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

[0001] This application claims the benefit of provisional application 60/259,039 filed on Dec. 29, 2000.

[0002] 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.

[0003] 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.

[0004] 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.

[0005] 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.

[0006] 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.

[0007] 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.

[0018] FIG. 1 illustrates a wireless communication system 100 in accordance with one embodiment of the present invention. As shown in FIG. 1, a base station 110 provides communication service to a geographic region known as a cell 103, 102. At least one user device 120 and 130 communicate with the base station 110. In one embodiment of the wireless communication system 100, at least zero external interference sources 140 share the same spectrum allocated to the base station 110 and subscriber devices 120 and 130. The external interference sources 140 represent an unwanted source of emissions that interferes with the communication process between the base station 110 and the user devices 120 and 130. The exact nature and number of the external interference sources 140 will depend on the specific embodiment of the wireless communication system 100. In the embodiment shown in FIG. 1, an external interference source will be another user device 140 (similar in construction and purpose to user device 120) that is communicating with another base station 112 in the same frequency spectrum allocated to base station 110 and user devices 120 and 130. As shown in FIG. 1, user devices 120 have a single transmit antenna 101, while user devices 130 have at least one antenna 101. One embodiment of the invention provides that the user devices 120 and 130, as well as the base station 110 may transmit, receive, or both from the at least one antenna 101. An example of this would be a typical cellular telephone. Additionally, one embodiment of the invention can be implemented as part of a base station 110 as well as part of a user device 120 or 130. 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.

[0019] FIG. 2, is a block diagram illustrating one embodiment of a receiving device 200 imbedded within the wireless communication system 100 of FIG. 1, in accordance with the present invention. The receiving device 200 includes at least one antenna 101 wherein the outputs of the antennas are each provided to a receiving unit 201. The outputs of the receiving units 201 are provided to at least one antenna combiner 202. The signals from the receiving units 201 are also fed into a combiner controller 210, which may regulate the operation of the at least one antenna combiner 202. The signals from the receiving units 201 may also be fed into a channel estimation device 208. A pilot symbol generator 212 generates pilot symbol information that is used by the combiner controller 210 to control the antenna combiner 202. The pilot symbol information generated by the pilot symbol generator 212 is also used by the channel estimation device 208 to estimate a time-varying frequency response of the transmitting devices of wireless communication system. The output of the antenna combiner 202 is fed into an information decoding unit 206, which decodes the antenna combiner output 204 and generates data information 213 that was received by the antennas 101.

[0020] FIG. 3 is a block diagram illustrating details of the antenna combiner 202 of receiving device 200 of FIG. 2. In one embodiment, antenna combiner 202 can be coupled to the receiving units 201, which in turn are coupled to the antennas 101. The receiving units 201 may include radio frequency pre-amplifiers, filters, and other devices that can be used to convert a radio frequency signal received by the antenna 101, to a digital stream of baseband equivalent complex symbols. As illustrated in FIG. 2, the output (y) of the i^th receiving unit 201 (where i is an integer between 1 and M inclusive, and M is the total number of antenna 101 elements) may be mathematically denoted by y_i(k), where k and i are integers, and is provided to the antenna combiner 202. The antenna combiner 202 can be in the form of at least one complex multipliers 302 that multiply the output of each receiving unit 201 by a weight (w) 304 mathematically denoted as w_i(k). A combiner 306 may sum the outputs of the at least one complex multipliers 302. For one embodiment of the invention, the combiner controller 210 of FIG. 2 controls the values of the weights 304.

[0021] FIG. 3A is a block diagram illustrating one embodiment of a transmitting device 3A00 imbedded within the wireless communication system 100 of FIG. 1, in accordance with the present invention. The transmitting device 3A00 may include at least one antenna 101 wherein the inputs to the antennas may be provided from at least one transmitting unit 3A01, and at least one transmit combiner 3A03. The inputs to the transmit combiner 3A03 may be provided from at least one transmit weighting unit 3A05. At least one Information Bit Stream 3A13 may be encoded by an Information Encoding Unit 3A07 and converted into a Data Stream 3A10. The Data Stream 3A10 is weighted by transmit weighting unit 3A05 and the outputs of the transmit weighting unit 3A05 may be sent to the transmit combiner 3A03. The Transmit Weighting Unit Controller 3A11 may be provided with channel state information 3A12, which may be used by the Transmit Weighting Unit Controller 3A10 to control the operation of transmit weighting units 3A05.

