UNIVERSITY
OF OSLO
Adaptive Array Processing
Adaptive beamforming
Algorithm’s characteristics depend on data it
receives
Dynamic adaptive beamforming
Algorithms that update themselves as each
observation is acquired
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/1
UNIVERSITY
OF OSLO
Estimation Theory
Measurement: y, Parameter: ξ
Bayes’s rule:
py,ξ (y, ξ) py|ξ (y|ξ)pξ(ξ)
=
pξ|y (ξ|y) =
py (y)
py (y)
1. A priori density: pξ(ξ)
2. A posteriori density: pξ|y (ξ|y)
3. Joint probability density: py,ξ (y, ξ)
4. Likelihood function: py|ξ (y|ξ)
Minimum mean-square error
estimator
The parameter’s a posteriori expected value:
ξˆM M SE (y) = E[ξ|y]
Maximum likelihood estimator
If ξ is not a random variable or its a priori
density is unknown, then maximize the
likelihood function.
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/2
UNIVERSITY
OF OSLO
Conventional Beamforming
Beamformer output: z(t, ~ζ) = e0W Y
Output power: P (e) = e0W RW 0e
W = diag(w1, . . . , wM ) - weights
e - steering vector
Y - vector of observations
R - estimate of correlation matrix
• Single frequency, no noise model assumed,
e determined from geometry only.
• For spatially white noise, Kn = σn2 I, and a
single signal present, it can be shown that
the conventional beamformer with W = I
is the maximum likelihood estimator of
direction and power: P = e0Re
• Beampattern: P (e) when the data, Y, i.e. R̂
varies. Steered direction is fixed.
• Steered response: P (e) when the assumed
~ varies. Data is fixed.
direction, i.e. e(ζ)
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/3
UNIVERSITY
OF OSLO
Minimum variance
beamforming
Capon (1967), Applebaum (1976), Howells
(1976), also called ‘MLM‘
Principle: minimize output of beamformer,
subject to a gain of 1 in the look-direction
min E[|w0y|2] = e0Re subject to Re[e0w] = 1
Weight vector: w =
R−1 e
e0 R−1e
Output power: P M V = w0Rw = [e0R−1e]−1
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/4
UNIVERSITY
OF OSLO
Linear prediction
beamforming
A time-series method adapted to an array, an
element’s output is predicted as a weighted
linear sum of the other elements:
X
∗
wm
Ym
Ym0 = −
m6=m0
Minimize mean-square prediction error
subject to unity gain on predicted element:
0
min E[|w0y|2] = e0Re subject to δm
w=1
0
Solution: w =
R−1 δm0
0 R−1 δ
δm
m0
0
{wm} are not beamformer weights =>
LP cannot be used as a beamformer, only as a
spatial spectrum estimator:
0
δm
R−1 δm0
LP
0
Output power: P = |δ0 R−1e|2
m0
Predicted element, m0: any element, typically
end or center.
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/5
UNIVERSITY
OF OSLO
Properties of correlation
matrix
• Hermitian symmetry: R = (R∗)t = R0
• Positive definite: x0Rx > 0, x 6= 0
This implies that
• The eigenvalues are positive
• The eigenvectors form an orthonormal set.
R=
M
X
λivivi0, R−1 =
i=1
DEPARTMENT OF INFORMATICS
M
X
0
λ−1
i vi vi
i=1
J&D 7, Oct 2000 SH/6
UNIVERSITY
OF OSLO
Eigenanalysis
M-dimensional spatial correlation matrix:
R = Kn + SCS 0
No signals, spatially white noise
R = Kn = σn2 I =
M
X
λivivi0
i=1
Multiple eigenvalues equal to noise variance
1 signal
R = σn2 I + A2ee0 =
M
X
λivivi0
i=1
λ1 = σn2 + M A2, v1 = e1/M
2 incoherent signals, C = diag[A21, A22]
R = σn2 I + A21e1e01 + A22e2e02 =
M
X
λivivi0
i=1
S+N subspace {v1, v2} contains {e1, e2}
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/7
UNIVERSITY
OF OSLO
MUSIC and EV methods
P M V (e) = [e0R−1e]−1
M
X
0 2 −1
=[
λ−1
|e
vi| ]
i
i=1
Signal+noise subspace + noise subspace:
Ns
M
X
X
0 2 −1
0 2 −1
P M V (e) = [
λ−1
+[
λ−1
i e vi | ]
i |e vi | ]
i=1
i=Ns +1
Eigenvector method, Johnson & DeGraaf
1982:
M
X
−1
0 −1
EV
−1 −1
REV
=[
λ−1
(e) = [e0REV
e]
i vi vi ] , P
i=Ns +1
MUltiple SIgnal Classification (MUSIC)
Schmidt 1979, Bienvenu & Kopp 1979
−1
RM
U SIC = [
M
X
−1
−1
vivi0 ]−1, P M U SIC (e) = [e0RM
e]
U SIC
i=Ns+1
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/8
UNIVERSITY
OF OSLO
Coherent signals
Signal+noise subspace has lower rank than
number of signals
Subarray averaging
Linear array of M sensors is divided into
M − Ms + 1 overlapping subarrays each with
Ms sensors. Correlation matrix averaging, OK
rank if Ms ≥ Ns, but loss of resolution
Forward-backward averaging
Only for two coherent signals and linear
array. Average R and RB = JR∗J where J is an
exchange matrix with ones on the
anti-diagonal.
DEPARTMENT OF INFORMATICS
J&D 7, Oct 2000 SH/9