General minimum Euclidean distance-based precoder for MIMO wireless systems

In this article, we investigate the linear precoder based on the maximization of the minimum Euclidean distance between two received data vectors. This new precoding matrix is expressed as the product of a power allocation matrix and an input-shaping matrix. The input-shaping matrix is selected as a normalized discrete Fourier transform-matrix, and the optimal power allocation depends on the channel characteristics. For each number of available datastreams, we propose a general form of the optimized precoding matrix. These forms are suitable for different transmit channels and especially for all rectangular quadrature amplitude modulation modulations. We show, in the simulation results, that the proposed precoder provides a significant improvement in terms of bit error rate performance compared to other traditional precoding strategies.


Introduction
In wireless communication, the systems that employ multiple antennas at both the transmitter and the receiver known as multiple-input multiple-output (MIMO) systems, not only offer the diversity and capacity gains, but also achieve higher link reliability in comparison with single antenna systems [1]. The idea of using multiple transceivers and receivers was first proposed by Bell Lab [2], and, then, has been utilized worldwide to adapt to various high-speed wireless transmissions.
In order to overcome the multi-path effect and improve the robustness of MIMO systems, a linear precoding transceiver can be used. Precoding is a processing technique that exploits the channel state information at transmitter (CSIT) by operating on the signal before transmission [3]. In fact, the transmitted vectors are premultiplied by a precoding matrix, which adapts to various forms of the channel knowledge. Various optimization criteria can be used to design a precoding matrix such as maximizing the output capacity [4], minimizing the bit error rate (BER) [5], maximizing the received signal-tonoise ratio (SNR) [6], minimizing the mean square error (MSE) [7], and maximizing the minimum singular value [8]. These precoders belong to an important set of linear *Correspondence: qngo@irisa.fr IRISA, University of Rennes 1, Rennes, France precoding techniques called as diagonal precoders. A specific precoding matrix, which follows the non-diagonal structure, was proposed in [9]. This design employ Schurconvex functions in order to optimize MSE-based or BER-based criteria.
In this article, we consider another non-diagonal linear precoder that optimizes the minimum Euclidean distance (max −d min ) between two received data vectors. This precoder improves the BER performance of the MIMO systems, especially when an ML detection is used at the receiver. It is because that the minimum distance is mutual information optimal for discrete input at hight SNR [10]. The optimal solution of max −d min precoder is proposed in [11,12] for two transmit datastreams with 4-QAM and 16-QAM modulations. By decomposing the channel into 2 × 2 eigen-channel matrices and optimizing the distance d min for each sub-system, Vrigneau et al. [13] proposed a sub-optimal precoder for large MIMO channels. However, this solution is also available for low-order Quadrature Amplitude Modulation (QAM) modulations. It is because the optimized solution for two-datastreams transmission depends on the symbol alphabet, the detection rule, and the characteristic of the transmit channel. Another suboptimal design of the max −d min precoder, which allows transmitting more than two independent datastreams and http://asp.eurasipjournals.com/content/2013/1/39 increasing the order of the modulations, is presented in [14]. But, the precoding scheme considers just a block-Toeplitz form of the channel matrix and, therefore, is only suitable for quasi-stationary MIMO channels. In addition, we can apply lattice invariant operations with the linear precoding in order to transform the transmit channel into a lattice generator matrix with large minimum distance separation [15].
The problem of high-order QAM modulations and the number of datastreams for minimum distance-based precoder was settled in this article. The precoding matrix is then factorized as the product of a diagonal power allocation matrix and an input-shaping matrix. The inputshaping matrix is selected as a discrete Fourier transform (DFT) matrix, and the power allocation matrix varies depending on the channel characteristics. At that time, the expression of the precoding matrix is less complex with only b variables corresponding to the b diagonal entries of the power allocation matrix. The idea of using the DFTbased matrix in precoding scheme was also proposed in [16], but this precoding design is only due to the power leakage suppression.
We propose, herein, a sub-optimal DFT-based precoding scheme which not only reduces the complexity, but also improves the minimum distance. A numerical approach is considered to indicate which difference vectors provide the minimum distances, and by equalizing these distances, it is possible to obtain the optimized precoding matrix. For each number of available datastreams, we propose a general precoding matrix for all rectangular QAM modulations. The simulation results confirm a significant BER improvement of our new precoder in comparison with other traditional precoding strategies.
The remainder of this article is organized as follows. Section "System overview" presents a brief introduction of virtual MIMO channel representations and linear precoding systems. The new parameterized form of the precoding matrix is described in "Parameterization of the precoding matrix". Section "Design of the precoding matrix" is devoted to the description of the new precoder which is based on the observation of the SNR-like matrix. In Section "Optimized precoder for rectangular QAM modulations", we propose general extensions of the precoder for large MIMO channels and all rectangular QAM modulations. Simulation results in comparison with other traditional precoders are presented in section "Simulation results". Finally, the article ends with "Conclusion" section.

