WO2008157646A2 - Mimo transmit beamforming under uniform elemental peak power constant - Google Patents

Abstract

Embodiments of the invention pertain to a MIMO transmit beamforming method that is under the uniform elemental peak power constraint. Embodiments of the MIMO transmit beamforming method can be implemented without requiring unbalanced transmit power allocations among the various transmit antennas. Traditionally, MIMO transmit beamformer design under the uniform elemental power constraint is a non-convex optimization problem. Embodiments of the invention can relax this problem to a convex optimization problem via Semi-Definite Relaxation (SDR). For the multi-input single-output (MISO) case, the globally optimal solution to the relaxed problem has a closed-form and is also optimal to the original problem. With minor modifications, the closed-form MISO solution can be extended to that for the MIMO case, and a suboptimal closed-form solution to the MIMO transmit beamformer design can be obtained.

Description

DESCRIPTION

MIMO TRANSMIT BEAMFORMING UNDER UNIFORM ELEMENTAL PEAK POWER CONSTRAINT

The subject invention was made with government support under a research project supported by National Science Foundation Contract Nos. ECS-0621879 and CCF-0634786 and Office of Naval Research Grant No. NOOO 14-07- 1-0193.

Cross-Reference to Related Application

The present application claims the benefit of U.S. Application Serial No 60/944,679, filed June 18, 2007, which is hereby incorporated by reference herein in its entirety, including any figures, tables, or drawings.

Background of Invention

Multi-input multi-output (MIMO) transmit beamforming is a most spectrally efficient way to combat channel fading in wireless communications by exploiting the channel state information (CSI) at both the transmitter and receiver. To implement MIMO transmit beamforming in a practical system, two main problems exist. First, optimal transmit beamformers obtained by the conventional transmit beamforming may require unbalanced transmit power allocations among the various transmit antennas, which is undesirable from the antenna amplifier design prospective. Second, the CSI needs to be acquired at the transmitter.

The current conventional MIMO transmit beamforming may cause a wide power variation among the various transmit antennas. In practice, each antenna usually uses the same power amplifier, i.e., each antenna has the same power dynamic range and peak power, which means that the conventional MIMO transmit beamforming can suffer from severe performance degradations since it makes the power clipping of the transmitted signals inevitable. Exploiting multi-input multi-output (MIMO) spatial diversity is a spectrally efficient way to combat channel fading in wireless communications. Although the theory and practice of receive diversity are well understood, transmit diversity has been attracting much attention only recently. Generally, the transmit diversity systems belong to two groups. In the first group, the channel state information (CSI) is available at the receiver, but not at the transmitter. Orthogonal space-time block codes (OSTBC) [I], [2] have been introduced to achieve the maximum possible spatial diversity order. In the second group, the CSI is exploited at both the transmitter and the receiver via MIMO transmit beamforming, which has recently attracted the attention of the researchers and practitioners alike, due to its much better performance compared to OSTBC [3], [4]. Compared to OSTBC, MIMO transmit beamforming can achieve the same spatial diversity order, full data rate, as well as additional array gains. However, implementing MIMO transmit beamforming schemes in a practical communication system requires additional considerations.

First, optimal transmit beamformers obtained by the conventional, i.e., the maximum ratio transmission (MRT), approach may require different elemental power allocations on the various transmit antennas, which is undesirable from the antenna amplifier design perspective. Especially in an orthogonal frequency division multiplexing (OFDM) system, this power imbalance can result in high peak-to-average power ratio (PAPR), and hencewise reduce the amplifier efficiency significantly [5]. These practical problems have been considered in [6], [7], [8] for new transmit beamformer designs, and have also been addressed for transmitter designs in a downlink multiuser system [9].

Second, consideration of how to acquire the CSI at the transmitter is needed. Recent focus has been on the finite-rate feedback techniques for the current conventional transmit beamforming [10], [11], [12], [13], [14], [15]. These techniques attempt to efficiently feed back the transmit beamformer (or the CSI) from the receiver to the transmitter via a finite- rate feedback channel, which is assumed to be delay and error free, but bandwidth-limited. The problem is formulated as a vector quantization (VQ) problem [16], [17] and the goal is to design a common codebook, which is maintained at both the transmitter and the receiver. For frequency-flat independently and identically distributed (i.i.d.) Raleigh fading channels, various codebook design criteria can be used and the theoretical performance (e.g., outage probability [12], operational rate-distortion [14], capacity loss [15]) can be analyzed for the multi-input single-output (MISO) case. The feedback schemes can be readily extended to the frequency-selective fading channel case via OFDM. The relationship among the OFDM subcarriers can also be exploited to reduce the overhead of feedback by vector interpolation [18]. Brief Summary

Embodiments of the invention pertain to a MIMO transmit beamforming method that is under the uniform elemental peak power constraint. Embodiments of the MIMO transmit beamforming method can be implemented without requiring unbalanced transmit power allocations among the various transmit antennas. Traditionally, MIMO transmit beamformer design under the uniform elemental power constraint is a non-convex optimization problem. Embodiments of the invention can relax this problem to a convex optimization problem via Semi-Definite Relaxation (SDR). For the multi-input single-output (MISO) case, the globally optimal solution to the relaxed problem has a closed-form and is also optimal to the original problem. With minor modifications, the closed- form MISO solution can be extended to that for the MIMO case, and a suboptimal closed-form solution to the MIMO transmit beamformer design can be obtained.

