Abstract
Rendezvous is a prerequisite and important process for secondary users (SUs) to establish data communications in cognitive radio networks (CRNs). Recently, there has been a proliferation of different channel hopping (CH) based schemes that can provide rendezvous without relying on any predetermined common control channel. However, the existing CH schemes were designed with omnidirectional antennas which can degrade their rendezvous performance when applied in CRNs that are highly crowded with primary users (PUs). In such networks, the large number of PUs may lead to the inexistence of any common available channel between neighboring SUs which result in a failure of their rendezvous process. In this paper, we consider the utilization of directional antennas in CRNs for tackling the issue. Firstly, we propose two coprimalitybased sector hopping (SH) schemes that can provide efficient pairwise sector rendezvous in directional antenna CRNs (DIRCRNs). Then, we propose an efficient CH scheme that can be combined within the SH schemes for providing a simultaneous sector and channel rendezvous. The guaranteed rendezvous of our schemes are proven by deriving the theoretical upper bounds of their rendezvous delay metrics. Furthermore, extensive simulation comparisons with other related rendezvous schemes are conducted to illustrate the significant outperformance of our schemes.
1. Introduction
The inefficiency of radio spectrum utilization due to the fixed assignment of spectrum bands, coupled with the huge increase in the number of wireless devices recently, leads to the emerging of cognitive radios (CRs). CRs have been introduced as an efficient solution for the spectrum scarcity issue. They allow unlicensed users, also known as secondary users (SUs), to opportunistically use licensed bands as long as this use does not create any interference to the bands licensed users, that is, primary users (PUs). In distributed CRNs, SUs need to meet each other on a common channel and exchange control messages in order to set up their data transmissions [1]. This operation is called rendezvous, which is a fundamental and a vital process for initiating the connection of the SUs data communications.
The use of a dedicated common control channel (CCC) is one widely adopted rendezvous approach in the literature [2–6]. Although this approach can simplify the rendezvous process, it has many drawbacks such as the CCC susceptibility to longtime blocking by PUs, early saturation by SUs, or jamming by attackers [7, 8]. Moreover, the spectrum heterogeneity among SUs, caused by the spatial and temporal variations of the spectrum opportunities (influenced by neighboring PU activities), makes this approach not practical. Therefore, channel hopping (CH) has been proposed as an alternative approach for rendezvous without the need for any predefined CCC. The existing CHbased rendezvous schemes were designed with omnidirectional antenna according to different mathematical structures (e.g., [9–18]) in order to provide a successful channel rendezvous between a pair of singlehop SUs. In the CH approach, each SU generates its CH sequence based on the available channels that are sensed to be idle from any PU activities within its omnidirectional range. Then, the SU keeps hopping over the channels according to the generated CH sequence for achieving channel rendezvous. The rendezvous occurs between a pair of communicating SUs when they hop during the same time slot over a channel that is commonly available for both of them. The existence of common available channels (at least one) between the pair of communicating SUs is a common and essential assumption made by all the existing CH schemes. Their designs rely mainly on this assumption to take place for ensuring the success of the rendezvous operation. However, none of the existing CH rendezvous designs were tailored for highdensity PU networks, where the number of active PUs in the network is larger than the number of the channels. In such networks, the channel availabilities may vary dramatically among the SUs within the singlehop rendezvous region itself. This can lead to the inexistence of any common available channel between a pair of SUs and hence the failure of the rendezvous process.
One approach to overcome this serious rendezvous problem is using directional antennas instead of the conventionally used omnidirectional ones [19]. Directional antennas have been utilized in emerging wireless networks such as the millimeterwave (MMW) networks for providing highly reliable transmission and multiGigabit data rates. This is mainly due to their capabilities in enlarging the transmission range and limiting the interference [20, 21]. However, In CRNs, equipping SUs with directional antennas for channel rendezvous can bring about many advantages. First of all, it allows SUs to transmit their rendezvous messages towards specific directions which can reduce the amount of interference to the PUs in the network as compared with the omnidirectional antennas. This is because the transmission of the SU that is equipped with directional antenna is directed to a particular region which can only cause interference to those PUs that are within this region. On the contrary, when using the omnidirectional antennas, the transmission of the SU is scattered towards all the directions. Thus, it can cause interference to all the surrounding PUs within the SU transmission range. Second, as related to the interference restriction imposed by the CR concept, SUs can use only the channels that are idle from any PU activities within their transmission range. According to that and due to its directed transmission range, using directional antenna will increase the number of available channels that can be utilized by the SU in each of its transmission sectors. As a consequence, the probability for any pair of neighbouring SUs to have at least a common available channel within their crossed sectors is high regardless of the number of PUs in the CRN. Note that, in omnidirectional antennas, this probability could be less than for some scenarios when the number of PUs is larger than the network channels [19]. Increasing the probability of existing common available channel between the SUs will significantly enhance the probability of successful channel rendezvous. Therefore, in this paper, we study the rendezvous problem in directional antenna CRNs where SUs are equipped with directional antennas.
While the use of directional antennas for channel rendezvous in CRNs can bring about the abovementioned advantages, it also imposes some unique challenges. Among these challenges is the sector rendezvous which must be achieved in advance between the pair of communicating SUs before they can achieve channel rendezvous. Specifically, the SUs need to steer their antennas towards each other in order to communicate over their commonly available channels. This necessitates the design of a deterministic sector hopping (SH) scheme that can guarantee that any pair of neighbouring SUs will steer their antennas to the proper directions (i.e., towards each other) at a certain time instance. So, when the SH scheme is combined with a suitable and deterministic CH scheme, SUs can achieve successful sector and channel rendezvous simultaneously within a bounded time. However, since the SUs do not know the relative locations of each other as well as the available channels before they rendezvous, the combined sector and channel hopping (SCH) scheme should be designed in fully distributed manner without any prior information or synchronization. Designing such combined SCH schemes for achieving rendezvous in DIRCRNs is intuitively more challenging as compared with the traditional omnidirectional antenna paradigm.
