MVDR, MPDR and LMMSE Beamformers

Bhaskar D RaoUniversity of California, San Diego

Email: brao@ucsd.edu

Reference Books

1. Optimum Array Processing, H. L. Van Trees

2. Stoica, P., & Moses, R. L. (2005). Spectral analysis of signals (Vol.1). Upper Saddle River, NJ: Pearson Prentice Hall.

Narrow-band Signals

x(ωc , n) = V(ωc , ks)Fs [n] +D−1∑l=1

V(ωc , kl)Fl [n] + Z[n]

Assumptions

I Fs [n], Fl [n], l = 1, ..,D − 1, and Z[n] are zero mean

I E (|Fs [n]|2) = ps , and E (|Fl [n]|2 = pl , l = 1, . . . ,D − 1, andE (Z[n]Z[n]) = σ2

z I

I All the signals/sources are uncorrelated with each other and overtime: E (Fl [n]F ∗

m[p]) = plδ[l −m]δ[n − p] and E (Fl [n]F ∗s [p]) = 0

I The sources are uncorrelated with the noise: E (Z[n]F ∗l [m]) = 0

Interference plus Noise signal Covariance

I[n] =D−1∑l=1

VlFl [n] + Z[n], ; where Vl = V(ωc , kl)

Properties of I[n]

I I[n] is zero mean

I The covariance of I[n] is given by

Sn = E (I[n]IH [n]) =D−1∑l=1

plVlVHl + σ2

z I

MVDR Beamformer

Distortionless constraint on beamformer W : W HVs = 1Implication:

W Hx[n] = W HVsFs [n] + W H I[n] (1)

= Fs [n]︸ ︷︷ ︸distortionless constraint

+q[n], where q[n] = W H I[n] (2)

Minimum Variance objective: Choose W to minimize

E (|q[n]|2) = W HSnW

MVDR BF design

minW

W HSnW subject to W HVs = 1.

MVDR beamformer

MVDR BF design

minW

W HSnW subject to W HVs = 1.

Solution: Wmvdr = 1VH

s S−1n Vs

S−1n Vs

Derivation: Can be Obtained using Lagrange multipliers or by maximizingthe SINR (Signal to Interference plus Noise Ratio)

Interpretation

Wmvdr = 1VH

s S−1n Vs

S−1n Vs tries to minimize

E (|q[n]|2) = E (|W H I[n]|2) = W HSnW , where Sn =∑D−1

l=1 plVlVHl + σ2

z I

It will try to place nulls at angular locations consistent with theinterference plane waves if σ2

z is small.

If the number of antennas is greater than or equal to D, i.e. N ≥ D, theMVDR BF can null out all the (D − 1) interferers.

If N < D, the MVDR BF will attempt to control the depth of the nulls tominimize interference.

If σ2z is large compared to the power in the interfering plane waves, then

Sn ≈ σ2z I and hence Wmvdr ∝ Vs

Array Gain

Input SINR at each sensor

SINRI =ps∑D−1

l=1 pl + σ2z

SINR at the output of the array after MVDR (MPDR) beamformer

SINRo =ps

E (|W Hmvdr I[n]|2)

=ps1

VHs S

−1n Vs

Array Gain

Amvdr =SINRo

SINRI=

∑D−1l=1 pl + σ2

z1

VHs S

−1n Vs

Challenges with MVDR

Wmvdr =1

VHs S

−1n Vs

S−1n Vs

The main challenge is estimating Sn?

This requires coordination and may not always be possible.

This leads to MPDR, minimum power distortionless responsebeamformer.

Most books refer to MPDR as MVDR.

MPDR, minimum power distortionless responsebeamformer

MPDR very similar to MVDR with respect to the constraint.

Distortionless constraint on beamformer W : W HVs = 1Implication:

W Hx[n] = W HVsFs [n] + W H I[n] (3)

= Fs [n]︸ ︷︷ ︸distortionless constraint

+q[n], where q[n] = W H I[n] (4)

Minimum Power objective: Choose W to minimize E (|W H |x[n]|2), thepower at the output of the beamformer

E (|W Hx[n]|2) = W HSxW , where Sx = psVsVHs + Sn

MPDR BF design

minW

W HSxW subject to W HVs = 1.