Embodiments of the invention relate to the use of finite-rate feedback methods for MIMO transmit beamforming to allow acquisition of the CSI at the transmitter. Embodiments of the invention relate to a simple scalar quantization method and a vector quantization method, which maximizes the Inner product of Maximum Eigenmode and is referred to as MIME. In order to analyze the MIME vector quantization performance for the MISO case, an approximate expression for the average degradation of the receive signal-to- noise ratio (SNR) caused by the MIME vector quantization is obtained. Numerical examples are provided to demonstrate the effectiveness of our transmit beamforming and feedback techniques.

Under the uniform elemental peak power constraint, numerical examples demonstrate that embodiments of the subject closed- form designs outperform the conventional MIMO transmit beamforming with peak power clipping. For the finite-rate feedback, embodiments of the subject scalar quantization method is shown to be quite effective with relatively large numbers of feedback bits (e.g., B=6, 8 for a (4, 1) or (4, T) system), and MIME can provide the same performance as the conventional sphere vector quantization (SVQ) when the number of feedback bits is small, even though the latter is not subject to the uniform elemental peak power constraint. For the (2, 1) system, embodiments of the subject finite-rate feedback schemes can achieve more than 2 dB in SNR improvement compared to the "Alamouti Code" at the cost of requiring only a 2-bit feedback. Embodiments of the invention pertain to an approximate closed-form expression for the average degradation of the receive SNR caused by MIME for the MISO case. This expression is quite accurate and can provide a good guideline to determine the number of feedback bits needed in a practical system.

Embodiments of the invention can be used for wireless communication systems. Embodiments of the subject MIMO transmit beamforming can be implemented in wireless communications and wireless local area networks (WLAN's). Embodiments can be used with frequency flat Rayleigh fading channels. Further embodiments of this invention can be incorporated with frequency selective fading channels and/or adopted in any orthogonal frequency division multiplexing (OFDM) based wireless systems with multiple transmit antennas. For example, embodiments of the invention can be used in IEEE 802.1 In wireless local area network (WLAN, or WiFi) systems and IEEE 802.16 wireless metropolitan area network (WMAN, or WiMAX) systems. Embodiments of the subject MIMO transmit beamformer designs can be utilized with frequency selective fading channels and used in, for example, MIMO-OFDM based WLAN systems.

Brief Description of Drawings

Figure 1 shows transmit power distribution across the index of the transmit antennas for a (4,1) system.

Figures 2A and B show performance comparison of various transmit beamformer designs with perfect CSI at the transmitter: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. Note that the (4,2) UEP TxBm and (4,2) Heuristic SDR curves almost coincide with each other in (b).

Figure 3 shows performance comparison of various transmit beamformer designs for the (8,8) MIMO case. Figures 4A and 4B show performance comparison of various transmit beamformer designs with 2-bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. Note that 2-bit CVQ and 2-bit VQ-UEP curves basically coincide with each other for both (4,1) and (4,2) systems, although the former is not under the uniform elemental power constraint while the latter is. Figures 5 A and 5B show performance comparison of various transmit beamformer designs with 4-bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. Note that 4-bit CVQ and 4-bit VQ-UEP curves basically coincide with each other for both (4,1) and (4,2) systems, although the former is not under the uniform elemental power constraint while the latter is.

Figures 6A and 6B show performance comparison of various transmit beamformer designs with 6-bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. Note that 6-bit CVQ and UEP TxBm with perfect feedback curves almost coincide with each other for both (4,1) and (4,2) systems.

Figures 7A and 7B show performance comparison of various transmit beamformer designs with 8-bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. Note that 8-bit SQ, 8-bit VQ-UEP and UEP TxBm with perfect feedback curves almost coincide with each other for both (4,1) and (4,2) systems.

Figure 8 shows performance comparison of various (2,1) MISO systems. Note that

CVQ, SQ, VQ-UEP and UEP TxBm with perfect feedback curves almost coincide with each other, although SQ and VQ-UEP are subject to both the uniform elemental power and 2-bit feedback rate constraints, while UEP TxBm assumes perfect feedback and CVQ is not subject to the uniform elemental power constraint.

Figure 9 shows average degradation of the receive SNR for a (4,1) MISO system.

