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Multiantenna transmission for underlay and overlay cognitive radio with explicit messagelearning phase
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 195 (2013)
Abstract
We consider the coexistence of a multipleinput multipleoutput secondary system with a multipleinput singleoutput primary link with different degrees of coordination between the systems. First, for the uncoordinated underlay cognitive radio scenario, we fully characterize the optimal parameters that maximize the secondary rate subject to a primary rate constraint for a transmission strategy that combines rate splitting and interference cancellation. Second, we establish a model for the coordinated overlay cognitive radio scenario that consists of a messagelearning phase followed by a communication phase. We then propose a transmission strategy that combines techniques for cooperative communication and for the classical cognitive radio channel. We optimize our system to maximize the rate of communication for the secondary users under a primaryuser rate constraint and find efficient algorithms to compute the optimal system parameters. Finally, we compare both cognitive radio strategies to assess their relative merits and to evaluate the effect of the messagelearning phase. We observe that for closely located transmitters, the overlay strategy outperforms the underlay strategy. In this situation, learning the primary message is very beneficial for the secondary systems, especially if they are interferencelimited rather than powerlimited. The situation is reversed when the distance between the transmitters is large. In either case, we observe that there is room for significant improvement if the transmitter implements both strategies and decides adaptively which one to use according to the channel conditions. We conclude our work with a discussion on the extension to the coexistence with multipleinput multipleoutput primaries.
1 Introduction
The scarcity of available spectrum for accommodating new services in combination with the underutilization of currently allocated spectrum has fueled research on alternative visions on communications over the last decade. It has been suggested that new, unlicensed (i.e. secondary) users could utilize portions of the spectrum licensed to primary users as long as the latter are not significantly affected. In this context, the concept of cognitive radio, with its promise of reconfigurability and adaptability to varying conditions, has emerged as a strong candidate for implementing communication systems that make a more efficient use of the spectrum.
Three major cognitive radio paradigms that consider different degrees of interaction between primary and secondary users have been identified: interweave, underlay and overlay[1, 2]. Interweave cognitive radio is conceptually the simplest one: the secondary devices sense the environment to detect the presence of primary users and transmit opportunistically only when these are silent. Underlay cognitive radio goes one step further and permits communication between secondary users as long as the disturbance created to the primary system is below some predefined threshold. Clearly, in this case, the secondary terminals need not only assess whether primary users are transmitting or not but also how much interference they will create and whether this will disrupt the primary communication. Finally, the overlay paradigm allows for a tight interaction between primary and secondary systems. Of course, this comes not only at the price of a higher degree of sophistication of the secondary terminals but also requires flexibility in the primary system. Nevertheless, in all three cases, it is necessary to assess the impact of the presence of secondary users on primary systems. Several measures have been discussed in the literature for this purpose, for example, the probabilities of miss detection and interference for interweave cognitive radio or, more in general, soft and peakpowershaping interference temperature constraints[3, 4]. An alternative is to consider directly the degradation suffered by the primary users, for example, in terms of the loss in rate[5].
Research on the physical layer has focused on establishing basic models for the different cognitive radio scenarios, deriving their fundamental limits, and designing practical transceivers that come close to these limits. From an information theoretic point of view, two channel models have been considered for the three cognitive radio paradigms: the Gaussian interference channel[6, 7] and the cognitive radio channel[8–10]. As described before, in the cases of interweave and underlay cognitive radio, there is no cooperation between primary and secondary systems. This is precisely the situation described by the interference channel. The interweave cognitive radio paradigm corresponds to time sharing in the interference channel[6], with a sharing parameter that is fixed by the activity of the primary users. In this case, the challenge lies almost exclusively in sensing accurately the primary activity, a topic that lies outside the scope of this paper (see, e.g.[11] and references therein). Therefore, interweave cognitive radio scenarios will not be considered here. On the other hand, in the case of underlay cognitive radio, primary and secondary systems can transmit at the same time and thus the scenario is richer from the point of view of the communication strategies that can be used. This is well characterized by the interference channel if one places some additional restrictions on the model. For example, one usually restricts the communication strategies used by the primary user pairs to consist of pointtopoint codes and singleuser decoding.
In contrast, overlay cognitive radio scenarios are not described properly by the interference channel. The main reason for this is that the interference channel does not allow for any active cooperation between the user pairs. With the aim of overcoming this limitation, the cognitive radio channel was introduced in[9]. This model extends the interference channel by assuming that the secondary transmitter has noncausal knowledge of the primary message. This additional knowledge allows for asymmetric cooperation in the sense that the secondary transmitter can help the primary users to carry their communication. In addition, it can combat the interference that the primary signal creates on the secondary receiver by means of interference cancellation or dirtypaper coding. This asymmetric cooperation was key for establishing the capacity of the cognitive radio channel with weak interference[8, 9].
A usual system design criterion is to maximize the rate of transmission for the secondary users while ensuring a minimum quality of service (QoS) for the primary users. A key observation is that multiple transmit antenna techniques are a powerful and efficient way of controlling the disturbance created by the secondary users[12]. Unfortunately, the use of such techniques often leads to complex matrix optimization problems. This has motivated the use of tools from optimization theory for the design of transceivers. For example, convex optimization tools were used in[13] to study underlay cognitive radio models with singleuser decoders. An underlay scenario with rate splitting and multipleuser decoding was considered in[14]. The problem of distributed beamforming and rate allocation in decentralized cognitive radio networks was treated in[15]. In a more general framework, the set of efficient strategies for multipleinput singleoutput (MISO) interference networks was characterized in[16, 17] in terms of beamformers. The extension of the cognitive radio channel to the multipleinput multipleoutput (MIMO) case was introduced in[18]. Overlay cognitive radio strategies for this channel with partial channel state information were considered in[19]. Optimal beamforming for the coexistence of a MIMO secondary user with a MISO primary user with noncausal knowledge of the primary message was considered in[20]. We studied the coexistence of a MISO secondary system with a singleinput singleoutput primary system in[21] for different levels of channel state information, and considered linear precoding strategies in[22].
A comparison of the results for underlay and overlay cognitive radio channel models suggests that the additional knowledge of the primary message at the secondary transmitter in the cognitive radio channel leads to significantly higher achievable rates[21]. However, a critical point is how the secondary transmitter can acquire such knowledge in practice. Clearly, requiring the secondary transmitter to learn actively the primary message before communicating will lead to an inevitable loss in rate for the secondary users, especially under practical constraints such as half duplex communication. Some authors have motivated practical scenarios in which the primary message is obtained causally. For example, the secondary users may overhear a primary automatic repeat request (ARQ) session and use their resources during the repetition phases to help the primaries finish their transmission earlier or to exploit the inefficiencies of the ARQ protocol[23, 24]. Similarly, in[25], the secondary system acquires the primary message and uses it to help the primary system finish the transmission earlier and then use the channel during the idle period. However, these schemes do not fully exploit the possibilities of overlay cognitive radio, in particular the possibility of interaction between primary and secondary systems. The use cooperative communication techniques[26–28] as an enabling technology for cognitive radio networks was surveyed in[29]. They were considered in[30] for singleantenna overlay cognitive radio and evaluated in terms of outage probabilities. The optimal secondary power allocation and phase split in a twophase spectrum sharing scenario was considered in[31]. In[32], the authors studied beamforming and power allocation for the coexistence of a primary singleinput singleoutput (SISO) user with a secondary singleinput multipleoutput or MISO that acquired the message in a causal fashion. However, as opposed to the work presented here, their work focused only on the second phase of communication, without considering explicitly the first, learning phase. In[33], beamforming and power allocation were studied for a system, where the secondary users relay the primary signal in an amplifyandforward fashion, and the performance of the proposed system was compared to an underlay cognitive radio scheme. The use of cooperative relaying mechanisms for spectrum sensing and secondary user transmission in cognitive radio systems was described in[34, 35].
1.1 Contributions and outline
We study physicallayer aspects of cognitive radio communications in a scenario, where a MISO primary system coexists with a halfduplex MIMO secondary system. We consider two approaches: on one hand, an underlay cognitive radio model without any cooperation between primary and secondary systems. On the other hand, an overlay cognitive radio model that allows for causal cooperation between the systems. Our goal is to compare both strategies and assess the potential advantages of each of them under conditions that are more realistic than the original cognitive radio channel model in[8, 9]. In particular, we require that the primary message be learned causally by the secondary system.
