FR2769373A1 - Method of detection of Doppler effect of low speed targets - Google Patents

Abstract

The method involves receiving the values of echoes in response to the train of successive coherent interrogation impulses. A high resolution spectral analysis of the echoes and validation via an auto regressive model is determined using coefficients of reflection and the obtained spectra are tested based on the calculation of the probability values.

Description

DOPPLER DETECTION METHOD
TARGETS WITH LOW OR NULL RADIATION SPEED
The present invention relates to the Doppler detection of targets with a low radial or zero velocity, with moving parts, generating echoes whose Doppler spectra are distinguished from those of clutter by minimal differences in shape, such as, for example, atmospheric turbulence or turbulence. helicopters hovering.

 Until now, an efficient Doppler radar detection of atmospheric turbulence has not been achieved, although such detection is of fundamental importance for the safety of aircraft flight because it is impossible, with current treatments by banks of Doppler filters, to distinguish the spectrum of atmospheric turbulence from that of a cloud in calm air.

 The detection of a helicopter by means of a Doppler radar poses a particular problem insofar as the helicopter constitutes for the radar a target which may have a low or zero radial velocity and which consequently can not be detected every time. , on static clutter (ground type, ...), by a simple shift analysis of the average Doppler frequency of its echo with respect to the zero frequency.

 Currently, helicopter detection is essentially from a phenomenon known as blade lightning. Such a phenomenon is constituted by a periodic intensification of the amplitude and the bandwidth of the Doppler spectrum of the echo returned by a helicopter caused by intense reflection of the blades of its rotor when they appear in the lobe of the rotor. radar antenna, perpendicular to the radar-helicopter direction.

The periodicity of the blade flashes is related to the speed of rotation and the number of rotor blades of the helicopter concerned. It is relatively long because the rotational speed of the rotor of a helicopter is limited by the speed of the airflow at the end of the blade. It requires, to reliably detect a blade flash when a helicopter is present, to illuminate each site-azimuth-distance box monitored by the radar over a long period of time, either by adopting a wide antenna beam in azimuth, or by adopting for the radar antenna, a low azimuth scanning speed.

 The object of the present invention is to enable the detection of atmospheric turbulence by means of a Doppler radar based on a weak broadening of the Doppler spectrum of the echoes returned by turbulence which results in a multi-dimensional appearance more pronounced than that of the Doppler spectrum. echoes returned by a cloud in calm air.

 It also aims to avoid the previous limitations of Doppler detection of a helicopter by lightning of blade by not basing more the detection of a helicopter only on the lightning of blades but especially on a weak widening of the Doppler spectrum of its echo with respect to the Doppler spectra of clutter echoes. Indeed, the Doppler spectrum of the echo of a helicopter outside the lightning of blade has a multiraie aspect more pronounced than that of the echoes returned by clutter of soil of rather monoraie nature, because of the presence of mechanical parts in Rotation contiguous to the rod of its rotor (blade incidence control rods).

 Since the Doppler spectrum extensions sought for echoes due to atmospheric turbulence or a helicopter outside a lightning strike are rather weak, it is proposed to rely on a high-resolution spectral analysis of radar echoes from site-azimuth-distance cases and to adopt, as a likelihood criterion for the presence of a turbulence or a helicopter, one or more spectrum-width tests based on the distribution of the eigenvalues of the decomposition in signal subspaces more noise and noise alone from the autocorrelation matrix of the sampled echo signal. Most of these tests are based on the test of equality of (N-p> smaller eigenvalues, where N is the number of samples analyzed and p is an integer smaller than N, proposed by LIGGET W. S.

[1] WS LIGGET, "Passive sonar: Fitting models to multiple time series." in Proc. NATO Signal Processing Institute, Loughborough, England, 1972, pp. 327-345, detailed and justified from a probabilistic point of view by Kopp L. and Bienvenu
G.: [2] WELCOME G., KOPP L., "Optimality of High Resolution Array
Processing Using the Eigensystem Approach, "EEE Trans., On ASSP, vol.31, n05, October 1983, pp. 1235-1248, but they are much more robust in the usual case where only a small number is available. echo samples.

The subject of the invention is a method for the Doppler detection of targets with low radial or zero velocity and with moving parts, consisting of
to collect the samples of N successive echoes coming from the same site-azimuth-distance box, in response to the emission of a train of N consecutive and coherent interrogation pulses,
to perform a high-resolution spectral analysis of the N successive echo samples by means of an autoregressive model of order n (n strictly less than N) defined by a series of n reflection coefficients k and leading to a spectrum modeling at n frequency lines of various amplitudes, the frequency lines of higher amplitudes, corresponding to frequency lines actually existing in the signal, while others of smaller amplitudes may be solely due to a modeling noise and not to have physical reality,
- to check the equality between the np lines of smaller amplitudes of the spectrum modeling (p being less than or equal to n) to ensure, either that they are due only to modeling noise, or that they belong to a spectrum broad enough to correspond to that of a target, this verification being effected by means
the likelihood value calculation

of the selection of the distribution law of probability of the
Khi-2 to (N + 2) (N + 1) / 2 degrees of freedom,
the choice of an arbitrary false alarm probability,
determining a given upper threshold value
by the distribution law of probability of Chi-2 based
the probability of false alarm adopted, and
of the likelihood value comparisonLog (A (p))
at the upper threshold value obtained, a value of
likelihoodLog (A (p)) less than said threshold value
superior confirming the likelihood of equality of np
lines of lower amplitudes of the modeling of the
spectrum.

 Advantageously, the number p is given the value 0. This makes it possible, by comparing the likelihood value Log (A (o)) with an associated higher threshold value SP1 determined by means of the distribution law of probability of Chi-2. from a chosen false alarm rate VP, to test the likelihood of an expanded signal spectrum corresponding to a blade flash, a value Log (A (O)) found to be lower than the associated higher threshold value SP1, showing the relative equality of the amplitudes of all the frequency lines of the spectrum modeling, which implies that the echo signal has a flat spectrum corresponding to that of a helicopter blade flash.

