# Residue Number System Arithmetic Assisted Coded Frequency-Hopped OFDMA

- Dalin Zhu
^{1}Email author and - Balasubramaniam Natarajan
^{1}

**2009**:263695

**DOI: **10.1155/2009/263695

© D. Zhu and B. Natarajan. 2009

**Received: **31 July 2008

**Accepted: **23 February 2009

**Published: **26 March 2009

## Abstract

We propose an RNS arithmetic-based 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, high-rate data streams can be broken down into a number of parallel lower-rate 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 FH-OFDMA 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 space-frequency coding in multiple-antenna systems. Orthogonal Latin squares (LSs) are presented as FH patterns in TCM/BICM coded OFDMA in [6]. In LS-aided FH-OFDMA 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. Welch-Costas array is introduced in [7] and evaluated in [8] for coded FH-OFDMA. 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 pilot-assisted channel estimation in FH-OFDMA 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 "top-down" 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 "bottom-up" method driven by grouping and indexing the subcarriers according to the RNS arithmetic. For practical OFDMA cellular systems, the proposed "bottom-up" 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 so-called two-stage and multistage selection algorithms to construct RNS-FH 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 RNS-FH OFDMA has significantly better BER performance relative to traditional OFDMA scheme without FH. Another aspect that makes RNS-FH 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 RNS-FH 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

where .

where is the Doppler frequency.

where is the Fourier transform of the noise vector.

### 2.2. Access Model

In this part, clustered and independent FH-OFDMA 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 FH-OFDMA

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 FH-OFDMA

where is the probability that subcarriers out of subcarriers occupied by each user experience collisions due to interfering users.

Theorem 1.

Proof.

Equation (16) represents the conditional probability that each of the subcarriers of the desired user collides given that the other subcarriers are collision-free. 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 Gaussian-distributed 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 FH-OFDMA are presented in the following.

#### 2.3.1. Clustered FH-OFDMA

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 FH-OFDMA

For a fully loaded system with independent hopping, is identical to becomes to one.

## 3. RNS-FH Pattern Design

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.

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 RNS-FH patterns are given in the following subsections. The first part describes the two-stage algorithm, while the second part introduces the multistage algorithm which can be considered as generalization of the two-stage algorithm. At the end of this section, we compare our proposed RNS-FH pattern design strategy with the method presented in [11].

### 3.1. Two-Stage Algorithm

- (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.

### 3.2. Multistage Algorithm

- (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 first-stage groups from 0 to .

- (2)
Index the second-stage groups in each first-stage group from 0 to . Then group the subcarriers in each second-stage group into subgroups.

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

- (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 second-stage group out of the th first-stage group, then similar selecting procedures continue on until the subcarrier at the th-stage 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 5-6.

- (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.

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.

- (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 RNS-FH patterns can be obtained.

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

- (4)
With knowledge of moduli and residue, the base station can regenerate the entire RNS-FH pattern using the CRT.

### 3.3. Comparison with [11]

In this section, we compare our proposed RNS-FH 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 "top-down" 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 MFSK-modulated 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 "bottom-up" 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 "bottom-up" 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 length-one 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 "bottom-up" 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

System parameters.

Transmission BW | 5 MHz |
---|---|

Carrier frequency | 2 GHz |

OFDM symbol duration | 100 s |

CP duration | 10 s |

Tone spacing | 11 KHz |

FFT size | 128 |

Occupied subcarriers | 110 |

Channel impulse response | Veh A/Veh B |

Channel coding | 1/2 convolutional code |

Modulation | QPSK |

Time slots | 10 |

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 time-averaged intercell interference-to-signal power ratio for a given user. Two situations are considered: (1) different RNS-FH patterns are allocated to the cell of interest and the interfering cell (denoted by the solid line); (2) the same RNS-FH 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 Gaussian-distributed 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 RNS-FH pattern to the interfering cell), the intercell interference floor can be significantly lowered relative to the scenario where all cells employ identical RNS-FH patterns.

