 Research Article
 Open Access
 Published:
Residue Number System Arithmetic Assisted Coded FrequencyHopped OFDMA
EURASIP Journal on Wireless Communications and Networking volume 2009, Article number: 263695 (2009)
Abstract
We propose an RNS arithmeticbased FH pattern design approach that is well suited and easy to implement for practical OFDMA systems. The proposed FH scheme guarantees orthogonality among intracell users while randomizing the intercell interferences and providing frequency diversity gains. We present detailed construction procedures and performance analysis for both independent and cluster hopping scenarios. Using simulation results, we demonstrate the gains due to frequency diversity and intercell interference diversity on the system bit error rate (BER) performance. Furthermore, the BER performance gain is consistent across all cells unlike other FH pattern design schemes such as the Latin squares (LSs)based FH pattern design where wide performance variations are observed across cells.
1. Introduction
Orthogonal frequency division multiplexing (OFDM) has been widely accepted as an enabling technology for next generation wireless communication systems. In OFDM, highrate data streams can be broken down into a number of parallel lowerrate streams, thereby avoiding the need for complex equalization. OFDM also forms the foundation for a multiple access scheme termed as orthogonal frequency division multiple access (OFDMA). In OFDMA, each user is assigned a fraction of available subcarriers based upon his/her demand for bandwidth. The advantages of OFDMA include (1) the flexibility in subcarriers' allocation; (2) the absence of multiuser interference due to subcarriers' orthogonality; (3) the simplicity of the receiver design [1].
In order to enhance system throughput and spectral efficiency, frequency hopping (FH) is generally used in OFDMA cellular systems. It is desirable for FH patterns to satisfy the following conditions [2]: (i) minimize intracell interference; (ii) average intercell interference; (iii) avoid ambiguity while identifying users; (iv) exploit frequency diversity by forcing hops to span a large bandwidth. The first aspect is relatively easy to achieve by using orthogonal hopping patterns within a cell. To average intercell interferences, hopping patterns are constructed in a way that two users in different cells interfere with each other only during a small fraction of all hops. The third condition requires base stations to have the capability of distinguishing different users efficiently according to their unique FH signatures. Finally, the last requirement not only ensures the security of the transmission, but also mitigates the effect of fading by exploiting frequency diversity.
Frequency hopping pattern design has received considerable attention in both commercial and military communication systems. There has been extensive work on designing FHOFDMA systems [3–10]. In [3], concepts of fast frequency hopping along with OFDM are illustrated. In [4], authors show that the expected number of collisions per symbol under both independent and cluster hopping does not depend on the hopping strategy. In their later work [5], it is shown that the number of collisions can be further reduced by using spacefrequency coding in multipleantenna systems. Orthogonal Latin squares (LSs) are presented as FH patterns in TCM/BICM coded OFDMA in [6]. In LSaided FHOFDMA systems, it is seen that there is a wide variability in the performance of users in different cells. Therefore, it is not an effective scheme if one considers fairness to be important. WelchCostas array is introduced in [7] and evaluated in [8] for coded FHOFDMA. Here, although users across cells experience significant performance improvements, users within a cell may not occupy all of the available bandwidth to exploit full frequency diversity. Other aspects focusing on preventing hostile jamming and pilotassisted channel estimation in FHOFDMA are explored in [9, 10], respectively.
In this paper, we propose a novel frequency hopping pattern design strategy based on RNS arithmetic for practical OFDMA cellular systems. We show that the resulting patterns are orthogonal within a cell and intersect only once across cells in a frequency hopping cycle. RNS arithmetic has found applications in many areas. However, its use in designing frequency hopping patterns is rarely considered [11, 12]. In [11], the design procedure can be visualized as a "topdown" approach where a given bandwidth is divided into multiple candidate subbands based on a predetermined moduli set. As a result, if the moduli set changes, the bandwidth of subcarriers varies. In this work, the division of bandwidth into candidate subcarriers is assumed to be given or determined in advance. Therefore, we can consider our proposed approach as a "bottomup" method driven by grouping and indexing the subcarriers according to the RNS arithmetic. For practical OFDMA cellular systems, the proposed "bottomup" approach is more feasible. For example, in downlink OFDMA cellular systems, a fixed number of subcarriers (e.g., 1024) with identical subcarrier bandwidth within each cell is usually assumed. Furthermore, for reducing intercell interference, [11] suggests the use of different moduli sets for adjacent cells. This approach results in adjacent cells employing different numbers of subcarriers with different bandwidths across cells. Once again, this is a stringent requirement that may not be feasible in practice. In this work, we invoke the use of the socalled twostage and multistage selection algorithms to construct RNSFH patterns such that (1) different users