Homotopy Manifold Optimization Algorithm for Extracting Multiplicative Phase Modulation Signals

Phase modulation jamming (PMJ) alters the characteristics of radar echo signals through multiplicative time-varying phase modulation, thereby degrading the performance of conventional radar signal processing. Recovering the multiplicative phase modulation sequence from radar echoes is of critical importance both for understanding the modulation characteristics and suppressing the interference. To address this issue, a manifold optimization algorithm incorporating a homotopy strategy is proposed to reconstruct the multiplicative phase modulation signal from PMJ-affected radar echoes. Initially, a deconvolution model is constructed for 1-bit and 2-bit PMJ, wherein total variation regularization and discrete phase constraints are integrated within a constant modulus framework. Subsequently, the complex combinatorial optimization problem is transformed into a continuous optimization task through alternating minimization and manifold optimization on the complex circle. Moreover, the homotopy strategy is progressively employed to enforce discrete constraints, thus avoiding entrapment in local minima. Finally, simulation tests based on linear frequency-modulated (LFM) signals demonstrate that the proposed algorithm outperforms MATLAB’s fmincon solver in terms of root-mean-square error (RMSE) and efficiency. Additionally, the 2-bit model is capable of extracting 1-bit sequences, thereby demonstrating the versatility of the proposed algorithm across different modulation bit configurations.