Title of the invention: Method for determining the detection threshold of a radar
suited to a given environment.
[0001] The present invention relates to a method for determining the detection
threshold suited to a given environment, it applies notably for the surveillance radars
that are provided with an air/sea detection mode. The invention is, for example,
intended for airborne radars.
[0002] The higher (finer) the distance resolution and the more grazing the observation
(case where the altitude of the carrier of the radar is low), the more the signal
returned by the sea, called clutter, deviates from the gaussian statistics. Indeed,
amplitude peaks, called spikes, give rise to so-called heavy-tailed distributions. In
other words, the more grazing the observation, the lower the level of sea clutter
perceived, but spikes are superimposed on the clutter. In this context, the K law is
commonly used to model the sea clutter.
[0003] These spikes lead to an increase in false alarms. In these conditions, it is
necessary to adapt the radar detection threshold to the environment in order to retain
an acceptable probability of false alarms (PFA).
[0004] One technical problem to be resolved is therefore how to determine a
detection threshold, suited to the environment, that is as efficient as possible, that is
to say committing the fewest possible errors on the estimated threshold value. In fact:
- If the threshold is over-evaluated, the PFA is satisfactory but the radar is
desensitized;
- If the threshold is under-evaluated, the PFA is unacceptable.
[0005] Several methods from the prior art are known for resolving this technical
problem.
[0006] A first method consists in uprating in a fixed manner the detection threshold
that would have been calculated in the presence of only thermal noise (purely
Gaussian disturbance). This new threshold is applied to the entire radar map. This
causes the fixed PFA to be maintained. However, the probability of detection is
greatly degraded as predicted by the Neyman-Pearson criterion.
2
[0007] Another known method breaks down the radar map into blocks. For each
block, the probability density of interferences (clutter and thermal noise) is estimated
by regression with a parametric model and the detection threshold is calculated for a
given PFA. More specifically, it is the CCDF (“Complementary Cumulative
Distribution Function”), or complementary distribution function, which is estimated.
This method is relatively independent of a precise clutter model. However, it requires
a large number of training samples per block. Because of this, the inevitable nonuniformities within each block degrade the performance of the method, particularly
when targets or objects other than sea clutter are present.
[0008] A third method could use a modelling of the sea clutter by an a priori law
(generally the K law) to which is added the thermal noise of the receiver. In order to
limit the rate of false alarms linked to this law, the processing proposes a local
adaptation of the detection threshold as a function of the environment encountered
on the basis of the corresponding sea clutter model. For this modelling, the
environment disturbing the detection is characterized by a distribution with several
characteristic parameters, including in particular:
- the total average power of the noise and of the clutter;
- the clutter-to-noise ratio, denoted CNR;
- the form factor of the K distribution.
[0009] One drawback with this approach is notably that the estimation of the
statistical properties can be biased or have a wide variance, for example within
certain CNR regions.
[0010] One aim of the invention is notably to mitigate the abovementioned drawbacks
of the prior art by determining an acceptable PFA threshold.
[0011] To this end, the subject of the invention is a method for determining the
detection threshold of a radar suited to a given environment, this method comprising
at least:
- a step in which a set of statistical quantities characterizing said
environment is selected;
- a step in which a set of functions is defined, each of said functions giving
an intermediate detection threshold that is a function of statistical
quantities taken from a subset of said set of statistical quantities;
3
- a step of combination of said intermediate detection thresholds, said
detection threshold being the result of said combination.
[0012] In a particular implementation, said set of statistical quantities comprising N
statistical quantities, said quantities forming a space with N dimensions, said
combination is such that said space is split into partitions of dimensions smaller than
or equal to N, each function being associated with a partition. Said intermediate
detection threshold is, for example, constant over a partition. The threshold is not
necessarily constant, it is the function which gives the intermediate threshold which
does not change. On the other hand, the function is not necessarily a constant
function and depends on the statistical quantities forming the subset.
[0013] When two partitions overlap, the detection threshold on the zone of overlap is,
for example, a function of the thresholds associated with said two partitions. The
detection threshold in the zone of overlap is, for example, a weighted sum of the
thresholds associated with said two partitions.
[0014] The number of statistical quantities contained within said set is strictly greater
than two. The statistical quantities characterize, for example, sea clutter and thermal
noise.
[0015] Said functions giving said intermediate thresholds are, for example,
determined for different environment configurations.
[0016] Another subject of the invention is a radar implementing the method as
previously described.
[0017] Other features and advantages will become apparent from the following
description, given in light of the attached drawings which represent:
[Fig.1] Figure 1, an illustration of the sea clutter in grazing observation mode;
[Fig.2] Figure 2, a presentation of the steps implementing the method according to
the invention;
[Fig.3] Figure 3, the principle of an example of contrast detection at distance;
[Fig.4] Figure 4, an illustration of the principle of deduction of the detection threshold
according to the invention.
4
[0018] Figure 1 illustrates the appearance of the signal 1 returned by the sea (clutter)
in the case of grazing radar observation, the carrier being at low altitude. The clutter
comprises amplitude peaks 2 (spikes) leading to an increase in false alarms, as
explained previously.
[0019] Figure 2 illustrates the different functions of implementation of the method
according to the invention. The latter is based on the third method of the prior art
presented previously. However, it differs from that in that there is no effort to explicitly
estimate the parameters which usually characterize the distribution, notably the form
factor of the K distribution of the clutter. The solution provided by the invention is
therefore different from the methods for characterizing the clutter distribution such as
those described notably in the document by D. Blacknell and R. J. A. Tough,
“Parameter Estimation for the K-distribution based on [zlog(z)]”, IEE Proceedings -
Radar, Sonar and Navigation, Vol. 148, No.6, pp. 309-311, 2001 (document D1). In
the present invention the objective is to directly deduce what detection threshold
should be put in place as a function of measurements of various statistical quantities
(parameters) obtained easily with a population of learning data. Without losing
generality, the simplifying assumption is made that the environment is made up of
sea clutter and of thermal noise, without the presence of disturbing phenomena of
precipitation cloud type or other phenomena. The invention is based on the use of
data sets, a limited number of samples being considered in order to respect the
uniformity of the samples and to limit the required computation power.
[0020] The signals 21 received by the radar are sampled in a conventional way for
digital processing. For the detection, the power of these received signals is
considered, the power of a sample being conventionally calculated on the basis of its
I and Q components. It is these power data which will be taken into account with the
thresholding function 10 to confirm a detection.
[0021] Obtaining the thresholding function 10 begins with a step 22 in which a set of
statistical quantities characteristic of the environment is defined. The statistical
quantities that are chosen are also a function of the signals, and therefore of the
power data 21 that are expected to be processed.
[0022] Thus, a set of N statistical quantities which make it possible to characterize the
environment is defined.
,…,
denotes these N characteristic quantities. In the
5
case of the invention, N is strictly greater than 2. Some of the parameters
can be,
notably:
- the clutter-to-noise ratio, denoted CNR,
- and the impulsivity of the received signals (characterizing the impulsivity
of the mixture of clutter and thermal noise), denoted IMP.
[0023] These two parameters characterize the appearance of the clutter and
therefore the environment. In the case of grazing observation, the appearance of the
clutter is as illustrated by Figure 1, with a low CNR and a high impulsivity
characterized by the spikes 2.
[0024] The impulsivity can be expressed by a quantity defined according to the
following relationship:

