CN111464217B - Improved SVD precoding algorithm for MIMO-OFDM - Google Patents

Improved SVD precoding algorithm for MIMO-OFDM Download PDF

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CN111464217B
CN111464217B CN202010154734.0A CN202010154734A CN111464217B CN 111464217 B CN111464217 B CN 111464217B CN 202010154734 A CN202010154734 A CN 202010154734A CN 111464217 B CN111464217 B CN 111464217B
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CN111464217A (en
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蒋轶
胡婉辰
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Fudan University
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    • 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
    • H04B7/0413—MIMO systems
    • H04B7/0456—Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00—Modulated-carrier systems
    • H04L27/26—Systems using multi-frequency codes
    • H04L27/2601—Multicarrier modulation systems
    • H04L27/2626—Arrangements specific to the transmitter only
    • H04L27/2627—Modulators
    • H04L27/2628—Inverse Fourier transform modulators, e.g. inverse fast Fourier transform [IFFT] or inverse discrete Fourier transform [IDFT] modulators
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L5/00—Arrangements affording multiple use of the transmission path
    • H04L5/0001—Arrangements for dividing the transmission path
    • H04L5/0003—Two-dimensional division
    • H04L5/0005—Time-frequency
    • H04L5/0007—Time-frequency the frequencies being orthogonal, e.g. OFDM(A), DMT
    • H04L5/001—Time-frequency the frequencies being orthogonal, e.g. OFDM(A), DMT the frequencies being arranged in component carriers

Abstract

The invention belongs to the technical field of MIMO communication, and particularly relates to an improved SVD precoding algorithm for MIMO-OFDM (multiple input multiple output-orthogonal frequency division multiplexing) under the background of a frequency domain selective fading channel. The core of the invention aims to optimize the SVD of the MIMO channel matrix under each subcarrier so as to avoid the smoothness of the channel in the frequency domain from being damaged by the traditional pre-coding matrix based on the SVD. The invention provides a modeling method for quantizing channel smoothness, and an algorithm for designing and improving an SVD pre-coding matrix. The optimization simulation result shows that the smoothness of an equivalent channel in the MIMO-OFDM system on the frequency domain can be obviously improved by the algorithm, and the channel estimation is facilitated.

Description

Improved SVD precoding algorithm for MIMO-OFDM
Technical Field
The invention belongs to the technical field of MIMO communication, and particularly relates to an improved SVD precoding algorithm for MIMO-OFDM.
Background
In 5G wireless systems, MIMO or massive MIMO and Orthogonal Frequency Division Multiplexing (OFDM) are often combined for frequency selective fading channels. Each subcarrier corresponds to a respective independent fading coefficient, but different SVD precoding is used for each subcarrier, which deteriorates the smoothness of the channel and makes channel estimation difficult.
Aiming at MIMO-OFDM, an improved SVD (singular value decomposition) precoding algorithm is designed to improve the smoothness of an equivalent channel in a frequency domain, and the method is a problem with practical significance. The existing related work can be seen [1] [2 ]. However, the focus of these efforts is to interpolate the precoding matrix, or to reduce the feedback amount of the precoding matrix [2], or to make the precoding matrix somewhat smooth [1 ]. The method of the invention is directly oriented to the equivalent channel matrix, improves the smoothness of the equivalent channel by utilizing the degree of freedom provided by the non-uniqueness of the SVD, and brings convenience to channel estimation.
Disclosure of Invention
The invention aims to provide a method for optimizing SVD (singular value decomposition) so as to avoid the phenomenon that the smoothness of a channel in a frequency domain is damaged by the traditional SVD precoding.
The core of the invention is that a plurality of orthogonal sub-channels are obtained by respectively using SVD pre-coding on MIMO channels on different sub-carriers, the phases of the orthogonal sub-channels are optimized on different sub-carriers, and each corresponding orthogonal sub-channel on all sub-carriers is transformed to a time domain through discrete inverse Fourier transform. By utilizing the dual property of time domain and frequency domain, the energy of each subchannel in the time domain is concentrated at the head of the time domain by optimizing the phase on different subcarriers, thereby improving the smoothness of the subchannel in the frequency domain. The method has low algorithm complexity, can obviously improve the smoothness of the channel in the frequency domain, and brings convenience to channel estimation.