We emphasize that this paper deals with idealized models. In particular, the overlay scenario requires a high degree of cooperation between primary and secondary systems. Similarly, quite often, the terminals have access to larger portion of channel state information than in practical systems. In spite of this idealization, we have decided to take this approach to quantify the benefits of having coordinated primary and secondary system (through the messagelearning phase) in a quite general way, as compared to the more ad hoc approaches in[23–25]. Moreover, these systems are, at least in theory, implementable, unlike the less realistic scenarios where the secondaries have noncausal knowledge of the primary messages.
This paper extends our previous work on the coexistence of a SISO primary system with a MISO secondary link for underlay[14] and overlay systems[36] to the case of coexisting MISO primary and MIMO secondary systems. The addition of multiple antennas at the primary transmitter and secondary receiver results in a model that is richer and substantially more complex. In particular, for the overlay scenario, the new model allows not only for MIMO communication between secondary users but also for MIMO intertransmitter communication. Moreover, this new channel configuration represents a departure from the interference network (e.g.[17]) as it also incorporates aspects from cooperative communications. Finally, note that the convex optimization framework developed in[13] for underlay cognitive radio is not directly applicable to the strategies presented here because they result in nonconvex problems.
The main contributions of this paper refer to the coexistence of a MIMO secondary link with a MISO primary system. They are the following: First (Section 3), we consider an underlay strategy that includes rate splitting and interference decoding at the secondary and characterize completely the set of transmission parameters that maximize the secondary rate subject to a constraint on the primary rate. Second (Section 4), we establish a transmission strategy for cognitive radio communication over an extended channel model that consists of an initial learning phase, followed by a communication phase. This strategy combines elements from cooperative communications and communication over a noncausal cognitive radio channel that exploit the special properties of the extended cognitive radio channel model. In addition, we characterize the set of parameters that maximize the rate of the secondary users under a primary rate constraint and formulate simple algorithms to find such parameters. Third (Section 5), using a simple geometrical model, we evaluate numerically the performance of the strategies and compare them to establish the regions in which each of them outperforms the other. To our knowledge, this is one of the few studies that try to quantify the advantages of the informationtheoretic cognitive radio channel models under realistic conditions (i.e. without assuming noncausal knowledge of the primary message). Finally (Section 6), we discuss the extension of all these contributions to MIMOMIMO coexistence scenarios. The last part (Section 7) concludes our work. For clarity of exposition, we present the proofs of all the results in the ‘Appendices’ Section.
2 Preliminaries
2.1 Notation
Column vectors and matrices are represented in lower case and upper case boldface letters, respectively. · is the absolute value of a scalar or the determinant of a matrix, · is the Frobenius norm of a vector or matrix, and (·)^{H} stands for Hermitian transpose. The trace of a square matrix is denoted by tr{·}.${\mathbf{\Pi}}_{\mathit{X}}\triangleq \mathit{X}{\left({\mathit{X}}^{H}\mathit{X}\right)}^{1}{\mathit{X}}^{H}$ denotes the orthogonal projection operator onto the column space of X, and${\mathbf{\Pi}}_{\mathit{X}}^{\perp}\triangleq \mathit{I}{\mathrm{\Pi}}_{\mathit{X}}$, where I is the identity matrix, denotes the orthogonal projection operator onto the orthogonal complement of the column space of X. The notation X ≽ 0 denotes that the matrix X is positive semidefinite. All logarithms in this paper are taken to the base of 2, and all rates are expressed in bits.
2.2 System model
We consider a MISO primary system with N_{T,1} transmit antennas that is willing to share its channel with a halfduplex MIMO secondary system with N_{T,2} antennas at the transmitter and N_{R,2} antennas at the receiver. Our goal is to compare basic communication strategies for underlay and overlay cognitive radio without assuming noncausal knowledge of the primary message at the secondary transmitter. For this purpose, we introduce the following two channel models.
2.2.1 Underlay cognitive radio
We use the Gaussian MIMO/MISO interference channel as a model to study the conflict between a primary and a secondary link in underlay cognitive radio. Each of the transmitters sends a signal that is observed by the intended receiver in the presence of interference (from the other transmitter) as well as white Gaussian noise. The t^{th} received sample from the matchedfiltered complex baseband model is
where x_{1}(t) and x_{2}(t) are the N_{T,1} × 1 and N_{T,2} × 1 signal vectors sent by the primary and secondary transmitters, respectively, h_{i1} is the N_{T,i} × 1 vector of the channel gains from transmitter i ∈ {1,2} to receiver 1, and H_{i2} is the N_{T,i} × N_{R,2} matrix of channel gains from transmitter i ∈ {1,2} to receiver 2. The scalar y_{1}(t) and the vector y_{2}(t) are the observations at the receivers, which are corrupted by the noise processes n_{1}(t) and n_{2}(t), respectively.
2.2.2 Overlay cognitive radio
Our model for communication with halfduplex devices in an overlay cognitive radio environment is illustrated in Figure1 and consists of two phases. In the first phase, the primary transmitter broadcasts its message to both its intended receiver and the secondary transmitter. The t^{th} received sample from the matchedfiltered complex baseband model in this phase is
where${\mathit{x}}_{1}^{(1)}(t)$ is the N_{T,1} × 1 signal vector sent by the primary transmitter, h_{11} is the N_{T,1} × 1 vector of channel coefficients between primary transmitter and receiver, and H_{t} is the N_{T,1} × N_{T,2} matrix of channel coefficients between both transmitters. The scalar${y}_{1}^{(1)}(t)$ and the N_{T,2} × 1 vector y_{st}(t) are the observations at the primary receiver and secondary transmitter, respectively, which are corrupted by the noise processes${n}_{1}^{(1)}(t)$ and n_{st}(t), respectively. Note that, in principle, the secondary receiver can also obtain its own observation${\mathit{y}}_{2}^{(1)}$ of the primary signal. However, as we shall see, this does not provide any gain for the transmission strategy proposed in Section 4.1.
The second phase corresponds to the setup which is known as the cognitive radio channel. In this phase, the secondary transmitter can make use of the knowledge of the primary message (obtained in a causal fashion in the first phase). The model in this phase is
where${\mathit{x}}_{1}^{(2)}(t)$ and${\mathit{x}}_{2}^{(2)}(t)$ are the N_{T,1} × 1 and N_{T,2} × 1 signal vectors sent by the primary and secondary transmitters, respectively, h_{i1} is the N_{T,i} × 1 vector of channel gains from transmitter i ∈ {1,2} to receiver 1, and H_{i2} is the N_{T,i} × N_{R,2} matrix of channel gains from transmitter i ∈ {1,2} to receiver 2. The scalar${y}_{1}^{(2)}(t)$ and the vector${\mathit{y}}_{2}^{(2)}(t)$ are the observations at the receivers, which are corrupted by the noise processes${n}_{1}^{(2)}(t)$ and n_{2}(t), respectively.
The entire transmission is carried out over n channel uses; k channel uses are consumed during the first transmission phase, and (n − k) channel uses during the second phase. The fraction of the channel uses in the first and the second phases is given by α = k/n and 1 − α, respectively. We will assume that the channels remain constant during the duration of the two phases.
Noise and channel statistics
For both underlay and overlay cognitive radio models, the noises at the receivers are modeled by independent circularly symmetric additive white Gaussian noise processes with unit variance:${n}_{1},{n}_{1}^{(1)},{n}_{1}^{(2)}\sim \mathcal{C}\mathcal{N}(0,1),{\mathit{n}}_{2},{\mathit{n}}_{\text{st}}\sim \mathcal{C}\mathcal{N}(0,\mathit{I})$. In this paper, we assume that all nodes have perfect channel knowledge on all links. In order to evaluate the average behavior of our transmission strategies for different realizations of the channel coefficients, we will model the entries of H_{t},h_{11},H_{12},h_{21}, and H_{22} as samples from independent circularly symmetric Gaussian processes with zero mean with appropriate variances.
3 Underlay cognitive radio
In this section, we introduce the transmission strategy that we consider for the underlay cognitive radio paradigm. Our goal is to maximize the communication rate of the secondary users while ensuring that the primary users have a minimum QoS, defined in terms of a minimum rate${R}_{1}^{\star}$.
