WO2019039980A1 - Beamforming with quadratic computational complexity in the antenna array size - Google Patents

Abstract

A beamforming method is provided. The method includes determining an initial parameter set, the initial parameter set including an initial time-frequency set (t₀, f₀) and an initial inverse total interference matrix (I.). The method includes, for a given time-frequency set (t, f), performing the following steps: (1) Performing uplink measurements. (2) Computing a channel error contribution (II.) based on the uplink measurements. (3) Computing an interference matrix contribution (III.) based on the uplink measurements. (4) Computing an inverse total interference matrix (IV. ) based on the inverse total interference matrix ((IV.) ) at a previously-computed time-frequency set (t', f) and at least one of (II.) and (III.). (5) Computing a beamforming transceiver solution (Wt,f) based on at least the inverse total interference matrix ((IV.) ).

Description

BEAMFORMING WITH QUADRATIC COMPUTATIONAL COMPLEXITY

IN THE ANTENNA ARRAY SIZE

TECHNICAL FIELD

[001] Disclosed are embodiments related to beamforming with improved performance, and in particular, beamforming with quadratic computational complexity in the antenna array size.

BACKGROUND

[002] In the emerging 5G cellular systems, beamforming and MIMO transmission will be central technologies. The reason is that spectral resources are running out at low carrier frequencies which leads to a gradual migration into higher frequency bands, like the mmw band. There, beamforming and a use of massive antenna arrays are needed to achieve a sufficient coverage. There is, for example, plenty of available spectrum around 28 GHz and 39 GHz in the US and other markets. This spectrum needs to be exploited to meet the increasing capacity requirements. The 5G frequency migration is expected to start at 3.5 - 5 GHz, and then continue to these 28 GHz and 39 GHz bands that are expected to become available in the not-too-distant future.

[003] In the following description, 3GPP terminology for the 4G LTE system is used

(unless otherwise noted), since the standardization of the 5G counterparts are not yet finalized. The operation of the 5G functionality is expected to be essentially the same as in the 4G system.

[004] Beamforming and MIMO transmission is a mature subject today. This section presents the basics.

[005] There are two main methods available for wireless beamforming. The first method relies on the downlink and uplink utilizing the same frequency band. Then channel reciprocity persists and a matrix channel estimated for the uplink can be used for optimal beamforming and MIMO transmission in the downlink, e.g. using Reciprocity Assisted

Interference aware Transmission (RAIT). The second method relies on reference signals being transmitted from the base station, and by feedback information on signal quality being sent back from at least one user equipment (UE) (i.e., a device capable of wireless communication, such as, for example, a smartphone, a smart appliance, a sensor, a tablet, a computer, etc.). In the 4G LTE standard, the UE typically measures the channel response and reports the result back to the base station in terms of channel quality indication (CQI) (based on, e.g., signal-to-noise ratio (SNR) estimates); rank indication (RI) (e.g., indicating channel matrix rank); and precoding matrix indicator (PMI). This reported feedback is denoted channel state information (CSI). A similar solution, with richer codebooks, is currently being standardized for the 5G mmw standards.

[006] To explain the beamforming concept, consider FIG. 1, which shows an idealized, one-dimensional beamforming case. If it is assumed that the UE 102 is located far away from the antenna array 104 (e.g., found at a base station (BS)), then it follows that the difference in travel distance from the base station to the UE, between adjacent antenna elements, is / =

H-sin(0), where k is the antenna element separation. Here k is the separation factor, which may for example be 0.5-0.7 in a typical correlated antenna element arrangement. This means that if a reference signal s_ie ^_-^/wt is transmitted from the z^':th antenna element, it will arrive at the UE antenna as a weighted sum:

[007] Here ω is the angular carrier frequency; z, is the complex channel from the z^':th antenna element; t is the time; and _c is the carrier frequency. In the above equation Θ and hi are unknown. In case of a feedback solution, the UE therefore needs to search for all complex channel coefficients /¾, and the unknown angle Θ. For this reason, the standard defines a codebook of beams in different directions given by steering vector coefficients like w_{m i} = _e -jf(rn,i) ^ _wh_{ere m} indicates a directional codebook entry. The UE then tests each codebook and estimates the channel coefficients. The information rate achieved for each codebook entry m is computed, and the best one defines the direction and channel coefficients. This is possible since Si is known. The result is then encoded and reported back to the base station. This provides the base station with a best direction (codebook entry) and information that allows it to build up a channel matrix H. This matrix represents the channel from each of the transmit antenna elements to each of the receive antenna elements. Typically, each element of H is represented by a complex number.

[008] The channel matrix can then be used for beamforming computations, or the direction represented by the reported codebook entry can be used directly. In case of MIMO transmission, the MIMO beamforming weight matrix W needs to be determined, so that a best match to the condition WH = I is achieved. / denotes the identity matrix. In case of an exact match, each layer will become independent of other layers. This concept can be applied for single users or multiple users.

[009] CSI reference signals (CSI-RS), which has been available since release 11 in

LTE, support beamforming. CSI-RS are assigned to a specific antenna port. CSI-RS may be transmitted to the whole cell or may be beamformed in a UE- specific manner. In 3 GPP, since release 13, two classes of CSI-RS reporting modes have been introduced: a class A CSI-RS mode uses fixed-beam codebook-based beamforming, while a class B CSI-RS mode may send beamformed CSI-RS in any manner.

[0010] A CSI-RS process in a UE comprises detection of selected CSI-RS signals, measuring interference and noise on CSI Interference Measurement (IM) (CSI-IM), and reporting of the related CSI information, in terms of CQI, RI, and PMI. A UE may report more than one set of CQI, RI, and PMI, i.e. information for more than one codebook entry. Since release 11, up to 4 CSI-RS processes can be set up for each UE.

[0011] The codebook of the 3GPP standard is defined to represent certain directions. In release 13, directions in both azimuth and elevation is defined, thereby allowing 2D

beamforming to be used. These 4G codebooks are specified in detail in 3GPP TR 36.897. A similar definition, but with finer granularity is expected for the 3GPP 5G standard.

[0012] To illustrate that the codebooks indeed define specific directions, it can be noted that the formula for the azimuth codebook is w_k = i k = 1,

K. It has the same structure as discussed above. Similarly, the vertical codebook is given by for / = 1, ..., L. Further details of these equations may be

found in the specification document 3GPP TR 36.897. [0013] A 2D beam is obtained by a multiplication of the two above equations (i.e.

equations for azimuth and vertical codebooks).

[0014] Releases 11 and 12 both support 4 CSI-RS processes per UE. In these releases, however, only one-dimensional codebooks, corresponding to 8 antenna ports, are supported, as compared for the support of 2D codebooks for 16 ports in release 13.

