CN109302224B - Hybrid beamforming algorithm for massive MIMO - Google Patents

Abstract

The invention belongs to the field of millimeter-wave massive MIMO systems, in particular to a hybrid beamforming algorithm for massive MIMO. This new algorithm aims to jointly optimize beamforming in the analog and digital domains to maximize spectral efficiency. The invention is divided into two parts: one is the algorithm of analog beamforming; the other is that when the analog beamformer is given, the digital beamformer is obtained according to the "water injection method". The algorithm is suitable for different types of analog networks, including continuously adjustable phase shifter networks, finite-bit adjustable phase shifter networks, and switching networks. The simulation results show that the performance of the algorithm is very close to the performance of the optimal all-digital beamforming; and the hybrid beamforming based on the switch network is close to the performance of the phase shifter network, which is more conducive to engineering implementation.

Description

Hybrid Beamforming Algorithm for Massive MIMO

technical field

The invention belongs to the field of MIMO communication, and in particular relates to a hybrid beamforming algorithm for massive MIMO.

Background technique

5G millimeter wave technology increases the communication bandwidth by hundreds of megahertz or even thousands of hertz, and installs dozens or even hundreds of antennas at the base station, so it can greatly improve the channel capacity of the cell. However, if a traditional communication receiver is used, the digital domain needs to process hundreds of high-speed code streams in real time, which is very difficult. The researchers proposed to perform analog beamforming (Analog Beamforming/ABF) before the digital-to-analog converter (ADC) to compress the high-dimensional signals received by M antennas into N dimensions (N<<M). ABF, the antenna gain of massive MIMO is preserved, and the signal dimension is greatly compressed, thereby greatly reducing the amount of operations in the digital domain and the number of ADCs required, and significantly reducing hardware costs. This technique is called hybrid beamforming (Hybrid Beamforming/HBF).

For the above-mentioned technology, some research work on hybrid beamforming (Hybrid Beamforming/HBF) has been published. The traditional analog domain network is mainly divided into phase shifters and switch structures. How can we solve different analog domain network structures (including switches, fixed phase phase shifters, and variable phase phase shifters with different resolutions) in one way? etc.), this is still a problem.

SUMMARY OF THE INVENTION

In order to overcome the deficiencies of the prior art, the purpose of the present invention is to provide a hybrid beamforming algorithm for different types of analog networks of massive MIMO. The algorithm of the invention is suitable for phase shifter network, switch network and phase shifter plus switch composite network in massive MIMO, so as to facilitate joint optimization of analog domain beamforming and digital domain beamforming and maximize spectrum efficiency.

The present invention provides a hybrid beamforming algorithm for massive MIMO, and the specific steps are as follows:

经过数字基带预编码器

和模拟预编码器

然后通过M_t个天线传输。发送天线为：The first step, the first consideration is the single-user mmWave MIMO system, the signal vector

digital baseband precoder

and analog precoder

It is then transmitted through M _t antennas. The transmitting antenna is:

x=F _RF F _BB s (1)

其中

表示期望，

是维度为N_s单位阵。通过一个平衰落信道后，我们将得到基带信号：assumed

in

express expectations,

is an identity matrix of dimension N _s . After passing through a flat fading channel, we will get the baseband signal:

Y=HF _RF F _BB s+z (2)

是信道矩阵，

是协方差矩阵为

的循环对称复高斯分布的白噪声，由于总发射功率是由噪声功率归一化的

因此输入信噪比是:

is the channel matrix,

is the covariance matrix of

cyclic symmetric complex Gaussian white noise, since the total transmit power is normalized by the noise power

So the input signal-to-noise ratio is:

The spectral efficiency is:

The purpose of the algorithm provided by the present invention is to improve the spectral efficiency, in order to solve the following problems:

Subject to

对于开关网络S＝{0,1}；对于移相器加开关网络S＝{e^jφ:φ∈[0,2π]}∪{0}或者

F_RF(i,j)∈S的约束将混合波束赋形与传统的全数字波束赋形区分开。where the first constraint is about the input SNR, and the set of S in the second constraint depends on the set of RF feedback networks: for infinite-resolution phase shifter networks S={e ^jφ :φ∈[0,2π ]}; for a phase shifter network with b-bit resolution

For switch network S={0,1}; for phase shifter plus switch network S={e ^jφ :φ∈[0,2π]}∪{0}or

The constraint of F _RF (i,j)∈S distinguishes hybrid beamforming from conventional all-digital beamforming.

