mvdr, mpdr and lmmse beamformersdsp.ucsd.edu/home/wp-content/uploads/ece251d... · fft based...
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MVDR, MPDR and LMMSE Beamformers
Bhaskar D RaoUniversity of California, San Diego
Email: [email protected]
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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.
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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
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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
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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.
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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)
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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
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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
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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.
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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.
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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.
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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
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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.
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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.
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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)
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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
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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
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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 .
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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]
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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
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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]
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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)
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