Construction of Optimal Sets of Frequency Hopping Sequences

Abstract

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the set-theoretic characterization of an optimal set of FH sequences.

1. Introduction

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are an integral part of spread-spectrum communication systems such as FH-CDMA systems (for a description of such systems, see [1]). In modern radar and communication systems, frequency-hopping (FH) spread-spectrum techniques have become popular (see [2], for example).

Assume that 𝐹={𝑓₀,𝑓₁,…,𝑓_𝑚−1} is a set of available frequencies, called an alphabet. Let 𝒳(𝑣;𝐹) be the set of all sequences of length 𝑣 over 𝐹. Any element of 𝒳(𝑣;𝐹) is called a frequency hopping sequence (FHS) of length 𝑣 over 𝐹. Given two FH sequences, 𝑋=(𝑥₀,𝑥₁,…,𝑥_𝑣−1) and 𝑌=(𝑦₀,𝑦₁,…,𝑦_𝑣−1), we define their Hamming correlation 𝐻_𝑋,𝑌(𝑤) to be 𝐻_𝑋,𝑌(𝑤)=_{0≤𝑖≤𝑣−1}ℎ𝑥_𝑖,𝑦_𝑖+𝑤, (1) where 0≤𝑤<𝑣 if 𝑋≠𝑌 and 0<𝑤<𝑣 if 𝑋=𝑌, and where ℎ𝑥,𝑦=1,if𝑥=𝑦,0,otherwise, (2) and all operations among position indices are performed modulo 𝑣. If 𝑋=𝑌, 𝐻_𝑋,𝑌(𝑤) is the Hamming autocorrelation. If 𝑋≠𝑌, 𝐻_𝑋,𝑌(𝑤) is the Hamming cross correlation.

2. Lower Bounds on the Correlations of FHSs

FH sequences for FH-CDMA systems are required to have good Hamming correlations and large linear span [3]; the linear span is defined to be the length of the shortest linear feedback shift register that can produce the sequence. FH sequences’ design normally involves six parameters: the size 𝑚 of the frequency library 𝐹, the sequence length 𝑣, the family size 𝑁 of the subset 𝒮⊂𝒳(𝑣;𝐹), the maximum out-of-phase Hamming autocorrelations 𝐻_𝑎, the maximum Hamming cross correlations 𝐻_𝑐, and the linear span. It is generally desired that the family 𝒮 of FH sequences has the following properties: (1)the maximum out-of-phase Hamming autocorrelations 𝐻_𝑎 should be as small as possible,(2)the maximum Hamming cross correlations 𝐻_𝑐 should be as small as possible,(3)the family size 𝑁=|𝒮| for given 𝐻_𝑎, 𝐻_𝑐, 𝑀, and 𝑣 should be as large as possible,(4)the linear span should be as large as possible.

In order to evaluate the theoretical performance of the FH sequences, it is important to find some theoretical bounds for these parameters. Given 𝑚, 𝑣, and 𝑁 of 𝒮, Lempel and Greenberger [4] and Peng and Fan [5] derived lower bounds on 𝐻_𝑎 and 𝐻_𝑐 of FH sequences in 𝒳(𝑣;𝐹). We restate their results in this section, which will be used later as the criteria to determine whether the new FH sequences constructed in this paper are optimal or not.

For any single FH sequence 𝑋∈𝒳(𝑣;𝐹), let 𝐻_𝑎(𝑋)=max_{1≤𝑡≤𝑣−1}{𝐻_𝑋,𝑋(𝑡)} be the maximum out-of-phase value of 𝐻_𝑋,𝑋(𝑡). If 𝐻_𝑎(𝑋^∗)≤𝐻_𝑎(𝑋) for all 𝑋∈𝒳(𝑣;𝐹), that is, if the value 𝐻_𝑎(𝑋^∗) is the least among all FH sequences of the same length 𝑣 and over the same frequency library 𝐹, 𝑋^∗ is called an optimal FH sequence. Lempel and Greenberger [4] developed the following lower bound for 𝐻_𝑎(𝑋).

