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1
Deterministic Maximum
Likelihood for DoA Estimation:
Algorithm and Implementation

2
Reference papers
• P. Hacker and B. Yang : Single snapshot DOA estimation”
• Rife, D. C. and Boorstyn, R. R.: Multiple Tone Parameter Estimation
from Discrete-Time Observations, The Bell System Technical Journal,
55, 1389–1411, 1976
• Viberg, M., Ottersten, B., and Nehorai, A.: Estimation Accuracy of
Maximum Likelihood Direction Finding Using Large Arrays, Signals,
Systems and Computers, 928–932, 1991b.

3
Algorithm descriptions
Let s(t) be the incoming waves after mixing to baseband, the sensor array
signal to be processed is given by
X(t) = A(θ)s(t) + n(t)
where A(θ) = (a(θ
1
), …a(θ
M
)) is the steering matrix
a(θ) = (e
j2πy
1
sin(θ)
, …, e
j2πy
N
sin(θ)
) is the steering vector
M is number of targets
y
n
is the sensor position normalized by wavelength
DML approach is
θ
DML
= argmax
θ
{trace(Π
A
(θ)*R
n
)}
where R
n
is the covariance matrix from the single snapshot antenna
samples.
Π
A
(θ) = A(θ) * [A(θ)
H
* A(θ)]
-1
* A(θ)
H

4
Implementation details
• Assumptions:
– M = 2 always detect 2 objests from antenna samples. Will also calculate confidence
metric for following module to filter results.
– Receive antennas are equally spaced
• Calculation of Π
A
(θ) can be significantly reduced with a(θ) constructed
for equally spaced antenna. It becomes an Hermitian matrix that is also
persymmetric.
• Given that R
n
is also Hermitian, calculation of {trace(Π
A
(θ)*R
n
) can also be
simplified.
• Total cycle cost depends on the length of steering vector a(θ).
– 2 stage search: first stage of coarse search, then zoom in with finer granularity in
second stage search.
• Output of normalized variance of per target (dimension) metrics
– The smaller the normalized variance, the stronger the signal from the
particular target
– Experimenting a threshold to determine single target.

5
Results from MATLAB simulator (1)
channel_target_initPosition = {[0.8682,
4.9240, 0], [-0.8682, 4.9240, 0]}; % m in [x,
y, z] -10/+10degree
Detection output:
angles(I)= 8.5000 -9.0000
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