Advantageously, after having given the number p the value 0 and tested the inequality of the amplitudes of the frequency lines of the spectrum modeling, the number p is given the value 1, which makes it possible, by comparison of the likelihood value Log (A (1)) with an associated upper threshold value SK determined by means of the Chi-2 probability distribution law from a chosen false alarm rate VK, to confirm the likelihood of a signal spectrum of single-line ground clutter echoes corresponding to the highest amplitude line of the spectrum modeling, a Log likelihood value (A (i)) found to be lower than the associated threshold value
SK, showing the equality of the n-1 lines of smaller amplitudes while the inequality of the amplitudes of all the frequency lines of the spectrum modeling has been noted elsewhere.

étant le k ième paramètre dit autorégressif du modèle autorégressif d'ordre n, définissant la loi de prédiction d'un i ième échantillon x, du signal dont on fait l'analyse spectrale haute résolution, à partir des (i-p) échantillons précédents, par une relation de la forme

étant étant une erreur de prédiction dans le sens direct, par
le modèle d'ordre p, de l'échantillon x,
à comparer la valeur obtenue du terme z1 avec une valeur de
seuil SP2 déterminée expérimentalement pour avoir un taux
de fausse alarme arbitraire, et
à admettre la présence d'un spectre Doppler de signal élargi
correspondant à un éclair de pale en cas d'une valeur
obtenue pour le terme n inférieure au seuil SP1. Advantageously, the comparison of the likelihood value Log (A (O)) with a threshold for detecting the presence of an expanded signal Doppler spectrum corresponding to a flash of a blade is completed by a test on the autoregressive filter residue having led to the modeling of the echo signal spectrum, this test consisting of
to evaluate the value of # equals

being the k th autoregressive parameter of the autoregressive model of order n, defining the prediction law of a i th sample x, of the signal of which the high resolution spectral analysis is made, starting from the (ip) preceding samples, by a relationship of form

being a prediction error in the direct sense, by
the model of order p, of the sample x,
to compare the value obtained from the term z1 with a value of
threshold SP2 determined experimentally to have a rate
arbitrary false alarm, and
to admit the presence of an expanded signal Doppler spectrum
corresponding to a blade flash in case of a value
obtained for the term n below the threshold SP1.

avec, par définition,

Advantageously, in the case of a signal frequency spectrum close to a monaural spectrum, ie leading to a likelihood value Log (# (1)) less than the upper threshold value set
SK, we test the distance between the n-order real spectrum model and the first-order monaural spectrum model by
a distance measurement d2 based on the parameters
cepstral Ck which are the Fourier coefficients of the
logarithm of the spectral density of the echo signal and which
are calculated from the autoregressive parameters

with, by definition,

akm being, as indicated previously, the k th
autoregressive parameter of the model of prediction of order m
regularized,
a comparison of the value obtained for the measurement of
distance d2 with a determined threshold value SL1
experimentally to have a false alarm rate
arbitrary, and
a presence deduction of an expanded signal spectrum
corresponding to a target as soon as the value obtained for the 2
distance measurement d2 is greater than the threshold value
SL1.

As a variant, in the case of a signal frequency spectrum close to a monaural spectrum, that is to say leading to a likelihood value Log (A (1)) below the set threshold SK, the distance between the n-order real-spectrum model and the first-order monamy spectrum model by:
# a distance measurement d2 defined according to the cosine
of the angle between cepstral vectors which are, as indicated
previously, the Fourier coefficients of the Logarithm of
the spectral density of the echoes

a comparison of the value obtained for the measurement of
distance d2 with a determined threshold value SL2
experimentally to have a false alarm rate
arbitrary, and
a presence deduction of an expanded signal spectrum
corresponding to a target as soon as the value obtained for the
distance measurement d2 is greater than the threshold value
SL2.

Other features and advantages of the invention will emerge from the following description of an embodiment of the invention given by way of example. This description will be made with reference to the drawing in which
FIG. 1 is a flowchart illustrating the succession of tests implemented, according to the invention, on the Doppler spectrum of the echoes to detect helicopters in the middle of the ambient clutter, and
FIG. 2 is a block diagram of the video portion of the receiver of a Doppler radar embodying the invention.

It is proposed to detect targets with low or zero radial velocity returning echoes having Doppler spectra of shapes similar to but different from that of the Doppler spectrum of the clutter in which they are embedded, this detection being done using a radar
Doppler with a short observation time, delivering, for each target, only a small number of successive echo samples. Examples of such targets include all low or zero radial velocity targets with relatively moving parts such as atmospheric turbulence or helicopters hovering, pop-up, or tangential flight outside the trivial case. Blade flash. This kind of targets is distinguished from the clutter only by differences in shape of their Doppler spectrum rather tenuous that it is difficult to identify the output banks of filters that usually equip the Doppler radars. Also, instead of the usual Fourier transform spectral analysis carried out using narrow band filter banks, it is proposed to use a high resolution spectral analysis which has the advantage of providing frequency spectra. more accurate by at least one order of magnitude.

 For a better understanding, we quickly recall the philosophy of high resolution spectral analysis methods.

où
i est l'indice de chaque échantillon. On suppose qu'il y a N échantillons et que l'indice i est un entier variant de 1 à N.
a1 est l'amplitude complexe de la léme sinusoïde complexe
ol est la fréquence angulaire de la léme sinusoïde complexe
v- est le iéme échantillon du bruit blanc. There are two approaches to high resolution spectral analysis methods: an eigenvalue and eigenvector approach of the signal autocorrelation matrix and an autoregressive modeling approach, which provide complementary lessons. Both approaches are based on the assumption that the analyzed time signal is sampled at a constant time interval and composed of L uncorrelated complex sinusoids, zero average values, and zero average white noise. Samples uj of such a signal are defined by a relation of the form

or
i is the index of each sample. It is assumed that there are N samples and that the index i is an integer ranging from 1 to N.
a1 is the complex amplitude of the complex sine-lemma
ol is the angular frequency of the complex sine-lemma
v- is the third sample of the white noise.

