDR Estimation, 2019 ICASSP.
![Page 8: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/8.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 9: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/9.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 10: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/10.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 11: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/11.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 12: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/12.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 13: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/13.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 14: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/14.jpg)
Outline
I Background
I Beamforming and Minimum Power Distortionless Response(MPDR) beamforming
I Sparse Bayesian Learning (SBL)
I Connection between SBL and MPDR beamforming
I Uniform Linear Arrays and Toeplitz Matrix Approximation
I Nested Arrays and more sources than sensors
![Page 15: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/15.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 16: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/16.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 17: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/17.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 18: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/18.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA):
Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 19: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/19.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 20: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/20.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2
So array manifold can be denoted by V(ωl).
![Page 21: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/21.jpg)
Beamforming and MPDR in Array Processing
Narrow-band signal model for an array with N sensors.
y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1
V(ωc , kl)xl [n] + Z[n], n = 1, .., L
ωc is the carrier frequency, kl is the source direction, and V(ωc , k) is thearray manifold and is the response of the array to a source at direction k.
Far-Field model: V(ωc , k) = [1, e−jωcτ1(k), . . . , e−jωcτN−1(k)] and afunction of array geometry.
Uniform Linear Array (ULA): Sensors placed on a line with a unformspacing of d .
e−jωcτm(kl) = e−jωlm, with ωm = 2π dλ cos(θl) where θl is the elevation
angle. Common to use dλ = 1
2So array manifold can be denoted by V(ωl).
![Page 22: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/22.jpg)
Assumptions
y(n) =D−1∑l=0
V(ωl)xl [n] + Z[n]
= [V0,V1, . . . ,VD−1]
x0[n]x1[n]
...xD−1[n]
+ Z[n]
= Vx[n] + Z[n]
where V ∈ CN×D , and x[n] ∈ CD×1.
Assumptions: E (x[n]) = 0D×1, E (x[n]ZH [n]) = 0D×N and temporallyuncorrelated.
![Page 23: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/23.jpg)
Assumptions
y(n) =D−1∑l=0
V(ωl)xl [n] + Z[n]
= [V0,V1, . . . ,VD−1]
x0[n]x1[n]
...xD−1[n]
+ Z[n]
= Vx[n] + Z[n]
where V ∈ CN×D , and x[n] ∈ CD×1.
Assumptions: E (x[n]) = 0D×1, E (x[n]ZH [n]) = 0D×N and temporallyuncorrelated.
![Page 24: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/24.jpg)
Assumptions
y(n) =D−1∑l=0
V(ωl)xl [n] + Z[n]
= [V0,V1, . . . ,VD−1]
x0[n]x1[n]
...xD−1[n]
+ Z[n]
= Vx[n] + Z[n]
where V ∈ CN×D , and x[n] ∈ CD×1.
Assumptions: E (x[n]) = 0D×1, E (x[n]ZH [n]) = 0D×N and temporallyuncorrelated.
![Page 25: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/25.jpg)
Covariance Matrices of interest
Source Covariance matrix: Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 26: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/26.jpg)
Covariance Matrices of interest
Source Covariance matrix:
Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 27: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/27.jpg)
Covariance Matrices of interest
Source Covariance matrix: Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 28: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/28.jpg)
Covariance Matrices of interest
Source Covariance matrix: Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 29: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/29.jpg)
Covariance Matrices of interest
Source Covariance matrix: Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 30: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/30.jpg)
Covariance Matrices of interest
Source Covariance matrix: Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers
For uncorrelated sources Rx is a diagonal matrix, i.e.Rx = diag(p0, p1, . . . , pD−1).
Data Covariance Matrix
Ry = E (y[n]yH [n]) = Rs + σ2z I = VRxVH + σ2
z I
Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D
For ULA and uncorrelated sources, Ry is a Toeplitz matrix.
![Page 31: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/31.jpg)
Sparse Signal Recovery (SSR)
I y is a N × 1 measurement vector and x is M × 1 desired vectorwhich is sparse with k non zero entries.
I Φ is N ×M dictionary matrix where M >> N.
For array processing,Φ is obtained by employing a grid, i.e. lth column is V(ωl).
![Page 32: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/32.jpg)
Sparse Signal Recovery (SSR)
I y is a N × 1 measurement vector and x is M × 1 desired vectorwhich is sparse with k non zero entries.
I Φ is N ×M dictionary matrix where M >> N. For array processing,Φ is obtained by employing a grid, i.e. lth column is V(ωl).
![Page 33: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/33.jpg)
Multiple Measurement Vectors (MMV)
I Multiple measurements: L measurements
I Common Sparsity Profile: k nonzero rows
![Page 34: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/34.jpg)
Beamforming(BF)
Pick a direction of interest ωs , and select the linear combining weights Wsuch that r [n] = W Hy[n] contains mostly the signal from direction ωs
Conventional beamforming: W = V(ωs) and the power in direction ωs isE (|r [n]|2) = V(ωs)HRyV(ωs).
If you scan the spatial angles and look for the direction with most power,it has similarity to the search step of OMP.
BF with Null constraints: Incorporate constraints, usually nulls in certaindirections, in the BF design.
![Page 35: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/35.jpg)
Beamforming(BF)
Pick a direction of interest ωs , and select the linear combining weights Wsuch that r [n] = W Hy[n] contains mostly the signal from direction ωs
Conventional beamforming: W = V(ωs) and the power in direction ωs isE (|r [n]|2) = V(ωs)HRyV(ωs).
