EP0620546B1 - Energy detection procedure for noisy signals - Google Patents

Description

The present invention relates to a method of energetic detection of signals embedded in noise.

When a signal model is available, tools for detecting this signal are widely available in the literature, the best known methods being based on the notion of adapted filter and, more generally, on the theory of decision in signal processing (PY ARQUES, Technical and Scientific Telecommunications Collection, MASSON). These techniques govern the development of coherent and non-coherent receivers in digital communications (Principle of Coherent Communication A.J. VITERBI, MacGraw-Hill).

By cons, the present invention is in the case where there is no model capable of allowing direct application of the theory of detection. It is assumed to be in the presence of background noise, and from time to time an "anomaly" occurs, which, depending on the context, may in fact represent a signal that it would be desirable to detect.

Examples of detection of a so-called "useful" signal in noise are widely found in the literature concerning speech detection. Indeed, the speech signal, by its great variability, does not lend itself to effective modeling and one of the most natural means to detect it consists in carrying out an energy thresholding.

Thus, a lot of current research relates either to the instantaneous amplitude by reference to a threshold determined experimentally (Speech-noise discrimination and its applications V. PETIT, F. DUMONT Technical Review THOMSON-CSF - Vol. 12 - N ° 4 - Dec. 1980), or on an empirical thresholding of energy ("Suppression of Acoustic Noise in Speech Using Spectral Substraction", SF BOLL, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.ASSP-27, N °. 2, April 1979), or on the energy of the total signal over a time slice of duration T, by thresholding, still experimentally, this energy using local histograms, for example ("Problem of detection of word boundaries in the presence of noise additives ",. P.WACRENIER, DEA thesis from the University of PARIS-SUD, Center d'ORSAY). Other techniques are presented in" A Study of Endpoint Detection Algorithms in Adverse Conditions: Incidence on a DTW and HMM Recognizer ", JC JUNQUA, B. REAVES, B. MAK EUROSPEECH 1991.

In all of these approaches, much is made of heuristics, and few powerful theoretical tools are used.

It should be recalled the work presented in "Evaluation of Linear and Non-Linear Spectral Substraction Methods for Enhancing Noisy Speech", A. LE FLOC'H, R. SALAMI, B. MOUY and J-P. ADOUL, Proceedings of "Speech Processing in Adverse Conditions", ESCA WORKSHOP, CANNES-MANDELIEU, 10-13 November 1992, where all the energies above a certain experimental threshold are considered as revealing the presence of useful signal and where all the energies below this threshold are considered as energies due to noise alone when the usual distance (absolute value of their difference) separating them is below a threshold, also experimental. However in this document by Le Floc'h et al, the authors work on the concept of distances between energies, but the distance used is a simple absolute value of the difference in energies and their work makes great use of heuristics.

EP-A-518742 discloses a method for detecting a noisy useful signal, but this assumes that the expected signal / noise ratio and the measurement of the estimated noise alone are known.

The subject of the present invention is a method for the energetic detection of useful signals embedded in noise, a method which essentially implements rigorous means without practically using heuristics, and which is optimized, that is to say which allows to detect practically all useful signals drowned in noise, even intense, with the lowest possible rate of false detections.

The method according to the invention for energetic detection of useful signals in the presence of noise, is a method comprising a preclassification step followed by a detection step, said preclassification step consisting, from a set of samples d '' a noisy signal grouped in successive frames, to be decided according to the ratios of the energies of the successive frames with each other, the decision being effectively based on the calculation of a distance which is the absolute value of the difference in the logarithms of the two energies, and on the comparison of this distance with an optimal threshold (Log s) so as to classify in a first class "noise only" at least some of the frames probably belonging to this class; said detection step consisting in detecting for the other frames, those having a sufficiently high energy compared to a reference energy (Eo) calculated from the energies of the "noise only" frames identified during the step of pre-classification of so that these detected frames have a high probability of belonging to a second class "noise + useful signal".

The method of the invention assumes that when the useful signal is present, the energy of the observed signal belongs to a certain class denoted C ₁ , and that when the useful signal is absent, the observed energy belongs to a class denoted C ₂ . One of the new characteristics of the present invention is to be able to demonstrate such energies of class C ₂ (therefore noise energies alone), then used so as to optimize the detection of energies of class C ₁ (therefore d energies revealing the presence of useful signal), by an optimized process.

We consider a distance between energies, but the distance in question in the invention, between two energies U and V, is not the usual distance | UV | but | Log (U / V) |, which amounts to considering that two energies U and V are close to each other when we have 1 / s <U / V <s, which is equivalent to | Log (U / V) | <Log (s). This distance and the threshold attached thereto are very advantageous. Indeed, consider the case where the useful signal s (n) and the noise x (n) are both white and Gaussian, s (n) being of variance σ _s ² and x (n) of variance σ _x ² . In the presence of s (n), we observe U = Σ _{0≤ n ≤ N -1} u ( n ) ² , with u (n) = s (n) + x (n). In the absence of s (n), we observe V = Σ _{0≤ n ≤ N -1} x ( n ) ² . Classical results in statistics allow us to write: and If U and V are considered independent, Let us denote by r = σ _s ² / σ _x ² the signal to noise ratio. We can still write: The result depends on σ _x ² and r, which shows that a thresholding of the distance IU-VI is not valid when one does not know σ _x ² . On the other hand, if we consider the U / V ratio, we demonstrate that the probability density of U / V now only depends on r, and is therefore independent of σ _x ² . This remarkable result validates the use of a threshold on U / V when we only know r.

