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This allows the synthesis of frequencies that are N/M times the reference frequency. This can be accomplished in a different manner by periodically changing the integer value of an integer-N frequency divider, effectively resulting in a multiplier with both whole number and fractional component. Such a multiplier is called a fractional-N ...
A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is fixed relative to the phase of an input signal. Keeping the input and output phase in lockstep also implies keeping the input and output frequencies the same, thus a phase-locked loop can also track an input frequency.
Thus it will produce an output of 100 kHz for a count of 1, 200 kHz for a count of 2, 1 MHz for a count of 10 and so on. Note that only whole multiples of the reference frequency can be obtained with the simplest integer N dividers. Fractional N dividers are readily available. [20]
The lower, yellow trace is the N counter output whose frequency corresponds to the channel spacing frequency of 30 kHz. The green trace is the output from the dual-modulus prescaler, which happens to correspond to 7.1714 MHz in the case that the prescaler is at 128 and 7.1158 when it is at 129.
A fractional-n frequency synthesizer can be constructed using two integer dividers, a divide-by-N, and a divide-by-(N + 1) frequency divider. With a modulus controller, N is toggled between the two values so that the VCO alternates between one locked frequency and the other. The VCO stabilizes at a frequency that is the time average of the two ...
A first linear mathematical model of second-order CP-PLL was suggested by F. Gardner in 1980. [2] A nonlinear model without the VCO overload was suggested by M. van Paemel in 1994 [3] and then refined by N. Kuznetsov et al. in 2019. [4] The closed form mathematical model of CP-PLL taking into account the VCO overload is derived in. [5]
In the classic books on phase-locked loops, [1] [2] published in 1966, such concepts as hold-in, pull-in, lock-in, and other frequency ranges for which PLL can achieve lock, were introduced. They are widely used nowadays (see, e.g. contemporary engineering literature [ 3 ] [ 4 ] and other publications).
Given a fractional cover, in which each set S i has weight w i, choose randomly the value of each 0–1 indicator variable x i to be 1 with probability w i × (ln n +1), and 0 otherwise. Then any element e j has probability less than 1/( e × n ) of remaining uncovered, so with constant probability all elements are covered.