The invention provides an improved SVD pre-coding algorithm, which comprises the following specific steps:
(1) problem resolution
(1.1) consideration of NfMIMO-OFDM system of a subcarrier with M transmitting antennastThe number of receiving antennas is MrOn the ith subcarrier, the received signal may be represented as:
y(i)＝H(i)s(i)+z(i) (1)
wherein the content of the first and second substances,
Figure BDA0002403681170000011
is a frequency domain channel coefficient matrix of rank K,
Figure BDA0002403681170000012
it is indicated that the signal is transmitted,
Figure BDA0002403681170000013
is the received signal.
Figure BDA0002403681170000014
Is zero-mean circularly symmetric complex gaussian noise. When the channel state is known at the transmitting end, it can be calculated
Figure BDA0002403681170000015
And use of ViOrthogonalizing a channel [1] as a coding matrix on the ith pre-subcarrier]Thus obtaining an equivalent channel
Figure BDA0002403681170000021
However, precoding each subcarrier will destroy the smoothness of the channel in the frequency domain, and bring difficulties for the subsequent channel estimation. The current solution is to use the same precoding matrix for multiple adjacent subcarriers, but this results in performance loss. We propose an improved SVD algorithm for this as follows.
First, note that SVD decomposition is not unique. In fact, SVD [4] is performed on H:
H＝UΛVH (2)
if order:
Figure BDA0002403681170000022
using diagonal matrices but reciprocity, H ═ UD Λ D-1VH(ii) a After uptake of D into U and V gives:
Figure BDA0002403681170000023
comprises the following steps:
Figure BDA0002403681170000024
note that for any phase
Figure BDA0002403681170000025
The expressions (4) are all SVD.
Order to
Figure BDA0002403681170000026
Further obtaining:
Figure BDA0002403681170000027
at this time, the ith subcarrier (i ═ 1, …, N)f) The equivalent channels above are:
Figure BDA0002403681170000028
therefore, we can adjust D(i)To improve the equivalent channel by the phase of the diagonal elements of
Figure BDA0002403681170000029
And the smoothness in the frequency domain brings convenience for subsequent channel estimation.
(1.2) in the formula (6)
Figure BDA00024036811700000210
There are K orthogonal subchannels available for transmitting the K data streams. All N arefThe k-th sub-channel on the sub-carrier is arranged together to obtain an Mr×NfOf (2) matrix
Figure BDA00024036811700000211
Figure BDA00024036811700000212
Wherein the content of the first and second substances,
Figure BDA00024036811700000213
Figure BDA00024036811700000214
while
Figure BDA00024036811700000215
Represents U(i)The kth column of (1);
Figure BDA00024036811700000216
represents H(i)The k-th singular value of (a);
Figure BDA00024036811700000217
represents D(i)The kth diagonal element of (1).
According to Fourier transform properties
Figure BDA0002403681170000031
Where F is an inverse Fourier transform (IDFT) matrix:
Figure BDA0002403681170000032
Figure BDA0002403681170000033
according to the duality of time domain and frequency domain, in order to let
Figure BDA0002403681170000034
It is smoother in the frequency domain and should be made such that its energy in the time domain is concentrated at the head. Without loss of generality, to
Figure BDA0002403681170000035
Performing N-point inverse Fourier transform (N is more than or equal to N)f). The number of points N determines the density of the sampling in the frequency domain.
