Probabilistic Co-Existence and Throughput of Cognitive Dual-Polarized Networks

Diversity techniques for cognitive radio networks are important since they enable the primary and secondary terminals to efﬁciently share the spectral resources in the same location simultaneously. In this paper, we investigate a simple, yet powerful, diversity scheme by exploiting the polarimetric dimension. More precisely, we evaluate a scenario where the cognitive terminals use cross-polarized communications with respect to the primary users. Our approach is network-centric, i


I. INTRODUCTION
Cognitive radio networks and, more generally, dynamic spectrum access networks are becoming a reality. These systems consist of primary nodes, which have guaranteed priority access to spectrum resources, and secondary (or cognitive) nodes, which can re-use the medium in an opportunistic manner [1]- [4]. Cognitive nodes are allowed to dynamically operate the secondary spectrum, provided that they do not degrade the primary users' transmissions [5]. From a practical viewpoint, this means that the secondary terminals must acquire a sufficient level of knowledge about the status of the primary network. This information can be gathered through the use of techniques such as energy detection [6], cyclostationary feature detection [7], and/or cooperative distributed detection [8]. Due to the complexity and drawbacks of the detection phase, the FCC recently issued the statement that all devices "must include a geolocation capability and provisions to access over the Internet a database of protected radio services and the locations and channels that may be used by the unlicensed devices at each location" [9]. Furthermore, the positions of the primary nodes and other meta-information can be shared in the same way. Though the locations of the nodes and their configurations can be obtained easily, the exploitation of such information remains an open problem. Considering that any diversity technique can be used by cognitive nodes, several approaches have been proposed to allow for the co-existence of primary and secondary networks [10]. These include, for example, the use of orthogonal codes (code division multiple access, CDMA) [11]; frequency multiplexing (frequency division multiple access, FDMA); directional antennas (spatial division medium access, SDMA) [12]; orthogonal frequencydivision multiple access (OFDMA) [13]; and time division multiple access (TDMA) [14]; among others.
In this paper, we investigate a simple, yet powerful, diversity scheme by exploiting the polarimetric dimension [15]- [17]. More specifically, a dual-polarized wireless channel enables the use of two distinct polarization modes, referred to as co-polar (symbol: ) and cross-polar (symbol: ⊥), respectively. Ideally, cross-polar transmissions (i.e., from a transmitting antenna on one channel to the receiving antenna on the corresponding orthogonal channel) should be impossible. In reality, this is not the case due to an imperfect antenna cross-polar isolation (XPI) and a depolarization mechanism that occurs as electromagnetic waves propagate (i.e., a signal sent on a given polarization "leaks" into the other). Both effects combine to yield a global phenomenon referred to as crosspolar discrimination (XPD) [18]- [20]. The scenario of interest for this work is shown in Fig. 1. The primary system consists of a single transmitter located at a distance of d 0 from its intended receiver. Without any loss of generality, the primary receiver is considered to be located at the origin of the coordinates system, leading to a receiver-centric analysis. The secondary (cognitive) terminals are deployed along with the primary ones. However, limitations on interference prevent them from entering a protected region around the receiver. This region, referred to as the "primary exclusive region" [21], is assumed to be circular and therefore, is completely characterized by its radius, denoted as d excl .
Since polarimetric diversity does not allow a perfect orthogonality between primary and secondary nodes' transmissions, its use is possible under the application of a so-called underlay paradigm [10], [22], [23]. This means that both cognitive and primary terminals carry out communications, provided that the capacity loss caused by cognitive users does not degrade communication quality for primary users. For this purpose, we can further characterize the underlay paradigm by requiring that the primary system must be guaranteed a minimum (transmission) capacity during a large fraction of time. As will be shown, this can, in turn, be formulated as a probabilistic co-existence problem under the constraint of a limited outage probability in the primary network.
We argue that using the polarimetric dimension allows dynamic spectrum sharing to be efficiently implemented in cognitive systems. To this end, we propose a theoretical model of interference in dual-polarized networks and derive a closed-form expression for the link probability of outage. We theoretically prove that polarimetric diversity can increase transmission rates for the secondary terminals while, at the same time, can significantly reducing the primary exclusive region.
First, we validated the expected (theoretical) performance gains analytically. To the best of our knowledge, none of the past studies in literature has investigated the behavior of the XPD under a complete range of propagation conditions, such as indoor-to-indoor and outdoor-to-indoor. In particular, we conducted a vast experimental campaign to provide relevant insights on the proper models and statistical distributions which would accurately represent the XPD. Based on these measures, the achievable performance of these dual-polarized cognitive networks, considering both half-duplex and full-duplex communications, will be determined.
