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For a high-order polynomial-phase signal (PPS), instantaneous frequency rate (IFR), which is defined as the second derivative of the phase, is estimated by using an estimator with only a second-order nonlinearity. Compared to high-order phase function (HPF), the proposed IFR estimator presents improved performance including smaller mean-squared error (MSE) and lower SNR threshold. Statistical analysis via a multivariate first-order perturbation analysis is derived for the estimate bias and MSE. Numerical results verify our analytical results.