If you scan the spatial angles and look for the direction with most power,it has similarity to the search step of OMP.
BF with Null constraints: Incorporate constraints, usually nulls in certaindirections, in the BF design.
![Page 36: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/36.jpg)
Beamforming(BF)
Pick a direction of interest ωs , and select the linear combining weights Wsuch that r [n] = W Hy[n] contains mostly the signal from direction ωs
Conventional beamforming: W = V(ωs) and the power in direction ωs isE (|r [n]|2) = V(ωs)HRyV(ωs).
If you scan the spatial angles and look for the direction with most power,it has similarity to the search step of OMP.
BF with Null constraints: Incorporate constraints, usually nulls in certaindirections, in the BF design.
![Page 37: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/37.jpg)
Beamforming(BF)
Pick a direction of interest ωs , and select the linear combining weights Wsuch that r [n] = W Hy[n] contains mostly the signal from direction ωs
Conventional beamforming: W = V(ωs) and the power in direction ωs isE (|r [n]|2) = V(ωs)HRyV(ωs).
If you scan the spatial angles and look for the direction with most power,it has similarity to the search step of OMP.
BF with Null constraints: Incorporate constraints, usually nulls in certaindirections, in the BF design.
![Page 38: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/38.jpg)
Beamforming(BF)
Pick a direction of interest ωs , and select the linear combining weights Wsuch that r [n] = W Hy[n] contains mostly the signal from direction ωs
Conventional beamforming: W = V(ωs) and the power in direction ωs isE (|r [n]|2) = V(ωs)HRyV(ωs).
If you scan the spatial angles and look for the direction with most power,it has similarity to the search step of OMP.
BF with Null constraints: Incorporate constraints, usually nulls in certaindirections, in the BF design.
![Page 39: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/39.jpg)
ULA and Beamforming in Pictures
Figure: ULA on the z-axis Figure: BF pattern(polar plot, N = 11)
![Page 40: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/40.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 41: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/41.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 42: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/42.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 43: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/43.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 44: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/44.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 45: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/45.jpg)
Minimum power distortionless response (MPDR)beamformer
Pick direction of interest ωs
Distortionless constraint on beamformer W : W HVs = 1, whereVs = V(ωs).
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2)
MPDR BF design
minW
W HRyW subject to W HVs = 1.
Solution: Wmpdr = 1VH
s R−1y Vs
R−1y Vs
![Page 46: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/46.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 47: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/47.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 48: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/48.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 49: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/49.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 50: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/50.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 51: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/51.jpg)
MPDR Spatial Power Spectrum
Wmpdr = 1VH
s R−1y Vs
R−1y Vs
Benefit:
I Ry is easier to determine making it computationally attractive
Ry ≈1
L
L−1∑n=1
y[n]yH [n]
I Same Ry is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.
Spatial Power Spectrum using MPDR:
Pmpdr (ωs) = E (|W Hmpdry[n]|2) = W H
mpdrRyWmpdr =1
VHs R−1
y Vs
![Page 52: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/52.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 53: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/53.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 54: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/54.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 55: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/55.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 56: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/56.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 57: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/57.jpg)
MPDR intuition
W Hy[n] = W HVsxs [n] + W H I[n]
= xs [n]︸︷︷︸distortionless constraint
+q[n], where q[n] = W H I[n]
Minimum Power objective: Choose W to minimize E (|W H |y[n]|2), thepower at the output of the beamformer
If the interference is uncorrelated with the desired signal, thenminimization of E (|q[n]|2) is achieved, i.e. interference is minimized.
Signal of interest has been isolated and interference minimized (SINRmaximized)
If the interference is correlated with the desired source, we havecancellation and the desired source is not preserved.
![Page 58: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/58.jpg)
MPDR: Uncorrelated sources
Figure: nsignals = 10 and nsensors = 12
![Page 59: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/59.jpg)
MPDR: Beamformer for uncorrelated sources
Figure: nsignals = 10 and nsensors = 12
![Page 60: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/60.jpg)
MPDR: correlated sources
Figure: nsignals = 10, two correlated, and nsensors = 12
![Page 61: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/61.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 62: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/62.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 63: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/63.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 64: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/64.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 65: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/65.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 66: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/66.jpg)
Sparse Bayesian Learning
y = Φx + v
SBL uses a Bayesian framework with a separable prior p(x) = Πp(xi ).
Gaussian Scale Mixtures (GSM)
p(xi ) =
∫p(xi |γi )p(γi )dγi =
∫N(xi ; 0, γi )p(γi )dγi
TheoremA density p(x) which is symmetric with respect to the origin, can berepresented by a GSM iff p(
√x) is completely monotonic on (0,∞).
Most of the interesting priors over x can be represented in this GSMform. [Palmer et al., 2006]
DefinitionA function f (x) is completely monotonic on (a, b) if(−1)nf (n)(x) ≥ 0, n = 0, 1, .., where f (n)(x) denotes the nth derivative
![Page 67: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/67.jpg)
GSM Hierarchy Viewpoint
γ︸︷︷︸p(γ)
→ X ∼ N(x ; 0, γ)
AlternativelyX =
√γG
where G ∼ N(g ; 0, 1) and γ and G are independent.
