ES2433695B1 - METHOD OF SPHERICAL DECODIFICATION OF SIGNS OF COMMUNICATION SYSTEMS OF MULTIPLE INPUTS AND MULTIPLE OUTPUTS (MIMO) OF MAXIMUM VEROSIMILITY. - Google Patents

Abstract

Spherical decoding method of signals from multiple input and multiple output communication systems (MIMO) of maximum likelihood. # In which a box minimization method is applied that accelerates spherical decoding and comprises a preprocessing stage and A search stage. It provides a new feasibility test included in the search stage, in which scr {sub, i} signals are obtained that when they are not in the box obtains the sck {sub, i} signal, which reduces and simplifies the number of minimizations, reducing the computation time and therefore the temporary cost of decoding. # Also use these scr {sub, i} and / or sck {sub, i} signals to select an initial signal x0 that is applied in the method of cash minimization, which also minimizes the temporary cost of cash minimizations.

Description

OBJECT OF THE INVENTION

The present invention relates to a method of spherical decoding of signals of communication systems of multiple inputs and multiple outputs (MIMO) of maximum likelihood, in which a method of dimensions by box minimization is applied which accelerates spherical decoding; and which aims to reduce the number of cash minimizations and simplify each of said minimizations, reducing the computation time and therefore the temporary cost of decoding.

In general, the invention is applicable in the telecommunications industry and more particularly in wireless communications. In addition the invention is applicable in the field of cryptography.

BACKGROUND OF THE INVENTION

In the last decade, one of the most significant technological developments that led to the new generation of wireless broadband is communication through multi-input and multi-output systems (multiple-input multiple-output, MIMO). The recent interest aroused by MIMO technologies is not only limited to the world of research but has also had a dramatic impact on the wireless communications industry, proof of this is the adoption of MIMO techniques in many wireless standards such as L TE, WiMAX and WLAN. MIMO systems increase maximum transmission rates and improve the reliability and coverage of current wireless communications without the need to use additional bandwidth or transmission power. This is undoubtedly a huge advantage, since spectral resources are very scarce and expensive. On the other hand, keeping the transmission power as low as possible is a crucial factor in the life of the battery of wireless communication devices and therefore we must try to minimize the consumption of the stages that make MIMO communication possible. In short, for all the reasons described above and for many others, MIMO is a field that attracts a large part of the highest level of international research.

Figure 1 shows the block diagram of a lipid MIMO communications system composed of a transmitter 1 equipped with M antennas for sending signals to a receiver 2 equipped with N antennas, through a MIMO channel 3 that generates a mixture of the transmitted signals, which is due to the appearance of multiple paths between each pair of transmitter-receiver antennas. The number of transmitting antennas is never greater than that of receiving antennas, that is, N 2 :: M. The receiver comprises a delector that processes the mix of received signals and estimates what the transmitted data were, generally following some probabilistic rule, obtaining the signal. decoded. It has been shown that the use of MIMO communication systems complicates the receiver and, especially, the data detection stage. Therefore, it is very important to get low complexity MIMO receivers that maintain a low detection error rate.

There are different types of detectors, with different error rate and complex complexity properties. The so-called "suboptimal" detectors are computationally efficient, but their error rate may not be acceptable, especially in the presence of a lot of transmission noise.

The detectors with the lowest error rate are the so-called optimal or maximum likelihood detectors, which in the decoding obtain the signal with a higher probability of being sent. The best known maximum likelihood detectors are those based on spherical decoding, which is described below to facilitate understanding of the invention.

The maximum likelihood detectors in general, and those of spherical decoding in particular, are very expensive computationally, that is, the computation time they require is high, especially when the noise in the communications channel is considerable. In addition, the cost of detecting a signal by these methods is quite unpredictable, depending on the noise of the channel. The invention described below is a modification of the spherical deoodification method aimed at reducing the cost of decoding calculation, and which has the additional effect that the detection complexity of a signal is practically independent of noise.

DESCRIPTION OF THE INVENTION

In order to achieve the objectives and solve the aforementioned drawbacks, the invention provides a new method of spherical decoding of signals from multiple input communication systems and

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multiple outputs (MIMO) of maximum likelihood, in which there is a transmitter with a plurality of

transmitting antennas M for sending signals through a MIMO channel to a plurality of antennas N of a spherical decoding receiver in which a method of dimensions by continuous minimization that accelerates spherical decoding is applied; and where it is known that the receiver's input data is:

-

a HE RN "M channel matrix, with N rows and M columns, that is the number of rows is equal to the number of receiving antennas and the number of columns is equal to the number of transmitting antennas, where the number of receiving antennas is greater than the number of transmitting antennas N ~ M.

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the constellation: D = (d1 • .. • dL} set of L symbols, all of them in IR.

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the received signal and € IRN;

From said input data, the receiver obtains the maximum likelihood signal, that is the decoded signal, which is selected from among all the M component signals that fulfill each of the M components of the signal belonging to the constellation D. The maximum likelihood signal means that one of these M component signals that, after being filtered by the H-channel matrix, has the minimum distance to the received signal and.

The conventional spherical decoding method comprises a preprocessing stage and a search stage.

In the preprocessing stage a new RE IRM "MY channel matrix is obtained, in addition a new received signal z E IR M is obtained, by orthogonal transfonation of the H channel matrix and the received signal y; where the new R channel matrix it is triangular superior by means of which the search stage is simplified to the pennite to organize the search sequentially.

Furthermore, in the preprocessing stage a first auxiliary signal sck is obtained, by solving a conventional triangular equation system described below to facilitate the understanding of the invention. Then it is verified if the condition that said first auxiliary signal sck has all its components within a box defined by the maximum and minimum values of the constellation D is fulfilled, so that if it is verified that said condition is met, then obtains an initial radius whose value is the distance between the sck signal after being quantized and filtered by the matrix R, and the received signal z. On the contrary, if this condition is not checked, then in the preprocessing stage an initial signal xO is selected which is used to calculate a second auxiliary signal, by solving a minimization problem with conventional restrictions, and in addition a initial radius as the distance between the signal being after being quantized and filtered by the matrix R, and the signal received z.

In the search stage it is known that in the first instance the square of an initial radius is established as the smallest total distance obtained so far, and then partial signals are listed (5;, 5; +1, "', 5M_I' 5101) And complete signals (51.52, ... • 5101_1, 5101), on which a feasibility test is applied, to decide whether each of the partial signals listed is feasible ° no, and when a certain signal partial (5;, 5; +1, ···. 5101_1, 5101) is feasible, the following signals, selected from partial signals and complete signals, called successors, of said partial signal, and when a certain partial signal are still listed (5;, 51+ 1 '"' '5101_1' 5101) it is not feasible to be discarded and no partial signal or complete signal is listed, successor to said partial signal.

Each time in the enumeration a complete signal is obtained that, after being filtered by the R channel matrix, its distance to the received signal z is less than the smallest total distance obtained so far, said complete signal is saved as a candidate for maximum likelihood solution, and the lowest total distance obtained so far becomes that distance to the received signal z. When the enumeration ends, the resulting candidate signal is the maximum likelihood signal corresponding to the decoded signal.

In the feasibility test, where it is decided whether a partial signal (51.51 + 1, "', 5101_1' sM) is feasible or not, it is first checked whether the partial distance of said partial signal is less than the smallest total distance obtained so far In the case of not checking, that is, when the partial distance of the partial signal is not less than the smallest distance obtained from the received signal, then said partial signal is established as not feasible. On the contrary, if said check is made, that is, when the partial distance of the partial signal is less than the smallest distance obtained from the received signal, then an initial signal xO is used which is used as the initial signal to apply a minimization method with conventional restrictions, whereby the second auxiliary signal is obtained will be associated with the partial signal (51.51 + 1, '", 5101_1' 5101) 'Next, said second auxiliary signal serl p is used To calculate a continuous and conventional minimization dimension, and the c-value is added to the partial distance of the partial signal (5;, 5i + I. · ", SM_l. SM) 'Then it is verified if said sum is greater than the smallest total distance obtained so far in which case the partial signal is declared not feasible; and if the sum is less than the smallest distance obtained so far, the partial signal is declared feasible.

Basically, the invention brings two improvements to spherical decoding. Making the second improvement requires having made the first improvement.

