**5.1. Null space projection**

allocation is given by the water-filling algorithm and according to [5] the information rate


1 ( ( )) log( ) *ZFBF*

users in *S* are allocated with nonzero power, the water level has the compact form [10], [18]:

Suboptimal throughput maximization in Gaussian BC channels has been proposed in several works [5], [18], [19] based on the QR-type decomposition of the channel matrix **H**(*S*)=**L**(*S*)**Q**(*S*) obtained by applying Gram-Schmidt orthogonalization (GSO) [20]. **L**(*S*) is a lower triangular matrix and **Q**(*S*) has orthonormal rows. The beamforming matrix given by **W**(*S*)=**Q**(*S*)<sup>H</sup>

*ii pi*

mation is sent to users *k* + 1, ⋯, *K*. In order to eliminate the interference component

paper encoding, where *Ii* is non-causally known. This precoding scheme was proposed in [5] and the authors showed that the precoding matrix forces to zero the interference caused by users *j* >*i* on each user *i*, therefore this scheme is called zero-forcing dirty-paper (ZFDP) coding. The information rate achieved with optimum **P** in (3) under the ZFDP scheme is given

*sj*


1 ( ( )) log( ) *ZFDP*

*i R d*


The sum rate maximization problem (4) can be solved by fixing the precoder structure and power allocation method. Under ZF-based precoding the performance strongly depends on

= <sup>=</sup> å S

*si* + ∑ *j*<*i l ij p <sup>j</sup>*

( )

m<sup>+</sup>

*i*

*i R b*

= <sup>=</sup> å S

( )

m<sup>+</sup>

*i*

*i*∈*S* (*μ* - *bi*


**H** S (6)

*sj* + *ni*

**H** S (7)

*ii* and *μ* is the solution to the water-filling

for *i* =1, ⋯, *k* are obtained by successive dirty


, *i* =1, ⋯, *k* while no infor‐

achieved with optimum **P** in (3) is given by :

30 Contemporary Issues in Wireless Communications

*ii*

<sup>2</sup> <sup>+</sup> *<sup>P</sup>*) <sup>=</sup> <sup>1</sup>

generates a set of interference channels *yi* =*l*

is *pi* =(*μbi* - 1)+, the water level *μ* is chosen to satisfy ∑

<sup>|</sup>*<sup>S</sup>* <sup>|</sup> (*P* + ∑ *i*∈*S bi* -1 ).

of the *i*th user, the signals *p <sup>j</sup>*

*ii*|2 is the squared absolute value of *l*

where *bi* =( **<sup>H</sup>**(*S*)**H**(*S*)<sup>H</sup>

<sup>|</sup>*<sup>S</sup>* <sup>|</sup> ( **<sup>W</sup>**¯(*S*) *<sup>F</sup>*

**4.2. Zero forcing dirty paper**

*<sup>μ</sup>* <sup>=</sup> <sup>1</sup>

*Ii* = ∑ *j*<*i l ij p <sup>j</sup> sj*

by [5]:

where *di* =|*l*

equation ∑

*i*∈*S*

(*μ* - *di*

**5. Metrics of spatial compatibility**

Considering a given set of users *S* with channels **h***<sup>i</sup>* ∀*i* ∈*S* and for ZFBF the effective channel gain of the *i*th selected user defined in Section 4.1 is given by [5]:

$$b\_i = \frac{1}{\left[\left(\mathbf{H}(\mathbf{S})\mathbf{H}(\mathbf{S})^H\right)^{-1}\right]\_{i,i}} = \left\|\mathbf{h}\_i \mathbf{Q}\_{\mathbf{V}\_i}\right\|^2 = \left\|\mathbf{h}\_i\right\|^2 \sin^2 \theta\_{\mathbf{V}\_i \mathbf{h}\_i} \tag{8}$$

