In cognitive radar systems (CRSs), target scattering coefficients (TSC) can be utilized to improve the performance of target identification and classification. This work considers the problem of TSC estimation for temporally correlated target. Multiple receive antennas are adopted to receive the echo waveforms, which are interfered by the signal-dependent clutter. Unlike existing estimation methods in time domain, a novel estimation method based on Kalman filtering (KF) is proposed in frequency domain to exploit the temporal TSC correlation, and reduce the complexity of subsequent waveform optimization. Additionally, to minimize the mean square error of estimated TSC at each KF iteration, in contrary to existing works, we directly model the design process as an optimization problem, which is non-convex and cannot be solved efficiently. Therefore, we propose a novel method, similar in some way to semi-definite programming (SDP), to convert the non-convex problem into a convex one. Simulation results demonstrate that the estimation performance can be significantly improved by the KF estimation with optimized waveform.