The medium access control (MAC) protocol considered is a variant of the slotted ALOHA protocol [24] such that in each time slot, the nodes transmit independently with a certain fixed probability [25]. This approach is supported by the observations in [26, p. 278] and [25], [27], where it is shown that the traffic generated by nodes using a slotted random access MAC protocol can be modeled by means of a Bernoulli distribution. In fact, in more sophisticated MAC schemes, the probability of transmission of a terminal's transmission can be modeled as a function of general parameters, such as: queuing statistics, the queue-dropping rate, the channel outage probability incurred by fading; etc. [28] Since the impact of these parameters is not the focus of the this study, for more details we refer the interested reader to the existing studies in the literature [29]- [31] for further details.
The remainder of this paper is organized as follows. In Section II, we demonstrate how the polarimetric dimension increases spectrum-utilization efficiency and supports the co-existence of primary and secondary users in a prob-abilistic sense, which requires guaranteed capacity for the primary network. After these theoretical developments, several insights are presented to move from the concept to practical implementation. First, Section III presents an experimental determination of the main parameters used to characterize cognitive dual-polarized networks in indoorto-indoor and outdoor-to-indoor situations. These results are then used in Section IV for analytical performance evaluation. Section V concludes the paper.

A. Probabilistic Co-existence and Interference
Consider the cognitive network shown in Fig. 1 with two types of users: primary and secondary (cognitive). The primary network is supposed to be co-polar and the cognitive network is cross-polar. Without cognitive users, the primary network would operate with background noise and with the usual interference generated by the other primary users. Let C p (dimension: [bit/s/Hz]) be the desired capacity for a user in the primary network 1 . We impose that the secondary network operates under the following outage constraint on a primary user: where 0 < ε < 1 and C (dimension: [bit/s/Hz]) is a minimum per-primary user capacity. Equivalently, this constraint guarantees a primary user a maximum transmission rate of at least C for at least a fraction (1 − ε) of the time.
Under the simplifying assumption of Gaussian signaling, 2 the rate of this primary user can be written as a function of the signal-to-noise and interference ratio (SINR) as follows: Using (2) into (1) yields and, by introducing θ 2 C − 1, one has where P {SINR > θ} can be interpreted as the primary link probability of successful transmission for an outage SINR value θ. This value depends on the receiver's characteristics, the modulation and coding scheme, among others [32]. The SINR at the end of a primary link with length d 0 can be written as where P 0 (d 0 ) is the instantaneous received power (dimension: [W]) at distance d 0 , N 0 /2 is the noise power spectral density of the noise (dimension: [W/Hz]), B is the channel bandwidth, and P int is the cumulated interference power (dimension: [W]) at the receiver, that is, the sum of the received powers from all the undesired transmitters. We now provide the reader with a series of theoretical results, which stem from the following theorem.
THEOREM 1 In a narrowband Rayleigh block faded dual-polarized network, where nodes transmit with probability q on the co-polar and the cross-polar channels, the probability that the SINR exceeds a given value θ on a primary transmission, given a fixed transmitter-receiver distance d 0 , N int co-polar interferers at distances with a cross-polar discrimination coefficient XPD 0 , is 1 In this manuscript, bold letters refer to random variables. 2 Note that this assumption is expedient for analytical purposes. However, in the following the analytical predictions will be confirmed by experimental results.
where P 0 is the transmit power, N 0 B is the average power of the background noise, θ is the SINR threshold, α is the path loss exponent, XPD 0 is the reference cross-polar discrimination of the antenna at a reference distance is a function that characterizes for the polarization loss over distance.
PROOF: We assume a narrowband Rayleigh block fading propagation channel. The instantaneous received power P(d) from a node is exponentially distributed [33] with temporal-average received power E t [P(d)] = P (d) = P · L(d), where P denotes the transmit power and L(d) ∝ d −α is the path loss at distance d (it accounts for the antenna gains and carrier frequency). The received power is then a random variable with the following probability density function: .