Detailed Disclosure MIMO transmit beamformer design under the uniform elemental power constraint is typically approached as a non-convex optimization problem, which is usually difficult to solve, and no globally optimal solution is guaranteed [6]. In accordance with embodiments of the invention, generally, the original problem can be relaxed to a convex optimization problem via Semi-Definite Relaxation (SDR). The relaxed problem can be solved via, for example, public domain software [19]. A solution to the original non-convex optimization problem can then be obtained from the solution to the relaxed one by, for example, a heuristic method [20] (referred to as the heuristic SDR solution). In the multi-input single-output (MISO) case, the optimal solution has a closed-form expression and can be referred to as the closed- form MISO transmit beamformer. (Similar results have appeared in [6], [7], [8] for equal gain transmission (EGT).) A cyclic algorithm for the MIMO case, which uses the closed-form MISO optimal solution iteratively, can then be implemented and the solution can be referred to as the cyclic MIMO transmit beamformer. The cyclic algorithm has a low computational complexity and is shown via numerical examples to converge quickly from a good initial point. The numerical examples also show that the transmit beamformmg approach outperforms the conventional one with peak power clipping. Meanwhile, the cyclic solution has a comparable performance to the heuristic SDR based design and outperforms the latter when the rank of the channel matrix increases.

Embodiments can utilize finite-rate feedback schemes for the transmit beamformer designs. A simple scalar quantization (SQ) method can be used by taking advantage of the property of the uniform elemental power constraint, where the number of parameters to be quantized can be reduced to less than one half of their conventional counterpart. VQ methods can also be implemented. Although the existing codebooks [10], [11], [12], [14], [15] can be used with some modifications by the MISO closed-form solution, the performance may not be optimal since they do not take into account the uniform elemental power constraint in the codebook construction. Embodiments of the invention relate to a VQ method for transmit beamformer designs whose codebook is constructed under the Uniform Elemental Power constraint (referred to as VQ-UEP). The generalized Lloyd algorithm [16] is adopted to construct the codebook. When the number of feedback bits is small, VQ-UEP performs similarly to the conventional VQ (CVQ) method without uniform elemental power constraint. For the MISO case, the performance of VQ-UEP can be further quantified by obtaining an approximate closed-form expression for the average degradation of the receive signal-to- noise ratio (SNR). It is shown that this approximate expression is quite tight and can be used use as a guideline to determine the number of feedback bits needed in practice, for a desired average degradation of the receive SNR.

Section 2 below describes the conventional MIMO transmit beamforming and its limitations. Section 3 presents embodiments of closed-form MISO and cyclic MIMO transmit beamformer designs under the uniform elemental power constraint, hi Section 4, the finite- rate feedback schemes are described, where embodiments of a simple SQ method and VQ- UEP are taught. In Section 5, the MISO case is described and the average degradation of the receive SNR caused by VQ-UEP is quantified by obtaining an approximate closed-form expression. Numerical examples are given in Section 6 to demonstrate the effectiveness of various designs. The following notations are adopted:

Notation: Bold upper and lower case letters denote matrices and vectors, respectively. (-)^τ is used to denote the transpose and (•)^* to denote the conjugate transpose. | ■ | stands for the absolute value of a scalar and [| • || denotes the two-norm of a vector. C is the complex set; C^M/N and R^MκN are the complex- and real-valued M x N matrices, respectively, tr(-) is the trace of a matrix. E{-} is the expectation, E_a{-} is the ensemble average and Var{-} denotes the variance. Zx is the vector formed by the phase angles of x and [J denotes the floor operation.

2. MIMO Transmit Beamforming

Consider an (N_t,N_r) MIMO communication system with N_t transmit and N₁. receive antennas in a quasi-static frequency flat fading channel. At the transmitter, the complex data symbol s e C is modulated by the beamformer , and then

transmitted into a MIMO channel. At the receiver, after processing with the combining vector

' ^me sampled combined baseband signal is given by

(1)

where H e C^{Nj xN}' is the channel matrix with its (i, j) th element h_tj denoting the fading coefficient between the y^'th transmit and z^' th receive antennas, and n e C^A'^xl is the noise vector with its entries being independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and variance

. Note that in the presence of interference, i.e., when n is colored with a known covariance matrix Q , we can use pre-whitening at the receiver to get

(2)

Hence (2) is equivalent to (1) except that H in (1) is now replaced by Q ¹H and the whitened noise has unit variance. Without loss of generality, we focus on (1) hereafter. The transmit beamformer w₍ and the receive combining vector w_r in (1) are usually chosen

to maximize the receive SNR. Without loss of generality, we assume that w - 1 ,

w_r = 1 , and E{| s |²} = 1 . Then the receive SNR is expressed as (3)

To maximize the receive SNR, the optimal transmit beamformer is chosen as the eigenvector corresponding to the largest eigenvalue of H^*H [14] (referred to as MRT in [6]), which is also the right singular vector of H corresponding to its dominant singular value. The optimal combining vector is given by , which can be shown to be the left singular vector of

H corresponding to its dominant singular value (referred to as maximum ratio combining (MRC) in [6]). Thus, the maximized receive SNR is ⁾ _; where /L_1113x (O denotes the

maximum eigenvalue of a matrix. The co variance matrix of the transmitted signal is

(4)

The average transmitted power for each antenna is

(5)

where R.. denotes the z th diagonal element of R . (Note that if the constellation of s is phase shift keying (PSK), P₁ represents the instantaneous power.) The average power P₁ may vary widely across the transmit antennas, as illustrated in Figure 1, which shows a typical example of transmit power distribution across the antennas. The wide power variation poses a severe constraint on power amplifier designs. In practice, each antenna usually uses the same power amplifier, i.e., each antenna has the same power dynamic range and peak power, which means that the conventional MIMO transmit beamforming can suffer from severe performance degradations since it makes the power clipping of the transmitted signals inevitable. 3. Transmit Beamformer Designs under Uniform Elemental Power Constraint

We consider below both MIMO and its degenerate MISO transmit beamformer designs under the uniform elemental power constraint.

