EP0518742A1 - Method for detecting a noisy wanted signal - Google Patents

Abstract

In order to detect a noisy wanted signal, a measurement of the expected S/N ratio of this signal is taken over one time slice, a measurement of the estimated white noise alone is taken over another time slice without wanted signal, the mean energy of the noise and of the noisy signal are calculated, each in its time slice, the theoretical detection threshold and the ratio of these two energies are calculated, and the ratio is compared with the calculated threshold, this threshold being greater than 1 (ideal threshold).

Description

The present invention relates to a method for detecting a noisy useful signal.

One of the important problems in signal processing, simple as for its statement, but all the more complex as for its resolution, consists in determining the presence or the absence of a useful signal drowned in an additive noise.

Various solutions are possible. The instantaneous amplitude of the signal received or processed can be used as a variable by reference to a threshold determined experimentally.

One can also use as variable the energy of the total signal on a time slice of duration T, by thresholding, always experimentally, this energy.

These thresholds allow a first presumption on the presence or absence of the signal. They are also applicable to any signal. Also, they are supplemented by "confirmation" systems, defining "almost certain" criteria, specific to the type of useful signal, when the nature of this is known a priori.

Such a complementary system is widely used in speech processing and can consist, for example, in a pitch extraction or in the evaluation of the minimum energy of a vowel.

The subject of the present invention is a method for detecting a noisy useful signal, determining as rigorously as possible the detection threshold, and which can operate in a self-adaptive manner.

According to the invention, there is the expected signal / noise ratio of the signal to be processed, and there is a measurement of the estimated noise alone, measurement digitized on M points, this noise being white or made white, the average energy is calculated. noise on these M points, we take a slice of N noisy signal points, we calculates the average energy of these N points, the theoretical detection threshold is calculated, the ratio of the two said average energies is calculated, and this ratio is compared with said threshold.

The present invention will be better understood on reading the detailed description of an embodiment taken by way of nonlimiting example.

We will first explain how to theoretically, in the ideal case, detect a noisy signal.

   n étant un nombre entier : 0 ≦ n ≦ N-1, s(n) étant un signal utile et x(n) un bruit. En outre, on dispose d'une autre information y(n), avec 0 ≦ n ≦ M-1, et M pouvant être égal à N ou différent de celui-ci. y(n) est une mesure du bruit x(n) sur une autre tranche temporelle exempte de signal utile.We have a first piece of information u (n) for a first time slot such as: $$\text{u (n) = s (n) + x (n)}$$

n being an integer: 0 ≦ n ≦ N-1, s (n) being a useful signal and x (n) a noise. In addition, there is other information y (n), with 0 ≦ n ≦ M-1, and M can be equal to N or different from it. y (n) is a measure of the noise x (n) over another time slot free of useful signal.

   et $$\text{V = (y(0)² + y(1)² + ... + y(M)²)/M}$$

   et $$\text{Z = U/V}$$

We ask: $$\text{U = (u (0) ² + u (1) ² + ... + u (N) ²) / N}$$

and $$\text{V = (y (0) ² + y (1) ² + ... + y (M) ²) / M}$$

and $$\text{Z = U / V}$$

et le simple critère de détection serait : $$\text{Z > 1 : présence de signal utile}$$

$$\text{Z < 1 : absence de signal utile}$$

Thus, in an ideal and unrealistic case, we would have, by noting RSB = signal to noise ratio: $$\text{Z = 1 + RSB}$$

and the simple detection criterion would be: $$\text{Z> 1: presence of useful signal}$$

$$\text{Z <1: absence of useful signal}$$

According to the present invention, the theoretical threshold of 1 is replaced by a threshold µ, calculated as explained below, which takes into account the fact that the signals available are not perfectly ergodic and that U and V are only estimates of the true values of the variances σ _u 2 and σ _x 2.

To perform this calculation of µ, we proceed as follows.

We start from the fact that the variables U and V are random in nature, and therefore Z is too. We then calculate the probability density of Z (which depends on the signal-to-noise ratio).

Next, using the maximum likelihood principle, determine the best estimate of the signal-to-noise ratio after calculating the variable Z.

To this end, the aforementioned variable U (n) is measured on a time slice, and the variable y (n) is measured on another time slice where it is certain that there is no useful signal, but only noise (independent and decorrelated from s (n)).