Lemma 1 (see [4]). For every FH sequence 𝑋 of length 𝑣 over a frequency alphabet 𝐹 of size 𝑚. One has 𝐻_𝑎(𝑋)≥(𝑣−𝜖)(𝑣+𝜖−𝑚)𝑚(),𝑣−1 (3) where 𝜖 is the least nonnegative residue of 𝑣 modulo 𝑚.

Corollary 2. For every FH sequence 𝑋 of length 𝑣 over a frequency alphabet 𝐹 of size 𝑚. One has 𝐻_𝑎(𝑋)≥𝑘,𝑖𝑓𝑚≠𝑣,0,𝑖𝑓𝑚=𝑣, (4) where 𝑣=𝑘𝑚+𝜀 with 0≤𝜀≤𝑚−1.

For any given subset 𝒮 of 𝒳(𝑣;𝐹) containing 𝑁 FH sequences, any two 𝑋,𝑌∈𝒮, 𝑋≠𝑌, we write 𝐻_𝑐(𝑋,𝑌)=max_{0≤𝑡≤𝑣−1}{𝐻_𝑋,𝑌(𝑡)}. We define the maximum out-of-phase Hamming autocorrelations 𝐻_𝑎(𝒮) and the maximum Hamming cross correlations 𝐻_𝑐(𝒮) as 𝐻_𝑎(𝒮)=max𝐻_𝑎(𝑋),𝐻∶𝑋∈𝒮_𝑐(𝒮)=max𝐻_𝑐(),𝑀(𝒮)𝐻𝑋,𝑌∶𝑋,𝑌∈𝒮,𝑋≠𝑌=max_𝑎(𝒮),𝐻_𝑐(𝒮). (5)

Peng and Fan [5] developed the following bounds on 𝑀(𝒮), which take into consideration the number of FH sequences in the set.

Lemma 3 (see [5]). Let 𝒮⊆𝒳(𝑣;𝐹) be a set of 𝑁 sequences of length 𝑣 over an alphabet of size 𝑚. Define 𝐼=⌊𝑣𝑁/𝑚⌋. Then 𝑀(𝒮)≥(𝑣𝑁−𝑚)𝑣()𝑚,𝑀(𝒮)≥()𝑣𝑁−12𝐼𝑣𝑁−𝐼+1𝑚𝐼()𝑁.𝑣𝑁−1 (6)(7)

A family 𝒮⊆𝒳(𝑣;𝐹) is an optimal set if the Peng-Fan lower bound (2) or (3) in Lemma 3 is met. Let 𝑁=𝑚, and we have the following corollary which is a useful tool to check the bound in Lemma 3.

Corollary 4 (see [6]). Let 𝒮⊆𝒳(𝑣;𝐹) be a set of 𝑚 sequences of length 𝑣 over an alphabet of size 𝑚. Then 𝑀(𝒮)≥𝑎,𝑖𝑓𝑚∣𝑣,𝑎+1,𝑖𝑓𝑚∤𝑣, (8) where 𝑣=𝑎𝑚+𝑏 with 0≤𝑏≤𝑚−1.

A number of authors have made contributions to the construction of optimal FH sequences. Both algebraic and combinatorial constructions of optimal FH sequences have been given (see, for example, [3, 4, 7–11]). Most of them are concentrated on single optimal FH sequences. The purpose of this paper is to present a construction of optimal sets of FH sequences based on the set-theoretic characterization of an optimal set of FH sequences. Throughout what follows, we use (𝑣,𝑚,𝜆)-FHS to denote an FH sequence 𝑋 of length 𝑣 over an alphabet of size 𝑚 whose Hamming autocorrelation 𝐻_𝑎(𝑋)=𝜆. We also call a set 𝒮 of 𝑁 FH sequences in 𝒳(𝑣;𝐹) a (𝑣,𝑁,𝜆;𝑚) set of FH sequences, where 𝜆=𝑀(𝒮).

