Signal Processing - March 2017 - 41

Processing sequence

Multiple target indication

A maximum likelihood framework
For the Fourier domain models in (4)-(6), and the spatial
domain model in (14), we want to decide between local signal models with K r = 1 and K r = 2 , and estimate the
respective unknown parameters. In a maximum likelihood
framework, an optimal solution for this task is a generalized likelihood ratio test (GLRT) [45]. The GLRT statistic
is given by the ratio of respective likelihood functions,
which have been maximized with respect to the unknown
parameters. A simplified version of the GLRT statistic can
be obtained by the ratio of the mean squared errors of the
respective models using the maximum likelihood parameter estimates.
The maximum likelihood framework is presented for the
model in (4) only. It can be obtained accordingly for the models
in (5), (6), and (14). Using the simplified version of the GLRT,
a decision for a two-target situation is made if
MSE m, 1
2 c,
MSE m, 2
where
2
1
z m - w (mt 0) ft0
lb - la + 1
1
MSE m, 2 =
z m - w (mt 1) ft1 - w (mt 2) ft2
lb - la + 1

MSE m, 1 =

(15)
2

(16)

are the mean squared errors of (4) with K r = 1 and K r = 2 ,
respectively, and c is a suitable threshold. Herein, mt 0 and ft0
are the maximum likelihood estimates in the single target
case, and mt 1, ft1, mt 2 , and ft2 are the maximum likelihood estimates for the two-target case.
The mean square errors corresponding to the models in (5), (6),
and (14), are denoted MSE n, 1, MSE n, 2, MSE o, 1, MSE o,2, MSE 1,
and MSE 2, respectively, and can be obtained accordingly.
A suitable threshold c can be obtained by fixing the false
alarm rate to a certain level, where false alarm refers to the
erroneous decision for the two-target case when only a single
target is present. This can be done empirically via simulations
and should be performed in a conservative way, such that the
two-target case is only detected when reliable parameter estimation is possible.
The maximum likelihood estimates in the single target case
are approximated using a look-up table or a quadratic interpolation approach around the maximum in the periodogram,
as suggested in the section "Practical Aspects." The required
calculation is simple and can typically be performed for every
detected processing cell. The maximum likelihood estimates
in the two-target case are described in the section "High-Resolution Algorithms." Here, the required calculation is computationally intensive and can only be performed for a selected
subset of detected processing cells. This selection should take
into account the deviation from the single target model and is
described next.

A realization of the described maximum likelihood framework
can be obtained by calculating the two-target maximum likelihood estimates only when the single target situation is unlikely
and a multiple target situation is indicated. This indication
is based on a goodness-of-fit test of the single target model
[25], [46]. A test with low computational cost is given by
MSE m,1 2 T,
where T is a suitable threshold that depends on the noise
power. In simple words, a single target situation is considered
if MSE m,1 is of similar magnitude as the estimated noise
power, which can be estimated from neighboring processing
cells without targets. A multiple target indication is considered if MSE m,1 is significantly larger than the estimated
noise power. A suitable threshold T can be obtained by fixing the false alarm rate to a certain level. This can be done
empirically via simulations, or using the approximate distribution of the test statistic under the single target model. Further practical considerations, taking into account model
deviations due to a weak secondary target or an imperfectly
calibrated array, are described in [25]. Note, that the threshold T depends not only on the noise power but also on the
sample support and applied window function in the resolution dimension. However, we omit this dependency for notational simplicity.

Optimal selection of the resolution dimension
The performance of high-resolution frequency estimation
mainly depends on the available signal-to-noise ratio and the
frequency separation normalized to the available sample support. For decoupled multidimensional frequency estimation,
the best result will thus be obtained when the dimension with
the largest frequency separation is selected as the resolution
dimension. We have found in simulations that this corresponds to the largest mean squared error of the single target
model, so that the resolution dimension is selected according
to the largest value among MSE m, 1, MSE n, 1, and MSE o,1 .

Overview
Figure 2(a) shows the signal flowchart for a flexible framework of high-resolution frequency estimation in the Fourier
domain with optimal selection of resolution dimension.
Figure 2(b) shows the flowchart for high-resolution frequency
estimation in the spatial domain, corresponding to the proposed approach in [25], in which the resolution dimension is
fixed to the spatial dimension.

High-resolution algorithms
Resolution dimension
The decoupled parameter estimation method is presented for
the model in (4) only. It can be obtained accordingly for the
models in (5), (6), and (14). In this section, we consider the
model in (4) with K r = 2

IEEE SIgnal ProcESSIng MagazInE

March 2017

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