In a dual-polarized system, the cross-polar discrimination (XPD) is defined as the ratio of the temporal-average power emitted on the cross-polar channel and the temporal-average power received in the co-polar channel [15], i.e.: where d is the transmission distance, P is the temporal-average value of the instantaneous leaked power P (⊥→ ) (d), and P ⊥ (d) E t P ⊥ (d) is the temporal-average value of the instantaneous cross-polar power P ⊥ (d). In a generic situation, the XPD is subject to spatial variability [19] and, therefore, in the context of this network-level analysis, we define the XPD in a spatial-average sense, i.e.: where the operator X denotes the average of the value X computed on multiple different locations at the same distance d. Note that, even though the XPD is considered here in a spatial-average sense, it is possible to accommodate its expected variability for the purpose of ensuring a required minimum cross-polar discrimination. This will be detailed in Section IV. Finally, it is shown in [17]- [19], that XPD(d), defined according to (7), can be expressed as follows: where XPD 0 ≥ 1 is the XPD value at a reference distance d ref and the function G(d, d ref ) ≤ 1 characterizes the de-polarization experienced over the distance. Let the traffic at the N int primary and N ⊥ int cognitive interfering nodes be modeled through the use of independent In other words, {Λ i }, {Λ j } are sequences of independent and identically distributed (iid) Bernoulli random variables: if, in a given time slot, one of these indicators is equal to 1, then the corresponding node is transmitting; if, on the other hand, the indicator is equal to 0, then the node is not transmitting. We also assume that the traffic distribution is the same at all interfering nodes of the network, i.e., ∀i, P {Λ i = 1} = q and ∀j, P{Λ j = 1} = q, which is supported by the analyses presented in [25], [27], [34]. The overall interference power at the receiver is the sum of the interference powers due to co-polarized and cross-polarized (leaked because of depolarization) interference powers, i.e.: } are the (instantaneous) interfering powers at the receiver. The probability that the SINR at the receiver exceeds θ can thus be expressed as follows: where, in the second passage, we have exploited the fact that, in a Rayleigh faded transmission, the SINR is also exponentially-distributed [33]. Since all terminals have an independent transmission behavior and are subject to non-correlated channel fading, i.e., }, and {Λ j } are independent sets of random variables, it then holds that The generic first expectation term at the right-hand side of (8) can be expressed as follows: The generic second expectation term in (8) can be expressed, by using (6), in a similar way: By plugging (9) and (10) into (8), one finally obtains expression (5) for the probability of successful transmission.
Theorem 1 gives interesting insights on the expected performance in a dual-polarized transmission subject to background and inter-node interference. First, the leftmost term of the expression at the right-hand side of (5) is relevant in a situation where the throughput is limited by the background (typically thermal) noise. In large and/or dense networks, the transmission is only limited by the interference and one can focus on the interference and polarization terms (i.e., the two other term of the expression, assuming N 0 B is negligible). The first exponential term can be easily evaluated if N 0 B = 0.
The second and the third terms of expression (5) relate to the interference generated by the surrounding nodes transmitting in co-and cross-polarized channels. These terms depend on (i) the polarization characteristics of the interfering nodes, (ii) the traffic statistics, and (iii) the topology of the network. Note that the impact of the topology has been largely investigated in [35] and we will limit our study to the effect of polarization.
Finally, channel correlation is neglected here, as often in the literature, for the purpose of analytical tractability and because these correlations do not change the scaling behavior of link-level performance. For the sake of completeness, we note that in [36] an analysis of the impact of channel correlation is carried out. The authors conclude that, when the traffic is limited (q < 0.3), the assumption of uncorrelation holds. On the other hand, when the traffic is intense (q ≥ 0.3), the link probability of success is higher in the correlated channel scenario than in the uncorrelated channel scenario.

B. Probabilistic Link Throughput
Referring back to our definition of the probabilistic co-existence of the primary and secondary terminals in (1), a transmission is said to be successful if and only if the primary terminal is not in an outage for a fraction of time longer than (1 − ε), i.e., if the (instantaneous) SINR of the cognitive terminal is above the threshold θ. Therefore, we denote the probability of successful transmission in a primary link as P s , i.e., The probabilistic link throughput [37] (adimensional) of a primary terminal is defined as follows: • in the full-duplex communication case, it corresponds to the product of (i) P s and (ii) the probability that the transmitter actually transmits (i.e., q); • in the half-duplex communication case, it corresponds to the product of (i) P s , (ii) the probability that the transmitter actually transmits (i.e., q), and (iii) the probability that the receiver actually receives (i.e., 1 − q). The probabilistic link throughput can be interpreted as the unconditioned reception probability which can be achieved with a simple automatic-repeat-request (ARQ) scheme with error-free feedback [38]. For the slotted ALOHA transmission scheme under consideration, the probabilistic throughput in the half-duplex mode is then τ (half) q(1 − q)P s and in full-duplex case τ (full) q P s .

C. Properties and Opportunities of Polarization Diversity
Theorem 1 expresses a network-wide condition to support the co-deployment of primary and cognitive terminals. In order to implement polarization diversity and make it work, proper considerations have to be carried out. In this section, we propose several lemmas, all derived from Theorem 1, that allow to design and operate dual-polarized systems.
LEMMA 2 In a dual-polarized system subject to probabilistic co-existence of primary and secondary networks, relocating a cognitive terminal from the co-polar channel to the cross-polar channel increases its probability of transmission while keeping intact the transmission capacity of the primary network.