System overview
We consider a MIMO system with n T transmit and n R receive antennas. For each Rayleigh fading channel used, b independent datastreams are transmitted, with b ≤ rank(H) ≤ min(n T , n R ). The received signal is expressed as where y is the b × 1 received symbols vector, s is the b × 1 transmitted symbols vector, η is the n R × 1 additive Gaussian noise vector, H is the n R × n T channel matrix, F is the n T × b precoding matrix, and G is the b × n R decoding matrix. When full channel state information (CSI) is available at both transmitter and receiver, the channel can be full-rank diagonalized by using a successive linear transformations presented in [11]. The precoding and decoding matrices are then decomposed as where One should note that the virtual channel matrix H v is now diagonal and defined by where ρ i stands for every subchannel gain and is sorted in decreasing order. In this article, an ML detection is considered at the receiver, and then, the decoding matrix G d has no effect on the performance. Hence, G d is consequently assumed to be an identity matrix of size b. The virtual system model can be then simplified as The precoding matrix F d is designed under the power constraint where E s is the average transmit power.

Parameterization of the precoding matrix
We now design a precoding matrix to improve the probability of error subject to the constraint of transmission powers. This design is difficult because it is rarely solvable in closed form: the solution depends on the symbol alphabet, the number of parallel datastreams, and channel characteristics. In general, the average error probability can be approximated by [17]  where N 0 is the variance of the white Gaussian noise η v , andd ij is the normalized Euclidean distance between two vector s i and s j at the receiver. Let us note N i the number of distancesd ij such thatd ij = d min , where d min denotes the minimum Euclidean distance and is defined by The probability of error in (6) can be now simplified as where M is the number of all possible transmitted vectors s, and It is observed that when an ML detection is considered at the receiver, a key to reduce the probability of error is maximizing the minimum Euclidean distance between received symbols. We can now formulate the design problem as follows By using a singular value decomposition, a linear precoder can be considered as a combination of an input shaper and a multimode beamformer with per-beam power allocation [3] where A and B * are b × b unitary matrices, and is a diagonal matrix. The orthogonal beam directions are the left singular matrix A, of which each column represents a beam direction (pattern). It is noted that the matrix A contains all eigenvectors of the matrix F d F * d , thus it is often referred to as eigen-beamforming. The matrix controls the power allocation on each beam. These powers correspond to the squared singular values of 2 . The right singular matrix B * concerns with the rotation and scaling of the input symbols on each beam and hence is referred to as the input-shaping matrix.
Let us definex a difference vector asx = s k − s l , with s k = s l , and the set which contains all possible difference vectors asX. The optimized criterion is then where R H denotes the channel covariance matrix and is given by . One should note that R H is a diagonal matrix since the virtual channel H v is already diagonalized.

Lemma 1.
Without loss of optimality, the left singular matrix A of the optimal precoder F d can always be chosen to coincide with an identity matrix.
From the result in Lemma 1, it follows that the max −d min precoder can be parameterized as where negative real numbers on the diagonal. The power constraint in (5) can be then rewritten as

Principle of the approach
Design optimizing the minimum Euclidean distance is difficult to deal with because of two reasons. First, the space of solution is large and exponentially proportional to the number of datastreams b. Second, the exact expression of max −d min precoder depends on many parameters such as the symbol alphabet or the characteristic of the virtual channel. Here, we propose a design that can come close to the desired goal. Based on (8), the formulation of the problem can be rewritten as Let us define an SNR-like matrix of F d as SNR (13), we can obtain a suboptimal but more general solution by realizing some properties of SNR(F d ). Scaglione et al. [8] proposed a suboptimal precoder which is based on the observation of the minimum eigenvalue of SNR(F d ). We present, herein, another suboptimal solution that considers the minimum diagonal element of the SNR-like matrix. Let us denote the diagonal elements of SNR(F d ) as δ k , we have In order to simplify the complexity of the solution, we assume that the function O(x i x j ) has little influence on the performance. Then, the design problem can be simplified as The criterion on the right-hand side of (15) has a lower bound where δ min denotes the minimum diagonal element of SNR(F d ). It is observed that increasing the minimum diagonal element δ min (SNR(F d )) to higher value will possibly obtain a suboptimal solution of the minimum distance criterion. Therefore, we first deal with δ min (SNR(F d )) and then maximize its value. By substituting (11) into the form of SNR(F d ), we obtain where is a diagonal matrix with non-negative real numbers on the diagonal. For any given ϒ, we first consider an optimal choice of the unitary matrix B which maximizes the minimum diagonal element of SNR(F d ). Such a matrix B is provided by the following lemma.