To determine the density of the random variable Z (which can be called the observed variable), we proceed as follows. Let X₁ belonging to N (m₁; σ₁²) and X₂ belonging to N (m₂; σ₂²) two independent Gaussian random variables for which the probabilities P _r {X₁ <o} and P _r {X₂ <o} are practically zero.
We ask: $$\text{m = m₁ / m₂, σ² = σ₁² / σ₂², α = m₂ / σ₂.}$$

où U(x) = 1 si x ≧ o et U(x) = o si x < o.The probability density f _x (x) of X is then:

where U (x) = 1 if x ≧ o and U (x) = o if x <o.

on a : P(x) = P_r {X < x} = F [h(x)], expression dans laquelle F(x) désigne la fonction caractéristique de la variable gaussienne normalisée.Yes $$\text{h (x) = α.}\frac{\text{x - m}}{{\text{(x² + σ²)}}^{\text{1/2}}}$$

we have: P (x) = P _r {X <x} = F [h (x)], expression in which F (x) denotes the characteristic function of the normalized Gaussian variable.

We now assume that the signals s (n), x (n) and y (n) are white, Gaussian and centered.

We pose

This last term is therefore also white, Gaussian and centered;
and we ask

Z est donc le rapport de deux variables gaussiennes indépendantes. On peut facilement démontrer que U et V sont indépendantes.
Avec : $${\text{m₁ = σ}}_{\text{u}}{\text{² , σ₁² = 2σ}}_{\text{u}}{\text{⁴/N , m₂ = σ}}_{\text{x}}{\text{² , σ₂² = 2σ}}_{\text{x}}\text{⁴/M}$$

il vient : $${\text{m = σ}}_{\text{u}}{\text{²/σ}}_{\text{x}}{\text{² , σ² = (M/N) (σ}}_{\text{u}}{\text{²/σ}}_{\text{x}}{\text{²)², α = (M/2)}}^{\text{1/2}}\text{.}$$

Or : σ_u²/σ_x²= 1+ r où r = σ_s²/σ_x² est le rapport signal à bruit. Soit k = M/N, il vient : m = r+1, σ²= k(r+1)².Since we define σ _s ² and σ _x ², we implicitly assume that the probability density is calculated at σ _s ² and σ _x ² known. We therefore evaluate the density of Z by knowing σ _s ² and σ _x ². In this case, U and V follow laws of chi-2, and, for N and M sufficiently large, U and V are approximated by Gaussian laws practically always positive:

Z is therefore the ratio of two independent Gaussian variables. We can easily demonstrate that U and V are independent.
With: $${\text{m₁ = σ}}_{\text{u}}{\text{², σ₁² = 2σ}}_{\text{u}}{\text{⁴ / N, m₂ = σ}}_{\text{x}}{\text{², σ₂² = 2σ}}_{\text{x}}\text{⁴ / M}$$

he comes : $${\text{m = σ}}_{\text{u}}{\text{² / σ}}_{\text{x}}{\text{², σ² = (M / N) (σ}}_{\text{u}}{\text{² / σ}}_{\text{x}}{\text{²) ², α = (M / 2)}}^{\text{1/2}}\text{.}$$

Now: σ _u ² / σ _x ² = 1+ r where r = σ _s ² / σ _x ² is the signal to noise ratio. Let k = M / N, it comes: m = r + 1, σ² = k (r + 1) ².

• 1°) x ≧ 0
  
• 2°) x ≦ 0 d'où: f_z(z : σ_s², σ_x²) = 0

On posera :

de sorte que : f_z(z:σ_s², σ_x²) = f_k,M(z,σ_s²/σ_x²)The probability density of Z, knowing σ _s ² and σ _x ², is therefore expressed by:

• 1 °) x ≧ 0
  
• 2 °) x ≦ 0 where: f _z (z: σ _s ², σ _x ²) = 0

We will ask:

so that: f _z (z: σ _s ², σ _x ²) = f _{k, M} (z, σ _s ² / σ _x ²)

From the above results relating to the probability density f _x (x), we deduce the probability $${\text{Pr {Z <z: σ}}_{\text{s}}{\text{², σ}}_{\text{x}}\text{²}.}$$

Il vient : $${\text{Pr {Z < z : σ}}_{\text{s}}{\text{² ; σ}}_{\text{x}}{\text{²} = F{h}}_{\text{k,M}}\text{(x,r)}.}$$

Is : $${\text{h}}_{\text{kM}}{\text{(x, r) = (M / 2)}}^{\text{1/2}}\frac{\text{x - (r + 1)}}{{\text{[x² + k (r + 1) ²]}}^{\text{1/2}}}$$

He comes : $${\text{Pr {Z <z: σ}}_{\text{s}}{\text{²; σ}}_{\text{x}}{\text{²} = F {h}}_{\text{k, M}}\text{(x, r)}.}$$

We will now examine the case of any signal s (n) and a white Gaussian noise.