Given MRC at the receiver, maximizing the receive SNR p in (3) under the uniform elemental power constraint is equivalent to: max IIHWJ

(6)

This is a non-convex opt

imization problem, which is usually difficult to solve, and no globally optimal solution is guaranteed [6]. [20], [21]. In an embodiment of the invention, the problem in (6) can be reformulated as

(7)

where G = H H e C^N'^*N' , R e C^N'^xN' , and the inequality R ^ O means that the matrix R is positive semi-definite. Note that in (7), the objective function is linear in R , the constraints on the diagonal elements of R are also linear in R , and the positive semi-definite constraint on R is convex. However, the rank-one constraint on R is non-convex. The problem in (7) can be relaxed to a convex optimization problem via Semi-Definite Relaxation (SDR), which amounts to omitting the rank-one constraint yielding the following Semi-Definite Program (SDP) [22]:

(8)

The dual form of (8) is given by [20]

(9)

where with 1_Λ, denoting an ^ -dimensional all one column vector, and

diag{x} is a diagonal matrix with x on its diagonal. The problem in (9) is also a SDP. Both (8) and (9) can be solved by using a public domain SDP solver [19]. The worst case complexity of solving (9) is O(N^⁵) [23]. We can obtain the optimal solution to (9), whose dual is also the optimal solution to (8). Assume that the optimal solution to (8) isR_opl . Then for any w,

under the uniform elemental power constraint. If the rank of R_opt is one, then we obtain the optimal solution w^° to (6) as the eigenvector corresponding to the non-zero eigenvalue of R_opt . If the rank of R_opt is greater than one, we can obtain a suboptimal solution w^* from R_opt via a rank reduction method. For example, the heuristic method in [20] chooses w^* as the eigenvector corresponding to the dominant eigenvalue of R_opt . The Newton-like algorithm presented in [24] uses the SDR solution as an initial solution and then uses the tangent-and-lift procedure to iteratively find the solution satisfying the rank-one constraint, and can be utilized in an embodiment of the invention. However, the approximate heuristic method is preferred, as shown in our later discussion, due to its simplicity and can be utilized in other embodiments of the invention.

The optimal solution to (6) has a closed-form expression for the MISO case. Moreover, a cyclic algorithm for the MIMO case, which uses the closed-form MISO optimal solution iteratively, can be used. The cyclic method has a low complexity and numerical examples in Section 6 show that it converges quickly given a good initial point. Furthermore, as shown in Section 6, the performance of the cyclic algorithm is comparable to that of the Heuristic SDR solution and in fact better when the rank of the channel matrix is large. Hence, the former may be preferred over the latter in practice. MISO Optimal Transmit Beamformer

Let

be the row channel vector for the MISO case. We consider the maximization problem in (6)

(10) where the equality holds when denoting the unit-

norm column vector having the angles of h^* , and φ e [0, 2π) . Note that the optimal solution is not unique due to the angle ambiguity; yet we may take w^° as the optimal solution to (6) for simplicity. (This result can also be found in [6], [7], [8] for EGT.)

The Cyclic Algorithm for MIMO Transmit Beamformer Design The original maximization problem for (6) is

(11)

Applying the cyclic method (see, e.g., [25]), the problem in (11) can be solved in a cyclic way for the MIMO case. The cyclic algorithm is summarized as follows: (1) Step 0: Set w, to an initial value (e.g., the left singular vector of H corresponding to its largest singular value). (2) Step 1 : Obtain the beamformer w, that maximizes (11) for W₇ fixed at its most recent value. By taking w^*H as the "effective MISO channel," this problem is equivalent to (6) for the MISO case. The problem is solved in (10) and the optimal solution is: C¹²)

(3) Step 2: Determine the combining vector w, that maximizes (11) for w₍ fixed at its most recent value. The optimal w_; is the MRC and has the form:

(13)

Iterate Steps 1 and 2 until a given stop criterion is satisfied. Examples of stop criteria that can be implemented include, but are not limited to, a certain number of repetitions of steps 1 and 2, such as 6, and a certain percentage improvement compared with prior results, such as less than 0.1% improvement.

An important advantage of the above algorithm is that both Steps 1 and 2 have simple closed- form optimal solutions. Also the cyclic algorithm is convergent under mild conditions [25].

The cyclic algorithm is flexible and more constraints on w_r or w_t can be added. A useful one is the uniform elemental power constraint on the receive antennas (or equal gain combining (EGC) [11], [6]), i.e., . Then we only have to modify

(13) as in Step 2 of each iteration. Given a good initial value (e.g., the one as

given in Step 0), the cyclic algorithm usually converges in a few iterations in our numerical examples, and the computational complexity of each iteration is very low, involving just (12) and (13).