The fact that the complex sinusoidal components of the signal are decorrelated means that we have

<tb><SEP> Pl, <SEP> if <SEP> k <SEP> = <SEP>
<tb><SEP> E [&alpha; k &alpha; l *] <SEP> = <SEP> {
<tb> 0, <SEP> if <SEP> k <SEP>#<SEP> l
<Tb>
E is the mathematical expectation. * denotes the conjugate complex operator and Pl.the average power of the first complex sinusoidal component.

The fact that the noise is white means that we have

# 2, <SEP> if <SEP> j <SEP> = <SEP> i
<tb> E [vivj *] <SEP> = <SEP> {
<tb><SEP> 0, <SEP> if <SEP> j <SEP>#<SEP> i
<tb> a2 is the variance of the noise.

The approach of the high-resolution spectral analysis methods according to the eigenvectors and eigenvalues of the autocorrelation matrix of the signal is based on the observation that the autocorrelation matrix Rm of order m of the signal defined and averaged over m successive samples (m being less than the number N of available samples) can be written in the form of the matrix relation
Rm = SmDSm ++ # 2 # m (1) where
+ denotes the transconjugated complex operator
Sm is a rectangular matrix with m lines and L columns,
said signal matrix, defined by

T designating the transposed operator
D is a diagonal matrix defined by
D = diag (P1, P2, ..., PL)
1m is the identity matrix of dimensions m by m.

Since the autocorrelation matrix Rm is Hermitian and positive definite, it has real and positive eigenvalues and orthogonal eigenvectors as well as the SmDSm + matrix. Let # m1 # # m2 .... # #mm the eigenvalues of the autocorrelation matrix Rm and vm1 # vm2 .... # vmm the eigenvalues of the SmDSm + matrix. We deduce from relation (1) that:
#mi = vmi + # 2, i = 1,2, ...., m
Since the signal matrix Sm is composed of L linearly independent non-zero vectors, the (ml) smaller eigenvalues of the SmDSm + matrix are zero. As a result, the (mL) smaller eigenvalues of the correlation matrix Rm are equal to a2

ou encore, d'après la relation (1)
SmDSm+qmi = 0
Comme tous les éléments de la matrice signal Sm ainsi que ceux de la diagonale de la matrice D sont différents de zéro, cette dernière relation implique que l'on ait smqm o, i=L+l,...,m où plus explicitement en tenant compte de la relation (2)

<tb> vmi <SEP> + <SEP># 2, <SEP> i <SEP> = <SEP> 1, ..., L
<tb><SEP>#mi<SEP><SEP> = <SEP> {
<tb># 2, <SEP> i <SEP> = <SEP> L <SEP> + <SEP> 1, ..., m
<Tb>
Let qm1, qm2, ..., qmm be the eigenvectors of the autocorrelation matrix Rm associated with the eigenvalues # m1, # m2, ..., # mm. The (mL) eigenvectors associated with (mL) smaller eigenvalues satisfy the relation:
Rmqmi = # 2qmi i = L + 1, ..., m which is written again:

or again, according to the relation (1)
SmDSm + qmi = 0
Since all the elements of the signal matrix Sm as well as those of the diagonal of the matrix D are different from zero, this last relation implies that we have smqm o, i = L + l, ..., m where more explicitly taking into account the relation (2)

<SEP> i <SEP> = <SEP> L <SEP> + <SEP> 1, ..., m
<tb> Sml + qi <SEP> = <SEP> 0, <SEP>#
<tb> l <SEP> = <SEP> 1,2, ..., L
<Tb>
Thus, the complex sinusoidal components of the analyzed signal are perpendicular to the eigenvectors corresponding to the (mL) smaller eigenvalues of the autocorrelation matrix of the signal.

A fundamental property of the eigenvectors of the autocorrelation matrix is, as we have seen previously, that they are orthogonal to each other. The eigenvectors qm1, ..., qm @ thus generate a subspace orthogonal to the subspace generated by the remaining eigenvectors q, L + i, ..., q, m. It follows then from the previous relation that the vectors 5m (<iSl> generates the same subspace as the eigenvectors qmi (<i <L>, so we can make the following two important remarks on the autocorrelation matrix Rm of m successive samples of a temporal signal composed of a linear combination of L (L <m) complex sinusoids and white noise
1. The space generated by the eigenvectors of this
autocorrelation matrix Rm is formed of two sub
disjointed orthogonal spaces: a first subspace
called signal subspace plus noise generated by
eigenvectors associated with the L largest values
own of Rm and a second subspace called sub
noise generated by the eigenvectors associated with
(mL) smaller eigenvalues of Rm.

2. The knowledge of the eigenvectors of the matrix
of autocorrelation Rm makes it possible to determine the frequencies
angular complex sinusoids composing the signal
analyzed by searching for vectors of
complex sinusoidal components orthogonal to the sub
noise space.

avec les vecteurs propres qmL+l, qmL+2,...,qmm du sous-espace bruit

An immediate consequence of the second remark is that the power spectrum of the signal can be expressed as a function of the angular frequency by the inverse of a combination of the squares of the scalar products of a frequency scanning vector em (O ) corresponding to a sinusoidal component of variable angular frequency

with the eigenvectors qmL + l, qmL + 2, ..., qmm of the noise subspace

<tb> Spectrum (w)
<tb><SEP> E <SEP> | qk + e <SEP> (w) |
<tb><SEP> k = L + 1
<Tb>
In practice, we do not have the autocorrelation matrix of the signal but an estimate Rm with eigenvectors xm1, xi2 ,, ..., xmm that can be ordered by eigenvalues # m'1, # m'2, ..., # m'm of decreasing values. Difficulties arise because of the finite number of samples available for the temporal signal to be analyzed, which makes the elements of the autocorrelation matrix are estimated all the more roughly because they correspond to a delay and therefore to a difference between correlated samples large. The orthogonality relations are not really respected any more and one is forced to look for the vectors of complex sinusoidal components closest to the orthogonality to the subspace noise. This leads to variations in the expression of the estimated frequency spectrum of the signal according to the various types of high resolution methods.

