Sparse Bayesian Learning: A Beamforming and Toeplitz...

Sparse Bayesian Learning: A Beamforming andToeplitz Approximation Perspective

Bhaskar D RaoUniversity of California, San Diego

Email: brao@ucsd.edu

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.

Some References

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

Outline

I Background

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I Background

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I Background

Outline

I Background

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I Background

Outline

I Background

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).

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

Uniform Linear Array (ULA):

Sensors placed on a line with a unformspacing of d .

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

So array manifold can be denoted by V(ωl).

y(ωc , n) = V(ωc , k0)x0[n] +D−1∑l=1

V(ωc , kl)xl [n] + Z[n], n = 1, .., L

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.

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]

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]

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

Ry ∈ CN×N , Rs ∈ CN×N , Rx ∈ CD×D , and V ∈ CN×D

For ULA and uncorrelated sources, Ry is a Toeplitz matrix.

Source Covariance matrix:

Rx = E (x[n]xH [n]) where Rx ∈ CD×D , andthe diagonal elements are p0, p1, ..., pD−1, the source powers

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).

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).

Multiple Measurement Vectors (MMV)

I Multiple measurements: L measurements

I Common Sparsity Profile: k nonzero rows

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.

Beamforming(BF)

ULA and Beamforming in Pictures

Figure: ULA on the z-axis Figure: BF pattern(polar plot, N = 11)

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

W HRyW subject to W HVs = 1.

Solution: Wmpdr = 1VH

s R−1y Vs

R−1y Vs

MPDR BF design

s R−1y Vs

R−1y Vs

MPDR BF design

s R−1y Vs

R−1y Vs

MPDR BF design

s R−1y Vs

R−1y Vs

MPDR BF design

s R−1y Vs

R−1y Vs

MPDR BF design

s R−1y Vs

R−1y Vs

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−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

Wmpdr = 1VH

s R−1y Vs

R−1y Vs

Benefit:

Ry ≈1

L−1∑n=1

y[n]yH [n]

mpdrRyWmpdr =1

VHs R−1

Wmpdr = 1VH

s R−1y Vs

R−1y Vs

Benefit:

Ry ≈1

L−1∑n=1

y[n]yH [n]

mpdrRyWmpdr =1

VHs R−1

Wmpdr = 1VH

s R−1y Vs

R−1y Vs

Benefit:

Ry ≈1

L−1∑n=1

y[n]yH [n]

mpdrRyWmpdr =1

VHs R−1

Wmpdr = 1VH

s R−1y Vs

R−1y Vs

Benefit:

Ry ≈1

L−1∑n=1

y[n]yH [n]

mpdrRyWmpdr =1

VHs R−1

Wmpdr = 1VH

s R−1y Vs

R−1y Vs

Benefit:

Ry ≈1

L−1∑n=1

y[n]yH [n]

mpdrRyWmpdr =1

VHs R−1

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.

MPDR intuition

MPDR: Uncorrelated sources

Figure: nsignals = 10 and nsensors = 12

MPDR: Beamformer for uncorrelated sources

MPDR: correlated sources

Figure: nsignals = 10, two correlated, and nsensors = 12

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

y = Φx + v

p(xi ) =

y = Φx + v

p(xi ) =

y = Φx + v

p(xi ) =

y = Φx + v

p(xi ) =

y = Φx + v

p(xi ) =

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

γ︸︷︷︸p(γ)

→ X ∼ N(x ; 0, γ)

AlternativelyX =

√γG

Kurtosis: Note

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

γ︸︷︷︸p(γ)

→ X ∼ N(x ; 0, γ)

AlternativelyX =

√γG

Kurtosis: Note

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

γ︸︷︷︸p(γ)

→ X ∼ N(x ; 0, γ)

AlternativelyX =

√γG

Kurtosis: Note

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

γ︸︷︷︸p(γ)

→ X ∼ N(x ; 0, γ)

AlternativelyX =

√γG

Kurtosis: Note

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

Examples of Gaussian Scale Mixtures

Laplacian density

p(x ;β) =1

2βexp(−|x |

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)

(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.