This first improvement is characterized in that after applying the feasibility test and verifying that a partial signal (S, .51 + 1. · .., 5M_I. S / <1) is feasible, since its partial distance is less than the smallest Total distance obtained so far, as described, comprises:

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calculate the first auxiliary signal sek / = (S] .S2 '... S / _ I) attached to the partial signal (sr, SI + I' ". SM_I. 514).

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verify if the first auxiliary signal sek, associated with the partial signal (s; .s / + 1 ... ·, SM_], SM) has all its components within the box defined by the maximum and minimum values of the constellation,

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when said verification is verified. that is, when the condition is given that all the components of the first auxiliary signal 5ek, are inside the box. then the pardal signal (5 "S '+ I ... ·. S"' _ I 's ",) is declared feasible;

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on the contrary when this verification is not verified. that is, there is no prior condition that all the components of the first auxiliary signal sek¡ are inside the box. Then the second auxiliary signal 5er¡ associated with the partial signal (S¡'S¡ + I ... ·. s ", _¡, S / oI) is calculated, and then the continuous minimization value e is calculated and it is decided via the feasibility test, if the partial signal (S /, Sl + l "", SM_I 'SM) is feasible or not.

Therefore the described method limits the number of minimizations in cash. using the position of the first auxiliary signal 5ek¡ relative to the box defined by the constellation. so that it is not always necessary to calculate the second auxiliary signal to be /. Consequently, with this first improvement proposal, the number of cash minimizations made in the state of the art is reduced.

The second improvement proposed by the invention is characterized in that after checking the feasibility of a partial signal (s "S '+ 1" ", S"' _1 's ") for an ith component with ¡<M. that verifies that its partial distance is less than the smallest total distance obtained so far, and for which the first auxiliary signal sek, associated with said partial signal has any of its components outside the box defined by the maximum and minimum values of The constellation, for which the second auxiliary signal has to be calculated, will be associated with the partial signal (S¡'S¡ + I. · ... S "'_I' S / oI). according to the method of the first improvement. and the continuous minimization level must be calculated e. It also includes:

-

Obtain the initial signal xO for the continuous minimization method with restrictions. as the first i -1 components of the second auxiliary signal Ser1 + 1 = (s], s ~, "', S, _I) when checking the feasibility of a predecessor partial signal (s / + 1 ... ·. S "'_ I' s",) of a partial signal (s ,, ",, S / oI_1'S",) the second auxiliary signal was calculated to be '+ 1' "(SI 'Sl' '' ', Si) associated to the partial signal (S¡ + I ... ·. S "'_ I' 5/01). and

Obtain the initial signal xO for the continuous minimization method with restrictions. as the first i-1 components of the first auxiliary signal sekl + l: (5], 52. ···. S / _I). when, when checking the feasibility of the predecessor partial signal (SI + I '..., s "' _ ¡. s",) the second auxiliary signal seri + l 'was not calculated and therefore the first auxiliary signal sek¡ + was calculated I '"(51.52 ..... Yes) associated with the partial signal (S¡ + I ... ·. S"' _ I '5/01)'

Therefore, through this second proposal, the initial signal is selected for the box minimization method. using the auxiliary signals be / and / or sek¡ obtained when examining the feasibility of the predecessor partial signal. which minimizes the temporary cost of each of the cash minimizations.

Additionally the invention is characterized in that:

-

The preprocessing step comprises storing the first 5ek auxiliary signal. and store the second auxiliary signal be when it has been calculated.

use the first auxiliary signal sek and the second auxiliary signal to be stored. in the possible minimizations that are applied to verify the feasibility of partial signals of the form (S / oI) with

i = M,

-

after checking the feasibility of a partial signal (SM) with i = M • verifying that its partial distance is

less than the lowest total distance obtained so far, and for which the first auxiliary signal sck M associated with said partial signal has some of its components outside the box defined by the maximum and minimum constellation values, so It is necessary to calculate the second auxiliary signal ser ", associated with the partial signal (SM) and the continuous minimization dimension c, also includes:

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obtain the initial signal xO for the method of continuous minimization with restrictions, such as the first M-1 components of being, when the auxiliary signal was calculated in the preprocessing stage, and

obtaining the initial signal xO for the method of continuous minimization with restrictions, such as the first M -1 components of sck, when the second auxiliary signal was not calculated in the preprocessing stage and only the first auxiliary sck signal was calculated.

This additional characterization allows to complement the advantages described for the first and second improvement, obtaining a more efficient method, in which the temporal cost of spherical decoding is practically independent of noise.

It should also be noted that the proposed method can be applied simultaneously (and with cumulative improvements) together with other well-known optimizations for spherical decoding (Schnorr-Euchner sorting, column array ordering, etc.)

It is common for the signal to be sent in the form of complex numbers, and for decoding methods to be written directly in complex form. The proposed method can also be adapted to spherical decoding methods for complex data; On the other hand, it is well known that the problem of complex decoding can be written as a problem in terms of real numbers.

In addition, it has been proven that the algorithm for box minimization proposed in the book Ake Bjorck, "Numerical Methods for Least Squares Problems ·, SIAM, Philadelphia, 1996 section 5.2, page 203, is perfectly adapted to solve the problems of continuous minimization of the invention, although other algorithms could also be used.

In general, spherical signal decoding methods also have application in the field of cryptography; The proposed method can also be used in that field.

In order to facilitate a better understanding of this descriptive report and forming an integral part thereof, a series of figures are attached in which the object of the invention has been shown as an illustrative and non-limiting nature.

BRIEF STATEMENT OF THE FIGURES

Figure 1.- Shows a conventional generic block diagram of a MIMO communications system with M transmitting antennas and N receiving antennas, which has been described in the background section of the invention.

Figure 2.- Shows a conventional flow chart of the preprocessing stage of spherical decoding.

Figure 3.- Shows a conventional flow chart of an improvement of the preprocessing stage of spherical decoding.

Figure 4.- Shows a conventional Hujo diagram of the search stage of spherical decoding.

Figure 5.- Shows a conventional flow chart of the feasibility test that is included in the search stage of the previous figure.

Figure 6.- Shows a flow chart of the improved conventional preprocessing stage with respect to the preprocessing of Figure 2, with respect to which it also includes a technique for obtaining a good approximation of the radius.

Figure 7.- Shows a conventional flow chart of the feasibility test of the partial signals with a continuous minimization level that replaces and improves the test of Figure 5.

Figure 8.- Shows a flow chart of the new preprocessing stage according to the invention.

Figure 9.- Shows a flow chart of the new feasibility test that includes the search stage according to the invention.

DESCRIPTION OF THE PREFERRED FORM OF REALlZAC IÓN

Below is a description of the invention based on the figures mentioned above.

First, to facilitate the understanding of the invention, the mathematical representation of the problem of signal decoding in conventional MIMO systems is described, which is shown in Figure 1 and which was described in the background section of the invention.

10 The transmission of data from the M transmitting antennas to the N receiving antennas is mathematically modeled by a HE IRN matrix> <M, with N rows, corresponding to the number of receiving antennas and M columns, corresponding to the number of transmitting antennas, being N 2: M, called the channel matrix, and is known to the receiver. The sending of an M-dimensional signal s = (SI. ··· .SM _ I.SM) through the channel is simulated as the product of the channel matrix by the signal sent: H · s and IRN. Thus. he

The result of a signal s being filtered by the channel matrix is the signal H. s E RN.

When a s signal is sent through the channel. It is not only filtered by the channel matrix. It is transformed by noise. Due. the received signal and E RN could be written as y = H. s + v; where v E IRN is the noise (unknown).

Further. it is known that the components (SI '··· .SM _ l'SM) of the signal sent s E RM belong to a

20 discreet set. finite D = {di '.... dd called constellation or alphabet. The elements of the constellation D are called symbols and therefore D is formed by the symbols. For example, the constellation 4-PAM (pulse amplitude modulation) is the set {-3. -1, 1.3}. represented in figure 3. which will be described in greater detail later.

The goal of the problem is to find the maximum likelihood solution ("Maximum Likelihood Solution", or

25 ML solution hereafter). This is the signal s = (SI "" .SM _ l, SM) E KM that verifies that the distance between said signal. filtered by the channel matrix H (H · s E to N) and the received signal and E IRN is the minimum. from among all the possible signals composed of constellation symbols D. Mathematically it is expressed as s = argminsllH. S -yll. where s (i) E D. i = l ..... M.

