WO2010026093A1

WO2010026093A1 - Method for decoding a signal subjected to a space/time encoding and corresponding decoding device

Info

Publication number: WO2010026093A1
Application number: PCT/EP2009/061018
Authority: WO
Inventors: Ghaya Rekaya-Ben Othman; Jean-Claude Belfiore; Laura Luzzi
Original assignee: Groupe Des Ecoles Des Telecommunications / Ecole Nationale Superieure Des Telecommunications
Priority date: 2008-09-02
Filing date: 2009-08-26
Publication date: 2010-03-11
Also published as: FR2935570B1; FR2935570A1

Abstract

The invention relates to a method for decoding a signal received on at least two reception antennas, wherein said signal has been subjected before transmission to a space/time encoding using a code built on a switching algebra over a field and is transmitted on at least two transmission antennas. According to the invention, the method comprises: - a step of pre-processing (21) said received signal (Y) that comprises the following sub-steps: determining a matrix representative of the transmission channel (H); standardising said matrix representative of the transmission channel and of the received signal; approximating in an algebraic manner said standardised channel matrix from a matrix (U) selected in said algebra and having a determinant equal to 1; - and the step of space/time decoding (22) the normalised received signal (Y ₁) using the algebraic approximation of said standardised channel matrix.

Description

METHOD FOR DECODING A SIGNAL HAVING SUCCESSFUL SPACE / TIME CODING AND

CORRESPONDING DECODING DEVICE

FIELD OF THE DISCLOSURE The field of the invention is that of wireless communications.

More specifically, the invention relates to the reception of signals in the context of MIMO systems (multiple-input multiple-output systems), using several transmission antennas and several antennas. reception, and having undergone space / time coding before transmission. The invention finds particular applications in the field of mobile radio.

2. Prior Art

Transmission techniques in multi-antenna systems have several advantages, and offer, for example, high-speed transmission possibilities with good reliability.

In particular, these techniques make it possible to achieve an increased transmission capacity by increasing the spectral efficiency, in particular by virtue of the use of space / time codes in transmission.

The space / time codes used in transmission make it possible to distribute the modulated symbols on the various degrees of freedom of the channel (in space on the antennas and in time). For example, these space / time codes can be constructed using an algebraic construct, ensuring that these codes are: full-yield, maximum-diversity, reaching channel capacity and having a minimal determinant that does not fade when efficiency spectral increases. The decoding of such algebraic space / time codes is then implemented using a point network representation of these codes.

It has, for example, in connection with Figure 1, a transmission system implementing space / time transmit codes in the context of MIMO transmission implementing n _T transmit and n _R receive antennas. On the transmission side, a binary signal 10 to be transmitted (possibly coded according to a source coding technique and / or channel coding) is modulated in a modulation block 11 intended to convert binary elements into complex symbols: such a block thus associates with a group of bits a complex symbol belonging to a constellation (of type QAM for example). We then proceed to a space-time coding 12 of each group of K symbols in the form of a matrix, also called code word X. The elements of the code word matrix X are then transmitted on the n _t transmit antennas IS ₁ to 13 _n .

The signal is then conveyed through a transmission channel and received on the N _r receive antennas 14 γ to IA _n. Each reception antenna receives a linear combination of the signals transmitted on each of the n _T transmit antennas.

Classically, we note:

¹ Hf Xt ^{~ n} Hf XHf Hf Xt ^" * ^{" YV} H _r Xt with:

- F the received signal on the n _τ receiving antennas; - X signal (or codeword) transmitted on the n _T transmit antennas;

The representative matrix of the transmission channel;

- the representative matrix of a Gaussian white additive noise; and

- t the temporal length of the space / time code.

The received signal is first decoded in a spatio-temporal decoding block 15, implementing a reverse processing at the space / time coding implemented in transmission.

For decoding, it is possible to use a representation of the received signal in a network of points, especially if the space / time codes used in transmission are constructed using an algebraic space / time coding technique. It is recalled that the representation in point networks of the space / time codes allows their decoding by decoders of point networks.

These decoders allow optimal decoding in the ML or sub-optimal direction, depending on the technique used. Thus, Schnorr-Euchner decoding algorithms of spherical decoding type, sequential (of the Fano or stack type, for example), etc., allow optimal decoding in the maximum likelihood (ML) sense. ). Linear decoding algorithms (ZF or "Zero-Forcing", MMSE or "Minimum Mean Square Error", etc.) or non-linear decoding (decision feedback decoding: ZF-DFE or "Zero-Forcing - Decision Feedback Equalization", MMSE -DFE or "Minimum Mean Square Error - Feedback Equalization Decision", etc.) allow sub-optimal decoding.

The signal at the output of the space / time decoding block 15 then supplies a demodulation block 16, delivering an estimated bit signal 17.

