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 typically evaluated using the total squared correlation metric. Traditional metrics such as total squared correlation and aperiodic total squared correlation are only applicable to synchronous communication systems and asynchronous systems with time shifts only, respectively. However, in modern high-speed mobile and satellite communications, the Doppler effect becomes significant, causing both time and Doppler shifts in the received signal and consequently leading to severe signal distortion. In communication scenarios considering only time shift, the one-dimensional correlation function is typically employed to measure interference within the system. However, in high-speed mobile environments, the Doppler effect is introduced during signal transmission, necessitating the simultaneous consideration of both time shift and Doppler shift of the sequence. In such cases, the two-dimensional ambiguity function should be used in place of the one-dimensional correlation function. To mitigate Doppler effects, the research community has increasingly focused on designing Doppler-resilient sequences to address the Doppler effects present in various mobile channels. Existing studies are primarily concentrated on the theoretical bounds of the ambiguity function, namely the maximum ambiguity magnitude, with sequence sets subsequently constructed that achieve or asymptotically achieve these bounds. This research, however, focuses on the overall ambiguity function performance of binary sequence sets in asynchronous communication, namely the ATSAF. The specific objectives are as follows:1. The theoretical lower bound for the ATSAF of binary sequence sets is derived.2. Based on the derived ATSAF lower bound, several classes of optimal binary sequence sets that achieve this theoretical bound are designed. 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 $, in order to account for both time shifts and Doppler shifts. This definition transforms the problem of computing the ATSAF for the set $ \boldsymbol{S} $ into that of calculating the total squared correlation of the matrix $ {\boldsymbol{S}}_{a} $. Subsequently, the theoretical lower bounds for the ATSAF of the binary sequence set $ \boldsymbol{S} $ are derived for different combinations of the set size $ K $, sequence length $ L $, and Doppler shift $ V $. To design binary sequence sets that achieve these derived ATSAF lower bounds, it is first proven that binary aperiodic complementary sets constitute optimal binary sequence sets with respect to the ATSAF. Furthermore, based on Hadamard matrices and specific sequences, two additional classes of optimal binary sequence sets are designed, which are shown to achieve the theoretical ATSAF lower bound. Results and Discussions Existing research primarily focuses on the maximum ambiguity magnitude of sequence sets, while this study emphasizes the overall ambiguity function performance. The one-dimensional aperiodic total squared correlation analysis for asynchronous communication with delay only, as investigated by Ganapathy et al., is extended in this work to the two-dimensional aperiodic total squared ambiguity function, which incorporates both time delay and Doppler shift. This paper first defines the aperiodic time-phase cycling extension matrix $ {\boldsymbol{S}}_{a} $ for a binary sequence set $ \boldsymbol{S} $ (Definition 3). Subsequently, the theoretical lower bounds for the ATSAF of the binary sequence set $ \boldsymbol{S} $ are derived for various parameters, including the set size$ K $, sequence length $ L $, and Doppler shift $ V $ (Theorem 1). When the Doppler shift $ V=1 $, the ATSAF theoretical bound derived in this paper reduces to the aperiodic total squared correlation theoretical bound. Binary sequence sets that achieve these ATSAF lower bounds maintain the overall cross interference energy in the two-dimensional delay-Doppler domain at its theoretical minimum. To design binary sequence sets that achieve these derived ATSAF bounds, it is first proven that binary aperiodic complementary sets are ATSAF-optimal binary sequence sets (Theorem 2). Furthermore, based on Hadamard matrices and specific sequences, two additional classes of ATSAF-optimal binary sequence sets are designed (Theorems 3 and 4). Finally, an example is provided in this paper to demonstrate that the sequence set constructed in Theorem 4 is an ATSAF-optimal binary sequence set (Example 1). Conclusions In high-speed mobile communication scenarios, Doppler effects lead to distortion in the received signal. Therefore, 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, which specifies the minimum theoretical value for 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. This study not only provides the theoretical ATSAF bound for binary sequence sets in the two-dimensional delay-Doppler domain but also designs several classes of optimal binary sequence sets that achieve this bound. These sets achieve the theoretical minimum for overall cross interference energy 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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