where **P***<sup>i</sup>* =**H**˜ *i H* (**H**˜ *i* **H**˜ *i <sup>H</sup>* )-1**H**˜ *i* is the projector matrix onto *<sup>i</sup>* <sup>=</sup>*Sp*(**<sup>H</sup>**˜ *i* ) the subspace spanned by the rows of the aggregate interference matrix **H**˜ *<sup>i</sup>* = **h**<sup>1</sup> *<sup>H</sup>* , <sup>⋯</sup>, **<sup>h</sup>***i*-1 *<sup>H</sup>* , **<sup>h</sup>***i*+1 *<sup>H</sup>* , <sup>⋯</sup>, **<sup>h</sup>**|*<sup>S</sup>* <sup>|</sup> *<sup>H</sup> <sup>H</sup>* associated with user *i*. **Q***<sup>i</sup>* =**I** - **P***<sup>i</sup>* is the projector matrix onto the null space of *i*. The operation in (8) is equivalent to the projection of **h***<sup>i</sup>* onto the null space spanned by the channel components of **H**˜ *i* illustrated in Fig. 2. Notice that **h***<sup>i</sup>* <sup>2</sup> in (8) is affected by the weight sin<sup>2</sup> *<sup>θ</sup>i***h***<sup>i</sup>* which is the squared sine of the angle between the channel vector **h***<sup>i</sup>* and the subspace spanned by the components of **H**˜ *i* . The weight sin<sup>2</sup> *<sup>θ</sup>i***h***<sup>i</sup>* is referred in the literature as the *projection power loss factor* since it will affect the effective amount of power that is transmitted over the *i*th link. Using the properties of water-filling and the strong relationship between the sum rate maximization and the maximization of the terms *bi* we elaborate a compact formulation of the maximization of metric (8). Theoretical results in [9] show that for the MISO BC system a meaningful metric to estimate the achievable performance of *S* is given by ∏ *i*∈*S bi* under certain constraints over the water level *μ*. However, for single antenna receivers the performance gap between the sum and the product of the terms *bi* is negligible and hereafter we analyze the metric ∑ *i*∈*S bi* which is equivalent to a matrix trace operation. Let **H**¯(*S*)=**H**(*S*)**H**(*S*)*H* and **<sup>H</sup>**˙ (*S*)=**H**¯(*S*)-1 be a Wishart matrix and its respective inverse which characterize the interaction of all user channels in *S*. Considering the definitions in (6) and (8) the set of users that achieves a suboptimal solution to problem (4) maximizing the sum of the effective channel gains is given by the set of users that optimally solves of the following combinatorial problem:

$$\mathbf{S}\_o = \arg\min\_{\mathbf{S} \in \Omega : \|\mathbf{S}\| = N\_t} \text{tr}(\dot{\mathbf{H}}(\mathbf{S})) \tag{9}$$

where tr(∙) denotes the trace operator. The optimum set *S<sup>ω</sup>* for the NSP metric is unique and it will contain the users that maximize ∑ *i*∈*S* **h***i* 2sin2 *<sup>θ</sup>i***h***<sup>i</sup>* . The selection of *S* based on the NSP in (9) yields a close-to-optimal solution to problem (4) for a ZF-based precoding under the following conditions: *P* >*P*0, *K* > *Nt*, or large values of both *K* and *Nt*. The term *P*0 is a critical SNR value that depends on **H**(*S*) in order to meet the cardinality constraint over *S* [5]. The evaluation of the metric (8) is not unique and several algorithms described in next section use different operations to compute or approximate it in order to define algorithms that solve (9) with less computational complexity than the exhaustive search, especially when *K* →*∞*. Next we present several methods to evaluate the NSP for ZFBF and ZFDP. The application of each one of these methods lies in the complexity required to evaluate them and the available CSI at the BS. In other words, each method represents a trade-off between accuracy of the NSP evaluation and the required CSI at the transmitter.

#### *5.1.1. Orthogonal projector for ZFBF*

The computation of **Q***<sup>i</sup>* is not unique and different forms to evaluate such matrix can be efficiently used in different contexts, i.e., depending on the available CSI at the transmitter, the required computational complexity, and the deployment. Let us decompose matrix **H**˜ *i* of the *i*th user by means of singular value decomposition (SVD) [17] as follows:

$$\tilde{\mathbf{H}}\_i = \mathbf{U}\_{\tilde{\mathbf{H}}\_i} \Sigma\_{\tilde{\mathbf{H}}\_i} \left[ \tilde{\mathbf{V}}\_{\tilde{\mathbf{H}}\_i} \mathbf{V}\_{\tilde{\mathbf{H}}\_i} \right]^H \tag{10}$$

where **V**˜ **H**˜ *i* contains the *Nt* - *<sup>r</sup>* basis of the null space of **H**˜ *i* and *<sup>r</sup>* <sup>=</sup>*rank*(**<sup>H</sup>**˜ *i* ). The orthogonal projector matrix is given by **Q***<sup>i</sup>* =**V**˜ **H**˜ *i* **V**˜ **H**˜ *i <sup>H</sup>* and the set *S<sup>ω</sup>* in (9) maximizes the objective function ∑ *i* **h***i* **Q***<sup>i</sup>* 2. In some scenarios described in the following sections, it is assumed that the BS knows the basis of *<sup>i</sup>* for any user *i* ∈*S*. Let {**v** *<sup>j</sup>* } *<sup>j</sup>*=1 *<sup>r</sup>* be the column vectors of **V**¯ **H**˜ *i* defined in (10) and the NSP in (8) for the *i*th user can be computed as follows:

maximization of metric (8). Theoretical results in [9] show that for the MISO BC system a

constraints over the water level *μ*. However, for single antenna receivers the performance gap

