METHODS, APPARATUSES, AND COMPUTER READABLE MEDIA FOR PRECODING IN MULTIPLE-INPUT MULTIPLE-OUTPUT SYSTEM BASED ON ARRAY OF SUBARRAY ARCHITECTURE

Disclosed are methods for precoding in a downlink multiple-input multiple-output system based on an array of subarray architecture. An example method may include: determining an analog precoding matrix for a plurality of radio frequency chains in the downlink multiple-input multiple-output system; determining a digital precoding matrix for the plurality of radio frequency chains based on the determined analog precoding matrix; and performing a hybrid precoding for a plurality of downlink data streams based on the determined digital precoding matrix and the determined analog precoding matrix. Related apparatuses and computer readable media are also disclosed.

TECHNICAL FIELD

Various example embodiments relate to methods, apparatuses, and computer readable media for precoding in a multiple-input multiple-output (MIMO) system based on an array of subarray (AoSA) architecture.

BACKGROUND

In a telecommunication system such as a sixth-generation mobile network or a sixth-generation wireless system after new radio (NR or 5G) system, Terahertz (THz) band with ultra-broad bandwidth may be used for a rapid growth of wireless data rates. A MIMO (for example, a massive MIMO or a multiple-user MIMO) solution may be utilized in such telecommunication system with ultra-short wavelength, for example to achieve better multiplexing gains, better diversity gains, improved energy efficiency, and so on, where a base station may be configured with a large number of antennas. Precoding may be applied to process downlink signals in a MIMO system, where for example channel status information (CSI) of the transmitter of the downlink signals may be unitized to transform modulated symbol streams to data streams suitable for current channels and signal energy may be focused to target users.

SUMMARY

In a first aspect, disclosed is a method precoding in a downlink multiple-input multiple-output system based on an array of subarray architecture. The method may include: determining an analog precoding matrix for a plurality of radio frequency chains in the downlink multiple-input multiple-output system; determining a digital precoding matrix for the plurality of radio frequency chains based on the determined analog precoding matrix; and performing a hybrid precoding for a plurality of downlink data streams based on the determined digital precoding matrix and the determined analog precoding matrix.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a plurality of physical channels associated with the downlink multiple-input multiple-output system and a switch matrix in the downlink multiple-input multiple-output system.

In some example embodiments, in a current iteration of the at least one iteration, the determination of the analog precoding matrix may include: determining a first approximation of an inverse of a Hessian matrix of a first objective function for the current iteration; determining angles of departure corresponding to the column based on the first approximation; and determining a second approximation of an inverse of a Hessian matrix of a second objective function for a next iteration of the at least one iteration through matrix plus and multiplication operations based on the first approximation, a phase difference between the current iteration and the next iteration, and a gradient difference between the current iteration and the next iteration.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a subset of a predetermined analog precoding matrix.

In some example embodiments, a carrier frequency of the downlink multiple-input multiple-output system is at or above Terahertz level.

In a second aspect, disclosed is an apparatus for precoding in a downlink multiple-input multiple-output system based on an array of subarray architecture. The apparatus may include: a plurality of transmitting antennas; a plurality of radio frequency chains; an analog precoder between the plurality of radio frequency chains and the plurality of transmitting antennas; and a digital precoder connecting to the plurality of radio frequency chains. An analog precoding matrix associated with the analog precoder is determined independently of a digital precoding matrix associated with the digital precoder, and the digital precoding matrix is determined based on the determined analog precoding matrix.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a plurality of physical channels associated with the downlink multiple-input multiple-output system and a switch matrix in the downlink multiple-input multiple-output system.

In some example embodiments, in a current iteration of the at least one iteration, the determination of the analog precoding matrix may include: determining a first approximation of an inverse of a Hessian matrix of a first objective function for the current iteration; determining angles of departure corresponding to the column based on the first approximation; and determining a second approximation of an inverse of a Hessian matrix of a second objective function for a next iteration of the at least one iteration through matrix plus and multiplication operations based on the first approximation, a phase difference between the current iteration and the next iteration, and a gradient difference between the current iteration and the next iteration.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a subset of a predetermined analog precoding matrix.