Lemma 2.
Given a b × b diagonal matrix ϒ whose diagonal elements are non-negative and a unitary matrix B of size b, then we have the following properties 2. The optimized value in (18) is provided by a normalized DFT-matrix where ω is a primitive b th root of unity, i.e., Proof. First, we prove that the right-hand side of (18) is the upper-bound for the left-hand side. Then, we show that the DFT-matrix D b can provide this upper bound.

Since B is a unitary matrix and
Furthermore, since the diagonal elements of ϒ are non-negative, those of BϒB * are non-negative, too. Given the set of b non-negative numbers that sum to M, the minimum number is obviously less than M/b. The left-hand side of (18) is, therefore, upper-bounded by If B * is selected as a DFT-matrix, i.e., the magnitude of each element of the DFT-matrix D b is equal to Lemma provides the key to obtain a sub-optimal solution for the problem of maximizing the minimum distance. One should note that the minimum Euclidean distances on the received constellation are always provided by some difference vectors. By equalizing these difference distances, we can obtain an analytical solution of the precoding matrix. Proposition 1. In order to equalize any difference distances, we can retain the input-shaping matrix B * , and change only the power allocation matrix .
Proof. We assume that, at the channelĤ v diag with real non-negative elements such that where κ is a constant. By substitutingσ i into the power constraint in (12), we obtain The Euclidean distance provided byx 1 is then Similarly, we get Since d 2 It is obvious that κ does not depend on the difference vectorsx 1 andx 2 . It means that for any number of difference vectors, we can equalize their difference distances by changing only the power allocation matrix .

Proposed model
Now, we present the key to design a new linear precoder. First, the precoding matrix F d is factorized as the product of the power allocation matrix and the scaling matrix B * .
Beside the role of controlling the power allocation on each stream, the matrix also determines how many virtual channels used to transmit signal. One should note the maximum number of activate virtual channels is upperbounded by the rank of matrix H. In other words, the non-null diagonal elements of are less than the number of datastreams b. We assume that the signal is transmitted on k subchannels, i.e., k ≤ b = rank(H). The matrix B * is then selected as a normalized DFT-matrix of size k. According to Proposition 1, the diagonal matrix depends on the channel characteristics, and has k positive real elements on the diagonal. We have total b different expressions of the precoding matrix F d corresponding to b precoders which pour powers on 1, 2, . . . , and b virtual subchannels.
The precoding system structure, which contains an input-shaping matrix and a power allocation matrix, is shown in Figure 1. Due to different forms of CSIT, the precoder first decides number of virtual subchannels used for transmission, and then maps the data-bits into k symbols. The method used for selecting the modulation will be discussed in Section "Range of definition". After that these symbols are pre-processed by a DFT block of size k. At the end of the precoder, the transmit signal is directly operated by a power distribution block, i.e., multiplied to a diagonal matrix .
The expression of the power allocation matrix depends on the symbol alphabet and the modulation used at the transmitter. In the following section, we propose a simple solution for one of the most common schemes: rectangular QAM.