The second step is to optimize the beamforming matrix F _RF in the analog domain

Before introducing the optimization method of the beamforming matrix in the analog domain, we first introduce two majorization theories.

被向量

所乘积主要化，定义为：

如果Definition 1: Vector

be vectored

The multiplied product is majorized, defined as:

if

Among them, x[i] and y[i] are the i-th largest elements of x and y, respectively.

叫做在

上的乘积Schur-convex，如果：Definition 2: A function φ:

called in

The product Schur-convex over , if:

Let's look at the optimization of beamforming in the analog domain, assuming that N _s =N _RF =N, for the case where ρ is large enough, it satisfies

It is easy to show that the spectral efficiency is

In the following discussion we use the two definitions of majorization mentioned above.

是主要化的Schur-convex。对于每个λ_i来说，C(λ)是非递减的函数。Theorem 1: Function

is the main Schur-convex. For each λ _i , C(λ) is a non-decreasing function.

由R的对角元素所组成，我们所知道的是R的对角元素被HU_RF的奇异值所乘积主要化，并且λ是HU_RF的奇异值的平方。因此，Given QR is decomposed into HU _RF = QR, define

Composed of the diagonal elements of R, what we know is that the diagonal elements of R are dominated by the product of the singular values of HU _RF , and λ is the square of the singular values of HU _RF . therefore,

According to Theorem 1, we get

C(λ)≥C(|r| ² ) (13)

Instead of maximizing C(λ), we maximize the lower bound C(|r| ² ) of C(λ). Next we discuss the internal structure of the R diagonal:

Theorem 2: Regarding the diagonal elements of R where HU _RF = QR in QR decomposition,

in,

U _i-1 represents the first i-1 column of U _RF , and _ui represents the i-th column of U _RF .

Theorem 3: For a column of U _RF ,

F_i-1表示F_RF的前i-1列，f_i表示F_RF的第i列。in,

F _i-1 represents the first i-1 column of F _RF , and f _i represents the i-th column of F _RF .

From Theorem 2 and Theorem 3, we can deduce

From this, we can see that R _ii only depends on the first i columns of F _RF , and has nothing to do with the last few columns of F _RF .

The above observation prompts us to consider an N-step iteration, where the ith step is used to maximize the ith diagonal element of R R _ii :

Split x into:

and define

Then the cost function can be rewritten as

For infinite precision phase shifters, x _n = e ^jθ , then we can rewrite the cost function as:

in,

时，我们可以对于不同集合限制S来优化x_n。where ∠ represents the phase of the complex number, when we fix

, we can optimize x _n for different set constraints S.

Theorem 4: The solution for θ _opt :

Equivalent to

subject to

in,

For a b-bit resolution phase shifter, the integer k is determined by:

and update θ _opt

Then,

x_n可得：For a phase shifter plus switching network, 0∈S, compare g(θ _opt ) with the ratio

x _n can be obtained:

For switch networks,

The specific steps of Algorithm 1 to solve problem (18) are summarized as follows:

(1) Input: Calculate A and B;

(2) Initialization: choose a random

(3) When the objective function value still increases, perform step (4);

通过(28)、(29)、(30)，根据不同S的约束计算x_n；(4) When n takes 1 to M _t , fixed

Through (28), (29), (30), calculate x _n according to the constraints of different S;

(5) When the objective function value remains unchanged, exit the loop;

(6) Return the final result

Furthermore, the specific steps of designing the algorithm 2 of the analog beamforming matrix F _RF are as follows:

(1) Input parameter H;

(2) First given

(3) For i, take 1 to N, and perform the following steps (4) to (8);

(4) Calculate A and B according to (20);

(5) Calculated according to Algorithm 1

(6) F _RF (:,i)←x;

(7) Calculation

(8) Calculation

(9) Return F _RF .

