On the Throughput and Energy Efficiency of Cognitive MIMO Transmissions

# On the Throughput and Energy Efficiency of Cognitive MIMO Transmissions

\authorblockNSami Akin and M. Cenk Gursoy
S. Akin is with the Institute of Communications Technology, Leibniz Universität Hannover, 30167 Hanover, Germany. M. C. Gursoy is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, 13244. (e-mail: sami.akin@ikt.uni-hannover.de, mcgursoy@syr.edu).This work was supported by the National Science Foundation under Grants CCF – 0546384 (CAREER) and CCF – 0917265. The material in this paper was presented in part at the IEEE Wireless Communications and Networking Conference (WCNC) in March 2011.
###### Abstract

In this paper, throughput and energy efficiency of cognitive multiple-input multiple-output (MIMO) systems operating under quality-of-service (QoS) constraints, interference limitations, and imperfect channel sensing, are studied. It is assumed that transmission power and covariance of the input signal vectors are varied depending on the sensed activities of primary users (PUs) in the system. Interference constraints are applied on the transmission power levels of cognitive radios (CRs) to provide protection for the PUs whose activities are modeled as a Markov chain. Considering the reliability of the transmissions and channel sensing results, a state-transition model is provided. Throughput is determined by formulating the effective capacity. First derivative of the effective capacity is derived in the low-power regime and the minimum bit energy requirements in the presence of QoS limitations and imperfect sensing results are identified. Minimum energy per bit is shown to be achieved by beamforming in the maximal-eigenvalue eigenspace of certain matrices related to the channel matrix. In a special case, wideband slope is determined for more refined analysis of energy efficiency. Numerical results are provided for the throughput for various levels of buffer constraints and different number of transmit and receive antennas. The impact of interference constraints and benefits of multiple-antenna transmissions are determined. It is shown that increasing the number of antennas when the interference power constraint is stringent is generally beneficial. On the other hand, it is shown that under relatively loose interference constraints, increasing the number of antennas beyond a certain level does not lead to much increase in the throughput.

{keywords}

cognitive radio, effective capacity, energy efficiency, minimum energy per bit, multiple-input multiple-output (MIMO), quality of service (QoS) constraints, throughput.

## I Introduction

Cognitive Radio (CR), which has emerged as a method to tackle the spectrum scarcity and variability in both time and space, calls for dynamic access strategies that adapt to the electromagnetic environment [1]. Performance of cognitive radio systems has been studied extensively in recent years, and a detailed description of different CR models and an overview of recent approaches can be found in [2], [3] and [4]. For instance, three different paradigms, namely underlay, overlay and interweave operation of cognitive radio systems, were discussed in [3]. In underlay CR networks, cognitive secondary users (SUs) can coexist with the primary users (PUs) and transmit concurrently as long as they adhere to strict limitations on the interference inflicted on the PUs. This model is also known as spectrum sharing. On the other hand, in interweave CR networks, SUs initially perform channel sensing and opportunistically access only the spectrum holes in which the primary users are inactive. These two methods of spectrum sharing and opportunistic spectrum access can also be combined for improved performance. For instance, Kang et al. in [5] analyzed a hybrid model in which SUs first sense the frequency bands and detect the PU activity. Subsequently, cognitive radio transmission is performed at two different power levels depending on the sensed PU activity. More specifically, if the PUs are sensed to be active, secondary transmission still occurs but with reduced power level in order to lower the interference within tolerable levels. In such modes of cognitive operation, sensing the activities of PUs is a critical issue that has been studied and analyzed extensively (see e.g., [6], [7]) since the inception of the CR concept.

Another advancement in communications technology is multiple-antenna communications. It is well-known that employing multiple antennas at the receiver and transmitter ends of a communication system can improve the performance levels by providing significant gains in the throughput and/or reliability of transmissions. Therefore, there has been much interest in understanding and analyzing multiple-input multiple-output (MIMO) channels and numerous comprehensive studies have been conducted [8], [9]. In most studies, ergodic Shannon capacity formulations are considered as the performance metrics [10], [11], [12]. For instance, the authors in [10] and [11] studied multiple-antenna ergodic channel capacity and provided analytical characterizations of the impact of certain factors such as antenna correlation, co-channel interference, Ricean factors, and polarization diversity. It should be noted that ergodic capacity generally does not take into account any delay, buffer, or queueing constraints at the transmitter.