As indicated above, in order to increase the transmission rate, the number of transmitting and / or receiving antennas in the MIMO systems is increased.

Unfortunately, the increase in the number of antennas leads to an increase in the size of the associated point network, making the decoding of the signals particularly difficult. complex.

The complexity of optimal decoders in the ML sense makes it practically impossible to implement them in mobile terminals for large point networks. It is recalled that a code word of a code space / time transmitted over n _T transmit antennas and received on n _r receiving antennas is represented by a real dimension of point network

2n _r t (or 2n _t if n _t = n _r = t).

Suboptimal decoders are less complex, but have average performance. In addition, these boxes do not allow obtaining the maximum diversity n _t n _r • provided by the MIMO system. There is therefore a real need for a new signal decoding technique in systems employing a plurality of transmit and receive antennas, having a reduced complexity and / or to maintain the order of diversity of the transmission system. In other words, there is a real need for a decoder offering a good compromise between complexity and performance. 3. Presentation of the invention

The invention proposes a new solution that does not have all of these disadvantages of the prior art, in the form of a method of decoding a signal received on at least two reception antennas, said signal having undergone before transmitting a space / time coding using a code constructed from an algebra on a commutative body K, and being transmitted on at least two transmitting antennas.

According to the invention, such a method implements:

a step of pretreatment of the received signal comprising the following sub-steps: determining a representative matrix of the transmission channel; o normalization of the representative matrix of the transmission channel and the received signal, respectively delivering a standard channel matrix and a standard received signal; o algebraic approximation of said normalized channel matrix, from a matrix (U) selected in said algebra and having a determinant equal to

1; a step of decoding space / time of the normalized received signal, using the algebraic approximation of the standardized channel matrix.

Thus, the invention proposes a novel and inventive technique for decoding a space / time coded signal in a MIMO system implementing a plurality of transmit and receive antennas. To do this, an algebraic pretreatment is applied to the received signal, consisting of absorbing a portion of the normalized channel matrix by the algebraic space / time code. In this way, the remaining part of the transmission channel matrix, which defines the base of the point network used for decoding, is quasi-orthogonal. To this end, it is recalled that algebraic tools have so far only been used to construct space / time codes. The algebraic structure of these codes has never been exploited for decoding, which constitutes an innovative aspect of the invention.

More precisely, it is sought according to the invention to approximate the representative matrix of the normalized transmission channel. For example, assuming that the representative matrix H of the transmission channel has a determinant different from zero, it is sought to approximate the normalized channel matrix H ₁ such that H ₁ = (det (H)) ⁿ H, with H ₁ G SL _n (C), using a matrix of the algebra on the commutative body K.

A matrix of the algebra is thus selected, and the normalized channel matrix is approximated from the matrix of the selected algebra.

The received normalized signal can then be decoded using the approximation of the normalized channel matrix H ₁ , using a conventional decoding technique that can be suboptimal (of linear or nonlinear type) or optimal in the sense maximum likelihood (ML). In particular, according to the invention, a decoding technique which would be suboptimal if it were applied directly to the received signal becomes quasi-optimal when it is applied to the normalized received signal expressed from the approximation of the channel matrix. normalized. It is thus possible, according to the invention, to obtain good performance while maintaining a reduced complexity, since suboptimal decoding techniques have a reduced complexity.

The invention also makes it possible to accelerate the implementation of the decoding by simplifying the complexity if an optimal decoding technique in the ML sense is implemented.

According to a particular embodiment of the invention, the selected matrix is a unit [/ of an order of the algebra (denoted O), and the algebraic approximation step weights the unit U by an error of approximation E, such that:

H ₁ = EU.

In this case, the order O is the order used to construct the space / time code. In other words, the selected unit U is weighted by an approximation error E to approximate the normalized channel matrix H ₁ . Because U is a unit of the order O, U is invertible in O.

In particular, said order can be a maximal order of the algebra.

For example, the algebra is divisional and the space / time code is of the form (Oa), with a element of the algebra and O a maximal order of the algebra (several maximal orders that can exist).

According to another aspect of the invention, the algebraic approximation step selects the unit U so that the approximation error E is quasi-orthogonal.

In this way, the invention makes it possible to absorb part of the normalized channel matrix (corresponding to the unit U) by the space / time code, and to obtain a base of the network of points used for the quasi-decoding. orthogonal (corresponding to the approximation error E).

According to yet another aspect of the invention, the normalized signal is written in the form F ₁ = HγX + W ₁ , with X a transmitted code word and W ₁ a Gaussian additive normalized white noise. After algebraic approximation of the normalized channel matrix, the normalized signal can be written as F ₁ = EUX + W ₁ , with UX being a code word.