In spite of the existing research in the literature, there have been some papers that addressed the utilization of directional antennas in CRNs. However, most of the proposed works investigated issues other than the rendezvous issue such as sensing [22, 23], routing [24, 25], and connectivity [26, 27]. To the best of our knowledge, the works in [19, 28] are the only proposals that tackle the sector and channel rendezvous issue in DIRCRNS. In [19], an efficient framework for beamforming was proposed to provide a pairwise sector and channel rendezvous in DIRCRNs. However, the SH scheme in [19] is not a blind design, where it assumes that the receiver must have prior knowledge of the sender number of sectors in order to achieve a successful sector rendezvous. This is not practical, since SUs do not know any information about each other before they rendezvous. On the other hand, Li and Xie in [28] proposed a blind primebased SH scheme, where the number of sectors is adjusted to primes for achieving a guaranteed sector rendezvous. However, since the scheme relies on the coprimality between prime numbers only, the authors added an initiating rotated SH sequence on the sender side. This is in order to ensure a successful rendezvous with the receiver when the sender and the receiver construct their sequences with the same prime. Accordingly, the extra sequence overhead incurs extensively long sector rendezvous latency and hence a long timetorendezvous (TTR) will result when the SH is combined with a CH sequence. Furthermore, the SH schemes in [19, 28] are designed only for asymmetricrole environment, where SUs have preassigned role (i.e., SU is either a sender or receiver). However, this design limits the applications of the schemes; for example, SUs cannot work as a forwarder (i.e., receive packets from one SU and then forward them to another SU) due to the permanent role assignment [29]. Another drawback of these works is their limitations to provide a complete solution for combining the SH scheme with a suitable and efficient CH scheme which consider the unique traits of DIRCRNs. They claimed that their SH schemes can work on top of any existing deterministic CH schemes which were designed for the omnidirectional antennas paradigm. However, the combined CH scheme must be designed appropriately by taking into account the unique traits of DIRCRNs such as the channels’ qualities in the different sectors.
In this paper, we propose efficient schemes for achieving rendezvous in DIRCRNs. The main contributions of this paper are summarized as follows:(i)We propose two asymmetric and symmetricrole coprimalitybased SH schemes, called PESSH and IPESSH, for pairwise sector rendezvous in DIRCRNs.(ii)We combined our SH schemes with an efficient gridquorumbased CH scheme, where the CH sequences are generated based on the bestquality channels in the sectors. The combined SCH schemes will provide guaranteed sector and channel rendezvous between the pair of SUs.(iii)We derive the upper bound of the rendezvous delay metrics for all of our schemes. Also, we conduct extensive simulations to study their performance under various network settings and compare them with the related existing works in [19, 28, 30].
The rest of this paper is organized as follows. The system model and problem formulation are presented in Section 2. The design and theoretical analysis of the proposed SH schemes are presented in Sections 3 and 4. In Section 5, we present the combined sector and channel hopping schemes. Using simulations, in Section 6, we evaluate the performance of our proposed schemes and compare their performance with other comparable schemes. Finally, we conclude the paper in Section 7.
2. Models and Problem Definition
In this section, we present the system model and rendezvous problem definition in DIRCRNs.
2.1. System Model
We consider a CRN consisting of SUs that coexist with several PUs in an area. There are totally primary channels which can be accessed opportunistically by the SUs in order to communicate with each other. Each SU is equipped with a directional antenna with beamwidth . Accordingly, the communication range of the SU is divided into nonoverlapping sectors that are indexed from 1 to . However, the number of sectors and the orientation of the sector indexing (i.e., either clockwise or anticlockwise) by each SU may be different from the others (see Figure 1). This is a very practical issue because each SU will configure its sectors based only on its own view to the surrounding environment. Each SU can transmit over any of its transmission sectors as long as it does not make any interference to any active PU transmission. Accordingly, we assume that the transmission sectors are also used as sensing sectors by which every SU can sense the appearance of the active PUs within them. Thus, the SU can obtain the channel availability information per each sector. We consider a timeslotted communication, where time is divided into discrete slots that have fixed and equal durations. During each time slot, we assume that a SU can only transmit in one sector over a single channel.
(a) Pair of and SUs
(b) Pair of and SUs
2.2. Definitions of Sector and Channel Rendezvous
In this paper, we mainly focus on the pairwise sector and channel rendezvous between any pair of SUs in a DIRCRN. Firstly, we define the sector rendezvous problem as follows.
2.2.1. The Sector Rendezvous Problem
For any two neighbour SUs (say and ) that are equipped with directional antennas and want to communicate with each other, assume that is located in the sector of and is situated in the sector of . The pair of the sectors is called the sector rendezvous pair. SUs are said to achieve a successful sector rendezvous if and only if they steer their antennas towards each other, where their transmission sectors can cover each other. Formally, let and denote the sector hopping sequences for and , with periods and , respectively. Also let denote the clock drift between and in the asynchronous scenario. The sector rendezvous problem can be formulated as follows: If and , then the sector rendezvous is achieved over the sector rendezvous pair .
2.2.2. The Sector and Channel Rendezvous Problem
When a SU steers its directional antenna to each sector of its transmission sectors, it performs CH over the available channels within the sector according to a CH sequence. This is in order to attempt channel rendezvous with another SU over a commonly available channel between them. The combined CH scheme should be designed in such a way that guarantees a successful channel rendezvous within bounded time whenever the communicating SUs already achieved a successful sector rendezvous. During the current sector, if the SU does not achieve channel rendezvous with its intended communicating SU within the bounded time slots, it then steers its antenna to the next sector of its SH sequence and executes the CH scheme based on the available channels within this sector.
One approach to view the problem is to look at it in a hierarchical way, in which time is divided into frames, where the SU tunes its antenna to one sector during each frame. In other words, the SH is implemented on the frame scale. Then, each frame is divided into number of time slots, where the combined CH sequence is executed (i.e., the CH is implemented on the time slot scale). The main goal of this paper is to design deterministic combined SCH scheme so as to tackle the sector and channel rendezvous problem.