Kurtosis: Note
E (X 2) = E (γ)E (G 2) = E (γ), and E (X 4) = E (γ2)E (G 4) = 3E (γ2)
Kurt(X ) = E (X 4)− 3(E (X 2))2 = 3E (γ2)− 3(E (γ))2
= 3(E (γ2)− (E (γ))2) = 3Var(γ) ≥ 0
![Page 68: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/68.jpg)
GSM Hierarchy Viewpoint
γ︸︷︷︸p(γ)
→ X ∼ N(x ; 0, γ)
AlternativelyX =
√γG
where G ∼ N(g ; 0, 1) and γ and G are independent.
Kurtosis: Note
E (X 2) = E (γ)E (G 2) = E (γ), and E (X 4) = E (γ2)E (G 4) = 3E (γ2)
Kurt(X ) = E (X 4)− 3(E (X 2))2 = 3E (γ2)− 3(E (γ))2
= 3(E (γ2)− (E (γ))2) = 3Var(γ) ≥ 0
![Page 69: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/69.jpg)
GSM Hierarchy Viewpoint
γ︸︷︷︸p(γ)
→ X ∼ N(x ; 0, γ)
AlternativelyX =
√γG
where G ∼ N(g ; 0, 1) and γ and G are independent.
Kurtosis: Note
E (X 2) = E (γ)E (G 2) = E (γ), and E (X 4) = E (γ2)E (G 4) = 3E (γ2)
Kurt(X ) = E (X 4)− 3(E (X 2))2 = 3E (γ2)− 3(E (γ))2
= 3(E (γ2)− (E (γ))2) = 3Var(γ) ≥ 0
![Page 70: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/70.jpg)
GSM Hierarchy Viewpoint
γ︸︷︷︸p(γ)
→ X ∼ N(x ; 0, γ)
AlternativelyX =
√γG
where G ∼ N(g ; 0, 1) and γ and G are independent.
Kurtosis: Note
E (X 2) = E (γ)E (G 2) = E (γ), and E (X 4) = E (γ2)E (G 4) = 3E (γ2)
Kurt(X ) = E (X 4)− 3(E (X 2))2 = 3E (γ2)− 3(E (γ))2
= 3(E (γ2)− (E (γ))2) = 3Var(γ) ≥ 0
![Page 71: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/71.jpg)
GSM Hierarchy Viewpoint
γ︸︷︷︸p(γ)
→ X ∼ N(x ; 0, γ)
AlternativelyX =
√γG
where G ∼ N(g ; 0, 1) and γ and G are independent.
Kurtosis: Note
E (X 2) = E (γ)E (G 2) = E (γ), and E (X 4) = E (γ2)E (G 4) = 3E (γ2)
Kurt(X ) = E (X 4)− 3(E (X 2))2 = 3E (γ2)− 3(E (γ))2
= 3(E (γ2)− (E (γ))2) = 3Var(γ) ≥ 0
![Page 72: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/72.jpg)
Examples of Gaussian Scale Mixtures
Laplacian density
p(x ;β) =1
2βexp(−|x |
β)
Scale mixing density: p(γ) = 12β2 exp(− 1
2β2 γ), γ ≥ 0.
Student-t Distribution
p(x ; a, b) =baΓ(a + 1/2)
(2π)0.5Γ(a)
1
(b + x2/2)a+1/2
Scale mixing density: Inverse-Gamma Distribution.
Generalized Gaussian
p(x ; p) =1
2Γ(1 + 1p )
e−|x|p
Scale mixing density: Positive alpha stable density of order p/2.
![Page 73: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/73.jpg)
Examples of Gaussian Scale Mixtures
Laplacian density
p(x ;β) =1
2βexp(−|x |
β)
Scale mixing density: p(γ) = 12β2 exp(− 1
2β2 γ), γ ≥ 0.
Student-t Distribution
p(x ; a, b) =baΓ(a + 1/2)
(2π)0.5Γ(a)
1
(b + x2/2)a+1/2
Scale mixing density: Inverse-Gamma Distribution.
Generalized Gaussian
p(x ; p) =1
2Γ(1 + 1p )
e−|x|p
Scale mixing density: Positive alpha stable density of order p/2.
![Page 74: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/74.jpg)
Examples of Gaussian Scale Mixtures
Laplacian density
p(x ;β) =1
2βexp(−|x |
β)
Scale mixing density: p(γ) = 12β2 exp(− 1
2β2 γ), γ ≥ 0.
Student-t Distribution
p(x ; a, b) =baΓ(a + 1/2)
(2π)0.5Γ(a)
1
(b + x2/2)a+1/2
Scale mixing density: Inverse-Gamma Distribution.
Generalized Gaussian
p(x ; p) =1
2Γ(1 + 1p )
e−|x|p
Scale mixing density: Positive alpha stable density of order p/2.
![Page 75: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/75.jpg)
Two Options for Estimation with GSM priors
MAP Estimation (Type I)
Evidence Maximization (Type II)
![Page 76: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/76.jpg)
Two Options for Estimation with GSM priors
MAP Estimation (Type I)
Evidence Maximization (Type II)
![Page 77: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/77.jpg)
Two Options for Estimation with GSM priors
MAP Estimation (Type I)
Evidence Maximization (Type II)
![Page 78: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/78.jpg)
MAP Estimation Framework (Type I)
Problem Statement
x̂ = arg maxx
p(x |y) = arg maxx
p(y |x)p(x) = arg max[log p(y |x)+log p(x)]
Can derive a general version that includes many past reweightedalgorithms using a generalized-t distribution in one setting1
1Giri, R., and Rao, B. D. (2016). Type I and Type II Bayesian Methods forSparse Signal Recovery Using Scale Mixtures. IEEE Trans. Signal Processing,64(13), 3418-3428.
