Method and device for multiple input/multiple output transmit and receive weights for equal-rate data streams

The invention provides a method of operating a communication system. A channel matrix of a gain and phase between each transmit antenna and each receive antenna of the communication system is provided. At least one receive weight vector is computed as a function of the channel matrix and at least one of transmit weight vectors. An updated transmit weight vector is computed as a function of the transmit weight vector, the receive weight vector, the channel matrix.

FIELD OF THE INVENTION

In general, the invention relates to the field of communication systems. More specifically, the invention relates to strategies for wireless Multiple-Input/Multiple-Output (MIMO) communications and in particular, to establishing transmit and receive weighting matrices for use at the transmitter antenna array and the receiver antenna array.

BACKGROUND OF THE INVENTION

In a wireless communication system, a major design challenge is to maximize system capacity and performance. One such communication design known in the art is Multiple-Input/Multiple Output (MIMO), and is a means of transmitting multiple data streams on the same time-frequency channel to a single receiver. The MIMO strategy involves deploying multiple antennas on both the transmitter and the receiver.

In environments having rich multipath scattering, large increases in capacity can be achieved through the use of MIMO and appropriate transmit and receive signal processing techniques. It is well known that MIMO wireless channels have significantly higher capacities than single-input single-output wireless communication channels, which has motivated the design of wireless communication systems with multiple antennas. Current system algorithms designed to achieve high capacity include spatial multiplexing, space-time coding and adaptive modulation.

These MIMO systems include the various BLAST (Bell-labs LAyered Space-Time) type techniques proposed by Lucent as a subset. The BLAST type techniques however, do not use the channel knowledge at the transmitter and thus are sub-optimal.

For MIMO systems, the maximum theoretical system capacity can be achieved using transmit and receive weights based on the Singular Value Decomposition (SVD) combined with a water-pouring (optimally assigning power to the individual data streams) strategy for determining the optimal data rates and power distribution. This strategy for maximizing capacity must adaptively control not only the number of independent data streams to be formed but also the choice of modulation and coding to be used on each stream. However, even for full-rank channels (as are known in the art), there are cases where a fixed number of data streams having the same modulation type is desirable, for example to avoid having to make a complex real-time decision on both the number of streams and the modulation/coding type on each stream. Consequently, using the SVD weights for a fixed number of equal-rate data streams is not the best option because fixing the modulation type will not take advantage of the unequal signal-to-noise ratios at the outputs of the receive array.

Therefore, it would be desirable to have a method and device for finding transmit and receive weights that are optimized for the use of equal-rate data streams. Further, it would be desirable that these weights have a lower Bit Error Rate (BER) than weights found using the SVD approach.

DETAILED DESCRIPTION OF THE PRESENTLY PREFERRED EMBODIMENTS

FIG. 1illustrates a wireless communication system100in accordance with one embodiment of the present invention. As shown inFIG. 1, a base station110provides communication service to a geographic region known as a cell103,102. At least one user device120and130communicate with the base station110. In one embodiment of the wireless communication system100, at least zero external interference sources140share the same spectrum allocated to the base station110and subscriber devices120and130. The external interference sources140represent an unwanted source of emissions that interferes with the communication process between the base station110and the user devices120and130. The exact nature and number of the external interference sources140will depend on the specific embodiment of the wireless communication system100. In the embodiment shown inFIG. 1, an external interference source will be another user device140(similar in construction and purpose to user device120) that is communicating with another base station112in the same frequency spectrum allocated to base station110and user devices120and130. As shown inFIG. 1, user devices120have a single transmit antenna101, while user devices130have at least one antenna101. One embodiment of the invention provides that the user devices120and130, as well as the base station110may transmit, receive, or both from the at least one antenna101. An example of this would be a typical cellular telephone. Additionally, one embodiment of the invention can be implemented as part of a base station110as well as part of a user device120or130. Furthermore, one embodiment provides that user devices as well as base stations may be referred to as transmitting units, receiving units, transmitters, receivers, transceivers, or any like term known in the art, and alternative transmitters and receivers known in the art may be used.

FIG. 2, is a block diagram illustrating one embodiment of a receiving device200imbedded within the wireless communication system100ofFIG. 1, in accordance with the present invention. The receiving device200includes at least one antenna101wherein the outputs of the antennas are each provided to a receiving unit201. The outputs of the receiving units201are provided to at least one antenna combiner202. The signals from the receiving units201are also fed into a combiner controller210, which may regulate the operation of the at least one antenna combiner202. The signals from the receiving units201may also be fed into a channel estimation device208. A pilot symbol generator212generates pilot symbol information that is used by the combiner controller210to control the antenna combiner202. The pilot symbol information generated by the pilot symbol generator212is also used by the channel estimation device208to estimate a time-varying frequency response of the transmitting devices of wireless communication system. The output of the antenna combiner202is fed into an information decoding unit206, which decodes the antenna combiner output204and generates data information213that was received by the antennas101.