In summary, according to the method of the invention, L * N samples u (n) of a signal are observed. Each set when i varies from 0 to L-1, is called a frame and is associated with an energy E (T _i ) denoted U _i = E (T _i ), which makes it possible to define When the useful signal is absent, the samples u (iN + k) are exactly equal to the samples of the noise noted x (iN + k) (u (iN + k) = x (iN + k)). When the useful signal (noted s (iN + k)) is present, the samples u (iN + k) are exactly equal to u (iN + k) = s (iN + k) + x (iN + k). A first method, described below (so-called pre-classification method), highlights a subset Δ of elements of E which are probably energies of class C ₂ . It is then possible to calculate an autoregressive model of the noise x (n) which will whiten the frames which will be treated subsequently, or an average spectrum of the noise x (n) which can be used to denoise the following frames (nor the bleaching or denoising are imperative but are used according to the particular context treated). We then use a second method (so-called detection method) described below, which will best detect among the elements of E (whitened or not, denoised or not) the energies of class C ₁ . Let N then be new samples, gathered in the form of a frame associated with a new energy. We can either use this new energy to come and update the whole Δ using the pre-classification process, or decide in the sense of a particular aspect of the process if this new energy belongs to C ₁ or not, after a possible denoising or possible laundering. This process is repeated for each frame of N samples acquired. The method of the invention is characterized by the use of new theoretical tools for signal processing and statistics. Thus, it calls upon a model of the statistical laws which follow the energies of the signals, that of the Positive Gaussian Random Variables (VAGP) described below. We then use an original property concerning the ratio of two VAGPs.

We will now define the "Positive" Gaussian Random Variables (VAGP) used by the invention.
A random variable X will be called positive when Pr {X <0} << 1. Let X _{0 be} the normalized centered variable associated with X, we have:
Pr {X <0} = Pr {X ₀ <-m / σ} where m = E [X] and σ ² = E [(Xm) ² ].

As soon as m / σ is large enough, X can be considered positive. When X is Gaussian, we denote by F (x) the distribution function of the normal Gaussian variable and we have: Pr {X <0} = F (-m / σ) for X ∈ N (m, σ ² ). For a Positive Gaussian Random Variable X ∈ N (m, σ ² ), we will define the parameter α of this variable by α = m / σ, so that we can still write X ∈ N (m, m ² / α ² ).

Energy models: examples of "positive" Gaussian variables Deterministic energy signal

Let be the samples x (0), ... x (N-1) of any signal, whose energy is deterministic and constant, or approximable by a deterministic or constant energy (as specified below).
So we have
U = Σ _{0≤n≤ N-1} x (n) ² ∈ N (Nµ, 0)
or
µ = (1 / N) Σ _0≤n≤N-1 x (n) ²

Let us take as an example the signal x (n) = A cos (n + θ) where θ is equidistributed between [0.2π].
For N sufficiently large, then: (1 / N) Σ _0≤n≤N-1 x (n) # ² E [x (n) ^2] = A ^2/2. For large enough N, U can be likened to NA ^2/2 and thus a constant energy.

We will now examine the case of the energy of any Gaussian Process. Consider a second order, but Gaussian, process x (n), with variance σ _x ² . We demonstrate the following result:
U = Σ _0≤n≤N-1 x (n) ² ∈ N (Tr (C _{x, N} ), 2Tr (C _{x, N} 2)), where C _{x, N} is the covariance matrix of the vector
X = ^t (x (0), ..., x (N-1)): C _{x, N} = E [X. ^t X]
As the process is stationary in second order, it comes
Tr (C _{x, N} ) = Nσ _x ² .
So U ∈ N (Nσ _x ² , 2Tr (C _{x, N} ² ))
A simple calculation leads to Tr (C _{x, N} 2) = Σ _{0≤i≤N-1,0≤j≤N-1} Γ _x (ij) ² where Γ _x (i) is the correlation function of the process. The parameter α is: $${\text{α = σ}}_{\text{x}}{}^{\text{2}}{\text{/ (2Tr (C}}_{\text{x, N}}{}^{\text{2}}{\text{)))}}^{\text{1/2}}{\text{= N / {2 Σ}}_{\text{0≤i≤N-1.0≤j≤N-1}}{\text{[Γ}}_{\text{x}}{\text{(ij) / Γ}}_{\text{x}}{\text{(0)]}}^{\text{2}}{\text{}}}^{\text{1/2}}$$ This variable will be a positive Gaussian variable if the correlation function allows it. There are special interesting cases described below, allowing access to this autocorrelation function.