où m est l'ordre du modèle, c'est-à-dire le nombre d'échantillons précédents pris en compte dans la prédiction, (akm) une suite de m coefficients complexes définissant le modèle et Ef,i,m une erreur de prédiction dans le sens direct, par le modèle d'ordre m, de l'échantillon u qui doit être assimilable à un bruit blanc si le modèle est fidèle,
- soit une relation de prédiction dans le sens rétrograde, de la forme

où {bkm} est une suite de m coefficients complexes définissant le modèle et #b,i-m,m une erreur de prédiction dans le sens rétrograde, par le modèle d'ordre m, de l'échantillon ui.m qui doit également être assimilable à un bruit blanc si le modèle est fidèle.The second approach of high resolution spectral analysis methods using autoregressive modeling is based on the fact that the u samples of a signal composed of a finite number L of complex sinusoids can be modeled by means of a prediction law allowing to deduce the value of a sample one from a linear combination of the nm samples that precede it or the m samples that follow it. Autoregressive modeling is like looking for
- either a prediction relation in the direct sense, of the form

where m is the order of the model, that is, the number of previous samples taken into account in the prediction, (akm) a sequence of m complex coefficients defining the model and Ef, i, m an error of prediction in the forward direction, by the model of order m, of the sample u which must be assimilated to a white noise if the model is faithful,
- a prediction relation in the retrograde sense, of the form

where {bkm} is a sequence of m complex coefficients defining the model and # b, im, m a prediction error in the retrograde direction, by the model of order m, of the sample ui.m which must also be assimilable to a white noise if the model is faithful.

We show that the coefficients {bkm} of the prediction law in the retrograde direction are deduced from the coefficients {akm} of the prediction law in the forward direction by the relation bkm = akm * that the signal power as a function of the frequency angular is expressed by the relation

pour une erreur quadratique minimum s'exprime par l'équation normale de Yule-Walker :
RmAm = -Cm (5) avec
Cm=[c1,...,cm]t et

<tb> Spectrum (O) = <SEP> = <SEP> 2
<tb><SEP> rn
<tb><SEP> E <SEP> ame-jtsk
<tb><SEP> k = 0
<tb> with am0 equal to 1 and P a noise power, and the vector Arn of the {akm} coefficients of the prediction law in the forward direction

for a minimum squared error is expressed by the normal Yule-Walker equation:
RmAm = -Cm (5) with
Cm = [c1, ..., cm] t and

 The interest of the coefficients of the direct prediction law akrn} also known as autoregressive coefficients is that recursive methods are known on the order of the model allowing to obtain them in real time by means of a lattice filter structure easy to implement.

 Here again, it is noted that the knowledge of a limited number of samples of the signal makes it possible to make only an approximate estimate of the correlation matrix of the signal leading to an approximate knowledge of the autoregressive coefficients.

 The autoregressive coefficients can be obtained from the normal Yule-walker equation by simple matrix inversion, but we are not assured of conditioning the autocorrelation matrix of the signal, which is available only estimate.

To turn this difficulty, Levinson proposed a recursive algorithm on the order. To do this, it uses the Toeplitz characteristic of the signal autocorrelation matrix to express the autocorrelation matrix for a model of order m as a function of the autocorrelation matrix for the immediately lower order model n 1 and draw two relationships
a recurrence relation called the Levinson equation making it possible to determine the autoregressive coefficients, with the exception of the last one, by a recurrence on the order of the model

+ désignant l'opérateur complexe transconjugué et (-) I'opérateur:
V(-) = JV* où J est la matrice unité antidiagonale.a relation of definition of the last autoregressive coefficient of a model of order m as a function of the autoregressive coefficients and terms of the autocorrelation matrix of the immediately lower order model

+ designating the transconjugated complex operator and (-) the operator:
V (-) = JV * where J is the antidiagonal unit matrix.

The first relationship is interesting because it is easy to implement using a lattice calculation structure. On the other hand, it is not the case of the second relation which makes depend the last autoregressive coefficient of a term cm whose estimation can present a great variance in the case where one only has few samples of the signal. To solve this problem, Burg proposes to preserve the first recurrence relation or Levinson equation and to replace the last autoregressive coefficient amm by a reflection coefficient m calculated so as to minimize the power of the errors of direct and retrograde linear predictions. These are defined by

conduit aux relations de récurrence :

Where N is the number of successive signal samples available, which, taking into account the Levinson equation

leads to recurrence relations:

<tb> | 6f <SEP>#<SEP> SJ, i, mI <SEP> +
## EQU1 ##
<tb> that determine a lattice filter structure for estimating forward and backward prediction errors.

avec, à l'initialisation, pour l'ordre 0
P(0)=c0 puisque #f,i,0 et #b,i,0, prennent la valeur de l'échantillon xi. It is also shown that there exists a third relation of recurrence between the mean P (m) and P (m-1) of the power of the errors of the predictions direct and retrograde with the order m and the order m-1 of the form

with, at initialization, for order 0
P (0) = c0 since # f, i, 0 and # b, i, 0, take the value of the sample xi.

To obtain the value of the reflection coefficient m, minimizing the power of the errors of the linear direct and retrograde predictions, it suffices to remark that this value cancels the gradient of the average power U (m) of the forward and backward prediction errors.

ce qui s'écrit en tenant compte des relations de récurrence:

d'où l'expression du coefficient de réflexion de Burg

II apparaît alors que le coefficient de réflexion de Burg est égal à l'opposée de la covariance entre l'erreur de prédiction directe et l'erreur de prédiction rétrograde. On peut montrer, en utilisant l'inégalité de
Gauchy, que le coefficient de réflexion de Burg est de module strictement inférieur à un, ce qui assure la stabilité du filtre inverse et rend possible une synthèse sous forme de filtre en treillis.The value of F & m which minimizes plm ', checks the relation

what is written taking into account the relations of recurrence:

hence the expression of the reflection coefficient of Burg

It appears then that the reflection coefficient of Burg is equal to the opposite of the covariance between the forward prediction error and the retrograde prediction error. We can show, using the inequality of
Gauchy, that the reflection coefficient of Burg is of modulus strictly lower than one, which ensures the stability of the inverse filter and makes possible a synthesis in the form of lattice filter.