can use the spectral resources simultaneously within each cell and (2) the same number of subcarriers can be employed from cell to cell. Additionally, the proposed FH sequences force the intracell interferences to zero and average out the intercell interferences. The performance of the proposed FH pattern incorporating with both independent and cluster hopping schemes is characterized. Simulation results show that RNSFH OFDMA has significantly better BER performance relative to traditional OFDMA scheme without FH. Another aspect that makes RNSFH pattern design outperforms other existing FH techniques is that user hopping patterns span a larger bandwidth. Therefore, the channel fades associated with consecutive hops become independent. Moreover, with the use of FEC codes over multiple hops, the system can correct errors due to subcarriers that experience deep fades or subcarriers that are severely interfered by others.
The rest of the paper is organized as follows. In Section 2, system model along with signal transmission scheme, access strategies, and interference models is introduced. In Section 3, detailed RNSFH pattern design procedures along with comparisons with the existing technique are presented. Simulation results with performance analysis are given in Section 4. Finally, we conclude this paper in Section 5.
2. System Model
In this section, we first describe the signal transmission scheme for each individual user in an OFDMA system. Then, we introduce the access model and interference model under both independent and cluster hopping schemes.
2.1. Signal Transmission Scheme
The block diagram of FEC coded FHOFDMA system is shown in Figure 1. Here, data bits of every user are first channel coded and then mapped to complex constellation points. We assume that there are users in the system, utilizing a total of OFDM subcarriers. Each user is assigned a specific set of subcarriers out of the total available subcarriers according to his/her data rates. Let be the number of subcarriers allocated to user . Then, user transmits the information symbols ( represents the transpose operation) on the assigned subcarriers. Therefore, the baseband transmitted signal of user can be expressed as
where represents the timedomain signal, and denotes one OFDM symbol duration. Since this is an OFDMA system, it is important to remember that every user is assigned a different set of subcarriers for transmission, and this allocation is dynamic in the case of frequency hopping OFDMA. That is, in the IFFT module, the frequency assignment follows a predetermined FH pattern. Moreover, each user transmits zeros on subcarriers which are not assigned to him/her.
For convenience, we note as the subcarrier that is assigned to user . Hence, information symbols vector of user can be written as
The discrete form of the transmitted signal is then given as,
where is the IFFT matrix defined as
where .
Let denote the channel impulse response vector, then its Fourier transform is
where represents the Hermitian transpose. In general, each channel impulse response is a function of time and access delay which can be modeled as a tapped delay line, that is,
where is the number of multipaths and is the time delay of the th path. The tap coefficients are independent, zero mean, circularly symmetric complex Gaussian random processes at each instant , that is, with the total power normalized to unity, that is, . In this work, we use Jakes' model to describe the time/frequency variation of each channel coefficient. Therefore, the spaced frequency () spaced time () correlation function of the channel frequency response can be expressed as [13]
where is the Doppler frequency.
At the receiver end, after FFT, the received signal corresponding to user on subcarrier is
Then, the overall received signal which is a superposition of the signals transmitted from all users is
where is the Fourier transform of the noise vector.
2.2. Access Model
In this part, clustered and independent FHOFDMA are introduced, and closed form expressions of the expected number of collisions per symbol under both of these two hopping strategies are presented.
2.2.1. Clustered FHOFDMA
In cluster hopping, each user selects a set of continuous subcarriers, termed cluster, to transmit the information symbols. Specifically, the hopping takes place among clusters of subcarriers based on predetermined FH patterns. Therefore, collisions occur among clusters first, and then across all OFDM subcarriers within that cluster. The expected number of symbol losses per cluster collision corresponds to [4]
where is the number of subcarriers per cluster and represents the probability that at least one interfering user collides with the desired user. For cluster hopping, we have hopping clusters. Therefore, the collision probability between the desired user and the interfering user in one cluster is . Hence, the probability that at least one of the users collides with the desired user can be expressed as
For convenience, throughout the rest of this paper, we assume that each user employs the same number of subcarriers () per cluster.
2.2.2. Independent FHOFDMA
In independent hopping, subcarriers occupied by a user are selected independently from all available subcarriers. In other words, subcarriers in one cluster are not continuous anymore, and they are chosen in a pseudorandom fashion across the frequency spectrum. With independent hopping, the expected number of symbols lost per symbol collision is given by [4]
where is the probability that subcarriers out of subcarriers occupied by each user experience collisions due to interfering users.
Theorem 1.