In some example embodiments, a carrier frequency of the downlink multiple-input multiple-output system is at or above Terahertz level.

In a third aspect, disclosed is a computer readable medium. The computer readable medium may include instructions stored thereon for causing an apparatus for precoding in a downlink multiple-input multiple-output system based on an array of subarray architecture to perform: determining an analog precoding matrix for a plurality of radio frequency chains in the downlink multiple-input multiple-output system; determining a digital precoding matrix for the plurality of radio frequency chains based on the determined analog precoding matrix; and performing a hybrid precoding for a plurality of downlink data streams based on the determined digital precoding matrix and the determined analog precoding matrix.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a plurality of physical channels associated with the downlink multiple-input multiple-output system and a switch matrix in the downlink multiple-input multiple-output system.

In some example embodiments, in a current iteration of the at least one iteration, the determination of the analog precoding matrix may include: determining a first approximation of an inverse of a Hessian matrix of a first objective function for the current iteration; determining angles of departure corresponding to the column based on the first approximation; and determining a second approximation of an inverse of a Hessian matrix of a second objective function for a next iteration of the at least one iteration through matrix plus and multiplication operations based on the first approximation, a phase difference between the current iteration and the next iteration, and a gradient difference between the current iteration and the next iteration.

In some example embodiments, the determination of the analog precoding matrix may include: for a radio frequency chain of the plurality of radio frequency chains, determining a column of the analog precoding matrix corresponding to the radio frequency chain in at least one iteration based on a subset of a predetermined analog precoding matrix.

In some example embodiments, a carrier frequency of the downlink multiple-input multiple-output system is at or above Terahertz level.

DETAILED DESCRIPTION

The ultra-short wavelength may allow the design of an antenna array including large antenna elements at transceivers, for example to provide a high beamforming gain to compensate pathloss, and multiple data streams may be supported to offer a multiplexing gain and further improve the spectral efficiency (SE) of the system. For example, a hybrid precoding may be adopted in the THz system, where a signal processing procedure may be divided into a digital baseband part followed by an analog radio frequency (RF) band part. In some implementations, the hybrid precoding is based on a fully connected (FC) architecture of analog precoder, which for example may be of power inefficiency since each RF chain needs to connect to all antennal elements. In some implementations, precoding matrices of the digital precoder and the analog precoder are determined and optimized jointly, where optimization problems of the digital precoder and the analog precoder are coupled with each other in the same iterative process, and for example an optimal metric of the communication system (for example, the maximal SE and energy efficiency) may be not obtained.

In one or more example embodiments of this disclosure, the hybrid precoding is based on an AoSA architecture where each RF chain connects to a part of antennas rather than being fully connected to all antennas, so that power consumption may be reduced. Further, in one or more example embodiments, the determination and optimization of analog precoder design and digital precoder design are decoupled, where an analog precoding matrix associated with the analog precoder may be determined and/or optimized independently of a digital precoding matrix associated with the digital precoder, and the digital precoding matrix may be determined and/or optimized based on the determined analog precoding matrix after the determination and/or optimization of the analog precoding matrix. Thus, for example, a better (for example, the highest) power of effective channel and SE may be achieved.

Throughout this disclosure, (⋅)Tand (⋅)Hdenote transpose and conjugate transpose respectively, ∥⋅∥Fdenotes a Frobenius norm, E(⋅) denotes an expectation, CN(μ,σ2) denotes complex Gaussian vector with mean μ and covariance σ2, tr(x) means a trace of x, Diag(x) means to reshape a vector x as a diagonal matrix, (o) denotes the Hadamard product, anddenotes an integer set.

FIG.1illustrates an example downlink MIMO system100in an example embodiment, which for example may be at least a part of a network node or apparatus such as a base station configured with MIMO. For example, a carrier frequency of the example downlink system100may be at or above THz level.