The problem is solved by a diagonal matrix D(i)The optimization problem of (2) is specifically as follows:
the first optimization problem is to extract P rows representing time domain headers from the F matrix and take the first N rowsfRow, composition of
Figure BDA0002403681170000036
FL＝F(1:P,1:Nf) (10)
Solving a maximization function:
Figure BDA0002403681170000037
the function is equivalent frequency domain channel, after inverse Fourier transform, the first P time samples of energy;
the second optimization problem is to extract the M-N-P rows of the F matrix excluding the head P row and take the first NfColumn, composition FH,
Figure BDA0002403681170000038
FH＝F(S+1:N,1:Nf) (12)
Solving the minimization function corresponding to equation (11):
Figure BDA0002403681170000039
the function is the energy of the tail M time samples of the equivalent frequency domain channel after the inverse Fourier transform.
(2) Solving of problems
I.e. solve the problems (11) and (13).
For problem (11), the property of vectorization according to the matrix is exploited:
Figure BDA0002403681170000041
Figure BDA0002403681170000042
is the Kronecker product, fnIs FLThe column vector of (2). To simplify the representation, A is represented as:
Figure BDA0002403681170000043
the optimization function (11) can be equivalently converted into:
Figure BDA0002403681170000044
Figure BDA0002403681170000045
wherein the content of the first and second substances,
Figure BDA0002403681170000046
according to the definition of F-norm, the following results are obtained:
Figure BDA0002403681170000047
order to
Figure BDA0002403681170000048
Further obtaining an optimization objective equation:
Figure BDA0002403681170000049
the solving method comprises the following steps: random initialization
Figure BDA00024036811700000411
Performing a number of iterations on x, using xiRepresenting the result of the ith iteration, then the solution for the (i + 1) th iteration is:
Figure BDA00024036811700000410
wherein, the angle represents the operation of taking the phase of the vector element by element; and iterating until convergence.
For problem (13):
similar to (16), the objective function is further expressed as
Figure BDA0002403681170000051
Figure BDA0002403681170000052
Wherein A is as defined for (13), but fnIs FH(instead of F)L) A column vector of (a);
the solution of equation (20) is more complicated than equation (16). The solution method is explained as follows:
order to
Figure BDA0002403681170000053
Further obtaining an optimization objective equation:
Figure BDA0002403681170000054
using the coordinate descent method, x is split into:
Figure BDA0002403681170000055
resolving the above formula (21) into:
Figure BDA0002403681170000056
fixing
Figure BDA0002403681170000057
And define
Figure BDA0002403681170000058
Rewrite (23) to obtain:
Figure BDA0002403681170000059
thus, we obtain:
Figure BDA00024036811700000510
(3)AHsimplified operation of A
For solving the objective function algorithm, A can be simplifiedHA operation is performed to reduce the operation amount, and the specific method is：
By using the basic properties of the Kronecker product, the following are obtained:
Figure BDA00024036811700000511
Figure BDA0002403681170000061
wherein (F)HF) It can be calculated in advance to obtain that the product is the Hadamard product of the matrix.
Compared with the prior art, the invention has the advantages that: the algorithm is suitable for an MIMO-OFDM system, and system simulation shows that the smoothness of a channel pre-coded by using the improved SVD is obviously superior to that of a channel pre-coded by using the common SVD.
Drawings
Fig. 1 is a time domain response of an element of an equivalent channel.
Fig. 2 shows the variation of the imaginary part of the element at a specific position of the equivalent frequency domain channel with the label of the subcarrier.
Detailed Description
The technical solution of the present invention will be described in detail with reference to the accompanying drawings and embodiments.
As an embodiment, the invention simulates the subcarrier number N by a computerfA 4 x 192 massive MIMO channel of 272. Performing improved SVD decomposition, and calculating diagonal matrix D according to given formula(i)To obtain 4 × 4 equivalent channels
Figure BDA0002403681170000062
Figure BDA0002403681170000063
Obtaining 4 x 4 equivalent channels using ordinary SVD decomposition
Figure BDA0002403681170000064
In FIG. 1, without loss of generality, at each
Figure BDA0002403681170000065
And each
Figure BDA0002403681170000066
All take elements with a certain fixed position to form two elements with the length of NfRespectively performing inverse Fourier transform on the sequences to obtain time domain graphs. These were compared. The results show that the channel using the improved SVD is seen to be significantly more energy concentrated at the head of the time domain than the channel obtained by the normal SVD decomposition, while the amplitudes at the middle and tail are significantly suppressed. Thus, the improved SVD is shown to make the equivalent channel smoother in the frequency domain.