![Page 79: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/79.jpg)
MAP Estimation Framework (Type I)
Problem Statement
x̂ = arg maxx
p(x |y) = arg maxx
p(y |x)p(x) = arg max[log p(y |x)+log p(x)]
Can derive a general version that includes many past reweightedalgorithms using a generalized-t distribution in one setting1
1Giri, R., and Rao, B. D. (2016). Type I and Type II Bayesian Methods forSparse Signal Recovery Using Scale Mixtures. IEEE Trans. Signal Processing,64(13), 3418-3428.
![Page 80: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/80.jpg)
Evidence Maximization (Type II)
Find a MAP estimate of γ, i.e. γ̂ = arg max p(γ|y).Estimate of the posterior distribution for x using estimated γ̂;i.e.p(x |y ; γ̂).
This leads to Sparse Bayesian Learning (SBL).
![Page 81: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/81.jpg)
Evidence Maximization (Type II)
Find a MAP estimate of γ, i.e. γ̂ = arg max p(γ|y).
Estimate of the posterior distribution for x using estimated γ̂;i.e.p(x |y ; γ̂).
This leads to Sparse Bayesian Learning (SBL).
![Page 82: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/82.jpg)
Evidence Maximization (Type II)
Find a MAP estimate of γ, i.e. γ̂ = arg max p(γ|y).Estimate of the posterior distribution for x using estimated γ̂;i.e.p(x |y ; γ̂).
This leads to Sparse Bayesian Learning (SBL).
![Page 83: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/83.jpg)
Evidence Maximization (Type II)
Find a MAP estimate of γ, i.e. γ̂ = arg max p(γ|y).Estimate of the posterior distribution for x using estimated γ̂;i.e.p(x |y ; γ̂).
This leads to Sparse Bayesian Learning (SBL).
![Page 84: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/84.jpg)
Evidence Maximization Framework
Potential Advantages
I Averaging over x leads to fewer minima in p(γ|y) =∫p(γ, x |y)dx .
I γ can tie several parameters, leading to fewer parameters. ForMMV, a single γi for row i .
I Maximizing the true posterior mass over the subspaces spanned bynon zero indexes instead of looking for the mode.
![Page 85: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/85.jpg)
Evidence Maximization Framework
Potential Advantages
I Averaging over x leads to fewer minima in p(γ|y) =∫p(γ, x |y)dx .
I γ can tie several parameters, leading to fewer parameters. ForMMV, a single γi for row i .
I Maximizing the true posterior mass over the subspaces spanned bynon zero indexes instead of looking for the mode.
![Page 86: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/86.jpg)
Evidence Maximization Framework
Potential Advantages
I Averaging over x leads to fewer minima in p(γ|y) =∫p(γ, x |y)dx .
I γ can tie several parameters, leading to fewer parameters.
ForMMV, a single γi for row i .
I Maximizing the true posterior mass over the subspaces spanned bynon zero indexes instead of looking for the mode.
![Page 87: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/87.jpg)
Evidence Maximization Framework
Potential Advantages
I Averaging over x leads to fewer minima in p(γ|y) =∫p(γ, x |y)dx .
I γ can tie several parameters, leading to fewer parameters. ForMMV, a single γi for row i .
I Maximizing the true posterior mass over the subspaces spanned bynon zero indexes instead of looking for the mode.
![Page 88: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/88.jpg)
Evidence Maximization Framework
Potential Advantages
I Averaging over x leads to fewer minima in p(γ|y) =∫p(γ, x |y)dx .
I γ can tie several parameters, leading to fewer parameters. ForMMV, a single γi for row i .
I Maximizing the true posterior mass over the subspaces spanned bynon zero indexes instead of looking for the mode.
![Page 89: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/89.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 90: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/90.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 91: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/91.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 92: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/92.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)
Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 93: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/93.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 94: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/94.jpg)
Sparse Bayesian Learning
y = Ax + v
Solving for MAP estimate of γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
What is p(y |γ)Given γ, x is Gaussian with mean zero and Covariance matrix Γ withΓ = diag(γ), i.e. p(x |γ) = N(x ; 0, Γ) = ΠN(xi ; 0, γi ).
Then p(y |γ) = N(y ; 0,Σy ), where Σy = σ2I + AΓAT ,
p(y |γ) =1√
(2π)N |Σy |e−
12 y
T Σ−1y y
![Page 95: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/95.jpg)
Solving for the optimal γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
= arg minγ
log |Σy |+ yTΣ−1y y − 2
∑i
log p(γi )
= arg minγ
log |Σy |+ Trace Σ−1y yyT − 2
∑i
log p(γi )
where, Σy = σ2I + AΓAT and Γ = diag(γ)
For some of the discussion, we will ignore the prior on γ, i.e. p(γ). Thiscan be viewed as a non-informative prior on γ or γ as deterministic butunknown.Given the separable nature of the prior, the prior is easy to incorporate.
For MMV
γ̂ = arg minγ
log |Σy |+ Trace Σ−1y R̂y −
2
L
∑i
log p(γi )
where R̂y = 1L
∑Ln=1 y[n]yT [n]
![Page 96: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/96.jpg)
Solving for the optimal γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
= arg minγ
log |Σy |+ yTΣ−1y y − 2
∑i
log p(γi )
= arg minγ
log |Σy |+ Trace Σ−1y yyT − 2
∑i
log p(γi )
where, Σy = σ2I + AΓAT and Γ = diag(γ)
For some of the discussion, we will ignore the prior on γ, i.e. p(γ). Thiscan be viewed as a non-informative prior on γ or γ as deterministic butunknown.Given the separable nature of the prior, the prior is easy to incorporate.