We always assume that the noises x (n) and y (n) are white, Gaussian with σ _x ² = E [x (n) ²] = E [y (n) ²]. The useful signal s (n) is assumed to be arbitrary, independent of noise.

The new hypothesis made here is to suppose that s (n) and x (n) are uncorrelated in the temporal sense of the term, that is to say that: $$\text{c =}\frac{\text{Σ 0 ≦ n ≦ N-1 s (n) x (n)}}{{\text{(Σ 0 ≦ n ≦ N-1 s (n) ²)}}^{\text{1/2}}{\text{(Σ 0 ≦ n ≦ N-1 x (n) ²)}}^{\text{1/2}}}\text{= 0}$$

$$\text{Z =}\frac{{\text{µ}}_{\text{s}}\text{² + (1/N) Σ 0 ≦ n ≦ N-1 x(n)²}}{\text{(1/M) Σ 0 ≦ n ≦ M-1 y(n)²}}$$

We then show that U can be approximated by: $${\text{U = µ ²}}_{\text{s}}\text{+ (1 / N) Σ 0 ≦ n ≦ N-1 x (n) ² and Z by:}$$

$$\text{Z =}\frac{{\text{µ}}_{\text{s}}\text{² + (1 / N) Σ 0 ≦ n ≦ N-1 x (n) ²}}{\text{(1 / M) Σ 0 ≦ n ≦ M-1 y (n) ²}}$$

, σ $\genfrac{}{}{0ex}{}{\text{2}}{\text{x}}$

).As below, the calculation of the density of Z was done by knowing σ _s ² and σ _x ², here the calculation will be done by knowing µ _s ² and σ _x ². The density to be calculated will be noted by f _z (z: µ $\genfrac{}{}{0ex}{}{\text{2}}{\text{s}}$

, σ $\genfrac{}{}{0ex}{}{\text{2}}{\text{x}}$

).

(µ_s² + σ_x²; (2/N) σ_x⁴). V appartient à

(σ_x² ; (2/M) σ_x⁴).Knowing µ _s ², U = µ _s ² + (1 / N) Σ 0 ≦ n ≦ N-1 x (n) ² belongs to

(µ _s ² + σ _x ²; (2 / N) σ _x ⁴). V belongs to

(σ _x ²; (2 / M) σ _x ⁴).

Donc : m = r+1, σ² = k, α = (M/2)^1/2, avec k = M/N et r = µ_s²/σ_x².Z = U / V is therefore approximated by the ratio of two independent Gaussian laws. As U and V are independent, we therefore apply the result concerning the probability density of X, with: $${\text{m₁ = µ}}_{\text{s}}{\text{² + σ}}_{\text{x}}{\text{², σ₁² = (2 / N) σ}}_{\text{x}}{\text{⁴, m₂ = σ}}_{\text{x}}{\text{², σ₂² = (2 / M) σ}}_{\text{x}}\text{⁴}$$

So: m = r + 1, σ² = k, α = (M / 2) ^1/2 , with k = M / N and r = µ _s ² / σ _x ².

On posera :

de sorte que : f_z(z:σ_s², σ_x²) = f_k,M(z, σ_s²/σ_x²)The probability density of Z, knowing µ _s ² and σ _x ² is therefore worth:

We will ask:

so that: f _z (z: σ _s ², σ _x ²) = f _{k, M} (z, σ _s ² / σ _x ²)

Soit : $${\text{h}}_{\text{k,M}}{\text{(x,r) = (M/2)}}^{\text{1/2}}\frac{\text{x - (r+1)}}{{\text{[x² + k]}}^{\text{1/2}}}$$

il vient : $${\text{Pr { Z < z : µ}}_{\text{s}}{\text{, σ}}_{\text{x}}{\text{² } = F (h}}_{\text{k,M}}\text{(x,r))}$$

From the above results concerning the probability density of X, we deduce the probability $${\text{Pr {Z <z: µ}}_{\text{s}}{\text{², σ}}_{\text{x}}\text{²}.}$$

Is : $${\text{h}}_{\text{k, M}}{\text{(x, r) = (M / 2)}}^{\text{1/2}}\frac{\text{x - (r + 1)}}{{\text{[x² + k]}}^{\text{1/2}}}$$

he comes : $${\text{Pr {Z <z: µ}}_{\text{s}}{\text{, σ}}_{\text{x}}{\text{²} = F (h}}_{\text{k, M}}\text{(x, r))}$$

According to the present invention, the activity detection is implemented by using maximum likelihood.