où n est la dimension de la matrice d'autocorrélation utilisée, N le rang du modèle de prédiction égal au nombre de composantes du vecteur échantillon, R,l la matrice d'autocorrélation du signal utilisée, Rn son estimée, R7z' son inverse et Tr() la fonction trace.For further details on Burg's maximum entropy method, reference may be made to the many articles published on the subject, and in particular to [3] Burg JP, #Maximum entropy spectral analysis in Proc. 37th Annual
I used to obtain a spectrum modeling as faithful as possible and that of the determination of the number of effective lines of the signal, or the dimension of the subspace signal plus noise of the autocorrelation matrix of the signal used. The second problem is solved classically from the determination of the dimension of the complementary noise subspace by asymptotically evaluating the number of equal minimum eigenvalues of the autocorrelation matrix of the signal used by means of a test on the equality of the smallest own values. This test is based on the maximization of the probability density function of the vector of signal samples taken into account in the considered autoregressive model, assumed to be stationary, of zero mean and Gaussian. Let {x} be the sample vector, this probability density function can be expressed by

where n is the size of the autocorrelation matrix used, N the rank of the prediction model equal to the number of components of the sample vector, R, the autocorrelation matrix of the signal used, Rn is estimated, R7z is its inverse and Tr () the trace function.

The maximization of this probability density function is done through the maximization of the likelihood function

de sorte que l'on est ramené à minimiser le déterminant de la matrice d'autocorrélation en fonction du nombre p de raies de fréquence supposées exister réellement dans le signal. On a par hypothèse

les #i étant les valeurs propres de la matrice d'autocorrélation Rn classées par ordre décroissant. En considérant l'hypothèse où le spectre du signal renferme p raies, les n-p plus petites valeurs propres #i ont une amplitude égale à la variance a2 du bruit de modélisation et en se ramenant à l'estimée de la matrice d'autocorrélation qui est la seule accessible physiquement, on peut écrire

But we show that

so that one is brought back to minimize the determinant of the autocorrelation matrix as a function of the number p of frequency lines assumed to actually exist in the signal. We hypothesize

the #i being the eigenvalues of the autocorrelation matrix Rn ranked in descending order. Considering the hypothesis that the signal spectrum contains p rays, the np smaller eigenvalues #i have an amplitude equal to the variance a2 of the modeling noise and can be compared to the estimate of the autocorrelation matrix that is the only physically accessible one can write

avec
L(p)=(#.e)-nN[F(p)]-N ou encore

A likelihood ratio A (p) is deduced between the hypothesis of having in the spectrum of the signal more frequency peaks and the hypothesis of having in the spectrum of the signal at most p-lines of

with
L (p) = (# e) -nN [F (p)] - N or else

This test called test on the equality of (np) smaller eigenvalues and its mathematical justification are described in detail by Georges BIENVENU and Laurent KOPP in the article [2] WELCOME G., KOPP L., "Optimality of High Resolution Array
Processing Using the Eigensystem Approach "IEEE Trans., On ASSP, vol.31, n05, October 1983, pp. 1235-1248,
Christine WATERNAUX showed, in the article [2] WATERNAUX CM, "Principal Components in the Nonnormal Case
The Test of Equality of Q Roots. "Journal of Multivariate Analysis, vol.

14, 1984, pp. 323-335, that the value Log (A (P)) follows a distribution law of probability of the
Khi-2 to (N + 2) (N + 1) / 2 degrees of freedom. We can therefore, from an arbitrary false alarm probability rate, determine, by means of the distribution law of probability of Khi-2 to (n + 2) (N + 1) / 2 degrees of freedom, a threshold value S, which if not crossed by the value
Log (A (p)) shows that it is likely that the signal whose spectrum has been modeled contains a maximum of frequency peaks.

Presented in order to determine the number of frequency lines actually belonging to the spectrum of an approached signal by autoregressive modeling, this test can also be used to assess the actual width of the signal spectrum, which is the criterion on which we seek to base a target detection having parts in relative motion and a radial velocity may differ from the ambient clutter. Unfortunately, this test is of limited effectiveness in the case of a small number of signal samples due to the large variance presented in this case, by estimating the eigenvalues of the autocorrelation matrix of the signal
To make this test more robust, we propose to calculate the value Log (A (p)) not from the estimates of the eigenvalues of the autocorrelation matrix but from the coefficients of reflection which have a variant of estimation more weak than the eigenvalues in case of few data samples and are regularizable.

et le signe + désignant l'opérateur complexe transconjugué.To achieve this, we take into account that the genesis of this test passes through the minimization of the determinant of the autocorrelation matrix. It is then sought to express this determinant as a function of the reflection coefficients by using a particular decomposition of the inverse autocorrelation matrix of the signal presented in the French patent application registered on May 7, 1996 under number 09 05689 in the name of the Applicant. This decomposition consists in expressing the inverse autocorrelation matrix R71 corresponding to an autoregressive model of rank n, as a function of the autocorrelation inverse matrix RN'II of the vector of autoregressive coefficients and of an immediately lower order term y-1.

and the sign + designating the transconjugated complex operator.

If we have a matrix determinant M such that

<tb><SEP> M11 <SEP> M12
<tb> M <SEP> = <SEP> []
<tb> M21 <SEP> M22
<tb> we can write M # = # M11 ## M22-M21.M11-1.M12 # so that we have, referring to the relation (7)

Since the determinant of the inverse of a matrix is equal to the inverse of the determinant of the matrix, we also have Rn # = &alpha; n-1-1. # Rn-1 #
From where

We also show that we can identify the inverse of the term an with the mean P ('of the powers of the errors of estimation of the autoregressive model of order n

et que le test sur l'égalité des (n-p) plus petites valeurs propres devient

However, in the case where the spectrum of the signal contains frequency ranges, the average powers of the estimation errors of the autoregressive models beyond the order p are all equal to the variance of the modeling noise so that one can write

and that the test on the equality of (np) smaller eigenvalues becomes

Using the recurrence relation (8) on the a he comes

Hence the new expression of the test on the equality of (np) smaller eigenvalues

 It is obvious that this new expression leads to a reduced calculation volume compared to the previous one (6) since it no longer relates to the eigenvalues of the autocorrelation matrix of the signal but to the reflection coefficients which are obtained by means of a lattice structure filter.

