FR2606233A1 - Method and apparatuses for multidimensional modulation and demodulation of binary signals - Google Patents

Abstract

In order to carry out the modulation, the method consists in transcoding 1 each binary signal of the one-dimensional vector space in order to obtain a signal which can be represented in an N-dimensional vector space. Each transcoded signal is represented with respect to a basis of the N-dimensional vector space in the form of a sequence of elementary signals each formed by one term of a linear combination of the elements of the basis. The set of transcoded signals is sorted 3 in this basis into signal families described by one representative, all the signals of the same family having the same energy level, and in multiplexing the elementary signals obtained in each family with a view to their transmission 5. In order to carry out the demodulation, the transcoded signal received is identified within its family by comparing its energy with those of the representatives. A decoding table addressed by the representative of the family which possesses the energy closest to that of the signal received and by the sign and amplitude characteristics of each of the elementary signals received makes it possible to recover each transmitted signal. Application: information transmission.

Description

Method and apparatus for modulation and
multidimensional demodulation of binary signals
The present invention relates to a method and apparatus for multidimensional modulation and demodulation of binary signals.

 Information theory, the bases of which were established by the mathematician SHANNON, makes it possible to evaluate the capacity of a transmission channel, that is to say the maximum bit rate of information which can be conveyed without error on a transmission channel in the presence of noise. This flow is called the capacity of the channel. However, this flow turns out to be a theoretical maximum impossible to achieve in practice. What is more, information theory does not provide a method for building communications systems that would even allow one to approach the estimated theoretical bit rates a little.

 However, this theory is remarkable in that it uses the notion of multidimensional space, pushed to its limit, that is to say having an infinite number of dimensions. Instead of considering one-dimensional binary signals, as is conventionally done for FSK type modulations, for example, FSK being the abbreviation of the English term "Frequency shift Keying", the signals considered are signals which can be chosen from a very large number M of possible signals, each signal having a very large number N of dimensions, that is to say composed of N distinct samples which can be independently disturbed by noise.

The use of a large number of dimensions leads on the theoretical plane to consider a spherical model of representation of the signals where the signals M are points located inside a sphere of radius R. In this case the minimum distance which separates two distinct points increases with the number of dimensions considered. We can also express this fact by saying that if we wish to have M points distant by at least a minimum distance d, in a sphere with
N dimensions, the radius R of the sphere decreases when the number
N of dimensions increases.

 Consequently, as the radius of the sphere increases with the average energy of the signal, and the probability of error representing the risk of confusion between two neighboring signals varies decreasingly as a function of the minimum distance d, we perceive that it is possible to improve the performance of a link by grouping together k elementary signals from a n-dimensional space, by giving everyone the possibility of taking m distinct values, to transform them into symbols with M = mk values in a space at N = nk dimensions.

 However, the whole difficulty consists in choosing an arrangement of the M points as compact as possible while ensuring that the N components of the received signal are well decorrelated from the point of view of noise otherwise the principle explained above is no longer valid: indeed the fact that it is the distance between distinct signals which determines the performance is closely linked to the presence of decorrelated Gaussian noise on each of the components. It is also necessary that the bandwidth of the signal remains unchanged and that the complexity of the system remains within reasonable limits.

 The object of the invention is to propose a solution to the problem posed.

 To this end, the subject of the invention is a method for multidimensional modulation and demodulation of binary signals, characterized in that it consists, in order to carry out the modulation, in transcoding each binary signal from a vector space to one dimension to obtain a signal representable in an N-dimensional vector space, each transcoded signal being represented with respect to a base of the N-dimensional vector space in the form of a series of elementary signals each formed by a term of a linear combination of the elements of the base, the set of transcoded signals being classified in this base in families of signals described by a representative, all the signals of the same family having the same energy level, and in multiplexing the elementary signals obtained in each family with a view to their transmission, the method consisting in demodulating the transcoded signal received, in swapping its components in order to arrange them in the order of decreasing absolute values and to identify its family by comparing its energy to that of the representatives, to identify the sign and the amplitude of each elementary signal received and to decode the signal received in one-dimensional vector space by an addressed decoding table by the representative of the family which has the energy closest to that of the received signal and by the sign and amplitude characteristics of each of the elementary signals received.