Case of the energy of a White Gaussian Process.

We consider the case of a Gaussian white process x (n) where n is between 0 and N-1. The samples are independent and all have the same variance σ _x ² = E [x (n) ² ].
We then have C _{x, N} = σ _x ² I _N , where I _N is the identity matrix of dimension NxN.

We deduce: Tr (C _{x, N} 2) = Nσ _x ⁴ so that: $${\text{U = Σ}}_{\text{0≤n≤N-1}}{\text{x (n)}}^{\text{2}}{\text{∈ N (Nσ}}_{\text{x}}{}^{\text{2}}{\text{; 2Nσ}}_{\text{x}}{}^{\text{4}}\text{).}$$ The parameter α is α = (N / 2) ^1/2

Case of the energy of a Narrow Band Gaussian Process.
We suppose that the digital signal x (n) comes from the sampling of the process x (t), itself resulting from the filtering of a white Gaussian noise b (t) by a bandpass filter h (t) of transfer function :
H (f) = U _{[-f0-B / 2, -f0 + B / 2]} (f) + U _{[f0-B / 2, f0 + B / 2]} (f), where U denotes the characteristic function of the interval in index and f ₀ the central frequency of the filter.
The correlation function Γ _x (τ) of x (t) is equal to Γ _x (τ) = Γ _x (0) cos (2πf ₀ τ) sin _c (πBτ) where sin _c (x) = sin (x) / x .
The correlation function of x (n) is then:
Γ _x (k) = Γ _x (0) cos (2πkf ₀ T _e ) .sin _c (πkBT _e ). Yes
g _{f0, B, Te} (k) = cos (2πkf ₀ T _e ) sin _c (πkBT _e ),
we have :
Tr (C _{x, N} 2) = Γ _x (0) ² Σ _{0≤i≤ N-1.0≤j≤N-1} g _{f0, B, Te (ij)} 2.
We have: U ∈ N (Nσ _x ² , 2σ _x ⁴ Σ _{0≤i≤ N-1.0≤j≤N-1} g _{f0, B, Te} (ij) ² ). This variable is a positive Gaussian random variable. The parameter α of this variable is
α = N / [2Σ _{0≤i≤N-1,0≤j≤N-1} g _{f0, B, Te} (ij) ² ] ^1/2

These relationships remain valid even if f ₀ = 0.

Case of the energy of any "subsampled" Gaussian process.
This model is more practical than theoretical. If the correlation function is unknown, we know however that:
lim _{k → + ∞} Γ _x (k) = 0. Therefore, for k large enough such that k> k ₀ , the correlation function tends to 0. Also, instead of processing the series of samples x (0). ..x (N-1), can we treat the subsequence x (0), x (k ₀ ), x (2k ₀ ), ..., and the energy associated with this sequence remains a random variable positive Gaussian, provided that there remain enough points in this subsequence to be able to apply the approximations due to the central-limit theorem.

This way of proceeding may allow in certain difficult cases to apply the decision rules which are described below.
Fundamental theoretical result.
If X = X ₁ / X ₂ where X ₁ and X ₂ are both Gaussian, independent, such as:
X ₁ ∈ N (m ₁ ; σ ₁ ² ) and X ₂ ∈ N (m ₂ ; σ ₂ ² ). We set m = m ₁ / m ₂ , α ₁ = m ₁ / σ ₁ , α ₂ = m ₂ / σ ₂ .

When α ₁ and α ₂ are large enough to assume that X ₁ and X ₂ are positive Gaussian random variables, the probability density f _X (x) of X = X ₁ / X ₂ can then be approximated by: where U (x) is the indicator function of R ⁺ :
U (x) = 1 if x ≤ 0 and U (x) = 0 if x <0.
Yes where F denotes the distribution function of the normal variable, and where
P ( x , m | α ₁ , α ₂ ) = Pr {X < x }
Moreover :

In the following, when we use VAGP couples characterized by the parameters α ₁ , α ₂ and m, we will suppose to know the values of these parameters fixed by a priori knowledge or by heuristics.

We will now describe the step of pre-classification of the process of the invention. It is assumed that C ₁ = N (m ₁ , σ ₁ ² ) represents the energies observable in the presence of useful signal and that C ₂ = N (m ₂ , σ ₂ ² ) represents the energies observable in the absence of useful signal. We set m = m ₁ / m ₂ ,
α ₁ = m ₁ / σ ₁ and α ₂ = m ₂ / σ ₂ and we assume α ₁ and α ₂ large enough for the elements of C ₁ and C _{2 to} be VAGP. is the set of energies available to us. Each of these energies U _{i is} worth U _i = Σ _{0≤ k ≤ N -1} u _i ( k ) ² , where the u _i (k) for k ranging from 0 to N-1 represent the samples of the frame T _i , N the number of these samples u _i (k), that is to say the length of the frames T _i . The energies U _i are assumed to be independent from each other. The pre-classification stage seeks to highlight only a few energies, which are probably energies of class C ₂ . This step uses the concepts presented below.