In the case of processed signals, the probability density of the variable Z, knowing the energies of the useful signal and of the noise, is expressed by a function of the form:
f _{k, M} (z, r) where r denotes the signal to noise ratio. This probability therefore depends on the signal to noise ratio. Also, the decision rule can only be given with an expected signal-to-noise ratio. Let r _{o be} this expected signal-to-noise ratio.

We suppose that the probability of absence of s (n) is π _o and that the probability of presence of s (n) is π₁.

$$\frac{{\text{π₁ f}}_{\text{k,M}}{\text{(z,r}}_{\text{o}}\text{)}}{{\text{π}}_{\text{o}}{\text{f}}_{\text{k,M}}\text{(z,0)}}\text{< 1 ⇒ D = 0}$$

Since we know the probability density f _{k, M} (z, r) the optimal decision rule is provided by the general theory of detection and is expressed by: $$\frac{{\text{π₁ f}}_{\text{k, M}}{\text{(z, r}}_{\text{o}}\text{)}}{{\text{π}}_{\text{o}}{\text{f}}_{\text{k, M}}\text{(z, 0)}}\text{> 1 ⇒ D = 1}$$

$$\frac{{\text{π₁ f}}_{\text{k, M}}{\text{(z, r}}_{\text{o}}\text{)}}{{\text{π}}_{\text{o}}{\text{f}}_{\text{k, M}}\text{(z, 0)}}\text{<1 ⇒ D = 0}$$

We can also express this decision rule in the form: $$\text{(Z <µ ⇒ D = 0) and (Z> µ ⇒ D = 1).}$$

We must then determine µ and solve the equation: $${\text{ln [f}}_{\text{k, M}}{\text{(z, r}}_{\text{o}}{\text{)] - ln [f}}_{\text{k, M}}{\text{(z, 0)] - ln (π}}_{\text{o}}\text{/ π₁) = 0.}$$

We then demonstrate that the probability of error is worth: $${\text{Pe = π}}_{\text{o}}{\text{[1 - F (h}}_{\text{k, M}}{\text{(µ, 0))] + π₁ F (h}}_{\text{k, M}}{\text{(µ, r}}_{\text{o}}\text{)).}$$

We will now examine the case of the detection of a white Gaussian signal in a white Gaussian noise itself.

The signals s (n), x (n) and y (n) are assumed to be white, Gaussian, centered. Let r _{o be} the expected signal-to-noise ratio, and k = M / N. the probability of absence of s (n) is π _o and the probability of presence of s (n) is π₁.

Décision D = 0 lorsque :

The decision rule is then:
Decision D = 1 when:

Decision D = 0 when:

The threshold being determined for equality (instead of inequality) between the terms of these two expressions.

On obtient par exemple pour µ, à M = N = 128, π_o = π₁ = 1/2 : r_o en dB µ -2 1,27 -1 1,34 0 1,41 1 1,50 2 1,68 We can also express this decision rule in the form: $$\text{(Z <µ ⇒ D = 0) and (Z> µ ⇒ D = 1).}$$

We obtain for example for µ, at M = N = 128, π _o = π₁ = 1/2: r _o in dB µ -2 1.27 -1 1.34 0 1.41 1 1.50 2 1.68

avec : $${\text{h}}_{\text{k,M}}{\text{(x,r) = (M/2)}}^{\text{1/2}}\frac{\text{x - (r+1)}}{{\text{[x² + k(r+1)²]}}^{\text{1/2}}}$$

The probability of error is: $${\text{Pe = π}}_{\text{o}}{\text{[1-F (h}}_{\text{k, M}}{\text{(µ, 0))] + π₁ F (h}}_{\text{k, M}}{\text{(µ, r}}_{\text{o}}\text{))}$$

with: $${\text{h}}_{\text{k, M}}{\text{(x, r) = (M / 2)}}^{\text{1/2}}\frac{\text{x - (r + 1)}}{{\text{[x² + k (r + 1) ²]}}^{\text{1/2}}}$$

