Detection method for lattice reduction-aided MIMO system receiver and iterative noise cancellation

Abstract

A detection method for a MIMO system receiver in which a linear detection is carried out in order to provide an equalised vector. This equalised vector is represented in a reduced basis obtained from the reduction of the channel matrix. It undergoes an iterative noise cancellation process in the representation according to the reduced basis. Upon each iteration, a search is carried out for the component of the equalised vector in the reduced basis located the furthest from an area unperturbed by noise surrounding the product constellation with a tolerance margin, and the point representative of the equalised vector of this area by subtracting therefrom a noise vector in the direction of this component, the module whereof is equal to a fraction of the tolerance margin. The iterative cancellation converges when the equalised vector belongs to the area unperturbed by noise.

Claims

1. A detection method for a Multiple Input Multiple Output (MIMO) system receiver with N.sub.T transmit antennas and N.sub.R receive antennas and a channel matrix H, each transmit antenna transmitting a symbol belonging to a modulation constellation, a vector of the transmitted symbols belonging to a product constellation in the space .sup.2N.sup.T, the method comprising: performing, on the channel matrix H, a Lenstra-Lenstra-Lovasz (LLL) reduction in order to provide a reduced channel matrix {tilde over (H)}, such that {tilde over (H)}=HT where T is a unimodular matrix, equalizing a vector of signals received by the receive antennas by means of a linear detection matrix {tilde over (W)}, obtained from the reduced channel matrix {tilde over (H)}, in order to provide an equalized vector r in a reduced basis formed by column vectors of the matrix T, the reduced-basis equalized vector being represented by c=Tr in the reduced basis formed by the column vectors of the matrix T, defining a receive area unperturbed by noise Z.sub.ε in a representation according to the reduced basis, the receive area surrounding the product constellation with a predetermined tolerance margin ε, performing, on the equalized vector, an iterative noise cancellation process wherein, upon each iteration, a search is carried out for a component () of the equalized vector in the reduced basis that is located furthest from the receive area unperturbed by noise, and a noise vector is subtracted from the equalized vector c, wherein a module of the noise vector is a fraction of a tolerance margin, and a direction is that of the component (), the noise cancellation process being iterated until a point representative of the equalized vector is located within said receive area unperturbed by noise or until a predetermined number of iterations is reached, and making a hard or soft decision on the equalized vector after an end of the iterative noise cancellation process to obtain a result of the detection method.

2. The detection method according to claim 1, wherein the linear detection matrix is selected from the Zero Forcing (ZF), Minimum Square Error (MMSE), or Matched Filter (MF) detection matrices respectively defined by:
{tilde over (W)}.sub.ZF={tilde over (H)}.sup.†=({tilde over (H)}.sup.H{tilde over (H)}).sup.−1{tilde over (H)}.sup.H
{tilde over (W)}.sub.MMSE=({tilde over (H)}.sup.H{tilde over (H)}+σ.sub.n.sup.2I.sub.N.sub.R).sup.−1{tilde over (H)}.sup.H
{tilde over (W)}.sub.MF={tilde over (H)}.sup.H where σ.sub.n.sup.2 is a variance of the noise affecting the signal received by a receive antenna and I.sub.N.sub.R is an identity matrix of size N.sub.R.

3. The detection method according to claim 1, wherein modulation constellations relative to the different transmit antennas are quadrature amplitude modulation (QAM) constellations.

4. The detection method according to claim 3, wherein the modulation constellations relative to the different antennas are identical.

5. The detection method according to claim 4, wherein in order to determine the component of the equalized vector in the reduced basis that is located furthest from the receive area unperturbed by noise, the method further comprises searching for the component such that $\ell =\underset{i}{\arg \max }\left(c_i-s_{\max }-\mathrm{.Math.}\text{/}{v}_i\right)$ where c.sub.i is an i.sup.th component of the equalized vector c, s.sub.max is a maximum amplitude of the QAM constellation, and $v_i=\underset{j\ne i}{\max }\left(\left|.Math.t_i|t_j.Math.\right|\right)$ where t.sub.i is an i.sup.th column vector of T.

