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Tracking a ballistic re-entry target from radar observations is a highly complex problem in nonlinear filtering. The paper adopts a one-dimensional vertical motion model with unknown ballistic coefficient, we present a square-root quadrature Kalman filter (SRQKF) algorithm for this ballistic target tracking problem. The proposed algorithm is the square-root implementation of the quadrature Kalman filter (QKF). The quadrature Kalman filter is a recursive, nonlinear filtering algorithm developed in the Kalman filtering framework and computes the mean and covariance of all conditional densities using the Gauss-Hermite quadrature rule. The square-root quadrature Kalman filter propagates the mean and the square root of the covariance. It guarantees the symmetry and positive semi-definiteness of the covariance matrix, improved numerical stability and the numerical accuracy, but at the expense of increased computational complexity slightly.