We give below some values of Pe as a function of r _o . π _o and π₁ are taken equal to 0.5. r _o in dB Pe -2 0.086 -1 0.052 0 0.028 1 0.013 2 0.005

In a simulation example, a white Gaussian noise with unit variance was generated. For each frame of 128 points (N = M = 128), it was randomly decided to generate an additive noise s (n), having a signal to noise ratio defined beforehand. The probabilities of appearance and absence (π _o and π₁) are equal to 0.5. We generated a second Gaussian white noise with unit variance, which was used to calculate the random variable V. For each frame, we calculated Z. We then applied the decision rule and we counted the number of errors . r _o in dB Number of errors in 1000 iterations -2 73 -1 43 0 18 1 10 2 2

These results corroborate those predicted by the theoretical calculation.

We will now examine the case of any signal s (n) and a white Gaussian noise.

Décision D = 0 lorsque : $$\text{ln}\frac{\text{(r+1) z+k}}{\text{z+k}}\text{< (M/4)}\frac{{\text{[z-(r}}_{\text{o}}\text{+1)]² - (z-1)²}}{\text{z²+k}}\text{+ ln}\frac{{\text{π}}_{\text{o}}}{\text{π₁}}$$

We always assume that the noises x (n) and y (n) are white, Gaussian with σ _x ² = E [x (n) ²] = E [y (n) ²]. The useful signal s (n) is assumed to be arbitrary, independent of noise. Let r _{o be} the expected signal-to-noise ratio, k = M / N. The probability of absence of s (n) is π _o and that of presence of s (n) is π₁.
The decision rule is then:
Decision D = 1 when: $$\text{ln}\frac{\text{(r + 1) z + k}}{\text{z + k}}\text{> (M / 4)}\frac{{\text{[z- (r}}_{\text{o}}\text{+1)] ² - (z-1) ²}}{\text{z² + k}}\text{+ ln}\frac{{\text{π}}_{\text{o}}}{\text{π₁}}$$

Decision D = 0 when: $$\text{ln}\frac{\text{(r + 1) z + k}}{\text{z + k}}\text{<(M / 4)}\frac{{\text{[z- (r}}_{\text{o}}\text{+1)] ² - (z-1) ²}}{\text{z² + k}}\text{+ ln}\frac{{\text{π}}_{\text{o}}}{\text{π₁}}$$

We can also express this decision rule in the form: $$\text{(Z <µ ⇒ D = 0) and (Z> µ ⇒ D = 1).}$$

The following values are obtained for µ as a function of r _o , for M = N = 128, π _o = π₁ = 1/2. r _o in dB µ -2 1.30 -1 1.38 0 1.48 1 1.60 2 1.76

avec : $${\text{h}}_{\text{k,M}}{\text{(x,r) = (M/2)}}^{\text{1/2}}\frac{\text{x - (r+1)}}{{\text{[x² + k ]}}^{\text{1/2}}}$$

In addition, we get: $${\text{Pe = π}}_{\text{o}}{\text{[1-F (h}}_{\text{k, M}}{\text{(µ, 0))] + π₁ F (h}}_{\text{k, M}}{\text{(µ, r}}_{\text{o}}\text{))}$$

with: $${\text{h}}_{\text{k, M}}{\text{(x, r) = (M / 2)}}^{\text{1/2}}\frac{\text{x - (r + 1)}}{{\text{[x² + k]}}^{\text{1/2}}}$$

We give below some values of Pe as a function of r _o . The probabilities π _o and π _o are taken equal to 0.5. r _o in dB Pe -2 0.062 -1 0.032 0 0.013 1 0.004 2 0.001

In a simulation example, for each frame of 128 points of white noise generated (N = M = 128), we decided randomly to add s (n), which is here, a sinusoid, presenting a signal to noise ratio defined beforehand. π₁ and π _o are taken equal to 0.5.

A second white Gaussian noise of unit variance was generated, used to calculate V. For each frame, Z was calculated and the above decision rule was applied. We counted the number of errors.

The following results were obtained r _o in dB Number of errors in 1000 iterations -2 70 -1 37 0 12 1 6 2 3

These results corroborate those predicted by the theoretical calculation.

The previous results, because they are very general, allow the detection of signals embedded in additive noise, even when the signal to noise ratio is low, close to 0 dB.