[0015] Given two nodes equipped with antenna arrays that communicate in a single frequency band, the channel reciprocity property means that at any given point in time, the complex channel coefficient between any transmitting antenna element in one node and any receiving antenna element in the other node, is the same in the uplink and the downlink (up to conjugate transpose). Channel reciprocity is a consequence of Maxwell's Equations. The channel matrix therefore remains the same (except for a conjugate transpose representing the change of direction) between the antenna arrays of the two nodes when the direction of the transmission is reversed. The two nodes may typically be a UE and an eNB, or in 5G a UE and a gNB, where gNB is the commonly accepted acronym for a 5G base station. The time is assumed to be essentially the same for the two directions of transmission.

[0016] To exploit reciprocity, the channel coefficients can be directly estimated by the base station from UE uplink transmission of known pilot signals, for example so-called sounding reference signals (SRSs). The estimated channel can then be used to compute the combining weight matrix with a selected principle, and then used for downlink transmission. This works since the uplink and downlink channels are the same (except for a conjugate transpose) when reciprocity is valid.

[0017] However, an important restriction is that the locations of the antenna arrays and the rest of the radio environment remains the same during the time reciprocity-based

transmission is applied. This restriction, for example, does not necessarily hold in the case of UE motion (mobility). As a rule of thumb, the channel de-correlates for UE movement of about 0.4 wavelengths. This means that for a given UE mobility, the higher the carrier frequency, the less is the duration of time during which reciprocity holds. After de-correlation, new SRS measurements are needed and a renewed beamforming solution needs to be computed. [0018] RAIT is a MIMO/beamforming technique that is applicable primarily for time division duplex (TDD) deployments, where channel reciprocity can be used.

[0019] Briefly, RAIT offers a unified approach to single-point techniques like multi-user

MEVIO (MU-MIMO) and beamforming, and to multi-point techniques like coordinated multipoint (CoMP) beamforming. The key to achieve this is availability of a high fidelity multi- antenna element spatial ID or 2D matrix channel estimate H. Since reciprocity holds, the channel matrix can be estimated by application of sounding reference signals (SRS) in the uplink. So-called demodulation reference signals (DMRS) can also be used. The channel matrix is then also valid for downlink transmission, during a short period of time. By formulation of a criterion that embeds the above techniques as special cases, a combining weight matrix W ean then be computed and used to steer the downlink transmit power in an optimal way. RAIT is capable of avoiding transmission in directions where interference is likely to be created between users.

[0020] A shortcoming with RAIT, addressed by this disclosure, can be explained from the beamforming criterion W = arg min^ || (tj W— (^) , where H denotes the estimated downlink channel matrix (of size n_eNB X n_UE), e.g. obtained by estimation of the uplink counterpart and reversing, assuming reciprocity to hold. Typically SRSs would be used as reference signals. Similarly, G denotes the neighbor-cell interference matrix obtained e.g. by listening on background transmissions. R is the desired channel matrix and W is the combining beamforming matrix. The operator || ||_F denotes the Frobenius norm. The solution to this problem becomes W = {H*H + A_H + /1_G) H*R, where A_G = G * G denotes the estimate of the neighbor-cell interference covariance matrix and where AH denotes the estimate of the channel error covariance matrix. The operator X* denotes the conjugate transpose of X.

[0021] The dominating multi-user access technology for 5G is expected to become some variant of orthogonal frequency division multiple access (OFDMA). As is well known, this access is associated with a resource grid, divided in time and frequency, as shown in FIG. 2. The resource grid provides a division in frequency defined by sub-carriers and a division in time by OFDM symbols. The product set of a subcarrier and an OFDM symbol forms a resource element and as in LTE, a time-frequency block of resource elements forms a resource block. The currently evolving 3GPP NR 5G standard recently also defined slots and mini-slots, giving additional addressing modes of time-frequency resources. When multi-layered (e.g., MIMO) transmission is used, there is one overlaid resource grid per layer, separated by spatial pre- coding.

SUMMARY

[0022] As seen, for example, from the solution to the RAIT equation above (W =

(H^*H + Λ_Η + Λ_β) H^*R), an inversion of a very large matrix is required. This is

computationally intensive, since matrix inversion complexity is proportional to the cube of the matrix order (i.e., n for matrix order n). For certain applications, particularly in emerging 5G use cases, existing RAIT implementations are too slow, and may simply be impractical to implement.

[0023] For example, as noted above, reciprocity does not hold when the channel de- correlates, and a general rule of thumb is that de-correlation may happen for UE movement of about 0.4 wavelengths. This means, for instance, that for mmw carrier frequencies at around 30 GHz, the beamforming solution may need to be computed 10 times faster than at the conventional cellular carrier frequencies below 3 GHz. The consequence is a corresponding increase of the computational complexity in terms of operations per second.

[0024] As another example, existing RAIT implementations have (as mentioned above) complexity that is cubic in the number of antennas. Therefore, as the number of antennas increases (e.g., for certain 5G applications that allow for a large number of antennas), the complexity of RAIT (or other algorithms similarly having cubic complexity in number of antennas), increases.

[0025] An important driving factor in the computational complexity of beamforming solutions, at both 5G mmw frequencies and 5G frequencies below 6GHz, is the need for an increased number of antenna elements. An additional problem at these carrier frequencies is the fast fading and channel-decorrelation time. This means that beamforming computations need to be finalized much quicker (e.g., as compared to low frequencies). Existing solutions cannot perform beamforming with the computational resources available in existing hardware. As an example, the low-frequency limit for RAIT in terms of computational complexity is currently somewhere in the range 32-64 antenna elements. At high carrier frequencies, configurations with 512 antenna elements are planned, i.e. 8-16 times the low-frequency configuration. This means a computational complexity increase of about a factor of 1000. In addition, due to the 10- times higher carrier frequencies and 10-times smaller channel-decorrelation time, the

consequence is that high carrier frequency beamforming (e.g., using RAIT) may need up to more than 10,000-times more computational resources than low carrier frequency (e.g., using RAIT). This is not feasible. Accordingly, there is therefore a lack of methods for systematic interference avoidance for 5G cellular systems with many antenna elements.