As illustrated inFIG.1, the example downlink MIMO system100may include Nttransmitting antennas (antenna1to antenna Nt), NrfRF chains (RF chain1to RF chain Nrf), a digital precoder101connecting to the RF chains, and an analog precoder102between the RF chains and the transmitting antennas. The example downlink MIMO system100is configured based on an AoSA architecture where each RF chain connects to a set of subarrays (subarray1to subarray Nrf), and each subarray may include Na=└Nt/Nrf┘ antenna elements, where └x┘ denotes the largest integer less than x. As illustrated inFIG.1, the example downlink MIMO system100may be configured to process NSdata streams (stream1to stream Ns), for example to map the data streams to suitable antenna ports.

If a user equipment (UE) implements Nrreceiving antennas, a transmitting signal is denoted as X=[x1, . . . , xNs]Tsuch that E[|xk|2]=1 for k=1, . . . , Ns, then a receiving signal may be

where N is a noise vector including Nrelements, each element of N follows a distribution CN(0,σ2). CAis an Nr×Nrfanalog combiner, which satisfies ∥CA(i,j)∥F2=1/Nr. CDis an Nrf×Nsdigital combiner, which satisfies ∥CA·CD∥F2=1. H represents Nr×Ntphysical channels.

PAis an Nt×Nrfanalog precoding matrix of the analog precoder102, which satisfies a constant modulus constraint

W is a switch matrix whose dimension is NaNrf×Nrf.

where

that is, wijis a Na×1 dimensional vector with all elements equal to one if RF chain j connects to subarray i, otherwise all of its elements are equal to zero.

PDis a Nrf×Nsdigital precoding matrix of the digital precoder101, which satisfies the power constraints

where

Assuming that uniform linear array is implemented at both base station and UE, and if a ray-cluster based spatial channel model is employed, then H may be represented as

where gl,ϕl,θlrepresent a complex path gain, an angle of arrival (AoA) and an angle of departure (AoD) of path l, with a total path number being L. ar(⋅),at(⋅) are array response vectors of the receiving and transmitting antenna arrays.

If denoting λ as a wavelength of a carrier frequency, Dt=dt/λ,Dr=Dr/λ as relative inter-element distances of the transmitting and receiving antenna array, where dt,drare absolute inter-element distances of the transmitting and receiving antenna array, then

For designing the analog precoding matrix PAof the analog precoder102, in some example embodiments, an objective function may be to maximize a power of an effective channel with the constant modulus constraint, for example as follows:

The objective function of (P1) may be rewritten as the following, so that a (P1) is transformed to (P2) which is a set of individual sub-problems.

where {tilde over (P)}Ajdenotes the column j of {tilde over (P)}A.

Further, (P2) may be transformed into a standard optimization formulation, and each sub-problem of (P2) may be denoted as the following:

With the joint consideration of AoSA architecture for each RF chain, the following equitation may be obtained:

where Peffjis the set of non-zeros elements of {tilde over (P)}Aj, and Heffis the effective channel matrix composed of columns of H corresponding to non-zeros elements of {tilde over (P)}Aj.

Assuming that the set of non-zeros elements of {tilde over (P)}Ajis Kj, and based on the constant modulus constraint (2) and the above equation (9), the following equation may be obtained:

When representing H as H=[H1, H2, . . . HNt], the following equations may be obtained:

where Am,nis an amplitude of HmHHn, and Bm,nis an angle of HmHHn.

Since Ntis a constant variable, (P4) may be transformed into (P4) as the following.

Based on the above deduction, the non-convex constraint (2) of (P3) is removed in (P4). Moreover, Am,nand Bm,nare known variables given H. Thus, (P4) is a non-constrained optimization problem only related to the angle of Peffj.

Further, if denoting the set of non-zeros elements of {tilde over (P)}Ajat iteration t as Ktj, from the above equation (10), we may have

Then, for (P4), an objective function at an iteration t may be

and a gradient function at the iteration t with respect to m may be

where

and the gradient function at the iteration t is

Then, given H and W, for the RF chain j, at an initial iteration t=0, θt(the set of non-zeros elements of the column j of {tilde over (P)}Aat iteration t) may be initialized as a set of random angles.

