CN109302224B - Hybrid beamforming algorithm for massive MIMO - Google Patents

Hybrid beamforming algorithm for massive MIMO Download PDF

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CN109302224B
CN109302224B CN201811213984.6A CN201811213984A CN109302224B CN 109302224 B CN109302224 B CN 109302224B CN 201811213984 A CN201811213984 A CN 201811213984A CN 109302224 B CN109302224 B CN 109302224B
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蒋轶
冯艺萌
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Fudan University
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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B7/00Radio transmission systems, i.e. using radiation field
    • H04B7/02Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0615Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal
    • H04B7/0617Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal for beam forming
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B7/00Radio transmission systems, i.e. using radiation field
    • H04B7/02Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413MIMO systems
    • H04B7/0456Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting

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Abstract

The invention belongs to the field of millimeter wave large-scale MIMO systems, and particularly relates to a hybrid beam forming algorithm for large-scale MIMO. This new algorithm aims at jointly optimizing the analog domain beamforming and the digital domain beamforming to maximize the spectral efficiency. The invention is divided into two parts: firstly, simulating a beam forming algorithm; and secondly, after the analog beam shaper is given, the digital beam shaper is obtained according to a water injection method. The algorithm is suitable for different types of analog networks, including continuous adjustable phase shifter networks, finite bit adjustable phase shifter networks, switching networks and the like. Simulation results show that the performance of the algorithm is very close to the optimal performance of all-digital beam forming; and the performance of the hybrid beam forming based on the switch network is close to that of the phase shifter network, thereby being more beneficial to the realization of engineering.

Description

Hybrid beamforming algorithm for massive MIMO
Technical Field
The invention belongs to the field of MIMO communication, and particularly relates to a hybrid beam forming algorithm for large-scale MIMO.
Background
The 5G millimeter wave technology improves the communication bandwidth by hundreds of mega even kilohertz, and installs dozens of even hundreds of antennas at the base station, thereby greatly improving the channel capacity of the cell. However, if a conventional communication receiver is used, the digital domain needs to process hundreds of high-speed code streams in real time, and the difficulty is very great. Researchers put forward that Analog domain Beamforming/ABF is performed before a digital-to-Analog converter (ADC), high-dimensional signals received by M antennas are compressed to N dimensions (N < M), antenna gains of large-scale MIMO are saved through ABF, and meanwhile, signal dimensions are greatly compressed, so that the operation amount of a digital domain and the required number of ADCs are greatly reduced, and hardware cost is obviously reduced. This technique is known as Hybrid Beamforming/HBF.
For the above-mentioned technologies, some research work on Hybrid Beamforming/HBF has been published. The traditional analog domain network is mainly divided into a phase shifter and a switch structure, and how to solve different analog domain network structures (including a switch, a phase shifter with fixed phase, a phase shifter with variable phase and different resolutions, and the like) in one way is still a difficult problem.
Disclosure of Invention
To overcome the disadvantages of the prior art, the present invention aims to provide a hybrid beamforming algorithm for different types of analog networks for massive MIMO. The algorithm is suitable for a phase shifter network, a switch network and a phase shifter and switch composite network in large-scale MIMO, so that the joint optimization of analog domain beam forming and digital domain beam forming is facilitated, and the maximization of the frequency spectrum efficiency is realized.
The invention provides a hybrid beam forming algorithm for large-scale MIMO, which comprises the following specific steps:
first, consider a single-user millimeter wave MIMO system, signal vector
Figure GDA0002938000450000011
Precoder via digital baseband
Figure GDA0002938000450000012
And analog precoder
Figure GDA0002938000450000013
Then pass throughMtAnd transmitting by using the antenna. The sending antenna is:
x=FRFFBBs (1)
suppose that
Figure GDA0002938000450000014
Wherein
Figure GDA0002938000450000015
It is shown that it is desirable to,
Figure GDA0002938000450000016
is dimension NsAnd (4) unit array. After passing through a flat fading channel, we will get the baseband signal:
Y=HFRFFBBs+z (2)
Figure GDA0002938000450000017
is a matrix of the channels and is,
Figure GDA0002938000450000018
is a covariance matrix of
Figure GDA0002938000450000019
Due to the fact that the total transmitted power is normalized by the noise power
Figure GDA00029380004500000110
The input signal-to-noise ratio is therefore:
Figure GDA0002938000450000021
the spectral efficiency is:
Figure GDA0002938000450000022
the purpose of the algorithm provided by the invention is to improve the spectrum efficiency, and the following problems are solved:
Figure GDA0002938000450000023
Subject to
Figure GDA0002938000450000024
Figure GDA0002938000450000025
wherein the first constraint is related to the input SNR and the set of S in the second constraint depends on the set of RF feedback networks: phase shifter network for infinite resolution S ═ e:φ∈[0,2π]}; for b-bit resolution phase shifter networks
Figure GDA0002938000450000026
For a switching network S ═ {0,1 }; for phase shifter plus switch network S ═ e:φ∈[0,2π]} {0} or
Figure GDA0002938000450000027
FRFThe constraint of (i, j) ∈ S distinguishes hybrid beamforming from traditional all-digital beamforming.