4. Finite-Rate Feedback for Transmit Beamforming Designs

In the aforementioned transmit beamformer designs, it has been assumed that the transmitter has perfect knowledge on the CSI. However, in many real systems, having the CSI known exactly at the transmitter is difficult, if even possible. The channel information is usually provided by the receiver through a bandwidth-limited finite-rate feedback channel, and SQ or VQ methods, which have been widely studied for source coding [16], [17], can be used to provide the feedback information. In specific embodiments, it is assumed herein that the receiver has perfect CSI, as usually done in the literatures [10], [11], [12], [14], [15]. Scalar Quantization

Note that the transmit beamformer W₁ under the uniform elemental power constraint can be expressed as

(14)

where the transmit beamformer w_t(θ₀,...,θ_{N A}) is a function of N₁ parameters [O₁, O₁ e [0, 2π)}^¹ . Via simple manipulations, we obtain

(15)

where

Since , we can reduce one parameter and

quantize

instead of .

Denote

(16)

where

, with N₁ = 2^s' and M₁ denoting the number of quantization levels and feedback index of Q₁ , respectively, and where B₁ is the number of feedback bits for . After obtaining the transmit beamformer from (10) or the cyclic

algorithm in Section 3.3, we quantize the parameters Q₁ to the "closest" (via round off) grid points . Hence for this scalar quantization scheme, we need to send the

index set (n_x, n₂,...,n_N ^) from the receiver to the transmitter, which requires

bits. The receive combing vector is w,

The choice of

is known as a counting problem [26], which has

combinations. The optimal set is the one that maximizes

• However, this exhaustive search is too complicated for practical

applications. One simple suboptimal approach is to make B₁ approximately equal.

Specifically, let

= B^

bits for the first N_s parameters

and

bits for the remaining (N₁ —Ϊ) — N_s

parameters

We remark here that for the conventional MIMO transmit beamformer without uniform elemental power constraint, the SQ requires about twice as many parameters. In this case, the transmit beamformer is expressed as

where A₁, A₁ e [0,l] is the / th amplitude and O₁, O₁ <≡ [§, 2π) is the / th phase of the transmit beamformer vector, respectively, and hence there are totally 2N_t parameters.

Ad-hoc Vector Quantization Vector quantization can be adopted to further reduce the feedback overhead. In this case, both the transmitter and the receiver have to maintain a common codebook with a finite number of codewords. The codebook can be constructed based on several criteria. One approach is to directly apply the existing codebooks (e.g., [10], [11], [12], [14], [15]) constructed for the conventional transmit beamformer designs obtained without the uniform elemental power constraint. Among them, the criteria (e.g., [10], [14], [15].) that can be implemented by the generalized Lloyd algorithm can always lead to a monotonically convergent codebook. The generalized Lloyd algorithm is based on two conditions: the nearest neighborhood condition (KN C) and the centroid condition (CC) [16], [14], [15]. NNC is to find the optimal partition region for a fixed codeword, while CC updates the optimal codeword for a fixed partition region. The monotonically convergent property is guaranteed due to obtaining an optimal solution for each condition. Maximizing the average receive SNR is a widely used criterion to design the codebook [10], [12], [14] and will also be adopted here for codebook construction. Some modifications are still needed as below when the uniform elemental power constraint is imposed. Let a codebook constructed for the conventional transmit beamforming be } ₅ where N_v = 2^B is the number of codewords in the codebook W , and

B is the number of feedback bits. The receiver first chooses the optimal codeword in the codebook as:

(18)

where the operator argmax returns a global maximizer. Then we can feed back the index of w° from the receiver to the transmitter, which requires B bits. The transmit beamformer satisfying the uniform elemental power constraint is obtained as:

(19)

and the receive combining vector is . However, the codebook

may ^J not be

optimal for some transmit beamformer designs, since it is ad-hocly constructed without the uniform elemental power constraint (referred to as the ad-hoc vector quantization (AVQ) method).

Vector Quantization under Uniform Elemental Power Constraint

Like AVQ, embodiments of the invention can maximize the average receive SNR, while the codebook is constructed under the uniform elemental power constraint (referred to as "VQ-UEP"). For a given codebook the receiver first chooses the

optimal transmit beamformer as: (20)

and the corresponding vector quantizer is denoted as

■ Then we can feedback the index of fr°^m the receiver to the transmitter with log, N_^ = B bits, and the receive

combining vector is

Now the design problem becomes finding the codebook, which can be constructed off-line as follows. First, we generate a training set {H_PH₂,...,H^} from a sufficiently large

number N of channel realizations. Next, starting from an initial codebook (e.g., a codebook obtained from the conventional transmit beamformer designs or one obtained via the splitting method [16]), we iteratively update the codebook according to the following two criteria until no further improvement is observed.

(1) NNC: for given codewords

, assign a training element H_n to the i th region

(21)

where S₁J = 1,2, ..., N_v, is the partition set for the i th codeword yv, ■

(2) CC: for a given partition S₁ , the updated optimum codewords {w,}^ satisfy

(22)

N₁ for be Hermitian square root

of R_; . A simple reformulation results in

R^{1 '2}w

(23)

This problem is identical to (6) (H is replaced by R,^{1 2}) and can be efficiently solved by the cyclic algorithm proposed in Section 3.3.