Laplacian density

p(x ;β) =1

2βexp(−|x |

2β2 γ), γ ≥ 0.

p(x ; a, b) =baΓ(a + 1/2)

(2π)0.5Γ(a)

(b + x2/2)a+1/2

p(x ; p) =1

2Γ(1 + 1p )

e−|x|p

Laplacian density

p(x ;β) =1

2βexp(−|x |

2β2 γ), γ ≥ 0.

p(x ; a, b) =baΓ(a + 1/2)

(2π)0.5Γ(a)

(b + x2/2)a+1/2

p(x ; p) =1

2Γ(1 + 1p )

e−|x|p

Two Options for Estimation with GSM priors

MAP Estimation (Type I)

Evidence Maximization (Type II)

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.

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.

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).

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).

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.

I γ can tie several parameters, leading to fewer parameters.

ForMMV, a single γi for row i .

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−

T Σ−1y y

y = Ax + v

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

p(y |γ) =1√

(2π)N |Σy |e−

T Σ−1y y

y = Ax + v

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

p(y |γ) =1√

(2π)N |Σy |e−

T Σ−1y y

y = Ax + v

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 ).

p(y |γ) =1√

(2π)N |Σy |e−

T Σ−1y y

y = Ax + v

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

p(y |γ) =1√

(2π)N |Σy |e−

T Σ−1y y

y = Ax + v

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

p(y |γ) =1√

(2π)N |Σy |e−

T Σ−1y y

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

log p(γi )

= arg minγ

log |Σy |+ Trace Σ−1y yyT − 2

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 −

log p(γi )

where R̂y = 1L

∑Ln=1 y[n]yT [n]

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

= arg minγ

log p(γi )

= arg minγ

log p(γi )

For MMV

γ̂ = arg minγ

log p(γi )

where R̂y = 1L

∑Ln=1 y[n]yT [n]

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

= arg minγ

log p(γi )

= arg minγ

log p(γi )

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 p(γi )

where R̂y = 1L

∑Ln=1 y[n]yT [n]

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

= arg minγ

log p(γi )

= arg minγ

log p(γi )

For MMV

γ̂ = arg minγ

log p(γi )

where R̂y = 1L

∑Ln=1 y[n]yT [n]

γ̂ = arg maxγ

p(γ|y) = arg maxγ

p(y |γ)p(γ)

= arg minγ

log p(γi )

= arg minγ

log p(γi )

For MMV

γ̂ = arg minγ

log p(γi )

where R̂y = 1L

∑Ln=1 y[n]yT [n]

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)

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!

y = Ax + v

µx can be used as a point estimate.

Sparsity is achieved by having many of the γi ’s have zero value.

y = Ax + v

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

2log γi

)− log p(γ)]

The optimization involves M scalar optimization problems of the form

J(γl) =E (x2

l |y, γk)

2log γl − log p(γl)

E step

M step

2log γi

)− log p(γ)]

J(γl) =E (x2

l |y, γk)

E step

M step

2log γi

)− log p(γ)]

J(γl) =E (x2

l |y, γk)

E step

M step

2log γi

)− log p(γ)]

J(γl) =E (x2

l |y, γk)

Scalar Optimization

J(γl) =E (x2

l |y, γk)

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)

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

Scalar Optimization

J(γl) =E (x2

l |y, γk)

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)

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

Connection between SBL and MPDR

Note: r [n] = xs [n] + q[n]

Connection between SBL and MPDR

Note: r [n] = xs [n] + q[n]

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.

MPDR view of SBL

MPDR versus SBL

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

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

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

MPDR versus SBL

1. y[n]→ R̂y = 1L

∑Ln=1 y[n]yH [n]

s R̂−1y Vs

R̂−1y Vs

s R̂−1y Vs

s Σ−1y Vs

Σ−1y Vs

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.

Σy =∑M

MPDR: Uncorrelated sources

MPDR: correlated sources

Figure: nsignals = 10, two correlated, and nsensors = 12

Correlated signals with MUSIC

Figure: MUSIC: nsignals = 10 (Last two correlated), nsensors = 12

SBL to the rescue!

Figure: SBL: nsignals = 10 (Last two correlated), nsensors = 12

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)

Options

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!

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.

order-N2+2N

4 Toeplitz matrix

Transformation:

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

order-N2+2N

4 Toeplitz matrix

Transformation:

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

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).

Nested Arrays: SBL robustness

Figure: MUSIC Figure: SBL

Nested Arrays: SBL efficiency

Figure: nsens = 12, nsignals = 25, SNR = 10dB, nsnapshots = 500

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

Summary

problems

Summary

problems

Summary

problems

Sparse Bayesian Learning: A Beamforming and Toeplitz...

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