System and method for communications using spatial multiplexing with incomplete channel information

A system and method for communications using spatial multiplexing with incomplete channel information are provided. A method for wireless communications includes receiving, at a controller, a reference signal transmitted by a communications device, computing channel statistics based on the received reference signal, computing a first beamforming vector and a second beamforming vector, and transmitting information to the communications device. The reference signal being transmitted using a subset of antennas used for data reception at the communications device, and the controller and the communications device utilize cross-polarized antennas. The computing being based on the channel statistics, the transmitting uses the first beamforming vector and the second beamforming vector, the first beamforming vector precodes information for a first antenna at the communications device, and the second beamforming vector precodes information for a second antenna at the communications device.

TECHNICAL FIELD

The present invention relates generally to wireless communications, and more particularly to a system and method for communications using spatial multiplexing with incomplete channel information.

BACKGROUND

Generally, in wireless communications systems knowledge of downlink (DL) channel information may be essential for efficient beamforming to one or more users. The transmission of information to two or more users is commonly referred to as spatial multiplexing.

DL channel information may be provided by a receiver (e.g., a mobile station, a user, a terminal, a User Equipment, and so on) to a transmitter (e.g., a base station, a NodeB, an enhanced NodeB, a base terminal station, a relay station, and so forth) over a feedback channel. In frequency division duplexing (FDD) communications systems, the receiver may estimate or measure the DL channel and then feed the DL channel information back to the transmitter. The DL channel information may be feedback in its raw form, a quantized version (a codeword from a codebook known by both the receiver and the transmitter), an index to the quantized version (e.g., an index to the codeword from the codebook), or so on.

In time-division duplexing (TDD) communications systems, when calibrated antenna arrays are used, uplink (UL) and DL channels may be almost identical. Channel reciprocity may be a commonly used term to describe this phenomenon. Since the UL and DL channels may be almost identical, it may be possible for a receiver to transmit a sounding reference signal in an UL channel to a transmitter, the transmitter may measure the UL channel using the sounding reference signal, and the transmitter may use the information about the UL channel in its DL transmission to the receiver.

However, in practical communications systems, the receiver may not have an equal number of receive radio frequency (RF) chains and transmit RF chains. For example, a receiver may have two receive antennas but only one transmit antenna, i.e., a first of the two receive antennas may also transmit, but a second of the two receive antennas may only receive. Therefore, only a portion of the channel state information is available through sounding reference signal measurement, providing incomplete channel state information (I-CSI).

SUMMARY

These technical advantages are generally achieved, by embodiments of a system and method for communications using spatial multiplexing with incomplete channel information.

In accordance with an embodiment, a method for wireless communications is provided. The method includes receiving, at a controller, a reference signal transmitted by a communications device, computing channel statistics based on the received reference signal, computing a first beamforming vector and a second beamforming vector, and transmitting information to the communications device. The reference signal being transmitted using a subset of antennas used for data reception at the communications device, and the controller and the communications device utilize cross-polarized antennas. The computing being based on the channel statistics, the transmitting uses the first beamforming vector and the second beamforming vector, the first beamforming vector precodes information for a first antenna at the communications device, and the second beamforming vector precodes information for a second antenna at the communications device.

In accordance with another embodiment, a method for wireless communications is provided. The method includes transmitting a reference signal using a subset of antennas used for receiving information to a controller, and receiving information from the controller. The information is precoded using a first beamforming vector and a second beamforming vector, the information precoded by the first beamforming vector is received by a first antenna, the information precoded by the second beamforming vector is received by a second antenna, and the first beamforming vector and the second beamforming vector are computed from estimates of statistical channel information.

In accordance with another embodiment, a communications controller is provided. The communications controller includes a receiver, a beamforming vector compute unit coupled to the receiver, and a beamforming unit coupled to the receiver and to the beamforming vector compute unit. The receiver receives a reference signal transmitted by a communications device. The communications device transmits with a subset of antennas used for receiving information. The beamforming vector compute unit computes beamforming vectors for a communications device based on estimates of elements of correlation matrices for antennas in the subset of antennas based on a transmission of the reference signal by the communications device, and the beamforming unit precodes data to be transmitted to the communications device with the beamforming vectors computed by the beamforming vector compute unit for the communications device.