MPDR beamformer

MPDR BF design

minW

W HSxW subject to W HVs = 1.

Solution: Wmpdr = 1VH

s S−1x Vs

S−1x Vs

Derivation: Same as MVDR with Sx replacing Sn

Benefit:

I Sx is easier to determine making it computationally attractive

Sx ≈1

L

L−1∑n=1

x[n]xH [n]

I Same Sx is needed if you change your mind on direction of interest.Can deal with multiple signals of interest with considerable ease.

Relationship between MPDR and MVDR

For uncorrelated sources Wmpdr = Wmvdr

Proof is based on the Matrix Inversion Lemma

(A + BCD)−1 = A−1 − A−1B(DA−1B + C−1)−1DA−1

Note thatSx = psVsV

Hs + Sn = Sn + VspsV

Hs

Using the matrix inversion lemma

S−1x = S−1

n − S−1n Vs(VH

s S−1n Vs +

1

ps)−1VH

s S−1n

S−1x Vs = βS−1

n Vs where β =

1ps

VHs S

−1n Vs + 1

ps

Hence

Wmpdr =1

VHs S

−1x Vs

S−1x Vs =

1

βVHs S

−1n Vs

βS−1n Vs =

1

VHs S

−1n Vs

S−1n Vs = Wmvdr

Robust MVDR/MPDR

Sx = psVsVHs +

D−1∑l=1

plVlVHl + σ2

z I

Sx could be close to singular, particular in a low noise (σ2z ) scenario,

making the inversion of Sx problematic.Regularization:

minW

W HSxW + λ‖W ‖2 = minW

W H(Sx + λI)W subject to W HVs = 1.

where λ ≥ 0.

Solution: W robustmpdr = 1

VHs (Sx+λI)−1Vs

(Sx + λI)−1Vs

SVD truncation: Replacing S−1n by S+

n , particularly with the smalleigenvalues set to zero.Not very effective (Bad Idea):

S−1n Vs =

M∑l=1

eHVs

λlel

Does not work because we want W to align with the eigenvectors elcorresponding to the small eigenvalues to null out interference.

More when we discuss subspace methods.

Spatial Power Spectrum using MPDR

I Beamsteering and measuring power at the output of BF, i.e.E (|(Wd �V(ψT ))Hx[n]|2) = E (|V(ψT )H(W ∗

d � x[n])|2). FFT basedprocessing for ULA

I MPDR based spatial power spectrum estimation: Measure power atthe output of the MPDR BF given by Wmpdr = 1

VHs S

−1x Vs

S−1x Vs

Pmpdr (ks) = E (|W Hmpdrx[n]|2) = W H

mpdrSxWmpdr

=

(1

VHs S

−1x Vs

VHs S

−1x

)Sx

(S−1x Vs

1

VHs S

−1x Vs

)=

1

VHs S

−1x Vs

(VH

s S−1x SxS

−1x Vs

) 1

VHs S

−1x Vs

=1

VHs S

−1x Vs

Can be efficiently computed for a ULA exploiting the Toeplitzstructure of Sx .

1

1Musicus, B. (1985). Fast MLM power spectrum estimation from uniformlyspaced correlations. IEEE Transactions on Acoustics, Speech, and SignalProcessing, 33(5), 1333-1335.

Bayesian Options

Problem: Estimate random vector X given measurements of randomvector Y

I Posterior Density Estimation

I Maximum Aposteriori Estimation (MAP)

I Minimum Mean Squared Estimation (MMSE)

I Linear Minimum Mean Squared Estimation (LMMSE)

Minimum Mean Squared Estimation (MMSE)

Objective: Compute an estimate of x as x̂ = g(y) to minimize the meansquared error E (‖x− x̂‖2)Optimum minimum mean squared estimate is given by the conditionalmean

x̂mmse = E (X|Y = y) =

∫xp(x|y)dx

Linear Minimum Mean Squared Estimation: Optimum estimate isconstrained to be an affine estimate X̂ = CY + D.