FIG. 3is a block diagram illustrating details of the antenna combiner202of receiving device200of FIG.2. In one embodiment, antenna combiner202can be coupled to the receiving units201, which in turn are coupled to the antennas101. The receiving units201may include radio frequency pre-amplifiers, filters, and other devices that can be used to convert a radio frequency signal received by the antenna101, to a digital stream of baseband equivalent complex symbols. As illustrated inFIG. 2, the output (y) of the ithreceiving unit201(where i is an integer between 1 and M inclusive, and M is the total number of antenna101elements) may be mathematically denoted by yi(k), where k and i are integers, and is provided to the antenna combiner202. The antenna combiner202can be in the form of at least one complex multipliers302that multiply the output of each receiving unit201by a weight (w)304mathematically denoted as wi(k). A combiner306may sum the outputs of the at least one complex multipliers302. For one embodiment of the invention, the combiner controller210ofFIG. 2controls the values of the weights304.

FIG. 3Ais a block diagram illustrating one embodiment of a transmitting device3A00imbedded within the wireless communication system100ofFIG. 1, in accordance with the present invention. The transmitting device3A00may include at least one antenna101wherein the inputs to the antennas may be provided from at least one transmitting unit3A01, and at least one transmit combiner3A03. The inputs to the transmit combiner3A03may be provided from at least one transmit weighting unit3A05. At least one Information Bit Stream3A13may be encoded by an Information Encoding Unit3A07and converted into a Data Stream3A10. The Data Stream3A10is weighted by transmit weighting unit3A05and the outputs of the transmit weighting unit3A05may be sent to the transmit combiner3A03. The Transmit Weighting Unit Controller3A11may be provided with channel state information3A12, which may be used by the Transmit Weighting Unit Controller3A10to control the operation of transmit weighting units3A05.

FIG. 4is a block diagram illustrating one embodiment of the transmitting weighting unit3A05ofFIG. 3A, in accordance with the present invention. The symbol (x) on data stream i at time k, xi(k)410is weighted403by a transmit weight404. The transmit weight404value may be dependant on which antenna the transmit weight404is associated with. In equation form, an antenna m's signal (v) for data stream i at time k420may be expressed as vi,m(k)=Vi,m(k)xi(k). The antenna m's signal at time k for data stream i may be sent to the transmit combiner3A03for antenna m in order to combine all the data streams before radiating the signal out of antenna m.

FIG. 5is a flow chart representation of a method for providing multiple input multiple output MIMO receive weights as can be performed by the antenna combiner ofFIG. 3, and the transmitting device3A00, in accordance with the invention.

One embodiment of the invention may transmit and receive Nsdata streams, wherein each data stream can have the same modulation type with MTtransmit antennas and MRreceive antennas. The present invention may provide an alternative to the SVD weights that are known in the art. One embodiment of the present invention can minimize the mean square error between the receive array outputs and the transmitted symbols. Using the SVD weights for a fixed number of equal-rate data streams may not be the best option because fixing the modulation type will not take advantage of the unequal signal-to-noise ratios at the outputs of the receive array. This is because the SVD weights diagonalize the channel, which means that there is no cross talk between the data streams. Thus MMSE weights based on the SVD approach will not take advantage of the performance gains possible by trading off the suppression of other streams with gains over noise. Further, if the SVD weights are used while changing each streams' transmitted power so that each data stream is received with equal energy, the optimal data stream will be penalized while an inadequate data stream is built-up, again indicating that this is not the best solution. One embodiment of the present invention computes linear MMSE weights that are not constrained to diagonalize the channel, thereby providing superior performance when all data streams have the same modulation type.

In order to compute the linear MMSE weights in accordance with one embodiment of the present invention, a single estimate of the channel gain and phase between each receive antenna and each transmit antenna may be provided to the invention (this is also called the channel). The channel is modeled as stationary and flat-faded. As is known in the art, such channels occur in systems that include, but are not limited to, single carrier systems operating in frequency non-selective channels, OFDM systems in which an embodiment of the invention operates on a set of frequency-domain subcarriers having identical frequency response characteristics, or single carrier system in which an embodiment of the invention operates in the frequency domain on a set of frequency-bins having identical frequency response characteristics, or other similar systems as is known in the art. Also provided is that both the transmitting weighting unit and receiving device may know the channel.

In one embodiment, the MR×MTchannel matrix, H, contains the gain and phase between each transmit and receive element. Also let the MT×Nsmatrix (also known as the transmit weighting matrix), V, denote the transmit weights for the Nsstreams (the uthcolumn of V is denoted vu), and finally let the Ns×1 vector, x, be the symbols on the Nsstreams (at one particular time). Then for this embodiment, a received MR×1 vector can be given as:
y=HVx+n(1)
where n is a MR×1 vector of noise with power of σn2each of the received antennas. In one embodiment, in order to keep an average transmit power equal to one, the transmit weighting matrix may be normalized as follows:
trace(VHV)=1  (2)
where trace(A) means the sum of the diagonal elements of the square matrix A.