## 5. Conclusions

In this paper, we propose an RNS arithmetic-based FH pattern design that is well suited and easy to implement for practical OFDMA cellular systems. RNS-FH 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 RNS-FH 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 LS-based method where wide performance variations are observed across cells. Simulation experiments demonstrate the superior performance of the RNS-FH scheme in terms of frequency diversity and intercell interference diversity under both independent and cluster hopping strategies.

## Authors’ Affiliations

## References

- Gault S, Hachem W, Ciblat P: Performance analysis of an OFDMA transmission system in a multicell environment.
*IEEE Transactions on Communications*2007, 55(4):740-751.View ArticleGoogle Scholar - Simon MK, Omura JK, Scoltz RA, Levitt BK:
*Spread Spectrum Communications*. Computer Science Press, Rockville, Md, USA; 1985.Google Scholar - Scholand T, Faber T, Seebens A,
*et al*.: Fast frequency hopping OFDM concept.*Electronics Letters*2005, 41(13):748-749. 10.1049/el:20050644View ArticleGoogle Scholar - Kurt T, Deliç H: On symbol collisions in FH-OFDMA.
*Proceedings of the 59th IEEE Vehicular Technology Conference (VTC '04), May 2004, Milan, Italy*4: 1859-1863.Google Scholar - Kurt T, Deliç H: Space-frequency coding reduces the collision rate in FH-OFDMA.
*IEEE Transactions on Wireless Communications*2005, 4(5):2045-2049.View ArticleGoogle Scholar - Stamatiou K, Proakis JG: A performance analysis of coded frequency-hopped OFDMA.
*Proceedings of IEEE Wireless Communications and Networking Conference (WCNC '05), March 2005, New Orleans, La, USA*2: 1132-1137.Google Scholar - Maric SV, Moreno O: Using costas arrays to construct frequency hop patterns for OFDM wireless systems.
*Proceedings of the 40th IEEE Conference on Information Sciences and Systems (CISS '06), March 2007, Princeton, NJ, USA*505-507.Google Scholar - Wang C, Zhang X, Yang D: Evaluation of welch-costas frequency hopping pattern for OFDM cellular system.
*Proceedings of the 18th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '07), September 2007, Athens, Greece*1-5.Google Scholar - Li T, Ling Q, Ren J: A spectrally efficient frequency hopping system.
*Proceedings of the 50th IEEE Global Telecommunications Conference (GLOBECOM '07), November 2007, Washington, DC, USA*2997-3001.Google Scholar - Popovic BM, Li Y: Frequency-hopping pilot patterns for OFDM cellular systems.
*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*2006, E89-A(9):2322-2328. 10.1093/ietfec/e89-a.9.2322View ArticleGoogle Scholar - Yang L-L, Hanzo L: Residue number system assisted fast frequency-hopped synchronous ultra-wideband spread-spectrum multiple-access: a design alternative to impulse radio.
*IEEE Journal on Selected Areas in Communications*2002, 20(9):1652-1663. 10.1109/JSAC.2002.805059View ArticleGoogle Scholar - Chen J, Lv T, Zheng H: Joint cross-layer design for wireless QoS content delivery.
*Proceedings of IEEE International Conference on Communications (ICC '04), June 2004, Paris, France*7: 4243-4247.Google Scholar - Proakis JG:
*Digital Communications*. 4th edition. McGraw Hill, New York, NY, USA; 2001.Google Scholar - Watson KW, Hastings CW: Self-checked computation using residue arithmetic.
*Poceeding of the IEEE*1966, 54(12):1920-1931.View ArticleGoogle Scholar - Yang L-L, Hanzo L: Redundant residue number system based error correction codes.
*Proceedings of the 54th IEEE Vehicular Technology Conference (VTC '01), October 2001, Atlantic City, NJ, USA*3: 1472-1476.Google Scholar -
*ETSI TR 101 112, UMTS 30.03, V3.1.0, Annex B, Std.*

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