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This technical note analyzes the estimation delay in a high gain observer, where the state estimates may lag behind the actual states due to the observer's non-zero phase response. The technical note proves that, for a slowly time-varying system subject to bounded noises, the estimation delay depends on the observer gain, but is independent of the variations of system parameters. Rather than estimating the delay, a novel method is proposed to calculate the delay from the observer's phase response. In terms of system identification, the delay is compensated by aligning other measurements with the lagged estimate so that they have the same lag. The simulation results of an aero engine model show significant improvements in estimation. On one hand, the proposed approach improves the estimation accuracy, and on the other hand, it removes the assumption of zero delay and gives a new insight into the high-gain observer design.