Further, for the RF chain j, at any iteration t, a search direction may be determined as

where Dtis an approximation of an inverse of a Hessian matrix of the objective function ƒtat the current iteration t, and may be initialized as an identity matrix I at the initial iteration (t=0), and a step αtmay be determined or updated for example by one dimension search method such as Wolfe-Powell method, and may satisfy the following conditions:

where ρ1and ρ2are two random variables which satisfy ρ1∈(0,0.5) and ρ2∈(ρ1,1).

Then, a phase difference stbetween the next iteration t+1 and the current iteration t may be determined as st=αtdt, and θt+1(the set of non-zeros elements of the column j of {tilde over (P)}Aat the next iteration t+1) may be determined as θt+1=θt+st.

If ∥gt+1∥F≤δ where δ is a positive constant, the iteration for the RF chain j may be stopped, and the column j of {tilde over (P)}A(further, a column j of precoding matrix PAof the analog precoder102) may be determined. Then, an iteration process may be performed for another RF chain (for example the RF chain j+1).

If ∥gt+1∥F>δ, Dt+1(an approximation of an inverse of a Hessian matrix of the objective function ƒt+1at the next iteration t+1) may be determined based on the following equation:

where ε is a positive constant, Q(w) is an updated coefficient which is related to the weighted coefficient w∈[0,1], for example Q(w)=wytTst+2(1−w)(ƒt+1−ƒt−gtTst). Then, the next iteration t+1 for the RF chain j may be proceeded to.

Thus, after a completion of all iterations for all RF chains, the analog precoding matrix of the analog precoder102may be determined, which process may be independent of the determination of the digital precoding matrix of the digital precoder101or the design of the digital precoder101.

FIG.2illustrates an example process200for determining the analog precoding matrix of the analog precoder102in an example embodiment.

As illustrated inFIG.2, inputs201of the example process200may include H and W. Then, for the RF chain j among the NrfRF chains in the example downlink MIMO system100, at least one iteration may be performed to determine a column of the analog precoding matrix corresponding to the RF chain j.

As illustrated inFIG.2, at an operation202, an initialization may be performed for the at least one iteration for the RF chain j, where a value of an iteration counter t may be initialized as 0 (t=0), θ0(the set of non-zeros elements of the column j of {tilde over (P)}Aat the iteration t=0) may initialized as a set of random angles, and D0(an approximation of an inverse of a Hessian matrix of the objective function ƒtat the iteration t=0).

Then, a search direction dtat the iteration t for the RF chain j may be determined for example based on the above equation (16) in an operation203, a step αtat the iteration t for the RF chain j may be determined for example based on above conditions (17) in an operation204. Then, in an operation205, a phase difference stbetween the next iteration t+1 and the current iteration t may be determined as st=αtdt, and θt+1(the set of non-zeros elements of the column j of {tilde over (P)}Aat the next iteration t+1) may be determined as θt+1=θt+st.

Then, in an operation206, it is checked whether ∥gt+1∥F≤δ. If the operation206returns “Yes” (∥gt+1∥F≤δ), as illustrated inFIG.2, the example process200may proceed to the operation202for another RF chain (for example, RF chain j+1).

If the operation206returns “No” (∥gt+1∥F>δ), an operation207may performed at the iteration t for the RF chain j, to determine Dt+1(an approximation of an inverse of a Hessian matrix of the objective function ƒt+1at the next iteration t+1) based on at least one of Dt, the phase difference st, and the gradient difference yt=gt+1−gtbetween the current iteration t and the next iteration t+1. For example, the operation207may be performed based on the above equation (18). Then, the example process200may proceed to the operation203by updating t as t+1, to perform the next iteration for the RF chain j.

After the analog precoding matrix of the analog precoder102is determined, the digital precoding matrix of the digital precoder101may be determined based on the determined analog precoding matrix by using any suitable method.