Concept of compatibility between energies:

Let (U, V) ∈ (C ₁ CU ₂ ) X (C ₁ CU ₂ ) and X = U / V. We define the following hypotheses:
H ₁ : (U, V) ∈ (C ₁ XC ₁ ) U (C ₂ UC ₂ ) and H ₂ : (U, V) ∈ (C ₁ XC ₂ ) U (C ₂ UC ₁ ). If we have: 1 / s <X <s ⇔ we decide that U and V belong to the same class, that is to say that H ₁ is considered to be true. We will say that U and V are compatible. This decision will be noted D ₁ . But if we have: X <1 / s or X> s ⇔ we decide that U and V do not belong to the same class, that is to say that H ₂ is considered to be true. We will say that U and V are incompatible. This decision will be noted D ₂ .
If I = [1 / s, s], the rule is expressed according to: x ∈ I ⇔ D = D ₁ , x ∈ R - I ⇔ D = D ₂ . We seek to optimize this decision rule which allows to associate the realizations of random variables between them. To do this, we calculate the optimal threshold s. This calculation is different depending on whether or not we know the probability p. When p is known, the maximum likelihood criterion is directly applied. When p is unknown, and since the hypotheses are reduced to 2, we use the Neyman-Pearson criterion.

Maximum likelihood criterion:

We show that the probability of correct decision is: The optimal threshold is verified $\frac{\text{∂}{\mathit{\text{P}}}_{\mathit{\text{vs}}}}{{\text{∂}}_{\mathit{\text{s}}}}$ = 0.

This equation is solved on a computer, once the values m, p, α ₁ and α _{2 have been} fixed.

Neyman-Pearson criterion:

When p is unknown, a Neyman-Pearson type approach is used. We will say that there is detection if we take the decision D ₁ , that is to say if we decide that the two random variables are of the same class. The probabilities of non-detection P _nd and of false alarm P _{fa are} then defined by: (probability of deciding on incompatibility, while the variables are of the same class) and P _fa = Pr {D ₁ | H ₂ } (probability of deciding on compatibility when the variables are incompatible). The Neyman-Pearson criterion consists in minimizing P _nd when P _fa is fixed (or vice versa). This type of criterion applies when one error is much more serious than the other. As it is a question here of knowing whether or not the observed random variables belong to the same class, it is obvious that we will seek to have, in the realizations retained as being realizations of variables of the same class, only few errors. It is therefore the P _fa that we will set so as to have very few false alarms. and so that when α ₁ ≠ α ₂ , P _nd depends on p, which is unknown and is inaccessible.

In the case where α ₁ = α ₂ = α, then P _nd = 2.P (s, 1 | α, α) - 1 and is therefore accessible. We can fix P _nd in this case. Having the expression of P _fa (or of P _nd ), we fix this probability, which allows us to obtain the corresponding threshold s.

Compatibilities between several energies.

When the threshold has been calculated according to one of the two aforementioned methods, it is interesting to generalize this notion of compatibility between several energies. So let U ₁ , ..., U _N , N energies, we will say that these energies are compatible with each other, if and only if, ∀ i and j, U _i and U _j are compatible in the sense mentioned above, otherwise says, if all these energies are compatible two by two.

• les énergies de la classe C₂ sont plus faibles statistiquement que celles de la classe C₁ ;
• la trame présentant l'énergie la plus faible est une trame de la classe C₂. Soit T_i0, cette trame.

For the implementation of the method, the following assumptions are made:

• the energies of class C ₂ are statistically lower than those of class C ₁ ;
• the frame with the lowest energy is a frame of class C ₂ . Let T _{i0 be} this frame.

The calculation then proceeds as follows:

The set Δ is initialized: Δ = {T _i0 };

The noise confirmation method provides a number of frames which can be considered as noise with very high probability. One computes, from the data of the temporal samples, an autoregressive model of the noise. If x (n) designates the noise samples, we model x (n) according to: x ( n ) = Σ _{1≤ i ≤ p} a _i x ( n - i ) + e ( n ), where p is the order of the model, the a _i are the coefficients of the model to be determined and e (n) is the modeling noise, assumed to be white and Gaussian if we follow a maximum likelihood approach. This type of modeling is widely described in the literature "Spectrum Analysis - A modem Perspective", SM KAY / SL MARPLE JR., Proceedings of the IEEE, Vol. 69, No. 11, November 1981, in particular. As for the calculation methods of the model, many methods are available (Burg, Levinson-Durbin, Kalman, Fast Kalman ...). Advantageously, methods of the Kalman and Fast Kalman type are used: "Transverse Adaptive Filtering", O. MACCHI, M. BELLANGER, Signal Processing, Vol. 5, N ° 3, 1988 and "Signal analysis and adaptive digital filtering", M. BELLANGER, CNET-ENST Collection, MASSON, which present very good real-time performance. Having an autoregressive model of noise, it is then easy to whiten this noise, which makes it possible to work on an easily manipulated white Gaussian noise.