PROOF: Let us consider a scenario with a single interferer located at distance d and transmitting with power P . For the ease of understanding, let us assume that if the terminal uses a polarized antenna, its probability of transmission will be denoted as q = q ⊥ , whereas if a classical (not dual-polarized) scenario is considered, then q = q .
If the cognitive terminal is using the co-polar mode, the probabilistic co-existence condition (1) can be written as whereas if the cognitive terminal is using the cross-polar mode it holds that Therefore, the maximum acceptable probability of transmission in the co-polar mode is Note that, on average, XPD(d) ≥ 1 according to definition (6) and for physical reasons-the power leaked on the co-polar dimension is at most equal to the power transmitted on the cross-polar channel. Finally, all other quantities in (12) are strictly positive and, therefore, one obtains that where the right-hand side expression for q ⊥ max derives directly from (11). Therefore, the thesis of the lemma holds.
Lemma 2 indicates that polarization can be exploited as a diversity technique. Indeed, the achievable transmission rate will always be increased if the secondary network uses a polarization state that is orthogonal to that of the primary network and, furthermore, this remains true regardless of the values taken by the other system parameters (e.g., transmission power, acceptable outage rate ε, SINR value, etc.).
LEMMA 3 There exists a region of space, referred to as the primary exclusive region, where the cognitive terminals are not allowed to transmit, can be reduced by means of polarimetric diversity.
PROOF: As previously anticipated in Section I, the primary exclusive region is completely characterized by the primary exclusive distance d excl , i.e., the minimum distance at which a cognitive terminal has to be, with respect to a primary receiver, so that it does not impact the capacity of the primary user (in a probabilistic sense) [21]. Starting from (5), in the presence of a single cross-polar interferer, one can write This relation is equivalent to where the definition at the right-hand side allows to express the minimum distance d excl as a function of the distance d 0 and the other main system parameters as follows: In Fig. 2, the normalized primary exclusive distance, defined as d excl /d 0 , is shown, as a function of the terminal probability of transmission q, with ε = 0.2. It can be observed that in the case without polarization, one always has d excl d 0 , i.e., the cognitive terminals must be located outside the transmission zone defined by the primary emitter-receiver distance. On the opposite, it is possible to operate a cognitive terminal inside this region (i.e., with d excl < d 0 ) when the polarimetric dimension is used. Furthermore, in both cases the exclusive distance increases as a function of the terminal probability of transmission but its gradient is smaller in the dual-polarized case.
It is interesting to observe that relation (14) can also be used to parameterize practical realizations of the antennas. Indeed, it yields that , it follows: Therefore, the quantity at the right-hand side of (16) represents the minimum amount of XPD that the antenna of the cognitive terminal must possess. This value depends on the network configuration but also on the propagation environment (through the de-polarization function G (d, d ref )).
LEMMA 4 If q < ε, polarization diversity is not required to achieve a probabilistic co-existence.
PROOF: As previously introduced, the coefficient XPD 0 is greater than or equal to 1. Therefore, the minimum value of XPD 0 to guarantee error-free transmissions on the cross-polar channel is In (16), all quantities are greater than zero. Therefore, if q < ε, the quantity q − ε is always negative and the solution of (17) is XPD 0 = 1.
Lemma 4 indicates that, if the desired throughput remains limited, then the outage is guaranteed on the primary system without summoning up the diversity of polarization on the secondary terminal. Therefore, the cross-polar channel can be kept available for other terminals that may require higher data rates. This can be observed in Fig. 2.
THEOREM 5 Besides being limited by probabilistic co-existence considerations, there exists an optimum probability of transmission by a terminal in the primary network, denoted as q opt , that maximizes the throughput.
PROOF: Let us define the optimal user probability of transmission as where the probabilistic throughput τ has been defined in Subsection II-B. We first focus on half-duplex systems, using polarization diversity: in this case, the link throughput is τ = q(1 − q)P s . Since ln(·) is a monotonically increasing function, finding the maximum of τ is equivalent to finding the maximum of ln(τ ), i.e., q opt = arg max q ln(τ ).
In order to find the maximum, we compute the partial derivative of ln(τ ) with respect to q:   where By using the approximation 3 ln(1 + x) ≈ x and setting ∂ ln(τ )/∂q = 0, one has The positive solution of this equation is given by which is the probability of transmission that maximizes the throughput. The same derivation can be applied in the case of a full duplex system and leads to the solution q opt ≈ η. If the approximation ln(1 + x) ≈ x is not used, then the optimal probability of transmission cannot be given in a closed-form expression but has to be numerically evaluated.