2.2.3. Metrics
The proposed SH schemes as well as the combined SCH schemes will be evaluated according to the following metrics:(i)Sector rendezvous latency (SRL): it is defined as the required latency (in number of sector frames) for a pair of SUs to achieve sector rendezvous. We consider the maximum and average latency (MSRL/ASRL). MSRL is defined as the upper bound of the SRL between the SUs for all the possible clock drifts between them(ii)Timetorendezvous (TTR): it is defined as the required time (in number of time slots) for a pair of SUs to achieve sector and channel rendezvous simultaneously. Maximum and average timetorendezvous (MTTR/ATTR) are considered. MTTR means the required time for a guaranteed rendezvous even in the worst case.
3. Prime and EvenBased Sequences Sector Hopping (PESSH) Scheme
In this section, we propose an asymmetricrole SH scheme, called prime and evenbased sequences sector hopping (PESSH) scheme, and then we analyze its theoretical performance.
3.1. Scheme Design
For ensuring a successful sector rendezvous between a pair of communicating SUs, the key for the deterministic design of our PESSH scheme is to construct two SH sequences based on two coprime numbers. However, the coprimality is maintained between a pair of prime and even numbers that are relatively prime (i.e., coprime). In our PESSH scheme, we assume that is a global variable known by all SUs, which indicates the maximum number of sectors by which any SU is allowed to divide its omnidirectional transmission range. The value of is obtained based on the number of channels in the network; for example, for number of channels, the value of could be . Accordingly, we consider two avoided sets:(1)The Avoided Prime Set (APS) which contains the primes (2)The Avoided Even Set (AES) which contains any even number in the range , such that this even number is dividable by any prime number in the range . is the smallest prime not less than . For example, if , then and hence AES
The APS and AES sets are avoided by the sender and receiver, respectively, when they construct their SH sequences. Without loss of generality, assume that sender and receiver have and antenna sectors, respectively, where The main idea for the rendezvous of our PESSH scheme is to let one SU (sender) adjust the number of its sectors to be the smallest prime number greater than . On the other hand, the other SU (receiver) adjusts the number of its sectors to the smallest even number which is not smaller than and not in the AES. Accordingly, the adjusted prime number of sectors for the sender and the adjusted even number of sectors for the receiver are guaranteed to be relatively prime.
Algorithms 1 and 2 are used by the sender and receiver, respectively, to construct each round of their SH sequences. The generating steps of the PESSH sequences are as follows.


3.1.1. Sender Sequence in PESSH
When a SU has data to transmit, it serves as a sender and hence generates a primebased (PSSH) sequence as follows. First, given the sector set , randomly select a starting sector index and rotate circularly starting from as . Second, adjust to be the smallest prime number which is not smaller than and is not in the APS set (i.e., should be when ). Third, construct each round of the PS sequence which has a length equal to as follows: the sector index of the PS round is if ; otherwise the sector is selected randomly form when . The sender will keep steering its antenna according to the PS sequence until it achieves sector rendezvous with its intended receiver.
Consider the cases for the sender SUs and in Figure 1; SU in Figure 1(a) has a prime number of sectors , where is not rotated. Then the first round of the SH sequence with a length is . This round is repeated many times, where continues steering its directional antenna into its sectors as shown in Figure 2(a) to achieve sector rendezvous with its receiver . On the other hand, the sender in Figure 1(b) has number of sectors, so it selects as the smallest prime which is larger than . is rotated to start from and hence . Then the first round of the SH sequence which has adjusted length equal to is , where indicates a randomly selected sector from . The SH sequence is constructed as many rounds as shown in Figure 2(b), where keeps hopping on its sectors to achieve sector rendezvous with the receiver .
(a) Sector rendezvous between the sender and the receiver on the rendezvous sector pair for the cases when starts its SH with slots later than , where
(b) Sector rendezvous between the sender and the receiver on the rendezvous sector pair for the cases when starts SH with slots earlier, where
3.1.2. Receiver Sequence in PESSH
If a SU has nothing to transmit, it serves as a receiver and generates an evenbased (ESSH) sequence as follows. First, randomly select a starting sector index and circularly rotate the sector set starting from as . Second, adjust to be the smallest even number which is not smaller than and is not in the Avoided Even Set (AES). Finally, construct each round of the ES sequence which has a length equal to as follows: the sector index of the ESSH round is if ; otherwise the sector is selected randomly form .
Consider the receiver in Figure 1(a) which has an even number of sectors which is not in AES and does not rotate its set. The round of the SH sequence with a length is . This round is repeated many times by in order to be paired by a sender as shown in Figure 2(a). On the other hand, the receiver in Figure 1(b) has sectors; hence it finds as the smallest even number larger than and not in AES. rotates its to start from sector and hence . Accordingly, a round of the SH sequence with a length adjusted to is , where is a randomly selected sector from . The SH sequence is constructed in rounds as shown in Figure 2(b).
3.2. Scheme Analysis
In this subsection, we study the theoretical performance of the PESSH, where the MSRL between any two arbitrary SUs performing the PESSH is derived.
Theorem 1. The MSRL under the PESSH scheme is , where and denote the adjusted prime and even numbers of the sectors for the sender and receiver SUs, respectively.
Proof. Suppose that the sector rendezvous pair is , where here indicates the Cartesian product of the two SUs sectors sets. Without loss of generality, we assume that . Also, we assume that the receiver starts its SH sequence with sector slots earlier than the start of the sender . Accordingly, in each PSSH period of , the ESSH sequence of is . In the first round of the SH, when the sender is on its sector, the receiver is on the mod sector, where . During the subsequent rounds, while the sender SU is on its sector, the receiver must be sequentially on , ). As and are relatively prime, it can be easily proven with the help of the Chinese Reminder Theorem [31] that these sectors where the receiver resides in the rounds are all different. Hence, there must exist a sector among them which is equal to . According to that and since every round of the PSSH contains sector hops, the SRL required to guarantee a sector rendezvous is upper bound by .
4. Interleaved Prime and EvenBased Sequences Sector Hopping (IPESSH) Scheme
In the previous section, the PESSH scheme was designed using the asymmetric approach which requires that each SU have a preassigned role as either a sender or a receiver. In this section, we introduce our interleaved prime and evenbased sequences (IPESSH) scheme, which is symmetric (i.e., no preassigned role assumption).