For MMV
γ̂ = arg minγ
log |Σy |+ Trace Σ−1y R̂y −
2
L
∑i
log p(γi )
where R̂y = 1L
∑Ln=1 y[n]yT [n]
![Page 97: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/97.jpg)
Solving for the optimal γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
= arg minγ
log |Σy |+ yTΣ−1y y − 2
∑i
log p(γi )
= arg minγ
log |Σy |+ Trace Σ−1y yyT − 2
∑i
log p(γi )
where, Σy = σ2I + AΓAT and Γ = diag(γ)
For some of the discussion, we will ignore the prior on γ, i.e. p(γ). Thiscan be viewed as a non-informative prior on γ or γ as deterministic butunknown.
Given the separable nature of the prior, the prior is easy to incorporate.
For MMV
γ̂ = arg minγ
log |Σy |+ Trace Σ−1y R̂y −
2
L
∑i
log p(γi )
where R̂y = 1L
∑Ln=1 y[n]yT [n]
![Page 98: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/98.jpg)
Solving for the optimal γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
= arg minγ
log |Σy |+ yTΣ−1y y − 2
∑i
log p(γi )
= arg minγ
log |Σy |+ Trace Σ−1y yyT − 2
∑i
log p(γi )
where, Σy = σ2I + AΓAT and Γ = diag(γ)
For some of the discussion, we will ignore the prior on γ, i.e. p(γ). Thiscan be viewed as a non-informative prior on γ or γ as deterministic butunknown.Given the separable nature of the prior, the prior is easy to incorporate.
For MMV
γ̂ = arg minγ
log |Σy |+ Trace Σ−1y R̂y −
2
L
∑i
log p(γi )
where R̂y = 1L
∑Ln=1 y[n]yT [n]
![Page 99: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/99.jpg)
Solving for the optimal γ
γ̂ = arg maxγ
p(γ|y) = arg maxγ
p(y |γ)p(γ)
= arg minγ
log |Σy |+ yTΣ−1y y − 2
∑i
log p(γi )
= arg minγ
log |Σy |+ Trace Σ−1y yyT − 2
∑i
log p(γi )
where, Σy = σ2I + AΓAT and Γ = diag(γ)
For some of the discussion, we will ignore the prior on γ, i.e. p(γ). Thiscan be viewed as a non-informative prior on γ or γ as deterministic butunknown.Given the separable nature of the prior, the prior is easy to incorporate.
For MMV
γ̂ = arg minγ
log |Σy |+ Trace Σ−1y R̂y −
2
L
∑i
log p(γi )
where R̂y = 1L
∑Ln=1 y[n]yT [n]
![Page 100: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/100.jpg)
Algorithmic Variants
I Fixed Point iteration based on setting the derivative of the objectivefunction to zero (Tipping)
I Expectation-Maximization (EM) Algorithm
I Sequential search for the significant γ’s (Tipping and Faul)
I Majorization-Minimization based approach (Wipf and Nagarajan)
I Reweighted `1 and `2 algorithms (Wipf and Nagarajan)
I Approximate Message Passing (AlShoukairi, Schniter and Rao)
![Page 101: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/101.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 102: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/102.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 103: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/103.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.
Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 104: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/104.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 105: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/105.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 106: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/106.jpg)
Computing Posterior p(x |y ; γ̂)
y = Ax + v
Now because of our convenient GSM choice, posterior can be easilycomputed, i.e, p(x |y ; γ̂) = N(µx ,Σx) where,
µx = E [x |y ; γ̂] = Γ̂ATΣ−1y y = Γ̂AT (σ2I + AΓ̂AT )−1y
Σx̃ = Cov [x |y ; γ̂] = Γ̂− Γ̂ATΣ−1y AΓ̂ = Γ̂− Γ̂AT (σ2I + AΓ̂AT )−1AΓ̂
µx can be used as a point estimate.Sparsity is achieved by having many of the γi ’s have zero value.
The conditional mean and variance computation constitute a LMMSEBF!
![Page 107: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/107.jpg)
EM algorithm: Updating γ
Treating (y, x) as complete data and vector x as hidden variable.
log p(y , x , γ) = log p(y |x) + log p(x |γ) + log p(γ)
E step
Q(γ|γk) = Ex|y ;γk [log p(y |x) + log p(x |γ) + log p(γ)]
M step
γk+1 = argmaxγQ(γ|γk) = argmaxγEx|y ;γk [log p(x |γ) + log p(γ)]
= argminγEx|y ;γk [M∑i=1
(x2i
2γi+
1
2log γi
)− log p(γ)]
The optimization involves M scalar optimization problems of the form
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
![Page 108: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/108.jpg)
EM algorithm: Updating γ
Treating (y, x) as complete data and vector x as hidden variable.
log p(y , x , γ) = log p(y |x) + log p(x |γ) + log p(γ)
E step
Q(γ|γk) = Ex|y ;γk [log p(y |x) + log p(x |γ) + log p(γ)]
M step
γk+1 = argmaxγQ(γ|γk) = argmaxγEx|y ;γk [log p(x |γ) + log p(γ)]
= argminγEx|y ;γk [M∑i=1
(x2i
2γi+
1
2log γi
)− log p(γ)]
The optimization involves M scalar optimization problems of the form
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
![Page 109: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/109.jpg)
EM algorithm: Updating γ
Treating (y, x) as complete data and vector x as hidden variable.