Expressions of the precoding matrix
For a rectangular 4 m -QAM modulation, the transmit symbols belong to the set where M s = 2 3 (4 m − 1) and a, b ∈ {1, 3, . . . , 2 m − 1}. In the new precoding scheme, the input-shaping matrix B * is given by a DFT-matrix of size b. Our objective becomes to determine the matrix subject to the power constraint (12) in order to improve the minimum distance performance. To derive the analytical solution of the power allocation matrix in (11), we have to follow three main steps: In the case of the 4 m -QAM modulation, we have (2 m+1 − 1) 2b difference vectors. Some of them cannot provide the minimum distance. It is due to the collinear properties, for example, For that reason, by eliminating all collinear difference vectors, we can reduce significantly the space of solutions.
In step (ii), we first parameterize the power allocation matrix as the form of some trigonometric elements, such as, = E s diag (cos ψ 1 , sin ψ 1 cos ψ 2 , . . . , For each channel H v , a numerical search over all angles ψ i ∈ (0, π) in order to maximize the minimum distance shows us which difference vectors providing the distance d min . And finally, the analytic solution of the matrix can be derived by equalizing the difference distances given by these vectors. One should note that the set of difference vectors, which provide d min , is not fixed. It changes according to the channel values, and therefore lead to varies expressions of the matrix . For higher order QAM modulations, the form of our precoding matrix is more complicated. We propose, in this section, some characterized expressions of the new precoding scheme. These expressions are only suitable for the small dispersive channels (i.e., there is no much difference of SNRs between each subchannel). However, we can use them for all transmit channels because of their large gain in the performance of d min .
The number of non-null diagonal elements in represents the number of virtual-subchannels used for transmission. Let us denote the characterized expression of the precoder which enables powers on k subchannels as F k with k = 1, . . . , b. According to (25) and (27), the diagonal entries of the power allocation matrix can be defined by where φ j denotes the power coefficient of the jth virtual subchannel. It is obvious that the diagonal elements of are linearly proportional with φ j . We note that the value of φ j depends on the set of the optimization vectors. By equalizing the difference distances obtained by these vectors, we can derive the analytic values of φ j . Some normalized coefficients φ j are described in Table 1. At the end of this section, we show how to obtain this coefficient and propose a method for the general case F k (Table 1).

Precoder F 1
It is actually the max-SNR design which pours power on only the strongest virtual subchannel, i.e., = diag{ √ E s , 0, . . . , 0}. In order to retain the data-rate, the precoder F 1 can use a higher-order QAM modulation. In other words, it can transforms 4 m -QAM signals on b virtual subchannels into a rectangular 4 b.m -QAM signal on the first subchannel (detailed in Section "Range of definition"). The minimum distance provided by F 1 is defined by

Precoder F 2
This is the second expression of the N − d min precoder which is presented in our previous work [18]. We observe that the minimum distance is provided by two difference vectorsx 1 By substituting the DFT-matrix of size b into (11), we have The normalized distances provided byx 1 andx 2 are given by (33) Figure 2 shows the received constellation provided by the precoder F 2 . It is observed that whenever two received vectors are close on one virtual subchannel, they are distant on the other (e.g., points A and B).
An exciting property of the precoding matrix F 2 is that the average number of neighbors providing d min is less than that of the optimized solution presented in [11]. For that reason, it provides a slight improvement in term of BER performance compared to the optimized max −d min precoder.

Precoder F 3
This precoder pours power on three virtual subchannels, and has a characterized expression which is defined by A numerical approach shows that the minimum distance is provided by three difference vectorsx 1 One should note that three difference vectors do not always give the distance d min . They are only available for small dispersive channel, i.e., ρ 1 is not too higher than ρ 2 , and ρ 2 is not much higher than ρ 3 . It is reasonable to choose these vectors, because when the channel is large dispersive (ρ 3 ρ 2 or ρ 2 ρ 1 , for example), we can use the precoding matrix F 2 or F 1 .
By equalizing three difference distances provided by these vectors, we obtain The distance d min obtained by F 3 is then (36) Figure 3 plots the received constellation provided by the precoder F 3 in the case of 4-QAM. Like the case of the precoder F 2 , we observe that two received vectors processed by F 3 are close on one virtual subchannel but can be distant on the others (e.g., points B and C).

Precoder F 4
The characterized expression of the precoding matrix F 4 is given by A numerical search shows that the minimum distance is provided byx 1 Like the case of the precoder F 3 , by equalizing their difference distances, we obtain The distance d min obtained by F 4 is given by M s E s 4 9/ρ 1 + 5/ρ 2 + 1/ρ 3 + 1/ρ 4 . (39)