≜ log
1

−
1

log

in which designates the number of learning samples and
the ith sample this
clutter 2. In other words, IMP is the difference of the logarithm of the arithmetic mean
of the learning samples minus the logarithm of the geometric mean of the same
learning samples.
It is known that this estimator IMP or even that of the document D1 “[zlog(z)]”,
provides an estimation of the optimal detection threshold, close to the criterion of
maximum likelihood in the absence of thermal noise or if the clutter-to-noise ratio is
very much greater than 1 (or if CNR >> 1). However, it is also known that, with low
CNR, the estimation is very inaccurate. In these conditions, the detection threshold to
be set is therefore = fIMP (IMP, CNR), a threshold that can be established by
calculation or by simulation (offline).
[0025] With low CNR ratio, it is also possible to use the statistical moments M1, M2,
M3 or even M4 (first, second, third or even fourth order moments) as statistical
quantities. The document D1 notably proposes five methods for estimating statistical
quantities based on the moments and log-moments in order to quantify the “impulsive”
nature of the sea clutter. It is also possible to use other statistical quantities deduced
from these moments (Kurtosis, Skewness, etc.). However, a problem specific to
these estimators is their great sensitivity to the possible presence of targets in the
6
population of samples, giving rise to a high overestimation of the threshold. These
known solutions are therefore not satisfactory.
[0026] According to the so-called “ordered statistics” approach, it is also possible to
define, as statistical parameter (estimator), the number of samples Qλ crossing one
or more power levels T = µλ, in which µ designates the local mean of the samples,
several mean operators being able to be used (arithmetic or geometrical in particular).
This estimator, much less sensitive to the presence of targets, therefore consists in
counting, over a range of resolution of the radar, the number of overshoots with
respect to a test level T = µλ. This test is facilitated by a step 23, the power of the
samples received 21 is standardized with respect to a local average, for example by
using the known CA-CFAR (“Cell Average – Constant False Alarm Rate”) method. A
set of standardized samples is thus obtained with a target average value of 1.
[0027] This test of level relative to a local mean can be performed after a contrast
detector of CA-CFAR type but also after any other known contrast detector. The test
therefore amounts to counting the number of contrasts greater than T/µ = λ. Thus, a
target whose level exceeds T counts as an overshoot regardless of the power of the
target, hence the lesser sensitivity of this estimator mentioned above.
[0028] Figure 3 illustrates the principle of contrast detection at distance, of CA-CFAR
type. In this example, the local average µ is calculated using reference averages
before the distance-under-test cell and after the cell under test. The test consists in
comparing the power PCUT of the cell under test relative to T = µλ. If PCUT > T, 1 is
counted, otherwise 0 is counted. The operation is repeated in a sliding manner over
all the resolution cells (distance cells) of the range being processed. The term λ is
ideally chosen so that the number of overshoots “at rest” over thermal noise is
greater than the maximum number of targets likely to be found within an interval of p
resolution cells of the radar, in practice of the order of ten or so. In the general case,
Qλ overshoots of the test value T are obtained.
[0029] According to the latter approach, it is possible to deduce the detection
threshold to be set S = fλ(Qλ, CNR). An improvement consists in adding IMP as third
argument and therefore S = fλ,IMP(Qλ, CNR, IMP). The functions fλ or fλ,IMP can be
obtained by calculation or by simulation (offline).
7
[0030] The statistical parameters presented above are not limiting. Other statistical
parameters can be used to characterize the environment in this step 22.
[0031] In another step 24, a set of functions

,…,
is defined, such that each
function provides a more or less accurate estimation of the detection
threshold,
being a set of statistical quantities. These functions
are, for
example, obtained by simulation. There is not necessarily any analytical expression
of these functions, so they can therefore be defined by tabulation. There is the