The third step is to optimize the digital beamforming matrix F _BB : After calculating F _RF , we solve F _BB according to the above-mentioned “water injection” method.

First, when we fix the analog domain beamforming matrix F _RF , problem (5) can be simplified to:

Subject to

we define

We can substitute formula (7) into formula (6) to get,

Subject to Tr(GG ^H )≤ρ

是一个半酉矩阵，表示F_RF的列空间。对于问题(8)的解为：in,

is a semi-unitary matrix representing the column space of F _RF . The solution to problem (8) is:

(1) First perform SVD decomposition:

其中，

是对角阵，γ_i可用“注水”功率分配方法获得(2)

in,

is a diagonal matrix, _γi can be obtained by the "water injection" power distribution method

λ_i表示Λ的第i个对角线元素。当

时，得到拉格朗日乘子μ。(3) of which,

λ _i denotes the ith diagonal element of Λ. when

, the Lagrange multiplier μ is obtained.

(4) According to formula (7), the digital domain beamforming matrix _FBB is obtained.

Compared with the prior art, the present invention has the beneficial effects that the algorithm is suitable for different types of analog networks, including continuously adjustable phase shifter networks, finite-bit adjustable phase shifter networks, switching networks, and the like. The simulation results show that the performance of the algorithm is very close to the performance of the optimal all-digital beamforming; and the hybrid beamforming based on the switch network is close to the performance of the phase shifter network, which is more conducive to engineering implementation.

Description of drawings

Figure 1 shows the hybrid beamforming structure at the transmitting end in a MIMO system.

Figure 2 shows three implementations of an analog precoder (beamformer): (a) a phase shifter network, (b) a switch network, and (c) a phase shifter plus switch network.

Figure 3 shows the spectral efficiency comparison of different RF network structures.

Detailed ways

The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

Figure 1 shows the hybrid beamforming structure at the transmitting end in a MIMO system.

Figure 2 shows three implementations of an analog precoder (beamformer), ie, three different RF network structures.

Example 1

The channel model we use is the narrowband millimeter wave cluster channel model:

where the multipath gain is α _l ~CN(0,1), at (θ _l ) and a _r (φ _l ) are the antenna array responses of the transmitter and receiver, respectively _; where θ _l is the departure angle, φ _l for the angle of arrival. Our simulations use a Uniform Linear Array (ULA) whose array response for angle θ is:

是天线间距，a_r(φ_l)也有类似的形式。where λ is the wavelength of the signal,

is the antenna spacing, and a _r (φ _l ) has a similar form.

开关网络(-*-)，移相器加开关网络(--)。结果表明，移相器加开关结构与无限精度的移相器结构相比，性能略微要好一点；而且2bit分辨率的移相器(例如：S＝{±1 ±j})与无限精度的移相器网络相比在性能上的差距小于1dB；开关网络(-*-)性能也是比较出色的与无限精度的移相器网络相比在性能上的差距要小于4dB。为了便于比较，我们也仿真了贪婪天线选择算法^[1](图3中的点线)；天线选择方法(每个RF链只与一个天线相连)的性能明显不如开关网络。The system we actually simulate is a 64×16 MIMO system (M _t =64, M _r =16), where N _RF =N _s =8, and the multipath number L is 15. The results in Figure 3 show the comparison of the spectral efficiency of our algorithm under the constraints of different RF chains. Including infinite precision phase shifter network (-◇-), 1bit resolution phase shifter (-○-), 2bit resolution phase shifter

Switch network (-*-), phase shifter plus switch network (--). The results show that the performance of the phase shifter plus switch structure is slightly better than that of the infinite precision phase shifter structure; Compared with the phase shifter network, the performance gap is less than 1dB; the performance of the switch network (-*-) is also excellent, and the performance gap compared with the infinite precision phase shifter network is less than 4dB. For comparison purposes, we also simulated the greedy antenna selection algorithm ^[1] (dotted line in Figure 3); the antenna selection method (each RF chain connected to only one antenna) performs significantly worse than the switch network.