5. Average Degradation of the Receive SNR

For frequency flat i.i.d. MISO Rayleigh fading channels, various analysis approaches have been taught to quantify the vector quantization effect (outage probability [12], operational rate-distortion [14], capacity loss [15], etc.). These analyses provide theoretical insights into the vector quantization methods and can serve as a guideline for determining the optimum number of feedback bits needed for the conventional transmit beamforming. We quantify below the effect of VQ-UEP with finite-bit feedback on an embodiment of the subject closed-form MISO transmit beamformer design. Let (0 ll ) . Without loss of

generality, we assume . The average degradation of the receive SNR is defined as:

(24)

where ^{2 2} is the partition set (or Voronoi cell) for the /th

codeword is the probability that a channel

realization h belongs to the partition ■_> ^and the last equality is due to the independence

between h and the normalized vector h/|h [14], [26]. Obviously, we have

Maximum Average Receive SNR E{\ hw^° |²}

Using w^° in (10), we get: (25)

where the last equality is due to the i.i.d. property of _r The | A₁ | in (25) has the

probability density function (pdf) as follows [27]:

(26)

The mean and variance of | Zz₁ | are, respectively,

(27)

and

(28)

Combining (27) and (28) into (25), we get

(29)

Approximate Value of

Note that the vector v, is considered as uniformly distributed on the unit hypersphere

Ω ' [10], [12], [14], [15]. For a fixed codeword , the random variable

has a beta distribution Beta(l,iV, -1) [15], with the pdf: (30)

Now we consider the conditional density ) . Generally, each Voronoi cell [10], [12],

[15], [16] obtained from the generalized Lloyd algorithm has a very complicated shape and it is difficult to obtain an exact closed-form expression for . We adopt herein the

approximate method used in [12], [15] to analyze the problem.

When N_v is reasonably large, we can approximate the probability as

The Voronoi cells can be considered as identical to each other. We then approximate each Voronoi cell §i as a spherical segment on the surface of a unit hypersphere:

(31)

where a = ^^^ = i_ζ + f ^r is the maximum average value of | v^* w, f achieved by perfect feedback in our MISO transmit beamformer design, and the parameter δ > 0 is the minimum value of | v^*w; I in each Voronoi cell. We need to solve the following equation related to B to obtain δ :

(³²⁾

Using the pdf in (30), we get (33)

Thus, for the Voronoi cell s_t , we approximate the conditional pdf of γ_: as

(34)

where

(35)

is the indication function.

From the conditional pdf f_{γ lv e5} (x) in (34), we obtain

(36)

Quantifying the Average Degradation of the Receive SNR

Now we quantify the average degradation of the receive SNR in (24) using the approximate conditional pdf /_{7 |v e}~ (x) . From (36), we observe that the average receive SNR γ₀ is

(37)

Combining (29) and (37) into (25), we obtain the following proposition:

Proposition For i.i.d. MISO Raleigh fading channels, the average degradation of the receive SNR, for an N_t -antenna transmit beamforming system with an N_v = 2^B -size VQ-UEP codebook, can be approximated as: D_v(B) = (N_t -T) - 2^» \ 2^→ + {l -a) - (l -a) σl -Nβ -cήσ, (38)

The average degradation of the receive SNR in (38) can be proven to be monotonically decreasing with respect to non-negative real number B (see Appendix). Given a degradation amount D₀ , this proposition provides a guideline to determine the necessary number of feedback bits. That is, we can always find the optimum integer number of feedback bits B (via, e.g., the Newton's method) with the average degradation D_v(B) of the receive SNR being less than or equal to D₀ . Similarly, the average receive SNR in (37) can be shown to be monotonically increasing with respect to B , and we can determine the needed number of feedback bits with the average receive SNR being less or equal to a desired -

Although our analysis shares some similar features to those in [7], [8], our results are more accurate (see Section 6). In [7], [8], the authors found the pdf of

via making more approximations. Under high-resolution approximations, the average degradation of the receive SNR given in [7], [8] has the form:

(39)

Both (38) and (39) are compared with numerically determined average receive SNR loss at the end of the next section and (38) is shown to be more accurate than (39).

6. Numerical Examples

Below, several numerical examples are presented to demonstrate the performance of embodiments of the subject MISO and MIMO transmit beamformer designs under the uniform elemental power constraint. We assume a frequency flat Rayleigh channel model with E{\ h_ϋ \²} = 1, i = 1,2,... ,N_r, j = 1,2,..., N_t . In the simulations, we use QPSK for the transmitted symbols. First, we consider the bit-error-rate (BER) performance of embodiments of the MISO and MIMO transmit beamformer with perfect CSI available at the transmitter. For comparison purposes, we also implement several other embodiments. The "Con TxBm" denotes the conventional transmit beamforming design without the uniform elemental power constraint. The "TxBm with Clipping" stands for the conventional design with peak power clipping, which means that for every transmit antenna, if

will be clipped by , TV^ . The "Heuristic SDR" refers to the Heuristic SDR

solution described in Section 3.1. We denote "UEP TxBm" as the closed-form MISO and the cyclic MIMO transmit beamformer designs under uniform elemental power constraint.