4.1. Scheme Design
In our symmetric IPESSH scheme, every SU generates its SH sequence with the help of a binary bit sequence such as its globally unique ID. The SH sequence is constructed as an interleaved sequence of several PS and ES SH sequences through a twostep approach. Firstly, each SU transforms its unique ID according to a certain design into another ID sequence so that it is cyclically unique; that is, the resulting ID sequences of any two SUs are cyclic rotationally distinct to each other. Secondly, the SU replaces any bit of 1 (or 0) in its new ID sequence with a PSbased (or ESbased) SH sequence. Accordingly, for any pair of SUs, the cyclic uniqueness property will guarantee the existence of some time instances, where the two SUs are playing different roles. Hence, the sector rendezvous is achieved when one SU is hopping according to its PS sequence while the other SU is hopping based on its ES sequence and vice versa.
Several methods have been proposed in the literature to transform the unique ID into another cyclically unique binary sequence. In [32], a method was proposed to expand the bit ID into another bit extended ID that is cyclically unique. The unique bit ID in that method is expanded by appending bit consecutive 0’s into the beginning of the ID and other bit consecutive 1’s at the end of the ID. Another method for generating cyclically unique IDs that are used for the construction of symmetric role SH sequences has been proposed in [30]. The method appends the original bit ID with other bits of consecutive 0’s and 1’s, which results in bits expanded ID sequence. However, the extended ID sequences generated by these methods are relatively long, especially for the former method. This results in long rendezvous delay when they are used for constructing symmetric role sequences. Therefore, we propose an alternative method for constructing cyclic rotationally distinct sequences, which provides shorter extended ID lengths than the methods in [30, 32]. So, it provides shorter sector rendezvous delay when it is used for constructing symmetric role sequences in our IPESSH scheme.
4.1.1. Constructing the Cyclic Unique ID Sequences
In our method, each bit ID is extended by appending only bits of 1’s and 0’s in the beginning and in the end of the ID.
Lemma 2. Given any two bit ID sequences and , let and be two bit expanded ID sequences generated from and as follows: and , where is a bit sequence composed of only 1’s and is a bit sequence composed of only 0’s. Then and are cyclic rotationally distinct from each other, which means that
Proof. To prove the above lemma, we consider all the possible cases that may happen when sequence is rotated with . We show in each case that bits in sequence and other bits in sequence have different values, even though these bits are in the same positions. Considering the five cases in Figure 3, it is sufficient to prove that and are cyclic rotationally distinct to each other.
Case 1 (). As indicated by the red arrow in Figure 3, it holds that and .
Case 2 (). As indicated by the red arrow in Figure 3, it holds that and .
Case 3 (). As indicated by the red arrows in Figure 3, the bits of sequence starting from are , while the bits of in the same positions are . On the other hand, as indicated by the purple arrows in Figure 3, the first bits of sequence are 1’s and the last are 0’s, whereas the first bits of are and the last bits are . And because , it holds that there must be at least a single bit that is different in and .
Case 4 (). As indicated by the red arrow in Figure 3, it holds that and .
Case 5 (). As indicated by the red arrow in Figure 3, it holds that and .
According to these possible cases, we conclude that In Figure 4, we give an example for two ID sequences of original lengths which are expanded to bits. The figure illustrates the cyclic uniqueness property between the IDs for all the rotation cases (i.e., ).
4.1.2. Generating the Interleaved Sector Hopping Sequence
We now explain how to generate the symmetric IPESSH sequence for any SU, say . Let the number of sectors for be and hence the sector set . Suppose that has bits globally unique ID sequence, , which is extended by our above method to bits sequence .(i)Select randomly a starting index and generate a rotated set by rotating circularly starting from as . Also, generate another sector indices set as a reverse of which is .(ii)Find as the smallest prime number that is not smaller than . Also find AES) as the smallest even number that is not smaller than .(iii)Define an empty matrix that has columns and rows.(iv)Fill the matrix by mapping each bit in to a certain column as described in Algorithm 3.(v)Generate the SH sequence by concatenating the matrix rows (row by row). keeps hopping according to this generated SH sequence and repeats it in order to rendezvous with its pair partner SU.

Figure 5 illustrates the IPESSH sequence construction matrices for and in our previous example in Section 3.1 (Figure 1(b)). In Figure 5, the ID sequence of that has sectors is which is expanded to . On the other side, the ID sequence of that has sectors is which is expanded also to . The constructed matrix for has rows and columns. Similarly, constructs its matrix with rows and columns. Each SU will assign its matrix columns with either PS sequence or ES sequence based on the corresponding bits of its extended ID sequence. For example, since the first bit of extended ID is one, firstly generates its first sector sets as and then invokes Algorithm 1 with and to construct a round of a PS sequence . This round is repeated for times and assigned to the first column of matrix. Similarly, since the second bit of extended ID is one, the second column of matrix is assigned with a repeated sequence of the PS sequence , which is constructed after rotating the sector set by one. On the other hand, assigns the third and fourth columns of its matrix with ES sequences because the corresponding and bits of its extended ID sequence are zeros. Specifically, invokes Algorithm 2 with and to construct a round of an ES sequence . This round of the ES sequence is then repeated for times and assigned to the third column of matrix. Similarly, the fourth column of matrix is filled with the ES sequence after repeating it for times. Note that will follow the same procedures as to construct its matrix with taking into account the different expanded ID sequence and , as well as the initial sector sets and .
Figure 6 shows partial sector slots of the IPESSH sequences for the two SUs ( and ). As depicted from the figure, the IPESSH sequence of each SU is generated by concatenating the rows of the SU constructed matrix in Figure 5 row by row. The sequences are generated as interleaved slots of PS (light blue) and ES (gray) slots according to the respective columns types in the matrices. The figure also illustrates the rendezvous sector occurrence between the SUs (indicated by the green circle) on the sector rendezvous pair .
4.2. Scheme Analysis
In this subsection, the theoretical performance of IPESSH scheme is studied where the MSRL is derived.
