1.

INTRODUCTION

The design of sequences with good correlation properties has remained a classical research subject [1]. The traditional design of sequence assumes that the entire frequency band is available and divide it uniformly into multiple frequency segments assigned to different applications. However, with the rise of the number of wireless devices and their related applications, leading to explosive growth in mobile data traffic. An increasingly prominent contradiction has resulted between the enormous demand for wireless spectrum resources and the limited availability of spectrum resources. Cognitive radio [2-3] (cognitive radar [4]) can effectively improve spectrum utilization by dynamically allocating spectrum resources and utilizing idle spectrum resources.The direct application of traditional sequences to cognitive radio systems would lead to significant degradation of their correlation properties [5]. Therefore, researching for designing sequences under spectral constraints is of significant theoretical importance. When designing SCSs, the specific performance demands of sequence design can be summarized as follows [5]: (1) good correlation performance, (2) flexibility in spectrum holes, and (3) low peak-to-average power ratio (PAPR), ideally equal to 1.

The correlation properties are the key focus in the design of SCSs. One aspect involves considering the low correlation properties of SCSs. He et al. [6] proposed a SCAN numerical optimization algorithm to construct SCSs with low sidelobe values in correlation function. Aubry et al. [7] addressed the non-convex quadratic optimization problem in SCS design. Building upon this work, the design methods of single-channel and multi-channel sequence under spectral constraints were proposed by Song et al. [8, 9]. However, designing long-period SCS and large sets of SCSs using numerical optimization algorithms incurs high computational complexity. Hence, the theoretical analytical methods for SCS design has attracted significant attention from experts both domestically and internationally. In [5], single-carrier binary sequences under dynamic spectral hole constraints were constructed based on the time-domain Kronecker product. Based on cyclic difference set and the principles of maximum-length linear shift register sequences, a design method for spectrally-constrained single binary sequences that meet the theoretical bounds was proposed in [10]. In [11], new spectrally-constrained complementary sequence sets with low PAPR values were generated by using orthogonal complementary sets as seed sequences. In [12], a family of spectrally-constrained single-channel sequences was constructed using theoretical analytical methods. On the other hand, sequences with ZCZ can be used as pilot sequences in multiple-input multiple-output (MIMO) channel estimation [13]. Considering the spectrum constraints, Popovi et al. [14] proposed a transform domain method for generating sets of single-channel SNC-ZCZ sequences. SNC-ZCZ sequences and SNC-Z complementary sequences were constructed in [15], filling the gap in spectrum-constrained Z-complementary sequences. In [16], a framework was introduced based on cyclic Florentine rectangles and interleaving techniques to construct sets of ZCZ-SCS sequences. The new theoretical bounds for evaluating ZCZ-SCS sequence sets were proposed, providing new design insights for ZCZ-SCS sequence research.

The research on SCSs is very limited and relatively challenging, and the space for the existence of SCSs is also relatively small. Motivated by this, in this paper, a new class of SCS called PZCZ-SCSs is proposed, and the properties of PZCZ-SCSs are analyzed and studied. Then, a construction for PZCZ_SCSs is proposed from the perspective of the transform domain. To the best of our knowledge, there is currently no relevant research defining PZCZ-SCS. The proposed PZCZ-SCSs can also be used as spreading sequences in communication systems[16]. This research provides a theoretical foundation for the design of PZCZ-SCSs and PZCZ-SCS sets in the future.

2.

PRELIMINARIES

2.1

Theory of Difference Set.

Definition 1[10]: Let C = {c₁, c₂,···, c_n–1} be subset of Z_N ={0,1, ···, N – 1}, and the difference function of C is denoted by d_C (τ) =| C ∩ (C + τ)|, where τ ∈ Z_N and C + τ = {c + τ | c ∈ C}. C is called an (N, n, σ) “cyclic difference set” if d_C (τ) takes on the value σ for N – 1 times when τ ranges over the nonzero elements of Z_N.

Definition 2 [18]: Let G be an mq = N -order multiplicative group, and Q be a q -order subset of G. The k -order subset D of G is called divisible difference set (DDS) with parameter (m, q, k,λ₁, λ₂) relative to Q if {(d_i·– d_j) mod N,0 ≤ i ≠ j ≤ k – 1 d_i ∈ D, d_j ∈ D} contains exactly λ₁ copies of each nonidentity element of Q and exactly λ₂ copies of each element of the set G – Q. If λ₁ = 0, then D is called a relative difference set in G relative to Q.

Lemma 1 [18]: A subset D of mq = N -order multiplicative group G is an (m, q, k, λ₁, λ₂) DDS. Furthermore, G\D is also an (m, q, mq – k, mq – λ₁, λ₂) DDS.