One embodiment of the invention finds the transmit weighting matrix and a MR×Nsreceive weighting matrix, W, such that an estimate of the transmitted symbols, r, is:
r=WHy(3)
(The uthcolumn of W is denoted wu) In one embodiment of the invention, the operation of the invention may be optimized for the case where each of the Nsdata streams is constrained to have the same modulation type. This embodiment of the invention minimizes the Mean Squared Error (MSE) between the estimated symbol on each stream and the transmitted symbol assuming linear receive and transmit weights. In equation form, this can be expressed as:minwu,vu⁢∑u=1Ns⁢⁢E⁢ru-xu2=minwu,vu⁢∑u=1Ns⁢⁢E⁢wuH⁡(∑l=1Ns⁢⁢Hvl⁢xl+n)-xu2(4)
where ruis the uthelement of the vector r and xuis uthelement of the vector x. Using E{xuxl*}=δ(u−l), (4) becomes:minwu,vu⁢{∑u=1Ns⁢wuH⁡(∑l=1Ns⁢⁢Hvl⁢vlH⁢HH+σn2⁢IMR)⁢wu-∑u=1Ns⁢wuH⁢Hvu-∑u=1Ns⁢vuH⁢HH⁢wu+Ns}(5)

In addition to the minimization of the objective function in (5), the constraints on the transmit weight vectors (the columns of the matrix V) given in (2) need to be met. One embodiment of the invention meets the constraints by modifying the objective function of (5) as follows:minwu,vu⁢{∑u=1Ns⁢wuH⁡(∑l=1Ns⁢⁢Hvl⁢vlH⁢HH+σn2⁢IMR)⁢wu-∑u=1Ns⁢wuH⁢Hvu-∑u=1Ns⁢vuH⁢HH⁢wu+Ns+γ⁢∑u=1Ns⁢vuH⁢vu-12}(6)
where γ is an arbitrary scaling placed on the constraint part of the objective function.

A closed form solution to (6) does not exist for Ns>1. However, (6) is in the form of an unconstrained optimization problem so a modified version of an iterative algorithm such as the gradient-based optimization approach is used by one embodiment of the invention to solve for wuand vu. The gradient of (6) with respect to wucan be shown to be:
∇wu=(HVVHHH+σn2IMR)wu−Hvu(7)
Because (HVVHHH+σn2IMR) is well conditioned, its inverse can be found. Therefore W can be found at each iteration step as follows (i.e., by setting equation (7) equal to zero):
W=(HVVHHH+σn2IMR)−1HV(8)
The gradient of (6) with respect to vucan be shown to be:
∇vu=(HHWWHH+2γ(trace(VHV)−1)IMT)vu−HHwu(9)
The invention lets G=[∇v1. . . ∇vNs], and then using equation (9), G can be expressed as:
G=(HHWWHH+2γ(trace(VHV)−1)IMT)V−HHW(10)

The above embodiment of the invention explicitly adds the constraint on the transmit weights in equation (2) to the minimization. Another option for ensuring the constraint is to use a Lagrangian multiplier, λ, as is known in the art. For this method, the gradient matrix in (10) is changed as follows:
G=(HHWWHH+λIMT)V−HHW(11)

One addition gradient is needed, the Lagrangian gradient which is:
gλ=trace(VHV−1)  (12)

The previous equations and calculations express a mathematical means for a method of providing linear MIMO transmit and receive weights.

The method of providing MIMO receive and transmit weights, shown in the flow chart500, may begin by being provided a channel matrix510. The channel matrix H, is a MR×MTmatrix that contains the gain and phase between each transmit antenna (MTis the total number of transmit antennas) and each receive antenna (MRis the total number or receive antennas). In one embodiment of the invention, the received vector y, is modeled as y(k)=HVx(k)+n(k), where V is a MT×Nsmatrix of the transmit weight vectors, x(k) is a Ns×1 vector of the transmitted symbols at index k (e.g., time or frequency), and n(k) is a MR×1 vector of noise at index k. The embodiment of the invention illustrated inFIG. 5may require the total transmit power to be limited, therefore the transmit weight vectors are constrained to trace(VHV)=1, where trace(A) means to sum the diagonal elements of the matrix A, and superscript H means the conjugate transpose (also known as the Hermitian) of the matrix.

Another embodiment of the invention may find a receive weighting matrix (also referred to as the receive weight vectors) W, that find an estimate, r(k), of x(k) as a function of r(k)=WHy(k). Next for the embodiment500, block520initializes an iteration number t=0. In one embodiment of the invention, transmit weight vectors can be initialized530asV0=1Ns⁡[INs0(MT-Ns)×Ns],
where INsis a Ns×Nsmatrix of all zeros except for the diagonal elements which are one, and 0a×bis an a×b matrix of all zeros. Utilizing the iteration number t, the receive weight vectors Wt, are found540as Wt=(HVtVtHHH+σn2IMR)−1HVt, where Vtare the transmit weight vectors at iteration number t, σn2is the noise power, and IMRis a MR×MRmatrix of all zeros except for the diagonal elements which are one. Again utilizing the iteration number t, the gradient matrix Gt, is found550as Gt=(HHWtWtHH+2γ(trace(VtHVt)−1)IMT)Vt−HHWt, where γ is the constraint weight (in one embodiment, γ=1), and IMTis a MT×MTmatrix of all zeros, except for the diagonal elements that are one. In another embodiment of the invention, the gradient matrix is: Gt=(HHWtWtHH+λtIMT)Vt−HHWt, the Lagrangian multiplier at iteration number t can be given as λt=λt−1+αgλ,t−1and the Lagrangian gradient at iteration number t is: gλ,t=trace(VtHVt−1).