following inclusion: ⊂ , that is to say that each estimation function uses only a
subset
of the total set of the statistical quantities . In the representation of
Figure 2, the function is applied to a subset !, … , "
, the function # is applied to
a subset $, … ,
and the Mth function is applied to a subset %, … , &.
[0032] The set of functions
is chosen such that, in the set of the functions ,
there is always at least one function which makes it possible to estimate the
threshold ' “optimally” (in the sense of the preceding criterion) at each point of the
useful range of the environment parameters (statistical quantities).
There is therefore at least one threshold ' = .
[0033] These different functions of the set

,…,
making it possible to directly
estimate the detection threshold as a function of certain quantities
, can be
determined either by Monte-Carlo methods, or by a deterministic method.
[0034] In another step 25, a function of combination of the functions
is defined to
define the final threshold.
The final threshold * is therefore obtained by a function of combination + of the
intermediate thresholds:
* = +,', … , '|, … ,
.
This combination function will be detailed below.
[0035] The steps 22, 24 and 25 are established beforehand in order to be
subsequently used for the processing of the power of the received signal samples 21.
In other words, the thresholding function defined from the definition of the
environment parameters (step 22), from the functions
(step 24) and from the
8
combination function (step 25) is calculated for all the input data 21 used during the
radar mission, at least for a mission period that can be characterized by the
environment parameters retained. The thresholding function S is thus pre-calculated
and it is applied in the thresholding step 10 to the power data 21 of the samples
received, after these data have been standardized 23 as described previously.
[0036] For the combination function, the principle of the invention is to use several
intermediate detection thresholds '
each determined as a function of various
statistical parameters
(by the functions
) in order to combine them to have a
detection threshold * that is the most robust to the different environment
configurations which can be encountered.
[0037] One possible solution is to partition the space of the statistical parameters
so as to assign a threshold calculation mode to each partition. In a very simple case
of the two threshold estimators */0 = /0,
, +23. and '4 = 4,54, +23.
described previously, an example of combination function C can be defined as
follows.
* = +,*/0, '4|+23.
With:
6
+ = /0 7 +23 > +239:9
+ = 4 7 +23 < +23"<=
+ = >,+23./0 + ,1 − >,+23.. 4 @AℎCDE7'C F
in which CNRhigh denotes the “boundary” of CNR above which the threshold sIMP is
optimal and CNRlow denotes the “boundary” of CNR above which the threshold sλ is
optimal with a zone of transition (overlap) between the two “boundaries”.
The function α(CNR) lies between 0 and 1 and makes it possible to switch
progressively from one estimator to the other in the zone of overlap. It is for example
possible to use a modified sigmoid function such that its value is 0 in CNRlow and 1 in
CNRhigh. Typical limit CNR values are, for example: CNRlow = -3 dB and
CNRhigh = +6 dB.
9
[0038] Figure 4 illustrates the partitioning of the space of the parameters in a wider
case, with four functions and four parameters. Based on this positioning, the
detection threshold is deduced therefrom. As an example, the four environment
parameters retained are:
- The clutter-to-noise ratio, CNR;
- The impulsivity, IMP;
- The number of overshoots of level 5G
for example for G = +6 IJ;
- The ratio of the second order statistical moment to that of the first order.
[0039] To select a function, a space is taken with N dimensions, N being the number
of environment parameters used. In the example of Figure 4, N is equal to 4. The
space with N dimensions is partitioned into subspaces. Depending on the values of
the parameters , #, K, L, the location is a given partition corresponding to a
function fi..
[0040] For simplicity of representation, the partitioning is represented only in two
dimensions, based on two characteristic quantities and # (for example = CNR
and # = IMP), whereas four characteristic quantities are used: , #, K, L. The
partitioning space therefore has four dimensions. The partitions can be contiguous or
exhibit a certain overlap. The number of partitions is equal to the cardinal M of