[0022] FIG. 4 is a block diagram illustrating one embodiment of the transmitting weighting unit 3A05 of FIG. 3A, in accordance with the present invention. The symbol (x) on data stream i at time k, x_i(k) 410 is weighted 403 by a transmit weight 404. The transmit weight 404 value may be dependant on which antenna the transmit weight 404 is associated with. In equation form, an antenna m's signal (v) for data stream i at time k 420 may be expressed as v_i,m(k)=V_i,m(k)x_i(k). The antenna m's signal at time k for data stream i may be sent to the transmit combiner 3A03 for antenna m in order to combine all the data streams before radiating the signal out of antenna m.

[0023] FIG. 5 is a flow chart representation of a method for providing multiple input multiple output MIMO receive weights as can be performed by the antenna combiner of FIG. 3, and the transmitting device 3A00, in accordance with the invention.

[0024] One embodiment of the invention may transmit and receive N_s data streams, wherein each data stream can have the same modulation type with M_T transmit antennas and M_R receive 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.

[0025] 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.

[0026] In one embodiment, the M_R×M_T channel matrix, H, contains the gain and phase between each transmit and receive element. Also let the M_T×N_s matrix (also known as the transmit weighting matrix), V, denote the transmit weights for the N_s streams (the u^th column of V is denoted v_u), and finally let the N_s×1 vector, x, be the symbols on the N_s streams (at one particular time). Then for this embodiment, a received M_R×1 vector can be given as:
y=HVx+n  (1)
where n is a M_R×1 vector of noise with power of σ_n² each 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(V^HV)=1  (2)
where trace(A) means the sum of the diagonal elements of the square matrix A.

[0027] One embodiment of the invention finds the transmit weighting matrix and a M_R×N_s receive weighting matrix, W, such that an estimate of the transmitted symbols, r, is:
r=W^Hy  (3)
(The u^th column of W is denoted w_u) In one embodiment of the invention, the operation of the invention may be optimized for the case where each of the N_s data 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: $\begin{array}{cc}\underset{w_u,v_u}{\min }\sum _{u=1}^{N_s}E{r_u-x_u}^2=\underset{w_u,v_u}{\min }\sum _{u=1}^{N_s}E{w_u^H\left(\sum _{l=1}^{N_s}{\mathrm{Hv}}_l{x}_l+n\right)-x_u}^2& \left(4\right)\end{array}$
where r_u is the u^th element of the vector r and x_u is u^th element of the vector x. Using E{x_ux_l^*}=δ(u−l), (4) becomes: $\begin{array}{cc}\underset{w_u,v_u}{\min }\left\{\begin{array}{c}\sum _{u=1}^{N_s}{w}_u^H\left(\sum _{l=1}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_u-\\ \sum _{u=1}^{N_s}{w}_u^H{\mathrm{Hv}}_u-\sum _{u=1}^{N_s}{v}_u^H{H}^H{w}_u+N_s\end{array}\right\}& \left(5\right)\end{array}$

[0028] 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: $\begin{array}{cc}\underset{w_u,v_u}{\min }\left\{\begin{array}{c}\sum _{u=1}^{N_s}{w}_u^H\left(\sum _{l=1}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_u-\\ \sum _{u=1}^{N_s}{w}_u^H{\mathrm{Hv}}_u-\sum _{u=1}^{N_s}{v}_u^H{H}^H{w}_u+N_s+\\ \gamma {\sum _{u=1}^{N_s}{v}_u^H{v}_u-1}^2\end{array}\right\}& \left(6\right)\end{array}$
where γ is an arbitrary scaling placed on the constraint part of the objective function.

[0029] A closed form solution to (6) does not exist for N_s>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 w_u and v_u. The gradient of (6) with respect to w_u can be shown to be:
∇w_u=(HVV^HH^H+σ_n²I_MR)w_u−Hv_u  (7)
Because (HVV^HH^H+σ_n²I_MR) 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=(HVV^HH^H+σ_n²I_MR)⁻¹HV  (8)
The gradient of (6) with respect to v_u can be shown to be:
∇v_u=(H^HWW^HH+2γ(trace(V^HV)−1)I_MT)v_u−H^Hw_u  (9)
The invention lets G=[∇v₁. . . ∇v_Ns], and then using equation (9), G can be expressed as:
G=(H^HWW^HH+2γ(trace(V^HV)−1)I_MT)V−H^HW  (10)

[0030] 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=(H^HWW^HH+λI_MT)V−H^HW  (11)

[0031] One addition gradient is needed, the Lagrangian gradient which is:
g_λ=trace(V^HV−1)  (12)

[0032] The previous equations and calculations express a mathematical means for a method of providing linear MIMO transmit and receive weights.

