CN108900448B - Low-complexity packet decoding method based on MIMO system - Google Patents

Low-complexity packet decoding method based on MIMO system Download PDF

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CN108900448B
CN108900448B CN201810678780.3A CN201810678780A CN108900448B CN 108900448 B CN108900448 B CN 108900448B CN 201810678780 A CN201810678780 A CN 201810678780A CN 108900448 B CN108900448 B CN 108900448B
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CN108900448A (en
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李国权
周湘云
徐勇军
林金朝
庞宇
王家城
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Chongqing University of Post and Telecommunications
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03891—Spatial equalizers
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0045—Arrangements at the receiver end
    • H04L1/0047—Decoding adapted to other signal detection operation
    • H04L1/0048—Decoding adapted to other signal detection operation in conjunction with detection of multiuser or interfering signals, e.g. iteration between CDMA or MIMO detector and FEC decoder
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H04L1/0057—Block codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03006—Arrangements for removing intersymbol interference
    • H04L25/03178—Arrangements involving sequence estimation techniques
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
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Abstract

The invention discloses a low-complexity packet decoding method based on an MIMO system, namely a packet rounding decoding method, belongs to the technical field of communication and is used for reducing the decoding complexity of the MIMO system. The algorithm is based on a forced integer decoding algorithm, the received signals are grouped, each group independently uses the forced integer algorithm to decode, and then each group of independently decoded sending code words is collected, so that the most original transmitting information before grouping is obtained. The packet rounding algorithm reduces the decoding complexity by grouping the received symbols, so that compared with the rounding detection algorithm, the packet rounding is superior in decoding complexity, and meanwhile, as can be seen from simulation results, the packet rounding is superior in system performance compared with the rounding algorithm, and the difference of system performance is larger and larger as the number of antennas increases.

Description

Low-complexity packet decoding method based on MIMO system
Technical Field
The invention belongs to the technical field of communication, and relates to design of a receiving end decoding algorithm in a Multiple-Input Multiple-Output (MIMO) system, in particular to design of a packet forced integer decoding algorithm based on a forced integer decoding algorithm structure.
Background
The MIMO technology can multiply the channel capacity of the system without increasing additional system bandwidth and antenna transmission power, and thus the MIMO technology is widely applied in the field of wireless mobile communication [1 ]. The performance of the MIMO system is determined by the quality of the decoding algorithm at the receiving end, so it is important to find a low-complexity high-performance algorithm [2 ]. Decoding algorithms of a receiving end can be divided into two types, one type is a Maximum Likelihood (ML) decoding algorithm, the ML decoding algorithm is an optimal decoding algorithm, but the decoding complexity of the ML decoding algorithm increases exponentially along with the increase of a transmitting antenna and a modulation constellation diagram [2] and [3 ]. The other is a traditional linear decoding algorithm, including Zero-Forcing (ZF) decoding algorithm and Minimum Mean Square Error (MMSE) decoding algorithm, which is much less complex than ML algorithm but inferior to ML [2], [4 ]. Recently, a new Linear decoding algorithm based on the MIMO system, called as integral-forming Linear Receivers (IF for short), has been widely studied [2], [5], and the IF decoding algorithm has a lower complexity than the ML algorithm, but its performance approaches the ML decoding algorithm.
The most prominent characteristic of the IF detection algorithm is that it uses the receiving antenna to generate an effective integer channel matrix, and recovers the integer combination of the transmitted code word by the effective channel matrix [2], unlike the traditional linear detection algorithm which directly recovers the transmitted code word. The effective integer channel matrix must be reversible and non-singular, and there are a number of algorithms that can be used to find this effective integer channel matrix, such as HKZ, Minkowski, LLL, and CLLL algorithms [6] - [8 ]. Although the rounding decoding algorithm has a lower decoding complexity, there is still a need to find a decoding algorithm to further reduce the complexity, so that the decoding algorithm can be better applied to the situation with a large number of antennas or requiring lower complexity and general system performance, and therefore, intensive analysis and research are performed on the basis of the rounding decoding algorithm.
[1]G.J.Foschini and M.J.Gans,“On limits of wireless communications in a fading environment when usingmultiple antennas,”Wireless Personal Communications,vol.6,no.3,pp.311–335,1998.