,…,
, corresponding in fact to the number of functions fi established.
[0041] Thus, by way of example, for X1 = X10 and X2 = X20, the threshold S is defined
by the threshold s1 = f1 (X1, X2) and for X1 = X11 and X2 = X21, the threshold S is defined
by the threshold s4 = f4 (X1, X2, X4).
[0042] Each partition is associated with a function
, but it is possible for partitions to
overlap. Within these overlaps, it is possible to provide a weighting between the
functions specific to each overlapping partition. Thus, in the zone of overlap 31
between the partition 32 associated with # and the partition 33 associated with K, it
is possible to provide a threshold s23 = α s2 + β s3, α and β being less than 1 and a
function, for example, of the respective areas of the partitions 32, 33. This principle
can be extended to an overlapping of more than two partitions.
10
[0043] The partition of Figure 4 is pre-established, it is matched to a given PFA. The
space of the environment parameters with N dimensions (N = 4 in the example of
Figure 4) thus partitioned is memorized in the processing means of the radar with the
associated functions
. These thresholding elements are for example stored in
tabulation form.
[0044] The final threshold defined by the combination function, for example according
to the partition in the space of the environment parameters, is the detection threshold
sought. The power of the signals received is compared with this threshold. With
respect to Figure 2, the powers of the samples received are, for example,
standardized 23 before comparison 10 with the threshold.

CLAIMS
1. Method for determining the detection threshold of a radar suited to a given
environment, characterized in that it comprises at least:
- a step (22) in which a set of statistical quantities characterizing said
environment is selected;
- a step (24) in which a set of functions is defined, each of said functions
giving an intermediate detection threshold that is a function of statistical
quantities taken from a subset of said set of statistical quantities;
- a step (25) of combination of said intermediate detection thresholds,
said detection threshold being the result of said combination;
the number of statistical quantities contained in said set being strictly
greater than two.
2. Method according to Claim 1, characterized in that, said set of statistical
quantities comprising N statistical quantities, said quantities forming a space
with N dimensions, said combination is such that said space is split into
partitions (32, 33) of dimensions smaller than or equal to N, each function
being associated with a partition.
3. Method according to Claim 2, characterized in that said intermediate detection
threshold is constant over a partition.
4. Method according to either one of Claims 2 and 3, characterized in that, when
two partitions overlap, the detection threshold on the zone of overlap (31) is a
function of the thresholds associated with said two partitions.
5. Method according to Claim 4, characterized in that the detection threshold in
the zone of overlap (31) is a weighted sum of the thresholds associated with
said two partitions.
6. Method according to Claims 1 and 2, characterized in that said set of
statistical quantities comprises at least two quantities from among the
following quantities:
- the impulsivity of the received signals;
- the signal-to-noise ratio;
- the number of overshoots of a preestablished power level.
12
7. Method according to any one of the preceding claims, characterized in that
said statistical quantities characterize sea clutter and thermal noise.
8. Method according to any one of the preceding claims, characterized in that
said functions giving said intermediate thresholds are determined for different
environment configurations.
9. Method according to any one of the preceding claims, characterized in that
said radar is airborne.
10. Radar, characterized in that it is capable of implementing the method
according to any one of the preceding claims.