In analyzing non-stationary noisy signals with time-varying frequency content, it's convenient to use distribution methods in joint, time and frequency, domains. Besides different adaptive data-driven time-frequency (TF) representations, the approach with multiple orthogonal and optimally concentrated Hermite window functions is an effective solution to achieve a good trade-off between low variance and minimized stable bias estimates. In this paper, we propose a novel spectrogram method with multiple optimally parameterized Hermite window functions, with parameterization which includes a pair of free parameters to regulate the shape of the window functions. The computation is performed in the optimization process to minimize the variable projection (VP) functional problem. The proposed parametrized distribution method improves TF concentration and instantaneous frequency (IF) estimation accuracy, as shown in experimental results for synthetic signals and real-life ship motion response signals. With the optimization of nonlinear least-squares approximation of the ship response signals, the Hermite spectra are centralized, and only up to 15 basis functions are sufficient for concentration improvement in the TF domain