An advantage of an embodiment is that available channel state information and channel statistical information are used to compute missing or incomplete channel state information to enable spatial multiplexing with beamforming vectors in a MIMO wireless communications system with communications devices using cross-polarized antenna arrays.

A further advantage of an embodiment is that channel statistical information is used to select the beamforming vectors, thereby yielding better beamforming performance than selecting beamforming vectors without having instantaneous channel information in a random manner.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The embodiments will be described in a specific context, namely a WiMAX compliant communications system with mobile stations (MSs) having more receive antennas than transmit antennas and antennas at both the MS and base stations (BSs) arranged in a cross-polarized configuration. The invention may also be applied, however, to other communications systems that support spatial multiplexing where user equipments (UEs) have more receive antennas than transmit antennas, such as Third Generation Partnership Project (3GPP) Long Term Evolution (LTE), 3GPP LTE-Advanced, and so forth, with antennas arranged in a cross-polarized configuration.

FIG. 1illustrates a wireless communications system100. Wireless communications system100includes a BS101and a MS105and MS106, which may be mobile or fixed. BS101and MS105and MS106may communicate using wireless communications. BS101has a plurality of transmit antennas115, while MS105and MS106may have one or more receive antennas. BS101sends control information and data to MS105through downlink (DL) channel120, while MS105sends control information and data to BS101through uplink (UL) channel125. BS101and MS106may also communicate over similar channels.

MS105may send control information on UL channel125to improve the quality of the transmission on DL channel120. BS101may send control information on DL channel120for the purpose of improving the quality of uplink channel125. A cell130is a conventional term for the coverage area of BS101. It is generally understood that in wireless communications system100there may be multiple cells corresponding to multiple BSs, as well as multiple MSs.

In general, spatial multiplexing allows for the transmitting of parallel data streams in wireless communications systems equipped with an array of cross-polarized antenna pairs. It is also known that in a multiple-input, multiple output (MIMO) link, the knowledge of the channel (whether instantaneous or statistical) can help increase the capacity of the MIMO link. When full instantaneous knowledge of the channel matrix is available at the transmitter, the right singular vectors of the channel matrix provide the optimal directions for transmitting parallel data streams. When power control is also employed among transmitted parallel streams, the method is known as the water-filling solution which provides the maximum achievable capacity when the complete instantaneous channel is known. Additionally, when statistical knowledge of the channel is available, the eigen directions of the spatial correlation matrix are known to be the optimal directions for parallel data transmission in MIMO channels.

BS101may exploit spatial multiplexing to increase data rate in wireless communications system100. For example, although DL channel120between BS101and MS105is shown as a single channel, DL channel120may actually be multiple parallel data streams with each parallel data stream transmitted by a transmit antenna in plurality of transmit antennas115. Similarly, BS101may also use spatial multiplexing in its transmissions to MS106.

FIG. 2illustrates a portion of a wireless communications system200. As shown inFIG. 2, an antenna205, such as an antenna of a BS210operating in the wireless communications system200, may be partitioned into three sectors, such as sector215and sector216. Although shown inFIG. 2as a single antenna, the antenna205may consist of three individual antennas, with one antenna per sector. The BS210may be capable of transmitting separate signals within the different sectors. Furthermore, the BS210may spatially divide the signals to multiple MSs, such as MS220and MS221, within their respective sectors.

The use of polarized antennas is a popular technique for realizing a large antenna when there are restrictions on the physical size of the base-station. In such a case, the antennas are placed in two planes (a V-plane and an H-plane, for example) and at each plane, the antennas form a uniform linear array.

