Aperiodic Total Squared Ambiguity Function: Theoretical Bounds for Binary Sequence Sets and Optimal Constructions

WEI Wenbo; SHEN Bingsheng; YANG Yang; ZHOU Zhengchun

doi:10.11999/JEIT251327

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WEI Wenbo, SHEN Bingsheng, YANG Yang, ZHOU Zhengchun. Aperiodic Total Squared Ambiguity Function: Theoretical Bounds for Binary Sequence Sets and Optimal Constructions[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251327

Citation:

WEI Wenbo, SHEN Bingsheng, YANG Yang, ZHOU Zhengchun. Aperiodic Total Squared Ambiguity Function: Theoretical Bounds for Binary Sequence Sets and Optimal Constructions[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251327

Citation:

WEI Wenbo, SHEN Bingsheng, YANG Yang, ZHOU Zhengchun. Aperiodic Total Squared Ambiguity Function: Theoretical Bounds for Binary Sequence Sets and Optimal Constructions[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251327

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Aperiodic Total Squared Ambiguity Function: Theoretical Bounds for Binary Sequence Sets and Optimal Constructions

doi: 10.11999/JEIT251327 cstr: 32379.14.JEIT251327

1.
School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China
2.
School of Information Science and Technology, Southwest Jiaotong University, Chengdu 611756, China

Funds: The National Natural Science Foundation of China (12401695), Sichuan Province Natural Science Foundation Project (2026NSFSC0776, 2025ZNSFSC0872), Central Government Guides Local Science and Technology Development Project in Sichuan Province (2025ZYD0151), Sichuan Provincial Science and Technology Program Project (2024NSFTD0015)

Received Date: 2025-12-16
Accepted Date: 2026-02-11
Rev Recd Date: 2026-02-05

Available Online: 2026-03-01

Abstract

Abstract

Objective In direct-sequence code division multiple access systems, the performance of spreading sequence sets is commonly evaluated using the total squared correlation metric. Traditional metrics such as total squared correlation and aperiodic total squared correlation are applicable only to synchronous communication systems and asynchronous systems with time shifts only, respectively. In modern high-speed mobile and satellite communications, the Doppler effect becomes significant. It causes both time and Doppler shifts in the received signal and leads to severe signal distortion. In communication scenarios that consider only time shift, the one-dimensional correlation function is typically used to measure system interference. However, in high-speed mobile environments the Doppler effect appears during signal transmission. Both time shift and Doppler shift of the sequence must therefore be considered simultaneously. In such cases, the two-dimensional ambiguity function should replace the one-dimensional correlation function. To mitigate Doppler effects, recent studies have focused on the design of Doppler-resilient sequences for mobile channels. Existing work mainly studies theoretical bounds of the ambiguity function, particularly the maximum ambiguity magnitude. Sequence sets are then constructed to achieve or asymptotically approach these bounds. This study instead examines the overall ambiguity function performance of binary sequence sets in asynchronous communication, namely the Aperiodic Total Squared Ambiguity Function (ATSAF). The objectives are as follows. First, the theoretical lower bound for the ATSAF of binary sequence sets is derived. Second, several classes of optimal binary sequence sets that achieve this bound are constructed based on the derived ATSAF bound. Methods The aperiodic time-phase cycling extension matrix $ {\boldsymbol{S}}_{a} $ is defined for a binary sequence set $ \boldsymbol{S} $ consisting of $ K $ sequences of length $ L $ to account for both time shifts and Doppler shifts. This definition converts the computation of the ATSAF for the sequence set $ \boldsymbol{S} $ into the calculation of the total squared correlation of the matrix $ {\boldsymbol{S}}_{a} $. The theoretical lower bounds for the ATSAF of the binary sequence set $ \boldsymbol{S} $ are then derived for different combinations of the set size $ K $, sequence length $ L $, and Doppler shift $ V $. To design binary sequence sets that achieve these ATSAF lower bounds, it is first proven that binary aperiodic complementary sets form ATSAF-optimal binary sequence sets. Furthermore, two additional classes of optimal binary sequence sets are constructed using Hadamard matrices and specific sequences. These sets are proven to achieve the theoretical ATSAF lower bound. Results and Discussions Existing studies mainly examine the maximum ambiguity magnitude of sequence sets, whereas this study analyzes the overall ambiguity function performance. The one-dimensional aperiodic total squared correlation analysis for asynchronous communication with delay only, studied by Ganapathy et al., is extended to the two-dimensional ATSAF, which considers both time delay and Doppler shift. First, the aperiodic time-phase cycling extension matrix $ {\boldsymbol{S}}_{a} $ is defined for a binary sequence set $ \boldsymbol{S} $ (Definition 3). The theoretical lower bounds for the ATSAF of the binary sequence set $ \boldsymbol{S} $ are then derived for different parameters, including set size $ K $, sequence length $ L $, and Doppler shift $ V $ (Theorem 1). When the Doppler shift $ V=1 $, the derived ATSAF bound reduces to the aperiodic total squared correlation bound. Binary sequence sets that achieve these ATSAF bounds maintain the overall cross-interference energy in the two-dimensional delay-Doppler domain at its theoretical minimum. To construct such sequence sets, it is first proven that binary aperiodic complementary sets are ATSAF-optimal binary sequence sets (Theorem 2). Furthermore, two further classes of ATSAF-optimal binary sequence sets are constructed using Hadamard matrices and specific sequences (Theorems 3 and 4). Finally, an example demonstrates that the sequence set constructed in Theorem 4 is ATSAF-optimal (Example 1). Conclusions In high-speed mobile communication scenarios, Doppler effects cause distortion in received signals. By defining the aperiodic time-phase cycling extension matrix $ {\boldsymbol{S}}_{a} $ for a binary sequence set $ \boldsymbol{S} $, the theoretical lower bound for the ATSAF is derived. This bound specifies the minimum theoretical value of the total energy of the binary sequence set S in the two-dimensional delay-Doppler domain. When Doppler shifts are not considered, the derived ATSAF bound reduces to the aperiodic total squared correlation bound. Furthermore, three classes of ATSAF-optimal binary sequence sets that achieve this theoretical bound are constructed using binary aperiodic complementary sets, Hadamard matrices, and specific sequences. These sequence sets maintain the overall cross-interference energy at the theoretical minimum in the two-dimensional delay-Doppler domain.
- Aperiodic total squared ambiguity function,
- Ambiguity function,
- Aperiodic total squared correlation

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References(25)

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