For independent FHOFDMA scheme described above, corresponds to
Proof.
is the probability that subcarriers of the desired user collide with the subcarriers of interfering user given that each user occupies a total of subcarriers. It is evident that the number of possible combinations of subcarriers that experience collisions is . Define as the probability that symbols are collisionfree given that each user occupies subcarriers. Furthermore, define as the conditional probability that symbols collide given that symbols are collisionfree. Therefore, we can write as
Here, corresponds to
Equation (15) denotes the probability that the desired user's remaining subcarriers are collisionfree while none of the other users within the same cell occupies these subcarriers. is expressed as [4]
Equation (16) represents the conditional probability that each of the subcarriers of the desired user collides given that the other subcarriers are collisionfree. By substituting (15) and (16) into (14), we obtain the result in (13).
2.3. Interference Model
In this paper, we model intercell interferences as additive complex Gaussiandistributed distortions. This model is accurate when interferences from adjacent cells are perfectly randomized with respect to the cell of interest. Models specific to clustered and independent FHOFDMA are presented in the following.
2.3.1. Clustered FHOFDMA
In clustered FHOFDMA, if interference occurs on any symbol on one subcarrier in the cluster, all other symbols in the same cluster will also experience interferences from adjacent cells. Hence, the interference for the th user can be modeled as [6]
where is an vector, representing the received signal of user ; is the transmitted signal vector; is an matrix that contains the frequency domain representations of channel impulse response; is an vector whose components are complex Gaussian random variables with zero mean and variance . Here, the vector is the interference vector that captures the interference from all adjacent cells. The components of are i.i.d complex Gaussian random variables independent of and with mean zero, variances . The variances correspond to
where denotes the symbol signaltointerference ratio and characterizes the presence/absence of a collision between users in different cells. That is, if there is a collision, equals to one; if there is no collision, is set to zero. Furthermore, can be modeled as Bernoulli's random variable with probability of collision equals to (i.e., and ), which can be expressed as
where is the number of active users. If the system is fully loaded, then . If there is a collision, that is, , then all subcarriers in the cluster will be affected by the intercell interference.
2.3.2. Independent FHOFDMA
In independent hopping, since subcarriers are selected independently of all other subcarriers according to predetermined FH patterns, collisions occur independently. Hence, for the th subcarrier of the th user,
Here, the interference power of the i.i.d complex Gaussian random variable corresponds to
where with probabilities and , respectively. The collision probability is given by
For a fully loaded system with independent hopping, is identical to becomes to one.
3. RNSFH Pattern Design
RNS is defined by the choice of number of positive integers (), referred to as moduli [14]. If all the moduli are pairwise relative primes to each other, any integer which falls in the range of can be uniquely and unambiguously represented by the residue sequence (), where and for . Here, is used to describe the th user FH address. To recover , or to distinguish users at the base station, Chinese remainder theorem (CRT) is generally used which is well known for its capability of solving a set of linear congruences, simultaneously. According to CRT, it can be shown that the numerical value of can be computed as [15]
where and for .
Theorem 2.
The residue sequences obtained using the RNS arithmetic as described above are orthogonal.
Proof.
In order to prove that the residue sequences are orthogonal, we need to show that every in the range of has a unique residue set that is different from residue sets generated by other integers within the same range. We will prove this by contradiction as follows.
Assuming that and are different integers which are in the same range of with the same residue set. That is,
Therefore, we have
Thus, we can conclude from (25) that is actually the least common multiple (LCM) of . Furthermore, if are pairwise relative primes to each other, their LCM is and it must be that is a multiple of . However, this statement does not hold since and . Therefore, by contradiction, and should not have the same residue set. In general, the residue set () generated by is unique and can be used to represent the integer if .
Following the RNS arithmetic presented above, we propose to design FH patterns that satisfy all the requirements described in Section 1 while avoiding the limitations in [11]. Detailed procedures of constructing RNSFH patterns are given in the following subsections. The first part describes the twostage algorithm, while the second part introduces the multistage algorithm which can be considered as generalization of the twostage algorithm. At the end of this section, we compare our proposed RNSFH pattern design strategy with the method presented in [11].
3.1. TwoStage Algorithm
In this part, the detailed procedures of constructing RNSFH patterns via the socalled twostage algorithm is introduced. We present the algorithm for a cluster hopping OFDMA system. It is straightforward to extend the algorithm to the independent hopping scenario. The steps involved in the twostage selection algorithm are given as follows.