 The subject of the invention is also a modulation apparatus and a demodulation apparatus for implementing the above method.

Other characteristics and advantages of the invention will appear below with the aid of the description which follows given with reference to the appended drawings which
- Figure 1 a table illustrating a geometric configuration of signals in a four-dimensional space
- Figure 2 a table illustrating a geometric configuration of signals obtained from the previous table by a change of base
- Figure 3 an embodiment of the modulation apparatus according to the invention
- Figure 4 an embodiment of a demodulation apparatus according to the invention.

 Multidimensional modulation of binary signals is carried out according to the invention by grouping the basic symbols, constituting all the information to be transmitted between transmitter / receiver devices, in sample packets N to form an alphabet represented by M = m signals to N dimensions.

The signal received by the receiving device is represented by a set of N samples of the form s (I) = 181 (I), s2 (I) if (I) sN ((1) with sl (I) = A1 ( I) + X1 (2) for i = 1 ... N
In expression (2) A1 (I) denotes the amplitude of the sample if (I) and Xi denotes the noise accompanying the sample when it is received by the receiving device. We consider in the following description that this noise is
Gaussian, centered, decorrelated and has a standard deviation 6
To carry out the distribution into packets, a transcoder transforms each binary signal representing the information to be transmitted and identified in a one-dimensional vector space, into a signal represented in an N-dimensional vector space with respect to a base of this space.

 The choice of the points of the vector space network thus constituted and representative of the M possible signals is determined so that the set of M points is represented inside a sphere of the smallest possible radius: one thus obtains a minimum average signal energy.

 In this network, the signals are classified by families, each described by a representative, the signals of the same family having the same energy.

This arrangement allows, in the execution of the demodulation process, to identify each signal by comparing its energy to that of all the representatives and to perform according to the signs of its components in the vector space to
N dimensions and their amplitudes a table search for the binary signal corresponding to it in one-dimensional vector space.

 An example of implementation of the method according to the invention is described below. In this example, the basic symbols are grouped in packs of 4 samples. The processing is carried out on an alphabet of M = m4 = 256 signals, in a vector space with N = 4 dimensions, each symbol carrying a byte of information.

In this case, each signal is according to the relation (1) representable by a set of 4 samples of the form s (I) = Isl (I), s2 (I), s3 (I), s4 (I) I (3)
such as if (I) = A1 (I) + X1 for i = 1, 2, 3, 4 on receipt.

 One thus defines for each of the 256 possible signals, a set of four amplitudes A1, A2, A3 and A4 called the components of the signal.

 The distribution of the M signals in the N-dimensional vector space is defined by the points of this space whose coordinates correspond to whole integers of even sum.

That is to say if in the network thus formed, 1x1, x2, x3, X3, designates the coordinates of a point belonging to the network, each of the coordinates xi verifies a relation of the form
Xi = ki
where ki is an integer and i = 1, 2, 3, 4,
and the sum x1 + x2 + x3 + x4 is even.

 This distribution is that which allows in a 4-dimensional vector space to obtain the most compact possible arrangement for which the signals M are closest to each other. It can be compared, for example, to hexagonal paving which, in a two-dimensional space, also gives the most compact distribution.

 A table illustrating a possible geometric configuration of these signals is represented in FIG. 1. In this table the different representative points of the signals are defined with respect to representatives. The set of points is deduced from the representatives by permutations and sign changes by ensuring that the sum of their coordinates is a multiple of 4. This last condition is necessary to eliminate exactly half of all the possibilities of sign change .

 We thus obtain seven groups or families numbered from 1 to 7 with a representative in each group or family. Groups 1 to 7 each include 8 points, groups 2, 4 and 5 each include 32 points, group 3 comprises 48 points and group 6 comprises 96 points, which in total represents the 256 points representative of the 256 signals in the network.

