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Modern implementations of Discrete-Time Phase-Locked Loops (DT-PLLs) often contain delayed feedback. The delays are usually a side effect to pipelining, filtering, or other inner-loop mechanisms. Each delay increases the order of the system by introducing an additional pole to the closed-loop transfer function, and in many cases, makes the traditional type- 2 loop equations obsolete. This paper describes how the second-order notions of damping and natural frequency can be applied to type-2 DT-PLLs in the presence of any number of delays. It provides equations for loop parameters that will provide a desired transient behavior based on damping and natural frequency, along with a test to ensure the accuracy of the results.
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