Secondly, optimizing the analog domain beam forming matrix FRF
Before introducing the optimization method of the analog domain beamforming matrix, we introduce two majorization theories first.
Definition 1: vector quantity
Figure GDA0002938000450000028
Is vector
Figure GDA0002938000450000029
The product is mainly defined as:
Figure GDA00029380004500000216
if it is not
Figure GDA00029380004500000210
Wherein x [ i ] and y [ i ] are the elements of x and y, respectively, which are the i-th elements.
Definition 2: a function phi:
Figure GDA00029380004500000211
is called at
Figure GDA00029380004500000212
The product of Schur-covex, if:
Figure GDA00029380004500000213
in the following, we look at the optimization of the analog domain beamforming, assuming Ns=NRFN, for the case where ρ is sufficiently large, satisfies
Figure GDA00029380004500000214
It is easy to prove that the spectrum efficiency is
Figure GDA00029380004500000215
We now discuss the two definitions that we have used for the dominance mentioned above.
Theorem 1: function(s)
Figure GDA0002938000450000031
Is the predominant Schur-covex. For each lambdaiIn other words, C (λ) is a non-decreasing function.
Given decomposition of QR into HUsRFQR, definition
Figure GDA0002938000450000032
Consisting of diagonal elements of R, which we know are HURFIs mainly multiplied by the singular value of, and λ is HURFSquare of the singular value of (a). Therefore, the temperature of the molten metal is controlled,
Figure GDA0002938000450000039
according to theorem 1, the method
C(λ)≥C(|r|2) (13)
By maximizing the lower bound C (| r! y of C (λ)2) Instead of maximizing C (λ). Next we discuss the structure of the interior of the R diagonal:
theorem 2: concerning HU in QR decompositionRFThe diagonal elements of R of QR,
Figure GDA0002938000450000033
wherein,
Figure GDA0002938000450000034
Ui-1represents URFFirst i-1 column, uiRepresents URFColumn i.
Theorem 3: for URFThe column (c) of (a),
Figure GDA0002938000450000035
wherein,
Figure GDA0002938000450000036
Fi-1is represented by FRFFirst i-1 column of (1), fiIs represented by FRFColumn i.
From theorems 2 and 3, we can deduce
Figure GDA0002938000450000037
From there, we can see RiiDependent only on FRFFirst i columns of, and FRFThe last few columns of (c) are irrelevant.
