Method for generating a signal, method for constructing a base sequence, and corresponding device and computer program

Abstract

The invention relates to a method for generating a signal, implemented in a digital communication system, comprising the following steps: - obtaining a base sequence of q complex values, denoted B, such that the autocorrelation function R BB of the base sequence B, satisfies formula (I), for i ∈ {0,...,c - 1} and R BB (τ) = 0; otherwise, - selecting Λ subsequences of p complex values in the base sequence B; and - using at least one of the selected subsequences to generate said signal.

Description

DESCRIPTION TITLE: Method for generating a signal, method for constructing a basic sequence, corresponding device and computer program. 1. Field of the invention The field of invention is digital communications. More specifically, the invention relates to the determination of a sequence of complex values ​​having particular properties, and its use in communications systems. The invention finds applications in particular in the field of wireless communications (for example by radio waves or unguided optical waves) or wired communications (for example by optical fiber or electric cable). In particular, the invention finds applications in the field of loT communications (in English "Internet of Things"). 2. Prior art To improve the robustness of communications over a communication channel, it is known to use error-correcting codes. For example, LDPC (Low Density Parity Check) codes, turbo codes, polar codes, or Reed-Salomon codes offer good performance in terms of error correction. In the paper "Non-Binary Low-Density Parity-Check coded Cyclic Code-Shift Keying" (2013 IEEE Wireless Communications and Networking Conference (WCNC)), E. Boutillon et al. proposed the association of a non-binary error-correcting code of the LDPC type with a cyclic code shift keying (CCSK). As illustrated in Figure 1, such a coding scheme takes as input a source message. For example, the source message has a size of kxm bits, where each m-tuple of bits corresponds to a symbol of a Galois field of cardinality q = 2 m, noted GFÇq). During a first coding step 11, the k symbols of the source message are coded by a non-binary LDPC code (NB-LDPC) with rate r0 = k / n, also called external code. A code word coming out of the NB-LDPC coder 11 therefore includes n symbols belonging to GF(q), or nxm bits. Each of the GF(q) symbols of the code word is then encoded during a second modulation step 12 by a CCSK code of efficiency = m / q, also called inner code. Thus, each GF(q) symbol in the codeword (represented by m bits) can be encoded, or modulated, by a sequence of q bits. A BPSK (“Binary Phase Shift Keying”) modulation can then be implemented during a third step 13 to transmit the n cyclically shifted versions of the sequence of q bits, each associated with a GF(q) symbol of the code word. More precisely, the CCSK modulation step 12 can implement different steps. In particular, a bijection can be defined between the q elements (0, cr°, ... , α q ~ 2 ) defined in the Galois field GF(q) and q integers between 0 and q — 1 (0, 1, 2, ... , q — 1). Such a bijection makes it possible to represent indifferently the elements of GF(q) and the integers between 0 and q — 1. A basic binary sequence of size q, denoted B, also called a basic sequence, can then be defined: B with B(i) G {0,1}. Such a sequence can notably be pseudo-random. CCSK 12 modulation associates in particular with α ieme element of GF(q) the binary sequence B a defined by a circular rotation (or cyclic shift) of α positions on the basic binary sequence B:B a = {B (i - α mod q)} i=0 q -i- For example, if q = 8 and the basic binary sequence B is defined by B = (1, 0, 0, 1, 1, 1, 0, 0), then the first symbol of GF(8) is associated with the sequence at B { — (0, 0, 1, 1, 1, 0, 0, 1), the second symbol of GF(8~) is associated with the sequence B2= (0, 1, 1, 1, 0, 0, 1, 0) and so on. In the given example, this is a circular rotation or a cyclic shift to the left. It is possible, symmetrically, to define a circular rotation or a cyclic shift to the right. In order to increase the spectral efficiency of the internal code (performance = m / q in the example above), it was notably proposed in the document “Rate-adaptive Inner Code for Non-Binary Decoders”, C. Marchand et al. (llth International Symposium on Topics in Coding, 2021) to truncate the CCSK modulation to the first p bits, p < q, to ​​adapt the rate to the transmission conditions. The efficiency iq of the inner code is then iq = m / p. For a truncation of size p, the α ieme element of GF(q) can then be encoded / modulated by a binary sequence B? defined by B? = (B(α), B(q + 1), ... , B(α + p — 1)) with the operations on the indices carried out modulo q. Subsequently, we call B p the set of sequences B? of length p, such that α G {0, ... , q — 1}. In the above examples, the basic sequence considered is a binary sequence. However, it is also possible to use non-binary basic sequences for CCSK modulations. Such non-binary basic sequences are also called NB-CCSK, q-ary CCSK or <?-ary CCSK. Tout comme les séquences CCSK binaires, il est possible de tronquer les séquences q-ary CCSK. As an example, Figure 2A illustrates a transmission scheme combining a Non-Binary code with a Truncated Non-Binary CCSK (NB-TCCSK) modulation. Such a coding scheme takes as input a source message comprising k information symbols from a Galois field of cardinality q = 2 m , denoted GF(q). The source message therefore has a size of kxm bits, where each m-tuple of bits corresponds to a symbol of GF(q). First, the k information symbols of the source message, for example (cr°, α 3 , α 7 , α 2 ) with k = 4, are encoded by a non-binary code NB. The code word obtained includes n symbols of GF(q), i.e. a code word of size nxm bits, for example (α 0 , α 3 , α 7 , α 2 , α 1 , α 4 , α°, α 5 , α 2 , α 6 , α 3 , α 4 ) with n = k 1 12. The coding efficiency of the external code is therefore r0= - for example r0= -. In a second step, each symbol of the codeword is encoded by the inner code. The inner code uses a Truncated Non-Binary CCSK (NB-TCCSK) modulation. Thus, each GF(q) symbol in the codeword (represented by m bits) can be encoded, or modulated, by a non-binary sequence of q elements, or "chips". The non-binary sequence is, for example, constructed from integers from 0 to q — 1, where each integer corresponds to a point on a constellation of size q (q-ary constellation). For example, if q = 8, the first symbol α 0 of the code word is associated with a first version of the non-binary sequence of 8 chips, for example with the