[0026] The present disclosure provides embodiments capable of exploiting beamforming opportunities that arise in both the high mmw frequency bands and the lower 5G bands (e.g., below 6 GHz). Embodiments are also advantageous, for example, at low carrier frequencies with many simultaneous users and antenna arrays with a very high number of antenna elements. In both of these usage scenarios, digital beamforming and MIMO algorithms based on RAIT become very computationally intense, thereby making implementation difficult and/or expensive and/or outright impossible. One reason for this shortcoming is that the RAIT optimization problems involve matrix inversion and/or singular value decompositions, which have

computational complexities that are cubic in the number of antenna elements. In addition, the 5G cellular standards provide for very low latency, and this is a requirement for certain important new use cases such as Critical Machine-Type Communications (C-MTC) and the tactile internet. For such low-latency applications, the RAIT computations need to be terminated much faster. This is particularly true at mmw frequencies where the channel decorrelation time in mobility scenarios is much shorter than at low carrier frequencies.

[0027] Embodiments provide advantages over the rapidly increasing computational complexity of existing beamforming solutions, e.g. using RAIT, where the complexity increases with (1) the number of antenna elements and also with (2) the inverse of the latency and channel- decorrelation time. Unless this complexity problem is solved, applicability of beamforming solutions such as RAIT may become limited to very small antenna arrays (too small for 5G systems), and to low carrier frequencies. Although RAIT has been used as an exemplar here, RAIT is not the only algorithm where the above complexity shortcomings are present, and for which embodiments provided herein may be relevant. Other variants of MIMO transmission and interference rejection combining (IRC) receivers may benefit from embodiments as well.

[0028] Embodiments take advantage of one or more of the following principles:

(1) Embodiments exploit a beamforming calculations where the main complexity is based on a matrix operation (e.g., inversion, decomposition) of a matrix that expresses channel uncertainty and interference.

(2) Embodiments reduce a number of computations required by utilizing interference matrix contributions having low rank (e.g., rank 1).

(3) Embodiments exploit time and/or frequency filtering, such that computation of an inverse of an interference matrix becomes a recursive problem.

(4) Embodiments perform a computation of the inverse channel interference matrix recursively.

[0029] These principles, and the matrix manipulations and transformations behind them, are highly non-trivial. In particular, embodiments rely on a careful and proper use of the so- called matrix inversion lemma (MIL), which must be used in the right way to achieve the advantages described herein. This becomes possible by the introduction of the low-rank filtering restriction in the buildup of the interference matrix. The advantage is that each recursive update of the inverse becomes only quadratic in the number of antenna elements, as compared to cubic for a direct inversion. In the case of 512 antenna elements, for example, this amounts to a decrease in complexity of much more than a factor of 100. This decrease in complexity allows RAIT to be implemented with very large antenna arrays.

[0030] Embodiments are applicable also to other MIMO schemes (besides RAIT) as well as to receiver structures such as IRC receivers, where matrix inversion and/or decomposition is also applied.

[0031] According to a first aspect, a beamforming method is provided. The method includes determining an initial parameter set, the initial parameter set including an initial time- frequency set (to, fo) and an initial inverse total interference matrix (A_Q ¹). The method further includes, for a given time-frequency set (t, f), performing the following steps. (1) Performing uplink measurements. (2) Computing a channel error contribution (AH_{t f}) based on the uplink measurements. (3) Computing an interference matrix contribution (AG_{t f}) based on the uplink measurements. (4) Computing an inverse total interference matrix ((A_{Ht f} + A_G ) ) based on the inverse total interference matrix ((A_H + A_G ) ) at a previously-computed time- frequency set (t', f ) and at least one of AH_{t f} and AG_{t f} . And (5) computing a beamforming transceiver solution (W_t,f) based on at least the inverse total interference matrix ((A_H +

Ac,_,,)^"1).

[0032] In embodiments, step (4) computing the inverse total interference matrix {^A_{Ht f} + ^G_t f ) ) i^s further based on a recursive filter constant a. In embodiments, a channel estimate (H_t ) is computed based on the uplink measurements; and the beamforming transceiver solution (W_t,f) is computed based on the channel estimate (H_t ). In embodiments, the beamforming transceiver solution is based on a desired channel matrix (R_t,f).

[0033] In embodiments, step (4) computing the inverse total interference matrix

{^A_{Ht f} + ^G_t f ^ ) further includes computing the equation:

(A_{Ht f} + A_{Gt f})^_1 := X - XA X (l_nuE + A^*XA)^-1 X A^*X, wherein X = (a(A_Ht, _f, + A_Gt, _f,) + (1 - a) (AH_tjfAH^* _f)) \ A = Vl - aAG_u, and A^

l— aAG_{j f}, and wherein AH^_f denotes the conjugate transpose of AH_{t f} and AG^ _f denotes the conjugate transpose of AG_{t f}.

[0034] In embodiments, step (4) computing the inverse total interference matrix

(( [_{Ht f} + A_{G{ f} ) ) further includes computing the equation:

X := AH^* _fY'

wherein X = (a(A_Ht,_f, + A_Gvfl) + (1 -

and wherein ΔΗ^ _f denotes the conjugate transpose of AH_{t f}.

[0035] In embodiments, step (5) computing the beamforming transceiver solution (W_t,f) further includes computing the equation:

W_t,f := Y - YH^* _f(l_nuE + H_t,fYH^* _f)^_1H_t,fYH^* _fR_t,f, wherein Y = (A_{HT F} + A_{GT F}) , H_{t f} is a channel estimate, Rt,f is a desired channel matrix, and wherein H^_f denotes the conjugate transpose of H_{t f}.

[0036] In embodiments, the uplink measurements are performed on uplink reference signals. In embodiments, the uplink reference signals are one or more of sounding reference signals and demodulation reference signals. In embodiments, the method further includes incrementing the time-frequency set (t, f) in the time domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t' + At, f ); and repeating the incrementing step while a condition is satisfied, where the condition is that a transmit data buffer is nonempty. In embodiments, the method further includes incrementing the time-frequency set (t, f) in the frequency domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t', f + λο); and repeating the incrementing step while a condition is satisfied, where the condition is that a transmit data buffer is nonempty. In embodiments, the method further includes, for the given time-frequency set (t, f), (6) using the beamforming solution to transmit data to a user equipment (102).

[0037] According to a second aspect, a device for beamforming is provided. The device may be, for example, a user equipment and/or a base station and/or another node in a

communications network. The device is adapted to determine an initial parameter set, the initial parameter set including an initial time-frequency set (tO, fO) and an initial inverse total interference matrix (A_Q ¹). The device is further adapted to, for a given time-frequency set (t, f), perform the following steps. (1) Perform uplink measurements. (2) Compute a channel error contribution (AH_{t f}) based on the uplink measurements. (3) Compute an interference matrix contribution (AG_{t f}) based on the uplink measurements. (4) Compute an inverse total interference matrix ((A_{HT F} + A_G ) ) based on the inverse total interference matrix ((A_H + A_G ) ) at a previously-computed time-frequency set (t', f ) and at least one of AH_{T f} and AG_{T f}. And (5) compute a beamforming transceiver solution (W_t,f) based on at least the inverse total interference matrix ((Λ_Η + A_G ) ). The device may further be adapted to perform any of the

embodiments of the first aspect.