The above observations motivated us to consider an iteration of N steps, where the ith step is the ith diagonal element R used to maximize Rii
Figure GDA0002938000450000038
Splitting x into:
Figure GDA0002938000450000041
and define
Figure GDA0002938000450000042
The cost function can be rewritten as
Figure GDA0002938000450000043
For infinite precision phase shifters, xn=eThen we can in turn rewrite the cost function:
Figure GDA0002938000450000044
wherein,
Figure GDA0002938000450000045
Figure GDA0002938000450000046
Figure GDA0002938000450000047
wherein, the angle represents the phase of the complex number, when we are fixed
Figure GDA0002938000450000048
Then, we can optimize x for different set constraints Sn
Theorem 4: about thetaoptThe solution of (a):
Figure GDA0002938000450000049
is equivalent to
Figure GDA00029380004500000410
subject to
Figure GDA00029380004500000411
Wherein,
Figure GDA00029380004500000412
for a b-bit resolution phase shifter, the integer k is determined by:
Figure GDA00029380004500000413
and updates θopt
Figure GDA0002938000450000051
Then, the user can use the device to perform the operation,
Figure GDA0002938000450000052
for a phase shifter plus switch network, 0 ∈ S, compare g (θ)opt) To the ratio of
Figure GDA0002938000450000053
xnThe following can be obtained:
Figure GDA0002938000450000054
in the case of a switching network,
Figure GDA0002938000450000055
the specific steps of the algorithm 1 to solve the problem (18) are summarized as follows:
(1) inputting: calculating A, B;
(2) initialization: selecting a random
Figure GDA0002938000450000056
(3) When the objective function value is still increased, executing the step (4);
(4) when n is 1 to MtAt the same time, fix
Figure GDA0002938000450000057
By (28), (29) and (30), x is calculated from the constraints of different Sn
(5) When the objective function value is not changed, the circulation is exited;
(6) return final result
Figure GDA0002938000450000058
Further, an analog beamforming matrix F is designedRFThe specific steps of algorithm 2 are as follows:
(1) inputting a parameter H;
(2) first, give
Figure GDA0002938000450000059
(3) Taking 1 to N for i, executing the following steps (4) to (8);
(4) calculating A and B according to (20);
(5) calculation according to Algorithm 1
Figure GDA00029380004500000510
(6)FRF(:,i)←x;
(7) Computing
Figure GDA00029380004500000511
(8) Computing
Figure GDA00029380004500000512
(9) Return to FRF
Thirdly, optimizing the digital beam forming matrix FBB: calculate FRFThen, we solve for F according to the "water filling" method described aboveBB
First, we fix the analog domain beamforming matrix FRFProblem (5) can be simplified as:
Figure GDA0002938000450000061
Subject to
Figure GDA0002938000450000062
we define
Figure GDA0002938000450000063
We can substitute equation (7) into equation (6),
Figure GDA0002938000450000064
Subject to Tr(GGH)≤ρ
wherein,
Figure GDA0002938000450000065
is a semi-unitary matrix representing FRFThe column space of (a). The solution to problem (8) is:
(1) firstly, SVD decomposition is carried out:
Figure GDA0002938000450000066
(2)
Figure GDA0002938000450000067
wherein,
Figure GDA0002938000450000068
is a diagonal matrix, gammaiCan be obtained by a water injection power distribution method
(3) Wherein,
Figure GDA0002938000450000069
λithe ith diagonal element of Λ is represented. When in use
Figure GDA00029380004500000610
Then, the lagrange multiplier μ is obtained.
(4) Obtaining a digital domain beam forming matrix F according to a formula (7)BB
Compared with the prior art, the invention has the beneficial effects that: the algorithm is suitable for different types of analog networks, including continuous adjustable phase shifter networks, finite bit adjustable phase shifter networks, switching networks and the like. Simulation results show that the performance of the algorithm is very close to the optimal performance of all-digital beam forming; and the performance of the hybrid beam forming based on the switch network is close to that of the phase shifter network, thereby being more beneficial to the realization of engineering.
Drawings
Fig. 1 is a hybrid beamforming structure at a transmitting end in a MIMO system.
Fig. 2 shows three implementations of an analog precoder (beamformer): (a) a phase shifter network, (b) a switching network, (c) a phase shifter plus a switching network.
Fig. 3 is a comparison of spectral efficiency for different RF network configurations.
Detailed Description
The technical solution of the present invention will be described in detail with reference to the accompanying drawings and embodiments.
Fig. 1 is a hybrid beamforming structure at a transmitting end in a MIMO system.
Fig. 2 shows three implementations of an analog precoder (beamformer), i.e. three different RF network architectures.
Example 1
The channel model we use is a narrowband millimeter wave cluster channel model:
Figure GDA0002938000450000071
in which the multipath gain is alphal~CN(0,1),atl) And arl) Antenna array responses for the transmitter and receiver, respectively; wherein theta islIs the angle of departure, philIs the angle of arrival. We simulate using a Uniform Linear Array (ULA) whose array response for angle θ is:
Figure GDA0002938000450000072
where, λ is the wavelength of the signal,
Figure GDA0002938000450000073
is the antenna spacing, arl) And in a similar fashion.