**<sup>H</sup>**˙ (*S*)=**H**¯(*S*)-1 be a Wishart matrix and its respective inverse which characterize the interaction of all user channels in *S*. Considering the definitions in (6) and (8) the set of users that achieves a suboptimal solution to problem (4) maximizing the sum of the effective channel gains is given

which is equivalent to a matrix trace operation. Let **H**¯(*S*)=**H**(*S*)**H**(*S*)*H* and

S S **H** (9)

is not unique and different forms to evaluate such matrix can be

*<sup>i</sup>* = S%% % % % **UHH H H VH <sup>V</sup>** (10)

and *<sup>r</sup>* <sup>=</sup>*rank*(**<sup>H</sup>**˜

*i* ). *i*∈*S bi*

. The selection of *S* based on the NSP

is negligible and hereafter we analyze the

under certain

*i* of the

meaningful metric to estimate the achievable performance of *S* is given by ∏

by the set of users that optimally solves of the following combinatorial problem:

*i*∈*S* **h***i*

ÌW =

w

:| | arg min tr( ( )) *Nt*

where tr(∙) denotes the trace operator. The optimum set *S<sup>ω</sup>* for the NSP metric is unique and

in (9) yields a close-to-optimal solution to problem (4) for a ZF-based precoding under the following conditions: *P* >*P*0, *K* > *Nt*, or large values of both *K* and *Nt*. The term *P*0 is a critical SNR value that depends on **H**(*S*) in order to meet the cardinality constraint over *S* [5]. The evaluation of the metric (8) is not unique and several algorithms described in next section use different operations to compute or approximate it in order to define algorithms that solve (9) with less computational complexity than the exhaustive search, especially when *K* →*∞*. Next we present several methods to evaluate the NSP for ZFBF and ZFDP. The application of each one of these methods lies in the complexity required to evaluate them and the available CSI at the BS. In other words, each method represents a trade-off between accuracy of the NSP

efficiently used in different contexts, i.e., depending on the available CSI at the transmitter, the required computational complexity, and the deployment. Let us decompose matrix **H**˜

[ ] *ii i i*

*H*

*i*

*i*th user by means of singular value decomposition (SVD) [17] as follows:

contains the *Nt* - *<sup>r</sup>* basis of the null space of **H**˜

2sin2 *<sup>θ</sup>i***h***<sup>i</sup>*

= & S S

between the sum and the product of the terms *bi*

32 Contemporary Issues in Wireless Communications

it will contain the users that maximize ∑

evaluation and the required CSI at the transmitter.

*5.1.1. Orthogonal projector for ZFBF*

The computation of **Q***<sup>i</sup>*

where **V**˜

**H**˜ *i*

metric ∑ *i*∈*S bi*

$$\left\|\mathbf{h}\_{i}\mathbf{Q}\_{\mathbf{V}\_{i}}\right\|^{2} = \left\|\mathbf{h}\_{i}\left(\mathbf{I} - \sum\_{j=1}^{r} \mathbf{v}\_{j}\mathbf{v}\_{j}^{H}\right)\right\|^{2} \tag{11}$$

The NSP operation in (11) can be also computed by applying GSO to **H**(*S*) as in [21] which represents a lower computational complexity than the SVD approach [22]. Using the basis of *i* the magnitude of the NSP operation is given by **h***<sup>i</sup>* **Q***<sup>i</sup>* <sup>2</sup> = **h***<sup>i</sup>* <sup>2</sup> - **h** ^ *i* 2 where **h** ^ *i* is the projection of **h***<sup>i</sup>* onto each one of the orthogonal basis of *i* given by [20]:

$$\hat{\mathbf{h}}\_i = \sum\_{j=1}^r \frac{\|\mathbf{h}\_i\|\cos\theta\_{\mathbf{h}\_i\mathbf{v}\_j}}{\|\mathbf{v}\_j\|} \mathbf{v}\_j^H \tag{12}$$

where the term cos *θ***h***<sup>i</sup>* **v** *j* represents the coefficient of correlation between the vectors **h***<sup>i</sup>* and **v** *<sup>j</sup>* defined as [17]:

$$\cos\theta\_{\mathbf{h}\_i\mathbf{v}\_j} = \frac{\mathbf{h}\_i\mathbf{v}\_j^H}{\|\mathbf{h}\_i\| \|\mathbf{v}\_j\|}\tag{13}$$

The domain of the coefficient is 0≤*η***h***<sup>i</sup>* **v** *j* =cos *θ***h***<sup>i</sup>* **v** *j* ≤1 and *θ***h***<sup>i</sup>* **v** *j* = *π* <sup>2</sup> means perfect spatial orthogonality. The NSP computation is not unique and different matrix operations can be use to evaluate it. Using the full channel matrix **H**(*S*) and **H**˜ *i* for all *i* ∈*S* the block matrix deter‐ minant formula to compute det(**H**(*S*)**H**(*S*)*<sup>H</sup>* ) reads [23]:

$$\det\left(\mathbf{H}(\mathbf{S})\mathbf{H}(\mathbf{S})^H\right) = \det\left(\tilde{\mathbf{H}}\_i\tilde{\mathbf{H}}\_i^H\right) \left\|\mathbf{h}\_i\mathbf{Q}\_{\mathbf{V}\_i}\right\|^2 \tag{14}$$

The orthogonal projection defined in (8) has a direct relationship with the correlation coeffi‐ cients defined in (13). The normalized power loss experienced by the *i*th channel when it interacts with the subspace *i* is called the coefficient of determination given by [17]:

$$
\Delta\_{\mathbb{Q}|\mathbf{h}\_i}^2 = \frac{\mathbf{h}\_i \mathbf{P}\_{\mathbb{Q}\_i} \mathbf{h}\_i^H}{\mathbf{h}\_i \mathbf{h}\_i^H} \tag{15}
$$

where ∆*i***h***<sup>i</sup>* 2 measures how much the vector **h***<sup>i</sup>* can be predicted from the channel vectors of **H**˜ *i* . Notice that from (8) and (15) the projection power loss factor of **h***<sup>i</sup>* due to the projection onto the null space of *i* is equivalent to 1 - <sup>∆</sup>*i***h***<sup>i</sup>* 2 which can be evaluated as follows [17]:

$$(1 - \Delta\_{\mathsf{V}\_{i}\mathbf{h}\_{i}}^{2} = (1 - \eta\_{\mathbf{h}\_{i}\pi(1)}^{2}) \dots (1 - \eta\_{\mathbf{h}\_{j}\pi(k)|\pi(1)\dots\pi(k-1)}^{2}) \tag{16}$$

where *π*(*k*) is the *k*th ordered element of **H**˜ *i* and *η***h***<sup>i</sup> <sup>π</sup>*(*<sup>k</sup>* )|*π*(1)⋯*π*(*<sup>k</sup>* -1) is the partial correlation between the channel vector **h***<sup>i</sup>* and the selected vector associated with *π*(*k*) eliminating the effects due to *π*(1)*π*(2)⋯*π*(*k* - 1). Using multiple regression analysis it is possible to evaluate **h***i* **Q***<sup>i</sup>* <sup>2</sup> = **h***<sup>i</sup>* 2 (1 - <sup>∆</sup>*i***h***<sup>i</sup>* <sup>2</sup> ) by extracting the partial correlation coefficients from the correlation coefficients choosing one user order *π* out of (|*S*| - 1)! permutations of the users in *S* [17]. A different approach can be applied if for a given set of channels **h** *<sup>j</sup>* ∀ *j* ∈*S* the orthogonal projector matrix of each channel is known so that **Q** *<sup>j</sup>* =**I** - **h** *<sup>j</sup> <sup>H</sup>* (**<sup>h</sup>** *<sup>j</sup>* **h** *j <sup>H</sup>* )-1**<sup>h</sup>** *<sup>j</sup>* .

**Figure 2.** The spatial relationship between the components of vector **h***<sup>i</sup>* and *<sup>i</sup>* .

From [24] we have the following result:

$$\mathbf{Q}\_{\mathbb{V}\_i} = \left( \prod\_{j \neq i, j \in \mathbf{S}} \mathbf{Q}\_j \right)^n, n \to \infty \tag{17}$$

which establishes that the orthogonal projector matrix onto *<sup>i</sup>* can be approximated by recurrently projecting onto independent orthogonal subspaces such that their intersection strongly converges to **Q***<sup>i</sup>* as *n* grows. The NSP measured by tr(**H**˙ (*S*)) has been used by several user selection algorithms either by avoiding the exhaustive search required to solve (9) (e.g., [18], [19], [21]) or by using approximations of metric (8) (e.g., [25], [26]).