We therefore work directly on the matrix form (and non-vector) of the normalized received signal to determine an approximation of the normalized channel matrix.

According to a particular embodiment of the invention, the decoding method comprises a step of transforming the normalized received signal F ₁ in vector form, such that: y ₁ = Ji ₁ Os + W ₁ with:

y ₁ the standard received signal in vector form;

h ₁ the standardized channel matrix in vector form;

- the vector of information symbols; A space / time coding matrix, also called the space / time code generating matrix;

- W ₁ Gaussian additive white noise in vector form.

The vectorization is therefore implemented after the approximation of the standardized channel matrix. In particular, the matrix generating the dot network, corresponding to the standardized channel matrix in vector form multiplied by the space / time coding matrix, is equal to: h _x Φ = E _Z U _Z Φ = E ₁ OT _1J with : - E; a linear transformation corresponding to a left multiplication by said approximation error E;

- U; a linear transformation corresponding to a multiplication on the left by said unit U; - You a unimodular matrix.

According to the invention, the term "unimodular matrix" refers to a matrix whose elements are in the ring of the integers R of the commutative body K, also called base body, and whose determinant is invertible in said ring of integers R.

The standardized received signal F ₁ , in vector form, can be written in the following form: y _: = E _/ OT ^ s + W ₁ = E _/ Os '+ W ₁ , with s' a vector of symbols.

Advantageously, the space / time code is constructed from a division algebra, possibly cyclic. This is for example a perfect code like a golden code, or a near-perfect code. The invention also relates to a computer program product downloadable from a communication network and / or recorded on a computer readable medium and / or executable by a processor, comprising program code instructions for implementing the steps of the decoding method described above, when the program is run on a computer. In another embodiment, the invention relates to a device for decoding a signal received on at least two reception antennas, said signal having undergone a space / time coding before transmission using a code built from an algebra on a commutative body K and being transmitted on at least two transmitting antennas.

According to the invention, such a device comprises: - preprocessing means of the received signal comprising: o means for determining a representative matrix of the transmission channel; o normalization means of the representative matrix of the transmission channel and the received signal, respectively delivering a standardized channel matrix and a standard received signal; o Algebraic approximation means of said normalized channel matrix, from a matrix (U) selected in said algebra and having a determinant equal to 1; means for decoding space / time of the normalized received signal, using the approximation algebraic of the normalized channel matrix.

Such a decoding device is particularly suitable for implementing the decoding method described above. It can be integrated into a receiver of a MIMO system, comprising at least two reception antennas, for Wi-Fi (IEEE 802.1 In), WiMax mobile (IEEE 802.16e), WiMax cooperative (IEE 802.16j), 3GPP applications. -LTE, etc.

This device may of course include the various features relating to the decoding method according to the invention. 4. List of figures

Other features and advantages of the invention will emerge more clearly on reading the following description of the general principle of the invention and of a particular embodiment, given as a simple illustrative and non-limiting example, and drawings appended, among which: FIG. 1 illustrates a MIMO transmission chain implementing a space / time coding; FIG. 2 presents the main steps of the decoding method according to the invention; FIG. 3 illustrates the performances of the invention, applied to the Golden Code; Figure 4 shows the structure of a decoding device according to a particular embodiment of the invention. 5. Description of an embodiment of the invention

5.1 Definitions

The following is the terminology used in this document. This is the classic vocabulary, well known to the skilled person.

A) Algebra: Let K be a commutative body. An algebra A on the field K is called a set provided with the following binary operations: a "+" addition operation of AxA in A; an internal multiplication operation "x" of AxA in A; a scalar multiplication operation ". KxA in A; as :

1. (A, +,.) Is a vector space on K;

2. (A, x) is a ring;

3. (λa) b = a (λb) = λ (ab) VA e K, Vα, εe A.

The multiplication in algebra A is not necessarily commutative. The elements of algebra A can be represented in matrix form.

B) Divisional algebra:

An algebra A is called a divisional algebra if any non-zero element of A has an inverse with respect to the internal multiplication operation "x".

C) Cyclic algebra:

Let L be an extension field of K of degree n such that its Galois group G = GaI (L I K) is cyclic with a generator σ. Let γ G K be a non-zero element. We consider the non commutative algebra denoted A = (L / K, σ, γ) defined by:

A = L @ eL @ ... @ e ^{n ~ l} L such that e satisfies: e ⁿ = γ, xe = eσ (x) Vx e L

A is called a cyclic algebra, and n is the index of algebra.

An element a = X _Q + exγ + ... + e ^{n ~} x _n - _\ GA, with x _o , ..., x _n _ι GL, can be represented by the following matrix n X n:

κ

If n = 2, the cyclic algebra is called quaternion algebra. For example, the Golden Code is based on a divisional cyclic algebra of index n = 2. The field K = <Q (i), and its extension L = <Q (i, θ), where θ = and σ: i i-4 x is such that σ (θ) = θ = l - θ, and σ leaves the elements of Q (i) invariant. The element γ is equal to /. The element e is

with j .2 = i.