6. The detection method according to claim 5, further comprising determining the noise vector that is subtracted from the equalized vector c by calculating $a_{\ell }\mathrm{sgn}\left(c_{\ell }\right).\mu .\left(\frac{\mathrm{.Math.}}{v_{\ell }}\right)$ where a is an -th column of a matrix A, where the matrix A is a Gram matrix of the reduced basis A=TT.sup.T, sgn(c) is a sign of the -th component of the equalized vector c, and μ is a value of the fraction.

7. The detection method according to claim 6, wherein μ=½.

8. The detection method according to claim 1, wherein the step of making the hard or soft decision comprises making the hard decision on the equalized vector at the end of the iterative process by searching for a point of the product constellation that is located closest to the equalized vector.

9. The detection method according to claim 6, wherein the method further comprises, prior to the iterative noise cancellation process, initializing variances σ.sub.n,m.sup.2 of the noise affecting each of components of the equalized vector c by ${\sigma }_{n,m}^2=\left({\sigma }_n^2\underset{i=1}{\overset{N_T}{.Math.}}{.Math.{\tilde{W}}_{m,i}.Math.}^2+{\sigma }_a^2\underset{j=1j\ne m}{\overset{N_T}{.Math.}}{.Math.{\tilde{\beta }}_{m,j}.Math.}^2\right)\text{/}{.Math.{\tilde{\beta }}_{m,m}.Math.}^2$ where σ.sub.n.sup.2 is a variance of noise affecting the signal received by a receive antenna, σ.sub.a.sup.2 is an average power of the modulation constellation, {tilde over (W)}.sub.m,i is an element of a matrix {tilde over (W)} in a row m and in a column i, {tilde over (β)}.sub.m,j is an element of a matrix β={tilde over (W)}H in the row m and in a column j and that at each iteration, the variances are updated by ${\sigma }_{n,m}^2={\sigma }_{n,m}^2+{{\mu }^2\left(\frac{\mathrm{.Math.}}{v_{\ell }}\right)}^2|A_{m,\ell }{|}^2$ where A.sub.m, is an element in the row m and in a column of the Gram matrix A.

10. The detection method according to claim 9, further comprising computing, for each symbol transmitted by a transmit antenna m, soft values of different bits b.sub.m,q, where b.sub.m,q is a q.sup.th bit of a word having been used to generate the transmitted symbol s.sub.m, the soft values being given by $L\left(b_{m,q}\right)=\frac{1}{2{\sigma }_{n,m}^2}\left(\underset{u\in S_{m,q}^1}{\min }\left({.Math.c-u.Math.}^2-\underset{u\in S_{m,q}^0}{\min }\left({.Math.c-u.Math.}^2\right)\right)\right)$ where σ.sub.n,m.sup.2 is a value of a variance affecting an m-th component of the equalized vector c at the end of the iterative process, S.sub.m,q.sup.0 is a set of points of the product constellation, assuming b.sub.m,q=0 and S.sub.m,q.sup.1 is a set of points of the product constellation, assuming b.sub.m,q=1.

Description

BRIEF DESCRIPTION OF THE FIGURES

(1) Other features and advantages of the invention will appear upon reading one preferred embodiment of the invention, described with reference to the accompanying figures, in which:

(2) FIG. 1 diagrammatically shows a known MIMO system of the prior art;

(3) FIG. 2 diagrammatically shows a flow chart of a lattice reduction-aided linear detection and successive noise reduction method according to a first embodiment of the invention;

(4) FIG. 3 diagrammatically shows one example of noise reduction in a step of the method in FIG. 2;

(5) FIG. 4 diagrammatically shows a flow chart of a lattice reduction-aided linear detection and successive noise reduction method according to a second embodiment of the invention;

(6) FIG. 5 shows the throughput of the MIMO system as a function of the signal-to-noise ratio for a detection method according to the second embodiment of the invention and for a detection method according to the prior art.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

(7) A MIMO system will be considered in the paragraphs below as described with reference to FIG. 1. The information symbols transmitted by the N.sub.T transmit antennas belong to QAM constellations. Without loss of generality, it is assumed that the modulation alphabet (or constellation) used for the different transmit antennas is identical, in other words a block of transmitted symbols belongs to a product constellation A.sup.N.sup.T. It is also assumed that the transmit power of each antenna is normalised to 1. In the absence of noise, the received signal can be represented by a point of the lattice Λ.sub.H produced by the column vectors of H in the space .sup.n.sup.T where n.sub.T=2N.sub.T. In the presence of noise, the received signal can be considered to be a point of .sup.n.sup.T obtained by means of a translation by a noise vector of a point of the lattice Λ.sub.H.