It should be noted that any spherical decoding algorithm consists of two stages: a preprocessing stage. followed by a search stage.

The conventional basic version of spherical decoding for obtaining the ML signal is described below. whose preprocessing stage is shown in the flowchart of Figure 2 (d11) and comprises.

1) (d111) Calculate the decomposition Q R of the H-channel matrix. In this process, two matrices QE gNxN and R E are calculated to N XM so that their matrix product Q. R is equal to H; where Q is a matrix

35 orthogonal (which verifies that the product of Q by its transposition QT is the identity; Q. QT = f. And therefore the inverse of Q is its transposition QT) and R is a superior triangular matrix. The new R E IRMXM • channel matrix is a superior triangular matrix. with M rows and M columns. it is obtained as the first M rows of R.

2) (d112) The signal received and is projected on the matrix R. This is done by multiplying QT by y. Selecting the first M components of eQT. and) we obtain the new received signal. That

40 called Z E RM. Therefore the problem s = argminsIlH · s-yll becomes the mathematically equivalent problem (with the same solution) s = aTg minsllR. s -Z1l2. where s (i) E D. i = 1, ..., M

Next, the search stage that is represented in Figure 4 (d12) and Figure 5 (d18) is performed.

The upper triangular structure of the new R-channel matrix allows you to organize the search sequentially, first looking for a value for the M-th component of the sent signal, using the M-th

45 row of matrix R; then a value is searched for the M-1-th signal component. and so on. The selection of values for the ith component of the ML solution signal. It is said to be done in the ith layer.

The operation R. s -z can be written in detail as:

You

T2.2 r2.3 T2.M

O r 3.3 (ec.1)

T ~: ~.

C 'e'M) (Ji) {i) O O O TM • M

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Then, the distance IIR · s -zllz, where s = (S¡, 5Z · ··. 5M_l'5M) is written in detail as

IIR .s -zll2 = distance (st, .., 51> 1_1 '5M)

= (rl.!. 51 + TU. 5z + rl, 3. 53 + ... + TI.M. SM -z¡) Z + (rZ.2. SZ + rZ.3. 53 + ... + TU_ '. SM -Z2) 2

+

(T3.3. 53 + ". + T3.M. SM -Z3) 2 +.,. + (TM_l.M_¡. 5101 _1 + TM _ l.M. SM -2101_1) 2

+

(rM.M · SM -ZM) 2

(ec.2)

The ML signal is searched among the serials received in the form s = (51 ..... 5.11 _1 .5.11) where the Si values

they belong to constellation D, discarding those whose distance to the received signal is greater than an R2 R2

certain radius; This initial radius must be chosen so that there is at least one signal s = (s¡, "·, SM_I'SM), cons (i) ED such that distance (s¡, ·· ·, 5101 _1, 5, \ 1) S r 2.

To properly describe the search in spherical decoding, a partial signal in the ith layer can be defined as a signal of M -i + 1 elements of the constellation, already selected for the components of the possible sent signal, from the layer I-th to M-th: (SI, Si + I. ···. SM_I, SM) 'This partial signal has a partial distance associated, which is calculated as:

partial distance (5¡.5¡ + 1 '". 5M_1 • 5M)

= (ru. 5¡ + rU + I. 5i + 1 + ... + ri.M. 5M -Z¡) ~ + (r¡ + U + I. 5¡ + 1 + ... + r¡ + 1.M. 5M -Zi + I) ~ +.

+ (rM_I.M_I · 5M_1 + rM_I.M · 5M -ZM _I) 2 + (rM.M · 5M -ZM) 2

(ec.3)

In contrast to the partial signal concept, it is useful to define the complete signals; these are the signals that have their M components defined; they are of the form s = (s¡ '· ··. SM_I'SM) and, as mentioned above, the ML signal is searched among the complete signals.

Clearly, any possible complete signal 5 = (51'52 .. ··, 51_1'5¡, 51 + 1 .. ·· '5M_I, 5M) whose components from i to M are the partial signal (5i.5i + l. ·· ·. 5M_I.5M) will comply that its distance to the received signal IIR · 5 -zll ~ = di5taTlcia (51.52 ... · .5i_1 .5¡, 5i + I, .. · • 5M_ I. 5M) is greater than the partial distance associated with said partial signal.

From the previous 10 it follows that if a partial signal (SI.S¡ + I ... ·, SM_l .SM) has an associated partial distance greater than the radius r 2 then said partial signal can be discarded, since any complete signal 5 obtained from said partial signal will have an associated total distance greater than the radius r 2.

In addition to facilitate the understanding of the search stage it is useful to define the predecessor signal of a given as follows; given a partial signal in layer i (i <M) (5,, 5, + 1. ·· ·, 5M _ 1.5M), its predecessor signal is the signal (5¡ + 1 ... ·, 5M_1. 5M), in layer i +1. The concept is also valid for complete signals: the predecessor signal of a complete signal (SI'S2 ... · .SM_I.SM) is the partial signal

(S2, "', SM_I, SM)'

Similarly, the successor signals of a given signal are defined as follows; given a partial signal in layer i (1) 1) (SI, SI + I, .. · • SM _I, SM), its successor signals are the possible L signals of i + 1 components, of fonna (SI_l, S ¡, S / + 1 "", SM_l, SM) And dondesl_l is one of the elements of the constellation.

The search stage of any spherical decoding algorithm has two stages: A) enumeration of the possible partial and complete signals, and B) feasibility test of each partial signal.

A) The enumeration of the partial signals is done sequentially by the different ~ dimensions'

or layers of signal 5, from the last (the M-th) to the first. The diagram in Figure 3 aims, by means of a simple example, to give an idea of how this enumeration proceeds.

Figure 3 shows how the enumeration of the possible signals in a case with M = 5 and constellation D = {- 3, -1, 1,3) would be. In this example, some partial signals have been designated arbitrarily as feasible and others as not feasible, to explain how the enumeration proceeds depending on whether the partial signals found are feasible or not. Instead of the arbitrariness mentioned, the feasibility test must be applied as will be described later.

The enumeration begins with the j = M layer (M = 5 in the example), and considers all possible constellation values for the M position of the signal. Each of the elements of the constellation constitutes a partial signal of level M. To each partial signal of level M the feasibility test is applied, and those that are feasible are still explored, generating their successor signals of level M -1; the feasibility of each of these is checked, and again only those that are proven feasible are still being explored (generating successors).

The partial signals considered in layer 5 are then (-3), (-1 l, (1) AND (3) Each of these partial signals is subjected to the feasibility test. In this example, we assume that the partial signals ( 3) and (1) are not feasible (partial signals not feasible are slightly obscured), while partial signals (-3) and (-1) are feasible. First, the partial signal (- 3), moving on to the next layer (i = 4).

In layer 4, the partial signals having the value already assigned (-3) in the fifth component are considered, and all possible constellation values for the fourth component are considered: These partial signals are: (-3, - 3), (-1, -3), (1, -3), Y (3, -3), and to all of them the feasibility test would be applied. Of these, only the partial signal (-1, -3) is considered (arbitrarily in the example) feasible, and it is the only one that is passed to the next layer, i = 3.

The method continues to generate partial signals until it reaches layer i = 1. In this layer the signals are no longer considered "partial", but "complete" because they already have the desired number of components (M = 5). In this last level, the feasibility test is not applied, but the distance to the received signal is calculated (calculating IIR · 5-zlll and that complete signal having a minimum distance IIR.5 -zll between those is saved as ML solution. that have reached the last layer (i = 1).

After finishing the four "complete" signals, successors of the partial signal (1,1, -1, -3) return to the previous layer to complete the enumeration of the successor signals of the signal (1, -1, -3).

The superindice of each partial and complete signal indicates the order in which the partial and complete signals are traversed in the search method of Figure 4 (d12), assuming that the elements of the constellation are traversed in the natural order (from less than higher).

Note, for example, that the signal predecessor of the signal in layer 4 (3, -1) is the signal in layer 5 (1); or, similarly, the signal predecessor signal (-3,1,1, -1, -3) is the signal (1,1, -1, -3). Note also that a partial signal (of level i <M) or a complete signal is examined only if its predecessor signal is feasible.