sequence (0,1, 6, 7, 4, 5, 2, 3), according to a (spreading) efficiency r s — 1 / q — 1 / 8. The following symbols [α 3 , α 7 , ... ) can be associated with other versions of the non-binary sequence, defined by the circular rotation of the non-binary sequence of q chips (i.e. q versions). After encoding a symbol of the codeword, the resulting non-binary sequence of q chips can be truncated to p chips. For example, the sequence (0,1, 6, 7, 4, 5, 2, 3) of size q = 8 is truncated to a sequence (0,1,6) of size p — 3, according to a (truncation) efficiency r t — q / p — 8 / 3. Each chip (i.e. each element of the truncated sequence of size p) is then modulated using a modulation of size q, according to a (modulation) efficiency r m = log2(<7), then transmitted in a communication channel. Figure 2B illustrates in particular the obtained truncated sequence (0,1,6), where each chip corresponds to a point on the 8-PSK constellation. It is known that the quality of a modulation scheme depends on the distance between constellation points. In the above examples based on binary or non-binary, truncated or non-truncated CCSK modulation, the quality of the modulation scheme depends in particular on the minimum Euclidean distance between two sequences. We are interested below in the determination of the minimum Euclidean distance between two sequences BQ and B&, binary or non-binary, obtained by applying a circular rotation of α positions, respectively b positions, to a base sequence. If the base sequence is not truncated, then p — q, if the base sequence is truncated, then p < q, and if the base sequence is extended, then p > q- We define the square of the distance between the two sequences ^ as: with R(Ba , B }) the real part of the expression of the scalar product between the two sequences and the squared norms of the vectors respectively. In the case where the points corresponding to the graphical representations of the complex values ​​of the sequences are placed on the unit circle, then | | 11 2 = p and | \B? 11 2 = p, or The minimum distance associated with this family B p of sequences of size p (i.e. associated with the basic sequence and its different versions obtained by circular rotation to the right or to the left) can thus be expressed in the following form: As previously indicated, the CCSK modulation defines in particular a bijection between the q elements (0, α°, ... , α q ~ 2) defined in the Galois field GF(q) and q integers between 0 and q — 1 (0, 1, 2, ... , q — 1). Therefore, α, b belong indifferently to GF(q) or to {0, ... , q — 1}. This minimum distance can be normalized to obtain an average minimum distance per chip (per element / symbol): We can thus deduce, in the case where the points corresponding to the graphical representations of the complex values ​​of the sequences B p and B% are located on the unit circle: When the sequence is not truncated, i.e. p = q, it is possible to maximize D(B q ) using a CAZAC (Constant Amplitude Zero Autocorrelation) sequence, for example a Zadoff-Chu sequence. For this type of sequence, the out-of-phase cyclic autocorrelations are equal to zero, 1.e. (B q , B£) = 0 for α r b. Consequently, the normalized minimum distance is optimal, with D(B P ) = 2. However, these properties are no longer verified when the sequence is truncated, i.e. p < q. As an example, Figure 3 illustrates the evolution of the normalized minimum distance D(B P ) as a function of the size p of the sequence after truncation. If p = q = 64, the normalized minimum distance is optimal with D(B P ) = 2. If p < q, the normalized minimum distance decreases, and collapses rapidly as p decreases. There is therefore a need for a new type of sequence which can be used in particular for the implementation of non-binary CCSK modulation, presenting good properties in terms of minimum distance, before or after truncation. 3. Statement of the invention The proposed solution is based on a signal generation method, implemented in a digital communications system, comprising the following steps: - obtaining a basic sequence of q complex values, noted B, such that the autocorrelation function R BB of said base sequence B, defined by R BB (T) = T mod q)*, with B((n + T) mod qY the complex conjugate of B((n + T) mod q), and T = 0,1, ... , q — 1, verifies: with q a multiple of c, c > 1, h and c coprime integers, and j 2 = — 1, and: otherwise, - selection of A subsequences of p complex (consecutive) values ​​in said basic sequence B, such that said subsequence A, denoted B; = B p (s(AY), with 2 e {0, . . , A — 1}, 1 < A, is equal to: with s(A) an integer belonging to {0, , . , q — 1} corresponding to a position in said base sequence B, and the complex value associated with said position s(A) in said base sequence B, - using at least one of said selected sub-sequences to generate said signal. According to the invention, a basic sequence having particular properties is used to generate subsequences also having particular properties, in particular in terms of orthogonality or minimum distance. In particular, the generation of the basic sequence can be carried out beforehand, and the basic sequence(s) can be stored for example in a database internal to the entity implementing the method below, or accessible by such an entity. For example, if c = 4, the representation of the autocorrelation function R BBin the complex plane has 4-fold symmetry. The family comprising the basic sequence and at least one cyclically shifted version of the basic sequence has a normalized minimum distance D(B) equal to 2, and the family comprising the selected subsequences also has a normalized minimum distance D(B P ) equal to 2 for certain values ​​of p, notably p — q / Y p — q / 2 , p = 3q / Y Thus, the base sequence and its cyclically shifted versions are orthogonal (as for Zadoff-Chu sequences). Moreover, the selected subsequences have a minimum distance equal to 2p (which is not the case for truncated Zadoff-Chu sequences). It is thus possible to adapt the length p of the subsequence used to generate the signal according to the needs of the communication system. It is recalled in particular that the basic sequence B has a size q, and the subsequence B^ a size p. In a particular embodiment, p < q. In this case, a subsequence is considered to be a truncation or a portion of the basic sequence, or of a cyclically shifted version of the basic sequence. In another embodiment, p = q. In this case, a subsequence is the basic sequence or a cyclically shifted version of the basic sequence. In yet another embodiment, p > q. In this case, since a global sequence is formed from the basic sequence and at least one partial repetition of