In [13], the throughput of MIMO systems in the presence of statistical queuing constraints was investigated. Effective capacity was employed as the metric to measure the performance under quality-of-service (QoS) constraints. Effective capacity characterizes the maximum constant arrival rate that can be supported by a system under statistical limitations on buffer violations [14]. There have been several studies on effective capacity in various communication settings [15], [16]. Recently, the authors in [17] considered the maximization of effective capacity in a single-user multi-antenna system with covariance knowledge, and the authors in [18] studied the effective capacity of a class of multiple-antenna wireless systems subject to Rayleigh flat fading.

Recently, cognitive MIMO radio models have also been considered since having multiple antennas can provide higher performance levels for the SUs and lead to better protection of PUs. Modeling a channel setting with a single licensed user and a single cognitive user, that is equivalent to an interference channel with degraded message sets, the authors in [19] focused on the fundamental performance limits of a cognitive MIMO radio network, and they showed that under certain conditions, the achievable region is optimal for a portion of the capacity region that includes the sum capacity. In [20], three scenarios, namely when the secondary transmitter (ST) has complete, partial, or no knowledge about the channels to the primary receivers (PRs), was considered, and maximization of the throughput of the SU, while keeping the interference temperature at the PRs below a certain threshold, was investigated. Furthermore, in [21], the authors proposed a practical CR transmission strategy consisting of three major stages, namely, environment learning that applies blind algorithms to estimate the spaces that are orthogonal to the channels from the PR, channel training that uses training signals and employs the linear-minimum-mean-square-error (LMMSE)-based estimator to estimate the effective channel, and data transmission. Considering imperfect estimations in both learning and training stages, they derived a lower bound on the ergodic capacity that is achievable by the CR in the data-transmission stage. In another study [22], the authors proposed a practical cognitive beamforming scheme that does not require any prior knowledge of the CR-PR channels, but exploits the time-division-duplex operation mode of the PR link and the channel reciprocities between CR and PR terminals, utilizing an idea called effective interference channel, that is estimated at the CR terminal via periodically observing the PR transmissions. It was also shown in [23] that the asymptotes of the achievable transmission rates of the opportunistic (secondary) link are obtained in the regime of large numbers of antennas. Another study of cognitive MIMO radios was conducted in [24].

The above-mentioned references have not addressed considerations related to energy efficiency and QoS provisioning in cognitive MIMO channels. In our prior work, we studied the impact of QoS requirements in single-antenna cognitive radio systems. In particular, we considered a CR model in which SUs transmit with two different transmission rates and power levels depending on the activities of PUs under QoS constraints. In [25], the ST senses only one channel and then depending on the channel sensing results, it chooses its transmission policy, whereas in [26] the ST senses more than one channel and chooses the best channel for transmission under interference power limits and QoS constraints. In [27], effective capacity limits of a CR model is analyzed with imperfect channel side information (CSI) at the transmitter and the receiver.

In this article, we focus on a cognitive MIMO system operating under QoS constraints. In particular, we investigate the achievable throughput levels and also study the performance in the low-power regime in order to address the energy efficiency. We analyze the impact of imperfect sensing results and interference limitations on the performance, and determine energy-efficient transmission strategies in the low-power regime. In the system model, we consider two different transmission policies depending on the activities of PUs and interference power threshold required to protect the PUs. Essentially, we have a hybrid, sensing-based spectrum sharing model of cognitive radio operation as described in [5]. We consider a general cognitive MIMO link where fading coefficients have arbitrary distributions and are possibly correlated across antennas. Moreover, we model the received interference signals from the primary transmitters correlated as well. We assume that the ST and secondary receiver (SR) have perfect side information regarding their own channels. The contributions of the paper can be summarized as follows:

1. We identify a joint state-transition model, considering the reliability of the transmissions and taking into account the channel sensing decisions and their correctness.