Figure 2 shows the bit-error-rate (BER) performance comparison of various transmit beamforming designs for both the (4,1) MISO and (4, 2) MIMO systems. The "Con TxBm" achieves the best performance since it is not under the uniform elemental power constraint. Under the uniform elemental power constraint, the "UEP TxBm" schemes have much better performance than the "TxBm with Clipping." At BER = 1(T³ , for example, the improvement is about 1.5 dB for the (4,2) MIMO system. In the MIMO system, we note that the "UEP TxBm" achieves almost the same performance as the "Heuristic SDR." Interestingly, if we increase both the transmit and receive antennas to 8, as shown in Figure 3, the "UEP TxBm" outperforms the "Heuristic SDR." The performance degradation of "Heuristic SDR" is caused by reducing the high rank optimal solution to (8) to a rank-one solution heuristically. We note here that the "UEP TxBm" is also much simpler than the "Heuristic SDR" (see the discussions in Section 3).

We examine next the effects of the two quantization methods (SQ and VQ) on the overall system performance. We use herein the suboptimal combination of described

in Section 4.1 for SQ due to its simplicity (although the optimal one can provide a better performance). We show in Figures 4-7 the BER performance of various quantization schemes for embodiments of the invention and conventional transmit beamformer designs, with various numbers of feedback bits (5 = 2, 4, 6,8 ). We note that VQ-UEP outperforms the AVQ for all cases. When the number of feedback bits is small (e.g., B = 2, 4 ), VQ-UEP can provide a similar performance as that of CVQ, even though the latter is not under the uniform elemental power constraint! The VQ-UEP performance approaches that of the perfect channel feedback for "UEP TxBm" when the number of feedback bits becomes larger (e.g., B = 8 ). However, CVQ needs more bits to approach the performance of its perfect channel feedback counterpart. By using relatively large numbers of feedback bits (e.g., B = 6,8 ), we can reduce the gap between the suboptimal SQ method and VQ-UEP, since we have already reduced the number of parameters to be quantized for the scalar method due to imposing the uniform elemental power constraint.

Moreover, Figure 8 shows the BER performance of various (2,1) MISO systems. In this case, we know that the "Alamouti Code" [1] has full rate and satisfies the uniform elemental power constraint. Compared to the "Alamouti Code," embodiments of the subject transmit beamformer design can achieve more than 2 dB SNR improvement using only a 2- bit feedback, via either the suboptimal SQ or VQ-UEP. embodiments of the subject transmit beamformer design with a 2-bit feedback also performs similarly to its CVQ counterpart.

Finally, we examine the accuracy of the approximate degradation D_v(B) of the receive SNR given in (38) for the MISO case. We carry out Monte-Carlo simulations for a (4,1) system and plot the numerically simulated degradation results in Figure 9. The training sequence size is set to

⁷ , and the channel variance is

= 1 . We observe that the approximate degradation given in (38) is very close to the numerically simulated one for any feedback bit number (or rate) B . However, the high-resolution approximation given in (39) has accurate prediction only at high feedback bit rates. Note also that the SQ and VQ-UEP perform similarly when the feedback bit number is relatively large, which means that the approximate degradation expression of the receive SNR given in (38), which is obtained for VQ-UEP, can also be used for SQ for large i> .

Appendix We prove that the average degradation D_v(B) of the receive SNR in (38) is a monotonically decreasing function of the non-negative real number B . We let . Then the first derivative of D_v (B) with respect to B is

6^ΛH ( m2) 2^" σ

(40)

where c = (JV, - 1) • In 2 • σ\ is a constant.

Note that (41)

Using the Taylor series expansion, we can expand

b"'^~l as

(42)

or

(43)

» 0

where

Since , we obtain the inequalities as follows:

(44)

For the 2 < (1 - a) ' case, we have

(45)

For the case, we have

(46)

Summarizing the above inequalities, we get D_v{B) < 0 . Thus, the average degradation of the receive SNR D_v(B) is a monotonically decreasing function of the non-negative real number B .

All patents, patent applications, provisional applications, and publications referred to or cited herein are incorporated by reference in their entirety, including all figures and tables, to the extent they are not inconsistent with the explicit teachings of this specification.

It should be understood that the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application.

References

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Claims

Claims

1. A multi-input multi-output (MIMO) communication system, comprising: a transmitter including N_t transmit antennas; and a receiver including N₁. receive antennas, wherein at the transmitter the complex data

symbol s e C is modulated by a beamformer w, = \ w_n w_{t 2} ... w_{t N} , and then

transmitted into a MIMO channel, wherein at the receiver, after processing with a combining vector a sampled combined baseband signal is

' is the channel matrix with its (z,y) th element h_tj denoting the fading coefficient between the j ϋx transmit and z^' th receive antennas, and n e C^N' ^xl is the noise vector with its entries being independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and variance σ_n ² , wherein w, is determined by solving max tr(RG),

(R)

subject to R₁₁ = Y, i = l, 2,...,N_t,

R h O,

rank(R) = 1, where

, and the inequality R ^ O means that the matrix R is positive semi-definite, where

2. The system according to claim 1, wherein the N₁ transmit antennas and N_r receive antennas are in a quasi-static frequency flat fading channel.