the basic sequence, a subsequence is a truncation or portion of the global sequence. In a particular embodiment, the cardinality of A is greater than or equal to 2, i.e. A > 2. In a first example of application, the subsequences are spreading sequences intended to be used by terminals, or groups of terminals, distinct from said communications system, for example when implementing an access protocol. For example, the method according to one embodiment implements a step of allocating one of said selected sub-sequences to a terminal of said communications system, and of transmitting said signal using said sub-sequence. For example, a base station may generate a number of subsequences, and allocate a distinct subsequence, or subsequence identifier, to terminals within its coverage area. Alternatively, a terminal in the communications system may allocate itself a subsequence from a set of subsequences obtained from the base sequence. A terminal can then use the subsequence allocated to it for its communications with the base station, which makes it possible to resolve, or at least limit, collision problems when several terminals wish to establish radio communication with the base station simultaneously. Thus, a first user may use a first set of subsequences to modulate a first data signal and a second user may use a second set of subsequences to modulate a second data signal. Even though the terminals of both users transmit the first data signal and the second data signal simultaneously, a decoder would be able to decode the different data signals thanks to the significant distance between the sub-sequences. The proposed solution makes it possible in particular to increase the number of users in the communications system. Indeed, it is recalled as an example that according to the prior art, if we consider a Zadoff-Chu sequence of length q = 16, it is possible to generate 16 orthogonal sequences of length q = 16. According to the invention, if we consider a basic sequence of length q = 64, it is possible to generate 64 distinct subsequences of length p = 16. The higher the number of distinct sequences, the higher the number of users that can communicate simultaneously in the communications system. In a second example application, said subsequences are code words of a dictionary intended to be used by an encoder of said communications system. For example, the method implements the association of at least one coded symbol, obtained at the end of a step of coding at least one source symbol, with one of said selected sub-sequences. Such sub-sequences can in particular be used in a communications system implementing a coding scheme with a binary or non-binary coder with a non-binary, truncated or non-truncated CCSK modulation. The length p of the subsequences can in particular be chosen taking into account the conditions of the transmission channel. In particular, if the transmission channel has little noise or interference, the length of the subsequences can be reduced by choosing p < q, which increases the transmission rate. Conversely, if the transmission channel has a lot of noise and / or interference, the length of the subsequences can be increased to achieve better immunity to noise and / or interference, with the counterpart of a reduction in the transmission rate. The rate is therefore adaptive (“rate-adaptive”). In a particular embodiment, the base sequence B is constructed from the following steps: - obtaining an initial vector 0 comprising q / c real elements, - for each real element 0(fc) of said initial vector 0, transformation of said real element 0(AQ into a complex element A( / c), according to the expression: - determination of a Krônecker product between the vector A formed from said complex elements A(Zc) and a binary vector A of size n A = c, such that A(i = h) = 1 and A(ih) = 0, with ie {0, . . , c — 1}, h and c being coprime integers, delivering a base vector X of size q, - transformation from the frequency domain to the time domain (using for example an Inverse Fast Fourier Transform of size q) of said base vector X, delivering said base sequence B of q complex values. We note in particular that the multiplicative coefficient (CQ) 1 / 2 used in the determination of the vector A makes it possible to obtain a basic sequence B having, on average, components whose energy (i.e. the square of their modulus) is equal to 1. We then obtain directly by construction 115| | 2= q. Such steps make it possible in particular to construct deterministically, in the frequency domain, a basic sequence and subsequences presenting good properties in terms of orthogonality or minimal distance. We recall that the Krônecker product between the vector A of size n & = q / c and the binary vector A of size n A = c, gives the vector X = Krônecker ÇA, A) of size n x = n & xn A = q, with XÇk), k = 8n A + α given by K(Æ) = A(A) x A(a), with <5 6 {0, ... , n â — 1} and α e {0, ... , n A — 1}. According to a first example, the q / c real elements of said initial vector 0 are chosen randomly. According to a second example, the q / c real elements of said initial vector 0 are chosen so that the points corresponding to the graphical representations of the q complex values ​​of the basic sequence B belong to at least one circle, for example to the unit circle (modulation <?-PSK, pour « Phase Shift Keying ») ou à des cercles concentriques (modulation q-APSK, pour « Amplitude Phase Shift Keying »). In this way, it is easier to modulate the subsequence, or the signal obtained from the subsequence. In this way, the basic sequence, or a subsequence obtained from the basic sequence, can have support on a q-APSK modulation (e.g. 16 APSK or 32 APSK). According to a particular embodiment, with c = 4, a modified initial vector 0' is obtained by adding a detailed vector V q / c, called the added vector, and the initial vector 0 of q / c real elements, said added vector V being equal to vx N, with ve [1, q — 1] and N a vector of size q / c constructed by choosing its first q / ( / 2.c) elements from the set {—1,0,1}, then copying the first q / Ç2c) elements so that NÇC) — N(i + q / (2c)), i — 0,1, ... , q / (2c) — 1. It is thus possible to generate a basic sequence maximizing the normalized minimum distance. According to a particular embodiment, the selection step selects a subset of subsequences, noted B À , of cardinality A, among the set B p of subsequences of size p, such that said subset F 2 has the largest normalized minimum distance : real tie of the expression of the scalar product between the subsequence and the e {0, . A — l} 2 .In a particular embodiment, c is equal to 3, 4 or 5. For example, if c = 3, then A =