3. Mixed Difference Functions

Fuji-Hara et al. [9] characterized a (𝑣,𝑚,𝜆)-FHS in terms of partition-type cyclic difference packings. Given a partition 𝒟={𝐷₀,𝐷₁,…,𝐷_𝑚−1} of 𝑍_𝑣 into 𝑚 subsets (called base blocks), we can define a difference function on ℤ^∗_𝑣=ℤ_𝑣⧵{0} given by Γ_𝒟(𝑤)=∑^𝑚−1_𝑖=0|(𝐷_𝑖+𝑤)∩𝐷_𝑖|. Let max{Γ_𝒟(𝑤)|𝑤∈ℤ_𝑣⧵{0}}=𝜆. Then 𝒟 is called a (𝑣,𝐾,𝜆)_𝑚-PCDP (partition-type cyclic difference packing). Here, 𝑚 is used in the notation to indicate the number of base blocks, and 𝐾={|𝐷_𝑖|∶0≤𝑖≤𝑚−1} is the list of the sizes of base blocks. This is to say that a (𝑣,𝐾,𝜆)_𝑚-PCDP is a partition of ℤ_𝑣 into 𝑚 base blocks which satisfies the following property. For any fixed nonzero residue 𝑤∈ℤ_𝑣, the equation 𝑥−𝑦=𝑤 has at most 𝜆 solutions (𝑥,𝑦) in the multiset union ⋃_𝐷∈𝒟(𝐷×𝐷).

If we label the positions of a (𝑣,𝑚,𝜆)-FHS 𝑋 by the elements of ℤ_𝑣, then, by the above definition, the sets of position indices of 𝑚 frequencies in 𝑋 form a (𝑣,𝐾,𝜆)_𝑚-PCDP 𝒟 with Γ_𝒟(𝑤)=𝐻_𝑋,𝑋(𝑤) for any nonzero 𝑤∈ℤ_𝑣. Conversely, if we label the 𝑚 base blocks of a (𝑣,𝐾,𝜆)_𝑚-PCDP by the elements of ℤ_𝑚 and identify the frequency alphabet 𝐹 with ℤ_𝑚, then the PCDP gives a (𝑣,𝑚,𝜆)-FHS in 𝒳(𝑣;𝐹).

This fact reveals that a single FHS can be constructed by a PCDP. Apparently, the smaller the index 𝜆 of a PCDP the lower the Hamming autocorrelation 𝐻_𝑎(𝑋) of its corresponding FH sequence. For an optimal FH sequence, we need to construct a (𝑣,𝐾,𝜆)_𝑚-PCDP so that its index 𝜆 is as small as possible for any given value of 𝑣 and 𝑚. Based on Lempel-Greenberger bound on 𝐻_𝑎(𝑋) in Lemma 1, Fuji-Hara et al. [9] proved the following result.

Lemma 5 (see [9]). There exists a (𝑣,𝑚,𝜆)-FHS over the alphabet 𝐹=ℤ_𝑚 if and only if there exists a (𝑣,𝐾,𝜆)_𝑚-PCDP in ℤ_𝑣. Furthermore, this FH sequence is optimal if 𝜆=⌊𝑣/𝑚⌋ for 𝑣>𝑚 and if 𝜆=0 for 𝑣=𝑚.