Note that the 4 successor signals of the partial signal (-1, -3) are (-3, -1, -3), (-1, -1, -3), (1, -1, -3) Y (3, -1, -3).

As can be seen in the example, in the i-th layer partial signals are considered in which their components from i +1 to M (Si + l '···' 5M_¡, su) have values already assigned or selected in the layers i + 1 to M, and in the ith layer the different elements of the constellation are tested for the ith component of the partial signal. We will distinguish the components with already assigned value from those with value to be assigned using a top bar, that is, if the ith component of the signal s has an assigned value, we denote it as Sj, while if it still does not have an assigned value we denote it as 5/.

The method is detailed in the flowchart of Figure 4 (d12), which is described in detail below:

1-lnicially (d121) the layer counter i is initialized with the initial value M (to start processing the layer M); the minor variable _dist_obtained with the value r2 is initialized.

2-The feasible partial signals and the complete signals whose distances are less than the minor variable _disclosed are traversed in a tree. When all partial and total signals have been scanned; that is to say i is equal to M (d122) the method tries to access the layer M + 1, which does not exist; This indicates the end of the method (d123).

3-For the first layer to be examined (i = M), we start searching among the elements of the constellation by starting the constellation enumeration (d124), verifying if the enumeration has been completed (d125), and in case of not being complete a new constellation element is selected (d127). In this first layer, the elements of constellation 5101 are selected those that verify that the partial signal (5101) is feasible, test (d131).

4-If said partial signal (~ · M) is not feasible (d131 l, it means that this partial signal should not be further explored and discarded, so it is returned to (d125) and, if there are elements of the constellation to explore, the next item is selected.

5-If said partial signal (5101) is feasible, then said current partial signal will continue to be scanned, setting

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the element SM = 5M. And preparing the exploration of the new layer: the partial distance is saved and the layer counter i = i -1 (d132) is decremented.

6-Next, we begin to examine the layer i = M -1; (d124, d125, d127) of the same form that was performed for the first layer. At this moment, the elements 5.11_1 belonging to the constellation will be selected and that verify that the partial signal (5.11_1.5 ",) is feasible again, test (d131).

7-If the partial signal (5 "'_ 1,5",) is not feasible, this partial signal can be discarded, and the next element of the constellation for the M-l layer is scanned.

S-If the partial signal (5.11_1.5 ",) is feasible, then the current partial signal will continue to be scanned, setting the element 5M_1 = 5.11_ 1 And the layer counter is decremented (i = i - 1) (d132).

9-If given the selected element S '"all possible values for S"' _ 1 are checked and none of the partial signals (S "'_ I'S") is feasible. that means that no feasible complete signal can be obtained from the partial signal (s ",), in this case, after examining the last element of the constellation for layer i = M -1. the choice in (d125) is positive . the layer counter is incremented (i = i + 1). (d126) returning to layer M, the element s "is discarded, and a new value is selected for

s ", (d127).

10-This process of enumerating candidate partial signals is repeated for all layers until (and including) the last layer (i = 1). As in the previous layers, the layer i = 1 is now traversed; in the layer i = 1 partial signals are no longer obtained. but complete signals are obtained.

11 -If in layer i = 1, for some element of the constellation IF it is verified that the distance of the complete signal obtained is less than the best distance obtained so far:

(d130). then YES = YES is selected and the signal (Sl. ··· .S "'_ I.S"') becomes the best candidate signal to be ML solution. of those found so far. The value of minor_disCobtenida is updated. taking as a new distance value (51 .. ••, 5 "'_ 1.5",). (d12a)

12-The process continues with the enumeration and will end when there is no possible signal within the sphere of minor radius_disclosed; at that moment. The optimal solution obtained is the ML solution. Note that the variable minor_disCobtenida decreases each time a better solution is found.

The order in which the elements of the constellation are traversed (order in which each new element of the constellation is selected. In (d127) influences the performance of the method. In the previous example the most common ordering is described which according to the Description consists in going through the elements of the constellation from minor to major according to a tree configuration, and which is known as Fincke-Pohst ordination, there are also other ordinations, such as Schnorr-Euchner ordination is known to be more efficient.

The invention proposed below works equally beneficial for any arrangement of the constellation elements.

It is also possible to change the way partial signals are traversed; the search stage (d12) runs through the tree of partial and complete signals looking first in depth and then in width. There are other ways to travel the tree, first in width and then in depth. The invention proposed below can be applied equally beneficial to any of these tree travel techniques.

B) Feasibility test of partial signals

In the basic spherical decoding algorithm. The feasibility test of a partial signal consists only of checking whether the partial distance associated with said partial signal is less than the smallest distance obtained so far. Further. If a partial signal is feasible. in the feasibility test the necessary calculations (convenient) are carried out to continue exploring this signal l.

In the case of the basic algorithm. These calculations consist of that. each time a certain element is selected in layer i.If it is computationally convenient to correct the received signal by canceling the effect of the selected element Yes, on said received signal z. This effect is canceled by subtracting the ith column of the matrix R multiplied by the selected element SI from the z signal. For this, a corrected signal will be used for each layer we call zaux¡. In the first layer (i = M). For a certain selected element S "', zaux", it is a signal of M -1 components that is calculated as:

zauxM (1) = z (l) -TI, M 'SM;

zauxJIj (2) = z (2) -T2.M · SM;

(ec.4)

zaux ", (M -1) = z (M -1) -TM_1, M 's",

In the next layer (i = M -1), with SM already selected, a new element of the M1 layer would be selected, be it 5 "'_ 1' To obtain zaux", _ !. the M -2 component signal corrected after canceling the effects of SM and SM_l. The following operation is done:

zaUXM_1 (1) = zauxM (l) -T1, M_l. 5111_ 1;

zauXM_l (2) = zauxM (2) -T2.M-l. 514 _1:

5 (ec.S)

zauxM_¡ (M -2) = zaux ", (M -2) -rM-2, M-l. 514 _1

In general, suppose the elements have already been selected (5i - + - 1. ··· '.5M _ 1.5M), the corresponding corrected signal zaux¡ + lY in the ¡-th layer the element Si 'is selected Then the signal corrected zaux¡, of i -1 components, was calculated as follows: zaux¡ (l) = zauxi - + - I (1) -r l.1. Yes; zaux¡ (2) = zaux¡ + 1 (2) -r2, í. Yes;

(ec.6)

zaux¡ (i -1) = zaUX / + l (i -1) -r / _l./. yes

10 Equivalently it can be verified that zaux¡ can be written as: zaux¡ (1) = z (1) -ru 's¡ -rU + I. S¡ + I - ". -Rl.M · SM; zaux¡ (2) = z (2) -r ~ .1. SI -r ~ .1 + 1. SI + I -... -r ~. M. SM;

(ec.7)

zaux¡ (i -1) = z (i -1) -r¡_u 's¡ -r¡_u + I' S¡ + I - ". -r¡_1.M · SM

Given a partial signal in the i + 1 layer (S¡ + I '".S,., _ I, SM), if the partial distance corresponding to this partial signal is calculated and stored, it is possible to easily calculate the distance partial corresponding to any partial signal of level i obtained by adding the element yes to the partial signal

15 (s¡ + 1, '", SM _I, SM) as follows:

With the help of figure 5 (d18) the commented feasibility test is detailed.

As mentioned above, in this case the partial distance of the partial signal to be checked is checked (d181); If the partial distance is greater than the smallest distance obtained so far, the signal 20 is not feasible (d182).

If the partial distance is less than the smallest distance obtained so far, the zuux signal (d183) is updated, as shown in equation 6, and the partial signal is feasible (d184).

Furthermore, the use of spherical accelerated decoding methods by continuous minimization is known in the state of the art.

25 In this regard we need to consider now two conventional associated auxiliary problems, the problem of continuous minimization and the problem of continuous minimization restricted or "in cash".

The auxiliary problem of continuous minimization (PA1) aims to find the sck signal E KM which verifies that the distance between the R · scke IR signal "and the signal received ze KM is the minimum: sck = arg minsllR. S -zll where sck (i) E R. i = 1, ..., M

30 This is not a discrete problem, as in the case of basic spherical decoding, but it is a "continuous" problem (the components of sck can be any real number) and its solution is easily obtained by solving the system of triangular linear equations R. sck = z.