FIG. 3aillustrates an antenna configuration of a BS of a wireless communications system. The BS includes four antennas numbered, antennas305through308. Antennas305through308are referred to as antennas #1through #4, respectively. Antennas305and307may have a similar polarization, while antennas306and308may have a different polarization. Antennas305and306may be spaced a distance D apart from antennas307and308. Antennas305through308may be used in both transmit and receive modes.

FIG. 3billustrates an antenna configuration of a MS of a wireless communications system. The MS includes two antennas, antennas320and321. Antenna320is also referred to as antenna #1and antenna321as antenna #2. Antennas320and321may have different polarizations. Both antennas320and321may be used in a receive mode by the MS, but only one antenna (antenna320, for example) may be used to transmit. Although the antennas of the MS may have different polarizations, operability of the embodiments presented herein does not rely on the polarization of antennas of the MS.

FIG. 4illustrates a model400of communications between a BS405and a MS410. For discussion purposes of model400, let BS405have Ntantennas and MS410have Nrantennas. Furthermore, let all Ntantennas at BS405be operable as both transmit and receive antennas, but for all Nrantennas at MS410only a subset (e.g., one) may operate as a transmit antenna while all may operate as receive antennas.

As shown inFIG. 4, BS405may have multiple antennas, such as antennas415through418that may operate as both receive and transmit antennas. Antennas415through418may also be referred to as antennas #1through #4. BS405is shown inFIG. 4as having four antennas (Nt=4). However, a BS may have any number of antennas, such as one, two, three, four, five, six, and so forth. Therefore, the illustrative example of BS405having four transmit/receive antennas should not be construed as being limiting to either the scope or the spirit of the embodiments.

Also as shown inFIG. 4, MS410may have multiple antennas (Nr=2), such as antennas420and421. Antenna420may also be referred to as antenna #1and antenna421may also be referred to as antenna #2. In order to simplify MS design, it may often be the case that a MS may not have the same number of transmit and receive antennas. For example, antenna420of MS410may be used as both a transmit antenna and a receive antenna, while antenna421may only be used as a receive antenna.

Consider a flat fading MIMO model expressible as
Y=H·x+n,
where H εCNt×Ntis a MIMO channel response. Let

H=[h1Hh2H]
be a downlink channel matrix where h1and h2are the uplink responses corresponding to antennas420and421of MS410(antennas #1and #2), respectively. Assume that only antenna420of MS410can send sounding reference signals, therefore h1is known at BS405. Since antenna421of MS410cannot send sounding reference signals, h2is unknown. However, a correlation of h2is known at BS405and is expressible as E[h2hH2]=R. Furthermore, let ĥ1=h1+e1be an estimate of h1, where e1is an estimation error. It is desired to find beamforming directions w1and w2to send two independent data streams so that overall throughput is maximized.

Since the rank of the downlink channel is two, it is possible to transmit up to two independent streams, expressible as
x=√{square root over (P1)}·s1·u1+√{square root over (P2)}·s2·u2,
such that P=P1+P2, where S1and s2are transmitted symbols in directions w1and w2, respectively, and E[|s1|2]=E[|s2|2]=1.

A model of the signal may be expressed as
Y=H·F·s+n,
where

F=[u1u2]·[P100p2]
is a precoder matrix and

s=[s1s2]
is a vector of transmitted symbols. According to an embodiment, (u1, u2) and (P1, P2) should be found so that overall capacity is maximized.

Consider an orthogonal frequency division multiplexed (OFDM) wireless communications system with a cyclic prefix such that a linear model for a received signal at a MS is expressible as

Yt,f=P2·Ht,f·U·s+n,
where Y is the received signal,

Ht,f=[h1H⁡(t,f)h2H⁡(t,f)]
represents the channel response at time t and sub-carrier f, U=[u1u2] is the precoding matrix comprised of two beamforming vectors u1and u2, each with unit norm, s=[s1s2. . . sm]Tare the vector of transmitted symbols E[|s1|2]=E[|s2|2]=1, n˜N(0,σ2IM) is the vector of additive Gaussian noise and P is the total transmit power and is expressible as E[xHx]≦P.