[0038] According to a third aspect a device for beamforming is provided. The device may be, for example, a user equipment and/or a base station and/or another node in a

communications network. The device includes a determining module (602) configured to determine an initial parameter set, the initial parameter set including an initial time-frequency set (tO, fO) and an initial inverse total interference matrix (A_Q ¹). The device further includes a measurement module (604) configured to (1) perform uplink measurements for a given time- frequency set (t, f). The device further includes a computing module (606) configured to perform the following steps for the given time-frequency set (t,f): Compute a channel error contribution (AH_{T f}) based on the uplink measurements. Compute an interference matrix contribution (AG_{T f}) based on the uplink measurements. Compute an inverse total interference matrix ((A_{HT F} + A_G ) ) based on the inverse total interference matrix ((A_H + A_G ) ) at a previously-computed time-frequency set (t', f ) and at least one of AH_{T f} and AG_{T f}. And compute a beamforming transceiver solution (W_t,f) based on at least the inverse total interference matrix

((A_{HT F} + A_{GT F}) ). The determining module, measurement module, and computing module may further be configured to perform any of the embodiments of the first aspect.

[0039] According to a fourth aspect, a computer program is provided. The computer program includes instructions which, when executed on at least one processor, causes the at least one processor to carry out the method according to any one of the embodiments of the first aspect.

[0040] According to a fifth aspect, a carrier is provided. The carrier includes the computer program of the fourth aspect. The carrier is one of an electronic signal, optical signal, radio signal or computer readable storage medium. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The accompanying drawings, which are incorporated herein and form part of the specification, illustrate various embodiments.

[0042] FIG. 1 illustrates one-dimensional beamforming.

[0043] FIG. 2 illustrates a resource grid.

[0044] FIG. 3 illustrates a flow chart according to some embodiments.

[0045] FIG. 4 illustrates a flow chart according to some embodiments.

[0046] FIG. 5 illustrates a flow chart according to some embodiments.

[0047] FIG. 6 is a diagram showing functional modules of device according to some embodiments.

[0048] FIG. 7 is a block diagram of a device according to some embodiments.

DETAILED DESCRIPTION [0049] RAIT-type algorithms are planned to form the reciprocity beamforming backbone of present and future base station products. Embodiments are directed to solutions to the problems discussed above. In particular, embodiments do this by decreasing the computational complexity associated with the covariance matrix inversion in RAIT, and by introducing covariance matrix filtering.

[0050] The need for MU-MIMO using RAIT-type algorithms is today an established fact at low frequency bands, for both 4G and 5G. The reason is that user access becomes more and more tight, thereby increasing the need for interference avoidance by MU-MIMO. At high mmw bands this has been debated, since the beamforming have used more narrow beams and since the cells are smaller. However, there are reasons and use cases that may motivate RAIT and MU- MIMO also at mmw frequencies. If it is made feasible, then RAIT would provide a very flexible solution that can be tuned in several ways, a fact that could make the low- and high-band functionality more similar. Such a step towards a single track has significant benefits for operators in terms of operation and maintenance.

[0051] There are several use cases where MU-MIMO may be needed. For example, one exemplary use case is for C-MTC, e.g. in industrial robot control. Feedback control over wireless 5G interfaces, in this context, requires very fast sampling, combined with transmission of only a few hundred bytes per device (e.g., robot). To get an efficient utilization of the air- interface capacity many devices (e.g., robots) must therefore be accessed within the same transmission time interval (TTI) by means of some kind of frequency- selective beamforming algorithm. Here, RAIT may be effectively employed, provided it can be computed in an efficient manner.

[0052] Another exemplary use case occurs indoors, since small indoor nodes will need to use in-band backhaul to reduce deployment costs. In this context, application of RAIT-type MU- MEVIO schemes are needed in order to avoid accidental interference, supporting a kind of self- organizing interference- avoiding deployment of nodes that is also interference adaptive with respect to normal users.

[0053] The RAIT beamforming equation (RAIT-BF-EQ ) is provided below:

(RAIT-BF-EQ)

[0054] Here the subscript t,f refers to the corresponding resource (or set of resources) on the time-frequency grid. H_tj (of size n_eNB X n_UE) is the downlink channel estimate, typically obtained e.g. by exploiting reciprocity. A_{Ht f}{oi size n_eNB X n_eNB) is the channel error covariance matrix, and A_{Gt f} = G^ G_t (of size n_eNB X n_eNB) is the neighbor cell interference covariance matrix. R_t (of size n_eNB X stream) is the desired channel matrix and W_t (of size n_eNB X stream) is the combining beamforming matrix. Here n_eNB is the number of antenna elements of the eNB, n_UE is the number of antenna elements of the UE, and stream is the number of antenna streams. Typically, n_eNB≥ n_UE, and sometimes n_eNB » n_UE.

[0055] The computation of W_{t j} requires an inversion of a matrix of a high dimension

(i.e. of size n_eNB X n_eNB). Matrix inversion has a complexity that is cubic in the matrix order. This order may typically be as high as the number of antenna elements of the array, for example, possibly 512 elements in a 28 GHz macro base station. It has been noted that in practice, an inversion of a matrix of the order of 64 is at the limit of what may be processed in real time for the 4G LTE system. At mmw frequencies, such as will be used for 5G systems, coverage is reduced due to the frequency depending pathloss. To counter this effect, massive antenna arrays need to be exploited. Accordingly, macro base stations with as many as 512 or 1024 (or more) antenna elements are being discussed. This alone (i.e. the increase in number of antennas) means a complexity increase as compared to LTE of, for example, an order of 8 = 512 (e.g., difference in 512 antennas compared to 64 antennas is a factor of 8, and complexity is cubic in number of antennas). In addition, 5G systems will operate at wavelengths that are 10 times less than the wavelength where 4G LTE is deployed. This in turn means that the channel estimated by reciprocity de-correlates 10 times faster than for LTE. Therefore, the computation of W_t needs to be done 10 times faster. In summary, for some implementations of 5G network

configurations, RAIT algorithms may require about 5000 times more operations per second for execution, as compared with 4G LTE. Such calculations are not feasible without utilizing embodiments disclosed herein for improving beamforming performance.

[0056] The RAIT beamforming equation may be transformed with the so-called matrix inversion lemma. The matrix inversion lemma (MIL) is given below:

04 - BD^C)-¹ = A^'1 + Α^Β ^'1 - CA^~1B)-¹CA-¹ (MIL) where A, B, C, and D, are matrices and/or vectors of compatible dimensions.