We can then write:

A = (Q (i, θ) IQ (O, σ, /) = Q (/, <9) θ jQ (i, θ) An element of A can be represented by a matrix in the following form:

QL Orders and ideals:

Let A be a K-algebra, and let R be the ring of integers of K. An ideal of A is a i? -Module to finite generation contained in A such that KI = A, where:

KI =

GK, x _t IM me NL

An order O of A is an ideal which is a sub-ring of A with the same unitary element. A maximum order is an order that is not contained in any other order of A.

E) Unit:

A unit of an order O is an invertible element UGO such that its inverse U ^~ with respect to the multiplication "x" in A also belongs to the order O.

For example, in the case of the golden code, a maximum order of A is:

O can be written O = Z [i, θ] ® Z [i, θ] j.

The golden code is a normalized version with a normalization constant equal to 1 / v5 of a two-sided ideal Oa = aθ with a = 1 + iθ. Each X-code word generated in this way takes the following form:

with xγ = s _\ + S ₂ Θ, X ₂ = S ₃ + s ₄ θ, and the symbols s _\ , S ₂ , Sj, S ₄ GZ [Z]. 5.2 Description of an embodiment A) General principle

The invention relates to the decoding of a received signal transmitted in a MIMO system implementing at least two transmitting antennas and at least two receiving antennas. Before transmission, the signal has undergone a space / time coding using a code constructed from an algebra on a commutative body K, also called basic body. For example, the algebraic space / time coding implemented in transmission uses a divisional algebra. Thus, the space / time codes used can be perfect codes, such as the golden code, or quasi-perfect codes. Other codes can also be used, as long as they are constructed from an algebra on a commutative body.

The general principle of the invention is based on the implementation of a pretreatment of the received signal to facilitate the decoding of this signal.

More precisely, the preprocessing implemented makes it possible to determine an algebraic approximation of the representative matrix of the transmission channel, in standardized form, using a matrix selected in the algebra used to construct the space / time code. For example, the selected matrix is a unit U of any order or order of the algebra, corresponding to the order used to construct the space / time code, and the approximation of the normalized channel matrix H ₁ corresponds to this unit weighted by an approximation error E: H ₁ = EU.

The matrix selected in the algebra is chosen in such a way that the approximation error E is quasi-orthogonal.

In this way, part of the normalized channel matrix (corresponding to the matrix U selected in the algebra) is absorbed by the algebraic code, since the matrix U belongs to the order of the algebra used for the space code. time, and so UX is a codeword.

The rest of the normalized channel matrix (corresponding to the approximation error E) is a quasi-orthogonal matrix, defining the base of the point network used for decoding. It is recalled that the representation in point networks of the space / time codes allows their decoding by decoders of point networks. It can therefore be considered that the invention makes it possible to reduce the grating of points of the transmission system, in reception, since the base of the network of points used for the decoding is constructed from the matrix E corresponding to the error of approximation. , and not from the channel matrix. This reduction, carried out using the properties of the algebra, thus makes it possible to accelerate the decoding of the network of points.

The main steps implemented for the decoding of the received signal Y _n × _t , with n _r ≥ 2, emitted by at least two transmitting antennas and having undergone coding before transmission are now presented in relation to FIG. space / time using a code constructed from an algebra on a commutative body K. The technique according to the invention can notably be implemented in the spatio-temporal decoding block 15 of FIG.

More specifically, during a first step 21, the method according to the invention implements a preprocessing step of the received signal Y _{n xt} . This pretreatment on the right uses, in reception, the algebraic tools to determine an approximation of the normalized channel matrix.

More specifically, the preprocessing step of the received signal comprises the following substeps:

determination of a representative matrix H of the transmission channel; - normalization of the representative matrix of the transmission channel and the received signal, respectively delivering a standardized channel matrix H ₁ and a standard received signal F ₁ ;

approximating said normalized channel matrix, from a matrix (U) selected in said algebra and having a determinant equal to 1. In this way, the normalized channel matrix is approximated by using a matrix of the space / time code algebra of determinant equal to 1. A large part of the normalized channel matrix (and therefore the transmission channel) can then be absorbed. by the algebraic space / time code. During a second step 22, a space / time decoding is applied to the normalized received signal, using the approximation of the normalized channel matrix. This space / time decoding step makes it possible to estimate the information symbol vector of the transmitted signal, denoted s.

It should also be noted that the method according to the invention can be implemented in various ways, in particular in hard-wired form or in software form.