[0033] The method of providing MIMO receive and transmit weights, shown in the flow chart 500, may begin by being provided a channel matrix 510. The channel matrix H, is a M_R×M_T matrix that contains the gain and phase between each transmit antenna (M_T is the total number of transmit antennas) and each receive antenna (M_R is 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 M_T×N_s matrix of the transmit weight vectors, x(k) is a N_s×1 vector of the transmitted symbols at index k (e.g., time or frequency), and n(k) is a M_R×1 vector of noise at index k. The embodiment of the invention illustrated in FIG. 5 may require the total transmit power to be limited, therefore the transmit weight vectors are constrained to trace(V^HV)=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.

[0034] 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)=W^Hy(k). Next for the embodiment 500, block 520 initializes an iteration number t=0. In one embodiment of the invention, transmit weight vectors can be initialized 530 as $V_0=\frac{1}{\sqrt{N_s}}\left[\begin{array}{c}{I}_{N_s}\\ 0_{\left(M_T-N_s\right)\times N_s}\end{array}\right],$
where I_Ns is a N_s×N_s matrix of all zeros except for the diagonal elements which are one, and 0_a×b is an a×b matrix of all zeros. Utilizing the iteration number t, the receive weight vectors W_t, are found 540 as W_t=(HV_tV_t^HH^H+σ_n²I_MR)⁻¹HV_t, where V_t are the transmit weight vectors at iteration number t, σ_n² is the noise power, and I_MR is a M_R×M_R matrix of all zeros except for the diagonal elements which are one. Again utilizing the iteration number t, the gradient matrix G_t, is found 550 as G_t=(H^HW_tW_t^HH+2γ(trace(V_t^HV_t)−1)I_MT)V_t−H^HW_t, where γ is the constraint weight (in one embodiment, γ=1), and I_MT is a M_T×M_T matrix of all zeros, except for the diagonal elements that are one. In another embodiment of the invention, the gradient matrix is: G_t=(H^HW_tW_t^HH+λ_tI_MT)V_t−H^HW_t, the Lagrangian multiplier at iteration number t can be given as λ_t=λ_t−1+αg_λ,t−1 and the Lagrangian gradient at iteration number t is: g_λ,t=trace(V_t^HV_t−1).

[0035] If the embodiment of the invention utilizes an adaptive step size 560, the step size α, at iteration time t may be found 565 as the argument that minimizes the following objective function, C(V_t,W_t,G_t,γ,α) where

 C(V_t,W_t,G_t, γ,α)=W_t^HH(V_t−αG_t)^HH^HW_t−W_t^H(V_t−αG_t)−(V_t−αG_t)^HH^HW_t+γtrace((V_t−αG_t)^H(V_t−αG_t))

[0036] After a step size has been established, the transmit weight vectors can be computed 570 at 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 V_t+1=V_t−αG_t. In another embodiment of the invention, the Lagrangian multiplier may be updated with: λ_t+1=λ_t+αg_λ,t. After computing the transmit weight vectors 570, the iteration number is incremented by one 575 as t=t+1. If the iteration number has reached an ‘end’ value 580, where t_end is an integer designating the maximum number of iterations, then the receive weight vectors are the receive weight vectors at iteration number t, (W=W_t) 590 and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=V_t) and this embodiment is completed. If the iteration number has not reached an ‘end’ value, 585 decides if trace(G_t−1^HG_t−1) <ε, and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number t, (W=W_t) and the transmit weight vectors are chosen to be the transmit weight vectors at iteration number t, (V=V_t) 590 again completing this embodiment. For decision block 585, trace(G_t−1^HG_t−1)<ε means the sum of the diagonal elements of the square matrix (G_t−1^HG_t−1) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(G_t−1^HG_t−1)<ε, the algorithm has almost converged because at the optimal solution Gt_t+1=0. If in decision block 585, trace(G_t−1^HG_t−1)<ε, is false, the process recedes to block 540 and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart 500.

[0037] 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 M_R×N_s weight matrix, W, as follows:
r=W^Hy  (13)

[0038] 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=U_HS_HZ_H^H  (14)
where M_R×M_R U_H and M_T×M_T Z_H are unitary matrices and M_R×M_T S_H is a matrix of all zeros except for the upper left r_H×r_H portion which is:
[S_H]_Ns=diag(s_H,1, . . . , s_H,rH)  (15)
where r_H is the rank of H (r_H≦min(M_T,M_R)), [A]_l means the first l rows and columns of A, and it is assumed that s_H,1≧s_H,2≧. . . ≧s_H,rH. The transmit weight matrix can also be expressed by a SVD as follows:
V=U_vS_vZ_v^H  (16)
where M_T×M_T U_v is unitary, N_s×N_s Z_v is unitary, and M_T×N_s S_v may be matrix of all zeros except for the upper N_s×N_s portion which contains the N_s non-zero singular values of the transmit weight matrix as follows:
[S_v]_Ns=diag(s_v,1, . . . , s_V,Ns)  (17)