FIG. 5aillustrates a BS501that makes use of channel statistics to compute beamforming vectors. Data500destined for a plurality of MSs being served by BS501, in the form of bits, symbols, or packets, for example, may be sent to a scheduler505, which may decide which UEs will transmit or receive in a given time/frequency opportunity. Scheduler505may use any of a wide range of known scheduling disciplines in the literature including round robin, maximum sum rate, proportional fair, minimum remaining processing time, or maximum weighted sum rate. Generally scheduling decisions are based on channel quality information feedback (in the form of channel quality indicators or other short term information, for example) feedback from a plurality of MSs.

Data from MSs selected for transmission may be processed by a modulation and coding block510to convert the data to transmitted symbols. Modulation and coding block510may also add redundancy for the purpose of assisting with error correction and/or error detection. A modulation and coding scheme implemented in modulation and coding block510may be chosen based in part on information about the channel quality information feedback (in the form of channel quality indicators or other short term information).

The output of modulation and coding block510may be passed to a transmit beamforming block520, which maps the output (a modulated and coded stream for each MS) onto a beamforming vector. The beamformed outputs may be coupled to antennas521through RF circuitry, which are not shown. Although shown inFIG. 5aas having only two antennas, it should be understood that BS501may have any number of antennas. The transmit beamforming vectors are input from a beamforming vector compute block540.

Beamforming vector compute block540produces beamforming vectors from the channel quality information feedback or from sounding signals received from the MSs. Due to incomplete channel state information, beamforming vector compute block540may make use of channel statistics recorded or computed by BS501as it operates, to compute additional channel state information needed to determine all of the needed transmit beamforming vectors.

Beamforming vector compute block540includes an estimate channel correlation matrix unit545, a phase difference unit546, a phase estimate unit547, and a beamforming vector compute unit548. Estimate channel correlation matrix unit545may be used to compute, record, and/or update channel statistics (such as a channel correlation matrix, an estimate of the channel correlation matrix, elements of the channel correlation matrix, or estimates of elements of the channel correlation matrix) based on measurements (e.g., samples) of a sounding signal transmitted by the MSs. The channel statistics may be used to supplement the incomplete channel state information (I-CSI) received by BS501.

Phase difference unit546may be used to compute a phase difference between two correlated antennas or two correlated antenna pairs. Phase difference unit546may make use of channel statistics, such as channel correlation matrices, estimates of channel correlation matrices, elements of channel correlation matrices, or estimates of elements of channel correlation matrices, produced by estimate channel correlation matrix unit545. The phase difference may be used to determine the beamforming directions (beamforming vectors) u1and u2.

Similarly, phase estimate unit547may be used to compute a phase of each correlated antenna pair. Again, phase estimate unit547may make use of channel statistics, such as channel correlation matrices, estimates of channel correlation matrices, elements of channel correlation matrices, or estimates of elements of channel correlation matrices, produced by estimate channel correlation matrix unit545. The phase of each correlated antenna pair may be used to determine the beamforming directions (beamforming vectors) u1and u2.

Beamforming vector compute unit548may compute the beamforming directions (beamforming vectors) u1and u2. Beamforming vector compute unit548may directly compute some of the transmit beamforming vectors from the CSI feedback by the MSs. For example, referencingFIG. 4, beamforming vector compute unit548may be able to compute beamforming vector w1from channel state information provided regarding the channel represented by arrowed line425.

However, beamforming vector compute block548may also be able to compute beamforming vectors from the phase difference between correlated antennas (from phase difference unit546) or the phase of the correlated antennas (from phase estimate unit547). Sample unit549may be used to take time-frequency samples of channels with complete channel state information for use in the selection of beamforming vectors using channel statistics.