3.1 Underlay transmission strategy
We consider the extension to MIMO secondary systems of the underlay transmission strategy introduced in[14]. The primary transmitter is oblivious to the presence of the secondary users and broadcasts its singlestream signal with power P_{1} using the covariance matrix K_{1} corresponding to the maximumratio transmit (MRT) beamformer, i.e.${\mathit{K}}_{1}={P}_{1}\frac{{\mathit{h}}_{11}{\mathit{h}}_{11}^{H}}{{\Vert {\mathit{h}}_{11}\Vert}^{2}}$. The primary receiver decodes the message in the presence of interference from the secondary system and noise. The secondary transmitter splits its message into two parts (i.e. rate splitting) using possibly different covariance matrices with possibly different powers for each of the parts: K_{2,1} and K_{2,2}, respectively. The secondary receiver performs successive/interference decoding to recover the first part of the secondary message, then the primary message (i.e. the interference), and finally the second part of the secondary message.
The communication rate for the primary users is
and the rate achieved by the secondary users is
The first term in (8) corresponds to the part of the secondary message decoded in the presence of interference (both from primary transmitter and selfinterference). The second term in (8) corresponds to the part of the secondary message recovered after decoding and subtracting the primary message. This adds the constraint that the secondary receiver must be able to decode the primary message as well. That is,
In addition, we have the constraint on the QoS for the primary user, i.e.${R}_{1}^{\text{und}}\ge {R}_{1}^{\star}$. Note that by setting appropriately K_{2,1} and K_{2,2}, we obtain the extreme cases, where the secondary receiver decodes first the primary message or does not decode it at all.
We remark that we do not make any assumption on the rank of the matrices K_{2,1} or K_{2,2}. Basic considerations on the number of transmit/receive antennas required for multiplestream transmission apply here, too (see e.g.[37]).
3.2 Problem formulation
The problem of finding the covariance matrices K_{2,1} and K_{2,2} that maximize the secondary rate under the aforementioned constraints is expressed as
where it is implicitly assumed that (10c) applies only if K_{22} ≠ 0. Note that this problem is not concave due to the constraints (10b) and (10c). Constraint (10b) can easily be transformed into a linear constraint. However, dealing with (10c) is more involved.
3.3 Optimal transmission parameters
The following proposition characterizes the solution to (10). This extends the result in[14] to MIMO secondaries.
Proposition 1
The optimization problem in (10) admits the following solution:
Case 1: If
then decoding the primary message at the secondary receiver is not possible at all. Without interference decoding, we have that K_{2,2} = 0, and K_{2,1} is the covariance matrix that maximizes
subject to the corresponding constraints. This is equivalent to solving the following concave problem:
where
Case 2: If
where Σ^{⋆} is the covariance matrix that solves the concave problem
with${P}_{\text{int}}^{\text{und}}$ as defined in (14), then it is possible to decode the interference directly, without using rate splitting. Thus, the optimal covariance matrices are K_{2,1} = 0 and K_{2,2} = Σ^{⋆}.
Case 3: In all other cases, i.e. if
the problem is solved by K_{2,1} = γ Δ^{⋆} and K_{2,2} = (1−γ)Δ^{⋆}, where γ ∈ [0,1] is chosen such that${R}_{1,2}^{\text{und}}={R}_{1}^{\star}$, and Δ^{⋆} is the matrix that solves the following concave problem
with${P}_{\text{int}}^{\text{und}}$ as defined in (14).
Proof
The proof is provided in Appendix 1. □
Remark 1
In all three cases, the solution can be efficiently obtained using convex optimization tools[38].
Remark 2
The preceding results for case 3 reveal that the same covariance matrix (up to a scaling factor) is used for both parts of the secondary message when using rate splitting. For the case of beamformers, which are optimal for MISO secondaries (see e.g.[17] or[39]), this means that it suffices to consider the same beamformer for both parts of the secondary message (cf.[14]).
4 Overlay cognitive radio with explicit messagelearning phase
In this section, we introduce the transmission strategy that we consider for the overlay cognitive radio paradigm. Our goal is again to maximize the communication rate of the secondary users while ensuring that the primary users have a minimum QoS, defined in terms of a minimum rate${R}_{1}^{\star}$.
4.1 Overlay transmission strategy
Our strategy for overlay cognitive radio combines cooperative communication techniques, in particular decodeandforward (DF)[26–28], with communication for cognitive radio channels[8, 9]. The strategy makes full use of the potential of overlay cognitive radio by establishing active asymmetric cooperation between the users. The protocol establishes transmission of the primary message in two phases. Moreover, the primary transmitter chooses the system parameters as to maximize the system efficiency while ensuring that its message is reliably communicated. The secondary transmitter, which only broadcasts during the second phase, not only sends its own message but also acts as a relay for the message of the primary users. In addition to this, some degree of cooperation in the process of channel estimation is required so that the transmitters obtain the relevant channel state information.
Let${R}_{1}^{\star}$ be the target rate of the primary users. In the first phase, of relative duration α, the primary transmitter broadcasts its message using the N_{T,1} antennas with transmit covariance matrix${\mathit{K}}_{1}^{(1)}\succcurlyeq 0$. The primary receiver and secondary transmitter listen to this transmission. Consider the rates
and let${P}_{1}^{(1)}$ denote the power spent by the primary transmitter in the first phase, i.e.${P}_{1}^{(1)}\triangleq tr\{{\mathit{K}}_{1}^{(1)}\}$. Expressions (19) and (20) correspond to the rates from the primary transmitter to the primary receiver and to the secondary transmitter in the first phase, respectively.
If the channel H_{t} is significantly better than h_{11} (e.g.$tr\{{\mathit{H}}_{\mathrm{t}}^{H}{\mathit{H}}_{\mathrm{t}}\}\gg {\Vert {\mathit{h}}_{11}\Vert}^{2}$), then the secondary transmitter will need less redundancy to decode the message. In particular, if
then the secondary transmitter can decode the primary message but the primary receiver cannot. Although it cannot decode, the primary receiver has collected useful observations of the primary signal. Roughly speaking, it only needs additional redundancy to resolve its uncertainty and be able to decode[26].
Once the secondary transmitter is able to decode, the system can switch to the second phase. The second phase has the duration 1−α and consists of two simultaneous transmissions. On one hand, primary and secondary transmitters cooperate to resolve the uncertainty of the primary receiver. They act as one single virtual transmitter that uses a virtual covariance matrix
to send the remaining part of the primary message over the extended channel${\mathit{h}}_{\text{ext}}^{H}=[{\mathit{h}}_{11}^{H},{\mathit{h}}_{21}^{H}]$ that consists of the concatenation of both channels to the primary receiver. The submatrices${\mathit{K}}_{1}^{(2)}$ and K_{r} correspond to actual the covariance matrices used by each transmitter, while the submatrix Ψ corresponds to correlation of the signals sent by each transmitter, so that they add constructively at the receiver (cf.[18], Eq. (3)). Note that while they act coordinately, each transmitter has an independent power constraint (i.e. on$tr\{{\mathit{K}}_{1}^{(2)}\}$ and tr{K_{r}}, respectively): the primary transmitter uses the power left after the first phase, while the secondary uses only a fraction of its available power. Simultaneously with this cooperative transmission, the secondary transmitter employs the remaining power and a different covariance matrix K_{p} for private communication to the secondary receiver. Moreover, it can use the knowledge of the primary message to predict the interference that the secondary receiver will experience and precode against it using dirty paper coding. Using this strategy, the rates
are achievable for transmitting information about the primary message and the secondary message during the second phase. The factor$\frac{1}{1\alpha}$ in front of the matrices K_{p} and K_{r} scales up the power to take into account the duration of the second phase.
Using DF relaying arguments (see e.g.[26, 27]), it is possible to show that the rate
is achievable for the primary users. Note that at this point, we do not make any assumption on the rank of the covariance matrices. In particular, K_{p} can incorporate multiple streams, subject to the usual constraints[37].
Remark 3
We stress that it is necessary that${R}_{\mathrm{t}}\ge {R}_{1}^{\star}$ to start the second phase. However, enforcing${R}_{\mathrm{t}}={R}_{1}^{\star}$ does not necessarily yield the largest secondary rate. As we will see, it is sometimes better to extend ‘artificially’ the duration of the first phase.