 This new expression (9) of the equality test of (np) smaller eigenvalues of the autocorrelation matrix of the signal can be used to estimate the likelihood of having a monaural signal spectrum by setting p equal to 1. However, when this test established with p equals 1, concludes with a spectrum of monaural signal and therefore with a spectrum of clutter echoes, we may have to seek to refine the analysis by supplementing it with a more precise test based on a measure of distance between a modeling of the spectrum of the signal of order 1, therefore necessarily monaural and the spectrum modeling of the signal to the order n adopted as the most faithful.

et d'adopter comme mesure effective de distance dq le Logarithme de la déviation spectrale définie par la norme L;: dq(Sl,S2)= Log(s1)- L0g(s2)|q
Dans la suite, on ne s'intéresse qu'à la distance quadratique, c'est-à-dire à q égal à deux, et on l'exprime en fonction des coefficients cepstraux. It is usual to consider a measure of distance between two spectra S1, s2 in terms of norm Lq defined as follows

and to adopt as effective measure of distance of the Logarithm of the spectral deviation defined by the norm L ;: dq (Sl, S2) = Log (s1) - L0g (s2) | q
In the following, we are interested only in the quadratic distance, that is to say at q equal to two, and we express it according to the cepstral coefficients.

les akm étant les coefficients autorégressifs correspondant à un modèle de prédiction d'ordre m. D'une manière équivalente, les coefficients cepstraux peuvent être définis à l'aide de la relation

ce qui permet de les calculer par les paramètres autorégressifs

Cepstral coefficients Ckm are defined as the Fourier coefficients of the Logarithm of the spectral density of the echo signal, which amounts to writing from relation (3):

the akm being the autoregressive coefficients corresponding to a prediction model of order m. Equivalently, the cepstral coefficients can be defined using the relation

which allows to calculate them by the autoregressive parameters

Applying Parseval's formula we show that we have

We also show that the finite sum, given below, is positive definite and converges when L tends to infinity towards the previously defined norm.

 Experimentally, it appears that this last norm is close to the asymptotic norm for small values of L. In practice, we will take L equal to 2p, where p is the order of autoregressive models. Spectral distance measurement can then be expressed in terms of a Euclidean norm on cepstral coefficients, and not, as one might intuitively think, on autoregressive parameters.

on peut écrire

Since, moreover, the cepstral parameters of order p cn (p) are linked to the poles kk (p) of the autoregressive spectrum modeling by the relation

we can write

It appears in this last relation a weighting term in k detrimental to the sensitivity of the criterion. Therefore, it is proposed to adopt a modified spectral distance measure of the shape

As a variant, it is also proposed to use another spectral distance measurement implementing the cosine of the angle between cepstral vectors formed by the cepstral parameters taken in ascending order with the exception of the parameter cO: d (s1 , s2) = # Cs1 # 2. (1-cos2 (#)) = # Cs1 # 2.sin2 (#) with

 Experimentally, this measurement of spectral distance is more robust than the previous one.

soit sur la mesure de distance spectrale

In summary, when the test on the equality of (np) smaller eigenvalues of the autocorrelation matrix of the signal applied with p equal to 1 leads to the conclusion that we are dealing with a modeling of a signal spectrum monoraie, one can fine-tune the target detection by the implementation of a criterion on the spectral distance existing between a spectrum modeling of the signal of order 1 which is monoraie by definition and a modeling of spectrum of the signal of an order higher p leading to a supposedly more faithful modeling, this spectral distance criterion being based either on the measurement of spectral distance

either on the spectral distance measurement

 So far, it has been accepted that the order of the autoregressive model adopted to model the signal spectrum was the one that led to the most faithful modeling. To find this optimum order, many statistical criteria have been proposed but they have the disadvantage of not being suitable in the case envisaged here where one has only a small number of signal samples. For this particular case of a small number of signal samples, it has been proposed by the applicant, in the French patent application FR-2,735,594 to adopt the maximum possible order (Nl) while imposing a constraint. of spectral softness in the determination of reflection coefficients by minimizing forward and backward linear prediction errors.

avec

The reflection coefficient jlp is then determined so as to minimize an expression EP formed of the sum of the average power U (p) of the forward and backward prediction errors, and of a spectral softness term.

with

Stopping at the first two coefficients y, and Y1, this leads to a new value of the reflection coefficient jlp given by the formula

 The modulus of the reflection coefficient must remain less than 1, which conditions the structure of the associated trellis filter. Regularization allows it to not explode when G and D become small together (Zero Zero Indetermination). Indeed, when the order of the model is optimal, the errors become very weak, as well as G and D, but conversely, the terms of regularization of the numerator and the denominator become predominant and prevent the degeneration of the coefficient of reflection.

cette valeur r étant d'autant plus faible que le spectre du signal est plat. The evolution of the terms of regularization makes it possible to have an indication on the flatness of the spectrum of the signal which one seeks to model. We can derive a criterion which is based on the regulated autoregressive filter residue, that is to say the covariance of the autoregressive parameters obtained for a maximum and regularized order and which consists in the calculation of the value TI of the autoregressive coefficient lip for which the term D1 is minimum

this r value being even lower than the spectrum of the signal is flat.

 FIG. 1 is a flowchart illustrating the implementation of the tests described above for detecting helicopters using a Doppler radar delivering only a small number of echo samples per site-azimuth-distance box, by example eight corresponding to the emission of repetitive bursts of eight coherent interrogation pulses.

 The problem is to discriminate a helicopter at zero radial speed compared to the Doppler radar, be it hovering, in tangential flight compared to the radar or in "pop-up", in the middle of a clutter of ground without additive clutter of the rain type for a signal-to-noise ratio of less than 1.