[2]J.Zhan,B.Nazer,U.Erez,and M.Gastpar,“Integer-forcing linear receivers,” IEEE Transactions on Information Theory,vol.60,no.12,pp.7661–7685,Dec 2014.
[3]M.O.Damen,H.E.Gamal,and G.Caire,“On maximum-likelihood detection and the search for the closest lattice point,”IEEE Transactions on Information Theory,vol.49,no.10,pp.2389–2402,Oct 2003.
[4]K.R.Kumar,G.Caire,and A.L.Moustakas,“Asymptotic performance of linear receivers in mimo fading channels,”IEEE Transactions on Information Theory, vol.55,no.10,pp.4398–4418,Oct 2009.
[5]J.Zhan,B.Nazer,U.Erez,and M.Gastpar,“Integer-forcing linear receivers: A new low-complexity mimo architecture,”in 2010IEEE 72nd Vehicular Technology Conference-Fall,Sept 2010,pp.1–5.
[6]W.Zhang,S.Qiao,and Y.Wei,“Hkz and minkowski reduction algorithms for lattice-reduction-aided mimo detection,”IEEE Transactions on Signal Processing, vol.60,no.11,pp.5963–5976,Nov 2012.
[7]A.Sakzad,J.Harshan,and E.Viterbo,“On complex lll algorithm for integer forcing linear receivers,”in 2013Australian Communications Theory Workshop (AusCTW),Jan 2013,pp.13–17.
[8]——,“Integer-forcing mimo linear receivers based on latticereduction,” IEEE Transactions on Wireless Communications,vol.12,no.10,pp.4905–4915, October 2013.
Disclosure of Invention
The present invention is directed to solving the above problems of the prior art. The block forced integer decoding algorithm is provided based on the forced integer decoding algorithm, and the decoding complexity of the forced integer decoding algorithm is mainly derived from the fact that a channel matrix is changed into an effective integer channel matrix by using a lattice reduction algorithm, so that the block forced integer decoding algorithm reduces the dimensionality of the channel matrix by grouping received symbols, and the decoding complexity is reduced. The technical scheme of the invention is as follows:
a low complexity packet decoding algorithm based on MIMO system includes the following steps:
firstly, grouping received symbols, decoding each group of received signals independently by using a rounding algorithm to obtain a transmitted code word, wherein the concept of the rounding algorithm is to find an effective integer matrix A and an equilibrium matrix B, each group decodes the symbols of the groups respectively, and the transmitted code words decoded independently by each group are aggregated to be the total transmitted symbols before grouping is absent, so that the most original transmitted information before grouping is obtained.
Further, the grouping of the received signals specifically includes: firstly, two columns of a channel matrix H are taken as a group, received symbols are divided into L groups, L is N/2, N is the number of receiving and transmitting antennas, and then the MIMO system equation is
Figure BDA0001710440650000031
Wherein
Figure BDA0001710440650000032
SNR represents the average signal-to-noise ratio, X, at each receive antenna q Representing the transmitted symbols of the q-th group, q representing the q-th group, and under the condition of not losing generality, only the first group is studied to find the effective integer matrix A and the equalization matrix B, when one group is studied, other groups are regarded as noise, and the signal model of the first group is
Figure BDA0001710440650000033
Although the signal model of equation (2) is the same as the system equation of equation (1), in equation (2), only
Figure BDA0001710440650000034
Is taken as an effective signal component, wherein
Figure BDA0001710440650000035
Represents an N x 2 complex matrix, an
Figure BDA0001710440650000036
Are treated as effective noise components.
Further, the using an equalization matrix for the first packet signal
Figure BDA0001710440650000037
Then obtain
Figure BDA0001710440650000038
Wherein
Figure BDA0001710440650000039
Is the effective integer channel matrix of the first packet;
by using
Figure BDA0001710440650000041
Respectively represent
Figure BDA0001710440650000042
B 1 ,A 1 Line m of
Figure BDA0001710440650000043
Let
Figure BDA0001710440650000044
Equation (4) becomes
Figure BDA0001710440650000045
Further, the decoding using the improved rounding algorithm specifically includes:
step one is
Figure BDA0001710440650000046
Upper lattice decoding: will be provided with
Figure BDA0001710440650000047
In which each element is decoded into an integer field
Figure BDA0001710440650000048
To obtain the closest point thereto, i.e.