Remark 4
The requirement of decoding the primary message at the secondary transmitter in combination with the use of dirty paper coding during the second phase renders ineffective the direct observation${\mathit{y}}_{2}^{(1)}$ of the primary message obtained by the secondary receiver obtained during the first phase, that is, the rate (24) is already free from interference.
4.2 Problem formulation
We are interested in finding the choice of phase splitting α, covariance matrices${\mathit{K}}_{1}^{(1)},{\mathit{K}}_{1}^{(2)},{\mathit{K}}_{\mathrm{p}}$ and K_{r}, and the correlation matrix Ψ that maximize the secondary rate R_{2} while ensuring a target rate${R}_{1}^{\star}$ for the primary user pair under average power constraints P_{1} and P_{2} at the primary and secondary transmitters, respectively. This is formulated mathematically as
We characterize the solution to (26) in the following section.
4.3 Optimal transmission parameters
The problem in (26) is not convex; in particular, dealing with constraint (26c) is problematic. An exhaustive search over the 6 parameters seems unfeasible too. Our approach is to study the properties of the optimal parameters through a series of propositions. Then, we use them to reduce the optimization problem to a simpler search over a small set of bounded realvalued parameters and to find efficient algorithms to calculate the numerical values of the system parameters.
4.3.1 Characterization of the solution
As it was discussed in Section 4.1, our transmission strategy is reasonable only if the secondary transmitter can decode the primary message earlier than the primary receiver. This condition appears in the characterization of the solution to (26) and is captured by the following definition:
Definition 1 (Cooperation condition)
Let
for some$\sigma \in {\mathbb{R}}^{+}$. We say that the cooperation condition is satisfied if
The matrix K^{WF}(σ) corresponds to the waterfilling (WF) solution with power constraint σ. Note that if the cooperation condition is not satisfied, the primary receiver may decode the message earlier than the secondary transmitter when the transmission is optimized for the latter. In addition, we assume that K^{WF}(σ) is never proportional to the MRT covariance matrix$\frac{{\mathit{h}}_{11}{\mathit{h}}_{11}^{H}}{{\Vert {\mathit{h}}_{11}\Vert}^{2}}$. This technical condition simply ensures that the transmission between transmitters is never strictly colinear with h_{11} because this case would virtually turn the primary transmitter into a singleantenna transmitter.
The first observation that we make regarding the solution to (26) concerns the power used by the transmitters. Over the two phases, the primary transmitter uses all its available power. Note that this power is in general distributed unequally over the phases. Similarly, the secondary transmitter also exhausts all its power, distributing it between the two simultaneous transmissions: cooperation and private communication. This is stated in the following proposition.
Proposition 2
The optimal transmission strategy in (26) makes use of all the available power at the primary and secondary transmitters, that is,

1.
tr{K _{p} + K _{r}} = P _{2},

2.
$\alpha \phantom{\rule{.3em}{0ex}}tr\{{\mathit{K}}_{1}^{(1)}\}+(1\alpha )tr\{{\mathit{K}}_{1}^{(2)}\}={P}_{1}$.
Proof
The proof is provided in Appendix 2. □
Our second observation is that the presence of the secondary transmitter always pushes the primary system to the limit of decodability as described by the following proposition:
Proposition 3
The set of parameters that solves the optimization problem in (26) satisfies
(i.e. constraint (26c) with equality) if the cooperation condition is satisfied.
Proof
The proof is provided in Appendix 3. □
This result is a consequence of the tight interaction between users allowed in overlay cognitive radio scenarios. On one hand, the secondary system makes use of its resources in the way that maximizes the rate R_{2}. At the same time, the primary transmitter cooperates towards this goal by distributing its resources between the two phases in the way that R_{2} is maximized. For example, it may choose a covariance matrix${\mathit{K}}_{1}^{(1)}$ that makes the first phase shorter if this is beneficial in terms of secondary rate.
We can make a similar observation with respect to the communication between transmitters in the first phase.
Proposition 4
The set of parameters that solve the optimization problem in (2) satisfies
(i.e. constraint (26b) with equality) unless the optimal covariance matrix${\mathit{K}}_{1}^{(1)}$ is proportional to the orthogonal projector onto h_{11}, that is, proportional to$\frac{{\mathit{h}}_{11}{\mathit{h}}_{11}^{H}}{\parallel {\mathit{h}}_{11}{\parallel}^{2}}$.
Proof
The proof is provided in Appendix 4. □
This result can be interpreted in terms of the duration of the phases. In the cases where (30) holds, the system switches from first phase to second phase as soon as the secondary transmitter can decode the primary message. However, (30) is not always satisfied; hence, this is not true in general. In fact, it is sometimes beneficial to extend ‘artificially’ the first phase in order to achieve a larger secondary rate. For example, if the primary transmitter only has one antenna, then we cannot find nontrivial conditions that ensure${R}_{\mathrm{t}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{R}_{1}^{\star}$. The reason for this is that with only one antenna, there is no way to distinguish directions, i.e. we always transmit in the direction to the primary receiver. Similarly, it was observed in[27] in the context of DF for singleantenna Gaussian relay channels that the optimal split of phases has to be found numerically.
Although Proposition 4 only gives a partial characterization of the covariance matrix${\mathit{K}}_{1}^{(1)}$, it turns out to be very useful when it comes to finding its value numerically. Combined with Proposition 2, it allows us to derive Algorithm 1 that efficiently finds${\mathit{K}}_{1}^{(1)}$ given the optimal values of the phase split α and the power used by the primary in the first phase (i.e.${P}_{1}^{(1)}\triangleq tr\{{\mathit{K}}_{1}^{(1)}\}$).
Algorithm 1 starts by verifying (line 4) if a solution to (26b) exists for the given level of power${P}_{1}^{(1)}$ by allocating it freely, as in K_{f}, to maximize the expression in line 3. Provided that such solution exists, the algorithm verifies if MRT beamforming to the primary receiver (i.e. in the direction of h_{11}, using the covariance matrix K_{h}) is sufficient for decoding at the secondary transmitter (26b) (line 9). If MRT does not satisfy (26b), then it uses the bisection method (Algorithm 2) to find the covariance matrix with largest component in the direction of h_{11} that satisfies (26b). The search finishes when the rate achieved for this choice of covariance matrix exceeds the target rate${R}_{1}^{\star}$ by less than a predefined threshold ε. The maximization in Algorithm 1 (line 3) and in the bisection method (Algorithm 2, line 8) can be written as standard waterfilling problems, which can be efficiently approximated or solved exactly (see e.g.[40]). The following corollary establishes the the optimality of Algorithm 1.
Corollary 1
Given the optimal values of α and power${P}_{1}^{(1)}$ used by the primary in the first phase, Algorithm 1 finds the optimal covariance matrix${\mathit{K}}_{1}^{(1)}$ if the cooperation condition is satisfied.
Proof
The proof is provided in Appendix 5. □
Remark 5
Note that, by construction, if a call to Algorithm 1 results in the MRT covariance matrix for some$(\alpha ,{P}_{1}^{(1)})$, then it will also result in the MRT covariance matrix for any$(\alpha ,{\stackrel{~}{P}}_{1}^{(1)})$ with${\stackrel{~}{P}}_{1}^{(1)}>{P}_{1}^{(1)}$.
We conclude this section by characterizing the optimal covariance matrices used in the second phase.
Proposition 5
The optimal covariance matrices in the second phase are given by
and K_{p} is the solution to the following concave problem:
where
for some P_{r} ∈ [0,P_{2}] such that P_{int} ≥ 0.
Proof
The proof is provided in Appendix 6. □
The interpretation of the optimal values for${\mathit{K}}_{1}^{(2)}$ and K_{r} is straightforward: they are adapted to their respective channels and combine coherently at the receiver. The matrix K_{p} used for the secondary communication is chosen to maximize the secondary rate without violating the interference constraint at the primary.
In the case of secondary MISO systems (i.e. h_{12} and H_{22} instead of h_{12} and H_{22}, respectively), there is no loss in restricting the covariance matrix K_{p} at the secondary transmitter to have rank 1, i.e.${\mathit{K}}_{\mathrm{p}}=({P}_{2}{P}_{\mathrm{r}}){\mathit{w}}_{\mathrm{p}}{\mathit{w}}_{\mathrm{p}}^{H}$. The following corollary characterizes the optimal beamforming vector w_{p}.