The auxiliary problem of continuous minimization in box (PA2) aims to find the signal to be RM that verifies that the distance between the signal R. be E IR '"And the signal z E KM is the minimum distance, and that in addition 35 verifies that the components of being are real numbers contained between the minimum and maximum constellation: ser = arg minsllR. s -zll; where min (D) <= scr (i) <= max (D) and scr (i) E IR, i = 1. n • M

•

This is also a continuous problem, but with the restriction that it must be contained in the box defined by the maximum and minimum values of the constellation. There are different methods to solve this problem. Being a relatively complex problem, for This is usually done by using computer programs already written for this purpose.These programs start from an initial xO signal and proceed by taking steps in which the initial solution moves towards the final solution.If an initial solution close to the chosen program is supplied to the chosen program final solution, these programs can be considerably faster.They can work without initial solution (in this case, the program selects an initial solution xO randomly), but the cost of calculation to obtain the solution will usually be much higher.

In the state of the art, a modification of the method of spherical accelerated decoding by elevation by continuous minimization is known; which we call modification 1. Said modification 1 to improve the performance of spherical decoding, aimed at obtaining a good

Initial radius estimation r, has been described in document (1) Xiao-Wen and Qing Han, ~ Solving Box Constrained Integer Least Squares Problems "IEEE Transactions on Wireless Communications, vol. 7, no. 1 (2008), 277-287 .

In this modification 1 for the search of Figure 4 (d12) to be successful, the initial radius r must be chosen so that there is at least one signal s = (s¡, ·· ·, s "" _¡, s ,,,,), with s (i) ED such that distanciu. (s¡, ···, SM_¡, SM) <r ~. If there was no signal verifying this condition, search (d12) would not find any solution.

On the other hand, for the search to conclude in a reasonable time, the initial radius must be as small as possible. A well known technique for selecting an initial radius is based on finding a signal that is a reasonable approximation to the ML solution and taking its distance from the signal received as the initial radius.

It is well known that the solution of the continuous auxiliary problem described (PA1) sck can be used r to obtain a possible signal, rounding (quantizing) each component of the sck signal to the closest value of the constellation D. The obtained signal is called zf ("zero-forcing"), which is one of the possible signals sent. Since it is possible that zf is close to the ML solution, and it is certainly a possible feasible signal, the initial value r (for the search algorithm) is usually used as the IIR value .zf -'1.11. This radio is sure that a signal is within the sphere (the possible zf solution).

This technique to obtain an initial radius is performed at the preprocessing stage, obtaining a new preprocessing method that includes the one described with the help of Figure 2 (d11) and also includes the technique described below with the help of the figure 6 (d21) to obtain a good approximation of the radius.

Thus, the preprocessing stage shown in the flowchart of Figure 6 (d21), contains the stages (d11 and d112) described in Figure 2, which have been referred to as (d211 and d212).

Next, sck is obtained by solving a system of triangular equations Rxsck = z (d213) (PA1).

However, in situations with a lot of noise, this radius estimate produces unsatisfactory results. An important feature of these situations is that the sck signal is "far" from any possible s signal, which can be detected by examining whether the components of the sck signal are contained in the range defined by the minimum values (mO) and maximum (MD) of the constellation.

To clearly define this situation, we will say that a signal s is in the box if all its components sU) are contained in the interval [mD, MD], and we will say that it is out of the box if any of its components s (i) has a value outside the range [mD, MD] (d214).

If the sck solution of the auxiliary problem of continuous minimization (PA1) is out of the box, then the initial radius is obtained using zf and by quantifying (rounding) sck the radius r is obtained as r = IIR · zf -zll (d215).

If the sck solution of the auxiliary problem of continuous minimization (PA1) is out of the box, then the solution is of the auxiliary problem of continuous minimization in box (PA2) (which, by definition, is inside the box) is often a better approach to the ML solution that the sck signal.

Consequently, if sck is out of the box, instead of calculating the initial radius using zf, the initial radius will be calculated using the following method:

1-get to be by some method of minimization in cash (d216), solving (PA2).

2-round (quantize) each component of the signal be to the closest value of the constellation D, to the

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The resulting signal will be called zfr (d216).

3-Then, the initial radius is calculated as r = IIR. zfr -zll (d217).

This technique was proposed in the document [1] mentioned above.

Also known in the prior art is the spherical decoding method with glue by continuous minimization; which we call modification 2.

Such modification 2 to improve the performance of the spherical decoding search stage was proposed in document [2] M. Stojnic, H. Vikalo, and B. Hassibi, "Speeding up the Sphere Decoder with H" "and SDP inspired lower bounds. ", IEEE Transactions on Signal Processing, vol. 56, no. 2 (2008), 712

726. These modifications 2, as proposed below, only affect the feasibility test of each partial signal.

Therefore, the enumeration method remains the one described with the help of Figure 4 (d12), changing the feasibility test (d131). The test in Figure 5 (d18) is replaced by that in Figure 7 (d28), which is explained below.

Suppose we are examining the i-es layer and in it we are examining the partial signal s = (S;, 5; + I '", 5M_I, 5M)' Recall that the partial distance associated with said partial signal is calculated as:

distance paTcia / (5i, 5i + l '"', 5M_I, 5M)

= (TU '5i + TU + 1. 5; +1 + ... + T; .M. SM -Z;) ~ + (T; + I.i + 1. 5i + 1 + .. + T; + IM. 5M -Z; + I) ~ + ..

+ (TM_I.M_I. 5M_1 + TM_I.M. 5M -ZM _I) 2 + (TM.M. SM -ZM /

(ec.10)

While the distance of any total signal s obtained from the previous partial signal will be calculated as:

distance (SI, S2, '., S; _I, Si, 5; + I' "', 5M_I, 5M) = (TU 'SI + TI.2. 52 + ... + TI.M. 5M -ZI) ~ + (T2.2. S2 + T ~ .3. S3 + ... + T ~ .M. 5M -Z2) 2 + ...

+ (TI_l.I_1. Yes_1 + TI_l.i. Yes + ... + T; _l.M. 5M -ZI_I /

-) '(-)'

+

(TU 'S; + T; .i + 1. 5i + 1 + ... + Ti.M · 5M -Z; + T; + I.i + I · S; + I + .. + T; + IM · SM -Z; + I +.

+

(TM_1.M_1. 5M_1 + TM_l.M · 5M -ZM _1) 2 + (TM.M · SM -ZM) 2

(ec.11)

or equivalent

distance (SI, S2, .., Si_I, S;, 5i + l '", 5M_I, 5M) = (TU · SI + TI.2' S2 + ... + TI.M '5M -ZI) ~ + (T2.2 · S2 + T2.3 'S3 + ... + T2.M' 5M _Z2) 2 + ...

+ (TI_1.1_1. SI_1 + TI ~ 1.i. 51 + ... + Ti_I.M. 5M -ZI_1) 2 + distance paTcia / (Yes, 5i + l '"', 5M_I, 5M)

(ec.12)

We will name the difference between distance (sl, s2 '", Si-l' S;, Si + l '", 5M_I, 5M) and distance paTcia / (yes, 5i + l' ", 5M_I, 5M) as Testodistancia ( sl, s2 '..., 5i_I):

Testodistancia (s I, S2 '", S; _I) = (TU' SI + TI.2 · S2 + ... + TI.M · 5114 -ZI) 2 + (T2.2. 52 + T2.3. S3 + ... + T2.M · 5M -Z2) 2 +.

+ (Ti_I.i_1. S; _I + Ti_l.i · 5i + ... + T; _I.M · 5M -Z; _1) 2

(ec.13)

Note that Testodistancia (sl, s2 '", S; _1) is necessarily a real number greater than or equal to O, and that trivially it is verified

distance (SI, S2, '., Si_l, Si, 5; + I' "', 5M_I, 5M) = distance paTcia / (S ;, 5; + I'", 5M_I, 5M) + Testodistancia (sl, s2 , ..., Si_I) '

(ec.14)

Therefore, of the condition

and from (ec.14), we have the condition is obtained

Given that (SI.5 / + I, ···, SM_I, sM) have already been selected and have assigned values, then it can be considered that remaining_distance is a function of the elements (5 1.52., 51_1) '

or ••

Given the partial signal (s / .51 - + - 1. · ··, $ / 01_1.5 / 01), if its partial distance is less than the variable menoT_disCobtenida, this partial signal is feasible in the test (d18). The proposal in document [2] is to try to obtain a positive number e 2 :: O that is a lower level of remainder (s¡, s2, ..., 51_1}: that is, for any combination of values (S ¡, S2, ..., Si_¡) 'with Sk ED, k = t ..... i -1, it is true that OS and S remainder (yes, s2' ••• 51_1).

