This realization of the classic phase-lock concept can be generalized to the situation where the phase is also allowed to vary over time. In this work, we continue using a discrete-time representation for the phase, and we consider feedback that only operates on the estimate. In this case, the output of the loop can be expressed as:
This equation reveals the feedback property by establishing the relationship between the output error,, and filter states,. (It is easy to verify in this case that. The feedback loop gains are used to control the loop bandwidth, and a loop filter is used to provide the output with low pass filtering while reducing the phase noise. In essence, the loop is generating an output signal deliberately synchronized to the input signal after the outputs of successive samples have been averaged through the filter.
Capacity must therefore be given to the PLL to correct for this time-varying noise. The most common approach is to assume that the filter is ideal, and at a rate of, for example, a Nyquist rate. In that case, the probability density function for the estimate has an independent phase noise term given by Equation 1 , where is the amplitude noise, and is the phase noise. The Gaussian error function for the estimate may be approximated as a Gaussian distribution, so becomes , where ’s are the noise-sampled outputs of the PLL that are temporally averaged (i.e., the outputs of a discrete time-average filter (DTAF)).
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