2. We provide a formulation of the throughput metric (effective capacity) in terms of transmission rates and state transition probabilities which depend on sensing reliability and primary user activity.

3. We obtain expressions for the first and second derivative of the effective capacity at , and determine the minimum energy per bit in the presence of QoS limitations and imperfect sensing results.

The organization of the paper is as follows. We provide the cognitive MIMO radio model and describe the transmission power and interference constraints in Section II. In Section III, we construct a state transition model for CR transmission and identify the throughput under QoS constraints, and show the relation between the effective capacity and ergodic capacity. Finding the first and second derivatives of effective capacity at , we analyze in Section IV the energy efficiency in the low-power regime. In Section V, we provide numerical results. We conclude in Section VI. Proofs are relegated to the Appendix.

## Ii Channel Model, Power Constraints, and Input Covariance

### Ii-a Channel Model

As seen in Figure 1, we consider a setting in which a single ST communicates with a single SR in the presence of possibly multiple PUs. We consider a cognitive MIMO radio model and assume that the ST and SR are equipped with and antennas, respectively. In a flat fading channel, we can express the channel input-output relation as

 y=Hx+n+s (1)

if the PUs are active in the channel, and as

 y=Hx+n (2)

if the PUs are absent. Above, denotes the dimensional transmitted signal vector of ST, and denotes the dimensional received signal vector at the SR. In (1) and (2), is an dimensional zero-mean Gaussian random vector with a covariance matrix where is the identity matrix. In (1), is an dimensional vector of the sum of active PUs’ faded signals arriving at the secondary receiver. Considering that the vector can have correlated components, we express its covariance matrix as where is the variance of each component of and . Finally, in (1) and (2), denotes the dimensional random channel matrix whose components are the fading coefficients between the corresponding antennas at the secondary transmitting and receiving ends. We consider a block-fading scenario and assume that the realization of the matrix remains fixed over a block duration of seconds and changes independently from one block to another.

### Ii-B Power and Interference Constraints

We assume that the SUs initially perform channel sensing to detect the activities of PUs, and then depending on the channel sensing results, they choose the transmission strategy. More specifically, if the channel is sensed as busy, the transmitted signal vector is . Otherwise, the signal is . When the channel is sensed as busy, the average energy of the channel input is

 E{||x1||2} =P1B. (3)

On the other hand, if the channel is detected to be idle, the average energy becomes

 E{||x2||2}=P2B. (4)

In (3) and (4), is the bandwidth of the system. Note that under the assumption that complex input vectors are transmitted every second, the above energy levels imply that the transmission powers are and , depending on the sensing results.

We first note that and are upper bounded by , which represents the maximum transmission power capabilities of cognitive transmitters. In a cognitive radio setting, transmission power levels are generally further restricted in order to limit the interference inflicted on the PUs. As a first measure, we assume that where . Hence, smaller transmission power is used when the channel is sensed as busy, and we basically have

 P1≤P2≤Pmax. (5)

Additionally, we consider a practical scenario in which errors such as miss-detections and false-alarms possibly occur in channel sensing. We denote the correct-detection and false-alarm probabilities by and , respectively. We note the following two cases. When PUs are active and this activity is sensed correctly (which happens with probability or equivalently fraction of the time on the average), then SUs transmit with average power . On the other hand, if the PU activity is missed in sensing (which occurs with probability ), SUs send the information with average power . In both cases, PUs experience interference proportional to the product of the transmission power, average fading power, and path loss in the channel between the ST and PUs. In order to limit the average interference, we impose the following constraint

 PdP1+(1−Pd)P2≤Pint (6)

where can be seen as the average interference constraint normalized by the average fading power and path loss111For instance, if average transmission power is limited by when the primary users are active, the average interference experienced at a given primary receiver will be limited by where is the magnitude square of the fading in the channel between the secondary transmitter and primary receiver, is the distance between them, is the path loss exponent, and is some constant related to the path loss model.. We note that a similar formulation for the average interference constraint was considered in [5]. Noting the assumption that for some , we can rewrite (6) as

 PdμP2+(1−Pd)P2≤Pint, (7)

which implies that Considering the maximum of the average power, we can write

 P2≤min{Pmax,PintPdμ+(1−Pd)}. (8)

Note that for given and detection probability , if the interference constraints are relatively relaxed and we have , then we can choose to operate at and . Otherwise, interference constraints will dictate the transmission power levels.