Corollary 2
The optimal beamformer w_{p} is
with
with P_{int} as defined in (34), for some P_{r} ∈ [0,P_{2}] such that P_{int} ≥ 0.
Proof
The proof is provided in Appendix 7. □
In the MISO case, we see more clearly that the beamformer w_{p} used for the secondary communication is chosen to be the one with largest projection over H_{22} that satisfies the interference constraint, which is determined by the projection over h_{21}[13, 16].
4.3.2 An algorithm to find the optimal parameters
The results from the previous section allow us to reduce the solution to (26) to a search over three realvalued parameters: the phase split α, the power spent by the primary in the first phase (i.e.${P}_{1}^{(1)}\triangleq tr\{{\mathit{K}}_{1}^{(1)}\}$), and the distribution of power between relaying and private communication at the secondary (e.g. P_{r} = tr{K_{r}}). Each of these parameters is defined in a closed and bounded interval. In contrast, solving (26) directly requires search over one realvalued parameter and five complexvalued matrices. We have summarized this simplified search in Algorithm 3, which we describe in the following:
To find the solution, we perform a search over the phase split α and the admissible power for the primary transmitter in the first phase${P}_{1}^{(1)}$. Given these two values, the matrix${\mathit{K}}_{1}^{(1)}$ is found using Algorithm 1, whereas${\mathit{K}}_{1}^{(2)}$ is readily determined. To obtain the remaining matrices K_{p},K_{r} and Ψ, we perform a search over the different splits of secondary power using the results in Proposition 5. The optimal choice of parameters is the one that yields the largest secondary rate R_{2}.
5 Numerical evaluation
5.1 Geometrical model
To present our results, we will use the simple geometrical model in Figure2, in which the different nodes are placed on a plane. The relative positioning of the nodes is summarized by the distance between each pair of nodes. We model the block flat fading channel coefficient between two nodes as
where d_{ij} is the distance between them, p is the path loss exponent, and${\stackrel{~}{h}}_{\mathrm{i}\mathrm{j}}\sim \mathcal{C}\mathcal{N}(0,1)$. In the case of channel vectors or matrices, each of the entries is independently modeled as in (38).
For convenience, we normalize all distances with respect to the distance between the primary users (i.e. d_{11} = 1). We will consider the square surface {(x,y) : x ∈ [0,1], y ∈ [0,1]}, and vary the position of the secondary nodes (relative to the primary nodes) over a regular square grid of size 11×11, that is, we will move the secondary transmitter and receiver over this grid, always parallel to the line between primary transmitter and receiver (as in Figure2). The primary transmitter and receiver will be fixed at positions (0,0.5) (black filled circle) and (1,0.5) (black filled box), respectively.
In the plots, a pair of coordinates (x,y) identifies the position of the secondary transmitter. All our results consider d_{22} = 1/4 while the remaining distances d_{12},d_{21} and d_{tt} vary as described before. This models a secondary middlerange communication in the presence of primary users.
5.2 Note on the strategies
The overlay strategy in Section 4.1 yields R_{2} = 0 for some channel realizations. The reason for this is that constraint (26b) cannot always be fulfilled for R_{2} > 0. In such a scenario, a cognitive radio system would switch to a different transmission strategy that can provide a nonzero secondary rate R_{2}. For example, it could switch to the underlay transmission mode presented here. In this way, the hybrid overlayunderlay strategy would never perform worse than the pure underlay strategy. However, including such a functionality in our experiments is against the nature of our work, which is to compare the underlay and overlay scenarios, and evaluate the effect of the learning phase. For this reason, we implement the strategies exactly as described in Sections 3.1 and 4.1.
5.3 Complexity of the strategies
The complexity of the underlay solution varies for the different cases in Proposition 1, which depend on the instantaneous channel conditions. For cases 1 and 2, the complexity is that of solving one concave problem ((13) and (16), respectively). For case 3, the complexity is that of solving two concave problems: (16) (to check the constraint) and (18), and finding the optimal split γ (e.g. using a loop or a bisection method). For MISO secondaries, the complexity can be lowered (e.g. using Remark 2 and[14]).
In contrast, Algorithm 3 finds the optimal overlay transmission parameters by searching over threereal valued parameters defined on a closed and bounded space. Up to a scaling factor that depends on the powers, the matrices${\mathit{K}}_{1}^{(2)},{\mathit{K}}_{\mathrm{r}}$ and Ψ can be determined before hand. The covariance matrix${\mathit{K}}_{1}^{(1)}$ needs to be determined for each pair (α, tr{K}) using Algorithm 1. This algorithm relies on the waterfilling and bisection methods that can be implemented very efficiently (see e.g.[40]). In addition, note that Remark 5 can be used to minimize the number of calls to Algorithm 1. The optimal K_{p} needs to be determined for each triple$(\alpha ,{P}_{1}^{(1)},{P}_{\mathrm{r}})$ by solving the concave problem in (32), which can also be implemented efficiently. Solving this last problem can be avoided in the case where K_{p} has rank 1 using the results in Corollary 2.
When compared, it is clear that the complexity of solving the overlay problem is significantly larger than that of the underlay problem, in particular for the case where K_{p} is not rank 1. Nevertheless, the solution to both problems reduces to solving concave problems, for which a large variety of efficient algorithms exist (see e.g.[38]).
5.4 Simulation results
We have performed extensive simulations of our underlay and overlay cognitive radio strategies to assess their individual performances and merits relative to each other. We show here results for a few representative cases and comment in the end on the differences for other system parameters.
In the results in Figures3,4 and5 the transmitters are equipped with N_{T,1} = N_{T,2} = 2 antennas, and the receivers with one single antenna. In contrast, in Figure6, we study the behavior for varying N_{T,1} and N_{T,2} and singleantenna receivers. In all cases, the path loss exponent is fixed to p = 3, and the primary power is set to P_{1} = 10 dB. The secondary power is P_{2} = 1 dB for the results in Figures3 to5 and variable for Figure6. We assume that the primary system has a target rate${R}_{1}^{\star}$ that corresponds to a fraction ρ of its instantaneous pointtopoint Shannon capacity, that is,
We refer to ρ as the load factor of the primary system. We consider ρ = 0.75 for Figures3 to5, and ρ = 1 for Figure6. Every point in the plots represents the average over 5 · 10^{4} independent realizations of the channels. We focus on the results for the overlay strategy and the comparison between the strategies because the results for the underlay strategy alone do not differ qualitatively from the singleantenna case in[14].
Figure3 shows the average of the secondary rate R_{2} (in bits per channel use, bpcu) achieved by our overlay cognitive radio strategy for N_{T,1} = N_{T,2} = 2,N_{R,2} = 1, P_{1} = 10 dB, P_{2} = 1 dB, p = 3 and ρ=0.75. To set the numerical values in the figure in a context note that if the secondaries were alone in the scenario, the ergodic capacity would be 6.96 bpcu. In comparison, the highest average secondary rate in Figure3 is R_{2} = 6.29 bpcu and is obtained when primary and secondary transmitters are closely located. This represents 90% of the aforementioned capacity. As one would expect, the average secondary rate becomes lower as the two transmitters are separated.
It is more interesting to look at the advantage in average rate over the underlay strategy. Figure4 shows the ratio between the average of the secondary rate for overlay R_{2} and the average of the secondary rate for underlay${R}_{2}^{\text{und}}$ for N_{T,1} = N_{T,2} = 2,N_{R,2} = 1, P_{1} = 10 dB, P_{2} = 1 dB, p = 3 and ρ = 0.75. The results are somewhat surprising in the sense that the largestadvantage region does not correspond to the largestsecondaryrate region, that is, the maximum in Figure4 is not obtained for (x,y) = (0,0.5) but rather for (x,y) ≈ (0.4,0.5). The reason for this is that for (x,y) = (0,0.5), the underlay strategy also benefits from closely located transmitters, thanks to the interference decoding functionalities. In fact, if one removes this functionality in the underlay transmission mode, the results change significantly. In that case, the overlay system is overwhelmingly better than the underlay strategy.