FIG. 5billustrates a MS569that provides channel information feedback to a BS in the form of channel state information. MS569may have one or a plurality of receive antennas506, connecting through RF circuitry (not shown) to a receiver signal processing block551. Although the antennas of the MS may have different polarizations, operability of the embodiments presented herein does not rely on the polarization of antennas of the MS. Some of the key functions performed by receiver signal processing block551may be channel estimation block556and estimate signal-to-interference-plus-noise ratio (SINR) block557. Channel estimation block556uses information inserted into the transmit signal in the form of training signals, training pilots, or a structure in the transmitted signal such as cyclostationarity to estimate coefficients of the channel between BS501and MS569, i.e., perform channel estimation.

The output of channel estimation block556(channel state information, for example) may be provided to statistical channel information estimate block570, which may estimate the statistical channel information from the output of channel estimation block556. The statistical channel information (estimated) may be feedback to BS501to be used to aid scheduling and transmit beamforming, for example. Prior to feeding back the statistical channel information, the statistical channel information may be quantized to reduce the amount of information being feedback. According to an embodiment, channel estimation block556may also perform an estimation of the statistical channel information, eliminating a need for statistical channel information estimate block570.

The feedback of the statistical channel information may only be required in a frequency division duplexing (FDD) communications system, wherein a BS may not be capable of computing statistical channel information on its own using channel reciprocity. In a time-division duplexing (TDD) communications system, the BS may be able to directly acquire channel information through channel reciprocity and compute statistical channel information on its own. Therefore, a MS, such as MS569, may not need to measure a channel and compute estimates and/or statistical channel information.

MS569may also include a transmitter580coupled to one or more transmit antennas506that may be used to transmit a sounding reference signal that may be used by BS501to compute estimates of UL communications between MS569and BS501. Although MS569may have multiple transmit antennas, generally, MS569may have more receive antennas than transmit antennas.

FIG. 6aillustrates a flow diagram of BS operations600in communicating with incomplete channel state information. BS operations600may be indicative of operations occurring in a BS, such as BS501, as the BS communicates to a MS, such as MS569, using spatial multiplexing. The BS may make use of a sounding signal transmitted by the MS to obtain channel state information about communications channels between the MS and itself. However, the MS may have more receive antennas than transmit antennas, therefore the BS may not be able to obtain complete channel information (such as statistical channel information) about the communications channels. The BS may make use of channel statistics, from measurements or historical information that it has recorded itself, to supplement the I-CSI. BS operations600may occur while the BS and the MS are in normal operations and are communicating using spatial multiplexing.

It is widely known that it is possible to estimate the beamforming directions (beamforming vectors) u1and u2at a BS for the downlink channel based on measurements of a sounding signal provided by a MS. However, since the MS has more receive antennas than transmit antennas, uplink channel statistics are not available for all channels. For a communications system, such as one shown inFIG. 4, let antenna pair #1and #3have similar polarization and antenna pair of antennas #2and #4have similar polarization, meaning that antenna pair of antennas #1and #3and antenna pair of antennas #2and #4may be correlated (they each may form a 2×1 linear array, but the two sets of antenna pairs are not necessarily correlated due to their different polarizations).

BS operations600may begin with the BS receiving a sounding reference signal transmitted by the MS (block605). The sounding reference signal may be a reference sequence unique to the MS that is periodically transmitted by the MS to allow a receiver of the transmission, e.g., the BS, to determine channel information (e.g., statistical channel information) of a communications channel between the BS and itself, i.e., an UL communications channel. Furthermore, in TDD communications systems, channel reciprocity may be used to determine channel information regarding a DL communications channel between the receiver and the MS without having to have the MS transmit the sounding reference signal.

The BS may use the received sounding reference signal to compute an estimate of a channel for a first antenna pair h13(block610). According to an embodiment,

h13=[h1⁡(1)h1⁡(3)],
where h1(1) is a channel between antenna #1of the BS and antenna #1of the MS and h1(3) is a channel between antenna #3of the BS and antenna #1of the MS. The BS may also compute an estimate of a channel for a second antenna pair h24(block615). According to an embodiment,

h24=[h1⁡(2)h1⁡(4)],
where h1(2) is a channel between antenna #2of the BS and antenna #1of the MS and h1(4) is a channel between antenna #4of the BS and antenna #1of the MS.