Let u (n) = s (n) + x (n) be the total signal, composed of the useful signal s (n) and the noise x (n). Let the filter H (z) = 1-Σ _{1≤ i ≤ p} ai z ^-i . Applied to the signal U (z), it comes H (z) U (z) = H (z) S (z) + H (z) X (z).
Now H (z) X (z) = E (z) ⇒ H (z) U (z) = H (z) S (z) + E (z). The rejector filter H (z) whitens the signal, so that the signal at the output of this filter is a useful signal (filtered therefore distorted), added to a generally white and Gaussian noise. Working on a bleached noise brings you closer to ideal hypotheses, especially when applying the detection process. However, this bleaching is not compulsory and the detection process can be applied directly without going through this intermediary.

Since after implementing the method of the invention, there are a certain number of frames confirmed as being noise frames, it is also possible to calculate an average spectrum of this noise, so as to implement spectral filtering of the spectral subtraction type or WIENER filtering, type widely described in the literature: "Suppression of Acoustic Noise in Speech Using Spectral Substraction" SF BOLL, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-27, No. 2, April 1979; "Enhancement and Bandwidth Compression of Noisy Speech", JS LIM, AV OPPENHEIM, Proceedings of the IEEE, Vol. 67, N ° 12, Dec. 1979, and "Noise Reduction For Speech Enhancement In Cars: Non-Linear Spectral Subtraction, Kalman Filtering", P. LOCKWOOD, C. BAILLARGEAT, JM GILLOT, J. BOUDY, G. FAUCON, EUROSPEECH 91. This aspect can be interesting in these certain applications, see for example: "Speech detection process", D. PASTOR, French patent application No. 92 12582, filed on 21.10.92.

Detection according to the method of the invention.

Having a set Δ whose components are probably energies of class C ₂ (after possible bleaching), we seek to detect using these references the energies of class C ₁ . If V represents the average value of the energies of the set Δ, this variable is also a VAGP. If Δ = {V ₁ , ..., V _M }, we have, with the notations already used, ∀ i ∈ {1, ..., M}, V _i ∈ N (m ₂ , σ ₂ ² ).
E _o = (1 / M) Σ _{1≤ i ≤ M} Vi ∈N ( m ₂ , (1 / M ) σ ₂ ² ) since the V _i are independent. We set m = m ₁ / m ₂ , α ₁ = m ₁ / σ and α ₂ = m ₂ / σ ₂ .

We then move on to the optimal decision rule.
Application of the maximum likelihood criterion (we know the probability p of correct decision):
let p = Pr {U ∈ C ₁ }. The optimal decision rule is then:

Application of the Neyman-Pearson criterion:

• soit la fixer arbitrairement par une approche heuristique,
• soit la fixer à p = 0,5, ce qui place dans le cas le plus défavorable,
• soit utiliser le critère de Neyman-Pearson ou le critère de la médiane qui consiste à avoir :
  probabilité de fausse alarme = probabilité de non détection.

When we do not know the value of p, we can:

• either fix it arbitrarily using a heuristic approach,
• either fix it at p = 0.5, which places it in the most unfavorable case,
• either use the Neyman-Pearson criterion or the median criterion which consists of having:
  probability of false alarm = probability of non-detection.

If we want to apply the Neyman-Pearson criterion or the median criterion, the detection rule will be of the form:

The threshold λ is fixed to reach an a priori value of the probability of false alarm (or the probability of correct decision).

This probability P _FA of false alarm is worth:

There is a priori no simple theoretical calculation of this expression, therefore no theoretical means of evaluating the threshold λ. The calculation of λ can however be done under simulation, depending on the particular case treated. The simplified decision rule set out below is more convenient to use in this case. Simplified decision rule.

This rule is: x > s⇔ U ∈C ₁ , x> s⇔U∈C ₂
Maximum likelihood criterion:
the probability P _c of correct decision is:

The optimal threshold is obtained for:

Case of the Neyman-Pearson criterion:

• soit la fixer arbitrairement par une approche heuristique,
• soit la fixer à p = 0,5, ce qui place dans le cas le plus défavorable,
• soit utiliser le critère de Neyman-Pearson ou le critère de la médiane qui consiste à avoir Probabilité de fausse alarme = Probabilité de non-détection.

When the probability p is unknown, we can:

• either fix it arbitrarily using a heuristic approach,
• either fix it at p = 0.5, which places it in the most unfavorable case,
• either use the Neyman-Pearson criterion or the median criterion which consists of having Probability of false alarm = Probability of non-detection.

To apply the Neyman-Pearson criterion or the median criterion, the probabilities of non-detection and false alarm are defined: We have :

We then set P _fa or P _nd , to determine the value of the threshold.

The median criterion leads to:

Implementation.

When the decision rule has been defined using the theoretical tools recalled above, and having a noise reference energy E ₀ , the detection is performed on E (T ₁ ), ..., E (T _n ), with: $$\mathit{\text{E}}\text{(}{\mathit{\text{T}}}_{\mathit{\text{i}}}{\text{) = Σ}}_{\text{0≤}}{}_{\mathit{\text{not}}}{}_{\text{≤}}{}_{\mathit{\text{NOT}}}{}_{\text{-1}}{\mathit{\text{u}}}_{\mathit{\text{i}}}\text{(}\mathit{\text{not}}{\text{)}}^{\text{2}}$$ where the u _i (n) are the N samples constituting the frame T _i .