Theorem 3. Under the IPESSH scheme, the MSRL between a pair of SUs and is .
Proof. To prove that IPESSH has , it is sufficient to prove that any arbitrary pair of IPESSH sequences, for example, and , can achieve a successful sector rendezvous within () sector slots.
Let and be the corresponding IPESSH sequences of IPES matrices and (i.e., and are generated by concatenating the rows of matrices and , resp.). For example, Figures 5(a) and 5(b) show the IPES matrices and , respectively. Without loss of generality, assume that starts its SH sequence sector slots earlier than the start of . In each SH period of , the SH sequence of is rotate(, ). Clearly, the rendezvous between this pair of SUs is the same as the rendezvous of and . Suppose that the sector rendezvous pair is . The rendezvous of the matrices is a pair of entries in the matrices whose value in is and its values in is . Without loss of generality, suppose that the sector slot shift , where , is the shift of row and is the shift of column. Below, we consider the two different cases for rendezvous of and .
Case 1 (). In this case, it is implied that there is no column shift. Recall that the expanded ID sequences of and are distinct. Since the columns types of the matrices are assigned based on these distinct expanded IDs, it is guaranteed that there must exist at least a single common column in both matrices which is a PScolumn in one matrix and an EScolumn in the other matrix, and vice versa. In other words, the column of contains a PSSH sequence (reps., ESSH), while the same position column in contains an ESSH (resp., PSSH). By Theorem 1, we know that when and repeat the PSSH (resp., ESSH) and the ESSH (resp., PSSH) sequences, respectively, they rendezvous within (resp., ). Meanwhile, since the elements of a specific column of each matrix appear in the corresponding IPESSH sequence every slots, for any value of the row shift , and are guaranteed to rendezvous on their columns no later than . Specifically, the rendezvous happens when an entry of the column in is , while the same position entry of the column in is . For example, Figure 5 shows the rendezvous of and for the synchronous case where the slots shift . The sector rendezvous occurs between the second columns of and which is indicated by the green entry in . However, Figure 7(a) shows the IPES matrix of when it is shifted with row shift . Hence, this means that . The rendezvous between in Figure 5 and in Figure 7(a) is achieved at their third columns.
Case 2 (). In this case, it holds that there is a column shift . Let and be the expanded ID sequences of and , respectively. By Lemma 2, we know that and , where are cyclic rotationally distinct to each other. Thus, for a column shift of matrix , there must exist a column in matrix which has different type than the same position column in the shifted matrix . Accordingly, the rendezvous is guaranteed to occur at the columns in the matrices, despite any value of the row shift . For example, Figure 7(b) shows the shifted IPES matrix of when the column shift and the row shift , which results in slots. The first rendezvous between in Figure 5 and in Figure 7(b) is achieved at the forth columns.
Summarizing the discussion above, we show that, in IPESSH, any two SUs are guaranteed to rendezvous on their sector rendezvous pair within SH period .
5. Combined Sector and Channel Hopping Schemes
While we explained in the previous sections the procedures of our SH schemes by which the SUs keep steering their antennas in order to achieve sector rendezvous with their communicating partners, the scheme for channel hopping (CH) exposed by the SUs in their sectors for achieving a channel rendezvous between them was not explained. In this section, the design of the exposed CH scheme and the procedure for the combined sector and channel hopping (SCH) schemes are presented. The key for the design of the exposed CH scheme is to construct two different types of CH sequences which will be combined within the two different types of PESSH sequences. These CH sequences must guarantee a successful channel rendezvous between the pair of SUs as long as they achieve a successful sector rendezvous (i.e., directing their antenna beams towards each other). Furthermore, the CH sequences should be designed in such a way that the pair of communicating SUs stay for a similar period (i.e., number of time slots) in each sector of their SH sequences. By doing this, the pair of communicating SUs are guaranteed to achieve successful sector and channel rendezvous within a bounded and short time, despite any asynchronous time offsets between their local clocks.
5.1. Asymmetric Grid QuorumBased Channel Hopping
In this subsection, we explain the design of our Asymmetric Grid Quorumbased Channel Hopping (AGQCH) scheme that is executed by the SUs in each sector of their SH sequences. The AGQCH scheme utilizes the grid quorum system (GQS) in asymmetricrole design, where two types of CH sequences are constructed. These types of CH sequences will be combined within the two different types of our SH sequences. Some preliminary definitions related to grid quorum systems are presented first to facilitate the understanding of the CH scheme.
Definition 4. For a set , a quorum system under is a group of nonempty subsets of , each called a quorum, such that and . Here, is the set of integers less than or equal to n.
Definition 5. A GQS arranges the elements of as a square grid array, where must be a square of a positive integer. Hence, a quorum is formed as a union of the elements of one column and one row of the grid. There are grid quorums in a GQS that is constructed under and each quorum has a size of .
For example, a GQS under is .
Definition 6. For a given integer and a grid quorum in a GQS under , we define to denote a cyclic rotation of quorum by .
Definition 7. A GQS under satisfies the rotation closure property because, and , .
GQSs have been utilized to construct CH sequences that achieve asynchronous communications because they satisfy the intersection and the rotation closure properties [33, 34]. In the conventional GQS, each grid quorum is formed as a union of the elements of one column and one row of the grid array. However, different from the conventional GQS, our asymmetric GQS (AGQS) uses semigrid quorums. One semiquorum, called the Grid Columnbased Quorum (GCQ), is formed from the elements of a single column of the grid. Meanwhile, the other semiquorum, called Grid Rowbased Quorum (GRQ), is formed from the elements of a single row in the grid. The nonempty intersection and rotation closure properties in the AGQS are inherited from the conventional GQS. Specifically, for any arbitrary pair of GCQ and GRQ, there must exist one common element between them despite any rotation of the grid.