If the embodiment of the invention utilizes an adaptive step size560, the step size α, at iteration time t may be found565as the argument that minimizes the following objective function, C(Vt,Wt,Gt,γ,α) where

After a step size has been established, the transmit weight vectors can be computed570at iteration t+1 as a function of the transmit weight vectors at iteration number t, the step size at iteration number t, and the gradient matrix at iteration number t, using the equation Vt+1=Vt−αGt. In another embodiment of the invention, the Lagrangian multiplier may be updated with: λt+1=λt+αgλ,t. After computing the transmit weight vectors570, the iteration number is incremented by one575as t=t+1. If the iteration number has reached an ‘end’ value580, where tendis an integer designating the maximum number of iterations, then the receive weight vectors are the receive weight vectors at iteration number t, (W=Wt)590and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=Vt) and this embodiment is completed. If the iteration number has not reached an ‘end’ value,585decides if trace(Gt−1HGt−1) <ε, and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number t, (W=Wt) and the transmit weight vectors are chosen to be the transmit weight vectors at iteration number t, (V=Vt)590again completing this embodiment. For decision block585, trace(Gt−1HGt−1)<ε means the sum of the diagonal elements of the square matrix (Gt−1HGt−1) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(Gt−1HGt−1)<ε, the algorithm has almost converged because at the optimal solution Gtt+1=0. If in decision block585, trace(Gt−1HGt−1)<ε, is false, the process recedes to block540and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart500.

At this point the transmit vectors are, if necessary, renormalized to satisfy the constraint in Equation (2). An alternate embodiment of the invention is a closed form solution to the linear transmit and receive weights that has an equal Mean Squared Error (MSE) on each or the received data streams. At the receiver, an estimate, of the transmitted symbols, r, can be found by using a linear MR×Nsweight matrix, W, as follows:
r=WHy  (13)

An MMSE approach can be used to find W and V as summarized next where a closed form expression for W and V are given. Using the SVD, H can be expressed as follows:
H=UHSHZHH(14)
where MR×MRUHand MT×MTZHare unitary matrices and MR×MTSHis a matrix of all zeros except for the upper left rH×rHportion which is:
[SH]Ns=diag(sH,1, . . . , sH,rH)  (15)
where rHis the rank of H (rH≦min(MT,MR)), [A]lmeans the first l rows and columns of A, and it is assumed that sH,1≧sH,2≧. . . ≧sH,rH. The transmit weight matrix can also be expressed by a SVD as follows:
V=UvSvZvH(16)
where MT×MTUvis unitary, Ns×NsZvis unitary, and MT×NsSvmay be matrix of all zeros except for the upper Ns×Nsportion which contains the Nsnon-zero singular values of the transmit weight matrix as follows:
[Sv]Ns=diag(sv,1, . . . , sV,Ns)  (17)

One way to find the components of (16) may be to set UV=ZHand the singular values can be found as the solution to:sV,l2=1λ⁢σn⁢sH,l-1-σn2⁢sH,l-2(18)
where λ is chosen so that the following equation is satisfied (this forces the transmit weights to have unit power):∑l=1Ns⁢⁢sV,l2=1(19)

The MSE averaged across the data streams may be unchanged regardless of the choice of the right singular vectors of the transmit weight matrix. One option is to use the identity matrix (i.e., Zv=I), however in general this choice gives unequal MSE on each data stream as will be seen. This means that the BER performance can be dominated by the data stream with the highest MSE. Thus if some unitary Zvcan be found that gives equal MSE on each stream, the BER performance will improve while the average MSE will remain unchanged.

Using UV=ZH, the received data vector from (1) can be expressed as:y=UH⁢SH⁢ZHH⁢ZH⁢SV⁢ZVH⁢x+n=UH⁢SH⁢SV⁢ZVH⁢x+n(20)

Let Ns×1 YNs={UHHy}Nswhere {a}imeans the first i elements of the vector a, then YNsis:YNs={UHH⁢UH⁢SH⁢SV⁢ZVH⁢x+UHH⁢n}Ns={SH⁢SV⁢ZVH⁢x}Ns+{UHH⁢n}Ns=DZVH⁢x+NNs(21)
where D is a real Ns×Nsdiagonal matrix equal to diag(sH,1sV,1, . . . , sH,NssV,Ns) and NNshas a covariance matrix equal to σn2INs. (A unitary matrix times a vector of uncorrelated Gaussian random variables does not change its covariance matrix).
The MSE on stream l can be shown to be:
MSEl=1−ZV,lH{tilde over (D)}ZV,l(22)
where ZV,lis the lthcolumn of Zv, and
{tilde over (D)}=D2(D2+σn2I)−1(23)

Thus unless all of the diagonal elements of {tilde over (D)} are equal, using ZV=I gives unequal MSEs on each data stream. The MSE averaged across all streams can be shown to be:
{overscore (MSE)}=1−trace({tilde over (D)})/Ns(24)