Obviously, the maximum value of q will be the minimum between (i) the optimum probability of transmission in a slotted transmission system (in a general sense), given by (18), and (ii) the maximum rate that can be achieved under the constraint of a probabilistic co-existence in (13). Therefore, before selecting its transmission rate, a cognitive terminal must evaluate these two quantities, on the basis of the available information stored in the databases (positions of the nodes, acceptable outage, etc.), and use the smallest one. In Fig. 3(a) and Fig. 3(b), the accessible and optimal terminal probabilities of transmission are presented as functions of d/d 0 , in the cases with (a) half duplex and (b) full duplex communications, respectively. In each case, two polarization strategies are considered: (i) no polarization and (ii) XPD 0 = 10 dB. The accessible regions are defined by means of the inequality (15). In particular, the leftmost border of each exclusive region, denoted as line q excl , is defined as the probability of transmission for a terminal at the boundary of the primary exclusive region, i.e., with d = d excl .
From these figures, it can be observed that the probability of transmission of dual-polarized cognitive systems is mainly limited by the interference bound imposed to protect the primary system. In fact, the transmission rate of the terminals will nearly always be lower than the optimal transmission rate, except when the cognitive terminal is distant. In that specific case, the optimum probability of transmission (13) in the accessible region (in a probabilistic sense) saturates, i.e., it reaches q opt ≈ 1/2 in the half-duplex case and q opt ≈ 1 in the full-duplex case. Note that these values correspond to the maximum achievable throughput observed in any half-duplex or full-duplex system. Indeed, the definitions of the probabilistic link throughput are τ (half) q(1 − q)P s and τ (full) q P s and the corresponding optimum terminal probabilities of transmission cannot exceed q = 1/2 and q = 1, respectively.
In the scenarios where polarimetric diversity is exploited, this crossover distance is smaller (d excl /d 0 ≈ 1.5) than in the classical case (d excl /d 0 ≈ 3.3). Comparing the results in Fig. 3(a) with those in Fig. 3(b), another observation can be carried out. In the half-duplex case, for each distance d > d excl , the optimal transmission probability q opt lies inside the accessible region. In other words, q has to be properly selected to maximize the throughput. In the full-duplex case, q opt ≈ 1 everywhere in the exclusive region. These observations will be confirmed by the results presented in Section IV.
Finally, it is confirmed that, in the accessible regions, one either has (i) (d j /d 0 ) α 1 with XPD 0 > 1 (i.e., q/η j 1) or (ii) q opt 0.3. Therefore, the approximation used in proof of Theorem 5 (i.e., ln(1 + x) ≈ x) holds and the value of q opt derived in Theorem 5 can be considered as an accurate approximation of the true value.

D. Considerations for Practical System Implementation
In the previous subsections, we have shown that the capacity of a primary user can be guaranteed, while, at the same time, allowing efficient spectrum access, if the polarimetric dimension is exploited. Moreover, dual-polarized terminals will benefit from an increase of capacity by means of a higher transmission rate and reduced terminalto-terminal interference. The efficiency of polarization diversity depends on the cross-polar discrimination of the antennas in use. More precisely, the value of the initial cross-polar discrimination (i.e., XPD 0 ) has to be as high as possible; yet, the XPD of well-designed antennas is typically on the order of 10 ÷ 20 dB [15], [39], which allows a significant discrimination between co-polar and cross-polar channels. Depending on the achievable value of XPD 0 , the outage rate of a primary terminal, and the location of the terminals, the transmission rate of a cognitive terminal can be adapted taking into account the relations (13) and (18). Finally, the primary exclusive region can be determined by means of (15) and notified to the cognitive terminals which, in turn, can use it as a constraint.

III. EXPERIMENTAL DETERMINATION OF THE INDOOR-TO-INDOOR AND OUTDOOR-TO-INDOOR XPD
Several previous works have been undertaken in order to model the XPD for different kinds of environment. In [20], a theoretical analysis is conducted for the small-scale variation of XPD in an indoor-to-indoor scenario and it is concluded that it has a doubly, non-central Fisher-Snedecor distribution. A mean-fitting (i.e., the pathloss) model of XPD as a function of the distance in an outdoor-to-outdoor scenario was studied in [16], [19]. The corresponding performance is analyzed in [11].
In this paper, we provide the reader with original measurements campaigns in both indoor-to-indoor and outdoorto-indoor scenarios. Indeed, these correspond to real-life situations where various technologies, such as WiFi, sensor networks, personal area networks (indoor-to-indoor scenarios) or WiMax, public WiFi, and 3G systems (outdoorto-indoor scenarios) are in use.