B) Detailed description

The implementation of the preprocessing step of the received signal is described in more detail below. To do this, we use the algebra of the space / time code used in transmission, and, for example, the units of an order of this algebra. Referring again to Figure 1, it is proposed according to the invention to approximate the channel matrix H, after normalization, by an element of an order of the algebra. In this way, we can consider that part of the channel is absorbed by the code.

The received signal is expressed:

* H _r Xt ⁼ - "H _f XHf ^ Hf Xt + ^ H _f Xt with: X the signal transmitted on the n _t transmit antennas;

H the representative matrix of the transmission channel; W the representative matrix of a Gaussian white additive noise; and t the time length of the space / time code.

For example, the code word X is considered to belong to a divisional algebra. In other words, the space / time code considered is a subset of Oa, with O an order of algebra. For example, such a code is a perfect code, as described in the document

"Perfect space-time codes for any number of antennas" (P. Elia et al., IEEE Transactions on Information Theory, Vol 53, No. 11, November 2007).

We consider the case n _t = n _r = t. The indices are therefore omitted in the following equations. We consider according to the transmission model that the representative matrix of the transmission channel H has a determinant different from zero with a probability equal to 1. After normalization of the transmission system, we can write: H ₁ = (det (H)) ^{~ 1 / w} H, with H ₁ e SL _n (C). The normalized received signal F ₁ can then be written: Y ₁ = H ₁ X + W ₁ = (det (H)) ^{"1 / w} HX + W ₁ and we try to approximate the normalized channel matrix H ₁ from a matrix of the order of algebra used to build the space / time code.

C) Perfect approximation We first assume that we can select a matrix U in the order O of the algebra having a determinant equal to 1, such that H ₁ = U. This case corresponds to a perfect approximation of the normalized channel matrix. The normalized received signal can then be written:

Y ₁ = UX + W ₁ where UX is always a code word. U is a unit of the order O of algebra.

In other words, since U is a unit of the order O of the algebra, then U is invertible in O, and we have:

The normalized received signal is then vectorized, expressed as a function of the approximation of the normalized channel matrix.

To do this, we apply a function φ to transform matrices under

2 vector shape. For example, for φ: M _n (C) -> C ⁿ , we have:

φ: (m _n , ..., m _nl , m ₁₂ , ..., m _{n 2} , ... m _ln , ..., m _nn )

After vectorization, the standardized received signal, in vector form, can be written in the following form:

Y ₁ = h _: Φs + W ₁ = £ / _/ Φs + W ₁ where:

yι = Φ ⁽ Xύ;

- W ₁ = ^ (W ₁ ); - Φs = φ (X);

- U; is the linear transformation corresponding to a multiplication on the left by the unit U;

- Φ is a matrix generating code, also called space / time coding matrix; - s is a vector of information symbols that can be for example from a quadrature amplitude modulation (QAM).

More specifically, the matrix Φ generating the code is a matrix whose columns are the vectors w _v - for / ranging from 1 to n, which form a base of the network of points. complex dimension n.

For example, we consider w,, ..., w ₂ r ^a base of Oa. Each code word X can

I write in this database:

* = Σ S (Wi with S = Ls ₁ , ..., s ₂ ^e R ⁿ i = \ with R the ring of integers of the base body K, and the generating matrix Φ is formed of the following columns: φ ( w _ι ), ..., φ (w ₂ )

The vectorized X code word can then be expressed in the following form: n ² χ = φ (X) = Σ _If φ ( _Wi ) = Φs i = \ Moreover, since U is a unit of O, we have U _/ Φ = ΦTy, with T _j7 a unimodular matrix, that is to say a matrix with elements in the ring of the integers R of the basic body K, whose determinant is invertible in the ring of integers R.

Indeed, since U is a unit of O, yJwγ, ..., Uw ₂ | is still a base of Oa. The code word X can be written in this new database:

2 X = ^ s) [Uw ₁ ) with s' fs ¹ _! , ..., * ¹ ₂ ) ei? "I = \

In vectorized form, this code word can also be written:

Φ (X)

υ _t Φs' i = \ i = \ i = \

So we have Φs = U; Φs'.

The matrix T _f7 = Φ ^~ U; Φ is the basic change matrix between the bases w ,, ..., w ₂ and \ Uwi, ..., Uw ₂ \ • In particular, since Tr ₇ sends each vector at L n JL nor coordinates in R in another coordinate vector in R, and det (T ₍ y) = det (U;) = det (t /) = 1, T ^ ₇ is unimodular.

Returning to the expression of the standard received signal in vector form, we have:

Y ₁ = t ^ Φs + W ₁ = ΦT _fj S + W ₁ = Φs '+ W ₁ where T _f7 S = s' is always a signal in R.