From (6), we can also, for given , and , obtain

 μ≤min{max{Pint−P2(1−Pd)P2Pd,0},1}. (9)

From above, we see that if , then and hence no transmission is performed by the ST when the channel is sensed as busy.

In order to illustrate some of the interactions between the parameters discussed above, we plot, in Fig. 2, the ratio as a function of , the power level adapted when the channel is sensed as idle, for different values of power interference constraints . In all cases, we have for small values of , while diminishes to zero as increases due to the presence of interference constraints. Note also that we reach at smaller values of under more stringent interference constraints.

### Ii-C Input Covariance Matrix

Finally, we note that in addition to having different levels of transmission power, directionality of the transmitted signal vectors might also be different depending on the channel sensing results. We define the normalized input covariance matrix of as

 Kx1=E{x1x†1}P1/B (10)

if the channel is busy, and that of as

 Kx2=E{x2x†2}P2/B (11)

if the channel is idle. Note that the traces of normalized covariance matrices are

 tr(Kx1)=1 (12)

and

 tr(Kx2)=1. (13)

## Iii State Transition Model and Channel Throughput

### Iii-a State Transition Model

Depending on channel sensing results and their correctness, we have four scenarios:

1. Channel is busy, and is detected as busy (correct detection),

2. Channel is busy, but is detected as idle (miss-detection),

3. Channel is idle, but is detected as busy (false alarm),

4. Channel is idle, and is detected as idle (correct detection).

Using the notation where , we can express the instantaneous channel capacities in the above four scenarios as follows:

 C1=BmaxKx1⪰0tr(Kx1)=1log2det[I+μP2Bσ2nHKx1H†K−1z]=BmaxKx1⪰0tr(Kx1)=1log2det[I+μN\textscsnrHKx1H†K−1z], C2=BmaxKx2⪰0tr(Kx2)=1log2det[I+P2Bσ2nHKx2H†K−1z]=BmaxKx2⪰0tr(Kx2)=1log2det[I+N\textscsnrHKx2H†K−1z], C3=BmaxKx1⪰0tr(Kx1)=1log2det[I+μP2Bσ2nHKx1H†]=BmaxKx1⪰0tr(Kx1)=1log2det[I+μN\textscsnrHKx1H†], C4=BmaxKx2⪰0tr(Kx2)=1log2det[I+P2Bσ2nHKx2H†]=BmaxKx2⪰0tr(Kx2)=1log2det[I+N\textscsnrHKx2H†]. (14)

Above, we define as the signal-to-noise ratio when the channel is sensed as idle. If, on the other hand, the channel is sensed as busy, signal-to-noise ratio is since the transmission power is . We also note that since is a positive definite matrix and its eigenvalues are greater than or equal to 1, is a positive definite matrix with eigenvalues .

The secondary transmitter is assumed to send the data at two different rates depending on the sensing results. If the channel is detected as busy as in scenarios 1 and 3, the transmission rate is

 r1=BmaxKx1⪰0tr(Kx1)=1log2det[I+μN\textscsnrHKx1H†K−1z], (15)

and if the channel is detected as idle as in scenarios 2 and 4, the transmission rate is

 r2=BmaxKx2⪰0tr(Kx2)=1log2det[I+N\textscsnrHKx2H†]. (16)

In scenarios 1 and 4, sensing decisions are correct and transmission rates match the channel capacities, i.e., we have in scenario 1, and in scenario 4. In these cases, we assume that reliable communication is achieved. On the other hand, sensing errors in scenarios 2 and 3 lead to mismatches. We first establish the following result. Note that and are are Hermitian matrices, they can be written as [29, Theorem 4.1.5]