Lemma 2 [18]: An (m, q, k, λ₁, λ₂) DDS D relative to Q = {0, m, 2m,···, (q – 1)m} is the subset of mq = N -order multiplicative group G. For τ ∈ {1,···, N – 1}, we have

2.2

Basic Knowledge.

is a (scaled) discrete Fourier transform (DFT) matrix of order N. The conjugate transpose of F_N is represented the inverse DFT. f_i,j is defined by

where . The frequency domain dual of the sequence b = [b₀, b₁,⋯b_i,⋯b_N–1]^T is calculated as B = bF_N = [B₀, B₁,⋯,B_d,⋯B_N–1]^T.

Definition 3[17]: For any two length- N sequences a and b, the periodic cross-correlation function is defined as

where b* denotes the complex conjugation of b. if a = b, it is called periodic auto-correlation function which is denoted by θ_b (τ). If θ_b (τ ≠ 0) = 0, b is referred to as optimal sequence.

Definition 4[17]: The PAPR value of length- N sequence b is calculated as

where |b_i| is the modulus of the i -th element of b. If |b₀| = |b₁| = ⋯ = |b_N–1|, b is referred to as constant modulus sequence with PAPR=1.

According to time-frequency duality, the relationship between the optimal sequence and the constant modulus sequence as follows.

Lemma 3 [20]: A sequence is optimal in the frequency domain, then it is a constant-modulus sequence in the time domain.

For a CR system whose entire frequency band is uniformly divided into N carriers, let M = [m₀, m₁, ⋯, m_N-1 ] denote “carrier marking vector”, which is capable of implying the dynamic state of N carriers. More specifically, m_t = 1 if the i -th carrier is not occupied, furthermore, m_i = 0. Ω = {i|m_i = 0} represents the set of all occupied carrier. Note that Ω is called the “spectral constraint,” and .

Definition 5[17]: For any sequence b, denote by B its frequency-domain dual. b is said to be a SCS if B_d = 0 for d ∈ Ω.

Definition 6[17]: b is an ideal SCS if both of the following conditions are met: 1) b is a constant-modulus sequence; 2) the spectral constraint Ω of b is a cyclic difference set with parameter (N, n, σ).

Definition 7 : The length-N SCS b is a PZCZ-SCS if the periodic auto-correlation function satisfies

where K is an integer and γ is a constant.

3.

ANALYSIS ON THE PROPERTY OF PZCZ-SCS

Based on the time-frequency synthesis analysis, this section will discuss the property that the spectral constraint of PZCZ-SCSs.

Without loss of generality, for a sequence b, we have , as is widely known from the Parseval’ s theorem that for the frquency-domain dual B. Let B has a uniformly distributed energy, i.e., . Recalling equation (7) in[17], the magnitude of periodic auto-correlation function of b can be further expressed as

where

Recalling Definition 7, for PZCZ-SCSs, the following identity needs to be satisfied.

Combining Lemma 2, b is a PZCZ-SCS if and only if of b is an (m, q, k, λ₁, λ₂) DDS relative to Q = {0, m, 2m,···, (q – 1)m}, λ₁ = k and λ₂ < λ₁. the magnitude of periodic auto-correlation function of b is calculated as

Taking into account the practical implications of nonlinear effects, the design of spectrum-limited sequences with an ideal PAPR value holds significant practical value. We have Q = {0, m, 2m,···, (q – 1)m}, λ₁ = k and λ₂ < λ₁.

Definition 8 : b is an ideal PZCZ-SCS if both of the following conditions are met: 1) b is a constant-modulus sequence; 2) the spectral constraint Ω of b is an (m, q, k, λ₁, λ₂) DDS relative to Q = {0, m, 2m,···, (q – 1)m}, λ₁ = k and λ₂ < λ₁

4.

THE CONSTRUCTION OF PZCZ-SCS

In order to obtain ideal PZCZ-SCS, a construction of PZCZ-SCS will be proposed from the transform domain in this section. The specific construction is as follows.

Construction 1 : Step 1, generate the Matrix M of order q × m based on sequences x_t1 = [x₀, x_l,⋯, x_m–l] and y_t2 = [y₀, y_l,⋯, y_q–l] where gcd(m, q) = 1.

where 0 ≤ j ≤ q – 1 and 0 ≤ i ≤ m – 1. The row vector and column vector of M are denoted as s_j(t₁) = y_jx_t1 and r_i(t₂) = x_iy_t2, respectively, where 0 ≤ t₁ ≤ m – 1 and 0 ≤ t₂ ≤ q – 1.