We consider three generic models to describe the variation of the XPD with respect to the distance. For instance, when the transmission ranges are long (several hundreds of meters or a few kilometers), the best expression for the path loss function is where β is a decay factor (0 < β ≤ 1). On the other hand, when distances are small (tens of meters) or in indoorto-indoor scenarios, the XPD value, in decibels, decreases linearly with respect to the distance. In other words, one can write which corresponds, in linear scale, to the following path loss function: Finally, in some indoor scenarios where the transmission distances are small, it was observed that the XPD remains constant, i.e., In the remainder of this section, we characterize the applicability of the three XPD models just introduced. In other words, we consider an experimental setup and, on the basis of an extensive measurement campaign, we determine which XPD model is to be preferred in each scenario of interest (indoor-to-indoor and outdoor-to-indoor).

A. Setup
The measurements were performed using a Vector Signal Generator (Rohde & Schwarz SMATE200A VSG) at the transmitter (Tx) side and a Signal Analyzer (Rohde & Schwarz FSG SA) at the receiver (Rx) side. The Tx chain was composed of the VSG and a directional antenna. The Rx antenna was a tri-polarized antenna, made of three co-located perpendicular antennas. Two of these antennas were selected to create a Vertical-Horizontal dual-polarized antenna. The three receiver antennas were selected one after another by means of a switch and were connected to the Signal Analyzer through a 25 dB, low-noise amplifier. The Rx antennas were fixed on an automatic positioner to create a virtual planar array of antennas. A continuous wave (CW) signal at the frequency of 3.5 GHz was transmitted and the corresponding frequency response was recorded at the receiver side. The antenna input power was 19 dBm.
The measurement site was the third floor of a building located on the campus of Brussels University (ULB) and referred to as "Building U." In the outdoor-to-indoor case, shown in Fig. 4(a), the transmitter was fixed on the rooftop of a neighboring building (referred to as "Building L"), at a height of 15 m and was directed toward the measurement site. A brick wall was separating the line-of-sight (LOS) direction between this measurement site and the transmitter. The measurements were performed in a total of 78 distinct locations and in seven successive rooms. The rooms were separated by brick walls and closed wooden doors. The distance between the transmitter and the measurement points was in the range between 30 m and 80 m. In the indoor-to-indoor case, shown in Fig. 4(b), the Tx antenna was fixed in the first room and was directed toward the seven next rooms, in which 65 measurement points were considered. The distance between the transmitter and the measurement points was in the range between 8 m and 55 m. In order to characterize the small scale statistics of XPD a total of 64 spatially separated measurements were taken at each Rx position and in an 8 × 8 grid. The spacing between grid points was λ/2 = 4 cm. At each grid point, 5 snapshots of the received signal were sampled and averaged to increase the signal-to-noise-ratio.

B. Experimental Results and Their Interpretation
The analysis of the collected experimental results have shown that the values of the XPD, for a given distance, present a location-dependent variability. Therefore, in the following figures, where the XPD is shown as a function of the distance d, the average value is shown along with the 1σ and 2σ confidence intervals. Since the spatial variations were found to be Gaussian, these intervals account for 68% and 95% of the observed sets, respectively.
The horizontal polarization was first used in an indoor-to-indoor scenario and is reported in Fig. 5. It was observed that the XDP can be accurately modeled by means of the propagation model G 2 (d, d ref ) where one has XPD 0 = 11.3 dB, d ref = 1 m, and γ = 0.16 dB/m. The variation around the average value was also analyzed and the corresponding cumulative distribution function (CDF) is shown in Fig. 6. This variation was found to fit with a zero-mean Gaussian random variable with standard deviation equal to 0.295 dB. It is interesting to note that, unlike the case of the outdoor-to-outdoor scenarios presented in [19], the behavior of the XDP depends on the initial polarization of the antenna. More precisely, the results in Fig. 6 correspond to a horizontal polarization while the results in Fig. 7 correspond to an initial vertical polarization. It can be seen that, in the latter scenario, the XPD is almost constant and equal to XPD 0 = 4 dB. In this case, the XPD variability can be modeled as a zero-mean Gaussian random variable with standard deviation equal to 2.75 dB.
Finally, the results collected in an outdoor-to-indoor scenario are presented in Fig. 8. As expected, the XPD is a decreasing function of the distance and is suitably modeled by using the propagation model G 2 (d, d ref ), with XPD 0 = 12.87 dB, d ref = 20 m, and γ = 0.13 dB/m . The spatial variability can be modeled as a zero-mean Gaussian random variable with standard deviation equal to 2.95 dB. Note that full de-polarization occurs after a hundred of meters and the two initial polarizations (i.e., horizontal and vertical) lead to the same behaviour.