log p(y , x , γ) = log p(y |x) + log p(x |γ) + log p(γ)
E step
Q(γ|γk) = Ex|y ;γk [log p(y |x) + log p(x |γ) + log p(γ)]
M step
γk+1 = argmaxγQ(γ|γk) = argmaxγEx|y ;γk [log p(x |γ) + log p(γ)]
= argminγEx|y ;γk [M∑i=1
(x2i
2γi+
1
2log γi
)− log p(γ)]
The optimization involves M scalar optimization problems of the form
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
![Page 110: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/110.jpg)
EM algorithm: Updating γ
Treating (y, x) as complete data and vector x as hidden variable.
log p(y , x , γ) = log p(y |x) + log p(x |γ) + log p(γ)
E step
Q(γ|γk) = Ex|y ;γk [log p(y |x) + log p(x |γ) + log p(γ)]
M step
γk+1 = argmaxγQ(γ|γk) = argmaxγEx|y ;γk [log p(x |γ) + log p(γ)]
= argminγEx|y ;γk [M∑i=1
(x2i
2γi+
1
2log γi
)− log p(γ)]
The optimization involves M scalar optimization problems of the form
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
![Page 111: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/111.jpg)
Scalar Optimization
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
Note that E (x2l |y, γk) = µ
(k)x (l)2 + Σ
(k)x̃ (l , l)
Non-informative prior results in an objective function
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl
Update equation
γk+1l = E (x2
l |y, γk) = µ(k)x (l)2 + Σ
(k)x̃ (l , l)
Source powers updated in an iterated manner
![Page 112: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/112.jpg)
Scalar Optimization
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl − log p(γl)
Note that E (x2l |y, γk) = µ
(k)x (l)2 + Σ
(k)x̃ (l , l)
Non-informative prior results in an objective function
J(γl) =E (x2
l |y, γk)
2γl+
1
2log γl
Update equation
γk+1l = E (x2
l |y, γk) = µ(k)x (l)2 + Σ
(k)x̃ (l , l)
Source powers updated in an iterated manner
![Page 113: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/113.jpg)
Connection between SBL and MPDR
Note: r [n] = xs [n] + q[n]
![Page 114: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/114.jpg)
Connection between SBL and MPDR
Note: r [n] = xs [n] + q[n]
![Page 115: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/115.jpg)
MPDR view of SBL
I When p(xn) = N (0, pn) the MPDR + MMSE steps are equivalentto the LMMSE step in the EM SBL and the two algorithm areequivalent.
I More general priors can be used within the MPDR framework. TheEM-SBL has a closed form solution for the GSM prior only.
![Page 116: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/116.jpg)
MPDR view of SBL
I When p(xn) = N (0, pn) the MPDR + MMSE steps are equivalentto the LMMSE step in the EM SBL and the two algorithm areequivalent.
I More general priors can be used within the MPDR framework. TheEM-SBL has a closed form solution for the GSM prior only.
![Page 117: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/117.jpg)
MPDR view of SBL
I When p(xn) = N (0, pn) the MPDR + MMSE steps are equivalentto the LMMSE step in the EM SBL and the two algorithm areequivalent.
I More general priors can be used within the MPDR framework. TheEM-SBL has a closed form solution for the GSM prior only.
![Page 118: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/118.jpg)
MPDR view of SBL
I When p(xn) = N (0, pn) the MPDR + MMSE steps are equivalentto the LMMSE step in the EM SBL and the two algorithm areequivalent.
I More general priors can be used within the MPDR framework. TheEM-SBL has a closed form solution for the GSM prior only.
![Page 119: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/119.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 120: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/120.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 121: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/121.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 122: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/122.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 123: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/123.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 124: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/124.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 125: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/125.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 126: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/126.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 127: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/127.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 128: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/128.jpg)
MPDR versus SBL
MPDR:
1. y[n]→ R̂y = 1L
∑Ln=1 y[n]yH [n]
2. MPDR BF using R̂y , i.e. Wmpdr = 1VH
s R̂−1y Vs
R̂−1y Vs
3. MPDR spatial power spectrum Pmpdr (ωs) = 1VH
s R̂−1y Vs
SBL
1. Guess power of sources γ and employ uncorrelated model forcorrelation matrix Σy = ΦΓΦT + λI, where Γ = diag(γ).
2. MPDR BF using Σy , i.e. Wmpdr = 1VH
s Σ−1y Vs
Σ−1y Vs
3. Apply BF to data y[n], use BF output to compute new sourcepower estimate, and update Σy
4. Repeat steps 2 and 3 till convergence
![Page 129: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/129.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 130: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/130.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 131: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/131.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 132: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/132.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 133: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/133.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 134: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/134.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 135: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/135.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 136: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/136.jpg)
Toeplitz Matrix Approximation: ULA
Ry = Rs + σ2I and Rs is low rank and Toeplitz for uncorrelated sources
Theorem: A low rank Toeplitz R, rank D, is uniquely represented asR =
∑Dl=1 λlV(ωl)VH(ωl)
Σy =∑M
l=1 γlV(ωl)VH(ωl) + λI in SBL has the appropriate structure.
ML estimate: As L→∞, KL(p∗||p) is minimized, where p is the modelassumed by SBL, i.e. p(y ; Σy ), and p∗ is the actual data density, i.e.p(y ; Ry ).