Step 2, the elements of length-N frequency domain sequence B = [B₀, B₁,⋯, B_d,⋯, B_mq–1] is defined as follows.

where

Step 3, performing the inverse discrete Fourier transform (IDFT) on B yields the time-domain sequence b = [b₀, b₁,⋯, b_i,⋯, b_mq–1].we have

where i = d

Theorem 1: b is an ideal PZCZ-SCS if and only if x_t1 = [x₀, x_l,⋯, x_m–l] is an ideal SCS with the spectral constraint C is an (N, n, σ) cyclic difference set and y_t2 = [y₀, y_l,⋯, y_q–l] is an optimal constant-modulus sequence. In addition, the spectral constraint of ideal PZCZ-SCS b is expressed as .

Proof : The properties of b are analyzed as follows.

Analyzing the magnitude homogeneity of b. Let t = (t₁, t₂), τ = (τ₁, τ₂), and note that τ = (τ₁, τ₂), (t₁, t₂) ∈ Z_m × Z_q, the following cases are to be considered.

Case 1: when τ₁ = 0, τ₂ = 0, the auto-correlation function of B can be obtained by summing the zero-shifted auto-correlation function of s_j(t₁) = y_jx_t1. The specific process is as follows.

Case 2: when τ₁ ≠ 0, τ₂=0, the auto-correlation function of B can be obtained by summing the non-zero-shifted auto-correlation function of s_j(t₁) = y_jx_t1. We have

since x_t1 = [x₀, x_l,⋯, x_m–l] is an ideal SCS, . Therefore, θ_B (τ_1, 0) = 0.

Case 3: when τ₁ ≠ 0, τ₂=0, the auto-correlation function of B can be obtained by summing the non-zero-shifted cross-correlation function of r_i(t₂) = x_iy_t2. We have

since y_t1 = [y₀, y_l,⋯, y_q–l] is an optimal constant-modulus sequence, we have θ_yt2 (τ₂) = 0. Therefore, θ_β (0, τ₂) = 0.

Case 4: when τ₁ ≠ 0, τ₂ ≠ 0, the auto-correlation function of B can be obtained by summing the non-zero-shifted cross-correlation function of s_j(t₁) = y_jx_t1 or r_i(t₂) = x_iy_t2. We have

since . Therefore, θ_B (τ_1, τ₂) = 0.

As summarized above, B is an optimal sequence, and b is a constant-modulus sequence.

Analyzing the spectral constraint of b. According to the mapping relationship from Z_mq to Z_m × Z_q, the spectral constraint of b is . The difference function of Ω is

The following four cases are to be considered.

Case 1 : when τ₁ = 0, τ₂ = 0, (17) can be calculated as

Case 2: when τ₁ = 0, τ₂ ≠ 0, (17) can be recalculated as

Case 3: when τ₁ ≠ 0, τ₂ = 0, (17) can be written as

since C is an (N, n, σ) cyclic difference set, (20) can be calculated as follows.

Case 4: when τ₁ ≠ 0, τ₂ ≠ 0, (17) can be written as

the reasoning is similar to that of Case 3. (22) can be calculated as follows.

In conclusion, we have

the spectral constraint Ω of b isan (m, q, nq, nq, σq) DDS relative to Q = {0, m, 2m,···, (q – 1)m}.

Combining the above two situations, it can be conclude that b is an ideal PZCZ-SCS.

Example 2 : Let x_t1 = [–1,0,0,1,0,1,1] and y_t1 = [–1, –1, –1] where the spectral constraint of x_t1 is as (7,3,1) cyclic difference set. According to Construction 1, M is written ast. (7,3,1)

Then we obtain sequence B = [–1,0,0,–1,0,–1,–1,1,0,0,–1,0,1,–1,1,0,0,–1,0,–1,1,10,0,1,0,– 1,– 1] with the spectral constraint Ω = {1,2,4,8,9,11,15,16,18,22,23,25}. By calculating (12), we obtain the time domain sequence b. The correlation property of B and b is illustrated in Figure 1.

Figure. 1

Auto-correlation function of sequence.

Clearly, B is optimal and the time domain dual is constant magnitude. The auto-correlation function value of b is . In summary, b is an ideal PZCZ-SCS.

5.

SUMMARY

SCSs with good correlation properties have significant application value in CR-MIMO systems [16]. In order to expand the existing space of SCSs, the concept of PZCZ-SCS is introduced in this paper. Furthermore, based on the time-frequency synthesis analysis, the condition for spectral constraint of PZCZ-SCS is derived. Considering the practical applications, we present the definition of ideal PZCZ-SCS. Finally, the construction of ideal PZCZ-SCS is proposed based on matrix operation. The study in this paper provides a new research idea for the study of SCSs, and lays a theoretical foundation for the subsequent study of PZCZ-SCSs and PZCZ-SCS sets with periodic zero correlation zone.