The correspondence between an individual FH sequence and a PCDP can be naturally extended to give a set-theoretic interpretation of a set of FH sequences. Let 𝑁 be a positive integer. Let 𝒞={𝒟⁽⁰⁾,𝒟⁽¹⁾,…,𝒟^(𝑁−1)} be a collection of partitions of ℤ_𝑣 into 𝑚 subsets (called base blocks). Write 𝒟^(𝑟)={𝐷₀^(𝑟),𝐷₁^(𝑟),…,𝐷^(𝑟)_𝑚−1}, 0≤𝑟≤𝑁−1. For any ordered pair (𝑖,𝑗) with 0≤𝑖<𝑗≤𝑁−1, we define a difference function on ℤ_𝑣 given by Γ_𝒞^(𝑖,𝑗)(𝑤)=∑^𝑚−1_𝑘=0|(𝐷_𝑘^(𝑖)+𝑤)⋂𝐷_𝑘^(𝑗)|. For any integer 𝑟 with 0≤𝑟≤𝑁−1, we define a difference function on ℤ_𝑣⧵{0}, as before, given by Γ_𝒞^(𝑟,𝑟)(𝑤)=Γ_𝒟^(𝑟)(𝑤)=∑^𝑚−1_𝑘=0|(𝐷_𝑘^(𝑟)+𝑤)∩𝐷_𝑘^(𝑟)|. We refer to these 𝑁(𝑁+1)/2 difference functions defined above as mixed difference functions with respect to the given collection 𝒞. Since each partition in 𝒞 determines uniquely an FH sequence, the collection 𝒞 gives a set of 𝑁 FH sequences in 𝒳(𝑣;𝐹), and vice versa, where the alphabet is regarded as ℤ_𝑚. For the optimality of the derived set of FH sequences from 𝒞, we define 𝜆(𝑟)=max_{1≤𝑤<𝑣}{Γ_𝒞^(𝑟,𝑟)(𝑤)}, 0≤𝑟≤𝑁−1, 𝜆(𝑖,𝑗)=max_{0≤𝑤<𝑣}{Γ_𝒞^(𝑖,𝑗)(𝑤)}, 0≤𝑖<𝑗≤𝑁−1, 𝜇(𝑖,𝑗)=max{𝜆(𝑖),𝜆(𝑗),𝜆(𝑖,𝑗)}, 0≤𝑖<𝑗≤𝑁−1, 𝜆=max{max_{0≤𝑟≤𝑁−1}𝜆(𝑟),max_{0≤𝑖<𝑗≤𝑁−1}𝜆(𝑖,𝑗)}. Then 𝒟^(𝑟) is a (𝑣,𝐾_𝑟,𝜆_𝑟)_𝑚-PCDP, according to the previous definition. We say that 𝒞 is a (𝑣,{𝐾₀,𝐾₁,…,𝐾_𝑁−1},{𝜆₀,𝜆₁,…,𝜆_𝑁−1};𝜆)_𝑚 collection of 𝑁 PCDPs. It turns out that there exists a (𝑣,𝑁,𝜆;𝑚) set of FH sequences in 𝒳(𝑣;𝐹) if and only if there exists a (𝑣,{𝐾₀,𝐾₁,…,𝐾_𝑁−1},{𝜆₀,𝜆₁,…,𝜆_𝑁−1};𝜆)_𝑚 collection of 𝑁 PCDPs in 𝑍_𝑣 under our notations. This gives us an interpretation for a set of FH sequences from set-theoretic perspective. As with individual optimal FH sequence, for an optimal (𝑣,𝑁,𝜆;𝑚) set of FH sequences, we are required to construct a (𝑣,{𝐾₀,𝐾₁,…,𝐾_𝑁−1},{𝜆₀,𝜆₁,…,𝜆_𝑁−1};𝜆)_𝑚 collection 𝒞 of 𝑁 PCDPs so that its index 𝜆 is as small as possible. Since the index 𝜆 of 𝒞 is the same as the Hamming correlation 𝑀(𝒮), the Peng-Fan bounds in Lemma 3 can be employed as our benchmarks. As noted in [12], a set of FH sequences meeting one of the Peng-Fan bounds must be optimal. We have the following theorem.

Theorem 6 (see [13]). Let 𝑁≥2 be an integer. Then there exists a (𝑣,𝑁,𝜆;𝑚) set of FH sequences in 𝒳(𝑣;𝐹) if and only if there exists a (𝑣,{𝐾₀,𝐾₁,…,𝐾_𝑁−1},{𝜆₀,𝜆₁,…,𝜆_𝑁−1};𝜆)_𝑚 collection of 𝑁 PCDPs in ℤ_𝑣. Furthermore, this set is optimal if 𝜆 meets one of the Peng-Fan lower bounds given in Lemma 3.