Uncorrelated sources: then true model and SBL model is the same andSBL is finding a ML estimate of a Toeplitz matrix.
Correlated sources: True model Ry is no longer Toeplitz and SBL isfinding the best Toeplitz matrix in the KL sense.
The uncorrelated source model allows SBL to be largely unaffected bycorrelated sources.
![Page 137: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/137.jpg)
MPDR: Uncorrelated sources
Figure: nsignals = 10 and nsensors = 12
![Page 138: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/138.jpg)
MPDR: correlated sources
Figure: nsignals = 10, two correlated, and nsensors = 12
![Page 139: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/139.jpg)
Correlated signals with MUSIC
Figure: MUSIC: nsignals = 10 (Last two correlated), nsensors = 12
![Page 140: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/140.jpg)
SBL to the rescue!
Figure: SBL: nsignals = 10 (Last two correlated), nsensors = 12
![Page 141: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/141.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 142: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/142.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 143: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/143.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 144: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/144.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 145: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/145.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 146: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/146.jpg)
Beam Space processing and Nested Arrays
Consider yna[n] = Sy[n], where S is user chosen and SP×N , with P ≤ N.
Options
I Could be a random matrix as in CS
I Could be chosen to cover a region in space
I Could be used to thin the array (low complexity sensor array)
![Page 147: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/147.jpg)
Inspiration from Sister Field: Array Signal Processing
Nested Arrays: Structure & Properties
I A nested array with N antennas has a filled difference set withO(N2) elements
I Drastically increases the spatial degrees of freedom
I For source localization, this results in estimating location of moresources than sensors!
![Page 148: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/148.jpg)
Inspiration from Sister Field: Array Signal Processing
Nested Arrays: Structure & Properties
I A nested array with N antennas has a filled difference set withO(N2) elements
I Drastically increases the spatial degrees of freedom
I For source localization, this results in estimating location of moresources than sensors!
![Page 149: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/149.jpg)
Inspiration from Sister Field: Array Signal Processing
Nested Arrays: Structure & Properties
I A nested array with N antennas has a filled difference set withO(N2) elements
I Drastically increases the spatial degrees of freedom
I For source localization, this results in estimating location of moresources than sensors!
![Page 150: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/150.jpg)
Inspiration from Sister Field: Array Signal Processing
Nested Arrays: Structure & Properties
I A nested array with N antennas has a filled difference set withO(N2) elements
I Drastically increases the spatial degrees of freedom
I For source localization, this results in estimating location of moresources than sensors!
![Page 151: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/151.jpg)
Inspiration from Sister Field: Array Signal Processing
Nested Arrays: Structure & Properties
I A nested array with N antennas has a filled difference set withO(N2) elements
I Drastically increases the spatial degrees of freedom
I For source localization, this results in estimating location of moresources than sensors!
![Page 152: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/152.jpg)
DoA Estimation with Nested Arrays2
With Nested Arrays, we can transform an order-N covariance matrix into
order-N2+2N
4 Toeplitz matrix
Transformation:
Consider as an example N = 4, nested array sensor positions: {1, 2, 3, 6}
R0 R−1 R−2 R−5
R1 R0 R−1 R−4
R2 R1 R0 R−3
R5 R4 R3 R0
→
R0 R−1 R−2 R−3 R−4 R−5
R1 R0 R−1 R−2 R−3 R−4
R2 R1 R0 R−1 R−2 R−3
R3 R2 R1 R0 R−1 R−2
R4 R3 R2 R1 R0 R−1
R5 R4 R3 R2 R1 R0
Number of DoAs that can be estimated is 5 > 3 (for ULA)
2Pal, P., and Vaidyanathan, P. P. (2010). Nested arrays: A novel approachto array processing with enhanced degrees of freedom. IEEE Transactions onSignal Processing, 58(8), 4167-4181.
![Page 153: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/153.jpg)
DoA Estimation with Nested Arrays2
With Nested Arrays, we can transform an order-N covariance matrix into
order-N2+2N
4 Toeplitz matrix
Transformation:
Consider as an example N = 4, nested array sensor positions: {1, 2, 3, 6}
R0 R−1 R−2 R−5
R1 R0 R−1 R−4
R2 R1 R0 R−3
R5 R4 R3 R0
→
R0 R−1 R−2 R−3 R−4 R−5
R1 R0 R−1 R−2 R−3 R−4
R2 R1 R0 R−1 R−2 R−3
R3 R2 R1 R0 R−1 R−2
R4 R3 R2 R1 R0 R−1
R5 R4 R3 R2 R1 R0
Number of DoAs that can be estimated is 5 > 3 (for ULA)
2Pal, P., and Vaidyanathan, P. P. (2010). Nested arrays: A novel approachto array processing with enhanced degrees of freedom. IEEE Transactions onSignal Processing, 58(8), 4167-4181.
![Page 154: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/154.jpg)
DoA Estimation with Nested Arrays2
With Nested Arrays, we can transform an order-N covariance matrix into
order-N2+2N
4 Toeplitz matrix
Transformation:
Consider as an example N = 4, nested array sensor positions: {1, 2, 3, 6}
R0 R−1 R−2 R−5
R1 R0 R−1 R−4
R2 R1 R0 R−3
R5 R4 R3 R0
→
R0 R−1 R−2 R−3 R−4 R−5
R1 R0 R−1 R−2 R−3 R−4
R2 R1 R0 R−1 R−2 R−3
R3 R2 R1 R0 R−1 R−2
R4 R3 R2 R1 R0 R−1
R5 R4 R3 R2 R1 R0
Number of DoAs that can be estimated is 5 > 3 (for ULA)
2Pal, P., and Vaidyanathan, P. P. (2010). Nested arrays: A novel approachto array processing with enhanced degrees of freedom. IEEE Transactions onSignal Processing, 58(8), 4167-4181.