For example, after the completion of the example process200through which {tilde over (P)}Acorresponding to the analog precoding matrix PAof the analog precoder102is determined, the digital precoding matrix PDof the digital precoder101may be determined by using a single value deduction (SVD) method, for example as follows:

where V is the first Nscolumns of a right singular matrix from the SVD of the effective channel H{tilde over (P)}A, and Γ is an Ns×Nsdimensional water filling power allocation matrix.

Further, the digital precoding matrix PDmay be normalized as:

FIG.3illustrates an example process300for precoding in the example downlink MIMO system100in an example embodiment.

In an operation301, the analog precoding matrix analog precoding matrix PA(or {tilde over (P)}=PAoW) of the analog precoder102may be determined for the NrfRF chains in the example downlink MIMO system100. For example, in the operation301, the above example process200may be performed.

Then, after the analog precoding matrix analog precoding matrix PA(or {tilde over (P)}A=PAoW) of the analog precoder102is determined in the operation301, in an operation302, the digital precoding matrix PDof the digital precoder101may be determined based on the determined analog precoding matrix PA(or {tilde over (P)}A=PAoW) of the analog precoder102. For example, in the operation302, the digital precoding matrix PDof the digital precoder101may be determined by using the SVD method, for example based on the above equations (19) and (20).

Then, in an operation303, the determined digital precoding matrix PDof the digital precoder101and the determined analog precoding matrix PA(or {tilde over (P)}A=PAoW) of the analog precoder102may be used to perform a hybrid precoding for the NSdata streams. Thus, the NSdata streams may be mapped to suitable antenna ports.

Consider SE as a metric of the above example process300, according Shannon's theory, the SE of a stream s may be Rs=log(1+SNRs) where SNRsis a signal noise ratio of the stream s, and the SE of the above example process300may be

In a simulation, the following parameters are selected: Nt=256, Nr=4, Nrf=[1,2,4,8,16], Ns=min(min(Nt,Nr),Nrf) where min( ) is an operation for obtaining a minimum value, δ=0.001, ε=0.1, w=0.9, ρ1=0.25, ρ2=0.75, and SNR=[0,30] dB.

Then, for example in a case of SNR=10 dB, the powers of effective channels with different number of RF chains are illustrate inFIG.4, and the spectral efficiencies with different number of RF chains are illustrate inFIG.5, where OMP is an orthogonal matching pursuit (OMP) hybrid precoding method which is an example of hybrid precoding method based on FC architecture, VU is a vectorization plus unitary (VU) hybrid precoding method which is another example of hybrid precoding method where optimization problems of the digital precoder and the analog precoder are coupled with each other in the same iterative process, and MS is the example process300. It can be seen that the example process300for precoding in the example embodiment achieves higher power of effective channel and SE compared with OMP and VU.

FIG.6illustrates SEs with different SNRs in a case where Nrf=8. As illustrated inFIG.6, the SEs of all curves increase when SNR increases, and compared with OMP and VU, the example process300for precoding in the example embodiment may achieves higher power of effective channel and SE at any SNR equipped with any number of RF chains.

It is appreciated that the implementations of the operations301and302in the above example process300are not limited to the above examples, and the metric is not limited to the power of effective channel and SE.

For example, if opening a RF chain cannot provide sufficient transmission throughput, this RF chain may be closed to save power. Further, the transmission rank number may be adaptive to the received SNR in order to provide spatial diversity or multiplexing gain. For example, the transmission rank number may equal to 1 if the received SNR is low since spatial diversity can improve the received SNR to provide good coverage. While the transmission rank number may equal to the maximal rank number if the received SNR is high since spatial multiplexing can improve the transmission throughput. Thus, in some example embodiments, the analog precoder102and the digital precoder101may be configured to optimize a RF chain utilization rate which is defined as a ratio of the SE by using the current RF chain number to the SE by using the maximal RF chain number.

For example, the inputs provided to the example process300may include information associated with physical channels H, the switch matrix W, a predetermined set of candidate analog precoding matrices PAS, and a RF chain utilization rate threshold γ.