The general case F k
In the case of k parallel datastreams, a first numerical approach is first consider to determine all optimization vectors. This numerical search is implemented for all angles ψ i ∈ (0, π) of the matrix in (29) to optimize the distance d min . Let us denotex 1 ,x 2 , . . . ,x k as k difference vectors providing the minimum distance. The distance dx i is given by . . . , λ k ), and vector u i is given by By equalizing k difference distances, we have (k − 1) equations below where v i,j = |u 1(j) | 2 − |u i(j) | 2 with i = 2, . . . , k. For a 4 m -QAM modulation, it is noted that the difference vectoȓ x 1 is often defined byx 1 =[ 0, . . . , 0, 2] T , i.e., |u 1(j) | 2 = 4 http://asp.eurasipjournals.com/content/2013/1/39 with j = 1 . . . k. The power constrain in (12) can be now rewritten as Let us define λ =[ λ 1 , . . . , λ k ] T , and v 1, In conclusion, the power coefficients φ i are proportional to the entries of the vector λ which can be defined by λ = V −1 . The condition of the existence of the vector λ is that the matrix V is invertible. Whenx 1 =[ 0, . . . , 0, 2] T is one of the difference vectors providing the minimum distance, the distance d min is then defined by

Range of definition
To improve the BER performance of a MIMO system, we choose from these precoding matrices above the precoder that provides the highest minimum Euclidean distance. One should note that the data-rate of a precoder F i is different to each other's. For example, if we both use 4-QAM modulation for the precoders F 1 and F 2 , the bit-rate of F 2 is twice as that of F 1 . Therefore, we have to consider the data-rate of the b precoders when comparing their distances d min . The error probability in (7) can be re-expressed as where M is the number of alternative modulation symbols, B is the bandwidth, and f s is the symbol rate. For a given modulation order, by comparing the right-hand side of (48) corresponding to b precoders, we can obtain the range of definition for each precoding scheme. Another simple method to retain the data-rate is using different modulation for each precoder. Lets us come back to the example of the precoders F 1 and F 2 . If the 4-QAM modulation is used for the precoder F 2 , it means that two 2-bits symbols are transferred on two subchannels. Instead of transmitting like this, we can transfer one 4bits symbols (16-QAM) on the first virtual subchannels. Then, two minimum distances that correspond to F 1 using 16-QAM and F 2 using 4-QAM are compared in order to determine the range of definition for two precoders F 1 and In other words, if d 2 F 1 > d 2 F 2 or ρ 1 /ρ 2 > 7: the precoder F 1 is chosen, and for ρ 1 /ρ 2 < 7: the precoder F 2 is selected. Other precoders can be implemented in a similar way.

Comparison of minimum Euclidean distance
In this section, we indicate the improvement of the proposed precoder in terms of the minimum Euclidean distance compared to diagonal precoders. Indeed, the minimum Euclidean distance provided by a diagonal precoder is One should note that the minimum Euclidean distance is obtained when the two vectors s and r are different from only a symbol. The minimum Euclidean distance of F d is then given by It is noted that the diagonal entries of H v = diag( √ ρ 1 , . . . , √ ρ b ) are sorted in decreasing order, i.e., By comparing all of elements on right-hand side of (51), the minimum distances corresponding to some traditional precoders such as beamforming, max −λ min [8], Water-filling [4], and MMSE [7], can be determined. Table 2 illutrates the distance d min obtained by these diagonal precoders in comparison with our proposed precoder, in which (x) + def = max(x, 0). The normalized minimum distances provided by these precoder above, in the case of b = 2 virtual subchannels and 4-QAM modulation, are illustrated in Figure 4. It is observed that our precoder provides a large improvement in terms of d min compared to the diagonal precoders. In comparison with the max −d min precoder presented in [11], the proposed precoder has a small difference in the minimum distance. However, its average number of neighbors providing the minimum distance is less than that of http://asp.eurasipjournals.com/content/2013/1/39 the max −d min precoder [18]. According to that improvement, an enhancement in terms of BER is expected for our new precoder (Table 2).