As the coordinates of the representatives of groups 1 to 7 have respectively the values (1,1,1,1); (-3.1,1.1); (3,3,1,1) (5,1,1,1); (-3,3,3,1); (-5.3.1.1); (3,3,3,3) the energies,
E2 = x21 + x22 + x23 + x24 conveyed by each of the points in groups 1 to 7 have the values 4, 12, 20, 28, 28, 36 and 36 respectively.

 The coordinates of the points of group 1 form in the four-dimensional vector space vectors orthogonal or opposite to each other 8 in number.

The first four of them correspond in fact to the four known sequences of WALSH of dimension equal to 4, namely
W1 = I + 1, + 1, + 1, + 1I
W2 = 1 + 1, -1, + 1, -1
w3 = 1 + 1, + 1, -I, -11
w4 = | + 1, -1, -1, + 1 |
These four orthogonal sequences form a basis in the four-dimensional vector space defined above.

The coordinates of each of the points representative of the signals in 4-dimensional space can then be expressed as a function of the components of each of the vectors of this base according to the relation xl, x2, X3, X41 = i = î YiXi (9)
By applying this relation to the set of points, the table of figure 1 is simplified in the way represented by the table of figure 2 which is equivalent to that of figure 1, with the difference however that all the permutations and all changes of contact sign of each representative are authorized.

To the seven starting groups correspond, in the table of figure 2, six arrival groups g numbered from 1 to 6, whose representatives y1 (g), y2 (g), y3 (g), y4 (g) 1 have the values respectively, 1000 for group 1, 1110 for group 2, 2100 for group 3, 2111 for group 4, 2210 for group 5, and 3000 for group 6. I1 exists under these conditions 8 points for arrival group 1, 32 points for arrival group 2, 48 points for arrival group 3, 64 points for arrival group 4, 96 points for arrival group 5 and 8 points for arrival group 6, which in total forms a representation of 256 points. Under these conditions each signal represented has an energy
E2 (g) = y21 (g) + y22 (g) + y23 (g) + y24 (g) equal to 1 for the eight points of group 1 and equal to 3 for the 32 points of group 2. Signal energies represented by each point in groups 3, 4, 5 and 6 have the values 5, 7, 9 and 9 respectively.

 Besides the fact that all the possible values of the permutations y1 (g) y2 (g) y3, g) and y4 (g) are allowed, we note that their number is exactly equal to a power of 2 (M = 256 = 28), which avoids rejecting some of them.

Indeed, the fact of rejecting certain signals would have the unfortunate consequence of posing a problem of choice on the emission, namely which ones should be kept to have a better signal to noise ratio, and at the reception a decision problem to demodulate these signals.

 The advantage of the coding represented by the table in FIG. 2 is that it corresponds to a set of points representative of the signal, which can be considered optimal in the sense that it is as compact as possible and the tool is complete, no element not to be added or removed.

 The demodulation of the signals represented by the table in FIG. 2 is carried out taking into account the noise which naturally affects in a transmission each received signal.

Under these conditions, to each signal Y transmitted and defined by the components yi where i = 1, 2, 3, 4 corresponds in reception a signal R defined by components ri such that
ri = Yi + Xi (10) th
Xi representing the noise on the i component which is considered below as a Gaussian noise, centered and uncorrelated between components.

 Consequently, the probability of finding the correct information I without error is that which maximizes the conditional probability of obtaining the correct information I from the signal R received.

Taking into account the assumptions made on noise, this probability can be expressed according to the Gauss normal law

 being the standard deviation.

 By maximizing the logarithm of this probability, it can be seen that it is sufficient for I to minimize the distance between the point representing the Ie signal and that corresponding to the received signal.