Figure BDA0001710440650000049
Wherein
Figure BDA00017104406500000417
Representing a rounding operation;
step two lattice code word projection:
Figure BDA00017104406500000410
for is to
Figure BDA00017104406500000411
Performing a mold-removing operation to obtain
Figure BDA00017104406500000412
J represents the constellation order;
decoupling the three-lattice code word: based on linear equations
Figure BDA00017104406500000413
Obtaining a decoded vector
Figure BDA00017104406500000414
After the above steps, the decoding symbols of the first packet are obtained, and then the decoding symbols of other packets are obtained in the same steps as those of the first packet in the remaining packets, and the original transmission information before the packets are not grouped is obtained by grouping the decoding symbols of all the packets.
Further, the optimal effective integer channel matrix A 1 Expression (2)
Figure BDA00017104406500000415
Optimal equalization matrix B 1 Expression of (2)
Figure BDA00017104406500000416
The invention has the following advantages and beneficial effects:
the invention reduces the dimensionality of the channel matrix by grouping the received signals, and further reduces the complexity of reducing the channel matrix into an effective integer channel matrix by using a lattice reduction algorithm, thereby reducing the overall complexity of a forced integer decoding algorithm. The packet round-robin decoding algorithm has lower decoding complexity compared with round-robin decoding, and has better decoding performance compared with the round-robin decoding algorithm, and the performance difference is larger and larger as the number of antennas increases. Based on the characteristics of the packet rounding algorithm, the method can be applied to the conditions of low requirement on decoding complexity, general decoding performance requirement or more antennas.
Drawings
FIG. 1 is a block diagram of a warping system in accordance with the present invention;
FIG. 2 is a schematic diagram illustrating the comparison of bit error rate performance of the packet rounding decoding algorithm and the rounding decoding algorithm in 4 × 4 channels using 4-QAM constellation according to the present invention;
FIG. 3 is a schematic diagram illustrating the comparison of bit error rate performance of the packet rounding decoding algorithm and the rounding decoding algorithm in 8 × 8 channels using 4-QAM constellation according to the present invention;
fig. 4 is a schematic diagram illustrating the comparison of the bit error rate performance of the packet rounding decoding algorithm and the rounding decoding algorithm in 12 × 12 channels using 4-QAM constellation according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described in detail and clearly in the following with reference to the accompanying drawings. The described embodiments are only some of the embodiments of the present invention.
The technical scheme for solving the technical problems is as follows:
a low complexity packet decoding algorithm based on MIMO system, at receiving end, includes following steps: firstly, two columns of a channel matrix H are taken as a group, received symbols are divided into L groups, L is N/2, N is the number of receiving and transmitting antennas, and then the MIMO system equation is
Figure BDA0001710440650000051
Wherein
Figure BDA0001710440650000052
SNR represents the average signal-to-noise ratio at each receive antenna and q represents the q-th group. Without loss of generality, only the first group was studied, whose signal model was
Figure BDA0001710440650000053
Although the signal model of equation (2) is the same as the system equation of equation (1), in equation (2), only
Figure BDA0001710440650000054
Is taken as an effective signal component, wherein
Figure BDA0001710440650000055
And then
Figure BDA0001710440650000056
Are treated as effective noise components.
Further, an equalization matrix is used for the first packet signal
Figure BDA0001710440650000061
Then obtain
Figure BDA0001710440650000062
Wherein
Figure BDA0001710440650000063
Is the effective integer channel matrix of the first packet.
Further, by
Figure BDA0001710440650000064
Respectively represent
Figure BDA0001710440650000065
B 1 ，A 1 Line m of
Figure BDA0001710440650000066
Let
Figure BDA0001710440650000067
Equation (4) becomes
Figure BDA0001710440650000068
Further, an effective noise variance of
Figure BDA0001710440650000069
Further, to b 1m Taking reciprocal
Figure BDA00017104406500000610
Making equation (7) equal to zero
Figure BDA0001710440650000071
An optimal equalization matrix B is obtained 1 And the resulting equalization matrix B 1 The effective noise can be minimized.