(1)
Divide the total available subcarriers into clusters with each cluster containing number of contiguous subcarriers.

(2)
If can be written as a product of two pairwise relative primes, for example, , we can first group clusters into groups with clusters in each group. Then, we index the groups from to .

(3)
Index the clusters in each group from 0 to .

(4)
At the 0th time slot, assign integer to user as its FH address according to its access order to the system, where .

(5)
If , then user selects the th cluster out of the th group for transmission.

(6)
At the th time slot, assign integer to user as its current FH address and repeat step 5.

(7)
Repeat steps 4–6 until one mutually orthogonal FH pattern is obtained.

(8)
If can be expressed as products of other combinations of two pairwise relative primes, for example, , then different orthogonal FH patterns can be obtained by repeating steps 2–7, times.
An example is given in Figure 2 to illustrate the twostage RNSassisted frequency hopping strategy. Here, 6 users access the system (); the total number of subcarriers is 30 () and they are divided into 6 clusters () with each cluster containing 5 contiguous subcarriers (). At the 0th time slot, the FH address assigned to the 5th user is 5 according to his/her access order to the system. Therefore, . User 5 will choose the 2nd cluster of subcarriers out of the 1st group of clusters to transmit. At the 1st time slot, the FH address assigned to this user becomes . Obviously, , then he/she will select the 0th cluster of subcarriers out of the 0th group of clusters for transmission at this time. This process continues until one FH sequence of length is constructed.
3.2. Multistage Algorithm
The multistage algorithm is an extension of the twostage algorithm. Introducing the multistage algorithm cannot only enhance the flexibility of the pattern design, but also strengthen the robustness of the entire FH scheme. We describe the multistage algorithm assuming an independent hopping scheme with each user employing the same number of subcarriers, that is, for . The steps involved in the multistage algorithm correspond to the following.

(1)
If can be written as a product of pairwise relative primes, for example, , we can first group subcarriers into groups with subgroups in each group. Then, we index the firststage groups from 0 to .

(2)
Index the secondstage groups in each firststage group from 0 to . Then group the subcarriers in each secondstage group into subgroups.

(3)
Similar steps continue on until all of the subcarriers are grouped and indexed at the thstage.

(4)
At the 0th time slot, assign integer set to user as its FH addresses, where is its access order to the system, .

(5)
If , then user first selects the th secondstage group out of the th firststage group, then similar selecting procedures continue on until the subcarrier at the thstage has been extracted out for transmission.

(6)
The process in step 5 is repeated on the other elements in the integer set of user until subcarriers have been extracted out for user to transmit.