From the computed estimates of channels for the first antenna pair h13and the second antenna pair h24, the BS may compute channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24. According to an embodiment, the BS may not need to actually compute the channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24using expressions
R13=E[h13hH13],
and
R24=E[h24h24H],
where E[ ] is an expected value function. Instead, only certain off-diagonal elements of the channel correlation matrices may be needed and therefore computed (block620). It may be possible to estimate these off-diagonal elements.

According to an embodiment, the estimates of off-diagonal elements of the channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24may be computed using expressions

R13(t)⁡(1,2)=1Ω⁢∑f∈Ω⁢h1(t,f)⁡(1)·(h1(t,f)⁡(3))*andR24(t)⁡(1,2)=1Ω⁢∑f∈Ω⁢h1(t,f)⁡(2)·(h1(t,f)⁡(4))*,
where ( )* is a conjugation operator, t is a time index of the sounding symbol, Q is a set of indices of sounding subcarriers in a t-th sounding symbol, |Ω| is a total number of sounding subcarriers, and f is a time-frequency index of sounding subcarriers within the t-th sounding symbol.

The techniques for computing of the beamforming vectors u1and u2presented herein rely on second order statistics of DL channels (e.g., the channel correlation matrices, estimates of the channel correlation matrices, elements of the channel correlation matrices, estimates of the elements of the channel correlation matrices, and so on) instead of instantaneous information (e.g., CSI) that may not be accurate or available, for example, due to a MS having more receive antennas than transmit antennas. The techniques presented herein do not require the forming of correlation matrices or solving eigen problems. Instead, only estimates of a cross-correlation of antenna pairs (antennas #1and #3and antennas #2and #4, respectively) are needed.

In case the sounding signal continues in time, it may be possible to use temporal averaging to smooth out estimated values of the cross-correlations for the two antenna pairs. According to an embodiment, exponential averaging filtering may be used. Exponential averaging of the estimate values may be expressed as
R13(t)(1,2)=β·R13(t-1)(1,2)+(1−β)·R13(t)(1,2)
and
R24(t)(1,2)=β·R24(t-1)(1,2)+(1−β)·R24(t)(1,2),
where β is a filter coefficient. The filtering may alternatively make use of an infinite impulse response (IIR), finite impulse response (FIR), or other types of filters. According to an embodiment, β may be optimally adapted to a Doppler frequency of each MS. As an example, β may be set to 31/32, 15/16, and 7/8 for low mobility (3 Km/h to 10 Km/h), medium mobility (10 Km/h to 30 Km/h), and high mobility (30 Km/h to 120 Km/h) MSs, respectively. Actual values of β may be adjusted based on performance evaluation, simulation, and so forth.

However, if the antenna spacing D is small, e.g., λ/2, where λ is the wavelength, then a spatial correlation structure of antenna pair of antennas #1and #3and antenna pair of antennas #2and #4may be very similar and may be determined by a location of the scatterers. Therefore, R13≈R24.

The BS may compute beamforming vectors u1and u2based on the channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24(block625). According to an embodiment, the channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24may be used to estimate a phase of the two antenna pairs or a phase difference between the two antenna pairs, which may then be used to compute the beamforming vectors u1and u2. A detailed discussion of computing the beamforming vectors u1and u2based on the estimated phase and estimated phase difference for the two antenna pairs is provided below.

With the beamforming vectors u1and u2computed, the BS may use the beamforming vectors to beamform a transmission to the MS (block630) and BS operations600may then terminate.

FIG. 6billustrates a flow diagram of MS operations650in communicating with incomplete channel state information. MS operations650may be indicative of operations occurring in a MS, such as MS569, as the MS communicates with a BS, such as BS501, using spatial multiplexing. MS operations650may occur while the MS and the BS are in normal operations and are communicating using spatial multiplexing.