For example, before applying the example process300, the optimal transmission rank number Nsmay be determined by utilizing a rank adaptation technology, and the optimal digital precoding matrix P* of the digital precoder101may be determined by using SVD of the physical channels H. For example, P* may be the first Nscolumns of the right singular matrix from the SVD of the physical channels H.

Then, in the operation301, for each RF chain, a column of PAScorresponding to the RF chain may be selected. The selection criteria may include, but not limited to, one or more of: a column with the maximal received power, a column with the maximal SNR, a column with the nearest spatial angle, a column with the minimal latency, and so on. Then, the selected columns may be combined and sorted according to a descending order of degrees that respective selected columns match to H, and thus an available analog precoding matrix PAAof the analog precoder102may be obtained in the operation301.

In the operation302, the SE of the current RF chain number may be calculated where the RF chain number Irfincreases from Nsto Nrf, and may be denoted as [RNs,RNs+1, . . . , RNrf]. Then, an unnormalized digital precoding matrix of the digital precoder101may be determined by least square algorithm PD=(PACHPAC)−1PACHP*. In another example, the unnormalized digital precoding matrix of the digital precoder101may also be determined by a unitary matrix algorithm PD=UVHwhere U and V are the left and right singular matrix from the SVD of P*PAC. Further, the normalized digital matrix of the digital precoder101may be PD=PD/∥PACPD∥F.

Further, in some example embodiments, for each Irf∈[Ns,Nrf], the SE of the current RF chain number Irfmay be evaluated as

For each RIrfwhere Irf∈[Ns,Nrf], RIrf/RNrfmay be calculated, and the smallest value of Irf, which is denoted as I* and satisfies RI*/RNrf≥γ, may be determined. Then, optimal analog and digital precoding matrices may be selected corresponding to I*.

Let Nt=256, Nr=4, Nrf=8, γ=0.9, and SNR is in the range 0 to 50 dB.FIG.7illustrates a probability density function (PDF) of RF utilization rate in a case where the digital precoding matrix is determined based on the least square algorithm, andFIG.8illustrates a PDF of RF utilization rate in a case where the digital precoding matrix is determined based on the unitary matrix algorithm. It can be seen that an average RF chain utilization rate in a case of least square algorithm (Least square) is 54.35%, which means only 54.35% RF chains are needed to open so that 90% SE can be achieved, compared with a case of opening all RF chains. Similarly, the average RF chain utilization rate in a case of the unitary matrix algorithm (Unitary) is 66.3%. Thus, great energy may be saved.

In one or more example embodiments, the hybrid precoding is based on the AoSA architecture so that power consumption may be reduced. Further, the determination and optimization of analog precoder102and digital precoder101are decoupled, where an analog precoding matrix associated with the analog precoder102may be determined and/or optimized independently of a digital precoding matrix associated with the digital precoder101, and the digital precoding matrix may be determined and/or optimized based on the determined analog precoding matrix after the determination and/or optimization of the analog precoding matrix. Thus, the design of analog precoder102and digital precoder101in a downlink MIMO system may be simplified. Further, according to simulation experiment results, better power of the effective channel, better SE, and/or better energy efficiency may be achieved through solutions in one or more example embodiments of this disclosure.

Another example embodiment may relate to computer program codes or instructions which may cause an apparatus (for example, a base station in a downlink MIMO system based on AoSA architecture) to perform at least respective methods described above. Another example embodiment may be related to a computer readable medium having such computer program codes or instructions stored thereon. In some example embodiments, such a computer readable medium may include at least one storage medium in various forms such as a volatile memory and/or a non-volatile memory. The volatile memory may include, but not limited to, for example, a RAM, a cache, and so on. The non-volatile memory may include, but not limited to, a ROM, a hard disk, a flash memory, and so on. The non-volatile memory may also include, but are not limited to, an electric, a magnetic, an optical, an electromagnetic, an infrared, or a semiconductor system, apparatus, or device or any combination of the above.

Further, modifiers such as “first”, “second” and so on throughout the description and claims are generally intended to distinguish different elements, operations, and so on, rather than emphasizing any importance, specific sequences, specific priorities, specific elements, and so on.