[0039] One way to find the components of (16) may be to set U_V=Z_H and the singular values can be found as the solution to: $\begin{array}{cc}{s}_{V,l}^2=\frac{1}{\sqrt{\lambda }}{\sigma }_n{s}_{H,l}^{-1}-{\sigma }_n^2{s}_{H,l}^{-2}& \left(18\right)\end{array}$
where λ is chosen so that the following equation is satisfied (this forces the transmit weights to have unit power): $\begin{array}{cc}\sum _{l=1}^{N_s}{s}_{V,l}^2=1& \left(19\right)\end{array}$

[0040] 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., Z_v=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 Z_v can be found that gives equal MSE on each stream, the BER performance will improve while the average MSE will remain unchanged.

[0041] Using U_V=Z_H, the received data vector from (1) can be expressed as: $\begin{array}{cc}\begin{array}{c}y=U_H{S}_H{Z}_H^H{Z}_H{S}_V{Z}_V^{H}x+n\\ =U_H{S}_H{S}_V{Z}_V^{H}x+n\end{array}& \left(20\right)\end{array}$

[0042] Let N_s×1 Y_Ns={U_H^Hy}_Ns where {a}_i means the first i elements of the vector a, then Y_Ns is: $\begin{array}{cc}\begin{array}{c}{Y}_{N_s}={\left\{U_H^H{U}_H{S}_H{S}_V{Z}_V^{H}x+U_H^{H}n\right\}}_{N_s}\\ ={\left\{S_H{S}_V{Z}_V^{H}x\right\}}_{N_s}+{\left\{U_H^{H}n\right\}}_{N_s}\\ ={\mathrm{DZ}}_V^{H}x+N_{N_s}\end{array}& \left(21\right)\end{array}$
where D is a real N_s×N_s diagonal matrix equal to diag(s_H,1s_V,1, . . . , s_H,Nss_V,Ns) and N_Ns has a covariance matrix equal to σ_n²I_Ns. (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:
MSE_l=1−Z_V,l^H{tilde over (D)}Z_V,l  (22)
where Z_V,l is the l^th column of Z_v, and
{tilde over (D)}=D²(D²+σ_n²I)⁻¹  (23)

[0043] Thus unless all of the diagonal elements of {tilde over (D)} are equal, using Z_V=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)})/N_s  (24)

[0044] Therefore in order to make all data streams have the same MSE, Z_V,l must satisfy:
Z_V,l^H{tilde over (D)}Z_V,l=1−{overscore (MSE)}=trace({tilde over (D)})/N_s
subject to Z_VZ_V^H=Z_V^HZ_V=I_Ns  (25)

[0045] One type of matrix that solves (25) is the DFT matrix (another choice for when N_s is 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=V_HS_VF_Ns  (26)
where S_V is given by (17) and (18) and F_Ns is the N_s×N_s DFT matrix normalized such that F_Ns^HF_Ns=I. With the transmit weights chosen according to (26), the N_s streams can be recovered without any inter-stream interference as:
r=F_Ns^HD⁻¹Y_Ns  (27)

[0046] 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 chart 600, 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.

[0047] In one embodiment, for the successive cancellation method, the estimated symbols are given as: $\begin{array}{cc}{r}_u=w_u^H\left(y-\sum _{l=1}^{u-1}{\mathrm{Hv}}_l{\hat{x}}_l\right),\mathrm{where}{\hat{x}}_l=\mathrm{slice}\left(r_l\right)& \left(28\right)\end{array}$
where in one embodiment slice(r_l) means to choose the closest signal constellation point to r_l. In another embodiment, slice(r_l) refers to any technique that involves decoding a plurality of received symbols r_l into 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: $\begin{array}{cc}\underset{w_u,v_u}{\min }\sum _{u=1}^{N_s}E{r_u-x_u}^2=\underset{w_u,v_u}{\min }\sum _{u=1}^{N_s}E{w_u^H\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_l{x}_l+n\right)-x_u}^2& \left(29\right)\end{array}$
Note that (29) differs from (4) due to the inner sum being performed from u to N_s instead of 1 to N_s as is done in (4). Using E{x_u(n)x_l^*(n)}=δ(u−l), (29) becomes: $\begin{array}{cc}\underset{w_u,v_u}{\min }\left\{\begin{array}{c}\sum _{u=1}^{N_s}{w}_u^H\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_u-\\ \sum _{u=1}^{N_s}{w}_u^H{\mathrm{Hv}}_u-\sum _{u=1}^{N_s}{v}_u^H{H}^H{w}_u+N_s\end{array}\right\}& \left(30\right)\end{array}$
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: $\underset{w_u,v_u}{\min }\begin{array}{cc}\left\{\begin{array}{c}\sum _{u=1}^{N_s}{w}_u^H\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_u-\\ \sum _{u=1}^{N_s}{w}_u^H{\mathrm{Hv}}_u-\sum _{u=1}^{N_s}{v}_u^H{H}^H{w}_u+N_s+\\ \gamma {\sum _{u=1}^N{v}_u^H{v}_u-1}^2\end{array}\right\}& \left(31\right)\end{array}$
where γ is an arbitrary scaling placed on the constraint part of the objective function.