The GCQs and GRQs in the AGQS are used to construct two asymmetric CH sequences that can be combined within the different types of sectors in our SH schemes. The constructed CH sequences are guaranteed to achieve a successful channel rendezvous between a pair of communicating SUs as long as there exists at least a single common channel between the available channel sets of the SUs. In our AGQCH scheme, to construct CH sequences for assigning channels, grid array is built. In the GCQbased CH sequence, the rows of the grid are used for assigning the channels into the CH sequence, where the elements of each row are used to map a single channel. On the other side, the columns of the grid are used for the assignment of the channels into the GRQbased CH sequence. For example, Let be the number of channels for a sender and a receiver . Suppose that has the channel set , while the channel set of is . Accordingly, a grid array of size is formed by both SUs as shown in Figure 8. Without loss of generality, assume that the sender uses the GCQs to map its channels into the time slots of its CH sequence as depicted in Figure 8(a), while the receiver maps its channels based on the GRQs as depicted in Figure 8(b). The CH sequences for both SUs are shown in Figure 8(c), where they map each channel of their channel sets into the time slots of their CH sequence according to the selected (GCQ or GRQ) semiquorum. As shown in Figure 8(c), the channel rendezvous is achieved on the common channel between the SUs despite the time offsets between their local clocks.
(a) GCQs for mapping
(b) GRQs for mapping
(c) Channel rendezvous between SUs on the common channel when start its CH with 0, 1, and 5 slots later than
5.2. Channel Ranking
As explained in the beginning of the section, the sector frame length which represents the number of time slots by which every SU spends on each sector for executing CH should be the same in all SUs. This is in order to guarantee channel rendezvous between any pair of neighbouring SUs side by side with the sector rendezvous despite any time offsets between their local clocks. To achieve this, we let the SUs construct their combined CH sequences based on the bestquality channels. The channel’s quality is generally represented by the quantity of the measured PU noise on the channel within the corresponding sector. However, for cases where some of the channels have the same measured PU noise, channels indices are used to differentiate between their ranking levels. Specifically, the bigger the channel index, the higher the quality. We highlight here that the similarity of the measured PU noises for some channels is a practical issue in DIRCRNs which happen when the channels are idle from any PU activity within the sector area.
The channel ranking is done by each SU before executing the rendezvous process based only on its own view of the available channels in its sectors. Each ranks the network channels in each sector of its sectors set based on their qualities within the sector . Thus, a ranking table of size is maintained by each which contains the ranked channels in all the sectors. This table is called Sectors Ranked Channels Table (SRCT), where each column of it indicates a sector of the sectors. These columns contain the network channels ranked decreasingly from the best to the worst according to their qualities within the corresponding sectors. We note here that since each SU in the DIRCRN performs the same ranking mechanism, this will increase the probability of having a similar ranked channel list among any pair of neighbouring SUs on their rendezvous sectors.
5.3. Combined SCH Sequence Generation
We now explain the generation procedures for the combined PESSCH and IPESSCH sequences.
5.3.1. Combined PESSCH Scheme
Since our PESSH scheme is designed based on two different types of coprimalitybased sequences (i.e., either PS or ES), we combine each sector of the SH sequences according to its type with a corresponding type from the two AGQCH sequences. Specifically, each sector of the sender PSSH sequence is combined with a GCQCH sequence, while each sector of the receiver ESSH sequence is combined with a GRQCH sequence. Algorithms 4 and 5 describe the SCH generation procedures for the sender and receiver SUs, respectively. As illustrated by the algorithms, each SU will keep hopping on its best quality channels in each sector of its SH sequence according to the corresponding CH sequence. This will guarantee that when a successful sector rendezvous occurs between a sector frame of the sender and a sector frame of its receiver , a successful channel rendezvous is achieved between and the . The channel rendezvous is guaranteed as long as there is a common channel between their best channels, where they can set up a link through the exchange of RTS/CTS messages.


For example, consider the pair of sender and receiver in our previous example in Figure 1(a). The SCH sequences constructions for the pair of SUs and the rendezvous among them when they construct their combined CH sequences based on bestquality channels are illustrated in Figure 9. As a sender, in Figure 9(a) combines each sector of its PSSH sequence with a grid column (GCQCH) sequence based on the two bestranked channels of the respected column for the sector in . For instance, the sector is combined with a GCQCH sequence generated based on the two best channels of the 1st column of ; the GCQCH is . On the other hand, each ES sector of the ESSH sequence for the receiver in Figure 9(b) is combined with a grid row (GRQCH) sequence based on the best two channels of the column which corresponds to the sector number. For instance, combine the GRQCH sequence within its sector which is constructed based on the best channels . The sector and channel rendezvous between the SUs when they start their PESSCH sequences synchronously is shown in Figure 9(c). The rendezvous is achieved during the 25th and 28th time slots on the sector rendezvous pair ( of , of ), as well as on the rendezvous channels . However, when starts its SCH time slots later than , the channel and sector rendezvous is achieved after time slots only from the SCH start time of .
(a) The combined PESSCH sequence construction for the sender
(b) The combined PESSCH sequence construction for the receiver
(c) Synchronous sector and channel rendezvous between and
(d) Asynchronous sector and channel rendezvous between the SUs when starts its hopping earlier than by 4 slots
5.3.2. Combined IPESSCH Scheme
For this symmetric role scheme, any SU can generate its SCH sequence according to Algorithm 6. During each sector frame of its interleaved SH sequence that is constructed using the method in Section 4.1.2, the SU executes one type of the AGQCH sequences according to the type of the frame. Specifically, if the sector frame is a primebased (Pframe), the SU generates a GCQCH based on the bestquality channels in the sector. On the other hand, when the sector is an evenbased (Eframe), the SU generates a GRQCH. This will ensure that whenever the sector rendezvous occurs between a Pframe of and an Eframe of , channel rendezvous is achieved between the CH sequence and the CH sequence, and vice versa.

5.4. Performance Analysis of the Combined SCH Schemes
In this subsection, we analyze the theoretical performance for the combined SCH schemes. Specifically, we prove the guaranteed rendezvous between any two SUs performing the combined SCH scheme by deriving the maximum timetorendezvous (MTTR).
Theorem 8. For any scheme of our SH schemes which is combined with the AGQCH scheme for bestquality channels, the MTTR of the combined SCH is . MSRL here denotes the maximum sector rendezvous latency for the SH scheme.