[0057] Taking A = Λ_Η + A_G , B = H_t ^* _f, C = H_{t f} , and D = -I_nuE, the MIL results in transforming the RAIT-BF-EQ into the following equation (Eq. 1):

[0058] The advantage achieved is that the original matrix inversion, which inverted a matrix of dimension n_eNB X n_eNB , is transformed into an inversion of a matrix of dimension of size n_UE X n_UE, which is usually much smaller. However, the gain in complexity is nullified by the requirement to perform the inversion of the sum of the covariance matrices of dimension neNB ^{x n}eNB■ At first glance, then, it seems that little has been gained. What is needed is an efficient way to update (^_Htf + A._Gt^j . The inversion, as it stands, has a complexity that is cubic in the matrix order that may be as high as the number of antenna elements of the eNB array, possibly 512 elements in a 28 GHz macro base station. This problem is addressed below. [0059] As will be shown, additional structural constraints on the covariances holds the key towards a low complexity covariance matrix inversion. In particular, covariance matrices may be formed by rank 1 terms. The channel covariance matrix (valid for the time frequency resource set t,f) is updated from channel measurement errors (on this set), for a set of subcarriers and symbols, as follows: A_Hf =∑_Scef∑symboiset AH_tj AH_t ^* . Here, AH_t denotes channel error matrices. Similarly, A_{Gt f} =∑scef∑symboiset ^G_{t f}AG_t ^* _f .

[0060] In case of consecutive channel error measurements in time, the following recursive updates of the covariance matrices may be formulated as:

A_{Ht f} = aA_H(__{i f} + (1 - a) f f

scef symbolset

Λ^ = aA_Gt__{i f} + (1 - a) ΔΟ^ΔΟ^

scef symbolset

[0061] Here, 0 < a < 1 is a filtering constant. These equations demonstrate a recursive update employing time filtering. It is also possible to employ frequency filtering (e.g., in case of consecutive channel error measurements in frequency).

[0062] Exploiting the time (and/or frequency) filtering from above, and further noting that it is, for example, possible to approximate the channel error over a small range of subcarriers and symbols with a single error vector (e.g., AH_tj and AG_tj), the following equalities result:

+ (1 - a) (AH_tifAHl_f + AG_tifAG_t ^* )

scef symbolset J

= ( (A_H{__{i f} + A_Gt__{i f}) + (1 - a) (AH_tifAHl_f + AG_tifAGl_f†j ' [0063] Applying the MIL to this last equality (by taking A = a A_{HT IF} + A_{Gt if} +

(1— a AH_tjAH_t ^* f, B = Vl— AG_t , C = Vl— AG_t ^* , and D =—I_nuE) yields the equation (Eq.2) below:

(A_HT,F + A_GTF) ¹

[0064] This equation can be simplified further by applying the MIL again (taking

[0065] To understand the simplification, the case with one UE antenna is considered, i.e. l_n = 1. This, for example, means that the middle factor in (Eq. 3) above becomes a scalar, i.e. the matrix inversion for the term including l_n is simplified to a scalar division. Therefore, given (A_Ht__{i f} + A_G{__{i f}^ , it follows from the last chain of equalities that the expression (A_Ht__{i f} + A_{Gt i f}^ + (1— a)AH_t fAH_{{ j}^J can be computed with a complexity proportional to the square of n_eNB. This follows since the multiplication of a matrix with a row or column vector is quadratic in terms of the dimension of the vector and the matrix. Next, by applying

(Eq. 2) above, it follows that ( _{Ht f} + A_{G{ j}^ ^can be computed from (a (A_Ht__{l f} + A_{Gt i f}^ + (1— with a computational complexity again proportional to the square of n_eNB.

This thus shows how ( _{Ht f} + _Gt^j can be updated from (A_{H{ i f} + A_Gt__{i f}^j with a complexity proportional to the square of n_eNB. This represents a saving of a factor of n_eNB as compared to a direct inversion. At mmw frequencies, for instance, this could save two orders of magnitude of computations.

[0066] FIG. 3 shows a flow chart according to some embodiments. In particular, the flow chart illustrates the computational steps described above, for rank 1 covariance matrix updates. As shown, process 300 includes a number of configuration steps (steps 302-308). For example, a is configured (step 302). The configuration of a, the time (or frequency) filtering constant, which will affect the recursive updating as described above. Also, Ao is configured (step 304). Ao denotes an initial value to the covariance matrix filtering (e.g., it may be identity in some embodiments). A frequency band/is selected (step 306). Time t is initialized to to (step 308). The covariance matrix is also initialized (step 308). For example, the covariance matrix is initialized as ( _{Ht f} + A_Gt^ ^{: =} ^ο ¹· Following step 308, a loop comprising steps 312-324 is entered. This loop comprises measuring certain parameters, and performing calculations based on those parameters. The measurements and/or calculations can be re-ordered in some embodiments. As shown, AH_t is measured (step 312). This value (channel error matrix) is needed for computing the equation (Eq. 3). Thus, after it is measured, (Eq. 3) may be computed, since all inputs are known (step 314). By this, it is meant that the value (a (A_Ht__{i f} + A_Gt__{i f}^ + (1— a)AH_t AH j is computed by using (Eq. 3) and the known inputs required by (Eq. 3).

AG_t is measured (step 316). Based on AG_tj and the result of step 316, it is possible to compute (Eq. 2), since all inputs are now known. Thus, after it is measured, (Eq. 2) may be computed

(step 318). That is, (^_{Ht f} + A._Gt^j may be calculated by using (Eq. 2) and the known inputs. From these values, it is then possible to compute (Eq. 1) (step 320). That is, W_t is calculated based on (Eq. 1).

[0067] Following the calculation of W_t j at step 320, the time variable t may be updated

(step 322). In some embodiments, instead of updating time, the frequency may be updated instead (e.g., if using frequency filtering instead of, or in addition to, time filtering). Then a decision is made at step 324. If no data is available for transmitting, the algorithm ends.

However, if more data is available for transmitting, then flow continues back to step 312, and the loop repeats. Any data that was ready for transmitting at step 324, may be transmitted according to the beamforming matrix W calculated at step 320. In some embodiments, data transmission continues until a channel de-correlation criterion is triggered, in which case flow resumes back to step 312 at that time.