[0048] A closed form solution to (31) does not exist for N_s>1, therefore a modified version of an iterative algorithm, such as a gradient-based optimization approach, is used to solve for w_u and v_u. The gradient of (31) with respect to w_u is: $\begin{array}{cc}\nabla w_u=\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_u-{\mathrm{Hv}}_u& \left(32\right)\end{array}$
Equation (32) cannot be put into the same form as (8) because of the sum from u to N_s in the parenthesis. Setting equation (32) equal to zero, w_u can be found at each iteration as: $\begin{array}{cc}{w}_u={\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_l{v}_l^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)}^{-1}{\mathrm{Hv}}_u& \left(33\right)\end{array}$
The gradient of (31) with respect to v_u is: $\begin{array}{cc}\nabla v_u=\left(\sum _{l=1}^u{H}^H{w}_l{w}_l^{H}H+2\gamma \left(\mathrm{trace}\left(V^{H}V\right)-1\right)I_{M_T}\right)v_u-H^H{w}_u& \left(34\right)\end{array}$
In another embodiment, the gradient is computed using Lagrangian multipliers as follows: $\begin{array}{cc}\nabla v_u=\left(\sum _{l=1}^u{H}^H{w}_l{w}_l^{H}H+\lambda I_{M_T}\right)v_u-H^H{w}_u& \left(35\right)\end{array}$
One addition gradient is needed, the Lagrangian gradient which is:
 g_λ=trace(V^HV−1)  (36)
The Lagrangian gradient is needed in order to update the Lagrangian multiplier, λ.

[0049] The method for calculating the successive cancellation weights is described in FIG. 6 as a flow chart representation 600 of a preferred embodiment of an alternative method performed by the antenna combiner of FIG. 3 and the transmit device of FIG. 3A, for finding the MMSE successive cancellation weights, in accordance with the invention. FIG. 6 begins by being provided 610 a channel matrix H that is a M_R×M_T matrix which contains the gain and phase between each transmit antenna (M_T is the total number of transmit antennas) and each receive antenna (M_R is 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 M_T×N_s matrix of the transmit weight vectors, x(k) is a N_s×1 vector of the transmitted symbols at index (e.g., time or frequency), k, and n(k) is a M_R×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(V^HV)=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.

[0050] 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) as $r_u\left(k\right)=w_u^H\left(y\left(k\right)-\sum _{l=1}^{u-1}{\mathrm{Hv}}_l{\hat{x}}_l\left(k\right)\right),$
where {circumflex over (x)}_l(k)=slice(r_l(k)) and r_u(k) is the u^th element of r(k) slice(r_l(k)) may mean to choose the closest signal constellation point to r_l(k).

[0051] In another embodiment of the invention, the method of FIG. 6 can initialize an iteration number t, to zero 620, and the transmit weight vectors 630 may be initialized as $V_0=\frac{1}{\sqrt{N_s}}\left[\begin{array}{c}{I}_{N_s}\\ 0_{\left(M_T-N_s\right)\times N_s}\end{array}\right],$
where I_Ns is a N_s×N_s matrix of all zeros except for the diagonal elements which are one, and 0_a×b is an a×b matrix of all zeros. With the initialization complete, the method of FIG. 6 computes the receive weight vectors W_t at iteration number t 640, are found, for u=1 to N_s, as $w_{t,u}={\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_{t,l}{v}_{t,l}^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)}^{-1}{\mathrm{Hv}}_{t,u}$
where w_t,u is the u^th column of W_t, v_t,u are the transmit weight vectors at iteration number t, σ_n² is the noise power, and I_MR is a M_R×M_R matrix of all zeros except for the diagonal elements which are one.