Proof. To prove that the combined PESSCH has MTTR = , it suffices to prove that the pair of combined PESSCH sequences, for example, and , can achieve successful sector and channel rendezvous within time slots.
Without loss of generality, assume that starts its SCH sequence time slots earlier than the start of . In each SCH period of , the SCH sequence of is . As we stated before, the SCH sequence can be viewed as a series of sector frames which are composed of channel time slots. Let the clock drift , where denotes the shift amount of the sector frames and denotes the shift of the channel time slots. Next, we prove the theorem in the following two cases.
Case 1 (). This case implies that the sector frames are perfectly aligned. In this case, since the SUs stay for similar number of time slots in each sector which is () for executing their corresponding AGQCH sequences, for any value of , there must exist an entire overlap between a PSframe of and an ESframe of where the sector rendezvous happened. By Theorem 1, we know that the sector rendezvous is guaranteed to occur between a PS sector frame and an ES sector frame within sector frames. As a consequence, the channel rendezvous is guaranteed to happen between the CH sequence of and the CH sequence of . Thus, the MTTR is multiplied by the sector frame length . Figures 9(c) and 9(d) show the sector and channel rendezvous between and when for two different values of and for . In Figure 9(c), when and hence the clock drift , can achieve the sector rendezvous with on the second round of its SH frames. The SUs rendezvous on their commonly available channels (i.e., and ). However, when and , can rendezvous with SU earlier where it rendezvous with during the first round of its SH frames.
Case 2 (). When there is a time slot shift , it implies that the sector frames are not aligned. For any value of , it is easily verified that each PS sector frame of is overlapped with partial channel time slots form two consecutive ES frames of . Specifically, the PS frame of overlaps with time slots from the former ES frame and time slots from the later ES frame of . Recall the proof of Theorem 1, when repeats each round of its sector frames for times, each occurrence of the sector frame must overlap with different sector frame from the sector frames of . The sector rendezvous is achieved when overlaps with . However, considering the time slot shift of the sequence, we can prove in the same way as case that, within the same rendezvous delay bound ( time slots), there must exist at least two occurrences of which overlaps with different partial time slots of . In one of these occurrences, the overlap happens between the first time slots of and the last time slots of . On the other occurrence, the overlap happens between the last time slots of and the first time slots of . Accordingly, channel rendezvous can be achieved in any of these two partial sector rendezvous chances where partial slots of the CH sequence in overlap with other partial slots from the CH sequence in . For example, Figure 10 illustrates the asynchronous rendezvous between and when for three different values of .
Summarizing the discussion above, we show that, under the combined PESSCH, any two SUs are guaranteed to achieve channel rendezvous within () SCH period. We highlight here that the theoretical MTTR for the combined IPESSCH is . It can be proven in a similar way to the proof of the PESSCH MTTR by considering the two alignment cases of the sector frames.
6. Results and Discussions
In this section, we evaluate the performance of our developed directional antenna rendezvous schemes and compare them with three related works in [19, 28, 30]. The former two selected schemes for comparisons are the only schemes in the literature which tackle the rendezvous issue in DIRCRNs. However, since both of them are asymmetricrole, we select the third scheme [30] which is proposed for rendezvous in the 60 GHz networks for the comparison with our symmetricrole IPES scheme. This scheme follows a similar IDbased approach to our IPESSH scheme. However, the extended ID in this scheme is relatively longer and the adjusted parameters of the number of sectors which are used for constructing the interleaved SH sequences are different.
6.1. Performance Comparison of the SH Schemes
We first compare the performance of our SH schemes with the two blind SH schemes in [28, 30]. The SH scheme in [19] is not compared here because it is not a blind scheme. Extensive simulation experiments are conducted to compute the sector rendezvous latency (SRL) between two neighboring SUs (a sender and a receiver ). In the simulations, we consider different antenna configurations for the SUs where we simulate several combinations for the number of sectors ( and ). The relative locations of and represented by (, ) as well as the clock drift are randomly generated. For the IDbased schemes, two different 7bit IDs are randomly assigned to the SUs. For each ( and ) combination, the simulation results are obtained by conducting independent runs where we accordingly compute the average and maximum SRL (ASRL and MSRL).
Figure 11 depicts the SRL results for the asymmetricrole SH schemes (our PESSH and Li and Xie’s SH [28]). The results illustrate the significant reduction on the SRL provided by our PESSH in terms of ASRL and MSRL under all the different antenna configurations of the pair of SUs. For some antenna configurations, PESSH can reduce the ASRL and MSRL significantly up to 80% and 84%, respectively. The shorter ASRL and MSRL provided by the PESSH scheme are due to its efficient design which relies on the coprimality between the adjusted prime and even numbers of and , respectively. And since is prime for all the sender antenna configurations while the values of are even numbers that are valid for use (i.e., not in the AES set), this allows the sender in our PESSH to achieve sector rendezvous with its receiver within SH period.
(a)
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On the other side, the design of Li and Xie’s SH scheme requires and to be adjusted to primes. Furthermore, the sender should perform an initiating roundrobin SH sequence with a length of before it converts to the regular primebased sequence. Accordingly, this will prolong the SH period needed for achieving the sector rendezvous to () which actually represents the theoretical upper bound for the MSRL of Li and Xie’s SH scheme when . We can notice that this upper bound is tight for some ( and ) combinations, while it is not for others. Specifically, when (% ) where is the adjusted prime for , the sender cannot rendezvous with its receiver during its initiating roundrobin SH sequence. However, when it starts following the regular primebased SH sequence, the sector rendezvous happens within the subsequent slots, which results in () MSRL. For example, when and in Figure 11(b), the sector rendezvous occurs in sector slots. The same thing can be noticed for the cases when in Figures 11(a), 11(b), and 11(c), where the MSRL is (). This explains the irregular increased results in the figures for such ( and ) combinations.