[0010] or A =

[0001] . If c = 4, then A =

[0100] or A =

[0001] , If c = 5, then A =

[1000] , A =

[0100] , A =

[0010] or A =

[0001] , Notably, when c is equal to 3 or equal to 4, the normalized minimum distance for the base sequence is optimal and is equal to 2, for the values ​​of p = q / 3 and p = 2q / 3, or p = q / A, p = q / 2 and p = 3q / A, respectively. The representation in the complex plane of the autocorrelation function R BB respectively presents a symmetry of order 3, 4 or 5. The invention also relates to a method for constructing a basic sequence of q complex values, denoted B, intended to be used in a digital communications system, implemented by a processor, characterized in that the autocorrelation function R BB of said base sequence B, defined by ) mod q)* the complex conjugate of B (( ) q^, , , , q , 2d hi R RR with q a multiple of c, c > 1, h and c coprime integers, and y 2 = — 1, and: R BB (T) = 0 otherwise. In particular, such a basic sequence can be constructed in different ways, notably in the time domain or in the frequency domain. In another embodiment, the invention relates to a device for generating a corresponding signal. Such a device comprises at least one processor configured to implement the steps described above. Such a device is particularly suitable for implementing the method of generating a signal or constructing a basic sequence described above. It is for example integrated into a base station or into a device for supervising a communications system (allowing for example to assign each user a spreading sequence), or to a terminal of the communication system. This device may of course include the various characteristics relating to the methods according to the invention, which may be combined or taken in isolation. Thus, the characteristics and advantages of this device are the same as those of the method described previously. Consequently, they are not detailed further. The invention also relates to one or more computer programs comprising instructions for implementing a method as described above when this or these programs are executed by at least one processor. The invention also relates to a computer-readable information medium, comprising instructions of a computer program as mentioned above. 4. List of figures Other characteristics and advantages of the invention will appear more clearly on reading the following description of a particular embodiment, given as a simple illustrative and non-limiting example, and the appended drawings, among which: Figure 1 illustrates a coding scheme associating a non-binary error correcting code of LDPC type with a CCSK cyclic code shift modulation according to the prior art, Figure 2A illustrates a transmission scheme combining a non-binary error correcting code with a Truncated Non-Binary CCSK (NB-TCCSK) modulation according to the prior art, Figure 2B illustrates an example of a truncated sequence obtained in the transmission scheme of Figure 2A, Figure 3 shows the evolution of the normalized minimum distance D(B P>) as a function of the size p of the sequence after truncation, according to the prior art, Figure 4 presents the main steps implemented by a method for generating a signal according to an embodiment of the invention, Figure 5 illustrates the main steps for the construction of a basic sequence in the frequency domain, Figure 6A illustrates an example of representation of a basic sequence with q = 75 and Figure 6B the evolution of the normalized minimum distance D(B p ) depending on the size p of the subsequences, Figure 7A illustrates another example of representation of a basic sequence with q = 75, Figure 7B the evolution of the normalized minimum distance D(B P ) depending on the size p of the subsequences, and Figure 7C illustrates the representation in the complex plane of the associated autocorrelation function, Figure 8 illustrates the representation in the complex plane of an example of a C4 sequence, Figure 9A and Figure 9B illustrate examples of representation of C4 sequences with q = 64, Figure 10A and Figure 10B illustrate other examples of representation of C4 sequences with q = 64, Figure 11 illustrates yet another example of representation of a C4 sequence, Figure 12 presents the simplified structure of an entity of a communications system implementing a technique for generating a signal according to an embodiment of the invention. 5. Description of an embodiment of the invention 5.1 General principle The general principle of the invention is based on the construction of a basic sequence of length q having particular properties, and the use of this basic sequence, or of a subsequence of length p obtained from this basic sequence, in a digital communications system. Figure 4 shows the main steps implemented by a method for generating a signal according to one embodiment of the invention. During a first step 41, a basic sequence of q complex values, denoted B, is obtained as input to the method. In particular, the autocorrelation function R BB of said base sequence B, defined by R BB (T) = S„~j5(n)5((n + T ) mod <?)* =,? = 0,1, ... , q — 1, vérifie : with q a multiple of c, c > 1, h and c coprime integers, and j 2 = — 1, and: 5 BB (T) = 0 otherwise. Different techniques, presented below, can be implemented for the construction of such a basic sequence. For example, the basic sequence is equal to B = (5(0), 5(1), ... , 5(15)), with q = 16. During a second step 42, A subsequences of p complex values ​​are selected from the base sequence 5, such that said subsequence 2, noted 5^ = 5 p (s(A)), with A e {0, . . , A — 1}, 1 < A, is equal to: 5^ = 5 p (s(Â)) = {5(s(Â)), B(s(A) + 1 mod q), ... , B(s(A) + p — 1 mod q)} with s(A) an integer belonging to {0, . . , q — 1} corresponding to a position in said base sequence B, and the complex value associated with said position s(A) in said base sequence B, For example, we select A = 3 subsequences of p = 8 complex values ​​in the base sequence B. with s(0) = 3, s(l) = 7, s(2) = 12. Thus, a first subsequence B>=(i corresponds to the 8 consecutive complexes from B(3), a second subsequence B^ =1 corresponds to the 8 consecutive complexes starting from B(7), and a third subsequence B?. =2 corresponds to the 8 consecutive complexes from B( 14), modulo q. In a third step 43, at least one of the selected subsequences is used r generate said signal. For example, each of the selected subsequences may be allocated to a separate terminal in the communications system, and used by the terminal to generate a useful signal, for example by encoding, spreading, or modulating the useful data with the selected subsequence. In this way, it is possible for a receiver to decorrelate the received signals. The different subsequences can also form the code words of a dictionary intended to be used for encoding a user's data. 5.2 Construction of the basic sequence Different techniques can be implemented for the construction of a basic sequence. 