Therefore in order to make all data streams have the same MSE, ZV,lmust satisfy:
ZV,lH{tilde over (D)}ZV,l=1−{overscore (MSE)}=trace({tilde over (D)})/Ns
subject to ZVZVH=ZVHZV=INs(25)

One type of matrix that solves (25) is the DFT matrix (another choice for when Nsis a power of four is the Hadamard matrix where the Hadamard matrix is known in the art). Thus the linear transmit weights that minimize the average MSE on the data streams and also have the same MSE on each stream are:
V=VHSVFNs(26)
where SVis given by (17) and (18) and FNsis the Ns×NsDFT matrix normalized such that FNsHFNs=I. With the transmit weights chosen according to (26), the Nsstreams can be recovered without any inter-stream interference as:
r=FNsHD−1YNs(27)

Another embodiment of the invention provides an alternative approach to calculating transmit and receive weights for MIMO equal rate data streamsand is illustrated as FIG.6. The embodiment illustrated as flow chart600, finds the successive cancellation weights that minimize the MSE between the estimated symbols and the true symbols on each stream. The advantage of successive cancellation weights over linear weights is that some of the receive array's degrees are freedom are recovered by sequentially subtracting out the estimated contribution from the decoded streams. For example if there are three streams and three receive antennas, when decoding the first stream, two degrees of freedom in the array are needed to null the other two streams. For the second stream, however, only one degree of freedom is needed to null the third stream since the decoded first stream is subtracted out. The extra degree of freedom then can be used in a max-ratio sense to provide an additional gain against noise. Finally, the third stream has all of the degrees of freedom in the array available to provide a gain against noise.

In one embodiment, for the successive cancellation method, the estimated symbols are given as:ru=wuH⁡(y-∑l=1u-1⁢⁢Hvl⁢x^l),where⁢⁢x^l=slice⁡(rl)(28)
where in one embodiment slice(rl) means to choose the closest signal constellation point to rl. In another embodiment, slice(rl) refers to any technique that involves decoding a plurality of received symbols rlinto a bit stream, and then mapping the resulting bit stream to a signal constellation point.
The MMSE successive cancellation weights can be found as the solution to the following minimization problem:minwu,vu⁢∑u=1Ns⁢⁢E⁢ru-xu2=minwu,vu⁢∑u=1Ns⁢⁢E⁢wuH⁡(∑l=uNs⁢⁢Hvl⁢xl+n)-xu2(29)
Note that (29) differs from (4) due to the inner sum being performed from u to Nsinstead of 1 to Nsas is done in (4). Using E{xu(n)xl*(n)}=δ(u−l), (29) becomes:minwu,vu⁢{∑u=1Ns⁢wuH⁡(∑l=uNs⁢Hvl⁢vlH⁢HH+σn2⁢IMR)⁢wu-∑u=1Ns⁢wuH⁢Hvu-∑u=1Ns⁢vuH⁢HH⁢wu+Ns}(30)
In addition to the minimization of the objective function in (30), the constraints on the transmit weight vectors given in (2) need to be met. One embodiment of the invention meets those constraints by modifying the objective function of (30) as follows:minwu,vu⁢{∑u=1Ns⁢wuH⁡(∑l=uNs⁢Hvl⁢vlH⁢HH+σn2⁢IMR)⁢wu-∑u=1Ns⁢wuH⁢Hvu-∑u=1Ns⁢vuH⁢HH⁢wu+Ns+γ⁢∑u=1N⁢vuH⁢vu-12}(31)
where γ is an arbitrary scaling placed on the constraint part of the objective function.

A closed form solution to (31) does not exist for Ns>1, therefore a modified version of an iterative algorithm, such as a gradient-based optimization approach, is used to solve for wuand vu. The gradient of (31) with respect to wuis:∇wu=(∑l=uNs⁢Hvl⁢vlH⁢HH+σn2⁢IMR)⁢wu-Hvu(32)
Equation (32) cannot be put into the same form as (8) because of the sum from u to Nsin the parenthesis. Setting equation (32) equal to zero, wucan be found at each iteration as:wu=(∑l=uNs⁢Hvl⁢vlH⁢HH+σn2⁢IMR)-1⁢Hvu(33)
The gradient of (31) with respect to vuis:∇vu=(∑l=1u⁢HH⁢wl⁢wlH⁢H+2⁢γ⁡(trace⁡(VH⁢V)-1)⁢IMT)⁢vu-HH⁢wu(34)
In another embodiment, the gradient is computed using Lagrangian multipliers as follows:∇vu=(∑l=1u⁢HH⁢wl⁢wlH⁢H+λ⁢⁢IMT)⁢vu-HH⁢wu(35)
One addition gradient is needed, the Lagrangian gradient which is:
gλ=trace(VHV−1)  (36)
The Lagrangian gradient is needed in order to update the Lagrangian multiplier, λ.