(7)
At the th time slot, assign integer set as the current FH addresses of user and repeat steps 56.

(8)
Repeat steps 4–7 until one mutually orthogonal FH pattern is obtained.

(9)
If can be expressed as products of other combinations of pairwise relative primes, for example, , then different orthogonal FH patterns can be obtained by repeating steps 1–8, times.
It is easy to visualize the multistage algorithm by using a tree diagram. An example is given in Figure 3. Here, 30 users access the system (); a total of 30 subcarriers are used, that is, . Two specific examples are illustrated as follows: (1) consider user 2. The subcarriers used by this user at the 0th time slot can be calculated as follows: ; that is, in the 0th firststage group, the 2nd subcarrier out of the 2nd secondstage group is selected for transmission. This is indicated with a solid line in Figure 3; (2) consider user 27. ; that is, in the 1st firststage group, the 2nd subcarrier out of the 0th secondstage group is selected by the 27th user for transmission at the 0th time slot. This is indicated with a dashed line in the figure. This procedure continues until an FH sequence of length is completed. We should note that in this example, the system is fully loaded (). For , each user is assigned a set of FH addresses rather than one unique FH signature. For example, consider user 2 in Figure 3, the 2nd subcarrier occupied by user 2 at the 0th time slot is determined starting from his/her current FH address and following the steps as before. These steps are repeated until subcarriers for user 2 are identified. Extrapolating the procedure across the time axis, an entire FH sequence of length is designed.
With respect to the design procedures, the major difference between independent hopping and cluster hopping is the following: in independent hopping, each FH address specifies a single subcarrier that can be used. Therefore, if users have very high bandwidth/rate or other QoS requirements, multiple FH addresses can be given to accommodate. In cluster hopping scenario, a user may demand only one unique FH address as a single address completely specifies all subcarriers required for transmission. Fully loaded independent hopping system is a special case of cluster hopping with one subcarrier in each cluster.
From Figures 2 and 3, it is evident that the proposed RNSFH patterns guarantee the orthogonality among different users within a cell. That is, users within the same cell will not interfere with each other when they simultaneously access the system. The next example, which is shown in Figure 4, demonstrates that if different RNSFH patterns are assigned to adjacent cells, intercell interferences can be perfectly averaged. In this example, is set to 10 while the moduli sets used to construct FH patterns in cells 1 and 2 are and , respectively. From Figure 4, it is evident that every user in cell 1 experiences interference from different users from cell 2 during each of his/her hops. For example, in the first OFDM symbol duration, user 1 in cell 1 is interfered by user 8 from cell 2; in the next OFDM symbol slot, user 1 is interfered by user 5 from cell 2 and so on. In general, users from different cells collide only once during a frequency hopping cycle under the proposed scheme. Therefore, full interference diversity is exploited in the case of RNSFH patterns.
The properties of the proposed RNSFH patterns can be summarized as follows.

(1)
At most, a size of mutually orthogonal FH pattern can be obtained for the independent hopping scheme. The size becomes for the cluster hopping.

(2)
If () can be written as a product of pairwise relative primes, then at least, different RNSFH patterns can be obtained.

(3)
With the use of the same moduli set, for independent hopping, RNSFH patterns constructed after frames ( for cluster hopping) are actually periodical extensions of the RNSFH pattern designed during the first () frames.