MS operations650may begin with the MS transmitting a sounding reference signal to the BS (block655). The sounding reference signal may be a reference sequence unique to the MS that is periodically transmitted by the MS to allow a receiver of the transmission, e.g., the BS, to determine channel state information of a communications channel between the MS and itself, i.e., an UL communications channel. Furthermore, in TDD communications systems, channel reciprocity may be used to determine channel state information regarding a DL communications channel between the receiver and the MS.

The MS may then receive a transmission from the BS, wherein the transmission has been beamformed with beamforming vectors computed based on the sounding reference signal and selected using channel statistics (block660). Some of the beamforming vectors may be directly computed based on the sounding reference signal, while some of the beamforming vectors may be selected using channel statistics determined by the BS. MS operations650may then terminate.

FIG. 7aillustrates a flow diagram of BS operations700in computing beamforming vectors from an estimate of phase difference. BS operations700may be indicative of operations in a BS, such as BS501, as the BS computes beamforming vectors from an estimate of phase difference between two antenna pairs, wherein the two antenna pairs are cross-polarized. BS operations700may occur while the BS is in a normal operating mode and while the BS has information to transmit to a MS. BS operations700may be an implementation of compute beamforming vectors u1and u2, block625ofFIG. 6a.

BS operations700may begin with the BS computing an estimate of a phase difference between the two antenna pairs (block705). According to an embodiment, the BS may be able to estimate a channel correlation matrix from noisy channel samples by averaging two correlation matrices, e.g., the channel correlation matrices for the channels for the first antenna pair R13and the second antenna pair R24. The estimation of the channel correlation matrix may be expressed as

R_=R_13+R_242.
In order to estimate the phase difference between the two antenna pairs, it may be necessary to find a vector

e=[1ⅇjθ]
such that eH·R·e is maximized. According to an embodiment, eH·R·e may be maximized when

ⅇjθ=(R_⁡(1,2))*R_⁡(1,2),
whereR(1,2) indicates to a first row and a second column ofR.

The BS may compute the beamforming vectors u1and u2from the estimated phase difference between the two antenna pairs (block710). According to an embodiment, the beamforming vectors u1and u2may be expressed as

FIG. 7billustrates a flow diagram of BS operations750in computing beamforming vectors from estimates of phases. BS operations750may be indicative of operations in a BS, such as BS501, as the BS computes beamforming vectors from estimates of phases of two antenna pairs, wherein the two antenna pairs are cross-polarized. BS operations750may occur while the BS is in a normal operating mode and while the BS has information to transmit to a MS. BS operations750may be an implementation of compute beamforming vectors u1and u2, block625ofFIG. 6a.

BS operations750may begin with the BS computing estimates of phases for the two antenna pairs (block755). According to an embodiment, the estimate of a phase for the first antenna pair (antennas #1and #3of the BS) may be expressible as

ⅇjθ13=(R_13⁡(1,2))*R_13⁡(1,2),
while the estimate of a phase for the second antenna pair (antennas #2and #4of the BS) may be expressible as

The BS may compute the beamforming vectors u1and u2from the estimated phases for the two antenna pairs (block760). According to an embodiment, the beamforming vectors u1and u2may be expressed as

FIGS. 8 and 9illustrate plots of spectral efficiency versus signal to noise ratio for a communications system comparing a variety of techniques for computing beamforming vectors with random frequency hopping on and off, respectively. The techniques for computing beamforming vectors range from eigen value precoding (a full CSI technique), to Union MRT (an I-CSI technique using channel statistics to estimate beamforming vector direction), to the techniques presented herein (I-CSI techniques using channel statistics to estimate phase differences and phases of antenna pairs) labeled as “New Algorithm 1” and “New Algorithm 2.” As shown inFIGS. 8 and 9, the full CSI technique provides the best performance over all SNR values. However, the techniques presented herein provide substantially equal performance and were better than other non-full CSI techniques.