The standard received signal, in vector form, can therefore be expressed in the following form:

Y ₁ = φs' + W ₁

The vector of estimated symbols s' can then be written in the following form, where [] denotes a hard decision, used in the case of a ZF or ZF-DFE type decoder for example: s' = [φ ^" Vi] = [ _s ' + ^{0" 1} W ₁ ] = r _s ' + (det ( H)) ^{"1 / w} φ-Svl

The estimation of the vector of information symbols s, is then obtained by multiplying the vector of symbols estimated s' by the inverse of the matrix T _j7 (ie T _j7 ^" (s ¹ )): S = Tf ₇ - ¹ S '.

In the case where we consider that the normalized channel matrix is perfectly equal to one unit, and if the generator matrix of the code Φ is a unitary matrix, as is the case for a perfect code for example, as the code in gold, the normalized received signal can be decoded using a ZF-type technique, which exhibits optimal performance in the ML sense. The estimation of the symbol vector is then performed by taking the component integral part by component, and does not require the implementation of a dot matrix decoder.

D) Imperfect approximation

If there is no matrix U in the order of the algebra equal to the normalized channel matrix H ₁ , the normalized channel matrix is approximated by a matrix U selected in the order of the algebra, weighted by an approximation error E, such that H ₁ = EU, with the determinant of U equal to 1.

By repeating the previous steps, we obtain:

Y ₁ = E _/ U _/ Φs + W ₁ = E _/ ΦT _fj S + W ₁ = E; Φs' + W ₁ with E _/ the linear transformation corresponding to the multiplication on the left by the approximation error E.

The vector of estimated symbols s' can then be written in the following form, where [] denotes a hard decision: s' = [O ^-1 EfV ₁ ] = U '+ (det (H)) ^{"1 / w} 0 ^"1 Ef ¹ Wl = [s '+ n] The estimation of the vector of information symbols s, is then obtained by multiplying the vector of estimated symbols s' by the inverse of the matrix T _j7 (ie T _j7 ^" (s ¹ )): S = Tf ₇ - ¹ S '.

E) Choice of U-unit to optimize ZF decoding

In order to obtain quasi-optimal decoding using a ZF type decoding technique, it is desirable to have a quasi-orthogonal approximation error E.

It is therefore desirable to choose a unit U so as to obtain an approximation error E = H ^ U ^~ quasi-orthogonal. To do this, the criterion to be respected is for example to choose the unit U which minimizes the mean squared error.

For example, one places oneself in a context MIMO including n transmit antennas and n reception antennas, implementing a space / time code per block, on a quasi- static channel, in which H = det (H) ⁿ Hγ, and one seeks to minimize the standard of Frobenius 1 KF ^•

This criterion amounts to minimizing the trace of the noise covariance matrix n

| Det (H) | ^ι \ ^ι I \ I

v ^; | Det (H) | U »F | det (H) | U WF | det (H) | U HF It is recalled that the criterion presented above applies in the case where the matrix Φ generator of the code is unitary. Of course, this matrix is not necessarily unitary, and other criteria may be used, depending on the form of the code generating matrix and / or the decoding technique used (ZF, ZF-DFE, MMSE, MMSE, DFE, spherical decoding, battery decoding, ...). 5.3 Application to the Golden Code

The application of the solution of the invention to the gold code is presented below. The golden code is notably described in the document "The Golden Code: A 2 x 2 FuIl-Rate Space-Time Code With Nonvanishing Determinants" (Jean-Claude Belfiore et al., IEEE Transactions on information theory, Vol 51, No 4, April 2005). For example, we place ourselves in the context of a divisional algebra, used to construct a space / time code of the form Oa, with a element of the algebra and O a maximal order of the algebra. Note that some perfect codes other than golden codes are not necessarily constructed from a maximal order.

Because of the properties of algebra, each unit of the order O can be written as a finite product of a finite number of generators of the group of units and their inverses. For example, the generators are eight in number for the golden code. The minimum number of generators depends on the chosen algebra.

As already indicated, the algebraic pretreatment according to the invention consists in approximating the matrix of the normalized channel H ₁ using a unit U of the order O of the algebra used to construct the space / time code, weighted by an approximation error F. The following is an example of an algorithm that can be implemented for the search for U units, in the case where n _t = n _r = 2.

As previously indicated, for a type ZF decoding, the criterion for the choice of the unit U which approximates the normalized channel matrix H ₁ is the following: one seeks the unit U such that the Frobenius norm of the error approximation E = H ^ U ^~ be minimized.

In the case n _t = n _r = 2, whatever the algebraic code used, we can implement a realization of the unit search algorithm that exploits the action of the group of standard 1 units of the order O on the hyperbolic space Η.