3. The system according to claim 1, wherein solving max fr(RG) is relaxed to a

convex optimization problem via Semi-Definite Relaxation (SDR), wherein w, is determined by solving

4. The system according to claim 1, wherein solving max Jr(RG) is relaxed to a

{R} convex optimization problem, wherein w, is determined by solving

where with 1_Λ, denoting an N₁ -dimensional all one column

vector, and diag{x} is a diagonal matrix with x on its diagonal.

5. The system according to claim 3, wherein the rank of R_opt is one, where R_opt is an optimal solution to

where w, is the eigenvalue corresponding to the non-zero eigenvalue of R_{0 1} .

6. The system according to claim 4, wherein the rank of R_opt is one, where R_{0 1} is an optimal solution to

where w_t is the eigenvalue corresponding to the non-zero eigenvalue of R_opt .

7. The system according to claim 3, wherein the rank of R_opt is greater than one, wherein w, is obtained from R_opt via a rank reduction method.

8. The system according to claim 4, wherein the rank of R_opt is greater than one, wherein w, is obtained from R_opt via a rank reduction method.

9. The system according to claim 7, wherein w₍ is the eigenvector corresponding to the dominant eigenvalue of R_opt .

10. The system according to claim 1, wherein the system incorporates frequency flat Rayleigh fading channels.

11. A multi-input multi-output (MIMO) communication system, comprising: a transmitter including N_t transmit antennas; and a receiver including N_r receive antennas, wherein at the transmitter the complex data

symbol s e C is modulated by a beamformer , and then

transmitted into a MIMO channel, wherein at the receiver, after processing with a combining vector , a sampled combined baseband signal is

, where is the channel matrix with its (i,j) th element h_tj

denoting the fading coefficient between the j^' th transmit and i th receive antennas, and

is the noise vector with its entries being independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and variance , wherein w_/ is

determined by solving

12. The system according to claim 1 1, wherein the N_t transmit antennas and N₁, receive antennas are in a quasi-static frequency flat fading channel.

13. The system according to claim 11, wherein subject to

is solved by applying the cyclic method, comprising:

a) set w _r to an initial value,

b) obtain w, , where

c) determine w_r , wherein w_r is the MRC and has the form

d) repeat b) and c) until a given criterion is satisfied.

14. The system according to claim 13, wherein w_r is set to an initial value of the left singular vector of H corresponding to its largest singular value.

15. The system according to claim 13, wherein the given criterion is a certain number of repetitions of b) and c).

16. The system according to claim 13, wherein the given criterion is a certain percentage improvement compared with the prior results.

17. The system according to claim 13, wherein H is known at the transmitter, wherein w, is determined at the transmitter.

18. The system according to claim 13, wherein H is known at the receiver, wherein w₍ is determined at the receiver.

19. The system according to claim 18, wherein w, is fed back from the receiver to the transmitter, wherein w, is denoted as

where

denoting the number of quantization levels and feedback index of

₁ , respectively, and where B₁ is the number of feedback bits for

_n wherein given B, B₁ is determined by:

let B_aι =

bHs for the first

N_s parameters and bits for the remaining (N₁ -I) - N₁ parameters .

20. The system according to claim 18, wherein w, is fed back from the receiver to the transmitter, wherein a code book is maintained at both the transmitter and receiver, wherein for a given code book } , the receiver first chooses the optimal

transmit beamformer as:

and the corresponding vector quantizer is denoted as w_opt = β(H) , the index of w_opt i^s fed

back from the receiver to the transmitter with log₂ N_v = B bits, where the receive combining

vector is w = ,^-V , where the codebook is found by:

generating a training set

(H₁, H₂,..., H^ } from a sufficiently large number

N of channel realizations, starting from an initial codebook iteratively update the codebook according to the following two criteria until no significant further improvement is observed: (1) NNC: for given codewords {w,}^ , assign a training element H_n to the z^' th region

^

{H JH^r ≥Kw^Vy ≠ z},

where S₁ , i = 1, 2, ... , N_v , is the partition set for the i th codeword w, •

(2) CC: for a given partition S₁ , the updated optimum codewords {w,}^ satisfy

for be Hermitian square root of R₁ ,

where , is determined by solving r

subject to

21. The system according to claim 20, wherein

subject to \ w, _m |\2 ^~ — ■, m = \, ..., N_t is solved by applying the cyclic method, comprising:

N_t a) set w_r to an initial value,

b) obtain , where

c) determine w_r , wherein w, is the MRC and has the form w, =

22. The system according to claim 20, wherein B is determined by for a givenD_v(i?) ,

wherein D_V(B) is the average degradation of the receive SNR,

is the variance of the channel, wherein the channel is a multi-input single output (MISO) channel.