In addition, note that the advantage of the overlay system diminishes as the two transmitters are separated. In fact, in some regions, using the underlay strategy is better in terms of average secondary rate. The reason for this is simple: in these regions, the first phase is relatively long (e.g. α > 0.5), and the higher sophistication of the secondary transmitter (i.e. dirtypaper coding, cooperative transmission) cannot compensate for the loss in secondary rate due to the passive first phase. Thus, the underlay approach, even if it has to transmit mainly in the zeroforcing direction to avoid interference, can make a more efficient use of the resources and provide a larger rate to the secondary users.
In order to implement a system that combines both strategies (as discussed in Section 5.2), it is desirable to know how often they outperform each other. This is shown in Figure5, in terms of the percentage of channel realizations for which the overlay strategy yields a larger rate than the underlay strategy for N_{T,1} = N_{T,2} = 2,N_{R,2} = 1, P_{1} = 10 dB, P_{2} = 1 dB, p = 3 and ρ = 0.75. Again, we observe that the region with largest rate corresponding to the overlay strategy does not correspond exactly to the collocation of transmitters. In the figure, we observe that, except for a small region where overlay is better over 90% of the time, there is room for significant improvement if the system implements both strategies and chooses the best one in each block.
Regarding variations in the scenario, we have observed the following general trends. The secondary rate (Figure3) increases with both the number of antennas and the secondary power as one would expect. More interestingly, as we increase the secondary power P_{2} or the number of antennas, the maximum in Figure4 (i.e. the advantage of overlay in terms of average rate) increases its value and shifts its position towards the primary transmitter. The load factor ρ is the parameter that has the most impact: the largest advantages of the overlay strategy are obtained for high primary load factors. For example, if ρ = 1, the maximum advantage corresponds to a factor of approximately 2.55. In contrast, for small loads, the advantage might be too small to compensate for the additional complexity when compared to the underlay strategy; for example, in the case of a singleantenna primary system, we observed an advantage factor of just 1.15 (see[36]). Similar conclusions can be drawn for Figure5: the maximum tends to move towards the primary transmitter as we increase the secondary power or the number of antennas and the region where overlay is better most of the time becomes larger. Finally, for larger path losses (e.g. p = 4), the results become more extreme: the positions of the maxima in Figures3 to5 remain the same, but their values are higher. In contrast, when the transmitters are separated, the underlay scheme yields a larger advantage than the one presented here.
Finally, Figure6 shows the behavior of the underlay and overlay strategies in terms of the average of the rates and${R}_{2}^{\text{und}}$ and R_{2}, respectively, as a function of the secondary power P_{2} for different transmit antenna configurations such that N_{T,1} + N_{T,2} = 5 and N_{R,2} = 1 for P_{1} = 10 dB in a fully loaded system, i.e. ρ = 1, with path loss exponent p = 3. The secondary transmitter is placed at position (x,y) = (0.3,0.5), i.e. on the line between the primary users. The main observation is that, in terms of secondary rate, it is better to deploy the antennas at the secondary transmitter rather than at the primary transmitter. In the underlay case, this is rather straightforward for the secondary system cannot benefit from the antennas at the primary. In the overlay case, this observation implies that the gains obtained via spatial diversity (i.e. larger N_{T,2}) increase faster than those obtained by shortening the learning phase (i.e. larger N_{T,1}). However, observe that increasing N_{T,2} suffers from a law of diminishing returns and that beyond a certain value the gains are minor. Regarding the changes in the behavior for varying secondary power P_{2}, we observe the following general trends. For very low P_{2}, all the strategies are powerconstrained, and thus the gap between underlay and overlay vanishes. This effect is more pronounced for ρ < 1, where the primary can tolerate some interference. The gap between the strategies widens as P_{2} increases, meaning, than when the secondary transmitter is no longer power limited, the use of spatial shaping alone fails to exploit the available resources. A special, extreme case is the underlay strategy with N_{T,2} = 1 : lacking spatial resources, it cannot make any use of a fully loaded primary channel, i.e. R_{2} = 0 independently of P_{2}.
6 Coexistence with MIMO primary systems
The discussion in this paper has been restricted to the coexistence of a MIMO secondary system with a MISO primary link. The results presented here cannot be extended in their totality to the case of MIMO primaries neither for underlay nor for overlay. However, as we will see in this section, under some reasonable assumptions, they carry over to scenarios with MIMO primary systems.
In the case of underlay cognitive radio, it is important to emphasize the underlying assumption that the primary users are oblivious to the presence of secondary users. This effectively decouples the design of the optimal secondary transmitter from the primary transmit parameters. Moreover, note that the effect of the primary users enters the optimization in (10) through constraints (10b) and (10c). The validity of Lemma 1 which plays a fundamental role in dealing with the nonconvexity of (10c) does not rely on any assumption about the primary transmit covariance matrix and thus applies to the primary MIMO case as well. In contrast, the simple transformation of (10b) into a linear constraint (i.e. (40b)) is no longer possible in the MIMO primary case. If, however, this constraint is replaced by a constraint that is linear or convex in (K_{21},K_{21}), then the results in Proposition 1 remain valid. For example, one may define a constraint analog to (10b) by considering the worstinterference direction in the span of h_{21}. Alternatively, if the primary system uses singlestream transmission with fixed receiver beamformer, the results presented here remain valid.
In the case of the overlay cognitive radio strategy, the problem is more involved. In addition to a similar problem regarding constraint (26c), the transmit strategies of primary and secondary systems are necessarily coupled by the very nature of the extended cognitive radio channel (i.e. by the messagelearning phase). Moreover, in the case of MIMO primaries, the optimization over the virtual joint covariance matrix K_{co} is more complex than in the case of MISO primaries, where beamforming was optimal, and thus K_{co} could be determined easily. This is issue is especially important when considering efficient algorithms to find the optimal parameters. Notwithstanding these considerations, the results in this paper remain valid if the primary system uses singlestream transmission with fixed receive beamformer, as in the case of underlay.
7 Conclusion
In this paper, we have studied the transmission strategies for underlay cognitive radio and overlay cognitive radio with an explicit learning phase, in which the secondary transmitter acquires the primary message. Our strategy for underlay uses interference decoding and exploits spatial resources using multiantenna methods. For the overlay case, we have combined cooperative communication techniques (decodeandforward relaying) with communication over a cognitive radio channel (cooperation and interference control at the primary receiver and interference precancellation at the secondary transmitter) using multiantenna methods. For both strategies, we have characterized the set of system parameters that maximize the secondary rate while ensuring a fixed rate for the primary system.
Finally, we have evaluated the performance of the strategies relative to each other in order to quantify the advantages and disadvantages of the degrees of coordination (i.e. uncoordinated for underlay vs. messagelearning phase and cooperative communication for overlay). We have observed that for a wide range of channel conditions, when the primary and secondary transmitters are close to each other, the overlay strategy provides a significant advantage over the underlay strategy. This gain is particularly relevant for those scenarios where the secondary is interferencelimited rather than powerlimited. However, as the distance between transmitters becomes larger, this advantage vanishes and in fact at some point underlay starts outperforming overlay. Our analysis reveals that a combination of underlay and overlay strategies is necessary to exploit best the available resources, especially if the users in the system do not have fixed positions.
Appendices
Appendix 1
Proof of proposition 1
We first prove an auxiliary lemma that will be used in the proof of Proposition 1. Note that using simple manipulations, the optimization problem in (10) can be reformulated as
with${P}_{\text{int}}^{\text{und}}$ as defined in (14).
We will show now that when considering case 3, there is no loss of generality in restricting constraint (40c) to be an equality.
Lemma 1
Any optimal point that falls within case 3 can be attained by a pair of covariance matrices$({\stackrel{~}{K}}_{2,1},{\stackrel{~}{K}}_{2,2})$, such that${\stackrel{~}{K}}_{2,2}$ satisfies constraint (40c) with equality.
Proof
Let K_{2,1} and K_{2,2} solve the optimization problem and assume that
where the notation${R}_{1,2}^{\text{und}}({\mathit{K}}_{2,2})$ stresses out the dependency of${R}_{1,2}^{\text{und}}$ on K_{2,2}. Similarly, the notation${R}_{2}^{\text{und}}({\mathit{K}}_{2,1},{\mathit{K}}_{2,2})$ will stress out the dependency of${R}_{2}^{\text{und}}$ on K_{2,1} and K_{2,2}.
