Ambiguity Function Analysis of OFDM Signals With Pilots and Data Payloads

Abstract

Practical orthogonal frequency division multiplexing (OFDM) communication frames contain both deterministic pilots and random data payloads, motivating the joint ambiguity function (AF) analysis of the two components when the entire frame is reused for integrated sensing and communication (ISAC). This paper characterizes two discrete AF formulations for different Doppler regimes, namely the discrete periodic AF (DP-AF) and fast-slow-time AF (FST-AF), and derives closed-form expressions for their expected squared values. For the FST-AF, the expected sidelobe level (ESL) is uniform over the delay-Doppler plane and depends only on the pilot count, constellation kurtosis and total number of time-frequency resources, but not on the pilot symbols or pattern. For the DP-AF, we establish attainable lower and upper ESL bounds and show that no pilot design can minimize all sidelobes simultaneously. We further prove that attaining the lower bound at non-zero Doppler requires a periodic pilot pattern, while equally spaced chirp pilots, including Zadoff-Chu (ZC) sequences, maximize the numbers of sidelobes attaining the lower and upper bounds simultaneously. Two representative ZC pilot patterns widely encountered in communication frames are then examined: contiguous placement produces delay-Doppler ridges described by squared Dirichlet kernels, whereas equally spaced placement generates periodic peak-and-notch structures. Both regular patterns exhibit pronounced high sidelobes, suggesting that communication-oriented pilot patterns should be re-designed for delay-Doppler estimation in the context of ISAC. Numerical results validate the analysis and show that irregular pilot placement can suppress high sidelobes and improve target estimation performance.