In this case, the condition can be replaced.

By condition:

distance paTcia / (S¡, SI + I. "', SM_I .S / oI) + c <T2 (ec.15)

This new condition is more restrictive (it causes more partial signals that do not comply with it) and therefore causes more partial signals to be eliminated. Consequently the search ends earlier. The larger the c (maintaining that it is a lower level of distance (sl, s2 '..., 5; _1), 1. the better the performance of the method.

Various techniques are proposed in the document [2] to calculate the lower bound c. One of them . on which the proposed present invention is based. It is based on applying continuous minimization techniques to the expression Tesrodisrancia (sl, s2 '.... S¡ _d.

First, to raise the problem of continuous minimization, we rewrite the term Testodistancia:

resrodisrancia (SI'S2 '.... s¡_t)

= (TI.1 · SI + T1.2 '52 +. · + T1.M' SM _ZI) 2 + (T2.2 'S2 + TV' 53 + ... + T2.M 'SM _Z2) 2 + ...

+ (Ti_t.¡_1. Yes_ 1 + Ti_l.i. Yes + ... + T¡_V.!. SM -Z¡_ 1) 2

-) '-)'

= (TU '51 + T1.2. 52 + ... + rl.i_l. Si_ 1 + Tl.i. S¡ + ... + T1.M. SM -ZI

+ (0 · S2 + T2.2 · S2 + ··· + TZ.¡_t · Si_I + TV · S¡ + ··· + T2.M · SM -Z2 + ...

-) '

+ (O · 51 + O · 52 + ... + Ti_l.i_I · S¡_I + Ti_l.i · S¡ + ... + Ti_I.M · 514 -Zi _1

(ec.16)

Recall the calculation of the corrected signal zaux¡, assuming that the values have been assigned (Si. Si + l '"', SM_I .SM) 'Using the expression (ec.7). You can verify that

Restoration (yes, S2 • ... • 51_1)

= (T1.1. SI + T1.2. 52 + ... + Tl.I_1. 5 / _1 -zaUX¡ (l) /

+ (O. S2 + T2.2. 52 + ... + TV_I. S¡_I -zaux¡ (2)) 2 +.

+ (O · YES + O · 52 + ... + Ti_l.i_1. 5i_1 -zaux¡ (i _ 1)) 2

(ec.17)

Written matrixally. if we define the RI matrix: 1_I. t: ¡_1 as the bmatrix fontated by the first i -1 rows and i -1 first columns of the R matrix:

ES 2 433 695 A2

Restoration (s¡. 5: z. ..... 5i_¡) =

Rl: i-l, l: i-l '- (

(: ~); ~~;: ~ g)

(ec.18)

,; ~. , a = ;;: -1)

Recall that each SI must belong to the constellation D. The constellation D is composed of real values and therefore has a maximum value Mp and a minimum value mi) -A way of obtaining a lower level of Testodistancia (sl's ~ '.. .. SI_l) consists in calculating the minimum value of the expression on the right side of

5 (ec.18), restricting the possible values of Si to which they can lomar can be any real number in the interval [mo. Mol.

The problem to solve then is:

") (, a = '(l))

be. = ar min R "". 52 _ zaux¡ (2)

(ec.19)

, 9 S, .5, •.,. S'_ 1

1., - l, l.I-I .... ...

(

5i_1 zaux¡ (¡-1)

Note that being; It is a vector or signal with i -1 components. We will call it to be; as a second auxiliary signal 10 associated with the partial signal (s;, SÚ¡, '' ', 5M_¡ "<; M)'

Once the second auxiliary signal, "solution of (ec.19), is calculated, the continuous minimization level can be obtained through the expression:

... _ (:::: m),

(ec.20)

c =

R1: 1_1,1: 1_1 ,, ~,.

(zaux; é ~ -1)

The problem (ec.19) is a problem of continuous minimization with box-type restrictions, such as the continuous auxiliary problem (PA2). Assuming that we have a method to efficiently solve problems of this type, it is possible to modify the feasibility test (d131) in the enumeration method of Figure 4 (d12) to obtain a potentially more efficient alternative method.

A detailed description of the feasibility test in Figure 7 (d28) follows.

1-As in the test of Figure 5 (d18), we start by checking if when the partial distance 20 corresponding to the partial signal (5;, 5; +1, "', 5M_ 1.5. </) Is smaller that less_discovered. (d281)

2-If it is not, the signal is not feasible and the test ends (d282).

3-If the partial distance corresponding to the partial signal (S;, 5; + I, "·, 5M_1, .5M) is less than less_disclosed, the new corrected signal 2auxI (d283) is calculated.

4-Next, the problem of continuous minimization (ec.19) is solved by obtaining the second auxiliary signal 25 associated with (5,, 51 + 1 '' '' 5M_1.5M), which is used to calculate the c-level (d284), using the equation (ec.20).

5-Next, it is checked whether the partial distance (s;, 5; + I '", .5M_I, 5M) plus the dimension c is less than less than_distended (d285).

6-If it is not fulfilled that partial distance (S ,, 51 + 1, ", 5M_1.5M) + c <lower_discovered, then this partial signal is not feasible and the test ends. (D286).

7-If it is fulfilled that partial distance (s "51 + 1 '", 5, </ _ 1.5, </) + e <lower_dist_distined, then the current partial signal is feasible and the test ends. (d287).

The method composed of the preprocess of figure 6 (d21) and the search of figure 4 (d12) with a feasibility test of figure 7 (d28) constitutes the known method with modifications proposed by 35 other authors, closer to the invention.

Next, the novelties that the invention brings to the state of the art, described up to this point, are described.

The invention provides modifications to the method described with the aid of Figures 4 (d12), 6 (d21) Y7 (d28), and which will finally be integrated into the method described in Figures 8 (d31), 4 (d12) and 9 (d38). 14

ES 2 433 695 A2

The modifications of the invention basically affect the feasibility lesl for partial signals, since the new preprocessing stage represented in Figure 8 (d31) undergoes changes with respect to that represented in Figure 6 (d21)) And they are the following:

1) The number of minimizations in box is limited, using the position of the auxiliary signal sck¡ relative to the box defined by the constellation.

2) The initial signal is selected for the box minimization method, using the auxiliary signals ser, and / or sck¡ obtained when examining the feasibility of the predecessor partial signal.

The already described described method composed of figures 4 (d12), 6 (d21) and 7 (d28), is very efficient in terms of nodes explored, but not in terms of computation time, because of the temporary cost of minimizations in box The proposals of the invention are aimed at reducing the number of cash minimizations and the cost of each of said minimizations, so that the algorithm obtained with the new method is globally more efficient. It is important to understand that the number of partial signals scanned should not change substantially from the method described to that proposed by the invention. However, the computation time cost of the invention is much better (smaller).

the modifications proposed below are those of figures 4 (d12), 6 (d21) and 7 (d28), and therefore will be described from said algorithm.

The first proposal of the invention is to limit the number of minimizations in the box, using the position of the signal s <: k¡ relative to the box defined by the constellation.

This first proposal aims to reduce the number of cash minimizations that were described with the help of Figure 7 (d28), in (d284); In this phase of the method, the second auxiliary signal is calculated, of the minimization problem restricted to the box (ec.19), and then it will be used to calculate the e-value as in the equation (ec.20).

Consider now the problem of continuous, unrestricted minimization (PA1) for the same subproblem

(with matrix RI: I_ t.l: I_1 And received signal zaux.J. We will call the solution for this subproblem sck¡. As mentioned when describing (PA 1), sck¡ is easily obtained by solving the triangular system of linear equations:

(:: ~: ~ g) (: ~~;: ~ g). sek · I ~ ·; _I) zaux; · Ú

R'H.'oH (ec.21)

= 1)

It is called sek; as the first auxiliary signal associated with the partial signal (S;, 5HI '···. 514_1,514). The calculation cost of the first auxiliary signal sekl for a partial signal (SI.5; + I. ···. 514_1,514) is usually considerably less than the calculation cost of the second auxiliary signal for the same partial signal .