In Figure 12, the SRL results for the symmetricrole SH schemes (our IPESSH and the Chen et al.’s SH scheme [30]) are shown. In the figure, it is shown that our IPESSH scheme outperforms the other compared scheme especially in terms of MSRL. This improvement is due to the shorter SH period provided by IPES for achieving the sector rendezvous. In both schemes, the SH period for sector rendezvous depends mainly on the extended ID length as well as the adjusted numbers for ( and ) that are used for constructing the interleaved sequences. However, the adjusted parameters used by our IPES scheme are smaller than those used by the other scheme. Firstly, in the IPESSH scheme, the extended ID length is which is relatively smaller than the extended ID used by the other scheme. Furthermore, the interleaved SH sequences in Chen et al.’s SH scheme are constructed by adjusting the number of sectors to a prime number as well as to a number that is a powermultiple of two (i.e., , where is the smallest integer satisfying ). However, these adjusted numbers must be coprime with the extended ID length in order to achieve a successful sector rendezvous. Meanwhile, in IPESSH, the interleaved PS and ES sequences are constructed based on the closest prime number and the closest even number which is not in the AES set. Hence, under most of the simulated antenna configurations, the adjusted and in IPES are smaller than the adjusted and parameters in the other scheme. For example, in our IPESSH scheme, for all the values, while the adjusted parameters for the values in Chen et al.’s SH scheme are . Moreover, when or is ≤5, the adjusted prime number used in IPES is 5, while it is 7 on the other scheme because 7 is the closest prime that is coprime with the 15bit extended ID. According to that, SUs in the IPES scheme can achieve the sector rendezvous faster than Chen et al.’s SH scheme.
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6.2. Performance Comparison of the Combined SCH Schemes
In this subsection, we compare the channel rendezvous performance of our SCH schemes with the other works under a practical DIRCRN. The simulation configuration parameters are shown in Table 1.
We start our SCH simulation by comparing the performance of our combined PESSCH with the combined SCH schemes of Song and Xie [19] and Li and Xie [28]. We evaluate our PESSCH for bestquality channels which correspond to a AGQCH period of 16 time slots in each sector (i.e., the frame length = 16). The other selected schemes for comparison are simulated with their proposed CH schemes. In [19], the SH scheme is combined with the asymmetricrole CH scheme in [35], where the sender SU stays for slots on each sector frame of its SH sequence performing a sequential fast CH on the () available channels within the sector. Meanwhile, the receiver SU stays for slots on each sector of its sectors performing a slow CH on all of its () available channels. On the other side, the SH scheme of Li and Xie [28] is combined with the EJSCH scheme [10]. However, due to the long period of the EJSCH scheme required to guarantee channel rendezvous, Li and Xie simulated their SCH scheme by letting SUs stay for smaller period on each sector of their SH frames. During this period, SUs execute the corresponding subsequences of their generated EJSCH sequence. Therefore, we simulate Li and Xie’s SCH scheme with a similar sector frame length to our PESSCH scheme (i.e., 16 slots). All the schemes are simulated under different asynchronous settings where the clock drifts amount between SUs is selected randomly in each experiment.
Figure 13 depicts the TTR simulation results for the three schemes when they are evaluated under different antenna configurations for the pair of SUs. Even though the combined SCH scheme of Song and Xie is not a blind scheme, it produces very long TTR results. This is due to the large sector frame length where the sender and receiver stay for and time slots, respectively, on each sector for performing their corresponding CH sequence. As a consequence, the SCH period needed for achieving rendezvous is which is very long as compared to the period of our PESSCH scheme. On the other side, Li and Xie’s SCH scheme can provide shorter ATTR than Song and Xie’s scheme because it is simulated with smaller frame length. However, the MTTR of this scheme cannot be traced because we observe that TTR in some cases cannot be achieved within the whole simulation duration which is set to slots. The reason for the indeterministic behaviour of Li and Xie’s SCH scheme is because the stay period in each sector (i.e., 16 slots) is insufficient to guarantee rendezvous while using the EJSCH scheme. According to the EJSCH [10], the rendezvous cannot be guaranteed unless the EJSCH sequences of the communicating SUs are overlapped for at least time slots. is the smallest prime number greater than , while is the number of common available channels between the SUs available channels. Hence, to ensure rendezvous regardless of any misalignment of their local clocks, the SUs should stay for at least on each sector of their SH sequences to perform their EJSCH sequences. This tradeoff between the bounded MTTR of Song and Xie’s SCH scheme and the small ATTR of Li and Xie’s SCH scheme illustrates the efficiency of our proposed AGQCH scheme, which allows PESSCH to guarantee rendezvous with the shortest TTR results.
In the second setting of the SCH simulation, we combine the SH schemes of Li and Xie [28] and Chen et al. [30] with our proposed AGQCH scheme. This is in order to evaluate their channel rendezvous performance when they are combined with an efficient and suitable CH scheme. Figures 14 and 15 show the TTR results of these schemes and our schemes when they are simulated for different values of the bestquality channel . In the simulation, we consider different values for the sender number of sectors , while the receiver number of sectors is set as . The figures show that our schemes always outperform the other schemes under all the different antenna configurations of the SUs. This is mainly due to the shorter SRL of our SH schemes and, as explained in Section 5.4, TTR depends on the SRL of the SH scheme coupled with the length of the sector frame (). Moreover, the figures illustrate the effect of (i.e., the number of bestquality channels) on the rendezvous performance where we can notice that as increases, the TTR results for all the schemes increase. This is because the larger , the longer the frame length . Accordingly, when SUs spend longer period in each sector, they may waste more time while attempting rendezvous on other sectors rather than the rendezvous sector. This will result in a longer TTR until they can achieve a successful sector and channel rendezvous.
Finally, to illustrate the performance gain brought by applying directional antennas for channel rendezvous, we compare the performance of our combined PESSCH scheme with the AAsyncCH omnidirectional antenna rendezvous scheme [16]. The AAsyncCH is selected among the other omnidirectional CH schemes since it is designed for asymmetricrole environment (i.e., SUs have preassigned role) similar to our PESSCH. The schemes are compared in terms of their probabilities for achieving a successful channel rendezvous under different number of PUs in the network area. In our PESSCH, the simulated sender and receiver SUs have and