Assumption Y has mean E (Y) = µy and CovarianceΣyy = E (Y − µy )(Y − µy )H = ΣH

yy

X has mean E (X) = µx and CovarianceΣxx = E (X− µx)(X− µx)H = ΣH

xx

The cross covariance is denoted by Σxy = E (X− µx)(Y − µy )H = ΣHyx

LMMSE Estimation

Solution: Co = ΣxyΣ−1yy , Do = µx − Coµy .

X̂lmmse = µx + Co(Y − µy ) = µx + ΣxyΣ−1yy (Y − µy )

Error X̃lmmse has Covariance matrix given byE (X̃lmmseX̃H

lmmse) = Σxx − ΣxyΣ−1yy Σyx

Properties

I For zero mean random variables X̂lmmse = ΣxyΣ−1yy Y

I X̂lmmse is an unbiased estimate, i.e. E (X̃lmmse) = 0.

I X̃lmmse ⊥ BY, i.e E (X̃lmmse(BY)H) = 0

I If Q = BX, then Q̂lmmse = BX̂lmmse

I If X and Y are jointly Gaussian, MMSE estimate = LMMSEestimate

LMMSE Beamformer

x(n) = V(ks)Fs [n] +D−1∑l=1

V(kl)Fl [n] + Z[n]

= [Vs ,V1, . . . ,VD−1]

Fs [n]F1[n]

...FD−1[n]

+ Z[n]

= VF[n] + Z[n]

where V ∈ CN×D , and F[n] ∈ CD×1.

Note that E (F[n]) = 0D×1, E (F[n]ZH [n]) = 0D×N .

Goal: LMMSE Estimate of F[n]

Additional assumption

SF = E (F[n]FH [n]) = SH =

SHs

SH1...

SHD−1

,where SF ∈ CD×D , SH

s ∈ C 1×D and the diagonal elements areps , p1, ..., pD−1.

For uncorrelated sources SF is a diagonal matrix, i.e.SF = diag(ps , p1, . . . , pD−1). and SH

s = [ps , 0, . . . , 0]

LMMSE Beamformer

LMMSE estimate of F[n] given array output x[n] is given by

F̂[n] = ΣFxΣ−1xx x[n]

where

ΣFx = E (F[n]xH [n]) = E (F[n](FH [n]VH + ZH [n])) = SFVH

andΣxx = Sx = VSFV

H + σ2z I

Hence

F̂[n] = SFVHS−1

x x[n] and F̂s [n] = W Hlmmsex[n] = SH

s VHS−1

x x[n]

The LMMSE BF is Wlmmse = S−1x VSs .

For uncorrelated sources, since Ss = [ps , 0, .., 0]H , we have

Wlmmse = psS−1x Vs

Relationship between LMMSE and MPDR beamformers

Wmpdr = 1VH

s S−1x Vs

S−1x Vs and Wlmmse = psS−1

x Vs

Hence

Wlmmse = psS−1x Vs = ps(VH

s S−1x Vs)

1

VHs S

−1x Vs

S−1x Vs

= ps(VHs S

−1x Vs)Wmpdr = cWmpdr

with c = ps(VHs S

−1x Vs) being real and a positive scalar.

The LMMSE BF operation can be viewed as MPDR BF followed byscaling with c = ps(VH

s S−1x Vs).

x[n]→ r [n] = W Hmpdrx[n]→ F̂s [n] = c r [n]

Interpretation of the scalar c

Note Wlmmse also maximizes the SINR because it is a scaled version ofWmpdr .

r [n] = W Hmpdrx[n] = Fs [n] + q[n], where q[n] = W H

mpdr I[n]

Note q[n] is zero mean and uncorrelated with Fs [n].

E (|r [n]|2) = ps + E (|q[n]|2) ≥ ps

So the MPDR overestimates the power in direction ks .

Now we find a LMMSE estimate of Fs [n] given r [n].

F̂s [n] = ΣFrΣ−1rr r [n]

where ΣFr = ps and Σrr = 1VH

s S−1x Vs

.

This results inF̂s [n] = c r [n]

with c = ps(VHs S

−1x Vs)