5.2.1 Construction in the frequency domain, with any c Figure 5 illustrates the main steps for constructing a basis sequence in the frequency domain. The basis sequence obtained using this technique can be represented in the complex domain by non-unit points, i.e. by points not all located on the unit circle. In a first step 51, we obtain an initial vector 0 comprising q / c real elements. We recall that q is a multiple of c, c > 1. For each real element 0(fc) of the initial vector 0, we transform the real element 0(Æ) into a complex element A(fc) during a second step 52, according to the expression: In a third step 53, a Krônecker product is determined between the vector A formed from the complex elements A( / c) and a binary vector A of size n A = c, such that A(i = h) = 1 and A(i #= h) = 0, with ie {0, . . , c — 1}, h and c being coprime integers, yielding a base vector X of size q. Finally, during a fourth step 54, a transformation from the frequency domain to the time domain of the basis vector X is applied (for example an inverse Fast Fourier Transform), delivering the basis sequence B of q complex values. An example of an algorithm for implementing such a construction method is proposed below: end for; A = zeros(l, c) (construction of a zero vector of size c) A(h) — 1 (with h such that h and c are coprime numbers) X = Kronecker(A,A') According to a first example, for c = 3 and q = 75, the initial vector 0 of size 25 (75 / 3) can be chosen by randomly taking numbers between 0 and q — 1 = 74: 0 = (8 49 37 58 53 67 66 25 52 14 2 55 37 35 67 45 46 64 60 43 13 17 66 2 36). Figure 6A illustrates the points corresponding to the graphical representations of the q = 75 complex values ​​of the basis sequence B obtained with such an initial vector 0 and h = 1. The representation in the complex plane of the autocorrelation function R BB of the basic sequence B has a symmetry of order 3. Figure 6B illustrates the evolution of the normalized minimum distance D(B P ) depending on the size p of the subsequences. If p = q / 3 = 25, if p = 2q / 3 = 50, or if p = q = 75, the normalized minimum distance is optimal with D(5 P ) = 2. According to a second example, for c = 5 and q = 75, the initial vector 0 of size 15 (75 / 5) can be chosen by randomly taking numbers between 0 and q — 1 = 74: 0 = (8 49 37 58 53 67 66 25 52 14 2 55 37 35 67) Figure 7A illustrates the points corresponding to the graphical representations of the q = 75 complex values ​​of the basic sequence B obtained with such an initial vector 0 and h = 3. The representation in the complex plane of the autocorrelation function R BB of the basic sequence B has a 5-fold symmetry. Figure 7B illustrates the evolution of the normalized minimum distance D(B P ) depending on the size p of the subsequences. If p = q / 5 = 15, if p = 2q / 5 = 30, if p = 3q / 5 = 45, if p = 4q / 5 = 60, or if p = q = 75, the normalized minimum distance is optimal with D(B P ) = 1.38. We note that the minimum distance D(B P) is not equal to 2 in this case, but slightly less than 1.38. This corresponds to the minimum distance between two points of a pentagon located on the unit circle, as illustrated in Figure 7C. 5.2.2 Special case c = 4 Below we are interested in the special case c = 4. A basic sequence constructed with c = 4 is also called a C4 sequence. As previously stated, the autocorrelation function R BB of a base sequence B verifies: 1,2,3} with q a multiple of 4, h and 4 being coprime, h takes its values ​​from the set {1, 3}. and: R BB (T) = 0 otherwise. A basic sequence B of size q of type C4 (i.e. with c = 4) can therefore be characterized by its circular autocorrelation function R BB (T~) which verifies the following property: The parameter h allows two types of C4 sequence to be defined: h = 1 corresponds to a C4 sequence whose points of the autocorrelation function for T = iq / 4 with i G {0, 1, 2, 3} are connected clockwise (C4 sequence "clockwise"), while h = 3 corresponds to a C4 sequence whose points of the autocorrelation function for T = iq / 4 with i G {0, 1, 2, 3} are connected counterclockwise (C4 sequence "anti-clockwise"). The value of h = 3 corresponds to h = — 1 because (—1 mod 4) = (3 mod 4). The graphical representation of the autocorrelation function of the C4-type basis sequence in the complex plane, also called the C4 constellation, draws a cross centered at 0 that connects the zero point with the points q, jq, —q, —jq, and therefore has a symmetry of order 4. The C4 sequence thus takes its name from the words "Constellation", "Cross", "Circular", "auto-Correlation" (in French constellation, cross, circular auto-correlation). Such a C4 sequence allows to obtain a normalized distance ) equal to 2 for different values ​​of p, in particular A generic algorithm for constructing a basic sequence in the frequency domain, with any c, has been detailed above. Different techniques for constructing C4 basic sequences are presented below. 5.2.3 Construction of the basic sequence by operational research, with c = 4 According to a first technique, we define a priori the starting constellation C as an ordered set of q points of the complex plane, C = (C(0), C(l), ... , C(q — 1)) and the basic sequence B by a permutation p of size q which associates to the i ieme element B(i) of the basic sequence B the complex point C(p(iy), ie B(i) = C(p(iy), for i = 0,1, ... , q — 1. The constellation C can be constructed as C = (C(i) = e 2 ^ m ^ q , i = 0.1, ... , q — 1) for example. It is then a matter of solving a known optimization problem: for a set ? of given values ​​of p (P = {q / 4} or P = {1, 2, 3, ... , q — 1} for example), find the permutation p opt which maximizes the sum of the distances D(B P ), pe Popt = arg This first technique allows finding C4 sequences for small sequences [q < 16]. For larger sequences (typically q > 16), this solution is difficult to implement due to the exponential increase in the size of the solution search space (the number of permutations of a set of q elements in a set of q elements varies as a factorial of q). 5.2.4 Construction of the basic sequence by recursion, with c = 4 According to a second technique, it is possible to construct C4 sequences so that the points corresponding to the graphical representations of the q complex values ​​of the C4 sequence (constellation) are uniformly distributed on the unit circle, using a recursion algorithm to determine the phase <p des points : Initialisation : <p(0) = 0 Recursion: for i — 1 to q — 1 <p(i + 1) = <p(i) x (l + 1) + α mod q avec α = 1 ou α = — 1 et l = 4. We deduce the sequence C4: B(i) = e ^in <P(i)lq i i= 0 1); Q_ i As an example, Figure 8 illustrates the representation in the complex plane of the C4 sequence thus constructed for α = 1, l = 4 and q = 64. Two successive points of the sequence are linked by a line, forming a 4-cup epicycloid. Other sets of recursions are known, for example, the astroid defined by the recursion rp(i + 1) = (—^(0 x (Z — 1) + α) mod q or by the recursion <p(i + 1) = tp(C) + 4 x Z + 1 mod q. It is also possible to modify this recursion to reduce the number of points in the constellation while maintaining a C4 constellation. For example, to go from a 64-PSK constellation (regular 64-point constellation) to a 32-point constellation (half as many points) or a 16-point constellation (four times as many points), simply transform the sequence using the following function: with respectively α — 2 and α — 4. 5.2.5 Construction of the basic sequence in the time domain, with c = 4 According to a third technique, it is possible to construct C4 sequences in the time domain. Below is an example of an algorithm that can be implemented to generate a class of C4 sequences of length q, with q a multiple of 4, such that the points corresponding to the graphical representations of the q complex values ​​of a C4 sequence are on the unit circle. We thus consider as input an offset vector O of size 4 taking its values ​​between 0 and q — 1 (this condition is sufficient to obtain a sequence C4 having points located on the q-PSK constellation defined by the set {e 2kn i / q , k = 0, 1, ... , q — 1}; this is not, however, a necessary condition: this constraint is given as an example only), p a permutation of the set (1, 2, 3, 4) and d a variable taking either the value +1 or — 1. From these inputs, the following algorithm generates a unit C4 sequence B of size q: for k = 0 to q / 4 — 1 for i = 0: 3 <p(4 x k + i) = O(t) + k x (4 + Sd + M(p(Z))) fin pour fin pour (p = (p mod q for i = 0 to q — 1 B(Z) = e Wi) / "end for. 5.2.6 Construction of the basic sequence in the frequency domain, with c = 4 According to a