Among the frames available at the start, the pre-classification algorithm has highlighted a set Δ of frames almost certainly of the "noise" class. The average of the energies of the frames of the set Δ makes it possible to obtain a reference value E _o which will be used by the detection algorithm so as to classify the energies of the frames other than those of the set Δ as well as the new frames acquired later.

Application examples.

• détection d'un bruit blanc gaussien dans un autre bruit blanc gaussien ;
• détection d'un bruit blanc gaussien dans un bruit gaussien bande étroite ;
• détection d'une énergie déterministe dans un bruit gaussien bande étroite...

Détection d'un signal d'énergie bornée dans un bruit gaussien bande étroite :
Hypothèse 1 : nous supposons ne pas connaître le signal utile dans sa forme, mais nous ferons l'hypothèse suivante : pour toute réalisation s(0), ..., s(N-1) de s(n), l'énergie S définie par :
S = (1/N)Σ_0≤n≤N-1 s(n)² est bornée par µ_s ², et ce, dès que N est suffisamment grand, de sorte que : S = Σ_0≤n≤N-1 s(n)²>Nµ_s ².
Hypothèse 2 : Le signal utile est perturbé par un bruit additif noté x(n), que l'on suppose gaussien et bande étroite. On suppose que le processus x(n) traité est obtenu par filtrage bande étroite d'un bruit blanc gaussien.It is possible to deal with a large number of examples making it possible to demonstrate the advantage of the process of the invention. In fact, there are as many examples as there are pairs of models that can be formed from those described above (see the examples of VAGP given above):

• detecting a white Gaussian noise in another white Gaussian noise;
• detection of a white Gaussian noise in a narrow band Gaussian noise;
• detection of a deterministic energy in a narrow band Gaussian noise ...

Detection of a bounded energy signal in a narrow band Gaussian noise:
Hypothesis 1: we suppose not to know the useful signal in its form, but we will make the following hypothesis: for any realization s (0), ..., s (N-1) of s (n), the energy S defined by:
S = (1 / N) Σ _0≤n≤N-1 s (n) ² is bounded by µ _s ² , as soon as N is sufficiently large, so that: S = Σ _{0≤n≤N- 1} s (n) ² > Nµ _s ² .
Hypothesis 2: The useful signal is disturbed by an additive noise denoted x (n), which is assumed to be Gaussian and narrow band. We assume that the process x (n) processed is obtained by narrow band filtering of white Gaussian noise.

The correlation function of such a process is then: $${\text{Γ}}_{\text{x}}{\text{(k) = Γ}}_{\text{x}}{\text{(0) cos (2πkf}}_{\text{0}}{\text{T}}_{\text{e}}{\text{) sin}}_{\text{vs}}{\text{(πkBT}}_{\text{e}}\text{)}$$ If we consider N samples x (n) of this noise, we then have: $${\text{V = (1 / N) Σ}}_{\text{0≤n≤N-1}}{\text{x (n)}}^{\text{2}}{\text{∈ N (Nσ}}_{\text{x}}{}^{\text{2}}{\text{, 2σ}}_{\text{x}}{}^{\text{4}}{\text{Σ}}_{\text{0≤i≤}}{}_{\text{N-1.0≤j≤N-1}}{\text{g}}_{\text{f0, B, Te}}{\text{(ij)}}^{\text{2}}\text{)}$$ with: g _{f0, B, Te} (k) = cos (2πkf ₀ T _e ) sin _c (πkBT _e )

The parameter α of this variable is: $${\text{α = N / [2Σ}}_{\text{0≤i≤N-1.0≤j≤N-1}}{\text{g}}_{\text{f0, B, Te}}{\text{(ij)}}^{\text{2}}{\text{]}}^{\text{1/2}}$$ Hypothesis 3: The signals s (n) and x (n) are assumed to be independent. We then assume that the independence between s (n) and x (n) implies decorrelation in the temporal sense of the term, that is to say that we can write: $$\mathit{\text{vs}}\text{=}\frac{{\text{Σ}}_{\text{0≤}\mathit{\text{not}}\text{≤}\mathit{\text{NOT}}\text{-1}}\mathit{\text{s}}\text{(}\mathit{\text{not}}\text{)}\mathit{\text{x}}\text{(}\mathit{\text{not}}\text{)}}{{\left(\text{Σ}{}_{\text{0≤}\mathit{\text{not}}\text{≤}\mathit{\text{NOT}}\text{-1}}\mathit{\text{s}}\text{(}\mathit{\text{not}}\text{)}{}^{\text{2}}\right)}^{\text{1/2}}{\left(\text{Σ}{}_{\text{0≤}\mathit{\text{not}}\text{≤}\mathit{\text{NOT}}\text{-1}}\mathit{\text{s}}\text{(}\mathit{\text{not}}\text{)}{}^{\text{2}}\right)}^{\text{1/2}}}\text{= 0}$$ This correlation coefficient is only the expression in the time domain of the spatial correlation coefficient defined by:
E [s (n) x (n)] / (E [s (n) ² ] E [x (n) ² ]) ^1/2 when the processes are ergodic.
Let u (n) = s (n) + x (n) be the total signal, and U = Σ _0≤n≤N-1 u (n) ² .
We approximate U by: U = Σ _0≤n≤N-1 s (n) ² + Σ _0≤n≤N-1 x (n) ²
As we have: Σ _0≤n≤N-1 s (n) ² ≥ µ _s ² we will have:
U ≥ Nµ _s ² + Σ _0≤n≤N-1 x (n) ² .
Hypothesis 4: As we suppose that the signal presents a bounded average energy, we will suppose that a process capable of detecting an energy µ _s ² , will be able to detect any signal of higher energy.