[0052] Flow chart 600 proceeds to compute the gradient vectors at iteration number t 650. The gradient vectors, g_t,u, are found using the equation $g_{t,u}=\left(\sum _{l=1}^u{H}^H{w}_{t,l}{w}_{t,l}^{H}H+2\gamma \left(\mathrm{trace}\left(V_t^H{V}_t\right)-1\right)I_{M_T}\right)v_{t,u}-H^H{w}_{t,u},$
where γ can be the constraint weight (in one embodiment of the invention, γ=1), and I_MT is a M_T×M_T matrix 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: $g_{t,u}=\left(\sum _{l=1}^u{H}^H{w}_{t,l}{w}_{t,l}^{H}H+{\lambda }_t{I}_{M_T}\right)v_{t,u}-H^H{w}_{t,u}.$
It is by computing the gradient vectors, g_t,u, instead of the gradient matrix, G_t, that distinguishes the successive cancellation weights method of FIG. 6 from the linear MMSE weights method of FIG. 5.

[0053] If a step size has not been calculated prior to 660, the step size α, at iteration time t may be found 665 as the argument that minimizes the following objective function, C(V_t,W_t,G_t,γ,α) where $C\left(V_t,W_t,G_t,\gamma ,\alpha \right)=\left\{\begin{array}{c}\begin{array}{c}\sum _{u=1}^{N_s}{w}_{t,u}^H\left(\sum _{l=u}^{N_s}H\left(v_{t,u}-\alpha g_{t,u}\right){\left(v_{t,u}-\alpha g_{t,u}\right)}^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_{t,u}-\\ \sum _{u=1}^{N_s}{w}_{t,u}^{H}H\left(v_{t,u}-\alpha g_{t,u}\right)-\sum _{u=1}^{N_s}{\left(v_{t,u}-\alpha g_{t,u}\right)}^H{H}^H{w}_{t,u}+N_s+\end{array}\\ \gamma {\sum _{u=1}^{N_s}{\left(v_{t,u}-\alpha g_{t,u}\right)}^H\left(v_{t,u}-\alpha g_{t,u}\right)-1}^2\end{array}\right\}.$

[0054] After a step size has been established, the transmit weight vectors are computed 670 at 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 v_t+1,u=v_t,u−αg_t,u, where u=1 to N_s. In another embodiment of the invention, the Lagrangian multiplier is updated with: λ_t+1=λ_t+αg_λ,t.

[0055] After computing the transmit weight vectors 670, the iteration number may be incremented by one 675 as t=t+1. If the iteration number has reached an ‘end’ value 680, where t_end is 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=W_t) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=V_t) 690 completing this embodiment. If the iteration number has not reached an ‘end’ value, 685 decides if trace(G_t−1^HG_t−1)<ε, and if yes, the receive weight vectors are the receive weight vectors at iteration number t, (W=W_t) and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=V_t) 690 again completing this embodiment. For decision block 685, trace(G_t−1^HG_t−1)<ε means the sum of the diagonal elements of the square matrix (G_t−1^HG_t−1) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(G_t−1^HG_t−1)<ε, the algorithm has almost converged because at the optimal solution, G_t−1=0. If in decision block 685, trace(G_t−1^HG_t−1)<ε, is false, the process recedes to block 640 and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart 600. At this point the transmit vectors may be renormalized, if necessary, to satisfy the constraint in Equation (2).