[0068] Returning to the general filtered "sum of rank 1 covariance expressions," one question is how the rank 1 update derived above can be generalized to an arbitrary number of terms. Towards that end, the following formulation are considered, where the MIL is applied repeatedly (taking A, B, C, D, in the MIL, as appropriate each time the MIL is applied). Further,

Nf

in the following, /represents a set of frequencies {/}} with index i =1, Nf, and {/} . refers to a subset of those frequencies {/}}, where index i=j, . .. Nf. This results in the equations (Eqs. 4A- 4E) below:

„ + A

{/£} ^bt-^l.{ £}^

(Eq.4A)

+ (1 - a) ^H_t AH_t ^* _if + AG_t AG_t ^* _if)

scef symbolset

N_f

-1

+ (1 - a)AH_tihAH_t ^* _ifi

{ V (λ_Η .

V

, - + A_G , + AA_GFF +ΔΛ N_f)

,{/;} ^Gt-i,{f_I}J t.fi t,{f}₂ ^F

- ( VC ( VA_{H R} , +A_G , ,~) +AA_GTF

Ht- !,{/;} ^Gt-l,{f_I}J t,f₁

+ /1_G + A/1_G_ +ΔΑ _NF)

^Gt-i,{ .fi}J ^f. i t,{/},V

[0069] Here, A_G^ = and ^ ^ = (1 -

[0070] Note that this parallels the elimination of a single rank 1 term above that resulted in (Eq.2). The next step is to eliminate AA_G from the inverse of the above equation, by applying the MIL again, resulting in:

-1

(CC(A_H , , +A_G . )+AA_Gff +ΔΑ Nf )

V V ^Ht- !,{/;} ^Gt-l,{f_I}J ^Gt,f₁ t,{f}₂'J

= (^a ( _lW ⁺ ,₍J ⁺ (Eq.5A)

i^ ^^^ ^+^J _{(Eq 5]}

+ ΔΛ^) ¹ (l - c AGt V(l - a)AG{_h (a + /1_G . + ΔΛ N_f)

[0071] The two above chains of equalities (Eqs.4A-4E and5A-5B) allows for an update fromf Vaf/l_tf . . + A_G . + ΔΛ _Nf] to ( a (A_{HF F} + A_{GF f} ) + ΔΛ _Nf] .

V ^Gt-i^,{f_i}J t,{f}₂ ^fJ V V ^t-¹^ ^t-^) t.lft J

[0072] This procedure can be generalized to provide the following update from

la(A_H , , +A_G , +ΔΑ Nf ) to ( (A_H†„ . + A_GF„ _f) + ΔΛ _Nf ) . [0073] In particular, the following two equations (Eqs.6 and 7) follow:

-1

a (Λ_Η , , +A_G , + ΔΛ

"t-i.lfi} ^Gt-i,{f_i}) t,{f}j^fJ

- ^Λ^ ^+Λ^)^+ΔΛ^

[0074] The procedure starts with (a (-½_t + A_{Gt i {f }}) ) and ends with

[0075] As can be seen, by updating the inverse instead of the sum of covariances, each update involves only multiplications between matrices of dimension n_eNB X n_eNB and n_eNB X n_UE where normally n_UE « n_eNB and therefore the computational complexity remains quadratic in n_eNB .

[0076] When computing the complexity, further reductions are obtained by using positive definiteness and symmetries of the involved matrices.

[0077] FIG. 4 shows a flow chart according to some embodiments. In particular, the flow chart illustrates the computational steps described above, for rank n sum of rank 1 covariance recursive matrix updates. As in process 300, initialization of various parameters may be performed (step 402). For example, a, Ao ,f, t, and/or the covariance matrix may be initialized. An index j is set to N/ (step 404). Following step 404, a loop comprising steps 406- 422 is entered. This loop comprises measuring certain parameters, and performing calculations based on those parameters. The measurements and/or calculations can be re-ordered in some embodiments. As shown,

is measured (step 406). This value is needed for computing (for the current value of index j) the equation (Eq. 7). Thus, after it is measured, (Eq. 7) may be computed, since all inputs are known (step 408). ΔΗ_{^. is measured (step 410). Based on ^H_{t i} and the result of step 408, it is possible to compute (for the current value of index j) equation (Eq. 6), since all inputs are now known. Thus, (Eq. 6) may be computed (step 412). After this, the index j is decremented (step 414). Next, at step 416, it is determined whether j==0. If not, then control is returned to step 406, so that the inner loop steps 406-416 may be executed again (for the next value of index j). On the other hand, if at step 416 it is determined that j==0, then control passes to step 418. In particular, from the values computed over the running index j, it is then possible to compute equation (Eq. 1) (step 418). That is, W_t is calculated based on (Eq.

1).

[0078] Following the calculation of W_t at step 418, the time variable t may be updated

(step 420). In some embodiments, instead of updating time, the frequency may be updated instead (e.g., if using frequency filtering instead of, or in addition to, time filtering). Then a decision is made at step 422. If no data is available for transmitting, the algorithm ends.

However, if more data is available for transmitting, then flow continues back to step 406, and the loop repeats. Any data that was ready for transmitting at step 422, may be transmitted according to the beamforming matrix ^calculated at step 418. In some embodiments, data transmission continues until a channel de-correlation criterion is triggered, in which case flow resumes back to step 406 at that time.

[0079] As described above, and shown in embodiments herein disclosed, the RAIT complexity reduction solution may be described in antenna-element space. However, the first transformation above (leading to (Eq. 1)), for example, could be done in other ways, such as e.g. to a reduced beamspace, such that embodiments can be applied to re-defined beamspace versions of AH_t and AG_t .

[0080] Advantages provided by some embodiments are illustrated with an example.

Consider a usage case with a significant amount of frequency-selective simultaneous users (n_user) on one symbol of the frequency grid. If RAIT is applied in standard form (e.g., without the complexity-reducing algorithms disclosed herein), the computational complexity associated with RAIT on the complete symbol equals c^s^n^ .

[0081] Using the rank-one embodiment (e.g., see Eqs. 2-3),the computational complexity becomes c₂n_USerⁿeNB - Here c_x and c₂ are constants of the order of 1. The saving in terms of computational complexity for such a use case is hence of the order of n_eNB . This saving can be, for example, as high as 512 (or higher), but will be at least a factor of 100 at mmw frequencies.

[0082] Using the rank-n-sum embodiment (e.g., see Eqs. 6-7), the computational complexity becomes ₃n_userNfn _NB . Note that n_userN^is constant, therefore the computational complexity for the whole frequency band in some embodiments may not be worse when the rank-n-sum update is used for a user.

[0083] A usage scenario with many users on a single OFDMA symbol is expected to be common in some 5G applications, such as, for example, C-MTC.

[0084] It should be mentioned that at low frequencies, the conventional mix of traffic applies. Hence there may again be many simultaneous users, e.g. using voice services. For these users, embodiments are also applicable. Since the number of antenna elements at low

frequencies is less than at mmw frequencies, the advantages may be smaller. However, the advantages are still substantial for n_eNB in the range, for example, of 16-64 antenna elements.