fourth technique, it is possible to construct C4 sequences in the frequency domain, as presented in the general case with any c. Below is an example of an algorithm that can be implemented to generate a class of C4 sequences of length q, with q a multiple of 4, such that the points corresponding to the graphical representations of the q complex values ​​of a C4 sequence are located on the unit circle or not. We consider as input a size q, an initial vector 0 comprising ^real elements, and h a variable taking either the value +1 to generate a C4 “clockwise” sequence, or —1 to generate a C4 “anti-clockwise” sequence. From these inputs, the following algorithm generates a unitary or non-unitary C4 sequence s of size q, denoted X in the frequency domain: X = Kronecker(A,A) B = IfftÇX}. For c = 4, the binary vector A is either equal to [0 1 0 0] if h = +1, or equal to [0 0 0 1] if h = — 1. As already indicated, (—1 mod 4) = (3 mod 4). This technique makes it possible in particular to construct non-unitary constellations. As an example, Figures 9A and 9B illustrate the graphical representations of the C4 sequences obtained for <7=64, an initial vector 0 = (0(0 = i 2 , i = 0, 1, ... , 15) and h = +1 for figure 9A, or h = — 1 for figure 9B. A C4 sequence constructed with a random initial vector 0 is not necessarily unitary, i.e. the points associated with its graphical representation in the form of a constellation are not necessarily located on a circle. It is however possible to constrain the choice of real values ​​of the initial vector 0 so that the sequence C4 is unitary by construction. Conversely, all unit sequences already found can be constructed from a specific initial vector 0. In order to obtain a unitary C4 sequence, according to a first example, if q = 2 2t , with t > 1, the q / 4 real elements of the initial vector 0 can be chosen such that: 0(2 f l u + r) = cZ(r) + uy(r)2 t+1 with: d a real vector of size 2 t-1 . Note that there is no constraint on the construction of the vector d. It can be chosen randomly or optimized according to a criterion specific to the application. For example, we can consider the optimization of the normalized distance D(B P ) for particular values ​​of p. For illustration, p = 1 will generate a C4 sequence associated with a constellation whose minimum distance between two points of the constellation will be maximized; y a permutation of the set (0, 1, ... , 2 t-1 — 1), 0 < r < 2t-1 and 0 < u < 2 t-1 . According to a second example, if q = 2 2t+1 = 2 5 , with t = 2, the q / 4 real elements of the initial vector 0 can be chosen such that: with: d a real vector of size 2 t-1 . Again, there is no constraint on the construction of the vector d. It can be chosen randomly or optimized according to an application-specific criterion. For example, we can consider the optimization of the normalized distance D(B P ) for particular values ​​of p. For illustration, p = 1 will generate a C4 sequence associated with a constellation whose minimum distance between two points of the constellation will be maximized; p a permutation of the set (0,1) (p = (0,1) or p = (1,0)). (e0, ei) e {- 1,1} 2 . According to a third example, if q = 2 2t+1 = 2 7, with t = 2, the q / 4 real elements of the initial vector 0 can be chosen such that: with: i&28 = i — (i mod 4), (M3)2 = 2i4+ £3 of a real vector of size 2 t-1 . Again, there is no constraint on the construction of the vector d, p a permutation of the set (0,1,2,3) 5.2.7 Construction of the basic sequence in the frequency domain minimizing D (B), with c = 4 A modified initial vector 0' can in particular be obtained from an initial vector 0 allowing to obtain a unitary C4 sequence, so that the points corresponding to the graphical representations of the q complex values ​​of a basic sequence obtained from the modified initial vector correspond to the points of a constellation diagram of a q-APSK modulation. In particular, it is possible to apply the frequency method to generate a q-APSK constellation and an associated C4 sequence maximizing the distance DÇB 1}, that is, the minimum distance between two points in the constellation. The modified initial vector 0' can notably be obtained by adding an added vector V of size q / c and the initial vector 0 of q / c real elements, such that the added vector V is equal to vx N, with v S [1, q — 1] and N a vector of size q / c constructed by choosing its first q / (2c) coordinates in the set {—1,0,1}, then copying the <? / (2c) premières coordonnées de telle sorte que N(i) = W + <? / (20), i = 0,1, ... , Q / (2C) — 1. Below is an example of an algorithm for obtaining a basic sequence that minimizes the PCB distance. 1 ), from a unit base sequence. We consider as an example c = 4 and q = 64. Note however that the proposed algorithm can be generalized to different values ​​of c or length q of the sequence. We therefore consider as input a unitary C4 sequence of size q = 64, noted B. In a first step, the vector representing the unitary C4 sequence B is transformed from the time domain to the frequency domain: X = fft( B). We then seek to extract, from the base vector X thus obtained, the initial vector 0 comprising = 64 — 4 = 16 real elements (used to construct sequence B), by determining the phases of the 16 unaffected coefficients (coefficients located at indices equal to h mod 4, with h = +1 for a “clockwise” sequence and h = —1 for an “anti-clockwise” sequence). In a next step, the initial vector 0 is modified by adding a vector V (added vector) of size 16 defined as follows: V = vx N = vx [TV (0), N(l), N(2)~ , N(3), N(4), TV(5), N(6), N(7), N(0), 2V(1), TV(2), N(3), N(4), N(5), N(6), N(7)], with ve [1, q — 1] a real number which can take for example integer values ​​in the interval [1, 63] and the coefficients JV(i) taking their value among -1, 0 or and 1. This makes it possible to obtain a modified initial vector 0' = 0 + V, and consequently to construct a new sequence B' using the technique for constructing C4 sequences in the frequency domain presented previously, from the modified initial vector 0'. It is then possible to determine the value of D(B rl ) of the sequence thus created. It is then sufficient to generate different sequences B' to keep only the one maximizing D^' 1 ). As an example, we consider as input a unit C4 sequence of size q = 64, noted B, obtained by the frequency method from an initial vector 0 = (0,61,52,44,0,45,20,60,0,29,52,12,0,13,20,28) and h = +1, which gives the unit C4 sequence illustrated in figure IDA. With v = 18 and N = (—1, +1.0, +1.0, — 1, +1, —1), the added vector V is equal to: V = (-18, +18, 0, +18, 0, -18, +18, -18, -18, +18, 0, +18, 0, -18, +18, -18) Adding the added vector and the initial vector gives: 0' = 0 + 7 = (-18,79,52,62,0, 27, 38, 42, -18,47,52,30,0, -5,38,10). The initial vector thus modified and the variable h = +1 can then be used to generate a new sequence B' such as the non-unitary sequence C4 illustrated in Figure 10B. For example, the sequence B' thus constructed has a minimum distance D(B') = 6.12 x 10 -2 . 