The system we actually simulated was a 64 x 16 MIMO system (M)t=64,Mr16), wherein N isRF=NsThe number of multipaths L is 15, 8. The results of fig. 3 demonstrate a comparison of spectral efficiency of our algorithm under the constraints of different RF chains. Comprises a phase shifter network with infinite precision (o-), a phase shifter with 1bit resolution (o-), and a phase shifter with 2bit resolution
Figure GDA0002938000450000074
A switching network (-), a phase shifter plus a switching network (-). The result shows that the performance of the phase shifter plus switch structure is slightly better than that of a phase shifter structure with infinite precision; and 2bit resolution shifters (e.g., S { ± 1 ± j }) have less than 1dB difference in performance compared to infinite precision shifter networks; the switching network performance is also excellent with less than 4dB difference in performance compared to an infinite precision phase shifter network. For ease of comparison, we also simulated a greedy antenna selection algorithm[1](dotted line in fig. 3); the antenna selection method (only one antenna per RF chain) does not perform as well as the switching network.
Reference to the literature
[1]Y.Jiang and M.K.Varanasi,“The RF-chain limited MIMO system-Part I:optimum diversity-multiplexing tradeoff,”IEEE Transactions on Wireless Communications,vol.8,no.10,pp.5238–5247,2009。

Claims (3)

1. A hybrid beamforming algorithm for massive MIMO characterized in that the hybrid beamforming will contain N of informationsDimensional signal vector
Figure FDA0003020443640000011
First passes through a digital baseband precoder, denoted as
Figure FDA0003020443640000012
Then the obtained NRFDimension signal FBBs is through NRFThe RF chain is converted to RF signals and then passed through an analog precoder in the RF domain, denoted as
Figure FDA0003020443640000013
Finally obtaining MtDimension signal x ═ FRFFBBs is through MtTransmitting by each antenna;
the analog precoder consists of different devices, including: a phase shifter network, a switching network and a phase shifter plus switching network; corresponding to different analog precoder forming devices, FRFThe set S to which the element (S) belongs is different: for an infinite resolution phase shifter network, S ═ e:φ∈[0,2π]}; for a b-bit resolution phase shifter network,
Figure FDA0003020443640000014
Figure FDA0003020443640000015
for a switching network, S ═ {0,1 }; for a phase shifter plus switch network, S ═ e:φ∈[0,2π]} {0} or
Figure FDA0003020443640000016
Aiming at different types of simulated precoders, calculating the optimal coefficient of each element of the simulated precoders, thereby obtaining a matrix FRF
The method comprises the following specific steps:
first step, consider a single-user millimeter wave MIMO system, signal vector
Figure FDA0003020443640000017
Precoder via digital baseband
Figure FDA0003020443640000018
And analog precoder
Figure FDA0003020443640000019
Then through MtTransmitting by each antenna, wherein the sending signals are as follows:
x=FRFFBBs (1)
suppose that
Figure FDA00030204436400000110
Wherein
Figure FDA00030204436400000111
It is shown that it is desirable to,
Figure FDA00030204436400000112
is dimension NsThe unit array of (1); after passing through a narrow-band block fading channel, obtaining a baseband signal:
y=HFRFFBBs+z (2)
Figure FDA00030204436400000113
h is a matrix of the channel and H is,
Figure FDA00030204436400000114
z is a covariance matrix of
Figure FDA00030204436400000115
The total transmission power is white noise due to noise power normalization
Figure FDA00030204436400000116
And the input signal-to-noise ratio is represented by ρ;
the spectral efficiency is:
Figure FDA00030204436400000117
the frequency spectrum efficiency is improved under the condition of limited input signal-to-noise ratio, and the purpose is to solve the following problems:
Figure FDA00030204436400000118
Figure FDA00030204436400000119
Figure FDA00030204436400000120
secondly, optimizing the analog domain beam forming matrix FRF: mixing HURFQR decomposition to obtain HURFIn the case of QR, where,
Figure FDA00030204436400000121
URFis a semi-unitary matrix representing FRFA column space of (a); the basic idea is to optimize the analog beamforming matrix FRFTo maximize the diagonal elements of R; it uses iterative method to maximize QR decomposition HU in iterative ith stepRFI-th diagonal element R of R in QRii