 HKx1H†=A=UAΛAU†AandK−1z=UK−1zΛK−1zU†K−1z (17)

where and are unitary matrices and and are real diagonal matrices, consisting of the eigenvalues of and , respectively. Now, we can write

 det[I+μN\textscsnrHKx1H†K−1z] =det[I+μN\textscsnrAK−1z] (18) =det⎡⎣(UA00UK−1z)(μN\textscsnrΛA−IIΛK−1z)⎛⎝U†A00U†K−1z⎞⎠⎤⎦ =det[UAUK−1z]det[I+μN\textscsnrΛAΛK−1z]det[U†AU†K−1z] (19) (20) =det[I+μN\textscsnrA]=det[I+μN\textscsnrHKx1H†]. (21)

The inequality in (20) follows from the following observation:

 =∏i(1+μN\textscsnrλA,iλK−1z,i) (22) ≤∏i(1+μN\textscsnrλA,i) (23) =det[I+μN\textscsnrΛA]. (24)

Above, and denote the eigenvalues of and , respectively. The inequality in (23) follows from the fact that the eigenvalues of are smaller than 1, i.e., as mentioned before, and the fact that which is due to the positive semi-definiteness of 222The positive semi-definiteness can be easily seen from the following simple argument. For any vector , we can write , where we have defined and used the fact that is positive semi-definite.. From the inequality established through (18) – (21), we see that, in scenario 3, the transmission rate is less than the capacity (i.e., ). Hence, although reliable transmission is achieved at the rate of , channel is not fully utilized due to the false alarm in channel sensing. On the other hand, in a similar manner, it can be shown that in scenario 2, we have the transmission rate exceeding the channel capacity because sensing has not led to the successful detection of the active PUs, and the PUs’ interference on the SUs’ signals is not taken into account. In this case, we assume that reliable communication cannot be achieved. Hence, the transmission rate is effectively zero, and retransmission is required in scenario 2. In the other three scenarios, communication is performed reliably. These four scenarios or equivalently states are depicted in Figure 3. Following the discussion above, we assume that the channel is ON in states 1,3, and 4, in which data is sent reliably, and is OFF in state 2.

Next, we determine the state-transition probabilities. We use to denote the transition probability from state to state as seen in Fig. 3. Due to the block fading assumption, state transitions occur every seconds. We also assume that PU activity does not change within each frame. We consider a two-state Markov model to describe the transition of the PU activity between the frames. This Markov model is depicted in Figure 4. Busy state indicates that the channel is occupied by the PUs, and idle state indicates that there is no PU present in the channel. Probability of transitioning from busy state to idle state is denoted by a, and the probability of transitioning from idle state to busy state is denoted by b. Let us first consider in detail the probability of staying in the topmost ON state in Fig. 3. This probability, denoted by , is given by

 p11 =Pr{channel is busy and is % detected busyin the lth frame∣∣channel is busy and is detected busyin the (l−1)th frame} (25) =Pr{channel is busy in the lth frame∣∣channel is busy in the (l−1)th frame}×Pr{channel is detected busyin the lth frame∣∣channel is busyin the lth frame} =(1−a)Pd (26)

where is the probability of detection in channel sensing. Channel being busy in the frame depends only on channel being busy in the frame and not on the other events in the condition. Moreover, since channel sensing is performed individually in each frame without any dependence on the channel sensing decision and PU activity in the previous frame, channel being detected as busy in the frame depends only on the event that the channel is actually busy in the frame.

Similarly, the probabilities for transitioning from any state to state 1 (topmost ON state) can be expressed as

 pb1=p11 =p21=(1−a)Pdandpi1=p31=p41=bPd. (27)

Note that we have common expressions for the transition probabilities in cases in which the originating state has a busy channel (i.e., states 1 and 2) and in cases in which the originating state has an idle channel (i.e., states 3 and 4).