In FIG. 2, again without loss of generality, at each
Figure BDA0002403681170000067
And each
Figure BDA0002403681170000068
Taking some fixed position element, in each original channel H without precoding(i)Also taking the same fixed position element. The three subgraphs respectively show the real part and the imaginary part of an original channel, an equivalent channel pre-coded by using the traditional SVD and the equivalent channel pre-coded by using the improved SVD provided by the invention in the frequency domain, and the horizontal axis is the serial number of the subcarrier. As can be seen from the figure, original H(i)The method has certain smoothness in a frequency domain, the smoothness is completely destroyed by ordinary SVD decomposition, and after the improved SVD algorithm is adopted, a point line graph presented by an obtained equivalent channel is smoother than an original channel, and the smoothness is beneficial to channel estimation.
Reference to the literature
[1]K.Schober,R.-A.Pitaval,and R.Wichman,“Improved User-Specific Channel Estimation Using Geodesical Interpolation at the Transmitter,”IEEE Wireless Communications Letters,vol.4,no.2,pp.165–168,Apr.2015.
[2]Jihoon Choi and R.W.Heath,"Interpolation based transmit beamforming for MIMO-OFDM with limited feedback,"in IEEE Transactions on Signal Processing,vol.53,no.11,pp.4125-4135,Nov.2005.
[3]Palomar,D.P.,et al.“Joint Tx-Rx Beamforming Design for Multicarrier MIMO Channels:A Unified Framework for Convex Optimization.”IEEE Transactions on Signal Processing,vol.51,no.9,2003,pp.2381–2401.
[4]G.H.Golub and C.F.Van Loan，“Matrix Computation.”ThirdEdition,The John Hopkins University Press,1996。

Claims (2)

1. An improved SVD precoding matrix algorithm for MIMO-OFDM is characterized by comprising the following specific steps:
(1) problem resolution
(1.1) for having NfMIMO-OFDM system of a subcarrier with M transmitting antennastThe number of receiving antennas is MrOn the ith subcarrier, the received signal is represented as:
y(i)＝H(i)s(i)+z(i) (1)
wherein the content of the first and second substances,
Figure FDA0003307077710000011
is a frequency domain channel coefficient matrix of rank K,
Figure FDA0003307077710000012
it is indicated that the signal is transmitted,
Figure FDA0003307077710000013
is a received signal;
Figure FDA0003307077710000014
is zero-mean circularly symmetric complex gaussian noise; when the channel state is known at the transmitting end, it can be calculated
Figure FDA0003307077710000015
And use of ViAs a firstCoding matrices on i subcarriers to orthogonalize the channel, thus obtaining an equivalent channel
Figure FDA0003307077710000016
The algorithm of the improved SVD precoding matrix is as follows;
first, note that SVD decomposition is not unique; in fact, the SVD decomposition is performed on H:
H＝UΛVH (2)
let the diagonal matrix:
Figure FDA0003307077710000017
using reciprocity of diagonal matrices, H ═ UD Λ D-1VH(ii) a After uptake of D into U and V gives:
Figure FDA0003307077710000018
thus, there are:
Figure FDA0003307077710000019
note that for any phase
Figure FDA00033070777100000110
The formulas (4) are all SVD;
order to
Figure FDA00033070777100000111
Further obtaining:
Figure FDA00033070777100000112
at this time, the ith subcarrier (i ═ 1, …, N)f) The equivalent channels above are:
Figure FDA00033070777100000113
therefore, we can adjust D(i)To improve the equivalent channel by the phase of the diagonal elements of
Figure FDA00033070777100000114
Smoothness in the frequency domain, thereby bringing convenience to subsequent channel estimation;
1.2) in the formula (6)
Figure FDA0003307077710000021
There are K orthogonal subchannels available for transmitting the K data streams; all N arefThe k-th sub-channel on the sub-carrier is arranged together to obtain an Mr×NfOf (2) matrix
Figure FDA0003307077710000022
Figure FDA0003307077710000023
Wherein the content of the first and second substances,
Figure FDA0003307077710000024
while
Figure FDA0003307077710000025