Further, make
Figure BDA0001710440650000072
a 1m ＝a 1 The formula (8) is changed into
Figure BDA0001710440650000073
B of formula (9) opt,1m Substituting into formula (6) to obtain
Figure BDA0001710440650000074
The optimal effective integer channel matrix A can be obtained 1 Expression (2)
Figure BDA0001710440650000075
Further, H in the formula (10) 1 By H 1 After Singular Value Decomposition (SVD) substitution
Figure BDA0001710440650000076
Wherein V 1 Is formed by
Figure BDA0001710440650000077
Of the feature vectors of (a), D 1 Is a diagonal matrix whose m-th term is represented as
Figure BDA0001710440650000078
p m Is H 1 The m-th singular value of the matrix.
Through the above steps, two important matrixes are obtained when the rounding detection is used after the received symbols are grouped, namely an equalization matrix B 1 And an effective integer channel matrix A 1 。
A block diagram of a MIMO system based on the concept of the forced integer decoding algorithm is shown in FIG. 1, and the systemThe number of transmit antennas of (a) is N, the number of receive antennas is N, and N is an even number. Suppose from
Figure BDA0001710440650000079
To generate N mutually independent, uniform information sequences w of length k 1 ,…,w N . For channel matrices
Figure BDA00017104406500000710
Meaning that elements of H are all subject to independent homodistribution
Figure BDA00017104406500000711
And set h to remain unchanged for M slots and independent of the next slot. This system model uses a horizontal coding scheme of N layers, i.e. different antenna-independent transmission information is used, the transmission information of the r (1 ≦ r ≦ N) th layer is fed into the trellis encoder ε:
Figure BDA00017104406500000712
i.e. a message
Figure BDA00017104406500000713
Mapping to trellis code words
Figure BDA00017104406500000714
Wherein Λ is
Figure BDA00017104406500000715
And the additive and reflection operations are enclosed in the cell. The code words of the trellis codebook are elements of the trellis and any linear combination of trellis codes is itself a trellis code. Use of
Figure BDA00017104406500000716
Indicating the matrix of the transmitted code word, then the received matrix
Figure BDA00017104406500000717
Can be expressed as
Figure BDA0001710440650000081
Figure BDA0001710440650000082
Representing a noise matrix whose elements are independent identically distributed gaussian random variables. The channel information can only be known by the receiver. In the forced integer system model, the equation (12) is multiplied by an equalization matrix B to obtain an effective integer channel matrix A, which must be reversible to obtain
Figure BDA0001710440650000083
Wherein
Figure BDA0001710440650000084
As a component of the signal, the signal component,
Figure BDA0001710440650000085
is effectively noise.
Optimal integer channel matrix A obtained in group forced integer 1 Namely formula (11)
Figure BDA0001710440650000086
Has a Gram matrix G 1 ＝V 1 D 1 V 1 H The shortest vector problem of the lattice of (2), and because of G 1 Is a symmetric positive definite matrix, G can be defined 1 Write to G 1 ＝L 1 L 1 H ，
Figure BDA0001710440650000087
And L is 1 ＝V 1 D 1 1/2 May generate a lattice. Using LLL algorithm to make L 1 Becomes a lattice generating matrix L' 1 ，L 1 And L' 1 Generate the same lattice Λ and then pass
Figure BDA00017104406500000810
Row ofThe vector can obtain the effective integer channel matrix A 1 The pseudo code for this step is as follows:
inputting: h, H 1
And (3) outputting: a. the 1 ,B 1
1:S←ρ -1 I+HH H
2:(U 1 ,Σ,V 1 )←SVD(H 1 ) Wherein Σ ═ diag (p) 1 ,p 2 )
3:
Figure BDA0001710440650000088
4:L 1 ←V 1 D 1 1/2
5:L 1 '←LLL(L 1 )
6, obtaining A 1 ＝L 1 'L 1 -1 And
Figure BDA0001710440650000089
as is apparent from the above description, in the block rounding algorithm, only the LLL algorithm needs to be used to find a 2 × 2 integer matrix, instead of finding an N × N integer matrix like the block rounding algorithm, so the block rounding has lower decoding complexity.