The method for calculating the successive cancellation weights is described inFIG. 6as a flow chart representation600of a preferred embodiment of an alternative method performed by the antenna combiner of FIG.3and the transmit device ofFIG. 3A, for finding the MMSE successive cancellation weights, in accordance with the invention.FIG. 6begins by being provided610a channel matrix H that is a MR×MTmatrix which contains the gain and phase between each transmit antenna (MTis the total number of transmit antennas) and each receive antenna (MRis the total number or receive antennas). The received vector, y, is modeled as y(k)=HVx(k)+n(k), where for one embodiment of the invention, V is a MT×Nsmatrix of the transmit weight vectors, x(k) is a Ns×1 vector of the transmitted symbols at index (e.g., time or frequency), k, and n(k) is a MR×1 vector of noise at index k. Because the total transmit power has to be limited, in one embodiment of the invention, the transmit weight vectors may be constrained as trace(VHV)=1, where trace (A) means to sum the diagonal elements of the matrix A, and superscript H means the conjugate transpose (also known as the Hermitian) of the matrix.

An additional embodiment of the invention finds receive weight vectors which are the columns of the receive weight matrix, W, that can be used to find an estimate, r(k), of x(k) asru⁢(k)=wuH⁢(y⁢(k)-∑l=1u-1⁢Hvl⁢x^l⁢(k)),
where {circumflex over (x)}l(k)=slice(rl(k)) and ru(k) is the uthelement of r(k) slice(rl(k)) may mean to choose the closest signal constellation point to rl(k).

In another embodiment of the invention, the method ofFIG. 6can initialize an iteration number t, to zero620, and the transmit weight vectors630may be initialized asV0=1Ns⁡[INs0(MT-Ns)×Ns],
where INsis a Ns×Nsmatrix of all zeros except for the diagonal elements which are one, and 0a×bis an a×b matrix of all zeros. With the initialization complete, the method ofFIG. 6computes the receive weight vectors Wtat iteration number t640, are found, for u=1 to Ns, aswt,u=(∑l=uNs⁢Hvt,l⁢vt,lH⁢HH+σn2⁢IMR)-1⁢Hvt,u
where wt,uis the uthcolumn of Wt, vt,uare the transmit weight vectors at iteration number t, σn2is the noise power, and IMRis a MR×MRmatrix of all zeros except for the diagonal elements which are one.

Flow chart600proceeds to compute the gradient vectors at iteration number t650. The gradient vectors, gt,u, are found using the equationgt,u=(∑l=1u⁢HH⁢wt,l⁢wt,lH⁢H+2⁢γ⁡(trace⁡(VtH⁢Vt)-1)⁢IMT)⁢vt,u-HH⁢wt,u,
where γ can be the constraint weight (in one embodiment of the invention, γ=1), and IMTis a MT×MTmatrix of all zeros, except for the diagonal elements which are one. In another embodiment of the invention, the gradient vectors may be found according to:gt,u=(∑l=1u⁢⁢HH⁢wt,l⁢wt,lH⁢H+λt⁢IMT)⁢vt,u-HH⁢wt,u.
It is by computing the gradient vectors, gt,u, instead of the gradient matrix, Gt, that distinguishes the successive cancellation weights method ofFIG. 6from the linear MMSE weights method of FIG.5.

If a step size has not been calculated prior to660, the step size α, at iteration time t may be found665as the argument that minimizes the following objective function, C(Vt,Wt,Gt,γ,α) whereC⁡(Vt,Wt,Gt,γ,α)={∑u=1Ns⁢⁢wt,uH⁡(∑l=uNs⁢⁢H⁡(vt,u-α⁢⁢gt,u)⁢(vt,u-α⁢⁢gt,u)H⁢HH+σn2⁢IMR)⁢wt,u-∑u=1Ns⁢⁢wt,uH⁢H⁡(vt,u-α⁢⁢gt,u)-∑u=1Ns⁢(vt,u-α⁢⁢gt,u)H⁢HH⁢wt,u+Ns+γ⁢∑u=1Ns⁢(vt,u-α⁢⁢gt,u)H⁢(vt,u-α⁢⁢gt,u)-12}.

After a step size has been established, the transmit weight vectors are computed670at iteration t+1 as a function of the transmit weight vectors at iteration number t, the step size at iteration number t, and the gradient vectors at iteration number t, using the equation vt+1,u=vt,u−αgt,u, where u=1 to Ns. In another embodiment of the invention, the Lagrangian multiplier is updated with: λt+1=λt+αgλ,t.

After computing the transmit weight vectors670, the iteration number may be incremented by one675as t=t+1. If the iteration number has reached an ‘end’ value680, where tendis an integer designating the maximum number of iterations, then the receive weight vectors may be chosen to be the receive weight vectors at iteration number t, (W=Wt) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=Vt)690completing this embodiment. If the iteration number has not reached an ‘end’ value,685decides if trace(Gt−1HGt−1)<ε, and if yes, the receive weight vectors are the receive weight vectors at iteration number t, (W=Wt) and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=Vt)690again completing this embodiment. For decision block685, trace(Gt−1HGt−1)<ε means the sum of the diagonal elements of the square matrix (Gt−1HGt−1) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(Gt−1HGt−1)<ε, the algorithm has almost converged because at the optimal solution, Gt−1=0. If in decision block685, trace(Gt−1HGt−1)<ε, is false, the process recedes to block640and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart600. At this point the transmit vectors may be renormalized, if necessary, to satisfy the constraint in Equation (2).