(4)
With knowledge of moduli and residue, the base station can regenerate the entire RNSFH pattern using the CRT.
3.3. Comparison with [11]
In this section, we compare our proposed RNSFH pattern design method with the technique presented in [11] (which also considers RNS as the design metric).
First of all, although both strategies (one proposed here and the other presented in [11]) use the RNS arithmetic as a basis, the mechanisms of determining the hopping sequence are different. In [11], the FH scheme can be visualized as a "topdown" approach where a given bandwidth is divided into multiple candidate subcarriers in multistages according to the predetermined moduli set (see [11, Figure 2]). That is, the choice of the moduli set (top level decision) determines the number of subcarriers that can be used (bottom level decision) for hopping. This scheme is driven in conjunction with MFSKmodulated signals and a reference register , which has the same length as the moduli set (), providing reference to each user in order to enable synchronous transmission. However, in our work, we assume that the division of the frequency bandwidth has already been done in advance. That is, the number of subcarriers that can be used for hopping is given (bottom level decision). Based on this number, we employ a proper moduli set to group and index each of the candidate subcarriers (top level decision). Therefore, we can interpret our proposed initialization process as a "bottomup" approach (see Figure 3). It is important to note that in practical OFDMA cellular systems, the division of the bandwidth within a cell is usually fixed and predetermined (e.g., 1024 subcarriers). Therefore, our "bottomup" approach is more suitable for such practical systems. Furthermore, unlike the length reference register that is used in [11], the FH scheme proposed in this paper invokes the use of only a lengthone register to store the time index which in turn can be used to calculate current FH address of each user at the base station.
Secondly, for reducing intercell interference, [11] suggests the use of different moduli sets for adjacent cells. Since the choice of the moduli set determines the number of subcarriers used for hopping, a different moduli set in adjacent cells will result in different number of subcarriers in adjacent cells. If the total bandwidth is the same for all cells, this approach translates into subcarriers in adjacent cells having different bandwidths. This may be an unrealistic assumption for practical OFDMA systems. If the method in [11] is applied to a practical scenario using fixed number of subcarriers (each with the same bandwidth), high intercell interference will result (as shown in Figure 8). Our proposed "bottomup" approach does not suffer from this drawback as it is built on the premise that the number of subcarriers and their bandwidths are fixed across cells.
In summary, the method proposed in this work is flexible and well suited for practical OFDMA cellular systems.
4. Simulation Results
Parameters of the simulated system are provided in Table 1. The cyclic prefix within one OFDM symbol duration is assumed long enough to eliminate ISI (intersymbol interference). Two 6ray channel pulse responses are considered following the UTRA vehicular test environment [16]. In Figure 5, the correlation functions of these two channels are plotted versus the variation of , while slot and . From Figure 5, we can conclude that if small hopping intervals occur frequently in an FH pattern, Veh B can provide more frequency diversity than Veh A.
Theoretical (see (10) and (12) and simulated expected number of collisions per symbol in RNSFH OFDMA are given in Figure 6. The high collision probability severely limits the number of active users that can be simultaneously supported by the FH system.
In Figure 7, bit error rate (BER) versus SNR of RNSFH OFDMA under both cluster and independent hopping is plotted. The main objective of this example is to characterize the effects of frequency diversity exploited by RNSFH patterns on system performance. Here, we assume that 10 users are in the system with 11 subcarriers assigned to each via the twostage RNS hopping strategy. For cluster hopping, the moduli set used is , while for independent hopping, it is . It is observed that both independent and clustered RNSFH OFDMA dramaticallyoutperforms the regular OFDMA scheme without hopping in both Veh A and Veh B environments. Another observation is that under both independent and cluster hopping, the system performs better in Veh A. That is, in the proposed RNSFH patterns, large hopping intervals occur more frequently than small hopping distances. This characteristic is very important since it reveals that users occupy a wide bandwidth during a small fraction of all hops. Furthermore, since independent hopping scheme results in a much larger FH pattern than cluster hopping, more frequency diversity can be exploited in the independent hopping case. This is also clearly reflected by the simulation results shown in Figure 7. For example, at a BER level of , nearly 8 dB gain is offered by independent hopping relative to cluster hopping in Veh A environment.