[0056] FIG. 7 shows a flowchart of another embodiment of the invention 700. 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, 710 one embodiment of the invention may be provided a channel matrix. The iteration number, t 720, may next be set to 0 and the transmit and receive vectors can be initialized 730. In one embodiment of the invention, the transmit and receive vectors can be initialized to: $V_0=\frac{1}{\sqrt{N_s}}\left[\begin{array}{c}{I}_{N_s}\\ 0_{\left(M_T-N_s\right)\times N_s}\end{array}\right]$
For the linear weights, the receive weights are initialized to:
W₀=(HV₀V₀^HH+σ_n²I_MR)⁻¹HV₀
For the successive cancellation weights, the receive weights are initialized to: $w_{0,u}={\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_{0,l}{v}_{0,l}^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)}^{-1}{\mathrm{Hv}}_{0,u}$
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 computed 740 as 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:
V_t+1=(H^HW_tW_t^HH+2γ(trace(V_t^HV_t)−1)I_MT)⁻¹H^HW_t
For the successive cancellation weights: $v_{t+1,u}={\left(\sum _{l=1}^u{H}^H{w}_{t,l}{w}_{t,l}^{H}H+2\gamma \left(\mathrm{trace}\left(V_t^H{V}_t\right)-1\right)I_{M_T}\right)}^{-1}{H}^H{w}_{t,u}$
Next, the receive weight vectors at iteration t+1 may be computed as a function of the transmit weight vectors 750 at iteration number t+1, the noise power, and the channel matrix as follows:
For the linear weights:
W_t=(HV_tV_t^HH+σ_n²I_MR)⁻¹HV_t
For the successive cancellation weights: $w_{t,u}={\left(\sum _{l=u}^{N_s}{\mathrm{Hv}}_{t,l}{v}_{t,l}^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)}^{-1}{\mathrm{Hv}}_{t,u}$
The objective function at iteration number t+1 may next be computed 760 as 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:
C_t+1=N_s+W_t+1^H(V_t+1)(V_t+1)^HH^HW_t+1−W_t+1^HH(V_t+1)−(V_t+1)^HH^HW_t+1+γtrace((V_t+1)^H(V_{t +1}))
For the successive cancellation weights: $C_{t+1}=\left\{\begin{array}{c}\begin{array}{c}\sum _{u=1}^{N_s}{w}_{t+1,u}^H\left(\sum _{l=u}^{N_s}H\left(v_{t+1,u}\right){\left(v_{t+1,u}\right)}^H{H}^H+{\sigma }_n^2{I}_{M_R}\right)w_{t+1,u}-\\ \sum _{u=1}^{N_s}{w}_{t+1,u}^{H}H\left(v_{t+1,u}\right)-\sum _{u=1}^{N_s}{\left(v_{t+1,u}\right)}^H{H}^H{w}_{t+1,u}+N_s+\end{array}\\ \gamma {\sum _{u=1}^{N_s}{\left(v_{t+1,u}\right)}^H\left(v_{t+1,u}\right)-1}^2\end{array}\right\}$
Next, 770 if 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=W_t) and the transmit weight vectors may be the transmit weight vectors at iteration number t, (V=V_t) 790 completing this embodiment. If the objective function at iteration t is not less than the objective function at iteration t+1, then in 775 the iteration number, t, can be incremented by one. If the iteration number has reached an ‘end’ value 780, where t_end may 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=W_t) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=V_t) 790 completing this embodiment. If the end value is not reached, then the iterative procedure returns back to 740 and performs another iteration. After the iterations are completed, the transmit vectors may need to be renormalized to meet the constraint in Equation (2).

[0057] 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: $\begin{array}{cc}C={\log }_2\left[\det \left(I+\frac{1}{{\sigma }_n^2}{\mathrm{HVV}}^H{H}^H\right)\right]& \left(37\right)\end{array}$

[0058] Using (14) and (16), the capacity equation in (37) becomes: $\begin{array}{cc}C={\log }_2\left[\det \left(I+\frac{1}{{\sigma }_n^2}{U}_H{S}_H{Z}_H^H{U}_V{S}_V{Z}_V^H{Z}_V{S}_V^T{U}_V^H{Z}_H{S}_H^T{U}_H^H\right)\right]& \left(38\right)\end{array}$

[0059] Using Z_V^HZ_V=I and making U_V=Z_H, (38) becomes: $\begin{array}{cc}C={\log }_2\left[\det \left(I+\frac{1}{{\sigma }_n^2}{U}_H{S}_H{S}_V{S}_V^T{S}_H^T{U}_H^H\right)\right]& \left(39\right)\end{array}$

[0060] 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 Z_V, different Z_V'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 Z_V and the resulting weights will not change the theoretical capacity but will improve the receiver performance given the algorithm employed.

[0061] To maximize the capacity, the r (r≠N_s in 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(V^HV)=1). If the water-pouring strategy says to transmit on more streams than N_s, then the singular values of the transmit weight vectors are selected according to the water-pouring strategy using only the largest N_s singular values of H.

[0062] To summarize, the transmit weights are expressed as:
V=Z_HS_VZ_V^H  (40)

[0063] Therefore the received data vector can be expressed as:
y=U_HS_HZ_H^HZ_HS_VZ_V^Hx+n=U_HS_HS_VZ_V^Hx+n  (41)

[0064] Let r×1 Y_r={U_H^Hy}_r where {a}_i means the first i elements of the vector a, then Y_r is: $\begin{array}{cc}\begin{array}{c}{Y}_r={\left\{U_H^H{U}_H{S}_H{S}_V{Z}_V^{H}x+U_H^{H}n\right\}}_r\\ ={\left\{S_H{S}_V{Z}_V^{H}x\right\}}_r+{\left\{U_H^{H}n\right\}}_r={\mathrm{DZ}}_V^{H}x+N_r\end{array}& \left(42\right)\end{array}$
where the diagonalized channel, D, is a real r×r diagonal matrix equal to diag(s_H,1s_V,1, . . . , s_H,rs_Vr) and N_r has a covariance matrix equal to σ_n²I_r. Note that s_H,r≠0 because the water-pouring strategy would never dictate sending power on a stream with a singular value of zero.