IV. NUMERICAL PERFORMANCE EVALUATION
In this section, a numerical analysis of the performance of the proposed dual-polarized cognitive systems is presented. In Section III, it has been shown that the XPD experiences spatial shadowing: more precisely, at a fixed distance different values of the XPD can be observed at different locations. The system parameters for performance analysis are selected by taking into account this normal fluctuation. Therefore, instead of using the average value for XPD 0 , it is preferable to use a value (denoted as XPD min 0 ) that can be observed with a confidence equal to a pre-defined value δ ∈ (0, 1). Taking into account that XPD 0 has a Gaussian distribution, it follows that where µ and σ are the average value and the standard deviation of the observed XPD 0 , respectively. Therefore, XPD min 0 can be expressed as XPD min 0 = µ + σ Q −1 (δ) . For instance, if a confidence level of 80% is required (i.e., δ = 0.8), one has to select XPD min 0 = µ − σ Q −1 (0.8) ≈ µ − 0.81 σ. This approach will be used to set the initial parameters in the following performance analysis.

A. Full Duplex Systems in an Outdoor-to-Indoor Scenario
Cellular system typically correspond to an outdoor-to-indoor scenario. Examples include WiMax base stations or cellular mobile phone systems. A typical scenario is presented in Fig. 9. Referring to the experimental results presented in Section III, we used in our simulations the model G 1 (d, d ref ) with parameters α = 3, β = 0.4, and XPD 0 = 4 ÷ 10 dB. Also, the measurements lead us to set XPD min 0 equal to 10.48 dB with an 80% confidence

B. Results and analysis
In Fig. 6, the XPD in the indoor-to-indoor scenario is presented as a function of the distance. It can be observed that the XDP decreases with respect to the distance. More precisely, this scenario corresponds to the propagation model G 2 (d, d ref ) where one has XPD 0 = 11.3dB, d ref = 1m, and γ = 0.16 dB/m. As it can be observed, the values of the XPD present a lognormal variability. More precisely, it can be seen in Fig. 7 that the cumulative distribution function (cdf) of the differences between the data and their average value does fit with the cdf. of a zero-mean Gaussian random variable. It is interesting to note that, unlike in the case of the outdoor-to-outdoor scenarios presented in [3], the behavior of the XDP depends on initial polarization of the antenna. More precisely, the results in Fig. 7 correspond to an horizontal polarization while the results in Fig. 8 correspond to an initial vertical polarization. It can be seen that, in the latest scenario, XPD is almost constant and equal to XPD 0 = 4 dB.
Finally, outdoor-to-indoor scenario was considered and is presented in Fig. 9. As expected, the XPD decreases with respect to the distance and a full de-polarization is achieved after a hundred of meters.   Two different polarization strategies are investigated: (i) the primary and the cognitive networks do not use polarimetric diversity (this scenario is referred to as no polarization) and (ii) the systems reduce their interference by using two orthogonal polarization states (this scenario is referred to as full polarization).
In Fig. 11, the performance of full duplex systems is presented. More specifically, in Fig. 11(a), the throughput of the system is shown as a function of the terminal probability of transmission. It can be seen that the throughput of the dual-polarized system is significantly higher, particularly when the probability of transmission is high. In Fig. 11(b), the corresponding link probability of success in the primary network is investigated. It can be seen that it confirms the conclusions of Lemma 2: for a given minimum value of the link probability of success, the achievable transmission rate is significantly higher in the dual-polarized mode with respect to the value observed with the classical approach. For instance, with ε = 0.8, one has q max = 0.15 while, by using the dual-polarized Terminals  approach, the maximum probability of transmission can be increased up to q max = 1.0. In other words, virtually any transmission rate is achievable with a limited impact on the primary system.

B. Half-Duplex System in an Indoor-to-Indoor Scenario
In a second scenario, the probabilistic co-existence is analyzed in the context of half-duplex systems, where indoor-to-indoor transmissions are typically used. Examples include wireless sensor networks (WSNs), ZigBee systems, and body area networks (BANs). A typical scenario is presented in Fig. 10. In our simulations, we considered a primary transmission at distance d 0 = 15m and subject to interference from 5 terminals located at d = 25 m from the central base station. This corresponds to d/d 0 ≈ 1.67 and it can be seen from Fig. 3(a) that this value is in the accessible region. The propagation model G 3 (d, d ref ) is used and the other relevant parameters are θ = 10 dB, XPD 0 = 4 − 10 dB, α = 3. The transmit power is the same at all nodes. Referring to the experimental analysis conducted in Section III, one can observe that the values of interest for XPD min 0 (with a 80% level of confidence) are 8.91 dB and 1.8 dB for horizontal and vertical polarizations, respectively.