Taking into account the preceding hypotheses, class C ₁ is defined as being the class of energies when the useful signal is present. According to hypothesis 3, U ≥ Nµ _s ² + Σ _0≤n≤N-1 x (n) ² , and according to hypothesis 4, if the energy Nµ _s ² + Σ _{0≤n≤N- is detected 1} x (n) ² , we will also be able to detect the total energy U.
According to hypothesis 2, Nµ _s ² + Σ _{0≤ n≤ N-1} x (n) ² ∈ N (Nµ _s ² + Nσ _x ² , 2σ _x ⁴ Σ _{0≤i≤N-1.0≤ j≤N-1} g _{f0, B, Te} (ij) ² ).
So C ₁ = N (Nµ _s ² + Nσ _x ² 2σ _x ⁴ Σ _{0≤i≤N-1,0≤j≤N-1} ^g _{f0, B, Te} (ij) ² ) and the parameter α of this variable is worth:
α ₁ = N (1 + r) / [2Σ _{0≤i≤N-1,0≤j≤N-1} g _{f0, B, Te (ij)} 2] ^1/2 , where r = µ _s ² / σx ² represents the signal to noise ratio.
C ₂ is the class of energies corresponding to noise alone. According to hypothesis 2, if the noise samples are x (0), ..., x (M-1), it comes:
V = (1 / M) Σ _0≤n≤M-1 x (n) ² ∈ N (Mσ _x ² , 2σ _x ⁴ Σ _{0≤i≤ M-1,0≤j≤M-1} g _{f0, B , Te} (ij) ² )
The parameter α of this variable is:
α ₂ = M / [2Σ _{0≤i≤M-1,0≤j≤M-1} g _{f0, B, Te} (ij) ² ] ^½
So we have :
C ₁ = N (m ₁ , σ ₁ ² ) and C ₂ = N (m ₂ , σ ₂ ² ), with: m ₁ = Nµ _s ² + Nσ _x ² , m ₂ = Mσ _x ² ,
σ ₁ = σ _x ² [2Σ _{0≤i≤ N-1.0≤j≤N-1} g _{f0, B, Te} (ij) ² ] ^1/2 and
σ ₂ = σ _x ² [2Σ _{0≤i≤ M-1.0≤j≤M-1} g _{f0, B, Te} (ij) ² ] ^1/2 Hence m = m ₁ / m ₂ = (N / M) (1 + r),
α ₁ = m ₁ / σ ₁ = N (1 + r) / [2Σ _{0≤i≤N-1,0≤j≤N-1} g _{f0, B, Te} (ij) ² ] ^1/2 and
α ₂ = m ₂ / σ ₂ = M / [2Σ _{0≤i≤M-1,0≤j≤M-1} g _{f0, B, Te} (ij) ² ] ^1/2 .
We can then implement the steps of the method of the invention set out above.

PN code detection

We consider a BPSK modulation spread by a PN code of length L very large in front of 1. The duration of transmission of a binary element d _n is T _b . The duration of transmission of a binary element of the PN code is T.

• Λ_{[kTc,(k+1)Tc]}(t) = 1 si t ∈ [kT',(k+1)T'] et
  Λ_{[kT',(k+1)T']}(t) = 0 si t ∉ [kT',(k+1)T'],
• K désigne le nombre d'échantillons du code PN vus sur cet intervalle, et.
• ϕ la phase aléatoire équirépartie sur [0,2π]

Over the interval of [nT _b , (n + 1) T _b ], the signal sent is:
m (t) = (2E _b / T _b ) ^1/2 dn Σ _0≤k≤K-1 c _k Λ _{[kT ', (k + 1) T']} (t) cos (ω ₀ t + ϕ) with:

• Λ _{[kTc, (k + 1) Tc]} (t) = 1 if t ∈ [kT ', (k + 1) T'] and
  Λ _{[kT ', (k + 1) T']} (t) = 0 if t ∉ [kT ', (k + 1) T'],
• K denotes the number of PN code samples seen over this interval, and.
• ϕ the random phase equally distributed over [0,2π]

This transmitted signal is embedded in a background noise which is b (t), which is assumed to be white and Gaussian.