5.3 Application examples 5.3.1 Construction of orthogonal basis sequences As indicated above, the basic sequences thus constructed may in particular be spreading sequences intended to be used by terminals, or groups of terminals, distinct from a communications system. Indeed, in a multi-user context, it is advantageous to associate specific sequences with users (or groups of users) in order to help the multi-user detection task. In particular, it is possible to choose a C4 “clockwise” sequence (with h = +1), denoted x, and a C4 “anti-clockwise” sequence (with h = —1), denoted y. Indeed, these two types of sequences are, by construction, orthogonal to each other. Indeed, the calculation of the intercorrelation between the sequences x and y in the frequency domain gives: R xy = ifftÇfftM.fftÇyY) = 0 Since X = fft(x) has non-zero frequencies at indices k = 1 mod 4, while Y = fft(y) has non-zero frequencies at indices k = 3 mod 4, we therefore deduce that R xy (r) = 0- P our any value of T. 5.3.2 Identification by subsequences of size p In a multi-user context, it may also be advantageous to associate subsequences of size p with users (or groups of users) to assist the multi-user detection task. For example, a base sequence of size q can be truncated into at least one subsequence of size p. As an example, we consider a communications system with q = 4p users (or group of users), in which we wish to associate with each user a sequence of size p which allows them to identify themselves in a multi-user protocol. In particular, it is possible to construct a sequence C4 x of size q and to associate with the k ieme user (or group of users) the sequence of size p defined by x%. Indeed, it is possible to show that: if (k- Z) =# 0 mod p, then {x?, x?) « q, if l = k + 2p mod q then (x^ xf) = —p, and therefore, d ( X fc - X k + 2 Q ) 2 = l l4 9 H 2 + \ \ X k + 2 P \ \ 2 + 2 P xf} = ±jp, and therefore, according to the equation This amounts to showing that p = <? / 4 et pour tout k, x? = ~x^ +2p , x? = j h x p , _ p , x^ = j~ h x^+2p with h indicating the direction of rotation of the C4 sequence (thus, j h = j and j~ h = —j if h = 1, and j h = —j and j~ h = j if h = —1). This subsampling technique can of course be extended to other sequence lengths (in particular, p = q / 2 or p = 3q / 4:, or other values ​​of p). Moreover, the number of allocated sequences can be less than q. For example, such a system is advantageous for initiating communication in a cellular network between a mobile and a base station. 5.3.3 Identification by subsequences and orthogonal sequences In particular, it is possible to increase the number of users, or groups of users, in a communications system by combining the use of subsequences of size p and basic sequences or orthogonal subsequences (i.e., where the distance between each sequence is significant), so as to increase the number of users who can be easily separated from each other. Various examples are presented below. As a first example, for q = p, it is possible to generate 2q sequences of size p = q from two C4 sequences x and y of size q, with h = +1 for sequence x, and h = —1 for sequence y. The 2q identifiers / spreading sequences of size p — q can then be defined by the union of the q sequences {x a , a = 0,1, . . . , q — 1} and {y b , b = 0.1, . . . , q — 1}. In particular, it is possible to choose the C4 x and y sequences so that the distance between the disjoint sequences is equal to 2q, thus allowing effective separation of users from each other. For example, for q = 64, it is possible to define a so-called epicycloid sequence x with a = +1 and an epicycloid sequence y with a = — 1. This makes it possible to create a set of 128 sequences of size 64 with a normalized distance between two distinct sequences equal to 2. According to a second example, for q = 2p, it is possible to generate 4p = 2q sequences of size p = from two C4 sequences x and y of size q, with for the sequence x, h = +1, and for the sequence y, h = —1. The 2q identifiers / spreading sequences of size p = q / 2 can then be defined by the union of the q sequences [x£, a = 0,1, ... , q — 1} and {y?, b = 0,1, . . . , q — 1}. In particular, it is possible to choose the C4 sequences x and y so that the distance between two disjoint sequences is equal to q, thus allowing effective separation of users from each other. For example, for q = 64, it is possible to define the sequence x from the initial vector 0 = (46, 62, 0, 38, 14, 62, 0, 6, 46, 62, 0, 38, 14, 62, 0, 6) and h = +1, and the sequence y from the initial vector 0 = (42, 44, 0, 38, 10, 44, 0, 6, 42, 44, 0, 38, 10, 44, 0, 6) and h = — 1, to obtain this property. This allows to create a set of 128 sequences of size 32, with a normalized distance between two distinct sequences equal to 2. According to a third example, for q = 4p, it is possible to generate 4p = q sequences of size p = - from two C4 sequences x and y of size q, with h = +1 for sequence x, and h = —1 for sequence y. The 2q identifiers / spreading sequences of size p — - can then be defined by the union of the q sequences [x?, a = 0,1, . . . , q — 1} and {y?, b = 0,1, . . . , q — 1}. For this case, it is possible to choose the C4 sequences x and y to maximize the minimum distance between two sequences. For example, for q = 64, it is possible to define the sequence x from the initial vector 0 = (1, 2, 3, 4, 1, 18, 35, 52, 1, 34, 67, 100, 1, 50, 99, 14) and h = -1. The sequence obtained is illustrated in particular in figure 11. To generate the sequence in reverse, it is possible to define £ (t) = x(q — i), i = 0,1, ... , q — 1 with the operations on the indices carried out modulo q. The minimum (unnormalized) distance between two subsequences from the base sequence x and the base sequence x is 14.5, in other words, whatever a and b, d(x^, x^) 2 > 14.5. It is of course possible to use two sequences constructed with a random initial vector and h = +1, and h = — 1. The minimum distance remains of the same order. 5.3.4 Coding Scheme As previously indicated, the basic sequences thus constructed can also define code words of a dictionary intended to be used by a coder of said communications system. In particular, such basic sequences may be used in a coding scheme combining a binary or non-binary error-correcting code with CCSK cyclic code-shift modulation, as illustrated in Figure 1, or in a coding scheme combining a binary or non-binary code with truncated binary or non-binary CCSK modulation, as illustrated in Figure 2A. 5.3.5 Other application examples Although motivated by the design of efficient NB-TCCSK coding scheme, the proposed basic sequences, especially C4 sequences, can also be used in other application areas. In particular, such sequences can be used instead of Zadoff-Chu sequences, for example in 5G protocols. 5.4 Corresponding device Finally, in relation to figure 12, we present the simplified structure of an entity of a communications system implementing a technique for generating a signal according to one of the particular embodiments described above. Such an entity comprises a memory 121 comprising a buffer memory, a processing unit 122, equipped for example with a processor P, and controlled by the computer program 123, implementing the method for generating a signal according to one embodiment. Upon initialization, the code instructions of the computer program 123 are for example loaded into a RAM memory before being executed by the processor of the processing unit 122. The processor of the processing unit 122 implements the steps of the method described previously, according to the instructions of the computer program 123, to: - obtain a basic sequence of q complex values, denoted B, such that the autocorrelation function R BB of said base sequence B, defined by R BB (J) + T) mod q)*, with B((n + T) mod q)* the complex conjugate of B(fn + T) mod q) and T = 0,1, ... , q — 1, verifies: with q a multiple of c, c > 1, h and c coprime integers, and j 2 = —1, and: 7? BB (T) = 0 otherwise, - select A subsequences of p complex values ​​in said basic sequence B, such that said subsequence 2, noted Bj = B p (s(A)), with 2 e {0, A — 1}, 1 < A, is equal to: with s(2) an integer belonging to {0, .., q — 1} corresponding to a position in said basic sequence B, and B(s(2)) the complex value associated with said position s(2) in said basic sequence B, - using at least one of said selected subsequences to generate said signal. Such an entity, or a separate entity, may also implement a technique for constructing such a basic sequence.