This distance is defined by the relation

il revient au même de rendre maximale la relation

As the energy of all the signals of a group is constant and equal to

it’s the same to maximize the relationship

To perform this operation on each of the combinations (r1, r2, r3, r4) characterizing each received signal, the elementary signals ri are arranged in order of decreasing values, and written to a memory. The corresponding permutation as well as the resulting sign changes for each signal are memorized. The method then consists in calculating for each group (g) represented by its representative the quantity

with 16 g 4 6
The most likely group is then the group which maximizes the quantity a (g), the member of the group most likely emitted being the one which, using the permutation (n1, n2, n3, n4), is transformed into the representative of the said group and whose coordinates have the same signs as the elementary signals ri received.

For example, assuming that on reception a signal has respectively values
r1 = 0.1 r2 = -0.9 r3 = 0.3 r4 = -0.15
We will successively obtain, by applying the aforementioned process - the memorized signs
+1 -1 +1 -1 - the order of the permutation
2 3 4 1 - classification of absolute values
0.9 0.2 0.15 0.1 - and for each of the groups in the table in Figure 2 the following Lig values group 1 a (1) = 0.9 - 1 = - 0.1 group 2 h ( 2) = 1.25 - 3 = - 1.75 group 3 t (3) = 2 - 5 = - 3 groups 4 and 5 t (4.5) = 2.25 - 7 = - 4.75 group 6 t (6) = 2.35 - 9 = - 6.65 group 7 A (7) = 2.7 - 9 = - 6.3
It appears, in this example, that group 1 gives the best result. Its representative is the group defined by the combination
g1 = 1000
By applying to this representative the order 2, 3, 4, 1 of the permutations and by assigning to it the corresponding memorized signs, the transmitted signal Y = IO, -1.01.01 is obtained.

 It then suffices to apply this combination to a correspondence table to obtain the binary value of the corresponding transmitted signal in one-dimensional space.

 An example of implementation of the method according to the invention is shown in FIGS. 3 and 4 which represent, respectively, an apparatus allowing the modulation of binary signals and a demodulation apparatus, signals modulated by the apparatus in FIG. 3.

 The apparatus represented in FIG. 3 comprises a transcoder 1, a permutator 2, a memory of the representatives 3, a matrix operator 4, and a modulation device 5.

 The transcoder 1, possibly constituted by a programmable read only memory, performs the transcoding represented by the table of FIG. 2. It provides on three outputs designated respectively by la, lb and îc elementary signals representative of groups 1 to 7 of the table of the FIG. 1, the order of the permutation of each member of a group, and the signs to be assigned to each elementary signal of each member of a group as a function of the binary signals (I) coded on 8 bits, representing the information to be transmit.

 The group signals provided by the output la are applied to the addressing inputs of the representatives' memory 3 which contains the table of amplitudes (y1, y2, y3, y4) of the elementary signals of each group from the table in Figure 2 .

 These elementary signals are applied to the inputs of the permutator 2 and are permuted as a function of the order provided by the output lb of the transcoder 1. The permuted signals are applied to first operand inputs of multiplier circuits 6a, 6b, 6c and 6d whose second operand inputs respectively receive the sign bits supplied by the output lc of the decoder 1. The results of the multiplications are then applied to the inputs of the matrix operator 4, constituted for example in a known manner, by a set of multiplier circuits adders to perform the matrix product of the components y1 to y4 with the coefficients Uij of a matrix - - (U (UI of signals applied to the matrix operator 4.

 The choice of the coefficients uij of the matrix ICI U U depends on the type of modulation implemented in the modulation device 5 and on the constraints imposed on the transmission, in particular those on the width of the emission spectrum. In the hypothesis or in the mode of transmission chosen each sample of the signal is decorrelated in noise and has the same variance, the coefficients Uu may constitute an orthonormal basis in the four-dimensional vector space of table 2. Otherwise, the coefficients Ulj will be those of a matrix of non-orthogonal vectors of this space.

In either case, the matrix operator 4 provides for
each modulated signal a signal formed by components read 1,
e2, e3, e4 which results from the matrix product
lUI (Y1. y2, y3, y4)
The components e1, e2, e3 and e4 are applied to the inputs of the modulation device 5 which ensures their multiplexing in a known manner, for example in frequency, in phase and / or in amplitude, before transmitting them to a demodulation apparatus, one of which exemplary embodiment is shown in FIG. 4.