Setting M to 1, receiving matrix Y 1 Linear vectorization can be obtained
Figure BDA0001710440650000091
It should be noted that it is preferable that,
Figure BDA0001710440650000092
respectively representing the received and transmitted symbol vectors in the first packet, and y 1 ,w 1 Respectively representing the first received and transmitted symbols, note
Figure BDA0001710440650000093
And y 1 ,w 1 Meaning the difference between the meanings. Thereafter utilize
Figure BDA0001710440650000094
And with
Figure BDA0001710440650000095
The received complex vector of the first group is changed into a real vector, and the formula (14) is changed into
Figure BDA0001710440650000096
Wherein
Figure BDA0001710440650000097
Limited ring
Figure BDA0001710440650000098
J is a power of 2. Under these settings, the decoding process using the packet rounding algorithm is as follows:
step one is
Figure BDA0001710440650000099
Upper lattice decoding: will be provided with
Figure BDA00017104406500000910
In which each element is decoded into an integer field
Figure BDA00017104406500000911
To obtain the point closest thereto, i.e.
Figure BDA00017104406500000912
Wherein
Figure BDA00017104406500000918
Representing a rounding operation.
Step two, code word projection:
Figure BDA00017104406500000913
to pair
Figure BDA00017104406500000914
Performing a mold-removing operation to obtain
Figure BDA00017104406500000915
Step three-lattice code word decoupling: based on linear equations
Figure BDA00017104406500000916
Obtaining a decoded vector
Figure BDA00017104406500000917
After the above steps, the decoding symbols of the first packet can be obtained, then the decoding symbols of other packets can be obtained by using the same steps as the first packet in the rest packets, and the original transmission information before the packets are not grouped can be obtained by collecting the decoding symbols of all the packets.
In the simulation platform of this embodiment, in a flat rayleigh fading channel, 4-QAM modulation is adopted, a codeword is generated from a two-dimensional integer trellis code, and the number of antennas is 4 transmission 4 reception, 8 transmission 8 reception, and 12 transmission 12 reception, respectively. In the present embodiment, under the above simulation platform, a packet forced rounding decoding algorithm and a forced zero decoding algorithm are simulated, and Bit Error Rate (BER for short) performance obtained by comparison and analysis is obtained. Fig. 2 shows BER performance using a packet rounding decoding algorithm and a zero-forcing decoding algorithm in 4 × 4 channels, respectively, fig. 3 shows BER performance using a packet rounding decoding algorithm and a zero-forcing decoding algorithm in 8 × 8 channels, respectively, and fig. 4 shows BER performance using a packet rounding decoding algorithm and a zero-forcing decoding algorithm in 12 × 12 channels, respectively. From these embodiments, it can be known that the packet rounding decoding algorithm has better BER performance than the rounding detection, and as the number of antennas increases, the advantage of the packet rounding in performance becomes more and more obvious.
The above examples are to be construed as merely illustrative and not limitative of the remainder of the disclosure. After reading the description of the invention, the skilled person can make various changes or modifications to the invention, and these equivalent changes and modifications also fall into the scope of the invention defined by the claims.

Claims (1)

1. A low complexity packet decoding method based on MIMO system is characterized in that the method comprises the following steps:
firstly, grouping received symbols, wherein each group of received signals are independently decoded by using a forced integer algorithm to obtain a transmitted code word, the concept of the forced integer algorithm is to find an effective integer matrix A and an equilibrium matrix B, each group decodes the symbols of the groups respectively, and the transmitted code words which are independently decoded by each group are aggregated and are the total transmitted symbols before no grouping, so that the most original transmitted information before no grouping is obtained;
the decoding by using the improved rounding algorithm specifically comprises:
step one is
Figure FDA0003683939540000011
Upper lattice decoding: will be provided with
Figure FDA0003683939540000012
In which each element is decoded into an integer field
Figure FDA0003683939540000013
To obtain the point closest thereto, i.e.