FIG. 7shows a flowchart of another embodiment of the invention700. In this embodiment the gradient matrix or vectors may never be calculated, but the transmit and receive linear and successive cancellation weights can be found through means of iteration. Initially,710one embodiment of the invention may be provided a channel matrix. The iteration number, t720, may next be set to 0 and the transmit and receive vectors can be initialized730. In one embodiment of the invention, the transmit and receive vectors can be initialized to:V0=1Ns⁡[INs0(MT-Ns)×Ns]
For the linear weights, the receive weights are initialized to:
W0=(HV0V0HH+σn2IMR)−1HV0
For the successive cancellation weights, the receive weights are initialized to:w0,u=(∑l=uNs⁢⁢Hv0,l⁢v0,lH⁢HH+σn2⁢IMR)-1⁢Hv0,u
Additionally the objective function at iteration 0, C0, is set to an arbitrary large number. (In one embodiment C0=infinity.)
The transmit weight vectors at iteration number t+1 may next be computed740as a function of the transmit weight vectors at time t, the receive weight vectors at iteration number t, and the channel matrix as follows (where γ is the constraint weight):
For the linear weights:
Vt+1=(HHWtWtHH+2γ(trace(VtHVt)−1)IMT)−1HHWt
For the successive cancellation weights:vt+1,u=(∑l=1u⁢⁢HH⁢wt,l⁢wt,lH⁢H+2⁢γ⁡(trace⁡(VtH⁢Vt)-1)⁢IMT)-1⁢HH⁢wt,u
Next, the receive weight vectors at iteration t+1 may be computed as a function of the transmit weight vectors750at iteration number t+1, the noise power, and the channel matrix as follows:
For the linear weights:
Wt=(HVtVtHH+σn2IMR)−1HVt
For the successive cancellation weights:wt,u=(∑l=uNs⁢⁢Hvt,l⁢vt,lH⁢HH+σn2⁢IMR)-1⁢Hvt,u
The objective function at iteration number t+1 may next be computed760as a function of the transmit and receive weight vectors at iteration number t+1, the noise power, and the channel matrix as follows:
For the linear weights:
Ct+1=Ns+Wt+1H(Vt+1)(Vt+1)HHHWt+1−Wt+1HH(Vt+1)−(Vt+1)HHHWt+1+γtrace((Vt+1)H(Vt +1))
For the successive cancellation weights:Ct+1={∑u=1Ns⁢⁢wt+1,uH⁡(∑l=uNs⁢⁢H⁡(vt+1,u)⁢(vt+1,u)H⁢HH+σn2⁢IMR)⁢wt+1,u-∑u=1Ns⁢⁢wt+1,uH⁢H⁡(vt+1,u)-∑u=1Ns⁢⁢(vt+1,u)H⁢HH⁢wt+1,u+Ns+γ⁢∑u=1Ns⁢(vt+1,u)H⁢(vt+1,u)-12}
Next,770if the objective function at iteration t is less than the objective function at iteration t+1, then the receive weight vectors may be the receive weight vectors at iteration number t, (W=Wt) and the transmit weight vectors may be the transmit weight vectors at iteration number t, (V=Vt)790completing this embodiment. If the objective function at iteration t is not less than the objective function at iteration t+1, then in775the iteration number, t, can be incremented by one. If the iteration number has reached an ‘end’ value780, where tendmay be an integer designating the maximum number of iterations, then the receive weight vectors may be chosen to be the receive weight vectors at iteration number t, (W=Wt) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=Vt)790completing this embodiment. If the end value is not reached, then the iterative procedure returns back to740and performs another iteration. After the iterations are completed, the transmit vectors may need to be renormalized to meet the constraint in Equation (2).

A further embodiment of the invention may provide for the successive cancellation weights to find weights that maximize the theoretical capacity as described next. The capacity of the channel for a given transmit weight matrix, V, can be shown to be:C=log2⁡[det⁡(I+1σn2⁢HVVH⁢HH)](37)

Using (14) and (16), the capacity equation in (37) becomes:C=log2⁡[det⁡(I+1σn2⁢UH⁢SH⁢ZHH⁢UV⁢SV⁢ZVH⁢ZV⁢SVT⁢UVH⁢ZH⁢SHT⁢UHH)](38)

In other words, the capacity is independent of the right singular vectors of the transmit weight matrix. However, even though the capacity is not affected by ZV, different ZV's will greatly affect the performance of a practical receiver depending on what type of receive algorithm is employed (e.g., successive cancellation or linear weights). Therefore a search can be performed only over ZVand the resulting weights will not change the theoretical capacity but will improve the receiver performance given the algorithm employed.

To maximize the capacity, the r (r≠Nsin general) singular values for the transmit weight vectors are selected according to the water-pouring strategy as known in the art with a total transmit power of one (this enforces the constraint trace(VHV)=1). If the water-pouring strategy says to transmit on more streams than Ns, then the singular values of the transmit weight vectors are selected according to the water-pouring strategy using only the largest Nssingular values of H.