4. The Construction of Optimal Sets of FH Sequences

To begin with, we state some preparatory knowledge that will be used in our constructions.

Let 𝑞=𝑝^𝑠 be a power of prime 𝑝, where 𝑠 is an arbitrary positive integer. Write Tr_𝑞/𝑝 for the absolute trace function from 𝔽_𝑞 to 𝔽_𝑝 defined by Tr_𝑞/𝑝(𝑥)=𝑥+𝑥^𝑝+𝑥^𝑝2+⋯+𝑥^𝑝𝑠−1. (9)

We will use the following known result.

Lemma 7 (see [14, Corollary 7.17]). For any 𝑎∈𝔽_𝑞 the number of elements 𝛼∈𝔽_𝑞𝑚 such that Tr_{𝑞𝑚/𝑞}(𝛼)=𝑎 is 𝑞^𝑚−1.

With the previous preparations, we are now ready to describe our construction of optimal sets of FH sequences.

Theorem 8. Let 𝑝 be an odd prime and 𝑚 a positive integer. Define 𝑓∶ℤ_𝑝𝑚−1→ℤ_𝑝 by 𝑓(𝑥)=Tr_{𝑝𝑚/𝑝}(𝛼^𝑥), where Tr_{𝑝𝑚/𝑝} is the trace function from 𝔽_𝑝𝑚 to 𝔽_𝑝, 𝛼 is a generator of 𝔽^∗_𝑝𝑚. Define 𝐷_𝑗=𝑓⁻¹𝑗=𝑥∈ℤ_𝑝𝑚−1(𝑥)𝐷∶𝑓=𝑗,0≤𝑗≤𝑝−1,_𝑗(𝑟)=𝐷_𝑗+𝑟𝒟,0≤𝑗≤𝑝−1,0≤𝑟≤𝑝−1,_(𝑟)=𝐷₀^(𝑟),𝐷₁^(𝑟),…,𝐷^(𝑟)_𝑝−1,0≤𝑟≤𝑝−1, (10) where all operations among the subscripts are performed modulo 𝑝. Then 𝒞={𝒟⁽⁰⁾,𝒟⁽¹⁾,…,𝒟^(𝑝−1)} is an optimal (𝑝^𝑚−1,𝑝,𝑝^𝑚−1;𝑝) set of FH sequences over ℤ_𝑝 of length 𝑝^𝑚−1. Furthermore, each sequence of the set 𝒞 is optimal with respect to the bound of Lemma 1.