 The demodulation apparatus shown in FIG. 4 comprises a demodulation device 7, a matrix operator 8, a sign and absolute value calculation device 9, a storage logic 10, a calculation device 11 of # (g) , a first representatives memory 12, an address counter 13, a group calculation logic 14, a second representatives memory 15, a permutator 16, a sign assignment logic 17 and a transcoding memory 18.

 The signals received by the demodulator device 7 are the frequency, phase or amplitude replica of each of the components e1 to e4 transmitted. Once demultiplexed, these signals are applied to the input of the matrix operator 8 to be decorrelated in noise. This operator is constituted in a known manner, by a set of multiplier and adder circuits, which carry out the product of the components of the signal demodulated by the demodulator 7 by the inverse matrix | U | 1 of the matrix | U |.

 The signals r1, r2, r3 and r4 decorrelated in noise are thus obtained at the output of the operator 7. These signals are applied to the inputs of the device for calculating signs and absolute value 9. The latter provides, on the one hand, the sign indicator signals (s1, S2, S3 and S4) accompanying each component (r1, r2, r3 and r4) and on the other hand, the corresponding absolute values | r1 |, | r2 | lr1ï, Ir2l, r3l and | r4 | . These absolute values are arranged in descending order by the storage logic 9 which provides on the one hand on the first outputs the order of the permutations n1, n2, n3 and n4 and on the other hand, on the second outputs the values absolute rows in the calculation unit 11.

 The calculation unit 11 calculates the value a (g) given by the relation (15). For this it is connected to a memory 12 optionally formed by a programmable read-only memory containing all of the 256 signals | y1 ... y4 | of the alphabet and for each of them the value E2 (g) of their energy. The 256 signals of memory 11 are addressed by an address counter 13.

The set of results a (g) obtained at the output of 1 calculation unit 11 is transmitted to the calculation logic 14 which determines the group for which the value of A (g) is maximum. This latter value is applied to the addressing inputs of the memory 15, possibly consisting of a programmable read-only memory or any equivalent device in which a table of group representatives is also registered. the representative's samples corresponding to the group selected by logic 14 are applied to the permutator 16. These are permuted in the order indicated by the signals n1, n2, n3 and n4 supplied by the storage logic 10. The permutation obtained is applied to the sign assignment logic 17 which attributes to each element of the permutation the corresponding sign provided by the calculation device 9.

 Each resulting signal supplied by the sign assignment logic 17 is applied to the addressing inputs of the transcoding memory 18 which reconstitutes the corresponding transmitted binary signal signal coded on 8 bits.

 The method and apparatuses for modulation and demodulation which have just been described in a four-dimensional space structure allow performance to be obtained which is superior to that which can be obtained with a one-dimensional space structure.

In the example of modulation and demodulation which has just been described, the number of bits conveyed by symbol is log2 M = 8, the minimum distance (dmin) between symbols represented by their component yi) is equal to 2 (dmin = distance between 110001 and 101001) and the average energy is worth

 The geometric measurement of performance is given by the ratio: = E2 / (d2min x log2M) or g = 0.42.

 Compared to a modulation of known type in a space with only one dimension of type 4FSK for example, where each symbol carries only log2m = 2 bits of information and or useful information can take only 4 values represented by the weights - 3, -1, +1 and +3, the minimum distance between symbols being equal to 2A and the average energy E2 being equal to SA2, and where the ratio obtained is equal to 0.625, it is clear that the proposed modulation improves by therefore performance of 0.675 times.

 Naturally, as the invention is not limited to the example of embodiment which has just been described in four-dimensional space, it will be appreciated that it will be possible to further improve these performances by extending the method described in multidimensional spaces. of any order greater than 4.