In Fig. 12, the performance of these half-duplex systems is presented. More particularly, in Fig. 12(a), the throughput is shown as a function of transmission rate of the terminals, in a scenario with co-polar interferers (i.e., without diversity of polarization) and under the dual-polarized scheme under study. It can be observed that the diversity of polarization drastically increases the throughput, even when the terminal probability of transmission is  small. Regarding the probabilistic co-existence, in Fig. 12(b) the link probability of success at the primary terminal is shown as a function of the transmission rate of the cognitive terminals. It can be observed that the use of polarization diversity gives a clear advantage in terms of interference limitation and available throughput for the cognitive terminals. For instance, with ε = 0.8 and an horizontal initial polarization, one has q max = 0.07 while, by using the dual-polarized approach, this quantity can be increased up to q max = 0.25 at each terminal. Finally, it can be seen the optimum probability of transmission with XPD=10 dB is approximately q opt ≈ 0.5, which matches with the value of q opt found in Fig. 3(a).

V. CONCLUSIONS
In this paper, we have presented a novel theoretical framework to demonstrate the network-level performance increase that can be achieved in a polarimetric diversity-oriented system subject to Rayleigh fading and probabilistic co-existence of primary and secondary (cognitive) networks. The theoretical approach was supported by an extensive measurement campaign. It has been shown that different mathematical expressions must be used in order to suitably model the dependence of the XPD on the distance between transmitter and receiver. These models depend not only on the scenario of interest, but also on the initial antenna polarization. For instance, in an indoor-to-indoor scenario, we have observed that the horizontal polarization provides a significant diversity (XPD 0 around 10 dB) while the vertical polarization leads to a more limited gain (XPD 0 around 4 dB).
Our results suggest that dual-polarized networks are of interest, even if orthogonality (indicated by the XPD value) is limited. Indeed, with respect to the classical implementation of probabilistic co-existence of primary and secondary network on the same (single polarization) channel, the use of polarization diversity allows to remarkably increase the per-link throughput and reduce the primary exclusive region. In some cases (i.e., at low transmission rates), it could even be possible to deploy a cognitive terminal closer to a primary receiver than the primary transmitter itself, i.e., inside the primary exclusive region. APPENDIX The performance analysis carried out throughout the paper applies to networking scenarios with narrowband fading. In this appendix, we present a preliminary, yet insightful, extension of our approach to encompass the presence of wideband fading.
In the presence of a transmission channel experiencing wideband fading, the transmitted symbols of the considered packet suffers from interference of the other symbols that have been delayed by multipath [33]. This phenomenon is referred to as Inter-Symbol-Interference (ISI) and it depends on the channel model, modulation format, and symbol sequence characteristics, among others [40]- [42]. Therefore, the expression of the ISI is hard to obtain and typically is not in closed form. In the network-level approach we follow in this paper, an approximation to SINR in presence of wideband fading can be obtained by treating the ISI as an additive, uncorrelated, Rayleigh-faded noise power proportional to the received power [41]. The expression of the link-level SINR introduced in (4) becomes: SINR wb P 0 (d 0 ) N 0 B + P int + P ISI where P ISI is noise power associated with the ISI. Its average value (noted P ISI = E [P ISI ]) is supposed to be proportional to the received power [41] and can be defined as P ISI ν P 0 (d 0 ) 0 ≤ ν < 1.
Note that ν = 0 refers to the narrowband scenario. The Theorem 1 can now be extended to incorporate the case of wideband Rayleigh fading as follows. The probability that the SINR at the receiver exceeds a given value θ is where the expectation of the term containing the ISI power becomes The definition of P ISI gives Following the derivation outlined in the proof of Theorem 1, the link probability of successful transmission (19) in the wideband fading case finally becomes By comparing (20) with (5), it can be observed that the presence of wideband fading reduces the probability of succesful link tranmission by the factor 1/(1 + νθ). Since this factor is lower than 1 for ν ∈ (0; 1], it can be concluded that the presence of ISI has a negative impact on the link probability of outage. Moreover, for a given value of ν, i.e., for a given level of ISI, this negative impact is relatively stronger the higher is the considered SINR threshold θ. This, in turn, results on (i) an increase of the primary exclusive region (i.e., a reduction of the accessible region) and (ii) a degradation of system throughput. More precisely, in Fig. 13(a) we clearly show the reduction of the a comparison between the accessible transmission regions in the presence of narrowband fading  (shown in Fig. 3(a)) and in the presence of wideband fading (with P 0 /P ISI = 20dB). As one can see, the presence of a limited ISI has a detrimental impact, significantly increasing the primary exclusive region. In Fig. 13(b), the throughput in the presence of ISI is shown in a scenario with half-duplex communications. In this case as well, the negative impact of wideband fading is evident. Although the impact of frequency selective fading is detrimental, from these figures it can concluded that, even in presence of wideband fading channel, the use of polarimetric diversity significantly increases the overall performance of the whole system and is thus of interest in the context of cognitive radio networks.