(8) It is furthermore assumed, as stated hereinabove, that the elements of the matrix H are known by the receiver, for example by means of a channel estimation carried out from pilot symbols.

(9) The idea on which the invention is founded is to carry out noise cancellation by iterative reduction of the noise on the different components of the equalised signal, in the reduced basis of the lattice.

(10) FIG. 2 diagrammatically shows a flow chart of a lattice reduction-aided linear detection and successive noise reduction method according to a first embodiment of the invention.

(11) In step 210, a LLL reduction of the lattice Λ.sub.H produced by the column vectors of the channel matrix is carried out, expressed in real form, i.e. of size n.sub.R×n.sub.T. This LLL reduction is known per se, it uses a QR decomposition of the channel matrix (H=QR where Q is an orthogonal matrix and R is an upper triangular matrix) and provides in return matrices {tilde over (Q)}, {tilde over (R)} and T with {tilde over (H)}={tilde over (Q)}{tilde over (R)} where {tilde over (H)} is the LLL basis (δ−) reduced by Λ.sub.H (or channel matrix in the reduced basis), {tilde over (Q)} is an orthogonal matrix and {tilde over (R)} is an upper triangular matrix and T is a unimodular transformation matrix of size n.sub.T×n.sub.T such that {tilde over (H)}=HT, the column vectors of T forming the reduced basis.

(12) In step 220, linear detection of the received signal is carried out by means of an equalisation matrix {tilde over (W)}.

(13) This linear detection can, for example, be a detection of the ZF type ({tilde over (W)}={tilde over (W)}.sub.ZF), of the MMSE type ({tilde over (W)}={tilde over (W)}.sub.MMSE) or of the MF type ({tilde over (W)}={tilde over (W)}.sub.MF), in a manner known per se. The vector received by the receive antennas is denoted as y and the equalised vector is denoted as r={tilde over (W)}y. The latter can be considered to be a real vector with n.sub.T=2N.sub.T elements.

(14) In step 230, a representation of the equalised signal in the reduced basis is obtained by computing the vector c=Tr of size n.sub.T.

(15) In step 240, the Gram matrix A=TT.sup.T of the reduced basis is computed and if the Gram matrix is non-diagonal, the method skips to step 250. The Gram matrix gives the scalar products of the column vectors of T.

(16) It should be noted that if the matrix A is diagonal, the column vectors of T are orthogonal to one another. In such a case, the detection is completed in step 290.

(17) In step 250, for each vector of the reduced basis, i.e. each column vector t.sub.i i=1, . . . , n.sub.T, of the matrix T, the column vector t.sub.k is determined such that:

(18) $\begin{array}{cc}k=\underset{j\ne i}{\mathrm{argmax}}\left(.Math..Math.t_i|t_j.Math..Math.\right)& \left(17\right)\end{array}$
where t.sub.i|t.sub.j represents the scalar product of the vectors t.sub.i and t.sub.j of the reduced basis. In other words, for each vector of the reduced basis, the column vector that least satisfies the orthogonality condition is determined. A vector ν of size n.sub.T is thus constructed, the elements whereof are

(19) $v_i=\underset{j\ne i}{\max }\left(.Math..Math.t_i|t_j.Math..Math.\right).$
A simple way of constructing this vector is to search each row of A for the non-diagonal element having the highest absolute value.

(20) An iterative noise-reduction loop is then begun in step 260.

(21) In step 260, it is determined whether the following condition is satisfied:
∃i∈{1, . . . ,n.sub.T} such that c.sub.i>s.sub.max+ε/ν.sub.i  (18)
where s.sub.max is the maximum amplitude of a point of the constellation and ε is a positive value representative of a predetermined error margin. The choice of the value of the error margin is the result of a compromise between the convergence speed and the error tolerance on the result of the detection; the lower this margin, the slower the convergence and the less erroneous the result.

(22) If the condition (18) is not satisfied, in other words if all of the components of the equalised vector satisfy c.sub.i≤s.sub.max+ε/ν.sub.i, the detection ends in step 290.