More precisely, we define the hyperbolic space Η as the set of points {(x, y, z), x, y, z, t, z> θ}. The hyperbolic distance p (P _\ , P ₂ ) between two points P ₁ = (x _\ , yι, Zι) and P ₂ ⁼ {* ₂ , y ₂ , Z ₂ ) is given by the following equation:

P {p ['^ 2) ^{= arccos} h 1 + -

We consider the base point / = (θ, O, l) in the hyperbolic space Η _T 3.

Let B be a complex 2 x 2 matrix such that det (5) = 1:

(a bλ ₃ B = \ \ We define the point B (J) in the hyperbolic space Η:

I cd

where x * denotes the complex conjugate of x.

Let H _{1 be} the matrix of the normalized channel, and let Ui, ..., U _{r be} a minimal set of generators for the group of norm 1 units of the order O. Such a set of generators is calculated for example using Swan algorithm as described in the document

"Presentations of the unit group of an order in a non-split quaternion algebra" (G. Corrales et al., Advances in Mathematics, 186 n.2 (2004) 498-524).

The eight generators of the group of units in the case of the golden code are for example:

We denote £ / _{r + 1} = C / f, ..., U _2r = U ^~ the inverse of these r = 8 generators: U ₉ = UI '

CZ ₁₀ = CZ ₂ ¹ . -. ^ 16 = ^ S ¹ -

These generators differ from one algebraic code to another. According to this algorithm, it is sought to determine the unit U of the order of the algebra making it possible to approximate the normalized channel matrix H ₁ , knowing the standardized channel matrix H ₁ .

It is considered that the points Ui (J), ..., U _r (J), U _{r +} ι (J), ..., U _2r (J), calculated according to equation (1), are stored in a memory.

Let H, U and z _{0 be} variables. To initialize the algorithm, we ask:

H = H ₁

Z ₀ = O The following steps are then repeated:

1. Calculate H (/) = (x, y, z) using equation (1);

2. For i = l, ..., 2r, calculate:

d _i = 2cosh (p (ïT, - (/))) = i ₊ ^{(x "} ^ ^{) 2 + (3;} ₂ ^ ^{) 2 + (z" Z /) 2} and J _{0 = 2} -shf pfIrV), /) ! ₌ i + ^ l ± Z ± kzlL

3. Choose z ₀ = argmin d ^. If several indices reach the minimum, we choose the most

/ e {0,, 2r} small.

4. Update U and H, such as: U <- UU) _Q , H <- HU) _Q.

5. If / ₀ ≠ 0, we return to step 1. Otherwise, if i ₀ = 0 then the searched unit U is equal to U (U = U), and the algorithm is interrupted.

Thus, according to this example, a unit U of the algebra can be selected which can be used to approximate the normalized channel matrix H ₁ , and such that the approximation error matrix E is quasi-orthogonal. By using a ZF type decoding technique after the algebraic reduction described above, only 3dB is obtained with respect to the performances obtained with an optimal decoder in the ML sense.

It is therefore possible, according to the invention, to simplify the decoding of a signal received in a multi-antenna system by using a so-called suboptimal decoding technique, while obtaining good performance.

In particular, it can be noted that in the case of a channel that is not very variable in time, a simple adjustment of the approximation is necessary at different times of transmission ("time block "), and there is no need to redo a new approximation.

FIG. 3 illustrates the performance of the invention in word error rate (FER) as a function of the signal-to-noise ratio (SNR) in dB.

More precisely: the curve ML illustrates the performances of a conventional decoder implementing an optimal decoding technique in the ML sense;

the MMSE-GDFE + LLL + ZF curve illustrates the performances of a conventional decoder implementing a ZF sub-optimal decoding technique and a LLL-type right pretreatment ("Lenstra Lenstra Lovasz"), and a pretreatment left of type MMSE-GDFE;

the MMSE-GDFE + LLL + ZF-DFE curve illustrates the performances of a conventional decoder implementing a ZF-DFE sub-optimal decoding technique and a LLL-type right-handed pretreatment ("Lenstra Lenstra Lovasz") , and left-handed pretreatment of MMSE-GDFE type; the MMSE-GDFE + AR + ZF curve illustrates the performances of a decoder according to the invention implementing a suboptimal ZF-type decoding technique and an algebraic reduction type right-handed pretreatment according to the invention, and a Left pretreatment type MMSE-GDFE;

the curve MMSE-GDFE + AR + ZF-DFE illustrates the performances of a decoder according to the invention implementing a sub-optimal decoding technique of the type

ZF-DFE and an algebraic reduction type right pretreatment according to the invention, and a left-hand pretreatment of the MMSE-GDFE type. 5.4 Structure of the decoding device

Finally, in connection with FIG. 4, the simplified structure of a decoding device according to a particular embodiment of the invention is presented.