To summarize, the transmit weights are expressed as:
V=ZHSVZVH(40)

Therefore the received data vector can be expressed as:
y=UHSHZHHZHSVZVHx+n=UHSHSVZVHx+n(41)

Let r×1 Yr={UHHy}rwhere {a}imeans the first i elements of the vector a, then Yris:Yr={UHH⁢UH⁢SH⁢SV⁢ZVH⁢x+UHH⁢n}r={SH⁢SV⁢ZVH⁢x}r+{UHH⁢n}r=DZVH⁢x+Nr(42)
where the diagonalized channel, D, is a real r×r diagonal matrix equal to diag(sH,1sV,1, . . . , sH,rsVr) and Nrhas a covariance matrix equal to σn2Ir. Note that sH,r≠0 because the water-pouring strategy would never dictate sending power on a stream with a singular value of zero.

For the Successive Cancellation MMSE weights, in order to maximize the capacity, r must equal the number of streams that the water-pouring strategy dictates and r must be less than or equal to Ns. Assuming this is true (i.e., r≦Ns), the Successive Cancellation MMSE weights that maximize capacity are found by solving:minW,T⁢∑l=1Ns⁢E⁢wlH⁡(Yr-D⁢∑p=1l-1⁢tp+xp)-xl2=minW,T⁢∑l=1Ns⁢E⁢wlH⁡(D⁢∑p=1Ns⁢tp⁢xp+Nr)-xl⁢⁢subject⁢⁢to:TTH=I(43)
where the r×Nsright singular matrix, T, is T=[t1, . . . , tNs], t1through tNsare the right singular vectors, T=ZVH, and MR×NsW=[w1, . . . wNs].

It can be shown that the receive weight vector for stream l is given by:wl=(D⁢∑p=lNs⁢tp⁢tpH⁢D+σn2⁢Ir)-1⁢Dtl(44)

Ignoring the constraint for now, the gradient of the objective function in (44) with respect to tl(the lthcolumn of T) can be shown to be:∇tl=D⁢∑p=1l⁢wp⁢wpH⁢Dtl-Dwl(45)

Using (45), T can be found through a gradient search where the constraint of TTH=I is enforced at each step. This procedure is described in a flow chart representation900of a preferred embodiment of the invention illustrated in FIG.8.

The embodiment of the invention illustrated inFIG. 8may be provided a channel matrix, H (910). Next the iteration number, m, can be set to 0 (920). The right singular matrix at iteration number 0, T0, may next be initialized by choosing any arbitrary r×Nsmatrix such that T0T0H=I (930). Next, the receive weight vectors at iteration number m, wm,l, can be computed as a function of the diagonalized channel, D, and the right singular vectors at iteration number m, tm,l, (940) aswm,l=(D⁢∑p=lNs⁢tm,p⁢tm,pH⁢D+σn2⁢I)-1⁢Dtm,l
for l=1, . . . , Ns. Then the gradient vectors at iteration number m, gm,l, may be computed as a function of the diagonalized channel, the receive weight vectors at iteration number m, and the right singular vectors at iteration number m (950) asgm,l=D⁢∑p=1l⁢wm,p⁢wm,pH⁢Dtm,l-Dwm,l
for l=1, . . . , Ns. If a step size, α, has not been calculated prior to (960) then a step size can be calculated that will minimize the objective function in (43) subject to TmTmH=I (965). The next step may compute the right singular vectors at iteration number m+1 as a function of the right singular vectors at iteration number m, the step size at iteration number m, and the gradient vectors at iteration number m (970) as tm,l=tm,l−αgm,lfor l=1, . . . , Ns. The iteration number may then be incremented by one (975) as m=m+1. Then (977) the right singular matrix at iteration number m, Tm, is formed by concatenating the right singular vectors at iteration number m together (Tm=[tm,l|, . . . , |tm,Ns]) and a Gram-Schmidt orthogonalization (as known in the art) can be performed on the rows of the right singular matrix at iteration number m (this enforces the constraint TmTmH=I). The flow chart representation900continues as an embodiment of the invention in FIG.8A.

InFIG. 8A, if the iteration number has reached an ‘end’ value980, where mendmay be an integer designating the maximum number of iterations, then the right singular matrix may be the right singular matrix at iteration number m, (T=Tm) and the receive weight vectors may be chosen to be the receive weight vectors at iteration number m, (wl=wm,l)990completing this embodiment. If the iteration number has not reached an ‘end’ value, block985can decide if trace(GmHGm)<ε where Gm=[gm,1|, . . . , |gm,Ns], and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number m, (wl=wm,l) and the right singular matrix is chosen to be the right singular matrix at iteration number m, (T=Tm)990again completing this embodiment. For decision block985, trace(GmHGm) means the sum of the diagonal elements of the square matrix (GmHGm)and where ε is a number indicating how small the gradient matrix at iteration number m should get. In other words, when trace(GmHGm)<ε, the algorithm has almost converged because at the optimal solution, Gm=0. If in decision block985, trace(GmHGm)<ε, is false, the process recedes to block940ofFIG. 8, and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart900. After completion of the technique described within flow chart900, the transmit weight vectors may be given as V=ZHSVT.

The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive.