Interference alignment based on adaptive eigenmodes in a multiple-input,multiple-output cognitive radio network

Abstract

This article aims to optimize the information rate of a cognitive radio network with multiple secondary users. A primary user rate optimization approach based on dichotomy of the degree of freedom is proposed, where the primary users’ eigenmodes are adjusted according to its rate requirement. In order to provide a higher sum rate of secondary users, two interference alignment schemes are presented. The first one is an interference sub-space alignment scheme, which aims to align the sub-spaces spanned by interference from other secondary users with the sub-space spanned by interference from primary user. However, interference sub-space alignment may not be favorable in low signal-to-interference ratio region due to the negligence of the influence of noise. Thus, an iterative interference alignment scheme which maximizes the secondary system sum rate based on Grassmann manifold is developed. To accelerate the convergence speed, the objective function in Grassmann manifold is transformed into two parts without the inversion operation using the extensions of the Minkowski inequality. Simulation results show that interference sub-space alignment is more effective than Grassmann manifold to mitigate interference in the system with more secondary users. We further validate the effectiveness of Grassmann manifold and interference sub-space alignment in comparison with the existing schemes employing a water filling algorithm.

Introduction

With the rapidly increasing data rate requirements of mobile users on multimedia applications, the accessible radio spectrum is becoming critically scarce. To address this issue, the cognitive radio (CR) concept was introduced to improve spectral efficiency in wireless systems.^1–5 CR allows secondary users (SUs or cognitive users) to share the spectrum with the primary users (PUs or noncognitive users) under the condition that there is no or limited interference from SUs to the PU receiver. It has been a challenging problem to improve the performance of SUs while guarantee the minimum rate requirement of PU in CR networks. Effective interference management scheme called interference alignment (IA) was proposed in Cadambe and Jafar. ⁶

The main idea of IA is to mitigate the interference by dividing the received signal space into two sub-spaces, one for desired signals and one for interfering signals. At the initial research stage, IA schemes were presented in the closed-form expressions in the literature.^6–9 However, these closed-form solutions were only achieved under certain conditions. As an alternative, iterative methods were developed to numerically solve the alignment problem.^10–15 In Gomadam et al., ¹⁰ an iterative algorithm called interference leakage minimization was proposed for the multiple-input, multiple-output (MIMO) interference channel, which aimed to minimize the sum of interference powers at each receiver. This algorithm was extended to MIMO cellular networks in Zhuang et al. ¹¹ According to Jafar, ¹³ the degree of freedom (DoF) was formulated based on a high signal-to-noise ratio (SNR) approximation. The authors proposed a rank-constrained rank minimization (RCRM) in Papailiopoulos and Dimakis, ¹⁴ which minimized the nuclear norm of the interfering space with a full rank constraint on the desired signal space. A weighted version of nuclear norm minimization for finding the optimal encoders was further presented in Sridharan and Yu. ¹⁵ Apart from the above schemes, the literature ¹⁶ analyzed the achievable DoF over space, time, and frequency. An IA scheme was proposed in Vahdani and Falahati, ¹⁷ which attained higher multiplexing gain using a lot lower number of channel extension.

Due to the signal separation between the interference and the desired information in multi-user interference channel, the IA scheme has been exploited in CR networks. Since the water filling algorithm (WFA) applied at a PU can leave some of its spatial dimensions unoccupied due to the power constraint, the secondary system reuses these eigenmodes for data transmission, as proposed in the literature.^18,19 However, a PU using WFA might always spread its power among all its available eigenmodes at the moderate-to-high SNR regime, leaving insufficient eigenmodes for SUs at the moderate-to-high SNR regime and thus restricting SUs in the low SNR regime. Actually, SUs could still transmit data in the moderate-to-high SNR regime as long as the interference imposed by SUs on the PU was under the tolerance of PU. To this end, SUs in Cumanan et al. ²⁰ optimized their transmit covariance matrix subject to an explicit PU rate constraints. Spatial shaping constraints considering the structure of interference was proposed by Lameiro et al.^21,22 Under the assumption that the interference imposed by PU at each SU could be neglected, Lameiro et al. ²¹ extended the interference leakage minimization algorithm to incorporate the spatial interference shaping constraints. Lameiro et al.^21,22 improved and presented a more general optimization framework to design the spatial shaping constraints. Guo et al. ²³ proposed an eigenmodes constraint with which the number of eigenmodes used by the PU was adaptively adjusted by the rate requirement. However, Guo et al. ²³ only studied the scenario where a MIMO PU shared its spectrum with one MIMO SU. In contrast to the above IA-based CR algorithms, a joint design of PU and SU was put forward in Men et al., ²⁴ where the CR system was considered as a $(K+1)$ interference network. Nevertheless, the proposed scheme in Men et al. ²⁴ did not consider the interference from PU to SUs. In Abdulkadir et al., ²⁵ a space-time opportunistic IA technique was proposed for the overlay CR network, where the PU used space-time WFA to optimize its transmission and, in the process, freed up unused eigenmodes that could be exploited by the SU. Noam and Goldsmith ²⁶ proposed an algorithm that enables the secondary transmitter to learn the interference channel of the primary receiver by measuring a monotonic function of the interference to the primary receiver. Tsinos and Berberidis ²⁷ put forward a blind opportunistic IA method with which the SU is able to compute blindly the required channel state information (CSI) without interaction with the primary network.

In this article, considering a system that a PU can coexist with multiple SUs, we propose a PU rate optimization approach based on dichotomy of DoF (DDA). As we have mentioned before, if the PU directly adopts WFA in the high SNR region, there would be less DoFs for the secondary system while the PU still transmits data with the maximum data rate. Considering that the PU just needs to meet the specific transmission rate or certain bit error rate to complete the communication process in some situation, the total DoFs of the CR network are obviously wasted. In order to solve this problem, this article presents the DDA algorithm in which the PU participates in part of the collaboration, leaving more DoFs for the secondary system. The secondary precoders in most IA schemes have been designed mainly to eliminate the interference from SUs to the PU. For any SU applying such an approach, the signals from other SUs cannot be aligned to the same sub-space. As a result, the interference among SUs can lead to poor performance of the secondary system. Taking this into consideration, this article employs a sub-matrix optimization method, where the precoding matrix is divided into two sub-matrices. The first sub-precoder eliminates the interference from SUs to the PU while the second sub-precoder alleviates the interference among SUs.

In order to suppress interference effectively, we propose an interference sub-space alignment (ISA) scheme to design the second sub-precoder. Different from the existing methods that minimizing the distances among the sub-spaces of different interfering signals, ISA aligns the sub-spaces spanned by interference from other SUs with the sub-space spanned by interference from PU and thus dispensed with the process of iteration. Although ISA can effectively mitigate the interference at each secondary receiver, it may not be favorable in the low SNR region due to the fact that the performance at low SNR is dominated by both interference and noise, whereas ISA ignores the influence of noise. To alleviate this drawback, this article proposes another approach for the secondary system, namely, secondary system sum rate maximization based on the Grassmann manifold (GSR). In GSR scheme, the second sub-matrices of precoders and decoders are designed to maximize the secondary system sum rate, and the optimization is introduced on GSR to obtain a faster convergence. However, if we directly take the sum rate as the cost function, then there exist plenty of inverse operations in the process of derivation. In order to reduce the complexity of GSR, we utilize the extensions of the Minkowski inequality to transform the sum rate formula into two parts without inversion operation. In some scenarios, the number of unused eigenmodes may not guarantee that there is at least one DoF obtained by each SU. To solve this problem, the proposed algorithms decrease the number of SUs and allow one or several “lucky” SUs to stay active for a long time whereas other SUs remain silent. Simulation results show that the achievable data rate of PU is higher than its minimum rate requirement by applying DDA. ISA is a more effective method in comparison with GSR for the system with more SUs. Numerical simulation results also validate the effectiveness of the GSR and ISA in comparison with the existed schemes employing WFA.

The rest of this article is organized as follows. Section “System model” introduces the system model. The details of the DDA scheme are described in section “DDA scheme for the PU.” Section “IA for secondary MIMO links” provides the ISA and GSR schemes for secondary MIMO links. Simulation results are given in section “Simulation results.” The article concludes in section “Conclusion.”

Notation

Throughout the article, we use bold uppercase and lowercase letters to denote matrices and vectors, respectively. ${\mathbf{A}}^H$ , $\mathrm{rank}(\mathbf{A})$ , $\Vert \mathbf{A}{\Vert }_F$ , $\mathrm{tr}\{\mathbf{A}\}$ , and $\mathcal{N}(\mathbf{A})$ denote the Hermitian transposition, rank, Frobenius norm, trace, and null space of the matrix $\mathbf{A}$ , respectively. $\Vert \mathbf{A},\mathbf{B}{\Vert }_M$ denotes the distance between the two sub-spaces spanned by $\mathbf{A}$ and $\mathbf{B}$ . ${\mathbf{I}}_n$ stands for the $n\times n$ identity matrix and $0_{m\times n}$ stands for the $m\times n$ null matrix. $\left|a\right|$ represents the absolute value of a. $\lceil a\rceil$ and $\lfloor a\rfloor$ represent $\mathrm{ceil}(a)$ and $\mathrm{floor}(a)$ , respectively. $(a{)}^{+}=max\{0,a\}$ and $E\{\}$ represents the statistical expectation.

System model

Consider a $(K+1)-\mathrm{user}$ MIMO interference channel as shown in Figure 1, which consists of one PU and K SUs. PU is given index 0. Definition for the symbols used in the figure are provided in Table 1.

Figure 1.

System model.

Table 1.

Symbols definition.

Symbols	Definition
$M_t$	The number of transmit antennas
$M_r$	The number of receive antennas
${\mathbf{H}}_{00}$	The channel coefficient between primary transmitter and primary receiver
${\mathbf{H}}_{0k}$	The channel coefficient between kth secondary transmitter and primary receiver
${\mathbf{H}}_{k0}$	The channel coefficient between primary transmitter and kth secondary receiver
${\mathbf{H}}_{\mathrm{kj}}$	The channel coefficient between jth secondary transmitter and kth secondary receiver

Assume that user k, $k=0,1,\dots ,K$ , sends signal ${\mathbf{x}}_k\in {\mathbb{C}}^{d_k\times 1}$ , where $d_k$ is the DoF for user k. ${\mathbf{x}}_k$ is independently identically distributed (i.i.d). The transmitted power by the kth transmitter is $P_k=\mathrm{tr}({\mathbf{P}}_k)$ , where ${\mathbf{P}}_k=E\{{\mathbf{x}}_k({\mathbf{x}}_k{)}^H\}$ is the source covariance matrix. This article assumes that each SU requires the same DoF d. The received signal at the PU is expressed as

$${\mathbf{y}}_0={\mathbf{U}}_{00}^H{\mathbf{H}}_{00}{\mathbf{V}}_{00}{\mathbf{x}}_0+{\mathbf{U}}_{00}^H{\displaystyle \sum _{j=1}^K}{\mathbf{H}}_{0j}{\mathbf{V}}_j{\mathbf{x}}_j+{\mathbf{U}}_{00}^H{\mathbf{n}}_{00}$$ (1)

where ${\mathbf{V}}_{00}\in {\mathbf{C}}^{M_t\times d_0}$ and ${\mathbf{U}}_{00}\in {\mathbf{C}}^{M_r\times d_0}$ stand for the precoding matrix and the decoding matrix of PU, respectively, whose columns are the orthonormal basis of the signal spaces. ${\mathbf{U}}_{00}^H{\mathbf{n}}_{00}$ is the complex additive white Gaussian noise (AWGN) with covariance ${\sigma }^2{\mathbf{I}}_{d_0}$ .

The received signal at the kth SU is expressed as

$$\begin{array}{l}{\mathbf{y}}_k={\mathbf{U}}_k^H{\mathbf{H}}_{kk}{\mathbf{V}}_k{\mathbf{x}}_k+{\mathbf{U}}_k^H{\displaystyle \sum _{j=1,j\ne k}^K}{\mathbf{H}}_{kj}{\mathbf{V}}_j{\mathbf{x}}_j\hfill \\ +{\mathbf{U}}_k^H{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\mathbf{x}}_0+{\mathbf{U}}_k^H{\mathbf{n}}_{k}k=1,\cdots ,K\hfill \end{array}$$ (2)

where ${\mathbf{V}}_k\in {\mathbf{C}}^{M_t\times d}$ and ${\mathbf{U}}_k\in {\mathbf{C}}^{M_r\times d}$ stand for the precoding matrix and the decoding matrix of SU, respectively, whose columns are the orthonormal basis of the signal spaces. ${\mathbf{U}}_k^H{\mathbf{n}}_k$ is the complex AWGN with covariance ${\sigma }^2{\mathbf{I}}_d$ . On the reciprocal channel, we assume ${\overleftarrow{\mathbf{V}}}_k$ and ${\overleftarrow{\mathbf{U}}}_k$ denote the precoding matrix and the interference suppression filter (decoders), respectively. We set ${\overleftarrow{\mathbf{V}}}_k={\mathbf{U}}_k$ and ${\overleftarrow{\mathbf{U}}}_k={\mathbf{V}}_k$ . ${\overleftarrow{\mathbf{H}}}_{\mathrm{kj}}$ is the channel gain between transmitter j and receiver k, and ${\overleftarrow{\mathbf{H}}}_{\mathrm{kj}}={\mathbf{H}}_{\mathrm{jk}}^H$

$${\overrightarrow{\mathbf{U}}}_k:M_t\times d,{\overrightarrow{\mathbf{V}}}_k:M_r\times d,{\overrightarrow{\mathbf{H}}}_{kj}:M_t\times M_r$$

In our article, it is assumed that the secondary system has full CSI of the primary link, primary to secondary, secondary to primary, as well as the link between the secondary transmitter and receiver.

DDA scheme for the PU

Assuming that the interference generated by SUs to the PU is eliminated, which will be discussed later in section “IA for secondary MIMO links,” we choose ${\mathbf{P}}_{00}$ of the PU to maximize its rate in the following

$$\begin{array}{l}\underset{{\mathbf{P}}_{00}}{{\displaystyle \max }}{{\displaystyle \log }}_2\left|{\mathbf{I}}_{d_0}+\frac{1}{{\sigma }^2}{\mathbf{U}}_{00}^H{\mathbf{H}}_{00}{\mathbf{V}}_{00}{\mathbf{P}}_{00}{V}_{00}^H{\mathbf{H}}_{00}^H{\mathbf{U}}_{00}\right|\hfill \\ s.t.{\mathbf{P}}_{00}\underset{\_}{\succ }\mathrm{0}\mathrm{,}tr({\mathbf{P}}_{00})\le P_0\hfill \end{array}$$ (3)

where ${\mathbf{P}}_{00}$ is a diagonal matrix, the elements of which reflect the power allocation. In this article, we set ${\mathbf{U}}_{00}={\tilde{\mathbf{U}}}_{00}$ and ${\mathbf{V}}_{00}={\tilde{\mathbf{V}}}_{00}$ , where ${\tilde{\mathbf{V}}}_{00}$ and ${\tilde{\mathbf{U}}}_{00}$ , respectively, denote the left and right singular vectors of ${\mathbf{H}}_{00}$ . The optimization problem (3) can be reformulated into equation (4)

$$\begin{array}{l}\underset{{\mathrm{P}}_{00}}{{\displaystyle \max }}{{\displaystyle \log }}_2\left|{\mathbf{I}}_{d_0}+\frac{1}{{\sigma }^2}{\mathbf{\Lambda }}_{00}{\mathbf{P}}_{00}{\mathbf{\Lambda }}_{00}^H\right|\hfill \\ s.t.{\mathbf{P}}_{00}\underset{\_}{\succ }0\mathrm{,}tr({\mathbf{P}}_{00})\le P_0\hfill \end{array}$$ (4)

where ${\mathbf{\Lambda }}_{00}$ denotes the singular value matrix of ${\mathbf{H}}_{00}$ , that is, ${\mathbf{\Lambda }}_{00}={\tilde{\mathbf{U}}}_{00}^H{\mathbf{H}}_{00}{\tilde{\mathbf{V}}}_{00}$ . The optimal solution of equation (4) can be obtained by the WFA, where the power of the lth symbol is given by spatio-temporal water filling over the nonzero values of ${\mathbf{\Lambda }}_{00}$

$${\mathrm{P}}_{00}(l)={\left({\beta }_0-\frac{{\sigma }^2}{{\lambda }_{00}^2(l)}\right)}^{+},k=1,\dots ,d_0$$ (5)

where ${\lambda }_{00}(l)$ represents the lth singular value of ${\mathbf{\Lambda }}_{00}$ , ${\beta }_0$ is the Lagrange multiplier chosen to satisfy the power constraint in equation (4). Then, we have $d_0=\mathrm{rank}({\mathbf{P}}_{00})$ and $d_0\le min\{M_t,M_r\}$ .

According to Vahdani and Falahati, ¹⁷ for a symmetric system with $M_t$ transmit antennas and $M_r$ receive antennas, the system is proper if

$$M_t+M_r-2d_0-(K+1)d\ge 0$$ (6)

From formula (5), PU may leave some of its spatial dimensions unused due to power limitations. Therefore, the unused spatial dimensions can be used by the secondary system operating in the same frequency band. However, with the increase in the SNR, there will be fewer or even none of the singular values left unused by the primary link. More specifically, if $M_t=M_r$ and $d_0=min\{M_t,M_r\}$ , we have $d=0$ . Hence, in this article, we propose a dichotomy of DoF scheme shown in Figure 2, where $d_{0\_\mathrm{WFA}}$ denotes the PU’s DoF calculated by WFA directly. PU adjusts its DoFs adaptively according to the predefined rate requirement.

Figure 2.

Dichotomy of PU’s DoF.

Let $R_p^o$ denote the predefined rate requirement of PU and num denote the number of dichotomy. After the first dichotomy, the DoF of PU decreases to $d_0^{\mathrm{num}}=d_{0\_\mathrm{WFA}}/2$ . The singular value matrix is then updated according to the new DoF, and the transmission rate of PU is calculated using WFA. As shown in Figure 2, when $R_p^{\mathrm{num}}<R_p^o$ , DDA performs dichotomy according to the direction of the red line. After the second dichotomy, the DoF of PU rises to $d_0^{\mathrm{num}}=(3/4)d_{0\_\mathrm{WFA}}$ . From Figure 2, although $R_p^{\mathrm{num}}\ge R_p^o$ , it is still not clear if or not $d_0^{\mathrm{num}}=(3/4)d_{0\_\mathrm{WFA}}$ is the minimum DoF that satisfies the lowest requirement of PU’s rate. After the third dichotomy, one can obtain $d_0^{\mathrm{num}}=(5/8)d_{0\_\mathrm{WFA}}$ . If $R_p^{\mathrm{num}}<R_p^o$ , DDA performs dichotomy along the red line; if $R_p^{\mathrm{num}}\ge R_p^o$ , DDA performs dichotomy along the green line. The above process continues till one of the following two conditions is met:

• Condition 1: if $R_p^{\mathrm{num}-1}<R_p^o$ , $R_p^{\mathrm{num}}\ge R_p^o$ , and $|d_0^{\mathrm{num}}-d_0^{\mathrm{num}-1}|=1$ , then $d_0=d_0^{\mathrm{num}}$ .

• Condition 2: if $R_p^{\mathrm{num}-1}\ge R_p^o$ , $R_p^{\mathrm{num}}<R_p^o$ , and $|d_0^{\mathrm{num}}-d_0^{\mathrm{num}-1}|=1$ , then $d_0=d_0^{\mathrm{num}-1}$ .

It is not difficult to find that

1. when $R_p^{\mathrm{num}}<R_p^o$ , DDA performs dichotomy along the red line. The DoF of PU is calculated as $d_0^{\mathrm{num}}=\lceil d_0^{\mathrm{num}-1}+d_{0\_\mathrm{WFA}}/(2\mathrm{num})\rceil$ .

2. when $R_p^{\mathrm{num}}\ge R_p^o$ , DDA performs dichotomy along the green line. The DoF of PU is calculated as $d_0^{\mathrm{num}}=\lfloor d_p^{\mathrm{num}-1}-d_{0\_\mathrm{WFA}}/(2\mathrm{num})\rfloor$ .

Note that when $d_0^{\mathrm{num}}=1$ and $R_p^{\mathrm{num}}\ge R_p^o$ , the updated DoF is still equal to 1 after performing the next dichotomy process. As a result, the algorithm falls into the deadlock cycle. Hence, once $d_0^{\mathrm{num}}=1$ , the DDA scheme stops running the dichotomy process.

As previously mentioned, after $d_0$ value is calculated, the singular value matrix needs to be restructured. Equation (7) shows the reconstructed singular value matrix, from which one can see that the new singular value matrix keeps the original $d_0^{\mathrm{num}}$ elements unchanged while the remaining elements become zero

$${\tilde{\mathbf{\Lambda }}}_{00\_\mathrm{new}}=\left[\begin{array}{cccccc}{\sigma }_1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& \ddots \hfill & \hfill & \hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & {\sigma }_{d_0^{\mathrm{num}}}\hfill & \hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & \hfill & 0\hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & \hfill & \hfill & \hfill & 0\hfill \end{array}\right]$$ (7)

The reconstructed channel matrix is ${\tilde{\mathbf{H}}}_{00\_\mathrm{new}}={\tilde{\mathbf{U}}}_{00\_\mathrm{new}}{\tilde{\mathbf{L}}}_{00\_\mathrm{new}}{\tilde{\mathbf{V}}}_{00\_\mathrm{new}}^H$ , where ${\tilde{\mathbf{U}}}_{00\_\mathrm{new}}$ and ${\tilde{\mathbf{V}}}_{00\_\mathrm{new}}$ are the respective first $d_0^{\mathrm{num}}$ columns of ${\tilde{\mathbf{U}}}_{00}$ and ${\tilde{\mathbf{V}}}_{00}$ . We perform WFA under the new ${\tilde{\mathbf{H}}}_{00\_\mathrm{new}}$ to acquire the source covariance matrix ${\mathbf{P}}_{00}^{\mathrm{num}}$ and the data rate $R_p^{\mathrm{num}}$ of PU. The steps of the DDA algorithm are given as follows.

Algorithm 1. PU rate optimization approach based on dichotomy of DoF (DDA).
(1) Calculate ${\mathbf{\Lambda }}_{00}$ , ${\tilde{\mathbf{U}}}_{00}$ , and ${\tilde{\mathbf{V}}}_{00}$ based on the singular value decomposition of ${\mathbf{H}}_{00}$ .
(2) Calculate ${\mathbf{P}}_{00\_\mathrm{WFA}}$ using WFA. Let $d_{0\_\mathrm{WFA}}=\mathrm{rank}({\mathbf{P}}_{00\_\mathrm{WFA}})$ .
(3) Set $\mathrm{num}=1$ and $d_0^{\mathrm{num}}=d_{0\_\mathrm{WFA}}/2$ . Calculate $R_p^{\mathrm{num}}$ using WFA.
(4) If $R_p^{\mathrm{num}}\ge R_p^o$
(4.1) If $d_0^{\mathrm{num}}=1$ , $d_0=d_0^{\mathrm{num}}$ and $R_p=R_p^{\mathrm{num}}$ , then go to step 6. If not, go to 4.2;
(4.2) Let $\mathrm{num}=\mathrm{num}+1$ . After the num-th dichotomy, the PU DoF decreases to $d_0^{\mathrm{num}}=\lfloor d_0^{\mathrm{num}-1}-d_{0\_\mathrm{WFA}}/(2^\mathrm{num})\rfloor$ ;
(4.3) Update the singular matrix according to (7) to get the new ${\tilde{\mathbf{H}}}_{00\_\mathrm{new}}$ . Then calculate $R_p^{\mathrm{num}}$ using WFA;
(4.4) If $R_p^{\mathrm{num}}<R_p^o$ and $|d_0^{\mathrm{num}}-d_0^{\mathrm{num}-1}|=1$ , let $d_0=d_0^{\mathrm{num}-1}$ and $R_p=R_p^{\mathrm{num}-1}$ . Go to step 6;
(4.5) If $R_p^{\mathrm{num}}<R_p^o$ and $|d_0^{\mathrm{num}}-d_0^{\mathrm{num}-1}|\ne 1$ , go to step 5. If not, go to step 4.1.
(5) If $R_p^{\mathrm{num}}<R_p^o$
(5.1) $\mathrm{num}=\mathrm{num}+1$ , after the num-th dichotomy, the DoF of PU increases to $d_0^{\mathrm{num}}=\lceil d_0^{\mathrm{num}-1}+d_{0\_\mathrm{WFA}}/(2^\mathrm{num})\rceil$ ;
(5.2) Update the singular matrix according to (7) to get the new ${\tilde{\mathbf{H}}}_{00\_\mathrm{new}}$ , then calculate $R_p^{\mathrm{num}}$ using WFA;
(5.3) If $R_p^{\mathrm{num}}\ge R_p^o$ and $|d_0^{\mathrm{num}}-d_0^{\mathrm{num}-1}|=1$ , then $d_0=d_0^{\mathrm{num}}$ and $R_p=R_p^{\mathrm{num}}$ , go to step 6;
(5.4) If $R_p^{\mathrm{num}}\ge R_p^o$ , go to step 4. If not, go to step 5.1.
(6) ${\mathbf{U}}_{00}$ and ${\mathbf{V}}_{00}$ are the respective first $d_0^{\mathrm{num}}$ columns of ${\tilde{\mathbf{U}}}_{00}$ and ${\tilde{\mathbf{V}}}_{00}$ .

IA for secondary MIMO links

ISA scheme for the SU

Figure 3 shows the perfect IA, if the signals are constrained into the same sub-space at the unintended receivers, the desired signal can be recovered by eliminating the aligned interferences. However, as seen in Figure 4, not all the interfering signals can be aligned with ${\mathbf{H}}_{\mathrm{ki}}{\mathbf{V}}_i$ in reality. There may exist interference leaked into the desired space even if the interfering signals have been processed by the interference suppression filter at each receiver. Therefore, to reduce the dimensions occupied by the interference signals, in this article, we propose ISA to minimize the sum of distances among the sub-spaces of different interfering signals over all receivers.

Figure 3.

Perfect interference alignment.

Figure 4.

Practical interference alignment.

Precoding matrix design

It is assumed that SUs have accessed to the PU precoding and decoding matrices. In order to protect PU from the interference imposed by SUs, the secondary precoders need to satisfy

$${\mathbf{U}}_{00}^H{\mathbf{H}}_{0k}{\mathbf{V}}_k=0_{d_0\times d},k=1,\dots ,K$$ (8)

or equivalently

$${\mathbf{V}}_k=\mathcal{N}\{{\mathbf{U}}_{00}^H{\mathbf{H}}_{0k}\}k=1,\dots ,K$$ (9)

For a system with multiple SUs, it is very difficult to align the received interference signals that are processed by the precoding matrices in equation (9) into the same space direction. Taking this into consideration, this article employs the sub-matrix optimization method, where the precoding matrix is divided into two sub-matrices, ${\mathbf{V}}_k={\mathbf{V}}_k^1{\mathbf{V}}_k^2$ . We design the first sub-precoder ${\mathbf{V}}_k^1$ to manage the interference generated by SUs to the PU, and the second sub-precoder ${\mathbf{V}}_k^2$ is designed to manage the mutual interference among SUs.

The first sub-precoder is designed to protect the PU from the interference imposed by SUs

$${\mathbf{U}}_{00}^H{\mathbf{H}}_{0k}{\mathbf{V}}_k^1=0,k=1,\dots ,K$$ (10)

The necessary condition for existence of solution for equation (10) is

$$M_t>d_0$$ (11)

The kth first sub-precoder is expressed as

$${\mathbf{V}}_k^1=\mathcal{N}\{{\mathbf{U}}_{00}^H{\mathbf{H}}_{0k}\},k=1,\dots ,K$$ (12)

where ${\mathbf{V}}_k^1\in {\mathbb{C}}^{M_t\times (M_t-d_0)}$ .

To reduce the space dimensions occupied by the interference signals, ISA minimizes the sum of sub-space distances between different interfering signals over all receivers. Let ${\mathbf{H}}_{\mathrm{kj}}^1={\mathbf{H}}_{\mathrm{kj}}\times {\mathbf{V}}_j^1,k,j=1,\dots ,K$

$$min\sum _{k=1}^K\{\sum _{i=1,i\ne k}^K\sum _{j=1,j\notin \{k,i\}}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{\mathrm{ki}}^1{\mathbf{V}}_i^2{\Vert }_M^2+\sum _{j=1,j\ne k}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2\}$$ (13)

Theoretically, IA progressively reduces the sub-space distances between different interfering signals. However, since ${\mathbf{V}}_{00}$ is pre-designed in section “DDA scheme for the PU,” the sub-space spanned by the interference from PU is fixed. In view of this premise, we align the sub-spaces spanned by interference from other SUs with the sub-space spanned by interference from PU

$$\underset{{\mathbf{V}}_j^2,j=1,\dots ,K}{min}\sum _{k=1}^K\sum _{j=1,j\ne k}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2$$ (14)

It is not difficult to find that

$$\begin{array}{c}\sum _{k=1}^K\sum _{j=1,j\ne k}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2\hfill \\ =\sum _{j=1}^K\{\Vert {\mathbf{H}}_{1j}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{10}{\mathbf{V}}_{00}{\Vert }_M^2+\cdots \hfill \\ +\Vert {\mathbf{H}}_{(j-1)j}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{(j-1)0}{\mathbf{V}}_{00}{\Vert }_M^2\hfill \\ +\Vert {\mathbf{H}}_{(j+1)j}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{(j+1)0}{\mathbf{V}}_{00}{\Vert }_M^2+\cdots \hfill \\ +\Vert {\mathbf{H}}_{\mathrm{Kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{K0}{\mathbf{V}}_{00}{\Vert }_M^2\}\}\hfill \\ =\sum _{j=1}^K\sum _{k=1,k\ne j}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2\hfill \end{array}$$

Thus, at each secondary transmitter, the optimization problem is

$$\underset{{\mathbf{V}}_j^2}{min}\sum _{k=1,k\ne j}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2$$ (15)

From the optimal solution, the majority of the interference experienced at the kth secondary receiver falls into the sub-space spanned by ${\mathbf{H}}_{k0}{\mathbf{V}}_{00}$ . Nevertheless, the received signal-to-interference-and-noise ratio (SINR) could still have a significant degradation if the sub-space spanned by the jth desired signal lies close to the sub-space spanned by ${\mathbf{H}}_{j0}{\mathbf{V}}_{00}$ . To address this, we perform a weighted optimization

$$\underset{{\mathbf{V}}_j^2}{min}\sum _{k=1,k\ne j}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2-\mu \Vert {\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{j0}{\mathbf{V}}_{00}{\Vert }_M^2$$ (16)

According to Kumar and Xue, ²⁸ the distance between two sub-spaces spanned by $\mathbf{A}$ and $\mathbf{B}$ can be calculated as $\Vert \mathbf{A},\mathbf{B}{\Vert }_M\stackrel{\Delta }{=}\Vert \mathbf{A}-\mathbf{B}{\mathbf{B}}^H\mathbf{A}{\Vert }_F$ . Thus, we have

$$\begin{array}{c}\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M\hfill \\ \triangleq \Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2-({\mathbf{H}}_{k0}{\mathbf{V}}_{00})({\mathbf{H}}_{k0}{\mathbf{V}}_{00}{)}^H{\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2{\Vert }_F\hfill \\ =\Vert ({\mathbf{I}}_{\mathrm{Mr}}-({\mathbf{H}}_{k0}{\mathbf{V}}_{00})({\mathbf{H}}_{k0}{\mathbf{V}}_{00}{)}^H){\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2{\Vert }_F\hfill \end{array}$$ (17)

$$\begin{array}{c}\Vert {\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{j0}{\mathbf{V}}_{00}{\Vert }_M\hfill \\ \triangleq \Vert {\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2-({\mathbf{H}}_{j0}{\mathbf{V}}_{00})({\mathbf{H}}_{j0}{\mathbf{V}}_{00}{)}^H{\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2{\Vert }_F\hfill \\ =\Vert ({\mathbf{I}}_{\mathrm{Mr}}-({\mathbf{H}}_{j0}{\mathbf{V}}_{00})({\mathbf{H}}_{j0}{\mathbf{V}}_{00}{)}^H){\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2{\Vert }_F\hfill \end{array}$$ (18)

Let $\mathbf{I}{\mathbf{s}}_{\mathrm{kj}}$ and $\mathbf{D}{\mathbf{s}}_{\mathrm{jj}}$ be defined as follows

$$\mathbf{I}{\mathbf{s}}_{\mathrm{kj}}=({\mathbf{I}}_{\mathrm{Mr}}-({\mathbf{H}}_{k0}{\mathbf{V}}_{00})({\mathbf{H}}_{k0}{\mathbf{V}}_{00}{)}^H){\mathbf{H}}_{\mathrm{kj}}^1$$ (19)

$$\mathbf{D}{\mathbf{s}}_{\mathrm{jj}}=({\mathbf{I}}_{\mathrm{Mr}}-({\mathbf{H}}_{j0}{\mathbf{V}}_{00})({\mathbf{H}}_{j0}{\mathbf{V}}_{00}{)}^H){\mathbf{H}}_{\mathrm{jj}}^1$$ (20)

We further have

$$\begin{array}{c}\underset{{\mathbf{V}}_j^2}{min}\sum _{k=1,k\ne j}^K\Vert {\mathbf{H}}_{\mathrm{kj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{k0}{\mathbf{V}}_{00}{\Vert }_M^2-\mu \Vert {\mathbf{H}}_{\mathrm{jj}}^1{\mathbf{V}}_j^2,{\mathbf{H}}_{j0}{\mathbf{V}}_{00}{\Vert }_M^2\hfill \\ \multicolumn{1}{c}{\triangleq \underset{{\mathbf{V}}_j^2}{min}\sum _{k=1,k\ne j}^K\Vert \mathbf{I}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{\Vert }_F^2-\mu \Vert \mathbf{D}{\mathbf{s}}_{\mathrm{jj}}{\mathbf{V}}_j^2{\Vert }_F^2}\\ \multicolumn{1}{c}{=\underset{{\mathbf{V}}_j^2}{min}\mathrm{tr}\left[{({\mathbf{V}}_j^2)}^H\left(\sum _{k=1,k\ne j}^K{\mathbf{Is}}_{\mathrm{kj}}^H\mathbf{I}{\mathbf{s}}_{\mathrm{kj}}\right){\mathbf{V}}_j^2\right]}\\ \multicolumn{1}{c}{-\mu \times \mathrm{tr}[({\mathbf{V}}_j^2{)}^H{\mathbf{Ds}}_{\mathrm{jj}}^H\mathbf{D}{\mathbf{s}}_{\mathrm{jj}}{\mathbf{V}}_j^2]}\\ \multicolumn{1}{c}{=\underset{{\mathbf{V}}_j^2}{min}\mathrm{tr}\left[{({\mathbf{V}}_j^2)}^H(\sum _{k=1,k\ne j}^K{\mathbf{Is}}_{\mathrm{kj}}^H\mathbf{I}{\mathbf{s}}_{\mathrm{kj}}-\mu {\mathbf{Ds}}_{\mathrm{jj}}^H\mathbf{D}{\mathbf{s}}_{\mathrm{jj}}){\mathbf{V}}_j^2\right]}\end{array}$$ (21)

$\mathrm{COV}\_\mathrm{distanc}{e}_j$ is defined as the distance covariance matrix, which is equal to $\sum _{k=1,k\ne j}^K({\mathbf{Is}}_{\mathrm{kj}}^H\mathbf{I}{\mathbf{s}}_{\mathrm{kj}}-\mu {\mathbf{Ds}}_{\mathrm{jj}}^H\mathbf{D}{\mathbf{s}}_{\mathrm{jj}})$ . The columns of ${\mathbf{V}}_j^2$ are the least d dominant eigenvectors of $\mathrm{COV}\_\mathrm{distanc}{e}_j$

$${\mathbf{V}}_{\mathrm{jl}}^2=v_d[\mathrm{COV}\_\mathrm{distanc}{e}_j],l=1,\dots ,d$$ (22)

Decoder design

The decoder design also employs the sub-matrix optimization method, where the first sub-decoder is designed to eliminate interference, and the second sub-decoder is designed to improve the desired received signal power. In section “Precoding matrix design,” at each secondary receiver, the sub-spaces spanned by interference imposed by other SUs are almost aligned to the sub-space spanned by interference from PU. Thus, the first sub-decoder can be designed as the matrix spanning the null space of the matrix $({\mathbf{H}}_{k0}{\mathbf{V}}_{00}{)}^H$

$$({\mathbf{U}}_k^1{)}^H{\mathbf{H}}_{k0}{\mathbf{V}}_{00}=0k=1,\dots ,K$$ (23)

Thus, the kth first sub-decoder is expressed as

$${\mathbf{U}}_k^1=\mathcal{N}\{{\mathbf{V}}_{00}^H{\mathbf{H}}_{k0}^H\}k=1,\dots ,K$$ (24)

where ${\mathbf{U}}_k^1\in {\mathbb{C}}^{M_r\times (M_r-d_0)}$ . The intended signal power at the kth secondary receiver is

$$\begin{array}{c}\mathrm{tr}[({\mathbf{U}}_k^1{\mathbf{U}}_k^2{)}^H{\mathbf{H}}_{\mathrm{kk}}{\mathbf{V}}_k{({\mathbf{H}}_{\mathrm{kk}}{\mathbf{V}}_k)}^H{\mathbf{U}}_k^1{\mathbf{U}}_k^2]\hfill \\ \multicolumn{1}{c}{=\mathrm{tr}[({\mathbf{U}}_k^2{)}^H{({\mathbf{U}}_k^1)}^H{\mathbf{H}}_{\mathrm{kk}}{\mathbf{V}}_k{({\mathbf{H}}_{\mathrm{kk}}{\mathbf{V}}_k)}^H{\mathbf{U}}_k^1{\mathbf{U}}_k^2]}\end{array}$$ (25)

To maximize the intended signal power, the columns in ${\mathbf{U}}_j^2$ are the $(M_r-d_0-d)$ dominant eigenvectors of $\mathrm{COV}\_\mathrm{desir}{e}_j$ , where $\mathrm{COV}\_\mathrm{desir}{e}_j=({\mathbf{U}}_j^1{)}^H{\mathbf{H}}_{\mathrm{jj}}{\mathbf{V}}_j({\mathbf{H}}_{\mathrm{jj}}{\mathbf{V}}_j{)}^H{\mathbf{U}}_j^1$ denotes the desired covariance matrix. The logical flow of the ISA algorithm is given as follows.

Algorithm 2. ISA.
Precoding Matrix Design:
At each secondary transmitter,
(1) calculate the first sub-precoder ${\mathbf{V}}_j^1,j=1,\dots ,K$ according to equation (12).
(2) compute the distance covariance matrix $\mathrm{COV}\_\mathrm{distanc}{e}_j,j=1,\dots ,K$ .
(3) the columns of ${\mathbf{V}}_j^2,j=1,\dots ,K$ are the least d dominant eigenvectors of $\mathrm{COV}\_\mathrm{distanc}{e}_j$ .
Decoder design:
At each secondary receiver,
(1) calculate the first sub-decoder ${\mathbf{U}}_j^1,$ according to equation (24).
(2) compute the desired covariance matrix $\mathrm{COV}\_\mathrm{desir}{e}_j$ .
(3) the columns of ${\mathbf{U}}_j^2,$ are the $M_r-d$ dominant eigenvectors of $\mathrm{COV}\_\mathrm{desir}{e}_j$ .

SU GSR scheme

ISA only focuses on suppressing interference with noise neglected. In this sub-section, we propose an iterative scheme where the total secondary system sum rate is considered.

GSR also employs the sub-matrix optimization method, where the design of ${\mathbf{V}}_k^1$ and ${\mathbf{U}}_k^1,k=1,\dots ,K$ , is the same as in ISA. Let $\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}=({\mathbf{U}}_k^1{)}^H{\mathbf{H}}_{\mathrm{kj}}{\mathbf{V}}_j^1$ , where $\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}$ is a $(M_r-d_0)\times (M_t-d_0)$ matrix. The secondary system can be regarded as a multi-user MIMO interference system, where the channel coefficient between jth secondary transmitter and kth secondary receiver is $\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}$ . The sum rate of the secondary system is computed as equation (26). Furthermore, since $({\mathbf{U}}_k^2{)}^H{\mathbf{U}}_k^2={\mathbf{I}}_d$ , equation (26) is equivalent to equation (27)

$$R={\displaystyle \sum _{k=1}^K}{\log }_2\det \{I_d+{[{\sigma }^2{I}_d+{\displaystyle \sum _{j=1,j\ne k}^K}{(U_k^2)}^{H}H{s}_{kj}{V}_j^2{(V_j^2)}^{H}H{s}_{kj}^H{U}_k^2]}^{-1}{(U_k^2)}^{H}H{s}_{kk}{V}_k^2{(V_k^2)}^{H}H{s}_{kk}^H{U}_k^2\}$$ (26)

$$\begin{array}{cc}\multicolumn{1}{c}{R}& =\sum _{k=1}^K{\log }_{2}det\{{\mathbf{I}}_d+{[{\sigma }^2{\mathbf{I}}_d+\sum _{j=1,j\ne k}^K{({\mathbf{U}}_k^2)}^H\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{({\mathbf{V}}_j^2)}^H{\mathbf{Hs}}_{\mathrm{kj}}^H{\mathbf{U}}_k^2]}^{-1}{({\mathbf{U}}_k^2)}^H\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H{\mathbf{U}}_k^2\}\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=1}^K{\log }_{2}det\{{\mathbf{I}}_d+{({\mathbf{U}}_k^2)}^H{[{\sigma }^2{\mathbf{I}}_{(M_r-d_0)}+\sum _{j=1,j\ne k}^K\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{({\mathbf{V}}_j^2)}^H{\mathbf{Hs}}_{\mathrm{kj}}^H]}^{-1}{\mathbf{U}}_k^2{({\mathbf{U}}_k^2)}^H\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H{\mathbf{U}}_k^2\}\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=1}^K{\log }_{2}det[({\mathbf{U}}_k^2{)}^H]det\{{\mathbf{I}}_{(M_r-d_0)}+{[{\sigma }^2{\mathbf{I}}_{(M_r-d_0)}+\sum _{j=1,j\ne k}^K\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{({\mathbf{V}}_j^2)}^H{\mathbf{Hs}}_{\mathrm{kj}}^H]}^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H\}det({\mathbf{U}}_k^2)\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=1}^K{\log }_{2}det\{{\mathbf{I}}_{(M_r-d_0)}+{[{\sigma }^2{\mathbf{I}}_{(M_r-d_0)}+\sum _{j=1,j\ne k}^K\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{({\mathbf{V}}_j^2)}^H{\mathbf{Hs}}_{\mathrm{kj}}^H]}^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H\}\hfill \end{array}$$ (27)

To improve the secondary system performance at all SNRs, one of the effective method is to maximize the system sum rate, as shown in equation (28)

$$\begin{array}{c}\underset{{\mathbf{V}}_k^2,k=1,\dots ,K}{max}R\hfill \\ s.t.({\mathbf{V}}_k^2{)}^H{\mathbf{V}}_k^2={\mathbf{I}}_d\hfill \end{array}$$ (28)

For simplicity, we define

$${\mathbf{A}}_k=\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{\mathbf{P}}_k({\mathbf{V}}_k^2{)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H$$ (29)

$${\mathbf{B}}_k={\sigma }^2{\mathbf{I}}_{(M_r-d_0)}+\sum _{j=1,j\ne k}^K\mathbf{H}{\mathbf{s}}_{\mathrm{kj}}{\mathbf{V}}_j^2{\mathbf{P}}_j({\mathbf{V}}_j^2{)}^H{\mathbf{Hs}}_{\mathrm{kj}}^H$$ (30)

Thus

$$R=\sum _{k=1}^K{\log }_{2}det({\mathbf{I}}_{(M_r-d_0)}+({\mathbf{B}}_k{)}^{-1}{\mathbf{A}}_k)$$ (31)

To solve the optimization problem in an iterative fashion, we first calculate the derivative of R in formula (31) with respect to $({\mathbf{V}}_k^2{)}^{*}$ . Since there are multiple inverse operations in formula (31) and $d(\ln det(\mathbf{X}))=\mathrm{tr}\{{\mathbf{X}}^{-1}d(\mathbf{X})\}$ , computing derivative of R introduces even more matrix inversion operations, which further complicates the computation. In order to reduce the number of inverse operations, we reconstitute the sum rate formula according to the extensions of Minkowski inequality

$$f({\mathbf{V}}_k^2,k=1,\dots ,K)=\sum _{k=1}^K{\log }_{2}det{\mathbf{A}}_k-{\log }_{2}det{\mathbf{B}}_k$$ (32)

Appendix 1 shows the proof that formula (31) can be transformed into formula (32).

From $\mathrm{tr}(\mathbf{C}(d{\mathbf{X}}^T/d\mathbf{X}))=\mathbf{C}$ , the derivative of the new objective function in formula (32) with respect to $({\mathbf{V}}_k^2{)}^{*}$ is given by formula (33)

$$\begin{array}{c}{\nabla }_{{({\mathbf{V}}_k^2)}^{*}}f=\frac{\mathrm{df}}{d{({\mathbf{V}}_k^2)}^{*}}\hfill \\ \multicolumn{1}{c}{=\frac{\frac{1}{\ln 2}\mathrm{tr}({\mathbf{A}}_k^{-1}d{\mathbf{A}}_k)}{d{({\mathbf{V}}_k^2)}^{*}}-\sum _{i=1,i\ne k}^K\frac{\frac{1}{\ln 2}\mathrm{tr}({\mathbf{B}}_i^{-1}d{\mathbf{B}}_i)}{d{({\mathbf{V}}_k^2)}^{*}}}\\ \multicolumn{1}{c}{=\frac{\frac{1}{\ln 2}\mathrm{tr}[{\mathbf{A}}_k^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{\mathbf{P}}_{k}d{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{kk}}^H]}{d{({\mathbf{V}}_k^2)}^{*}}}\\ \multicolumn{1}{c}{-\sum _{i=1,i\ne k}^K\frac{\frac{1}{\ln 2}\mathrm{tr}[{\mathbf{B}}_i^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{ik}}{\mathbf{V}}_k^2{\mathbf{P}}_{k}d{({\mathbf{V}}_k^2)}^H{\mathbf{Hs}}_{\mathrm{ik}}^H]}{d{({\mathbf{V}}_k^2)}^{*}}}\\ \multicolumn{1}{c}{=\frac{\frac{1}{\ln 2}\mathrm{tr}[{\mathbf{Hs}}_{\mathrm{kk}}^H{\mathbf{A}}_k^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{\mathbf{P}}_{k}d{({\mathbf{V}}_k^2)}^H]}{d{({\mathbf{V}}_k^2)}^{*}}}\\ \multicolumn{1}{c}{-\sum _{i=1,i\ne k}^K\frac{\frac{1}{\ln 2}\mathrm{tr}[{\mathbf{Hs}}_{\mathrm{ik}}^H{\mathbf{B}}_i^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{ik}}{\mathbf{V}}_k^2{\mathbf{P}}_{k}d{({\mathbf{V}}_k^2)}^H]}{d{({\mathbf{V}}_k^2)}^{*}}}\\ \multicolumn{1}{c}{=\frac{1}{\ln 2}{\mathbf{Hs}}_{\mathrm{kk}}^H{\mathbf{A}}_k^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{kk}}{\mathbf{V}}_k^2{\mathbf{P}}_k}\\ \multicolumn{1}{c}{-\frac{1}{\ln 2}\sum _{i=1,i\ne k}^K{\mathbf{Hs}}_{\mathrm{ik}}^H{\mathbf{B}}_i^{-1}\mathbf{H}{\mathbf{s}}_{\mathrm{ik}}{\mathbf{V}}_k^2{\mathbf{P}}_k}\end{array}$$ (33)

Intuitively, the steepest descent (SD) approach can be employed to search the local optimal point for $f({\mathbf{V}}_k^2,k=1,\dots ,K)$ . However, the SD algorithm converges slowly in the high-dimensional complex space. In recent years, the SD method based on the GSR, which can effectively reduce the space dimensions, has been proposed.

Traditional optimization schemes in the multi-dimensional space ${\mathbb{C}}^{(M_t-d_0)\times d}$ is shown to have the following dimension ²⁹

$$\dim ({\mathbb{C}}^{(M_t-d_0)\times d})=(M_t-d_0)\times d$$ (34)

In Manton, ²⁹ the dimension of the schemes based on the GSR is shown to be

$$\dim \{\mathrm{Gr}[(M_t-d_0),d]\}=d\times [(M_t-d_0)-d]$$ (35)

From equations (34) and (35), it can be clearly observed that the optimization on the GSR is effective in reducing the dimensions. Therefore a faster convergence can be achieved in the GSR scheme using GSR. Nevertheless, the objective function needs to be unitarily invariant in the precoders when GSR is used. Hence, we first need to prove that the objective function in formula (32) is unitarily invariant in ${\mathbf{V}}_k^2$ (the detailed proof is in Appendix 2).

As a result, the objective function and the constraint can be simplified into

$$\begin{array}{c}\underset{{\mathbf{V}}_k^2,k=1,\dots ,K}{max}f=\sum _{k=1}^K{\log }_{2}det{\mathbf{A}}_k-{\log }_{2}det{\mathbf{B}}_k\hfill \\ \multicolumn{1}{c}{s.t.{({\mathbf{V}}_k^2)}^H{\mathbf{V}}_k^2={\mathbf{I}}_d,f({\mathbf{V}}_k^2)=f({\mathbf{V}}_k^2\mathbf{Q})}\\ \multicolumn{1}{c}{{\mathbf{Q}}^H\mathbf{Q}=\mathbf{Q}{\mathbf{Q}}^H={\mathbf{I}}_d}\end{array}$$ (36)

The tangent vector on the GSR is denoted as

$${\mathbf{Z}}_k=({\mathbf{I}}_{(M_t-d_0)}-{\mathbf{V}}_k^2{({\mathbf{V}}_k^2)}^H){\nabla }_{{\left({\mathbf{V}}_k^2\right)}^{*}}f$$ (37)

Assuming the step size for each iteration is t, the iteration trajectory of the independent variable ${\mathbf{V}}_k^2$ is expressed as

$${\mathbf{V}}_k^2=({\mathbf{V}}_k^2{\mathbf{N}}_k(\cos ({\mathbf{\Lambda }}_{\mathbf{k}}t)){\mathbf{N}}_k^H)+({\mathbf{M}}_k(\sin ({\mathbf{\Lambda }}_{\mathbf{k}}t)){\mathbf{N}}_k^H)$$ (38)

where ${\mathbf{M}}_k$ and ${\mathbf{N}}_k$ denote the left and right singular vectors of ${\mathbf{Z}}_k$ , respectively. ${\mathbf{\Lambda }}_{\mathbf{k}}$ is the singular matrix. In order to further improve the computation speed, we take ${\mathbf{V}}_k^2$ to maximize the received SINR as the initialization dot in our iterative computing method. Note that this step is calculated in the reciprocal channel. The unit vector ${\mathbf{V}}_{\mathrm{kl}}^2({\overleftarrow{\mathbf{U}}}_{\mathrm{kl}}^2)$ to maximize the SINR of the lth stream of the kth destination is computed as ¹⁰

$${\mathbf{V}}_{\mathrm{kl}}^2({\overleftarrow{\mathbf{U}}}_{\mathrm{kl}}^2)=\frac{{({\mathbf{c}}_{\mathrm{kl}})}^{-1}{\overleftarrow{\mathbf{Hs}}}_{\mathrm{kk}}{\overleftarrow{\mathbf{V}}}_{\mathrm{kl}}^2}{\Vert {({\mathbf{c}}_{\mathrm{kl}})}^{-1}{\overleftarrow{\mathbf{Hs}}}_{\mathrm{kk}}{\overleftarrow{\mathbf{V}}}_{\mathrm{kl}}^2\Vert }$$ (39)

where

$${\mathbf{c}}_{\mathrm{kl}}=\sum _{j=1}^K[\frac{P_j}{d}\sum _{\mathrm{dd}=1}^d{\overleftarrow{\mathbf{Hs}}}_{\mathrm{kj}}{\overleftarrow{\mathbf{V}}}_{\mathrm{jdd}}^2{({\overleftarrow{\mathbf{Hs}}}_{\mathrm{kj}}{\overleftarrow{\mathbf{V}}}_{\mathrm{jdd}}^2)}^H-\frac{P_k}{d}{\overleftarrow{\mathbf{Hs}}}_{\mathrm{kk}}{\overleftarrow{\mathbf{V}}}_{\mathrm{kl}}^2{({\overleftarrow{\mathbf{Hs}}}_{\mathrm{kk}}{\overleftarrow{\mathbf{V}}}_{\mathrm{kl}}^2)}^H+{\sigma }^2{\mathbf{I}}_{(\mathrm{Mr}-d_0)}]$$ (40)

The logical flow of the GSR algorithm is summarized as follows.

Algorithm 3. GSR.
(1) Calculate ${\mathbf{V}}_j^1,{\mathbf{U}}_j^1,j=1,\dots ,K$ according to (12) and (24), and reconstitute the channel transfer matrix $\mathbf{Hs}$ .
(2) Start with arbitrary ${\mathbf{V}}_k^2,k=1,\dots ,K$ , set $n=1$ .
(3) Calculate ${\mathbf{U}}_k^2$ according to (39) and (40), set ${\overleftarrow{\mathbf{V}}}_k^2={\mathbf{U}}_k^2$ .
(4) Reverse the communication direction and turn to the reciprocal network for k = 1: K
(4.1) Calculate the derivative of the objective function with respect to $({\overleftarrow{\mathbf{V}}}_k^2{)}^{*}$ according to formula (33);
(4.2) Compute ${\mathbf{Z}}_k=({\mathbf{I}}_{(M_r-d_0)}-{\overleftarrow{\mathbf{V}}}_k^2({\overleftarrow{\mathbf{V}}}_k^2{)}^H){\nabla }_{{({\overleftarrow{\mathbf{V}}}_k^2)}^{*}}f$ ;
(4.3) Using singular value decomposition to acquire ${\mathbf{M}}_k$ , ${\mathbf{N}}_k$ , and ${\Lambda }_k$ ;
(4.4) Move the independent variable ${\overleftarrow{\mathbf{V}}}_k^2$ along the iteration trajectory to update ${\overleftarrow{V}}_k^2$ according to equation (38).
end
(5) Calculate ${\overleftarrow{\mathbf{U}}}_k^2$ according to equations (39) and (40), set ${\mathbf{V}}_k^2={\overleftarrow{\mathbf{U}}}_k^2$ .
(6) Reverse the communication direction and turn to the original network for k = 1: K
(6.1) Calculate the derivative of the objective function with respect to $({\mathbf{V}}_k^2{)}^{*}$ according to formula (33);
(6.2) Compute Compute ${\mathbf{Z}}_k=({\mathbf{I}}_{(M_t-d_0)}-{\mathbf{V}}_k^2({\mathbf{V}}_k^2{)}^H){\nabla }_{{({\mathbf{V}}_k^2)}^{*}}f$ ;
(6.3) Using singular value decomposition to acquire ${\mathbf{M}}_k$ , ${\mathbf{N}}_k$ , and ${\mathbf{\Lambda }}_{\mathbf{k}}$ ;
(6.4) Move the independent variable ${\mathbf{V}}_k^2$ along the iteration trajectory to update ${\mathbf{V}}_k^2$ according to equation (38).
end
(7) Set $n=n+1$ and calculate the secondary system sum rate $R(n)$ according to equation (26).
(8) Repeat steps 3 to 7 until $R(n)-R(n-1)\le \xi$ .

Remark

To guarantee that there is at least one eigenmode available to each SU, the DoF of PU needs to satisfy the following condition

$$\begin{array}{c}{M}_t+M_r-2d_0\ge (K+1)d\hfill \\ \Rightarrow \frac{M_t+M_r-2d_0}{K+1}\ge d\hfill \\ \Rightarrow \frac{M_t+M_r-2d_0}{K+1}\ge 1\hfill \\ \Rightarrow M_t+M_r-2d_0\ge K+1\hfill \\ \Rightarrow d_0\le \frac{M_t+M_r-(K+1)}{2}\hfill \end{array}$$ (41)

However, the required DoF of PU to satisfy the minimum rate requirement could increase with the increase in the SNR. As a result, the unused eigenmodes are not enough for all the SUs. Thus, in this circumstance, we decrease the number of SUs which transmit concurrently with the PU. In order to allow more users to communicate at the same frequency band, we allocate 1 DoF to each SU. Therefore, according to equation (6), the permitted number of SUs is $K=M_t+M_r-2d_0-1$ .

Simulation results

In this section, we evaluate and compare the performance of ISA, GSR, OCIA, ¹⁷ SS, ²² ANEB, ²³ PIA-SU, ²⁴ and PIA-NSU. ²⁴ Assume that one PU and K SUs coexist in the same frequency band. The DoFs of SUs are calculated according to equation (6). The system is represented by $(M_t,M_r,K)$ . Simulation results are averaged over 500 channel realizations. The channel model is Rayleigh fading channel. The noise power is normalized to one. The convergence threshold is given as $\xi ={10}^{-4}$ . We select $\mu$ corresponding to SNR values [0,3,6,9,12,15,18,21,24,27,30] is [0.02,0.02,0.02,0.02, 0.02, 0.02,0.02,0.01,0.01,0.01,0.002]. ³⁰ $R_p^o$ is supposed to be $\delta$ percent of $R_{\mathrm{WFA}}$ , which denotes the transmission rate of the primary transmitter with the WFA scheme.

The impact of $\delta$ on the performance of all seven schemes is shown in Figures 5 and 6. The simulation curves represented by 0.6 and 0.8 WFA, respectively, display $60\%$ and $80\%$ of a MIMO link capacity without interference. It can be seen that the secondary system sum rate in OCIA is approximately equal to zero when SNR is greater than 6 dB. This is because PU with the WFA scheme tends to spread its power among all its available eigenmodes in the moderate-to-high SNR regime. As a result, there is few or even no singular values left unused by the primary link. In Men et al., ²⁴ the maximum DoFs for each user in PIA-SU and PIA-NSU algorithms are $M_r/(K+1)+K\times M_t/(K+1{)}^2$ and $(M_r+K\times M_t)/(2\times K+1)$ , respectively. Therefore, the number of DoFs in PIA-SU and PIA-NSU algorithm is two in the $(4,4,2)$ system. Due to the same achievable DoFs in the two algorithms, the PU performance of PIA-SU is approximately equal to that of PIA-NSU scheme. Although DDA and ANEB have a matching PU performance for all different SNR values, DDA is still superior to the ANEB scheme due to the fact that the search steps for finding the minimum PU DoF in the DDA scheme is about half that of ANEB.

Figure 5.

Achievable rate of the system versus SNR, $M_t=M_r=4,K=2,\delta =0.6$ .

Figure 6.

Achievable rate of the system versus SNR, $M_t=M_r=4,K=2,\delta =0.8$ .

In Figure 5, GSR achieves the best secondary system performance compared with other six schemes. Under the same conditions, the transmission rate of SUs in the GSR scheme is about 2.7 bps/Hz higher than ANEB. For SNRs greater than 3 dB, the ISA scheme benefits from higher transmission rates compared with the ANEB approach. SS is superior to ISA for SNRs greater than 9 dB, whereas the ISA scheme performs better in the moderate-to-high SNR region. It can be seen in the figure that the achievable rate of SUs in GSR and ISA attains the maximum at 6 dB. It then keeps deceasing till SNR reaches 15 dB due to the fact that the obtained DoFs of secondary system reduce with the increase in the PU’s DoF. As we have known that the PU’s DoFs of ANEB and DDA have the same trend, correspondingly, the SU’s DoFs of ANEB and DDA have the same trend. Hence, the ANEB scheme also obtains its local maximum value at 6 dB. From Figure 5, it can be observed that the secondary system sum rate achieved by SS decreases for SNRs greater than 12 dB. The performance at moderate-to-high SNR region is mainly dominated by interference, whereas SS only addresses the optimization of a MIMO SU. As a result, the interference among SUs lead to poor performance of the secondary system when SNR gets higher. As shown in Figure 6, GSR can improve the performance of SUs significantly compared with other six algorithms for SNRs lower than 21 dB. Moreover, the transmission rate of SUs in the GSR scheme is about 3 bps/Hz higher than ANEB. With the increase in the SNR, GSR and ISA obtain their local maximum value at 12 dB and local minimum value at 18 dB. Different from Figure 5, for SNRs higher than 21 dB, higher achievable rates for the GSR and ISA schemes are achieved by the secondary links. Moreover, GSR and ISA present similar performance. On the contrary, the achievable rate of SUs in ANEB declines after SNR exceeds 21 dB, which shows that the ANEB scheme does not perform well when the total DoFs of the secondary system are less than K.

Figure 7 presents the achievable rates of PU and SUs when there are six transmit antennas, six receive antennas, two SUs, and $\delta =0.8$ . GSR achieves the best secondary system performance compared with other five schemes. Under the same condition, the achievable rate of SUs in the GSR scheme is about 4 bps/Hz higher than ANEB. GSR and ISA obtain their local maximum value at 12 dB and local minimum value at 21 dB. Compared with Figure 6, DDA does not need to decrease the number of SUs for SNRs lower than 30 dB. This is because there are more unused eigenmodes for the secondary system with a higher number of antennas. From the DoF formula, the numbers of DoFs in the PIA-SU and PIA-NSU algorithms both equal 3. Therefore, in Figure 7, the PU performance of PIA-SU is approximately equal to that of PIA-NSU scheme. Since an increase in the number of antennas results in an increase in the number of unused eigenmodes, it can be seen that the performance is improved for both PU and SUs when the number of antennas increases.

Figure 7.

Achievable rate of the system versus SNR, $M_t=M_r=6,K=2,\delta =0.8$ .

In Figure 8, the secondary system sum rate is plotted when the number of transmit antennas and the number of receive antennas are different. The performance of PU in the DDA and ANEB schemes are similar, both of which are greater than $80\%$ of the capacity of a MIMO link without interference. GSR achieves the best secondary system performance compared with OCIA, ANEB, PIA-SU, and PIA-NSU schemes. Under the same condition, the transmission rate of SUs in the GSR scheme is about 4 bps/Hz higher than ANEB. The secondary system performance of ISA is lower than that of GSR, whereas it increases gradually and approaches the GSR curve when SNR is greater than 24 dB. It can be observed that GSR and ISA obtain their local maximum value at 3 dB and local minimum value at 9 dB. Note that the PU performance of PIA-SU is lower than that of PIA-NSU, due to the fact that the DoF of PU in PIA-NSU is 4 while the DoF of PU in PIA-SU is only two. Although the number of the achievable SU’s DoFs in PIA-NSU scheme is twice as large as that of PIA-SU, the latter still achieves better SU performance than the former. This is because an SU in the PIA-NSU scheme suffers not only the interference imposed by the PU but also the interference imposed by other SUs. It can be observed that the PU’s rate achieved by PIA-NSU is the highest. This is because PU in the PIA-NSU algorithm obtains the greatest DoFs compared with other schemes. Figure 9 shows the achievable rate of PU and SUs when there are eight transmit antennas, four receive antennas, six SUs, and $\delta =0.8$ . Compared with Figure 8, there are more SUs in Figure 9. When the SNR increases, the number of unused eigenmodes for the secondary system could decline. Thus, ISA and GSR schemes need to be applied in a limited antennas system. Furthermore, since there are more users, each secondary receiver may suffer from more serious interference imposed by other SUs. However, notice that ISA outperforms other schemes even at low SNR region, which indicates that the ISA scheme has a stronger ability to tackle multi-user interference than GSR. The SS scheme achieves its maximum rate at 21 dB in Figure 8 while 6 dB in Figure 9, from which one can infer that SS algorithm has a weak ability to tackle multi-user interference. This is mainly because SS only addresses the optimization of a MIMO SU.

Figure 8.

Achievable rate of the system versus SNR, $M_t=8,M_r=4,K=3,\delta =0.8$ .

Figure 9.

Achievable rate of the system versus SNR, $M_t=8,M_r=4,K=6,\delta =0.8$ .

The previous figures show that the secondary system achieves a superior performance by applying GSR. We further evaluate the computation complexity of GSR in Figure 10, in which the total achievable sum rate versus different iteration number is presented. A $(8,4,3)$ system is considered with $P=15\mathrm{dB}$ . $F_c(n)=R(n)-R(n-1)$ is defined as the difference of the objective function between two adjacent iterations. ³¹ In Figure 10, the difference of the objective function between the fifth iteration and the sixth iteration is no more than ${10}^{-5}$ . From the small figure in Figure 10, the value of objective function remains approximately constant after seven iterations. The number of iterations of OCIA, PIA-SU, and PIA-NSU is far more than seven, as confirmed in Gomadam et al. ¹⁰ Thus, GSR achieves a better performance with a faster convergence.

Figure 10.

Convergence of the GSR algorithm.

As mentioned earlier, the simulation curves of GSR and ISA are not monotonically increasing with SNR, the reason of which is explained through the analysis of K and $d_0$ . Two cases are considered, one with $M_t=4$ , $M_r=4$ , and $K=2$ , and the other one with $M_t=8$ , $M_r=4$ , and $K=3$ . Figures 11 and 12 show the results. Tables 2 and 3 show the number of PU’s DoFs and the number of the “active” SUs, respectively. In Figure 11, if PU maintains two DoFs for transmission when SNR equals 21 dB, its lowest rate requirement cannot be met. Thus, the PU increases the number of DoFs at 21 dB. Due to the PU’s DoF increasing, the number of “active” SUs decreases and thus the secondary performance of GSR and ISA achieves their local maximum value at 18 dB and local minimum value at 21 dB. According to equation (6), the number of the unused eigenmodes cannot satisfy the requirement that there is at least one DoF available to each SU. In this case, the permitted number of active SUs drops from 2 to 1. Similarly, in Figure 12, considering the minimum data rate requirement of PU, there is a local maximum value at 3 dB and local minimum value at 6 dB. Theoretically, the maximum point should be adjacent to the minimum point. However, since the simulation results in Figures 5–9 are averaged over 500 channel realizations, the maximum point may not necessarily be adjacent to the minimum point.

Figure 11.

Achievable rate with one simulation, $M_t=4,M_r=4,K=2,\delta =0.8$ .

Figure 12.

Achievable rate with one simulation, $M_t=8,M_r=4,K=3,\delta =0.8$ .

Table 2.

Achievable DoFs of PU versus SNR.

SNR (dB)	0	3	6	9	12	15	18	21	24	27	30
Primary DoF $(d_0)$ in $(4,4,2)$	1	2	2	2	2	2	2	3	3	3	3
Primary DoF $(d_0)$ in $(8,4,3)$	2	2	3	3	3	3	3	3	3	3	3

DoF: degree of freedom; PU: primary user; SNR: signal-to-noise ratio.

Table 3.

Number of “active” SUs.

SNR (dB)	0	3	6	9	12	15	18	21	24	27	30
Secondary users $(K)$ in $(4,4,2)$	2	2	2	2	2	2	2	1	1	1	1
Secondary users $(K)$ in $(8,4,3)$	3	3	3	3	3	3	3	3	3	3	3

SNR: signal-to-noise ratio.

Conclusion

In this article, we consider a cognitive network that comprises one primary link and multiple secondary links. We first propose a DDA scheme, which allows the PU to release some eigenmodes with the rate requirement still met. Although DDA achieves the same rate as ANEB, it benefits from a lower complexity compared with the ANEB approach, particularly in high SNR regions. To eliminate the received interference at each SU, this article proposes an ISA method, where the sub-spaces spanned by interference imposed by other SUs are constructed to cast overlapping shadows in the sub-space spanned by interference imposed by PU. Since ISA ignores the influence of noise, it may not work well at low SNRs. To tackle this problem, we further present an iterative algorithm to maximize the sum rate of SUs. Nevertheless, if we directly take the sum rate as the objective function, there will exist excessive inverse operations during derivation. Hence, in order to decrease the complexity of GSR, the objective function is transformed into two parts without inversion operation using the extensions of the Minkowski inequality. Besides, GSR is optimized to further improve the convergence speed of GSR. Simulation results show that DDA protects PU transmission while providing more unused eigenmodes for SUs. Simulations also indicate that ISA can be more effective than GSR to mitigate interference for the system with more SUs. When considering the physical layer aspects of the problem, the secondary links achieve higher data rates using GSR and ISA no matter whether the numbers of transmit/receive antennas are different or not.