Figure FDA0003020443640000021
Thirdly, optimizing the digital beam forming matrix FBB: given analog domain beamforming matrix FRFProblem (5) to
Figure FDA0003020443640000022
Figure FDA0003020443640000023
The method utilizes a water injection power distribution method; wherein:
in the second step, the first step is carried out,
in respect of QRHU in decompositionRFThe diagonal elements of R of QR,
Figure FDA0003020443640000024
wherein,
Figure FDA0003020443640000025
Ui-1represents URFFirst i-1 column, uiRepresents URFThe ith column;
for URFThe column (c) of (a),
Figure FDA0003020443640000026
wherein,
Figure FDA0003020443640000027
Fi-1is represented by FRFFirst i-1 column of (1), fiIs represented by FRFThe ith column;
deducing according to the formulas (15) and (16)
Figure FDA0003020443640000028
2. The hybrid beamforming algorithm according to claim 1, wherein in the third step, the water-filling power allocation method specifically includes the following steps:
Figure FDA0003020443640000029
the formula (7) is substituted into the formula (6),
Figure FDA00030204436400000210
Subject to Tr(GGH)≤ρ
wherein,
Figure FDA00030204436400000211
the solution to problem (8) is:
firstly, SVD decomposition is carried out:
Figure FDA00030204436400000212
② order
Figure FDA00030204436400000213
Wherein,
Figure FDA00030204436400000214
gamma is a diagonal matrix, gammaiThe power is obtained by a water injection power distribution method;
(iii) wherein,
Figure FDA0003020443640000031
λithe ith diagonal element of Λ is represented when
Figure FDA0003020443640000032
Then, obtaining a Lagrange multiplier mu;
fourthly, obtaining a digital domain beam forming matrix F according to a formula (7)BB
3. The hybrid beamforming algorithm of claim 1, wherein in the second step, an iteration of N steps is considered first, wherein the ith step is the ith diagonal element R used to maximize Rii
Figure FDA0003020443640000033
Splitting x into:
Figure FDA0003020443640000034
and define
Figure FDA0003020443640000035
The cost function is rewritten as
Figure FDA0003020443640000036
(one) for infinite precision phase shifters, xn=eThen the cost function is rewritten:
Figure FDA0003020443640000037
wherein,
Figure FDA0003020443640000038
Figure FDA0003020443640000039
Figure FDA00030204436400000310
wherein the angle represents the phase of the complex number, when fixed
Figure FDA00030204436400000311
Time of day, limit for different setsS to optimize xn
About thetaoptThe solution of (a):
Figure FDA00030204436400000312
is equivalent to
Figure FDA00030204436400000313
Figure FDA00030204436400000314
Wherein:
Figure FDA0003020443640000041
(ii) for a b-bit resolution phase shifter, the integer k is determined by:
Figure FDA0003020443640000042
and updates θopt
Figure FDA0003020443640000043
Then, the user can use the device to perform the operation,
Figure FDA0003020443640000044
(III) for the phase shifter plus switch network, 0 ∈ S, compare g (theta)opt) To the ratio of
Figure FDA0003020443640000045
Thus, xnThe following can be obtained:
Figure FDA0003020443640000046
(IV) for the switching network,
Figure FDA0003020443640000047
further, algorithm 1 for solving the problem (18) for infinite precision phase shifter networks, b-bit resolution phase shifters, and for phase shifter plus switching networks and switching networks is as follows:
firstly, inputting: A. b, provided by algorithm 2 below;
initializing: selecting a random
Figure FDA0003020443640000048
Thirdly, when the objective function value is still increased, executing the fourth step;
when n is from 1 to MtAt the same time, fix
Figure FDA0003020443640000049
Calculating x according to the constraints of different S through formulas (28), (29) and (30)n
Quitting circulation when the objective function value is not changed;
sixthly, returning the final result
Figure FDA00030204436400000410
Further, an analog beamforming matrix FRFAlgorithm 2 as follows:
inputting a parameter H;
first, give
Figure FDA00030204436400000411
③ from 1 to N for iRFExecuting the following steps of ((r) - (r));
fourthly, calculating A and B according to a formula (20);
calculating according to algorithm 1
Figure FDA0003020443640000051
⑥FRF(:,i)←x;
C calculation of
Figure FDA0003020443640000052
Calculation of
Figure FDA0003020443640000053
Ninthly returns to FRF
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