We then seek to detect the signal s (t) from the received signal r (t) = m (t) + b (t), assuming that we do not know the PN code, therefore neither the values c _k neither the duration L, nor the time T _b , nor the frequency ω ₀ .

• T est une durée d'intégration assez grande pour que les échantillons du code PN vus sur cet intervalle soient suffisamment nombreux et décorrélés, tout en restant assez faibles pour qu'on n'atteigne pas la prériodicité L du code PN. Si K est le nombre d'éléments binaires du code PN vu sur cet intervalle, on suppose donc: L >> 1, K << L et K >>1. T vérifie en outre ω₀T >>1
• ω est une fréquence qui sert à essayer de récupérer la porteuse, telle que ωT >> 1

Let then be the random variable: or :

• T is a sufficiently long integration time for the samples of the PN code seen over this interval to be sufficiently numerous and decorrelated, while remaining low enough so that the pre-periodicity L of the PN code is not reached. If K is the number of binary elements of the PN code seen over this interval, we therefore assume: L >> 1, K << L and K >> 1. T also checks ω ₀ T >> 1
• ω is a frequency which is used to try to recover the carrier, such as ωT >> 1

We ask: and:

Using the central-limit theorem, and after calculations as described in "Performance of a Direct Sequence Spread Spectrum System with Long Period and Short Period Code Sequences", R. SINGH, IEEE Transactions on Communications, Vol. Com-31, N ° 3, March 1983, we show that s (n) is a Gaussian variable, with zero mean and variance: σ _s ² = (T _b / T) (E _b / 2K) sin _c ² ( π (ω - ω ₀ ) / K).

In practice, we suppose that the s (n) are independent, so that the sequence of samples s (n) constitutes a discrete white Gaussian process.

Likewise, the sequence of samples x (n) constitutes a white Gaussian process with zero mean and variance σ _s ² = σ _b ² . To detect the PN code is to detect s (n), therefore to detect a white Gaussian noise embedded in another white Gaussian noise.

Let then be the variable U = Σ _0≤n≤N-1 u (n) ² . Using the results set out above with regard to VAGP, we have:
U ∈ N (N (σ _s ² + σ _x ² ); 2N (σ _s ² + σ _x ² ) ² ).
The parameter α of this variable is α ₁ = (N / 2) ^1/2

Let then be the variable V = Σ _0≤n≤M-1 x (n) ² . We have :
V ∈ N (Mσ _x ² ; 2Mσ _x ⁴ ).

The parameter α of this variable is α ₂ = (M / 2) ^1/2 . We can therefore model as in the case of a detection between the two classes:
C ₁ = N (N (σ _s ² + σ _x ² ); 2N (σ _s ² + σ _x ² ) ² ) and C ₂ = N (Mσ _x ² ; 2Mσ _x ⁴ ).
We then have: m = (N / M) (1 + r), α ₁ = (N / 2) ^1/2 , α ₂ = (M / 2) ^1/2
Note that if N = M ⇒ α ₁ = α ₂ = (N / 2) ^1/2
The method described above is therefore applicable to this problem.

Claims (8)

1. Energy-based method for detecting useful signals in the presence of noise, comprising a preclassification step followed by a detection step, the said preclassification step consisting, on the basis of a set of samples of a noisy signal which are grouped into successive frames, in deciding as a function of the mutual ratios of the energies of the successive frames, the decision being actually based on the calculation of a distance which is the absolute value of the difference of the logarithms of the two energies, and on the comparison of this distance with an optimal threshold (Log(s)) so as to class within a first "noise only" class at least some of the frames likely to be members of this class; the said detection step consisting in the detection for the other frames, of those exhibiting a sufficiently high energy relative to a reference energy (Eo) calculated on the basis of the energies of the "noise only" frames identified during the preclassification step so that these detected frames exhibit a high probability of membership of a second "noise + useful signal" class.
2. Method according to Claim 1, characterized in that the calculation of an optimal threshold is done by applying the maximum likelihood criterion when the correct decision probability is known.
3. Method according to Claim 1, characterized in that the calculation of the optimal threshold is done by applying the Neyman-Pearson criterion when the correct decision probability is not known.
4. Method according to one of the preceding claims, characterized in that membership of the second class is determined by applying the maximum likelihood criterion when the correct decision probability is known.
5. Method according to one of Claims 1 to 3, characterized in that membership of the second class is determined by applying the Neyman-Pearson criterion when the correct decision probability is not known.
6. Method according to one of the preceding claims, characterized in that after having processed a first frame of samples, an autoregressive model of the noise is calculated and is used to whiten the succeeding frames, on the basis of a set of frames consisting, with high probability, of noise alone.
7. Method according to one of Claims 1 to 5, characterized in that after having highlighted a set of frames consisting, with high probability, of noise only, a mean spectrum of the noise is calculated and noise-suppression is applied to the succeeding frames.
8. Method according to one of the preceding claims, characterized in that the results acquired for a frame of samples are utilized for reupdating the results of the previous frame.