(23) Failing this, in step 270 the component ϕ of the equalised vector that deviates the furthest from the product constellation is determined, more specifically that which deviates the furthest from a secure area (also referred to as a receive area unperturbed by noise) Z.sub.ε in the space .sup.n.sup.T surrounding the product constellation with a noise margin of width ε:

(24) $\begin{array}{cc}\ell =\underset{i}{\mathrm{argmax}}\left(c_i-s_{\max }-\mathrm{.Math.}/v_i\right)& \left(19\right)\end{array}$

(25) In other words, the maximum error direction is sought relative to the product constellation. It must be noted that this search is carried out in the reduced basis, in other words in a non-orthogonal basis.

(26) Noise reduction is then carried out in step 280 by subtracting a fraction μ of the error margin in the direction of the component determined in the previous step from the equalised vector:

(27) $\begin{array}{cc}c=c-a_{\ell }\mathrm{sgn}\left(c_{\ell }\right).Math.\mu \left(\frac{\mathrm{.Math.}}{v_{\ell }}\right)& \left(20\right)\end{array}$
where a is the column vector of index of the Gram matrix A. μ=½ is chosen for example.

(28) In other words, the equalised vector is brought closer to the area unperturbed by noise Z.sub.ε by reducing the distance to this area in the maximum error direction. A fraction of ½ is preferably used, however it is evident that other fractions can be used.

(29) It should be noted that not only the component of the equalised vector updated by means of the relation (20), is modified since A is not a diagonal matrix.

(30) Thus, the detection algorithm can pass several times in the same direction, by correcting an error introduced in another direction.

(31) The algorithm continues by returning to the test step 260.

(32) The iterations end when the equalised vector is located in the receive area unperturbed by noise Z.sub.ε (in which case the noise will have been cancelled) or, failing this, when a maximum number of iterations is reached (without convergence). In the case of convergence, the equalised vector is, at worst, located at a distance (1−μ)ε from the constellation and in the case where μ=½, in the middle of the error margin.

(33) In step 290, quantisation of the equalised vector having undergone iterative noise reduction is carried out. This quantisation is carried out in the same manner as in the prior art, by searching for the point of the constellation located the closest to the point representing the equalised vector.

(34) FIG. 3 shows on example of the noise-reduction step 280. In this example, it is assumed that only one transmit antenna (n.sub.T=2) is used and that the modulation constellation used was a 4-QAM constellation. The product constellation is thus reduced to this 4-QAM constellation.

(35) The product constellation has been shown in the reduced basis, in other words the figure shows the image of the modulation alphabet A by the transformation T.sup.−1.

(36) The reduced basis is formed by vectors

(37) $t_1=\left(\begin{array}{c}0\\ 1\end{array}\right)\mathrm{and}{t}_2=\left(\begin{array}{c}1\\ 1\end{array}\right)$
and thus the transformation matrix is

(38) $T=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right).$
The Gram matrix is thus

(39) $A=\left(\begin{array}{cc}1& 1\\ 1& 2\end{array}\right).$

(40) The modulation symbols unperturbed by noise, i.e. the points of the modulation constellation are shown in A, B, C and D. The received signal, after equalisation, i.e. the vector r, is shown by the point M. The vector c is assumed in this case to be

(41) 0$\left(\begin{array}{c}-2\\ -1\end{array}\right).$

(42) The receive area unperturbed by noise, in other words the area Z.sub.ε surrounds the modulation constellation with an error margin ε.

(43) Depending on the direction of the vector t.sub.2, the point M belongs to the area Z.sub.ε and thus noise is not reduced in this direction.

(44) However, depending on the direction of the vector t.sub.1, the point M does not belong to the area Z.sub.ε and thus is brought towards this area in this direction by means of the expression (16), where μ=½.

(45) $\begin{array}{cc}c=c-a_1\left(-1\right).Math.\frac{1}{2}\left(\frac{\mathrm{.Math.}}{1}\right)=c+\frac{\mathrm{.Math.}}{2}{a}_1=\left(\begin{array}{c}-2+\frac{\mathrm{.Math.}}{2}\\ -1+\frac{\mathrm{.Math.}}{2}\end{array}\right)& \left(21\right)\end{array}$

(46) The new equalised vector can thus be seen to be closer to the area Z.sub.ε by

(47) $\frac{\mathrm{.Math.}}{2}$
in the direction t.sub.1, however that the component thereof in the direction t.sub.2 has been modified. This explains that, in some cases, the area Z.sub.ε can be left in the direction t.sub.2 and that an iteration in the direction t.sub.2 must be repeated.