Represents U(i)The kth column of (1);
Figure FDA0003307077710000026
represents H(i)The k-th singular value of (a);
Figure FDA0003307077710000027
represents D(i)The kth diagonal element of (1);
according to the fourier transform property:
Figure FDA0003307077710000028
where F is an inverse Fourier transform (IDFT) matrix:
Figure FDA0003307077710000029
Figure FDA00033070777100000210
according to the duality of time domain and frequency domain, in order to let
Figure FDA00033070777100000211
The energy in the time domain is concentrated on the head part by being smoother in the frequency domain; to pair
Figure FDA00033070777100000212
Performing N-point inverse Fourier transform (N is more than or equal to N)f) (ii) a The point number N determines the sampling density on the frequency domain;
diagonal matrix D(i)The problem is solved by the following two equivalent optimization problems:
the first optimization problem is to extract P rows representing time domain headers from the F matrix and take the first N rowsfColumn, composition FL,
Figure FDA00033070777100000213
FL＝F(1:P,1:Nf) (10)
Solving a maximization function:
Figure FDA00033070777100000214
the function is equivalent frequency domain channel, after inverse Fourier transform, the first P time samples of energy;
the second optimization problem is to extract the M-N-P rows of the F matrix excluding the head P row and take the first NfColumn, composition FH,
Figure FDA00033070777100000215
FH＝F(P+1:N,1:Nf) (12)
Solving the minimization function corresponding to equation (11):
Figure FDA0003307077710000031
the function is the energy of tail M time samples of the equivalent frequency domain channel after inverse Fourier transform;
(2) solving of problems
For problem (11), the property of vectorization according to the matrix is exploited:
Figure FDA0003307077710000032
Figure FDA0003307077710000033
is the Kronecker product, fnIs FLThe column vector of (a) is represented as:
Figure FDA0003307077710000034
the optimization function (11) is equivalently transformed into:
Figure FDA0003307077710000035
Figure FDA0003307077710000036
wherein the content of the first and second substances,
Figure FDA0003307077710000037
according to the definition of F-norm, the following results are obtained:
Figure FDA0003307077710000038
order to
Figure FDA0003307077710000039
Further obtaining an optimization objective equation:
Figure FDA00033070777100000310
the solving method comprises the following steps: random initialization
Figure FDA00033070777100000311
Performing a number of iterations on x, using xiRepresenting the result of the ith iteration, then the solution for the (i + 1) th iteration is:
Figure FDA0003307077710000041
wherein, the angle represents the operation of taking the phase of the vector element by element; iterating until convergence;
for problem (13):
similar to (16), the objective function is further expressed as
Figure FDA0003307077710000042
Figure FDA0003307077710000043
Wherein A is as defined for formula (15), but fnIs FHA column vector of (a);
the solution method of equation (20) is explained as follows:
order to
Figure FDA0003307077710000044
Further obtaining an optimization objective equation:
Figure FDA0003307077710000045
using the coordinate descent method, x is split into:
Figure FDA0003307077710000046
resolving the above formula (21) into:
Figure FDA0003307077710000047
fixing
Figure FDA0003307077710000048
And define
Figure FDA0003307077710000049
Rewrite (23) to obtain:
Figure FDA00033070777100000410
thus, we obtain:
Figure FDA00033070777100000411
2. the improved SVD precoding matrix algorithm of claim 1, wherein for AHA, carrying out simplified operation, wherein the specific method comprises the following steps: by using the basic properties of the Kronecker product, the following are obtained:
Figure FDA00033070777100000412
Figure FDA0003307077710000051
wherein (F)HF) It can be calculated in advance to obtain that the product is the Hadamard product of the matrix.
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