BER performance
In this section, the BER performance of the proposed precoder is considered in comparison with other traditional precoding strategies. The proposed precoder obtains a significant improvement of BER in comparison with the diagonal precoders: Water-filling, MMSE, and max −λ min . A gain about 6 dB can be observed at high SNR. Furthermore, as discussed above, our precoder has the number of neighbors providing d min less than that of the optimal solution max −d min in [11], although it has a small difference in terms of d min . Therefore, the new precoder provides a slight BER improvement compared to the max −d min solution. The BER performance with respect to SNR for two transmit datastreams and 4-QAM modulation is plotted in Figure 5. The optimal solution for max −d min precoder is presented in [11,12], but it is only available for two transmit datastreams with 4-QAM and 16-QAM modulations. By decomposing the channel into 2 × 2 eigen-channel matrices and optimize the distance d min for each pair of datastreams, Vrigneau et al. [13] proposed a sub-optimal precoder for large MIMO channels. This extension is split into four steps: virtual diagonalization of the channel, combination in pairs of sub-channels, application of the optimal 2D max −d min solution, and power allocation on each sub-system. However, this solution is also suitable for low-order QAM modulations. A main advantage of our new precoder is that the solution is available for all rectangular QAM-modulations and for any number of datastreams.
For large MIMO simulations, we consider a system with n T = 5 transmit and n R = 4 receive antennas. The bitstreams are separated into b = 4 independent virtual subchannels, and the channel matrix H is i.i.d. zero-mean complex Gaussian. For each SNR, the precoders are optimized for about 30,000 random matrices H. It is observed in Figure 5 that the BER performance of the max −λ min solution is better than those of MMSE and Water-filling. Therefore, the max −λ min precoder is chosen to compare with our proposed precoder. Beside that some sophisticated transceivers such as: the Schur-convex ARITH-BER design [9], the linear precoder using Decision Feedback Equalization (DFE) [19], and the linear transceiver with bit allocation [20] are also mentioned in the comparison with our precoder. Figure 6 illustrates the BER performance for MIMO (5,4) systems using 4-QAM modulation. The comparison of the proposed precoder and other schemes for b = 4 transmit datastreams shows that the performance is significantly enhanced in terms of BER. It is obvious that our precoder performs much better than diagonal  DFE [18] ARITH−BER [9] max−λ min [8] MBitRate [19]  precoders such as max −λ min and Water-filling. Furthermore, we observe that the new precoder also presents a significant improvement of BER compared to the DFE, the Schur-convex ARITH-BER, and the maximum bit-rate solutions, especially when the SNR is high. The new precoder was found to be better than E − d min schemes and this is due to the fact that not only the minimum distance, but also the number of neighbors providing d min is taken into consideration. We also consider, in this section, the impact of imperfect CSI estimation on the BER performance of the proposed precoder. Figure 7 illustrates the BER performance with respect to SNR in the case of perfect CSI and imperfect CSI estimation. The estimated channel matrix of imperfect CSI system can be modeled as H est = H+H err , where H err represents the channel estimation error. The optimal training signals for the MIMO-OFDM channel estimation can be found in [21]. One should note that if the transmit channel is quasi-stationary, i.e., remains constant for several symbol periods, the precoding across time could be replaced by precoding across subcarriers [14]. In this simulation, we assume that the entries of H err are complex Gaussian i.i.d random with mean zero and variance σ err = 0.3 σ , where σ is the variance of the complex Gaussian entries of H. It is observed that the BER performance of our precoder decreases at high SNR, but it still remains better than the other precoding strategies. Furthermore, the BER enhancement obtained by the proposed precoder is much better than the case of full CSI in comparison with the E-d min solution: a gain of 2 dB in SNR can be observed at BER = 10 −5 .

Conclusion
We proposed, in this article, a new linear precoder that is based on the maximization of the minimum Euclidean distance between two received data vectors. The suboptimal design was obtained by observing the SNR-like matrix of the precoding matrix. An approximation of the minimum distance is derived, and its maximum value was obtained by maximizing the minimum diagonal element of the SNR-like matrix. We then showed that the minimum diagonal element can be attained by a specific set of the precoder. Firstly, the precoding matrix is parameterized as the product of a diagonal power allocation matrix and an input-shaping matrix. The input-shaping matrix concerns with the rotation and scaling of the input symbols on each virtual subchannel. We demonstrated that the minimum diagonal entry of the SNR-like matrix is obtained from a special choice of the input-shaping matrix, i.e., a DFT-matrix, and our objective becomes determining the power allocation matrix . As its name implies, the matrix decides how many subchannels are used by the precoder for data transmission. For each number of available datastreams, we proposed a simple characterized expression of the precoding matrix for all rectangular QAM modulations.
We also presented some performance comparisons to demonstrate that the proposed precoder obtains a significant improvement in terms of BER. The improvement may be more than several dB at reasonable BER levels. In comparison with the optimal max −d min solution, our proposed precoder also provides a slight BER improvement. One main advantage of our design is that the solution can be available for all rectangular QAMmodulations and for any number of datastreams. It is because that the precoder has a simple analytic form, and the space of the solution is smaller than the full design of minimum distance based precoders.