[0085] In embodiments, the computational complexity of the beamforming RAIT algorithm (or similar algorithms) may be reduced by at least two orders of magnitude (e.g., a factor of 100) for antenna arrays with more than about 100 antenna elements. This further entails an advantage in cost saving in terms of hardware manufacturing cost, which can be quite significant.

[0086] At 5G mmw frequencies, embodiments provide the difference between a possible implementation of RAIT-like algorithms, and the impossibility of such implementation.

[0087] FIG. 5 illustrates a flow chart according to some embodiments. According to process 500, a beamforming method includes determining an initial parameter set, the initial parameter set including an initial time-frequency set (tO, fO) and an initial inverse total interference matrix (/I_Q ¹) (step 502). The beamforming method further includes, for a given time-frequency set (t, f), performing the following steps (step 504): (1) performing uplink measurements (step 506); (2) computing a channel error contribution (AH_t ) based on the uplink measurements (step 508); (3) computing an interference matrix contribution (AG_t ) based on the uplink measurements (step 510); (4) computing an inverse total interference matrix ((^i_{Ht f} +

A-G_tf) ) based on the inverse total interference matrix {(^A_{Htl fl} + A_{Gtl fl} ^j ) at a previously- computed time-frequency set (t', f ) and at least one of AH_t and AG_t (step 512); and (5) computing a beamforming transceiver solution (W_t,f) based on at least the inverse total interference matrix ((/[_{Ht f} + A_{G{ f} ) ) (step 514). [0088] In embodiments, step (4) computing the inverse total interference matrix ((Λ_Η

■^G_{t f}) ) is further based on a recursive filter constant a.

[0089] In embodiments, for a given time-frequency set (t, f), (6) a channel estimate

(H_t ) is computed based on the uplink measurements; and (7) the beamforming transceiver solution (W_t,f) is computed based on the channel estimate (H_t ). In embodiments, the beamforming transceiver solution is based on a desired channel matrix (R_t,f).

[0090] In embodiments, step (4) computing the inverse total interference matrix

((^H_t f + ^G_t ) ) further includes computing the equation: (A_Ht + A_Gtf) := X— XA X (l_nuE + A^*XA)^-1 X A^*X, wherein X = (a(A_Ht,_f, + A_Gt,_f,) + (1 - A =

Vl— aAG_{t f}, and A* = l— aAG_{j f}, and wherein AH^_f denotes the conjugate transpose of AH_{t f} and AG_{j f} denotes the conjugate transpose of AG_{t f}.

[0091] In embodiments, step (4) computing the inverse total interference matrix

((^H_t f + ^G_t f ) ) further comprises computing the equation X := ^ Y' - Y ^« AH_t,f I_nuE +

Y'

= (A_{Hti fi} + A_{Gti fi}) , and wherein AH^_f denotes the conjugate transpose of AH_{t f}.

[0092] In embodiments, step (5) computing the beamforming transceiver solution (W_{t f}) further comprises computing the equation: W_{t f} := Y— YH_{j f}(l_nuE + H_{t f}YH_{j f}) H_{t f}YH_{j f}R_{t f}, wherein Y = (A_Htf + A_Gtf) , H_{t f} is a channel estimate, R_{t f} is a desired channel matrix, and wherein H^_f denotes the conjugate transpose of H_{t f}.

[0093] In embodiments, the uplink measurements are performed on uplink reference signals. In embodiments, the uplink reference signals are one or more of sounding reference signals and demodulation reference signals. In embodiments, the method further includes incrementing the time-frequency set (t, f) in the time domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t' + At, f ); and repeating the incrementing step while a condition is satisfied, wherein the condition is that a transmit data buffer is nonempty. In embodiments, the method further includes incrementing the time-frequency set (t, f) in the frequency domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t', f + λο); and repeating the incrementing step while a condition is satisfied, wherein the condition is that a transmit data buffer is nonempty. In embodiments, the method further includes, for the given time-frequency set (t, f), (6) using the beamforming solution to transmit data to a user equipment.

[0094] FIG. 6 is a diagram showing functional modules of a device (e.g., UE 102, BS

104) according to some embodiments. As shown in FIG. 6, the device includes a determining module 602, a measurement module 604, and a computing module 606. The determining module 602 is configured to determine an initial parameter set, the initial parameter set including an initial time-frequency set (tO, fO) and an initial inverse total interference matrix (A_Q ¹) . The measurement module is configured to, for a given time-frequency set (t, f), perform uplink measurements. The computing module is configured to compute a channel error contribution (AH_tf) based on the uplink measurements; compute an interference matrix contribution (A G_t ) based on the uplink measurements; compute an inverse total interference matrix {^A_{HT F} +

A-G_tf) ) based on the inverse total interference matrix ((^A_{HTL FL} + A_{Gtl fl} ^j ) at a previously- computed time-frequency set (t', f ) and at least one of AH_t and A G_t and compute a beamforming transceiver solution (W_tif) based on at least the inverse total interference matrix

[0095] FIG. 7 is a block diagram of a device (e.g. UE 102 and/or BS 104) according to some embodiments. As shown in FIG. 7, the device 102 and/or 104 may comprise: a data processing apparatus (DP A) 702, which may include one or more processors (P) 755 (e.g., a general purpose microprocessor and/or one or more other processors, such as an application specific integrated circuit (ASIC), field-programmable gate arrays (FPGAs), and the like); a network interface 748 comprising a transmitter (Tx) 745 and a receiver (Rx) 747 for enabling device 102 and/or 104 to transmit data to and receive data from other nodes connected to a network 710 (e.g., an Internet Protocol (IP) network) to which network interface 748 is connected; circuitry 703 (e.g., radio transceiver circuitry (RTC)) coupled to an antenna system 704 for wireless communication); and local storage unit (a.k.a., "data storage system") 708, which may include one or more non- volatile storage devices and/or one or more volatile storage devices (e.g., random access memory (RAM)). In embodiments where device 102 and/or 104 includes a general purpose microprocessor, a computer program product (CPP) 741 may be provided. CPP 741 includes a computer readable medium (CRM) 742 storing a computer program (CP) 743 comprising computer readable instructions (CRI) 744. CRM 742 may be a non-transitory computer readable medium, such as, but not limited, to magnetic media (e.g., a hard disk), optical media, memory devices (e.g., random access memory), and the like. In some embodiments, the CRI 744 of computer program 743 is configured such that when executed by data processing apparatus 702, the CRI causes device 102 and/or 104 to perform steps described above (e.g., steps described above with reference to the flow charts). In other embodiments, device 102 and/or 104 may be configured to perform steps described herein without the need for code. That is, for example, data processing apparatus 702 may consist merely of one or more ASICs. Hence, the features of the embodiments described herein may be implemented in hardware and/or software.