In a similar manner, the remaining transition probabilities are given by the following:

For all and ,

 pb2=(1−a)(1−Pd),andpi2=b(1−Pd),pb3=aPf,andpi3=(1−b)Pf,pb4=a(1−Pf),andpi4=(1−b)(1−Pf). (28)

Now, we can easily see that the state transition matrix can be expressed as

 R=⎛⎜ ⎜ ⎜⎝p11..p14p21..p24p31..p34p41..p44⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜⎝pb1..pb4pb1..pb4pi1..pi4pi1..pi4⎞⎟ ⎟ ⎟⎠. (29)

### Iii-B Effective Capacity

In [14], Wu and Negi defined the effective capacity as the maximum constant arrival rate that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent . If we define as the stationary queue length, then is defined as the decay rate of the tail distribution of the queue length :

 limq→∞logPr(Q≥q)q=−θ. (30)

Hence, we have the following approximation for the buffer violation probability for large : . Therefore, larger corresponds to more strict QoS constraints, while the smaller implies looser constraints. In certain settings, constraints on the queue length can be linked to limitations on the delay and hence delay-QoS constraints. It is shown in [18] that for constant arrival rates, where denotes the steady-state delay experienced in the buffer. In the above formulation, is a positive constant, and is the source arrival rate. Therefore, effective capacity provides the maximum arrival rate when the system is subject to statistical queue length or delay constraints in the forms of or , respectively, for large thresholds and . Since the average arrival rate is equal to the average departure rate when the queue is in steady-state [28], effective capacity can also be seen as the maximum throughput in the presence of such constraints.

The effective capacity for a given QoS exponent is formulated as

 −limt→∞1θtlogeE{e−θS(t)}=−Λ(−θ)θ (31)

where is a function that depends on the logarithm of the moment generating function of , is the time-accumulated service process, and is defined as the discrete-time, stationary and ergodic stochastic service process. Note that the service rate in each transmission block is if the cognitive system is in Scenario 1 or 3 at time . Similarly, the service rate is in Scenario 4. In the OFF state in Scenario 2, the service rate is effectively zero.

Considering the effective rates in each scenario and the probabilities of the scenarios, we have the following theorem.

###### Theorem 1

For the CR channel with the aforementioned state transition model , the normalized effective capacity in bits/s/Hz/dimension is given by

 CE(\textscsnr,θ) =max0≤μ≤10≤P2≤min{Pmax,PintPdμ+(1−Pd)}−1θTBNlogeE{12[(pb1+pi3)e−θTr1+pi4e−θTr2+pb2] 12{[(pb1−pi3)e−θTr1−pi4e−θTr2+pb2]2+4(pi1e−θTr1+pi2)(pb3e−θTr1+pb4e−θTr2)}1/2} (32)

where is the frame duration over which the fading stays constant, and are the transmission rates given in (15) and (16), and for are the state transition probabilities given in (27) and (28).

Proof: See Appendix -A.

Note that above we have assumed that is perfectly known at the transmitter, which, equipped with this knowledge, can choose the input covariance matrices to maximize the instantaneous channel capacities as seen in (15) and (16). If, on the other hand, only statistical information related to are known at the transmitter, then the input covariance matrix can be chosen to maximize the effective capacity. In that case, the normalized effective capacity will be expressed as

 CE (\textscsnr,θ)=max0≤μ≤10≤P2≤min{Pmax,PintPdμ+(1−Pd)}maxKx1,Kx2⪰0tr(Kx1)=tr(Kx2)=1−1θTBNlogeE{12[(pb1+pi3)Θr1+pi4Θr2+pb2] +12{[(pb1−pi3)Θr1−pi4Θr2+pb2]2+4(pi1Θr1+pi2)(pb3Θr1+pb4Θr2)}1/2}bits/s/Hz/dimension (33)

where and . Now, the input covariance matrices are selected to maximize the effective rate. For given and , and for given input covariance matrices and , we express the effective rate as

 RE (\textscsnr,θ)=−1θTBNlogeE{12[(pb1+pi3)Θr1+pi4Θr2+pb2] +12{[(pb1−pi3)Θr1−pi4Θr2+pb2]2+4(pi1Θr1+pi2)(pb3Θr1+pb4Θr2)}1/2}bits/s/Hz/dimension. (34)