(48) In a second embodiment of the invention, a soft detection of the transmitted symbols is carried out. This will be, for example, the case when the successive transmitted symbols are the result of a coding of a bit sequence, for example turbo coding of a bit packet. In such a case, the convolutional decoder (for example using a SOVA algorithm or a conventional Viterbi algorithm) requires the receipt, on the input side, of the soft values of the bits of the sequence taking into account the received symbols.

(49) The case of a SISO system is firstly described, i.e. that of a single transmit antenna and of a single receive antenna.

(50) If b.sub.m,q is denoted as the q.sup.th bit of the word used to generate the transmitted symbol s.sub.m belonging to the modulation alphabet A, the receiver must assess the soft value or LLR (Log Likelihood Ratio), L(b.sub.m,q) defined by:

(51) $\begin{array}{cc}L\left(b_{m,q}\right)=\ln \frac{P\left(b_{m,q}=1|y\right)}{P\left(b_{m,q}=0|y\right)}=\ln \frac{P\left(y|b_{m,q}=1\right)}{P\left(y|b_{m,q}=0\right)}& \left(22\right)\end{array}$
where it is reminded that the vector y represents the received signal (and where it is assumed that the prior probabilities of the two binary values were identical).

(52) The conditional probabilities P(y|b.sub.m,q=X) (where X=0 or 1) are determined from:

(53) $\begin{array}{cc}P\left(y|b_{m,q}=X\right)=\underset{u\in S_{m,q}^X}{.Math.}P\left(y|s=u\right)& \left(23\right)\end{array}$
where P(y|s=u) is the conditional probability of receiving the vector y given that the transmitted symbol is u∈S.sub.m,q.sup.X, S.sub.m,q.sup.X representing the set of symbols of the alphabet A assuming b.sub.m,q=X.

(54) In a conventional manner, the sum of the terms appearing on the right of the relation (23) can be approximated by the determinant term (referred to as the Max-Log approximation):

(55) $\begin{array}{cc}P\left(y|b_{m,q}=X\right)\approx \underset{u\in S_{m,q}^X}{\max }\left(P\left(y|s=u\right)\right)& \left(24\right)\end{array}$

(56) Given the Gaussian density of the conditional probability, the soft value of the bit b.sub.m,q is expressed in the following form:

(57) $\begin{array}{cc}L\left(b_{m,q}\right)=\frac{1}{2{\sigma }_n^2}\left(\underset{u\in S_{m,q}^1}{\min }\left({.Math.y-\mathrm{Hu}.Math.}^2-\underset{u\in S_{m,q}^0}{\min }\left({.Math.y-\mathrm{Hu}.Math.}^2\right)\right)\right)& \left(25\right)\end{array}$
where H is the matrix of the channel (in this case reduced to a single element since the system is SISO).

(58) The relation (25) easily extends to a MIMO system using a linear detection:

(59) $\begin{array}{cc}L\left(b_{m,q}\right)=\frac{1}{2{\sigma }_{n,m}^2}\left(\underset{u\in S_{m,q}^1}{\min }\left({.Math.r-u.Math.}^2-\underset{u\in S_{m,q}^0}{\min }\left({.Math.r-u.Math.}^2\right)\right)\right)& \left(26\right)\end{array}$
the vector u thus representing the vector of the transmitted symbols (and thus an element of the product constellation), H being the complex matrix of the channel of size N.sub.T×N.sub.T, r=Wy being the equalised vector where y represents the vector of size N.sub.R of the signals received by the different receive antennas, and where σ.sub.n,m.sup.2 represents the noise variance affecting the component r.sub.m of the equalised vector. As a result of this equalisation, the noise variance is not identical for the different components:

(60) $\begin{array}{cc}{\sigma }_{n,m}^2=\left({\sigma }_n^2\underset{i=1}{\overset{N_T}{.Math.}}|{\tilde{W}}_{m,i}{|}^2+{\sigma }_a^2\underset{j=1j\ne m}{\overset{N_T}{.Math.}}|{\tilde{\beta }}_{m,j}{|}^2\right)\text{/}|{\tilde{\beta }}_{m,m}{|}^2& \left(27\right)\end{array}$
where {tilde over (W)}.sub.m,i is the element of the matrix {tilde over (W)} in the row m and in the column i, in other words corresponding to the receive antenna m and to the transmit antenna i, {tilde over (β)}.sub.m,j is the element of the matrix β={tilde over (W)}H in the row in and in the column j, σ.sub.a.sup.2 is the average power of the modulation constellation (assumed to be identical for all transmit antennas) and σ.sub.n.sup.2 is the noise variance affecting the signal received by a receive antenna (the variance is assumed to be identical for all receive antennas).