[0096] While various embodiments of the present disclosure are described herein

(including the appendices, if any), it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of the present disclosure should not be limited by any of the above-described exemplary embodiments. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the disclosure unless otherwise indicated herein or otherwise clearly contradicted by context.

[0097] Additionally, while the processes described above and illustrated in the drawings are shown as a sequence of steps, this was done solely for the sake of illustration. Accordingly, it is contemplated that some steps may be added, some steps may be omitted, the order of the steps may be re-arranged, and some steps may be performed in parallel.

Claims

CLAIMS:

1. A beamforming method comprising:

determining an initial parameter set, the initial parameter set including an initial time- frequency set (to, fo) and an initial inverse total interference matrix (A_Q ¹) ;

for a given time-frequency set (t, f), performing the following steps:

(1) performing uplink measurements;

(2) computing a channel error contribution (AH_t ) based on the uplink measurements;

(3) computing an interference matrix contribution (A G_t ) based on the uplink measurements;

(4) computing an invers ((A_H + A_G ) based on the inverse total interference matrix a previously-computed time-frequency

set (t', f ) and at least one of A H_t and A G_t ; and

(5) computing a beamforming transceiver solution (W_t,f) based on at least the inverse

2. The beamforming method of claim 1, wherein step (4) computing the inverse total interference matrix (\ A_{HT F} + A_G{ A ) is further based on a recursive filter constant a.

3. The beamforming method of any one of the methods of claims 1-2, wherein

(6) a channel estimate (H_t ) is computed based on the uplink measurements; and

(7) the beamforming transceiver solution (W_t,f) is computed based on the channel estimate (H_t ).

4. The beamforming method of claim 3, wherein the beamforming transceiver solution is based on a desired channel matrix (R_t,f).

5. The beamforming method of any one of claims 1-4, wherein step (4) computing the inverse total interference matrix {^A_Htf + A_Gt^j ) further comprises computing the equation: X-XAX

Q_nuE + A^*XA)^→ X A^*X, wherein X = (a (A_Htlfl + A_Gtlfl) + (1 - a)(AH_t AH_t ^* _f†j , A = Vl - AG_t , and A^'

= Vl— cAGf, and wherein A\i_t ^* _j denotes the conjugate transpose of AH_t and AG_t ^* _j denotes the conjugate transpose of AG_t .

6. The beamforming method of any one of claims 1-5, wherein step (4) computing the inverse total interference matrix ((A_H{ + A_GT ) further comprises computing the equation: AHl_fY'AH_tif AHl_fY'

wherein X = (a (A_Htlifl + A_Gtif) + (1 - a)(AH_t AHl_f)) \ Y' = (A_Htlifl + A_GTIF)^~ and wherein AH^ denotes the conjugate transpose of AH_t .

7. The beamforming method of any one of claims 1-6, wherein step (5) computing the beamforming transceiver solution (W_tif) further comprises computing the equation:

W_tJ =Y- YHl_f(l_nuE + H_t YHl ) ^{~ 1} H_t,f YHt.f R t,f _> wherein Y = (^_HTF + A_G{F^ _» H_tj is a channel estimate, R_tif is a desired channel matrix, and wherein denotes the conjugate transpose of H_t .

8. The beamforming method of any one of claims 1-7, wherein the uplink measurements are performed on uplink reference signals.

9. The beamforming method of claim 8, wherein the uplink reference signals are one or more of sounding reference signals and demodulation reference signals.

10. The beamforming method of any one of claims 1-9, further comprising: incrementing the time-frequency set (t, f) in the time domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t' + At, f); and

repeating the incrementing step while a condition is satisfied, wherein the condition is that a transmit data buffer is nonempty.

11. The beamforming method of any one of claims 1-9, further comprising: incrementing the time-frequency set (t, f) in the frequency domain such that after performing steps (l)-(5) for the given set (t', f ), steps (l)-(5) are performed for the given set (t,f) = (t\ f + λο); and

repeating the incrementing step while a condition is satisfied, wherein the condition is that a transmit data buffer is nonempty.

12. The beamforming method of any one of claims 1-1 1, further comprising, for the given time-frequency set (t, f), (6) using the beamforming solution to transmit data to a user equipment (102).

13. A device (102, 104) for beamforming, the device (102, 104) being adapted to:

determine an initial parameter set, the initial parameter set including an initial time- frequency set (to, fo) and an initial inverse total interference matrix (A_Q ¹); for a given time-frequency set (t, f), perform the following steps:

(1) perform uplink measurements;

(2) compute a channel error contribution (AH_t ) based on the uplink measurements;

(3) compute an interference matrix contribution (AG_t ) based on the uplink

measurements; (4) compute an inverse total interference matrix ( A_{H( f} + A_{Gt f} ) based on the inverse total interference matrix (^Λ_{Η{Ι fi} + A_{Gti f}^ ) at a previously-computed time-frequency set (t', f) and at least one of AH_t and AG_t and

(5) compute a beamforming transceiver solution (W_t,f) based on at least the inverse total interference matrix (^A_{Ht f} + A_{G{ f}^ ).

14. The device (102, 104) of claim 13, being further adapted to perform any one of the methods of claims 2-12.

15. A device (102, 104) for beamforming, the device (102, 104) comprising:

a determining module (602) configured to determine an initial parameter set, the initial parameter set including an initial time-frequency set (to, fo) and an initial inverse total interference matrix (A_Q ¹) ;

a measurement module (604) configured to perform uplink measurements for a given time-frequency set (t, f); and

a computing module (606) configured to perform the following steps for the given time- frequency set (t,f):

compute a channel error contribution (AH_t ) based on the uplink measurements; compute an interference matrix contribution (AG_tj) based on the uplink measurements; compute an inverse total interference matrix {^A_{Ht f} + A_{Gt f}J^ ) based on the inverse total interference matrix {{A_{HV FI} + A_{Gti f}^ ) at a previously-computed time-frequency set (t', f) and at least one of AH_t and AG_t and compute a beamforming transceiver solution (W_tif) based on at least the inverse total interference matrix ((A_{Ht f} + A_{G{ f} ) ).

16. A computer program (743), comprising instructions (744) which, when executed on at least one processor (755), causes the at least one processor (755) to carry out the method according to any one of claims 1 to 12.

17. A carrier comprising the computer program (743) of claim 16, wherein the carrier is one of an electronic signal, optical signal, radio signal or computer readable storage medium (742).