Method to Compute Pointing Displacement, Smear, and Jitter Covariances for Optical Payloads

This paper presents a method to assess the pointing and image motion performance of optical payloads in the presence of image displacement (shift), smear, and jitter. The method assumes the motion is a stationary random process over an image exposure interval. Displacement, smear, and jitter covariances are computed from the solution to a Lyapunov differential equation. These covariances parameterize statistical image motion modulation transfer functions (MTFs), and they can be used to verify pointing and image motion MTF requirements. The method in the present paper extends a previous method to include smear, as well as displacement, and hence jitter. The approach in the present paper also leads, as a special case, to a more efficient method to compute the displacement covariance than the previous method. Numerical examples illustrate the proposed method.

💡 Research Summary

The paper introduces a systematic method for quantifying the image motion of optical payloads during an exposure interval by separating the motion into three distinct components—displacement (shift), smear, and jitter—and computing the covariance matrices associated with each. The authors begin by modeling the pointing dynamics of the bus‑payload system as a continuous‑time, linear time‑invariant (LTI) system driven by zero‑mean white noise. In state‑space form, the dynamics are expressed as (\dot{x}=Ax+Bu) and the image motion output as (p=Cx). Assuming the system has reached statistical steady‑state, the state covariance (P) satisfies the algebraic Lyapunov equation (AP+PA^{!T}+BB^{!T}=0).

Displacement and smear are defined as the time‑averaged position and the average linear trend of the image over an exposure of duration (T). By introducing a double integrator, the authors define auxiliary variables (z_{1}(t)=\int_{0}^{t}p(\alpha)d\alpha) and (z_{2}(t)=\int_{0}^{t}z_{1}(\tau)d\tau). The displacement (\bar p(T)) and smear vector (\bar s(T)=T\bar v(T)) can be written as linear combinations of (z_{1}(T)) and (z_{2}(T)). Consequently, the covariances of displacement and smear are directly related to the covariances of these auxiliary states.

To obtain these covariances, the authors extend the Lyapunov differential equation (LDE) used in earlier work. They augment the original state vector with (z_{1}) and (z_{2}) to form (\tilde{x}=