We will describe below an application in which this type of detection can be very useful.

The algorithms presented apply to the case of speech, as a pre-system for detecting voice activity.

The choice of detection threshold depends on the context.

With regard to the audio tapes used, a preliminary characterization of noise and speech, using measurements based on the maximum likelihood estimation shows that the speech signal to be detected has a signal-to-noise ratio of at least minus 6 dB.

On the other hand, the processing system uses 128 point signal frames, the sampling frequency being 10 kHz.

The variables U and V are both evaluated on 128 points so that M = N = 128.

From the above, the theoretical detection threshold is deduced from 3.

However, we cannot be satisfied with this single threshold. In fact, if the noise is relatively stationary, it has instabilities to be taken into account to renew the variable V, which makes the algorithm partially adaptive.

A second threshold is therefore introduced, which makes it possible to decide whether the variable V will be renewed or not.

This second threshold is chosen at 1.25, which corresponds to noise additive to stationary noise having a signal to noise ratio of -2 dB.

The decision rule is then:

If Z <1.25 :

Then the processed frame is composed of the same noise as that used as a reference. The variable V is replaced by the value of the energy of the processed frame.

Note that since the decision is to consider the frame processed as representative noise, we could renew the variable V by averaging the old value of V and the energy of the frame considered. Which brings to change the value of M (number of points on which V is evaluated) but this operation can induce a malfunction of the algorithm.

If 1.25 <z> 3 :

The frame is considered to contain non-stationarity of the noise, and free of speech.

If 3 <Z :

The frame is considered to be speech.

Tests carried out on samples of noisy signals have validated this detection.

However, remember that this voice detection can be improved by the use of criteria specific to the speech signal, such as the calculation of "pitch".

The algorithm proposed here concerns the study of some examples of signals. It is obvious that for other speech signals having different signal to noise ratios, a new choice of thresholds is necessary.

The use of two thresholds is generally preferable.

An application of this algorithm makes it possible to create correct reference files for the voice recognition system studied. A precise segmentation of speech is then necessary.

In one application, a micro alternation (micro opening and closing) was used which provides rough segmentation of speech.

The previous algorithm was used to refine this alternation. A first pass of the algorithm made it possible to specify the start of the speech. A second pass consisted in reading the speech file "upside down", that is to say starting from the microphone closure towards the microphone opening. This then made it possible to specify the end of the speech.

This non-causal use of the algorithm is necessary, since the activity detection is precise enough to detect, within words, the presence of rests, which is detrimental to the establishment of segmentation for the learning.

The same type of application also makes it possible to segment the speech files on which a recognition is carried out.

However, this algorithm is obviously not causal, which is detrimental to real-time use. Hence the need to complete this algorithm with a calculation specific to speech processing.

We have demonstrated the existence of optimal detection thresholds, which provides a theoretical approach to the problem of estimating the signal-to-noise ratio and, especially of detection, in the case of white noise and a signal known only by its energy on N points when it remains relatively stationary.

Claims (5)

Method for detecting a noisy useful signal, for which there is the expected signal / noise ratio of the signal to be processed, and a measurement of the estimated noise alone, measurement digitized on M points, characterized by the fact that l average energy of the noise on these M points, we take a slice of N noisy signal points, we calculate the average energy of these N points, we calculate the theoretical detection threshold, we calculate the ratio (Z) of the two so-called average energies, and this ratio is compared to said threshold. Method according to claim 1, characterized in that the only estimated noise is white or made white. Method according to one of the preceding claims, characterized in that the theoretical detection threshold is determined for: $$\text{ln}\frac{\text{(r + 1) z + k}}{\text{z + k}}\text{= (M / 4)}\frac{{\text{[z- (r}}_{\text{o}}\text{+1)] ² - (z-1) ²}}{\text{z² + k}}\text{+ ln}\frac{{\text{π}}_{\text{o}}}{\text{π₁}}$$

r _o being the expected signal to noise ratio, k = M / N, π _o being the probability of absence of the useful signal and π₁ its probability of presence. Method according to one of claims 1 or 2 for the detection of a white Gaussian signal, characterized in that the theoretical detection threshold is determined for:

Method according to one of claims 1 to 3, for the detection of speech, characterized in that, in addition to the theoretical detection threshold, a second decision threshold is used to update the measured portion of the estimated noise alone , in order to take account of the instarity of the noise.