Doppler compensation technique for M-ary spread spectrum signals with low complexity and high resolution in mobile underwater acoustic communication

In underwater mobile spread spectrum communications, time-varying Doppler effects cause severe carrier phase jumps, significantly degrading spreading gain. Addressing the limitations of conventional compensation techniques under high relative velocities, this study investigates the correlation cost factor (CCF) during de-spreading in M-ary spread spectrum systems. Theoretical analysis reveals that CCF serves as an effective loss function for frequency offset compensation. Leveraging this, a high-resolution Doppler estimation method (HCCF-MSS) is proposed. HCCF-MSS employs the adaptive learning rate of the Adam optimizer to update step sizes, enabling rapid, high-precision convergence of the CCF loss function to the accurate frequency offset. To mitigate convergence to local minima in highly dynamic environments, the Multi-starting Point Parallel Gradient Search (MPGS) algorithm is introduced. MPGS initiates concurrent gradient searches from diverse starting points and selects the optimal solution by comparing CCF values across convergent paths. Numerical simulations and sea trials demonstrate that HCCF-MSS-9 (9 iterations) achieves performance comparable to CCF-MSS-0.02 (±6 Hz range, 0.02 Hz precision) with superior computational efficiency. Under accelerations up to 3 m/s2, HCCF-MSS-9+MPGS-6 (6 starting points) attains a Doppler estimation mean squared error of approximately -3.3 dB, outperforming other algorithms by 0.5-1.5 dB, demonstrating its effectiveness for precise frequency offset estimation under complex variable-motion conditions.