Claims

CLAIMS 1. Method for generating a signal, implemented in a digital communications system, comprising the following steps: - obtaining a basic sequence of q complex values, noted B, such that the autocorrelation function R BB of said base sequence B, defined by R BB (T) = r mod qy, with BQÇn + r) mod qY the complex conjugate of B((n + T) mod q) and T = 0,1, ... , q — 1, verifies: with q a multiple of c, c > 1, h and c coprime integers, and j 2 = — 1, and: otherwise, - selection of A subsequences of p complex values ​​in said basic sequence B, such that said subsequence A, noted B^ = B p (s(AY), with AG {0, . . , A — 1}, 1 < A, is equal to: with s(A) an integer belonging to {0, , . , q — 1} corresponding to a position in said base sequence B, and B(s(A)) the complex value associated with said position s(A) in said base sequence B, - using at least one of said selected sub-sequences to generate said signal.

2. Method according to claim 1, characterized in that said sub-sequences are spreading sequences intended to be used by terminals, or groups of terminals, distinct from said communications system.

3. Method according to any one of claims 1 and 2, characterized in that it comprises a step of allocating one of said selected sub-sequences to a terminal of said communications system, and of transmitting said signal using said sub-sequence.

4. Method according to claim 1, characterized in that said sub-sequences are code words of a dictionary intended to be used by a coder of said communications system.

5. Method according to claim 4, characterized in that it implements the association of at least one coded symbol, obtained at the end of a step of coding at least one source symbol, with one of said selected sub-sequences.

6. Method according to any one of claims 1 to 5, characterized in that said base sequence B is obtained from the following steps: - obtaining an initial vector 0 comprising qjc real elements, for each real element 0( / c) of said initial vector 0, transformation of said real element 0( / c) into a complex element A( / c), according to the expression: - determination of a Krônecker product between the vector A formed from said complex elements A(Æ) and a binary vector A of size nA = c, such that A(i = h) = 1 and A(ih) = 0, with ie {0, . . , c — 1}, h and c being coprime integers, delivering a base vector X of size q, - transformation from the frequency domain to the time domain of said base vector X, delivering said base sequence B of q complex values.

7. Method according to claim 6, characterized in that said q / c real elements of said initial vector 0 are chosen randomly.

8. Method according to claim 6, characterized in that said q / c real elements of said initial vector 0 are chosen so that the points corresponding to the graphical representations of the q complex values ​​of said basic sequence B belong to at least one circle.

9. Method according to claim 6, characterized in that said q / c real elements of said initial vector 0 are chosen so that the points corresponding to the graphical representations of the q complex values ​​of said basic sequence B correspond to the points of a constellation diagram of a q-PSK or q-APSK modulation.

10. Method according to claim 6, characterized in that c = 4 and in that a modified initial vector 0' is obtained by adding a vector V of size q / c, called the added vector, and said initial vector 0 of q / c real elements, said added vector V being equal to vx N, with v G [1, q — 1] and N a vector of size q / c constructed by choosing its first q / (2c) elements from the set {—1,0,1}, then copying the first q / (2c) elements so that Af(i) = N(i + q / (2c)), i = 04. q / (2c) - l.

11. Method according to any one of claims 1 to 10, characterized in that said selection step selects a subset of subsequences, noted B Â , of cardinality A, among the set B p of subsequences of size p, such that said subset has the largest standardized minimum distance DÇB*'): with R ({Bx ,Bj i ')) the real part of the expression of the scalar product between the subsequence B^ and the subsequence 12. Method according to any one of claims 1 to 11, characterized in that c is equal to 3, 4 or 5.

13. Method for constructing a basic sequence of q complex values, denoted B, intended to be used in a digital communications system, said method being implemented by a processor, characterized in that the autocorrelation function R BBof said base sequence B, defined by / ? BB (T) S(n)fî((n + T) mod q), with B ((n + r) mod q) the complex conjugate of B((n + T) mod q), and T = 0,1, ..., q — 1, verifies: with q a multiple of c, c > 1, h and c coprime integers, and j 2 = — 1, and: Otherwise.

14. Device for generating a signal, implemented in a digital communications system, comprising at least one processor configured to: - obtain a basic sequence of q complex values, denoted B, such that the autocorrelation function R BB of said base sequence B, defined by B BB (T) = JXj BfnJSf / n + T) mod q)*, with B((n + T) mod q)* the complex conjugate of BÇ(n + r) mod q~) and T = 0,1, ... , q — 1, verifies: with q a multiple of c, c > 1, h and c coprime integers, and j 2 — — 1, and: RBBQÔ = 0 otherwise, - select A subsequences of p complex values ​​in said basic sequence B, such that said subsequence A, noted B^ = B p (s(À.y), with A e {0, . . , A — 1}, 1 < A < q, is equal to: with s(A) an integer belonging to {0, .. , q — 1} corresponding to a position in said base sequence B, and B(s(A)) the complex value associated with said position s(A) in said base sequence B, - using at least one of said selected sub-sequences to generate said signal.

15. 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 implementing a method according to at least one of claims 1 to 13.

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