[0065] 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 N_s. Assuming this is true (i.e., r≦N_s), the Successive Cancellation MMSE weights that maximize capacity are found by solving: $\begin{array}{cc}\underset{W,T}{\min }\sum _{l=1}^{N_s}E{w_l^H\left(Y_r-D\sum _{p=1}^{l-1}{t}_p+x_p\right)-x_l}^2=\underset{W,T}{\min }\sum _{l=1}^{N_s}Ew_l^H\left(D\sum _{p=1}^{N_s}{t}_p{x}_p+N_r\right)-x_l\mathrm{subject}\mathrm{to}:{\mathrm{TT}}^H=I& \left(43\right)\end{array}$
where the r×N_s right singular matrix, T, is T=[t₁, . . . , t_Ns], t₁ through t_Ns are the right singular vectors, T=Z_V^H, and M_R×N_s W=[w₁, . . . w_Ns].

[0066] It can be shown that the receive weight vector for stream l is given by: $\begin{array}{cc}{w}_l={\left(D\sum _{p=l}^{N_s}{t}_p{t}_p^{H}D+{\sigma }_n^2{I}_r\right)}^{-1}{\mathrm{Dt}}_l& \left(44\right)\end{array}$

[0067] Ignoring the constraint for now, the gradient of the objective function in (44) with respect to t_l (the l^th column of T) can be shown to be: $\begin{array}{cc}\nabla t_l=D\sum _{p=1}^l{w}_p{w}_p^H{\mathrm{Dt}}_l-{\mathrm{Dw}}_l& \left(45\right)\end{array}$

[0068] Using (45), T can be found through a gradient search where the constraint of TT^H=I is enforced at each step. This procedure is described in a flow chart representation 900 of a preferred embodiment of the invention illustrated in FIG. 8.

[0069] The embodiment of the invention illustrated in FIG. 8 may 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×N_s matrix such that T₀T₀^H=I (930). Next, the receive weight vectors at iteration number m, w_m,l, can be computed as a function of the diagonalized channel, D, and the right singular vectors at iteration number m, t_m,l, (940) as $w_{m,l}={\left(D\sum _{p=l}^{N_s}{t}_{m,p}{t}_{m,p}^{H}D+{\sigma }_n^{2}I\right)}^{-1}{\mathrm{Dt}}_{m,l}$
for l=1, . . . , N_s. Then the gradient vectors at iteration number m, g_m,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) as $g_{m,l}=D\sum _{p=1}^l{w}_{m,p}{w}_{m,p}^H{\mathrm{Dt}}_{m,l}-{\mathrm{Dw}}_{m,l}$
for l=1, . . . , N_s. 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 T_mT_m^H=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 t_m,l=t_m,l−αg_m,l for l=1, . . . , N_s. The iteration number may then be incremented by one (975) as m=m+1. Then (977) the right singular matrix at iteration number m, T_m, is formed by concatenating the right singular vectors at iteration number m together (T_m=[t_m,l|, . . . , |t_m,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 T_mT_m^H=I). The flow chart representation 900 continues as an embodiment of the invention in FIG. 8A.

[0070] In FIG. 8A, if the iteration number has reached an ‘end’ value 980, where m_end may 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=T_m) and the receive weight vectors may be chosen to be the receive weight vectors at iteration number m, (w_l=w_m,l) 990 completing this embodiment. If the iteration number has not reached an ‘end’ value, block 985 can decide if trace(G_m^HG_m)<ε where G_m=[g_m,1|, . . . , |g_m,Ns], and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number m, (w_l=w_m,l) and the right singular matrix is chosen to be the right singular matrix at iteration number m, (T=T_m) 990 again completing this embodiment. For decision block 985, trace(G_m^HG_m) means the sum of the diagonal elements of the square matrix (G_m^HG_m)and where ε is a number indicating how small the gradient matrix at iteration number m should get. In other words, when trace(G_m^HG_m)<ε, the algorithm has almost converged because at the optimal solution, G_m=0. If in decision block 985, trace(G_m^HG_m)<ε, is false, the process recedes to block 940 of FIG. 8, and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart 900. After completion of the technique described within flow chart 900, the transmit weight vectors may be given as V=Z_HS_VT.

[0071] 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.