 In absolute terms, a soil clutter returns echoes with a Doppler distribution in zero with no variance in the phenomenon.

Its Doppler spectrum is, as a first approximation, monoraie. It can then be modeled in a high-resolution spectral analysis using an autoregressive model of order 1. With regard to a helicopter, it returns, apart from the blade flashes, echoes with a narrow Doppler statistical distribution. around the zero frequency symmetrically, due to the energy backscattered by rotating parts of the rotor. Its Doppler spectrum is, in first approximation multiraie therefore slightly wider than that of the clutter in the middle of which one seeks to detect it. It is then sought to detect a helicopter, either by the echoes of pale flashes it returns, or by the slightly enlarged appearance of the Doppler spectrum of its echo with respect to the Doppler spectrum of clutter echoes.

A high-resolution spectral analysis is first performed on each collection of eight echo samples corresponding to the same site-azimuth-distance box. This consists in determining the autoregressive coefficients of the prediction model by the method of
Burg advantageously regulated according to the teaching of the aforementioned French patent FR-2,735,594 so as to be able to opt for the maximum order model equal to the number of available echo samples minus one.

Once the autoregressive coefficients and the reflection coefficients made available by the high-resolution spectral analysis, we begin by performing a detection of flashes of blades 1. This consists in finding out if one is in the presence of echoes to wide spectrum so flat. It is done by means of two tests on the form of the modeled spectrum which must be passed one and the other with success:
a first test ci whose criterion relates to the coefficients
of the prediction model used for the analysis
spectral high resolution and which is relative to the equality of
np eigenvalues of the autocorrelation matrix of the signal
with p taken equal to zero and with a higher threshold value
SP1 determined by application of a distribution law of
probability of Khi-2 to N + 2) (N + 1) / 2 degrees of freedom to
a false alarm probability value VP arbitrarily chosen criterion c1

N being the number of echo samples and n the order of the
prediction model. a second test which is a test based on the
autoregressive coefficients of the prediction model and which
implements the criterion based on the filter residue
Regressed autoregressive with a higher threshold value
SP2 experimentally chosen to have a false rate
arbitrary alarm criterion cib

the ak being the autoregressive coefficients of the model of
prediction.

 The experimental choice of the upper threshold value is done in the usual way, by determining the values taken by the variable rl for site-azimuth-distance boxes which one knows that they are occupied only by clutter and for boxes site-azimuth-distance which is known to be occupied by helicopters, and adopting as the upper threshold value SP2 the limit between the two ranges occupied by the TI variable for the huts with and without a helicopter, assuming a certain probability of false alarm.

 If the two previous tests are passed successfully, we conclude with echoes of lightning of blades and thus with the presence of a helicopter in the site-azimuth-distance box concerned. If no, proceed to the next step 2 of the monaural test.

This second step 2 of searching for a monaural spectrum consists in carrying out a test c2 whose criterion relates to the reflection coefficients of the prediction model used for the high resolution spectral analysis and which is relative to the equality of the n eigenvalues. of the autocorrelation matrix of the signal with p taken equal to one and with a higher threshold value SK determined by applying a distribution law of probability of Khi-2 to N + 2) (N + 1) / 2 degrees of freedoms to a false alarm probability value VK arbitrarily chosen criterion c2

 If the test is negative, it is considered that there are multirespectral echoes corresponding to the detection of a helicopter.

 If the test is positive, it is considered that it is presumably in the hypothesis of a monaural spectrum and one goes on to a third stage of refinement of the detection 3.

This third step 3 consists in finding out if one is not in the presence of a wider spectrum than would be a monorey spectrum of clutter. It is done using two tests on a possible enlargement of the modeled spectrum which must be passed together successfully to definitively conclude clutter
a first test c3 consisting in measuring a first
spectral distance between a spectrum modeling to
order 1 and spectrum modeling at maximum order
regularized and compared to a higher threshold value
SL1: criterion c3

et la valeur de seuil supérieur SL1 étant déterminée expérimentalement en fonction d'une valeur recherchée de taux de fausse alarme. un deuxième test c3b consistant à mesurer une deuxième
distance spectrale entre une modélisation du spectre à
l'ordre 1 et une modélisation du spectre à l'ordre maximum
régularisé et à la comparer à une valeur de seuil supérieur SL2 critère c3b

L being equal to 2n, ck) being a cepstral parameter corresponding to a Fourier coefficient of the Logarithm of the spectral density of the echo signal and calculated from the autoregressive parameters by means of the relations

and the upper threshold value SL1 being determined experimentally as a function of a desired value of false alarm rate. a second test c3b consisting of measuring a second
spectral distance between a spectrum modeling to
order 1 and spectrum modeling at maximum order
regularized and compared to a higher threshold value SL2 criterion c3b

the upper threshold value SL2 being determined
experimentally based on a desired value of
false alarm rate.

 If the two previous tests are passed successfully, it is concluded that there is a monaural spectrum of clutter echoes. If no, we admit the presence of a helicopter in the box site-azimutdistance concerned.

The detection of turbulence is the same as that of helicopters by eliminating step 1 of blade flash detection that no longer needs to be and adjusting the threshold values. Indeed, the turbulence that is important for the safety of an aircraft flight translates to a radar by a broadening of the spectrum
Doppler echoes returned relative to the spectrum of cloud echoes in calm air. The monoria test at step 2 of the helicopter detection flowchart as well as the two spectrum broadening tests proposed in step 3 of this flowchart are then relevant for characterizing turbulence in the atmosphere, with accuracy of high resolution methods, and represent a definite progress over the turbulence detection methods used in the prior art which rely solely on the detection of the variation of the shape of the Doppler spectrum which appears very degraded at the outputs of a bench Doppler filters.