Once you calculate the sek side, you can easily and efficiently check if sek is in the box defined by the constellation. Recall that a signal s is in the box if all its components s (i) are contained in the intelValo [mD, MD] where mD and MD are respectively the minimum and maximum values of the constellation

D.

Yes sek; is in the box, then it would coincide with being, so it would no longer be necessary to calculate SCT. Also, in that case. the value of the dimension e would be O, which would not have any effect. Therefore, it only makes sense to use box minimization if sckl is "out of the box".

Therefore, the first proposal includes the following steps for the feasibility test, which are described with the help of Figure 9 (d38)

1-As in the tests of Figure 5 (d18) and 7 (d28), it is started by checking if when the partial distance corresponding to the partial signal (SI, SI + I • ..., SM_I, SM) is less than minor_discovered. (d381).

2-If it is not, the signal is not feasible and the test ends (d382).

3-If the partial distance corresponding to the partial signal (S;, 5; + I '···' S14_1.5M) is less than less_disclosed, the new corrected signal Zaltx¡ (d383) is calculated and the first auxiliary signal sek¡ (d384) solving the system of triangular equations (ec. 21). 4-If the first auxiliary signal sck¡ is in the box (d385), the first auxiliary signal of the

ES 2 433 695 A2

sck ¡for possible use as an initial signal xO for minimization problems in layer i -1, the partial signal is feasible and the test ends (d386).

S-If the first auxiliary signal sck¡ is not in the box, the second auxiliary signal SCTI is obtained by solving the problem of continuous minimization (ec.19), for which the one obtained with the first components of the first components of the second auxiliary signal associated with the predecessor signal scr; +1 (if scri + 1 was calculated) or with the first i -1 components of the first auxiliary signal associated with the predecessor signal sckj + 1 (if SCT / +! was not calculated). Once the second auxiliary signal S CTj is obtained. ser¡ is used to calculate the e-dimension, using the echo (20). (d387)

S-Next, it is checked whether the partial distance (s ,, 5 '+ I' '', 5M_1.5M) plus the dimension e is less than less_distended (d388).

7-If it is not fulfilled that partial distance (S;, 5 '+ I' ", 5M_1.5M) + e <less _discovered, then this partial signal is not feasible and the test ends. (D389)

8-If it is fulfilled that partial distance (S¡, 5; + I '' '', 5M_I, 5M) + e <enhancedT_dissected, then the current partial signal is feasible, the auxiliary signal is saved to be, for possible use as initial signal xO for minimization problems in layer i -1 and the test ends (d390)

This proposal has the effect of reducing the number of cash minimizations.

An additional effect of this proposition is as follows: Suppose the method is considering the partial signal (S'_I, S ,, 5, + I, .. ·. 5M_I, 5M) 'Then, it is assured that the feasibility test has been applied to the predecessor partial signal (S, .5; + I ... ·, 5M_1.5M), and therefore sck will have been calculated; (solving (ec.21 ») and possibly also being (if sek¡ is out of the box).

It is important to note that, when processing a partial signal, the feasibility of the predecessor partial signal will necessarily have been previously checked, and in that case at least one of the sek¡ or SCT¡ auxiliary signals corresponding to the predecessor partial signal will have been calculated. This fact is necessary to be able to apply the second proposal of the invention described below.

The second proposal of the invention consists in making the selection of the initial signal for the minimization in box.

Assume that a partial signal is being examined (5,, 5, + 1, · ... 5M_I, 5M) and that it is necessary to apply box minimization. This second proposal aims to minimize the temporary cost of each of the cash minimizations, and is achieved by careful choice of the initial signal used for cash minimization.

The restriction that the components of the second auxiliary signal be, are contained between mD and MD guarantees that it is contained in a "box", which gives name to the problem. The "faces" of the box are the subsets of the box that limit with the outside: that is to say, they are the vectors or signals of the box that, for at least one component, fulfill that their value is MD or 11ID.

The fact that the solution of the unrestricted problem sek is out of the box guarantees that the solution of the restricted problem, SCT "will be on one side of the box; that is, at least one component j meets scr, (;) = MD oseT, U) = mD.

In general, box minimization methods are iterative methods that start from an initial signal (usually random) located in the box, which we will call xO. These methods carry out iterations that bring the initial signal xO closer to the final solution, SCT,.

We will name the face of the box where the SCT solution is the optimal face. If the initial xO signal is not on the optimal face, the continuous minimization processes "move" the xO signal from face to face towards the optimum face, and therefore towards the SCT solution; If there is a lot of distance between the optimal face and the face on which the initial signal xO is, then the minimization process can be made too long and expensive. Therefore, it is very important to select the initial signal xO so that it is on the optimum face, or "near" it.

It has been proven experimentally that the box minimization problems that arise when examining partial signals of layer i are strongly related to the problems that arose in the previous layer (i + 1). In particular, suppose the feasibility of the partial signal is being examined (S, .5; + I ... ·, 5M_1.5M), and that, after evaluating its partial distance, it is found to be less than the variable less_disclosed. According to the first proposal, sek¡ would now be calculated (solving the system of triangular equations ec.21 »and it would be checked if sek¡ is in the box; suppose then that sek¡ is

ES 2 433 695 A2

out of the box and, therefore, the next step is to solve the problem of minimization in cash (ec.19), obtaining SCT /.

Previously, in the previous layer i + 1, the current predecessor partial signal will have been examined: (5; + 1 "'" 5101_1, SM); when examining this partial signal in the i + 1 layer, the auxiliary signal scki + ¡will have been calculated safely. and possibly also the auxi liar signal SCTi + l '

It has been proven experimentally that it is very likely that the solution of the new minimization problem in layer i, to obtain SCTj. is on the same side of the box as the first i1 components of the second auxiliary signal SCTI_I, which was obtained by solving the problem of minimization of the i + 1 layer. Therefore, it is proposed:

Suppose SCT¡ +! is the solution of the minimization problem in the box when examining the partial signal (s¡ + 1. · ··. SM_l, SM) in the layer (i + 1) -th. Then, if in the next layer (i) when examining the partial signal (S¡ .. 5i + 1. ···. SM_l, SM) it is necessary to solve a new box minimization problem, it will be used as the initial signal xO for This new problem of minimization of the first i -1 components of SCTi + l:

srr ,,, (l))

xO = sCT¡ + 1 (2)

(SCTi + l (1 -1)

It is possible that for a certain partial signal (S¡.SI + 1. ···. SM_l, SM) in the layer i need to solve a box minimization problem. but that for the partial predecessor signal (s¡ + 1. ···. SM_l, SM) it was not necessary to apply box minimization. and therefore SCT has not been calculated; + I 'In that case. The initial i-1 components of SCk would be used as the initial signal; + I 'that was previously calculated when checking the feasibility of the predecessor signal (5; +1, .. ·. 5M_1.5M). Due to the first proposal. it is known with certainty that SCkl + 1 was calculated.

At first. This technique could not be applied when checking the feasibility of partial signals in the layer (i = M). since these have no predecessor signal. But nevertheless. remember also that. due to the first existing proposal. in the preprocessing stage of Figure 6 (d21). Before starting the search. the sck signals (analogous to the first auxiliary signal sck¡) are calculated and. If this is out of the box. it is calculated to be (analogous to the second auxiliary signal to be;). In the first proposal already existing. sck and ser are calculated to obtain a better approximation to the initial radius. But nevertheless. the signal to be (and. if it has not been calculated. the sck signal) can also be used to obtain an initial signal for possible box minimizations in the layer i = M. that is. for possible minimizations that may be

required at the initial layer (layer M) will be taken as the initial signal xO = (~~. ~~~). if being has been

be (M -1)

calculated. 'i otherwise it will be taken as the initial signal xO = (~~~. ~~~)

sck (M -1)

The preprocessing step of Figure 8 (d31) is very similar to that of Figure 6 (d21); with the difference that the signals sck or be. calculated in (d215). (d216) are stored in (d315) and (d316) for later use as an initial signal for minimization problems that may arise in layer M.

