Ambiguity Function Shaping for ISAC Using DFT-s-OFDM Signals

Abstract

This paper addresses the challenge of waveform synthesis in Integrated Sensing and Communication (ISAC) systems utilizing Discrete Fourier Transform spread Orthogonal Frequency Division Multiplexing (DFT-s-OFDM). A DFT-s-OFDM sensing signal is generated from a length-<inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> modulation sequence and a size-<inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> Inverse DFT (IDFT). In practical networks, the receiver utilizes a sampling rate where <inline-formula> <tex-math notation="LaTeX">${N}\gt {M}$ </tex-math></inline-formula>, resulting in an oversampled signal that alters the correlation properties of the underlying modulation sequence. Because the shape of the Ambiguity Function (AF) is critical for sensing performance, we first derive a closed-form analytical expression for the discrete-time Periodic AF (PAF) of the oversampled DFT-s-OFDM signal as a function of the modulation sequence’s PAF. We derive oversampled PAFs for widely used modulation sequences, including Generalized Chirp-Like (GCL) and Frank sequences. To shape the resulting PAF, we introduce the concept of transmitter-side sequence precoding and propose a Least-Squares (LS) precoder that minimizes the mean squared error between the actual and a target ideal signal. This is extended to a Weighted LS (WLS) precoder paired with a greedy iterative weight-selection algorithm, which we prove converges and reduces PAF sidelobes by several dB. Furthermore, we formulate a non-linear least-squares (NLS) optimization problem to minimize the error between the actual and a desired target PAF, solved via gradient descent method. Finally, we derive non-iterative, computationally efficient precoding methods leveraging singular value decomposition (SVD) and lower-upper (LU) decomposition of the transmitter matrix.