Rethinking Ergodic LMMSE of Random ISAC Signals

This paper studies the sensing performance of a monostatic integrated sensing and communication (ISAC) system using random signals. We quantify sensing accuracy via the ergodic linear minimum mean squared error (ELMMSE), which represents the average estimation error for the target impulse response (TIR). Two scenarios are analyzed based on the covariance matrix structure of the TIR: one with a covariance matrix diagonalizable by the discrete Fourier transform (DFT), allowing exact closed-form ELMMSE computation; and another with a general positive definite Hermitian covariance matrix, requiring analytical bounds. In the general scenario, we propose a novel bound, the normalized approximation error (NAE), which outperforms Jensen’s lower bound (JLB). Additional upper and lower bounds based on eigenvalue inequalities are also derived. Closed-form expressions of these bounds for Gaussian signals are derived. The performance of NAE and JLB is further evaluated under Gaussian, phase-shift keying (PSK), and quadrature amplitude modulation (QAM) constellations. Numerical simulations validate our analytical findings, and demonstrate that NAE improves estimation accuracy by approximately 15% for Gaussian signals and 5% for 16-QAM constellation compared to JLB, highlighting the effectiveness of the proposed bounds in characterizing the sensing performance of random ISAC signals.