Claims (3)

1. METHOD OF SPHERICAL DECODIFICATION OF SIGNS OF COMMUNICATION SYSTEMS OF MULTIPLE INPUTS AND MULTIPLE OUTPUTS (MIMO) OF MAXIMUM VEROSIMILITUDE. where there is a transmitter with a plurality of M transmitting antennas to send them by a MIMO callal and a 5 plurality of N antennas of a receiver that uses spherical decoding that is accelerated by applying a method of dimensions by continuous minimization; and where the receiver's input data is:

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an array of channel 11 E RN'XM, with N rows and M columns. where N 2 = M.

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the constellation: D = (di '., _, dd set of L symbols, all of them in R.

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the ser'lal received and E IRN;

10 Calculating, based on these data, the maximum likelihood corresponding to the decoded sel'lal, which is selected from among all the saMias of M components that meet that each of the M components of the signal belongs to the constellation O: with the maximum likelihood signal being understood as the one of these signals which, after being filled by the H-channel matrix, has a minimum distance from the received signal and: 15 comprising the method: • a preprocessing stage (d21) for the assignment of a new channel matrix R e R, M "M and a new signal received ze ~" 1, by orthogonal transformation of the channel matrix H and the received material and : where the new R channel matrix is triangular upper; also comprising the preprocessing stage: • Obtain a first auxiliary signal 5ch, solving a system of conventional triangular equations 20 (PA1) and • verify whether said first auxiliary signal has all its components within a box defined by the maximum and minimum values of the constellation D. that 25 if said verification is carried out, it comprises obtaining an initial radius such as the distance between the signal sc: h after being quantified and filtered by the matrix R. and the signal received z: and in case of not fulfilling said verification, It comprises selecting an initial signal xO and using it to calculate a second auxiliary signal being by solving a minimization problem with conventional restrictions (PA2). and also get an initial radius like the 30 distance between the signal being after being raised and filled by the matrix R. And the signal received z. • a search stage (d12) comprising: • set the square of an initial radius as the smallest total distance obtained so far. Y 35 • list partial sandals (S {. ~ I + l '".)' '' '_ 1' 1",) and complete setlales (SI, S2. ··· S, \ {_ I. S "l on which • a feasibility test is applied (d28). to decide whether each of the partial signals listed is feasible or not, and when a certain partial setlal U; 'S; + I ... · • S "' _ I 'sH) is feasible, 40 the following signals, selected between partial and complete set signals, called successors, of said partial sElflal, and when a certain partial signal are still listed. 1 (Yes, S; .I '···. I Nl' S ",) it is not feasible to be discarded and no partial or complete signal is listed, successor to said partial signal; • every time in the enumeration a complete signal is obtained so that, after being filtered by the R channel matrix, its distance to the received signal z is less than the smallest IOtal distance obtained so far, said complete signal is saved as a candidate for maximum likelihood solution, and the smallest distance tOlal obtained so far becomes said distance to the signal received z; and 50 When the enumeration is finished, the resulting candidate signal is the maximum likelihood signal corresponding to the decoded signal, and ES 2 433 695 A2 Understanding said feasibility test (d28), where it is decided whether one is "'partial (Yes, Yes")' ··· 'sN-L sNes feasible or not. first check if its partial distance is less than the smallest total distance obtained so far; Y 5 • in case of not doing this check. said partial signal is established as not feasible: and • in the case of such a check, it comprises selecting an initial signal xO and using it as the Initial signal to apply a conventional minimization method with restrictions, whereby the second auxiliary signal is obtained; associated to the signal 10 partial (S¡, SI.q. ", 5" '_ 1' SAl): and using said second auxiliary signal, a continuous and conventional minimization level will be calculated; adding the dimension ca the partial distance of the partial signal (s ;, 5; + \, ', • 5 "1_ \' 5/01) And then it is verified if this sum is greater than the smallest total distance obtained until the moment (ec. 15), in which case the partial signal is declared not feasible; and if the sum is less than the smallest distance obtained so far, 15 the partial signal is declared feasible: characterized by:

-

After applying the feasibility tesl and verifying that a partial signal (SI, SH¡,, ... 5.'1_1 '5 ...,) is feasible. Being its partial distance less than the smallest total distance obtained so far, it comprises:

-

calculate the first auxiliary signal sek; = (SI'S ~, ", Si-l) associated with the partial signal (SI, 51 + 1" ", 5, '1" _1, 510 /),

20 -check If the first auxiliary signal sek¡ associated with the partial signal (Si, Si + ['"'. 5/01 _ \, SIoI) has all its components within the box defined by the maximum and minimum values of the Constellation .

-

when said verification is checked, the partial signal (s ;, If + l ...., $. \. 1 _1 '5101) is declared feasible;

. when such verification is not verified, it comprises calculating the second auxiliary signal will be associated with the partial signal (S /, SI + I ''. ', S / oI_ I, SIoI). and then calculate the level of continuous minimization and decide, 25 using the feasibility test. if the partial signal (Yes, SI + I '.,., 5/01 _1. SM) is feasible or not. 2. METHODS OF STRATEGIC DECODIFICATION OF SIGNS OF COMMUNICATION SYSTEMS OF MULTIPLE INPUTS AND MULTIPLE OUTPUTS (MIMO) OF MAXIMUM VEROSIMILITY. according to claim 1, characterized in that:

-

after checking the feasibility of a partial signal (YES, YES .. I '' ', 5101_1, S / oI) for an i-th component, with

30 i <M. to verify that its partial distance is less than the smallest total distance obtained so far. and for which the first auxiliary signal sek, associated with said partial signal has some of its components outside the box defined by the maximum and minimum values of the constellation, so the second auxiliary signal must be calculated. associated to the partial signal (SI, S / + I '' '', S / oI-¡'S / oI), and the continuous minimization level e, 35 also includes:

-

obtain the initial signal xO for the continuous minimization method with restrictions, such as the first i -1 components of the second auxiliary signal scri + l: (SI 'S2' "'' Si_¡) when checking the feasibility of a signal ' The partial predecessor (5 / + 1, "', 5 / 01-1' s ..,) of a partial signal (SI," ', S / o1-¡, S / oI) the second signal was calculated auxiliary scri + ¡= (~ I '~ 2' "'' s¡) associated with the partial signal (51 + 1, ,,., 5i \ 1-¡'S./oI), and

40 obtain the initial signal xO for the continuous minimization method with restrictions, such as the first j-1 components of the first auxiliary signal sehl .. l: (51.52 .... S¡ _ I), when checking the feasibility of the predecessor partial signal (Si + l '"', SIol -l 'S / oI) the second auxiliary signal seri + l' was not calculated and therefore the first auxiliary signal sek '+ I = (5 \ .52 was calculated "", SI) associated with the partial signal (5¡ + 1, "', $ / 01_ 1' 5, ..,), 45 3. METHOD OF SPHERICAL DECODIFICATION OF SIGNS OF COMMUNICATION SYSTEMS OF MULTIPLE INPUTS AND MULTIPLE SAVINGS (MIMO) OF MAXIMUM VEROSIMILITUDE. according to claims 1 or 2. characterized in that:

-

the preprocessing step comprises storing the first auxiliary signal seh, and storing the second signal

ES 2 433 695 A2

auxiliary be when it has been calculated,

-emple the first auxiliary signal sch and the second auxiliary signal be in the possible minimizations that are applied to verify the feasibility of partial signals of the form (s..¡) with i = M.

5

-after checking the feasibility of a partial signal (5.1.1) with i "" M • to verify that its partial distance is less than the lowest total distance obtained so far, and for which the first auxiliary signal sc k M associated to said seal it has some of its components outside the box defined by the maximum and minimum values of the constellation. Therefore, the second auxiliary signal SeTA-! associated with the partial signal (SM) and the continuous minimization queue e,:

; It also includes:

10

"Obtain the initial signal xO for the method of continuous minimization with restrictions, such as the first ones: before being, when the auxiliary signal was calculated at the preprocessing stage, and M one

• obtain the initial signal xO for the method of continuous minimization with restrictions, such as the M -1 first sek components, when the second auxiliary signal was not calculated in the preprocessing stage and only the first sek auxiliary signal was calculated.

'5