### Iii-C Ergodic Capacity

As vanishes, the QoS constraints become loose and it can be easily verified that the effective capacity approaches the ergodic channel capacity, i.e.,

 limθ→0CE(\textscsnr,θ) =1Nmax0≤μ≤10≤P2≤min{Pmax,PintPdμ+(1−Pd)}bPd+aPfa+bE⎧⎪ ⎪⎨⎪ ⎪⎩maxKx1⪰0tr(Kx1)=1log2det[I+μN\textscsnrHKx1H†K−1z]⎫⎪ ⎪⎬⎪ ⎪⎭ +a(1−Pf)a+bE⎧⎪ ⎪⎨⎪ ⎪⎩maxKx2⪰0tr(Kx2)=1log2det[I+N\textscsnrHKx2H†]⎫⎪ ⎪⎬⎪ ⎪⎭. (35)

In order to gain further insight on the ergodic capacity expression, we note the following:

 Pr{channel isdetected busy} =Pr{channelis busy}Pr{channel is detected busy∣channelis busy}+Pr{channelis idle}Pr{channel is detected busy∣channelis idle} (36) =ba+bPd+aa+bPf (37) =bPd+aPfa+b (38)

where we used the fact that for the two-state Markov model of the PU activity depicted in Fig. 4, the probability of being in the busy state is . Similarly, we have

 Pr{channel is idle andis detected idle}=Pr{channelis idle}Pr{channel is detected idle∣channelis idle}=aa+b(1−Pf). (39)

Recall that when the channel is detected busy, the transmitter sends the data at the rate given in (15), and the transmission is successful because we are in either state 1 or 3 (of the state transition model in Fig. 3) which are both ON. If the channel is idle and is detected idle, then we are in state 4, which is also ON, and data is transmitted successfully at the rate given in (16). On the other hand, when the channel is busy but is detected idle, the rate cannot be supported by the channel and reliable communication cannot be achieved. Consequently, in this scenario (which is state 2 in Fig. 3), the successful transmission rate is zero. From this discussion, we immediately realize that the ergodic capacity in (III-C) is proportional to the average of these transmission rates weighted by the probabilities of the corresponding scenarios.

## Iv Energy Efficiency in the Low-Power Regime

In this section, we investigate the performance of cognitive MIMO transmissions in the low-power regime. For this analysis, we consider the following second-order low-snr expansion of the effective capacity:

 CE(\textscsnr,θ)=˙CE(0,θ)\textscsnr+¨CE(0,θ)\textscsnr22+o(\textscsnr2) (40)

where and denote the first and second derivatives of the effective capacity with respect to snr at . Note that the above expansion provides an accurate approximation of the effective capacity at low snr levels.

The benefits of a low-snr analysis are mainly twofold. First, operating at low power levels limits the interference inflicted on the PUs which is an important consideration in practice. Secondly, as will be seen below, energy efficiency improves as one lowers the transmission power. Hence, in this section, we consider a practically appealing and ambitious scenario in which cognitive users, in addition to their primary goal of efficiently utilizing the spectrum by filling in the spectrum holes, strive to operate energy efficiently while at the same time severely limiting the interference they cause on the PUs.

For the energy efficiency analysis, we adopt the energy per bit given by

 EbN0=\textscsnrCE(\textscsnr,θ), (41)

as the performance metric. It is shown in [32] that the bit energy requirements diminish as snr is lowered and the minimum energy per bit is achieved as snr vanishes, i.e.,

 EbN0min=lim\textscsnr→0\textscsnrCE(\textscsnr,θ)=1˙CE(0,θ). (42)

Note that is characterized only by the first derivative . At , the slope of the effective capacity versus (in dB) curve is defined as [32]

 S0=limEbN0↓EbN0minCE(EbN0)10log10EbN0−10log10EbN0min10log102. (43)

Considering the expression for the effective capacity, the wideband slope can be found from [32]

Hence, the wideband slope is obtained from both the first and second derivatives at . The wideband slope together with the minimum energy per bit provide a linear approximation of the effective capacity as a function of the energy per bit in the low-snr regime and enable us to gain insight on the energy efficiency of cognitive transmissions.

The next result identifies the first derivative of the effective capacity and the minimum bit energy.