First, we consider the case K_{2,1} = 0. Let Σ^{⋆} be the solution to problem (16) (in case 2) and recall that
for case 3. Now, construct the new covariance matrix
Note that for any γ ∈ [0,1], this matrix satisfies constrains (40b), (40d) and (40e), and
by the concavity property of the logdeterminant.${R}_{1,2}^{\text{und}}({\stackrel{~}{K}}_{2,2})$ is a continuous function of γ that satisfies
Thus, by choosing λ appropriately, we construct either an admissible matrix that yields a higher secondary rate or a matrix yielding the same secondary rate, and such that (40c) is satisfied with equality.
We now consider the case K_{2,1} ≠ 0. Construct the following two covariance matrices
for γ ∈ [0,1]. Note that by construction, both${\stackrel{~}{K}}_{2,1}$ and${\stackrel{~}{K}}_{2,2}$ are positive semidefinite. Moreover, this choice of covariance matrices satisfies,
and thus the constraints (40b), (40d) and (40e) are satisfied, and the first term in the objective function (40a) remains unchanged. However, noting that
for A ≽ 0,C ≽ 0 and B ≻ 0, we see that
for any γ ∈ [0,1]. Moreover,${R}_{1,2}^{\text{und}}({\stackrel{~}{K}}_{2,2})$ is a nonincreasing and continuous function of γ. If, for any γ ∈ (0,1], we have that
then we have contradicted our initial hypothesis. Otherwise, by the nonincreasing property, the pair of matrices${\stackrel{~}{K}}_{2,2}={\mathit{K}}_{2,2}+{\mathit{K}}_{2,1}$ and${\stackrel{~}{K}}_{2,1}=0$ (i.e. γ = 1) must also be a valid solution. Thus, we can use the first part of the proof to show that there is no loss of generality in restricting (40c) to be an equality. □
We now proceed to prove Proposition 1.
Proof of Proposition 1. The proof for case 1 follows from the fact that it is not possible for the secondary receiver to decode the primary message (for the case of equality in (11), any K_{2,2} ≠ 0 would render decoding of the primary message impossible). Thus, the best that the transmitter can do is to choose the covariance matrix that maximizes (12). The formulation in (13) follows by noting that the denominator in (12) is independent from the covariance matrix.
The proof for case 2 follows easily by noting that the solution to (16) is the best the secondary system can do given the power and interference constraints.
To prove the solution for case 3, we make use of Lemma 1 to rewrite the optimization problem in (40) as
Note that only the first term in the objective function is relevant for the optimization. Moreover, except for (53c), the maximization only depends on K_{2,1},K_{2,2} through their sum, which we denote by Δ. The general solution (K_{2,1},K_{2,2}) can be obtained by computing the optimal Δ^{⋆} disregarding constraint (53c) and then setting
with γ ∈ [0,1], such that${R}_{1,2}^{\text{und}}={R}_{1}^{\star}$. Note that such γ must exist because${R}_{1,2}^{\text{und}}$ is continuous in γ and
by assumption for case 3. □
Appendix 2
Proof of proposition 2
We shall make use of the following wellknown Lemma in our arguments:
Lemma 2
The function
defined for β ∈ (0,1], any B and any C ≽ 0 (with appropriate dimensions) is strictly increasing in β.
Proof
We have that
where λ_{i} and r are the singular values and the rank of B^{H}CB, respectively. It is easy to check that the first derivative of each of the terms in the sum is positive for β > 0, proving that (57) is strictly increasing in β. □
Proof of proposition 2. First, we prove statement 1 by contradiction. Assume that the set of parameters that attains the optimum satisfies
Consider two new covariance matrices
Since${R}_{1}^{(2)}$ is a continuous function of both tr{K_{p}} and tr{K_{r}}, we can find (sufficiently small) γ_{p} > 1 and γ_{r} > 1 that do not violate constraint (26d) and such that${R}_{1}^{(2)}$ evaluated for${\stackrel{~}{\mathit{K}}}_{\mathrm{p}}$ and${\stackrel{~}{\mathit{K}}}_{\mathrm{r}}$ remains unchanged (and hence satisfy (26c)). However, using${\stackrel{~}{\mathit{K}}}_{\mathrm{p}}$ yields a larger secondary rate R_{2}, which contradicts our assumption that the set of parameters solved the optimization problem.
We now prove statement 2 also by contradiction. Assume that the optimal choice of parameters yields
where${\mathit{K}}_{1}^{(1)}$ is the optimal choice of covariance matrix. Now, define the matrix${\stackrel{~}{K}}_{1}^{(1)}=\gamma {\mathit{K}}_{1}^{(1)}$ for some γ > 1, such that
This choice of matrix yields
and
where λ_{i} and r are the singular values and the rank of${\mathit{H}}_{\mathrm{t}}{\mathit{K}}_{1}^{(1)}{\mathit{H}}_{\mathrm{t}}^{H}$, respectively. Thus, we have that
and we can find a shorter duration of the first phase$\stackrel{~}{\alpha}<\alpha $ such that the rates, evaluated at$\stackrel{~}{\alpha}$, satisfy
At the same time, we have increased the secondary rate by Lemma 2, thus contradicting our hypothesis on the optimality of the set of parameters.
Appendix 3
Proof of Proposition 3
Assume that the set of parameters that attains the maximum in (26) satisfies
where${\mathit{K}}_{1}^{(1)}$ is the optimal covariance matrix. The notation remarks the dependency of${R}_{1}^{(1)}$ and R_{t} on the covariance matrix${\mathit{K}}_{1}^{(1)}$. Let σ^{⋆} denote the power used by this covariance matrix, i.e.${\sigma}^{\star}\triangleq tr\{{\mathit{K}}_{1}^{(1)}\}$. We divide the proof into two cases.
First, consider the case${\mathit{K}}_{1}^{(1)}\ne {\mathit{K}}^{\text{WF}}({\sigma}^{\star})$ with K^{WF}(σ^{⋆}) as defined in (27). Both${R}_{1}^{(1)}$ and R_{t} are continuous functions of the entries of the covariance matrix, and the logdet operator is concave on the set of Hermitian positive semidefinite matrices with bounded trace. Therefore, we can find a Hermitian positive semidefinite covariance matrix${\stackrel{~}{\mathit{K}}}_{11}$, with$\parallel {\stackrel{~}{\mathit{K}}}_{11}{\mathit{K}}_{1}^{(1)}\parallel $ small enough such that
Now, since${R}_{1}^{(1)}$, R_{t}, and${R}_{1}^{(2)}$ are all continuous in α, we can find a shorter duration for the first phase, i.e.$\stackrel{~}{\alpha}<\alpha $, such that the two constraints are still satisfied. However, by Lemma 2 in Appendix 2, shortening the first phase strictly increases the secondary rate R_{2}, contradicting our assumption on the optimality of the set of parameters.
In the case where${\mathit{K}}_{1}^{(1)}={\mathit{K}}^{\text{WF}}$, the rate R_{t} is already maximum. In this case, if either${\mathit{K}}_{1}^{(2)}\ne 0$ or K_{r} ≠ 0, we can use similar arguments to those used in the proof of Proposition 2 to arrive at a contradiction. In contrast, if${\mathit{K}}_{1}^{(2)}=0$ and K_{r} = 0, we cannot always ensure that (26c) is satisfied with equality. However, in the cases where we cannot reach a contradiction, we can use that${R}_{1}^{(2)}=0$ and$tr\{{\mathit{K}}_{1}^{(1)}\}=\frac{{P}_{1}}{\alpha}$ (cf. Proposition 2). Combined with the fact that the solution to (26) must satisfy${R}_{1}^{(1)}\ge {R}_{1}^{\star}$, we can show that
thus violating the cooperation condition.