 It will also be appreciated that other embodiments of the invention different from the one just described are also possible, notably by changing the structure of the circuits used. By way of example, it will be possible to envisage carrying out the various calculations described above using a microprogrammed architecture implementing one or more microprocessors.

Claims (10)

 1. A method of multidimensional modulation and demodulation of binary signals characterized in that it consists in carrying out the modulation, in transcoding (1) each binary signal of the vector space in one dimension to obtain a signal representable in a vector space in N dimensions, each transcoded signal being represented with respect to a base of the N-dimensional vector space in the form of a series of elementary signals each formed by a term of a linear combination of the elements of the base, the set transcoded signals being classified (3) in this base in families of signals described by a representative, all the signals of the same family having the same energy level, and in multiplexing the elementary signals obtained in each family with a view to their transmission (5).  2. Method according to claim 1, characterized in that it consists, in carrying out the demodulation of the transcoded signal received, in identifying (11, 12, 13, 14) his family by comparing his energy with those of the representatives, in identifying ( 9) the sign and the amplitude of each elementary signal received and to decode (18) the signal received in one-dimensional vector space by a decoding table addressed by the representative of the family which has the closest energy from that of the received signal and by the permutation characteristics of sign and amplitude of each of the elementary signals received.  3. Method according to any one of claims 1 and 2, characterized in that the vector space is a four-dimensional space formed of 256 signals, each signal being represented by a set of four samples.  4. Method according to claim 3, characterized in that the distribution of the 256 signals is defined by the points of the four-dimensional space whose coordinates correspond to half-integers of even sum.  5. Method according to claim 4, characterized in that the distribution of the 256 signals is carried out in 7 groups having a determined energy.  6. Method according to claim 5, characterized in that the coordinates of the points of one of the groups form in the vector space vectors orthogonal or opposite to each other, four of them forming a Walsh sequence.  7. Method according to claim 6, characterized in that the set of Walsh sequences is used as the basis for the set of 256 signals of the four-dimensional space.  8. Method according to any one of claims 1 to 7, characterized in that each signal sample is decorrelated in noise (4, 8).  9. Modulation apparatus for implementing the method according to any one of claims 1 to 8, characterized in that it comprises a transcoder 1 for transcoding each binary signal to be transmitted in the form of an identification signal of the group to which the signal belongs, of an order of permutation of its components with respect to that of the representative, and of a series of signs to be assigned to each component, a memory of representatives (3) addressed by the identification signal group, a permutator (2) to permute the components of the representative identified in the representatives' memory as a function of the permutation order provided by the transcoder (1), a set of multiplier circuits (6a, 6b, 6c, 6d) to apply to each signal component supplied by the permutator (2) a corresponding sign supplied by the transcoder (1), a modulation device (5) for transmitting each resulting signal supplied by the set of multiplier circuits to d estination of a demodulation apparatus, the modulation device (5) being connected to the set of multiplier circuits (6a, 6b, 6c, 6d) through a matrix operator (4) to decorrelate the transmitted signals in noise by the modulator (5).  10. Demodulation apparatus for implementing the above method and intended to be coupled to the modulation apparatus according to claim 9, characterized in that it comprises a demodulation device (7), a matrix operator (8) to decorrelate in noise the components of each signal supplied by the demodulation device (7), a device for calculating signs and absolute values (9) of each demodulated signal component, a logic for storing (10) the absolute values of the components of the demodulated signal in decreasing order, a calculating member (11) coupled to a logic for calculating the group to which the received signal belongs to calculate from the energies of the set of signals the most favorable group corresponding to l energy of the signal received, a memory of the representatives (15) to designate a representative corresponding to the most favorable group found, a permutator (16) controlled by the storage logic (10) to restore the order of the sign al received from the representative obtained from the memory of the representatives (15), a sign assignment logic (17) for assigning a sign to each component of the signal obtained at the output of the permutator (17) as a function of the signs provided by the sign calculation logic (9), and a transcoding memory (18) for reconstructing each signal transmitted as a function of the signal supplied by the sign assignment logic (17).