Such a device comprises a memory 41 consisting of a buffer memory, a processing unit 42, equipped for example with a microprocessor μP, and driven by the computer program 43, implementing the decoding method according to the invention .

At initialization, the code instructions of the computer program 43 are for example loaded into a RAM before being executed by the processor of the processing unit 42. The processing unit 42 receives as input a signal , received on at least two receiving antennas, and having undergone before transmission an algebraic space / time coding. The microprocessor of the processing unit 42 implements the steps of the decoding method described above, according to the instructions of the computer program 43, to decode the received signal. For this, the decoding device comprises, in addition to the buffer memory 41, preprocessing means of the received signal, making it possible to determine an approximation of the normalized channel matrix from a matrix selected in the algebra used to construct the code. space / time, and space / time decoding means of the normalized received signal, using the approximation of the normalized channel matrix. These means are controlled by the microprocessor of the processing unit 42.

The processing unit 42 thus outputs one or more estimated symbol vectors corresponding to an estimate of the signal transmitted on at least two transmitting antennas.

Such a decoder notably finds applications in MIMO systems for Wi-Fi (IEEE 802.1In), WiMax mobile (IEEE 802.16e), WiMax cooperative (IEE 802.16J), 3GPP-LTE, etc. applications.

Claims

1. A method of decoding a signal received on at least two reception antennas, said signal having undergone a space / time coding before transmission using a code constructed from an algebra on a commutative body K and being transmitted on at least two transmitting antennas, characterized in that it implements:

a preprocessing step (21) of said received signal (F) comprising the following sub-steps: determining a representative matrix of the transmission channel (H); o normalization of said representative matrix of the transmission channel and of said received signal, respectively delivering a normalized channel matrix (H ₁ ) and a normalized received signal (F ₁ ); o algebraic approximation of said normalized channel matrix, from a matrix (U) selected in said algebra and having a determinant equal to

1; a space / time decoding step (22) of said normalized received signal (F ₁ ), using the algebraic approximation of said normalized channel matrix.

2. Decoding method according to claim 1, characterized in that said selected matrix is a unit U of an order of said algebra, and in that said algebraic approximation step weights said unit by an approximation error E, such that: H ₁ = EU.

3. decoding method according to claim 2, characterized in that said order is a maximum order of said algebra.

4. decoding method according to any one of claims 2 and 3, characterized in that said algebraic approximation step selects the unit U so that said approximation error E is quasi-orthogonal.

5. decoding method according to any one of claims 2 to 4, characterized in that, said normalized signal being written in the form F ₁ = H ₁ X + W ₁ with X a transmitted code word and W ₁ a standardized Gaussian additive white noise, said normalized signal, after algebraic approximation of said normalized channel matrix, is expressed in the following form: Y ₁ = EUX + W ₁ with UX a code word.

6. Decoding method according to any one of claims 2 to 5, characterized in that it comprises a step of transforming said standard received signal F ₁ in vector form, such that: y ₁ = Ji ₁ Os + W ₁ with:

y ₁ the standard received signal in vector form;

- Ji ₁ the standardized channel matrix in vector form; - a vector of information symbols;

- a space / time coding matrix;

- W ₁ a Gaussian additive white noise in vector form.

7. Decoding method according to claim 6, characterized in that said standardized channel matrix, in vector form, multiplied by the space / time coding matrix is equal to:

1I ₁ O = E _/ U _/ Φ = E _/ ΦT _j y with:

E _/ a linear transformation corresponding to a multiplication on the left by said approximation error E; - U; a linear transformation corresponding to a multiplication on the left by said unit U;

- T _j7 a unimodular matrix; and in that said normalized received signal F ₁ , in vector form, is equal to: y ₁ = E; ΦruS + W ₁ = E; Φs '+ W ₁ , with s' a symbol vector.

8. decoding method according to any one of claims 1 to 7, characterized in that said space / time code is constructed from a divisional algebra.

9. Computer program product downloadable from a communication network and / or recorded on a computer readable medium and / or executable by a processor, characterized in that it comprises program code instructions for the implementation of steps of the decoding method according to at least one of claims 1 to 8, when said program is executed on a computer.

10. A device for decoding a signal received on at least two reception antennas, said signal having undergone a space / time coding before transmission using a code constructed from an algebra on a commutative body K and being transmitted on at least two transmitting antennas, characterized in that it comprises: pretreatment means (21) for said received signal comprising: means for determining a representative matrix of the transmission channel; transmission; o normalization means of said matrix representative of the transmission channel and said received signal, respectively delivering a normalized channel matrix (H ₁ ) and a normalized received signal (F ₁ ); o Algebraic approximation means of said normalized channel matrix, from a matrix (U) selected in said algebra and having a determinant equal to 1; space / time decoding means (22) of said normalized received signal, using the algebraic approximation of said normalized channel matrix.