references

[1] Y. Jiang and M.K. Varanasi, "The RF-chain limited MIMO system-Part I: optimal 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 information_sDimensional signal vector

First passes through a digital baseband precoder, denoted as

Then the obtained N_RFDimension signal F_BBs is through N_RFThe RF chain is converted to RF signals and then passed through an analog precoder in the RF domain, denoted as

Finally obtaining M_tDimension signal x ═ F_RFF_BBs is through M_tTransmitting 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, F_RFThe set S to which the element (S) belongs is different: for an infinite resolution phase shifter network, S ═ e^jφ:φ∈[0,2π]}; for a b-bit resolution phase shifter network,

for a switching network, S ═ {0,1 }; for a phase shifter plus switch network, S ═ e^jφ:φ∈[0,2π]} {0} or

Aiming at different types of simulated precoders, calculating the optimal coefficient of each element of the simulated precoders, thereby obtaining a matrix F_RF；

The method comprises the following specific steps:

first step, consider a single-user millimeter wave MIMO system, signal vector

Precoder via digital baseband

And analog precoder

Then through M_tTransmitting by each antenna, wherein the sending signals are as follows:

x＝F_RFF_BBs (1)

suppose that

Wherein

It is shown that it is desirable to,

is dimension N_sThe unit array of (1); after passing through a narrow-band block fading channel, obtaining a baseband signal:

y＝HF_RFF_BBs+z (2)

h is a matrix of the channel and H is,

z is a covariance matrix of

The total transmission power is white noise due to noise power normalization

And the input signal-to-noise ratio is represented by ρ;

the spectral efficiency is:

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:

secondly, optimizing the analog domain beam forming matrix F_RF: mixing HU_RFQR decomposition to obtain HU_RFIn the case of QR, where,

U_RFis a semi-unitary matrix representing F_RFA column space of (a); the basic idea is to optimize the analog beamforming matrix F_RFTo maximize the diagonal elements of R; it uses iterative method to maximize QR decomposition HU in iterative ith step_RFI-th diagonal element R of R in QR_ii：

Thirdly, optimizing the digital beam forming matrix F_BB: given analog domain beamforming matrix F_RFProblem (5) to

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 decomposition_RFThe diagonal elements of R of QR,

wherein,

U_i-1represents U_RFFirst i-1 column, u_iRepresents U_RFThe ith column;

for U_RFThe column (c) of (a),

wherein,

F_i-1is represented by F_RFFirst i-1 column of (1), f_iIs represented by F_RFThe ith column;

deducing according to the formulas (15) and (16)

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:

the formula (7) is substituted into the formula (6),

Subject to Tr(GG^H)≤ρ

wherein,

the solution to problem (8) is:

firstly, SVD decomposition is carried out:

② order

Wherein,

gamma is a diagonal matrix, gamma_iThe power is obtained by a water injection power distribution method;

(iii) wherein,

λ_ithe ith diagonal element of Λ is represented when

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 R_ii：

Splitting x into:

and define

The cost function is rewritten as

(one) for infinite precision phase shifters, x_n＝e^jθThen the cost function is rewritten:

wherein,

wherein the angle represents the phase of the complex number, when fixed

Time of day, limit for different setsS to optimize x_n；

About theta_optThe solution of (a):

is equivalent to

Wherein:

(ii) for a b-bit resolution phase shifter, the integer k is determined by:

and updates θ_opt

Then, the user can use the device to perform the operation,

(III) for the phase shifter plus switch network, 0 ∈ S, compare g (theta)_opt) To the ratio of

Thus, x_nThe following can be obtained:

(IV) for the switching network,

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

Thirdly, when the objective function value is still increased, executing the fourth step;

when n is from 1 to M_tAt the same time, fix

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

Further, an analog beamforming matrix F_RFAlgorithm 2 as follows:

inputting a parameter H;

first, give

③ from 1 to N for i_RFExecuting the following steps of ((r) - (r));

fourthly, calculating A and B according to a formula (20);

calculating according to algorithm 1

⑥F_RF(:,i)←x；

C calculation of

Calculation of

Ninthly returns to F_RF。

Images