## 1 Introduction  
In recent years, as a state estimation tool, a Kalman filter (KF) has been successfully applied in the target tracking system. However, in practice, the Gaussian assumption of KF is an over-simplification. The unknown maneuverings or outliers will produce non-Gaussian noises. To estimate systems with non-Gaussian noises, Huber-based Gaussian filters (HGFs) have been proposed. This kind of filter obtains the optimal solution by minimizing a Huber cost function, such as a Huber-based divided difference filter [1], Huber-based unscented Kalman filter [2], Huber-based cubature Kalman filter [3]. However, the influence function does not redescend, which may affect the accuracy of HGFs. The maximum correntropy criterion based Gaussian filters (MCCGFs) are also common methods for non-Gaussian noise [4; 5; 6]. However, the development of estimation error covariance matrix of MCCGFs lacks a theoretical basis, and the estimation accuracy may be degraded [7]. Particle filter (PF) can also deal with non-Gaussian noise and the estimation accuracy is higher than the filters in a GF framework [8; 9]. However, the particle degeneracy problem and the high computational cost restrict the application of PF in practical engineering. Moreover, the shadowing filter is a novel method to estimate the nonlinear system, in recent years [10; 11; 12], and it needs to obtain all the observations simultaneously to estimate the trajectory. The random time-delayed measurement may affect the accuracy of the shadowing filter. The Student's t distribution (STD) is a generalized Gaussian distribution with a heavier tail. As a result, it is more suitable to describe heavy-tailed noises than Gaussian distribution. Many works about Student's t based filters (STFs) are proposed [13; 14; 15] as well. The STD can only describe the distribution of noises. To describe the time-correlation of noises, two approaches have been proposed. One assumes that the noise is Gaussian and time-correlated noise with known covariance, and augments the original state with the noise to adapt the standard KF [16]. In Reference [17], the algorithm has been improved to reduce the computational complexity. The other assumes that the noise is a linear recursive model, and uses a measurement differencing approach to eliminate the time-correlated part of the noise [18; 19]. In addition, a randomly delayed measurement may occur, when the information is transferred to the system in practical application. To solve the randomly delayed problem, [PERSON] [20] proposed an unscented filter for two-step randomly delayed measurement. [PERSON] [21] proposed a filter based on one-step delayed measurement, which expanded the state vector with measurement noises to contain the correlation information between the noise and the one-step delayed measurement state. Based on Gaussian mixture approximation, [PERSON] [22] designed a filter to deal with the d-step state-delay problem. By state augmentation and projection theory, an optimal filter was designed for systems with one-step delayed measurement [23]. However, all the algorithms above mentioned are based on GF, and cannot handle the heavy-tailed noises. Summarizing the above discussions, the objective of the work was to estimate the states of the systems with one-step delayed measurement and time-correlated heavy-tailed noises accurately. Firstly, we used the linear recursive model and STD to construct the measurement noise, and the Bernoulli random variables to describe the one-step delayed measurement. Secondly, based on the measurement differencing approach, the time-correlated part of noises was eliminated. Then, the state space was expanded, and the measurement noise was regarded as a new state. Finally, the unscented transform (UT) was applied to compute the Student's t weighted integral. The rest is organized as follows. The preliminary and the problem statement are introduced in Section 2, and the main result of this paper is listed in Section 3. In Section 4, ISTUF is designed.