Appendix 4
Proof of Proposition 4
We prove the first part of the claim by contradiction. Assume that the optimal choice of parameters yields
where${\mathit{K}}_{1}^{(1)}$ is the optimal covariance matrix. Note that we can express${\mathit{K}}_{1}^{(1)}$ as
where${\beta}_{1}=\parallel {\mathrm{\Pi}}_{{\mathit{h}}_{11}}{\mathit{K}}_{1}^{(1)}\parallel ,{\beta}_{2}=\parallel {\mathrm{\Pi}}_{{\mathit{h}}_{11}}^{\perp}{\mathit{K}}_{1}^{(1)}\parallel ,{\mathbf{\Sigma}}_{1}={\beta}_{1}^{1}{\mathrm{\Pi}}_{{\mathit{h}}_{11}}{\mathit{K}}_{1}^{(1)}$, and${\mathbf{\Sigma}}_{2}={\beta}_{2}^{1}{\mathrm{\Pi}}_{{\mathit{h}}_{11}}^{\perp}{\mathit{K}}_{1}^{(1)}$ for i ∈ {1,2} with β_{i} > 0. Otherwise, set Σ_{i} = 0 for i such that β_{i} = 0. Assuming β_{i} > 0 for i ∈ {1,2}, both Σ_{1} and Σ_{2} have unit norm. Now, let
Note that${\mathit{K}}_{\parallel}={\mathrm{\Pi}}_{{\mathit{h}}_{11}}$. Thus, we have
Now define a new matrix
where ε = (1−γ)(β_{1} + β_{2}). Note that${\stackrel{~}{K}}_{1}^{(1)}$ is a valid choice of covariance matrix because it is the sum of positive semidefinite Hermitian matrices and satisfies$tr\{\underset{1}{\overset{(1)}{\stackrel{~}{K}}}\}=tr\{\underset{1}{\overset{(1)}{\mathit{K}}}\}$. Since the determinant is a continuous function of the entries of the matrix, and the logarithm is a continuous function of its argument, we can find 0 < γ < 1 such that
This choice of${\stackrel{~}{\mathit{K}}}_{1}^{(1)}$ yields
The inequality in (94) is due to the fact that
The inequality in (96) follows if β_{2} > 0 by the fact that 0 < γ < 1. Hence, for this new choice of covariance matrix${\stackrel{~}{K}}_{1}^{(1)}$, we have
Now, we can find a shorter duration of the first phase$\stackrel{~}{\alpha}<\alpha $, such that the rates evaluated at$\stackrel{~}{\alpha}$ satisfy
At the same time, we have increased the secondary rate by Lemma 2 in Appendix 2, thus contradicting our hypothesis on the optimality of the set of parameters.
Finally, note that β_{2} = 0 implies that
so that${\mathit{K}}_{1}^{(1)}$ is a Hermitian rankone covariance matrix. Therefore, we must have
for some$\rho \in \mathbb{R}$. This concludes the proof.
Appendix 5
Proof of Corollary 1
Assume that${\mathit{K}}_{1}^{(1)}$ is the optimal covariance matrix in (26), and let${\widehat{\mathit{K}}}_{1}^{(1)}$ be the output of Algorithm 1. Note that by construction of the algorithm$tr\{{\widehat{\mathit{K}}}_{1}^{(1)}\}=tr\{{\mathit{K}}_{1}^{(1)}\}$. We divide the proof into two parts:
If${\mathit{K}}_{1}^{(1)}=\rho \frac{{\mathit{h}}_{11}{\mathit{h}}_{11}^{H}}{\parallel {\mathit{h}}_{11}{\parallel}^{2}}$ for some$\rho \in \mathbb{R}$ (i.e. it corresponds to the MRT beamformer to receiver 1), then trivially${\widehat{\mathit{K}}}_{1}^{(1)}={\mathit{K}}_{1}^{(1)}$ as this is the initial guess of the algorithm (lines 7 and 8) and it satisfies
Thus, this is the output of the algorithm (lines 9 and 10).
For the case when${\mathit{K}}_{1}^{(1)}$ does not correspond to the MRT beamformer, we prove the optimality of the algorithm by contradiction. Assume${\widehat{\mathit{K}}}_{1}^{(1)}\ne {\mathit{K}}_{1}^{(1)}$ and note that
The equality in (106) comes from Proposition 4 and the fact that${\mathit{K}}_{1}^{(1)}$ is the optimal covariance matrix. The equality in (107) is ensured by construction of the algorithm in the limit of arbitrary numerical precision in the bisection method, i.e. ε → 0 (lines 9 to 17 in Algorithm 2). In addition, we have
because by construction, Algorithm 1 finds the matrix with largest component in the direction of h_{11} that satisfies (26b) with equality. Thus,
We can now proceed as in Proposition 3 to contradict our initial hypothesis on the optimality of${\mathit{K}}_{1}^{(1)}$. Thus, we must have${\widehat{\mathit{K}}}_{1}^{(1)}={\mathit{K}}_{1}^{(1)}$ in this case as well.
Appendix 6
Proof of Proposition 5
The matrix K_{co} and its submatrices${\mathit{K}}_{1}^{(2)},{\mathit{K}}_{\mathrm{r}}$ and Ψ only appear in the expression for${R}_{1}^{(2)}$ through the expression
It is easy to see that the optimal K_{co} has rank 1, i.e.${\mathit{K}}_{\text{co}}={\mathit{v}}_{\text{co}}{\mathit{v}}_{\text{co}}^{H}$. The vector v_{co} is chosen as to maximize the projection${\mathit{v}}_{\text{co}}^{H}{\mathit{h}}_{\text{ext}}$ while satisfying the constraints on the traces of${\mathit{K}}_{1}^{(2)}$ and K_{r}. Simple calculus shows that the optimal v_{co} is given, up to a common factor, by
The desired${\mathit{K}}_{1}^{(2)},{\mathit{K}}_{\mathrm{r}}$ and Ψ are readily obtained from K_{co}.
Using these results, it is straightforward to establish the identity
From (113), we see that the effect of Ψ is to correlate the primary and secondary transmissions so that their signals add constructively at the receiver. Finally, given the matrices${\mathit{K}}_{1}^{(2)},{\mathit{K}}_{\mathrm{r}}$ and Ψ, the characterization of K_{p} in terms of the concave problem in (32) follows immediately (see[20] as well).
Appendix 7
Proof of Corollary 2
The beamformer w_{p} appears both in the objective function (26a) and in constraint (26c) through${R}_{1}^{(2)}$. First, note that if P_{int} < 0, the problem has no valid solution. For a given second phase (that is, given α and${P}_{1}^{(2)}$), using Propositions 2 and 3, the optimization problem is reduced to finding w_{p} and P_{r} and can be reformulated, for P_{int} > 0, as
with 0 ≤ P_{r} ≤ P_{2},‖w_{p} ‖ = 1. For a fixed P_{r}, the objective function (114) is monotonically increasing in${\mathit{h}}_{22}^{H}{\mathit{w}}_{\mathrm{p}}{}^{2}$ and monotonically decreasing in${\mathit{h}}_{21}^{H}{\mathit{w}}_{\mathrm{p}}{}^{2}$. Thus, for given P_{r}, the optimal beamformer w_{p} can be parametrized as
for some λ ∈ [0,1]. Using this parametrization, we define$f(\lambda )\triangleq {\mathit{h}}_{22}^{H}{\mathit{w}}_{\mathrm{p}}(\lambda ){}^{2}$ and note that${\mathit{h}}_{21}^{H}{\mathit{w}}_{\mathrm{p}}{}^{2}=\lambda {\Vert {\mathit{h}}_{21}\Vert}^{2}$ to write, for fixed P_{r}, the optimization problem as
The function f(λ) is unimodal with maximum value attained for λ = λ_{MRT}, that is, when w_{p}(λ) is in the direction of H_{22} (i.e. MRT). Thus, if
then λ = λ_{MRT} yields the optimum value. Otherwise, the basic calculus shows that (114) is maximized for
Using this parametrization, we can find the optimal beamformer by varying P_{r} from P_{2} to 0 to find the maximum value of (114).
For P_{int} = 0, the primary receiver is already at the limit of decodability and cannot tolerate any interference. Thus, the secondary transmitter must transmit in the ZF direction. This special case is already considered by our parametrization (i.e. setting λ = 0).
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Acknowledgements
Part of this work has been performed in the framework of Network of Excellence ACROPOLIS, which is partly funded by the European Union under its FP7 ICT Objective 1.1  The Network of the Future. The authors would like to thank Adrian Kliks for the interesting discussions and valuable comments.
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BlascoSerrano, R., Lv, J., Thobaben, R. et al. Multiantenna transmission for underlay and overlay cognitive radio with explicit messagelearning phase. J Wireless Com Network 2013, 195 (2013) doi:10.1186/168714992013195
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Keywords
 Cognitive radio
 Underlay
 Overlay
 Multiple antennas
 Message learning
 Cooperation