###### Theorem 2

In the cognitive MIMO channel considered in this paper, the first derivative of the effective capacity with respect to snr at is

 ˙CE(0,θ)=1loge2{bPd+aPfa+bE[λmax(H†K−1zH)]+a(1−Pf)a+bE[λmax(H†H)]}. (45)

Consequently, the minimum energy per bit is given by

 EbN0min=loge2bPd+aPfa+bE[λmax(H†K−1zH)]+a(1−Pf)a+bE[λmax(H†H)]. (46)

Proof: See Appendix -B.

###### Remark 1

As detailed in the proof of Theorem 2, the first derivative of the effective capacity at and hence the minimum energy per bit is achieved by transmitting in the maximal-eigenvalue eigenspaces of and , when the channel is sensed as busy and idle, respectively. For instance, input covariance matrices in the cases of busy- and idle-sensed channels can be chosen, respectively, as

 Kx1=u1u†1 and Kx2=u2u†2 (47)

where and are the unit-norm eigenvectors associated with the maximum eigenvalues and , respectively. Hence, beamforming in the eigenvector directions corresponding to the maximum eigenvalues of and is optimal in terms of energy efficiency. Note that when the channel is sensed as busy, the possible interference arising from the primary users’ transmissions is taken into account by incorporating into the transmission strategy. Note further that as shown in (38) and (39), is the probability of detecting the channel as busy, and is the probability that channel is idle and is detected as idle.

###### Remark 2

The expressions in (45) and (46) do not depend on the QoS exponent , indicating that the performance in the low power regime as does not get affected by the presence of QoS requirements. Indeed, in (46) is the minimum energy per bit attained when no QoS constraints are imposed.

###### Remark 3

It is also interesting to note that the sensing performance has an impact on the energy efficiency. In particular, we can immediately notice that decreases with increasing detection probability . Similarly, decreases as the false alarm probability decreases. This can be seen by noticing that decreasing leads to an increased weight on and a decreased weight on , and noting that using Ostrowski’s Theorem [29, Theorem 4.5.9 and Corollary 4.5.11] and its extension to non-square transforming matrices in [30, Theorems 3.2 and 3.4], we have

 λmax(H†K−1zH)≤λmax(K−1z)λmax(H†H)≤λmax(H†H) (48)

where the last inequality follows from the property that .

Since minimum energy per bit is a metric in the asymptotic regime in which snr vanishes, we next consider the wideband slope in order to identify the performance at low but nonzero snr levels. Wideband slope in (44) depends on the both the first and second derivatives of the effective capacity at . In obtaining the second derivative, we essentially make use of the fact that the optimal input covariance matrices in the low snr regime, which are required to achieve the minimum bit energy and hence the wideband slope, can be expressed as

 Kx1=m1∑i=1κ1iu1,iu†1,i and Kx2=m2∑i=1κ2iu2,iu†2,i (49)

where are the weights satisfying and , and and are the multiplicities of and , respectively. Moreover, and are the orthonormal eigenvectors that span the maximal-eigenvalue eigenspaces of and , respectively. Despite this characterization, obtaining a general closed-form expression for the second-derivative seems intractable and we concentrate on the special case in which . Note that this case represents a scenario where there is no memory in the two-state Markov model for the PU activity. Hence, for instance, transitioning from busy state to busy state has the same probability as transitioning from idle state to busy state.

###### Theorem 3

In the special case in which the transition probabilities satisfy in the two-state model for the PU activity, the second derivative of the effective capacity with respect to snr at is

 ¨CE(0,θ) =θTBNlog2e2E2[ℓ1λmax(H†K−1zH)+ℓ2λmax(H†H)] −θTBNlog2e2E[ℓ1λ2max(H†K−1zH)+ℓ2λ2max(H†H)] −Nloge2E[ℓ1λ2max(H†K−1zH)m1+ℓ2λ2max(H†H)m2]

where and are the multiplicities of the eigenvalues and , respectively, and we have defined and . The wideband slope is

 S0=2E2[ℓ1λmax,1+ℓ2λmax,2]θTBN{E[ℓ1λ2max,1+ℓ2λ2max,2]−E2[ℓ1λmax,1+ℓ2λmax,2]}+NE[ℓ1λ2max,1m1+ℓ2λ2max,2m2]loge2. (50)

where we used the notation and