Proof. For any 𝑟 with 0≤𝑟≤𝑒−1 and any 𝑤∈ℤ^∗_𝑝𝑚−1, we have Γ_𝒞^(𝑟,𝑟)(𝑤)=^𝑝−1_𝑘=0𝐷_𝑘^(𝑟)+𝑤𝐷_𝑘^(𝑟)=^𝑝−1_𝑘=0𝐷_𝑘+𝑟+𝑤𝐷_𝑘+𝑟=^𝑝−1_𝑘=0𝐷_𝑘+𝑤𝐷_𝑘=𝑥∈ℤ_𝑝𝑚−1()(𝑥)=∣𝑓𝑥+𝑤−𝑓=0𝑥∈ℤ_𝑝𝑚−1∣Tr_{𝑝𝑚/𝑝}𝛼^𝑤−1𝛼^𝑥=0=𝑝^𝑚−1−1, (11) where the last equality follows from Lemma 7. Next we will show that Γ_𝒞^(𝑖,𝑗)(𝑤)=𝑝^𝑚−1 for any 𝑖,𝑗 with 0≤𝑖<𝑗≤𝑝−1 and any 𝑤∈ℤ^∗_𝑝𝑚−1. For any 𝑖,𝑗 with 0≤𝑖<𝑗≤𝑒−1 and any 𝑤∈ℤ^∗_𝑝𝑚−1, we have Γ_𝒞^(𝑖,𝑗)(𝑤)=^𝑒−1_𝑘=0𝐷_𝑘^(𝑖)+𝑤𝐷_𝑘^(𝑗)=^𝑒−1_𝑘=0𝐷_𝑘+𝑖+𝑤𝐷_𝑘+𝑗=^𝑒−1_𝑘=0𝐷_𝑘+𝑖+𝑤𝐷_{(𝑘+𝑖)+(𝑗−𝑖)}=^𝑒−1_𝑘=0𝐷_𝑘+𝑤𝐷_{𝑘+(𝑗−𝑖)}=^𝑒−1_𝑘=0𝐷_𝑘+𝑤𝐷_𝑘+𝑏=,𝑏=𝑗−𝑖𝑥∈ℤ_𝑝𝑚−1()(𝑥)=∣𝑓𝑥+𝑤−𝑓=𝑏𝑥∈ℤ_𝑝𝑚−1∣Tr_{𝑝𝑚/𝑝}𝛼^𝑤−1𝛼^𝑥=𝑏=𝑝^𝑚−1, (12) where the last equality follows from Lemma 7. It follows that 𝒞 is a (𝑝^𝑚−1,{𝐾₀,𝐾₁,…,𝐾_𝑝−1},{𝜆₀,𝜆₁,…,𝜆_𝑝−1};𝑝^𝑚−1)_𝑝 collection of 𝑝 PCDPs in ℤ_𝑝, where 𝜆₀=𝜆₁=⋯=𝜆_𝑚−1=𝑝^𝑚−1−1. By Theorem 6, it derives a (𝑝^𝑚−1,𝑝,𝑝^𝑚−1;𝑝) set of FH sequences in 𝒳(𝑣;𝐹). For this set, we have 𝑁=𝑝 and both lower bounds in Lemma 3 are equal to 𝑝^𝑚−1 by Corollary 4. Hence, it is optimal. Furthermore, each sequence of the set 𝒞 is optimal with respect to the bound of Lemma 1. The proof is then complete.

Example 9. Take 𝑝=3 and 𝑚=2. It is readily calculated that 𝐷₀={2,6},𝐷₁={1,3,4},𝐷₂={0,5,7}. (13) We then have 𝒟⁽⁰⁾=𝐷₀⁽⁰⁾={2,6},𝐷₁⁽⁰⁾={1,3,4},𝐷₂⁽⁰⁾={,𝒟0,5,7}₍₁₎=𝐷₀⁽¹⁾={1,3,4},𝐷₁⁽¹⁾={0,5,7},𝐷₂⁽¹⁾={,𝒟2,6}₍₂₎=𝐷₀⁽²⁾={0,5,7},𝐷₁⁽²⁾={2,6},𝐷₂⁽²⁾={1,3,4}, (14)

The family 𝒞={𝒟⁽⁰⁾,𝒟⁽¹⁾,𝒟⁽²⁾} is a (𝑣,𝑚,{𝐾₀,𝐾₁,𝐾₂};{𝜆₀,𝜆₁,𝜆₂})₃ collection of 3 PCDPs in 𝑍₉, where 𝐾₀={2,3,3}, 𝐾₁={3,3,2}, 𝐾₂={3,2,3}, and 𝜆₀=𝜆₁=𝜆₂=2. By Theorem 6, it corresponds a (8,3,3;3) set 𝒮 consisting of the following 3 FH sequences over 𝑍₃: (),(),(),𝑋=2,1,0,1,1,2,0,2𝑌=1,0,2,0,0,1,2,1𝑍=0,2,1,2,2,0,1,0 (15) with 𝑀(𝒞)=3. It is optimal meeting the lower bounds in Lemma 3.

Acknowledgment

The Paper was supported by the Tian Yuan Special Funds of the National Natural Science Foundation of China (Grant no. 11226282).