(61) The computation of the soft values according to (26) and (27) must be adapted to the case whereby an iterative noise reduction is carried out in addition to equalisation, as shown with reference to FIG. 4.

(62) FIG. 4 diagrammatically shows a flow chart of a lattice reduction-aided linear detection and successive noise reduction method according to a second embodiment of the invention.

(63) Steps 410 to 480 are identical to steps 210 to 280 described hereinabove with reference to FIG. 2 and the description thereof will not be repeated here.

(64) However, an initialisation step has been added in step 455, in addition to a step of updating the noise power in step 475.

(65) More specifically, step 455 comprises the initialisation of the noise variances (or noise powers) affecting the different components of the equalised vector by the relation (27).

(66) Then, upon each iteration, the noise variances σ.sub.n,m.sup.2, are updated in step 475 by:

(67) $\begin{array}{cc}{\sigma }_{n,m}^2={\sigma }_{n,m}^2+{{\mu }^2\left(\frac{\mathrm{.Math.}}{v_{\ell }}\right)}^2|A_{m,\ell }{|}^2& \left(28\right)\end{array}$
where is the element in the row m and in the column of the Gram matrix A. It is understood from the expression (28) that only the variances relative to the components affected by the noise reduction are updated.

(68) Finally, in step 490, instead of making a hard decision such as that in step 290, the distances of the equalised vector c to the different points of the product constellation are computed, then the soft values of the different bits are computed by:

(69) 0$\begin{array}{cc}L\left(b_{m,q}\right)=\frac{1}{2{\sigma }_{n,m}^2}\left(\underset{u\in S_{m,q}^1}{\min }\left({.Math.c-u.Math.}^2-\underset{u\in S_{m,q}^0}{\min }\left({.Math.c-u.Math.}^2\right)\right)\right)& \left(29\right)\end{array}$
the noise variances σ.sub.n,m.sup.2 affecting the different components of the equalised vector having been updated during successive iterations according to (28).

(70) If a turbo coding was carried out upon transmission in order to generate symbols on each of the antennas, decoding of the BCJR type can be carried out using, as an input, the soft values provided by step 490.

(71) FIG. 5 shows the throughput of the MIMO system as a function of the signal-to-noise ratio for a detection method according to the second embodiment of the invention and for a detection method according to the prior art.

(72) The MIMO system used was a 2×2 system with 16-QAM modulation. Turbo coding of rate R=1/2 for 6144-bit packets was carried out upon transmission before 16-QAM modulation for each antenna.

(73) The throughput of the system is given by ρ(1−η).sup.Q where ρ is the transmission speed in bit/s, Q is the size of the packet and η is the bit error rate.

(74) The curve 510 represents the throughput of the system when, upon receipt, a linear detection of the ZF type and a computation of soft values according to (22) and (23), i.e. without the Max-Log approximation, are carried out before carrying out BCJR decoding.

(75) The curve 520 represents the throughput of the system when, upon receipt, a linear detection of the ZF type, iterative noise cancellation according to the second embodiment of the invention, and computation of the soft values according to (22) and (23) are carried out before carrying out BCJR decoding.

(76) The curve 530 represents the throughput of the system under the same conditions as 510, however using the Max-Log approximation, in other words using the soft value expressions according to (26) and (27).

(77) The curve 540 represents the throughput of the system according to the second embodiment of the invention, however using the Max-Log approximation, in other words using the soft value expressions according to (26) and (27) and the iterative updating of the variances according to (28), before carrying out channel decoding.

(78) A significant improvement to the throughput is noted in the case wherein the second embodiment of the invention is used, see curves (520) and (540), instead of a detection method of the prior art, see curves (510) and (530).