![Page 155: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/155.jpg)
SBL and Nested Arrays
Simply apply SBL3 to yna[n], where yna[n] = Sy[n]
Actual covariance SRySH , and SBL covariance model is SΣySH , whereΣy is Toeplitz
The ML estimation minimizes the KL distance between the actual densityand model density assumed by SBL and so tries to best match the twocovariance matrices.
Figure: Nested array with 12 sensors
3Nannuru, S., and Gerstoft, P. (2019, May). 2D Beamforming on SparseArrays with Sparse Bayesian Learning. In ICASSP 2019-2019 IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 4355-4359).
![Page 156: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/156.jpg)
SBL and Nested Arrays
Simply apply SBL3 to yna[n], where yna[n] = Sy[n]
Actual covariance SRySH , and SBL covariance model is SΣySH , whereΣy is Toeplitz
The ML estimation minimizes the KL distance between the actual densityand model density assumed by SBL and so tries to best match the twocovariance matrices.
Figure: Nested array with 12 sensors
3Nannuru, S., and Gerstoft, P. (2019, May). 2D Beamforming on SparseArrays with Sparse Bayesian Learning. In ICASSP 2019-2019 IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 4355-4359).
![Page 157: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/157.jpg)
SBL and Nested Arrays
Simply apply SBL3 to yna[n], where yna[n] = Sy[n]
Actual covariance SRySH , and SBL covariance model is SΣySH , whereΣy is Toeplitz
The ML estimation minimizes the KL distance between the actual densityand model density assumed by SBL and so tries to best match the twocovariance matrices.
Figure: Nested array with 12 sensors
3Nannuru, S., and Gerstoft, P. (2019, May). 2D Beamforming on SparseArrays with Sparse Bayesian Learning. In ICASSP 2019-2019 IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 4355-4359).
![Page 158: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/158.jpg)
SBL and Nested Arrays
Simply apply SBL3 to yna[n], where yna[n] = Sy[n]
Actual covariance SRySH , and SBL covariance model is SΣySH , whereΣy is Toeplitz
The ML estimation minimizes the KL distance between the actual densityand model density assumed by SBL and so tries to best match the twocovariance matrices.
Figure: Nested array with 12 sensors
3Nannuru, S., and Gerstoft, P. (2019, May). 2D Beamforming on SparseArrays with Sparse Bayesian Learning. In ICASSP 2019-2019 IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 4355-4359).
![Page 159: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/159.jpg)
SBL and Nested Arrays
Simply apply SBL3 to yna[n], where yna[n] = Sy[n]
Actual covariance SRySH , and SBL covariance model is SΣySH , whereΣy is Toeplitz
The ML estimation minimizes the KL distance between the actual densityand model density assumed by SBL and so tries to best match the twocovariance matrices.
Figure: Nested array with 12 sensors
3Nannuru, S., and Gerstoft, P. (2019, May). 2D Beamforming on SparseArrays with Sparse Bayesian Learning. In ICASSP 2019-2019 IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 4355-4359).
![Page 160: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/160.jpg)
Nested Arrays: SBL robustness
Figure: MUSIC Figure: SBL
![Page 161: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/161.jpg)
Nested Arrays: SBL efficiency
Figure: nsens = 12, nsignals = 25, SNR = 10dB, nsnapshots = 500
![Page 162: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/162.jpg)
Summary
I Established a connection between SBL and MPDR beamforming.
I Provides better insight into effective BFI Enables an approach to deal with more intractable inference
problems
I Discussed Uniform Linear Arrays and Toeplitz Matrix Approximationproperty of SBL
I Shown effectiveness of SBL for Nested Arrays to identify moresources than sensors
![Page 163: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/163.jpg)
Summary
I Established a connection between SBL and MPDR beamforming.
I Provides better insight into effective BFI Enables an approach to deal with more intractable inference
problems
I Discussed Uniform Linear Arrays and Toeplitz Matrix Approximationproperty of SBL
I Shown effectiveness of SBL for Nested Arrays to identify moresources than sensors
![Page 164: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/164.jpg)
Summary
I Established a connection between SBL and MPDR beamforming.
I Provides better insight into effective BFI Enables an approach to deal with more intractable inference
problems
I Discussed Uniform Linear Arrays and Toeplitz Matrix Approximationproperty of SBL
I Shown effectiveness of SBL for Nested Arrays to identify moresources than sensors
![Page 165: Sparse Bayesian Learning: A Beamforming and Toeplitz ...dobigeon.perso.enseeiht.fr/SPARS2019/Rao_SPARS_2019.pdfBhaskar D Rao University of California, San Diego Email: brao@ucsd.edu](https://reader033.vdocuments.site/reader033/viewer/2022060807/608c2ab7a0072365841650ab/html5/thumbnails/165.jpg)
Summary
I Established a connection between SBL and MPDR beamforming.
I Provides better insight into effective BFI Enables an approach to deal with more intractable inference
problems
I Discussed Uniform Linear Arrays and Toeplitz Matrix Approximationproperty of SBL
I Shown effectiveness of SBL for Nested Arrays to identify moresources than sensors