 FIG. 2 shows a block diagram of the video portion of the receiver of a Doppler radar implementing the low or zero radial velocity target detection method which has just been described. The reception video signal is inputted to a quadrature demodulator recalled in the figure by two demodulators 20, 21 which receive two quadrature versions of the same demodulation carrier in addition to the reception video signal and which are followed by two low-pass filters 22, 23. The quadrature demodulator delivers in parallel the I-phase and Q-quadrature components of the demodulated reception signal which are sampled and converted into digital by two analog-to-digital converters 24, 25. These digital samples phase I and quadrature Q components of the demodulated received signal are sorted by distance cells and applied, in bursts corresponding to the bursts of interrogated coherent interrogation pulses, to a Doppler analysis circuit 26.

 The Doppler analysis circuit 26 performs a high-resolution spectral analysis of the bursts of digital samples occupying each distance slot and delivers, to a detection circuit 27, information on the shape of the spectrum modeled in the form of autoregressive coefficients and cepstral parameters. To do this, it uses the method of the maximum entropy Burg regularized which is described in detail in the French patent application FR 2,735,594 previously cited.

 The detection circuit 27 uses the autoregressive coefficients and cepstral parameters received from the Doppier analysis circuit 26, the detection method whose main steps have been described with respect to the flowchart of FIG. 1 in order to attribute the spectrum Doppler delivered by the Doppler analysis circuit 26 either to a target or to clutter.

Claims (6)

 1. Doppler detection method for targets with low or zero radial speed, characterized in that it consists  to collect the samples of N successive echoes coming from the same site-azimuth-distance box, in response to the emission of a train of N consecutive and coherent interrogation pulses,  to perform a high resolution spectral analysis of the N successive echo samples by means of an autoregressive model of order n (n strictly less than N) defined by a series of n reflection coefficients pk and leading to a spectrum modeling at n frequency lines of various amplitudes, the frequency lines of higher amplitudes, corresponding to frequency lines actually existing in the signal, while others of smaller amplitudes may be solely due to a modeling noise and not to have physical reality,  - to check the equality between the np lines of smaller amplitudes of the spectrum modeling (p being less than or equal to n) to ensure, either that they are due only to modeling noise, or that they belong to a spectrum broad enough to correspond to that of a target, this verification being effected by means  the likelihood value calculation

 spectrum.  lines of lower amplitudes of the modeling of the  superior confirming the likelihood of equality of n-p  likelihoodLog (A (p)) less than said threshold value  at the upper threshold value obtained, a value of  of the likelihood value comparisonLog (A (P))  the probability of false alarm adopted, and  by the distribution law of probability of Chi-2 based  determining a given upper threshold value  the choice of an arbitrary false alarm probability,  2 to (N + 2) (N + 1) / 2 degrees of freedom,  of the selection of the distribution law of probability of chi  2. Method according to claim 1, characterized in that the number p is taken equal to zero, the comparison of the likelihood value Log (A (O)) with a higher threshold value associated SP1 determined by means of the Chi-2 probability distribution law from an arbitrary false alarm rate VP, for testing the likelihood of an expanded signal spectrum corresponding to a helicopter blade flash, a Log value (A (O)) found to be lower than the associated upper threshold value SP1, showing the relative equality of the amplitudes of all frequency lines of the echo signal spectrum modeling and confirming that the echo signal originates from a helicopter blade flash.  3. Method according to claim 1, characterized in that the number p is taken equal to one, the comparison of the likelihood value Log (A (1)) with an associated higher threshold value SK determined by means of the law of chi-2 probability distribution from an arbitrary false alarm rate VK, for testing the likelihood of a monaural echo signal spectrum corresponding to clutter, a log likelihood value (N (1)) found to be lower than the associated upper threshold value SK, showing the equality of the n-1 lines of smaller amplitudes.  4. Method according to claim 2, characterized in that the comparison of the likelihood value Log (A (O)) to a higher threshold value SP1 to detect the presence of an expanded spectrum of signal corresponding to a flash of a blade is complemented by a test on the autoregressive filtering residue having led to the modeling of the echo signal spectrum, this test consisting in evaluating the value of the term rz equal to

 being the kth so-called autoregressive parameter of the autoregressive model of order n, defining the prediction law of an ith sample xi of the signal whose high resolution spectral analysis is made, starting from the (ip) preceding samples, by a relationship of form

 obtained for the term rj less than the threshold SP1.  corresponding to a blade flash in case of a value  to admit the presence of an expanded spectrum of echo signal  arbitrary false alarm, and  threshold SP2 determined experimentally to have a rate  to compare the obtained value of the term r with a value of  the model of order n, of the sample xj,  # f, i, n being a prediction error in the forward direction, by  5. Method according to claim 3, characterized in that the comparison of the likelihood value Log (# (1)) with a higher threshold value SK for detecting the presence of a monaural spectrum of echo signal is refined, when the likelihood value Log (A (1)) is found to be lower than the upper threshold value SK by a distance test between the obtained n-order spectrum modeling and a necessarily monaural order-of-magnitude spectrum modeling, this distance test consisting of: # 2  # to perform a distance measurement d2 based on the  (P)  cepstral parameters ck which are the coeficients of  Fourier of the logarithm of the spectral density of the echo signal and which are calculated from the autoregressive parameters

 with, by definition,

 SL1.  distance measurement d2 is greater than the threshold value  corresponding to a target as soon as the value obtained for the  to deduce the presence of an expanded signal spectrum  arbitrary, and  experimentally to have a false alarm rate  comparing the value obtained for the distance measurement # 22 with a determined threshold value SL1  autoregressive parameter of the autoregressive model of order m,  akm being, as indicated previously, the k th  Method according to claim 3, characterized in that the comparison of the likelihood value LoA (l)) with a higher threshold value SK for detecting the presence of a monaural echo signal spectrum is refined, when the likelihood value Log (A (1)) proves to be lower than the upper threshold value SK by a distance test between the n-order spectrum modeling obtained and a necessarily monaural order-of-magnitude spectrum modeling, this test consistent distance  to measure a distance d2 defined according to the cosine of  the angle between cepstral vectors

 SL2.  distance measurement d2 is greater than the threshold value  corresponding to a target as soon as the value obtained for the  to deduce the presence of an expanded signal spectrum  arbitrary, and  experimentally to have a false alarm rate  d2 with a determined threshold value SL2  to compare the value obtained for the distance measurement