Figure 8 quantifies the intercell interferences experienced by different users in the cell of interest, averaged across time. The axis represents the indices of the users within the cell of interest while the axis characterizes the timeaveraged intercell interferencetosignal power ratio for a given user. Two situations are considered: (1) different RNSFH patterns are allocated to the cell of interest and the interfering cell (denoted by the solid line); (2) the same RNSFH pattern as the cell of interest is assigned to the interfering cell (denoted by the dashed line). Here, we model the intercell interference as additive Gaussiandistributed distortion. Therefore, in scenario (1), users in the cell of interest will experience different interferences from the interfering cell across all hops, which in turn induces interference diversity. Figure 8 clearly demonstrates that by employing the proposed method (i.e., allocating a different RNSFH pattern to the interfering cell), the intercell interference floor can be significantly lowered relative to the scenario where all cells employ identical RNSFH patterns.
Figures 9 and 10 show the effects of intercell interference diversity on system performance. BER versus signaltointerference ratio (SIR) is plotted under cluster and independent hopping in Figures 9 and 10, respectively. For cluster hopping, the FH pattern assigned to the interfering cell is constructed by using while it is for the independent hopping scenario. We simulate the case where the same RNSFH pattern used in the cell of interest is assigned to adjacent interfering cells. Thus, users in the cell of interest will be affected by the same interferences from adjacent cells during all hops. Therefore, no interference diversity is exploited. Simulation results also reflect this feature. When the same RNSFH pattern is assigned, frequency diversity as a result of hopping reduces the interference floor. Therefore, the no hopping case still exhibits the worst BER performance. When different patterns are allocated to interfering cells, the interference diversity along with frequency diversity further improves system BER performance. For example, in cluster hopping (Figure 9), with different pattern assignments, nearly 3 dB gain at a BER level of is achieved relative to the system employing identical hopping. This gain grows to 5 dB under independent hopping scenario (Veh B environment).
BER versus user loads is plotted in Figures 11 and 12 under cluster and independent hopping, respectively, in both Veh A and Veh B. Effects of frequency and interference diversities on system performance are explored at given SNR and SIR. It is evident that the system throughput can be significantly enhanced by assigning different RNSFH patterns to different cells, while it is severely limited if no hopping occurs. Furthermore, the performance gap between the identical hopping and the different hopping decreases with the increase in user loads. That is, the benefit of intercell interference diversity is greater for lower user loads.
Figure 13 illustrates that by increasing the cluster size (the number of subcarriers in one cluster), or the number of active users, the number of collisions increases. This in turn induces degradation in BER performance as can be seen from Figure 13.
Finally, we compare our proposed RNSFH pattern design strategy with stateoftheart FH pattern designs. Specifically, our benchmark for comparison is the Latin squares (LSs)aided FH pattern design presented in [6]. In our proposed RNSFH pattern, the spacing between hops in time and frequency is far enough that subcarriers employed in a single time slot are weakly correlated. This feature provides remarkable performance improvements that are consistent across all cells. However, in Latin squares (LSs)aided FH pattern design, performances in different cells may vary a lot. Relative comparisons are given in Figure 14, where two Latin squaresbased FH patterns and [6] are employed. In LS , smaller hops happen more frequently, and for such smaller hops, Veh B exploits more frequency diversity than Veh A. The opposite is also true for LS . Using simulation results, we first observe that in RNSaided FHOFDMA, different RNSFH patterns provide nearly the same BER performance, while it varies a lot in LSaided FHOFDMA; the second observation is that our proposed RNSFH patterns have similar BER performances to LS while outperforming LS . Although there may exist LSaided FH pattern that has better performance than the proposed scheme, the performance variations in LSaided FH pattern design really limit their applications.
5. Conclusions
In this paper, we propose an RNS arithmeticbased FH pattern design that is well suited and easy to implement for practical OFDMA cellular systems. RNSFH patterns not only guarantee zero collision within a cell, but also average the intercell interferences by assigning different FH patterns to adjacent cells. Additionally, by having a large spacing between the hopping frequencies, the RNSFH patterns exploit frequency diversity effectively and provide significant improvement in BER performance. The BER performance gain is consistent across all cells unlike other FH pattern design schemes such as the LSbased method where wide performance variations are observed across cells. Simulation experiments demonstrate the superior performance of the RNSFH scheme in terms of frequency diversity and intercell interference diversity under both independent and cluster hopping strategies.
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Keywords
 Orthogonal Frequency Division Multiplex
 Orthogonal Frequency Division Multiple Access
 Orthogonal Frequency Division Multiple Access System
 Intercell Interference
 Multistage Algorithm