Figure FDA0003683939540000014
Wherein
Figure FDA0003683939540000015
Representing a rounding operation;
Figure FDA0003683939540000016
in order to equalize the matrix, the matrix is,
Figure FDA0003683939540000017
a vector of received symbols for a first packet;
Figure FDA0003683939540000018
receiving a signal matrix Y for a first packet 1 Is represented by a vectorization of (a),
Figure FDA0003683939540000019
to represent
Figure FDA00036839395400000110
Vector after getting whole according to element;
step two, code word projection:
Figure FDA00036839395400000111
to pair
Figure FDA00036839395400000112
Performing a mold-taking operation to obtain
Figure FDA00036839395400000113
J represents the constellation order;
step three-lattice code word decoupling: based on linear equations
Figure FDA00036839395400000114
Obtaining a decoded vector
Figure FDA00036839395400000115
Obtaining the decoding symbols of the first grouping through the steps, then obtaining the decoding symbols of other groups in the rest grouping by using the same steps as the first grouping, and collecting the decoding symbols of all the groupings to obtain the original sending information before the grouping;
optimal effective integer matrix A 1 Expression (2)
Figure FDA00036839395400000116
Optimal equalization matrix B 1 Expression (2)
Figure FDA00036839395400000117
Figure FDA00036839395400000118
For the optimal equalization matrix B 1 Row m;
the grouping of the received signals specifically includes: firstly, two columns of a channel matrix H are taken as a group, received symbols are divided into L groups, L is N/2, N is the number of receiving and transmitting antennas, and then the MIMO system equation is
Figure FDA00036839395400000119
Figure FDA0003683939540000021
Is a receiving matrix; h q Represents the q-th group after the channel matrix H grouping, wherein
Figure FDA0003683939540000022
SNR represents the average signal-to-noise ratio, X, at each receive antenna q Representing the transmitted symbols of the q-th group, q representing the q-th group, and under the condition of no loss of generality, only the first group is studied to find the effective integer matrix A and the equalization matrix B, and when one group is studied, other groups are used as noise, and the signal model of the first group is
Figure FDA0003683939540000023
H 1 、X 1 Representing the first grouping of the channel matrix H and the transmit codeword matrix X, respectively, although the signal model of equation (2) is the same as the system equation of equation (1), in equation (2), only
Figure FDA0003683939540000024
Is taken as an effective signal component, wherein
Figure FDA0003683939540000025
Figure FDA0003683939540000026
Represents an N x 2 complex matrix, and
Figure FDA0003683939540000027
are all considered as effective noise components;
the use of an equalization matrix for the first packet signal
Figure FDA0003683939540000028
Then obtain
Figure FDA0003683939540000029
Wherein
Figure FDA00036839395400000210
Is the effective integer channel matrix of the first packet;
by using
Figure FDA00036839395400000211
Respectively represent
Figure FDA00036839395400000212
B 1 ,A 1 Line m of
Figure FDA00036839395400000213
Let
Figure FDA00036839395400000214
Equation (4) becomes
Figure FDA00036839395400000215
Using a horizontal coding scheme of N layers, i.e., using different antenna-independent transmission information, the transmission information of the r (1 ≦ r ≦ N) th layer is fed into the trellis encoder ε:
Figure FDA0003683939540000031
i.e. a message
Figure FDA0003683939540000032
Mapping to trellis code words
Figure FDA0003683939540000033
Wherein Λ is
Figure FDA0003683939540000034
And the additive operation and the reflection operation are enclosed in the cell; the code words of the trellis code book are elements of the trellis, and any linear combination of the trellis codes is the trellis code; use of
Figure FDA0003683939540000035
Indicating the matrix of the transmitted code word, then the received matrix
Figure FDA0003683939540000036
Can be expressed as
Figure FDA0003683939540000037
Figure FDA0003683939540000038
Representing a noise matrix, wherein elements of the noise matrix are independent and identically distributed Gaussian random variables; channel information can only beIs known by the receiver; in the forced integer system model, the equation (12) is multiplied by an equalization matrix B to obtain an effective integer channel matrix A, which must be reversible to obtain
Figure FDA0003683939540000039
Wherein
Figure FDA00036839395400000310
As a component of the signal, the signal component,
Figure FDA00036839395400000311
is effective noise;
optimal integer channel matrix A obtained in group forced integer 1 Namely formula (11)
Figure FDA00036839395400000312
Has a Gram matrix G 1 ＝V 1 D 1 V 1 H The shortest vector problem of the lattice of (2), and because of G 1 Is a symmetric positive definite matrix, G can be set 1 Write to G 1 ＝L 1 L 1 H ，
Figure FDA00036839395400000313
L 1 Represents an intermediate variable, which is L 1 ＝V 1 D 1 1/2 May generate a lattice Λ; using LLL algorithm to make L 1 Becomes a lattice generating matrix L' 1 ，L 1 And L' 1 Generate the same lattice Λ and then pass
Figure FDA00036839395400000314
The row vector of the channel matrix A is obtained 1 。
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