Stability and performance of classical digital phase-locked loop in discrete-time settings

Many classical phase-locked loops (PLLs) are treated as linear continuous-time systems in the locked condition. While this is useful for sinusoidal input and output signals, the PLLs are more accurately modeled in the z-domain particularly when (i) digital phase detector is used; and (ii) the inputs and outputs are in digital form such as those mostly used in data communications. This paper presents a stability and performance analysis of classical digital phase-locked loops (CDPLLs) in discrete-time domain which is based on the induced ℓ₂ norm objective. The result is formulated in the form of a linear matrix inequality (LMI) search which is computationally tractable. A simple application to a CDPLL consisting of a Butterworth filter is presented to illustrate the effectiveness of the result compared to the existing one.