The need frequently arises to design a discrete-time digital signal processing (DSP) algorithm that closely mimics a conventional continuous-time phase-locked loop (CTPLL). Similarly, it is often desirable to talk about the DSP implementation in terms of traditional quantities like damping factor and equivalent noise bandwidth. This memorandum examines this interplay between discrete and continuous time systems in the context of the classical continuous time type-2 PLL. It also provides a plethora of detailed design formulas for implementing CTPLL designs with finite-difference equations for digital implementation.
For readers who prefer to skip the exhaustive material and jump right to the preferred re-design method, go to Section 7 which envelops the bilinear transform method for redesign directly.
| 2 April 2005
Introduction to Digital Re-Design of Analog Phase-Locked Loops
Discrete-time and continuous-time phase-locked loop (DTPLL and CTPLL) implementations are fundamentally different. The behavioral differences between these different systems becomes almost indistinguishable however, as the over-sampling rate (OSR) in the digital signal processing (DSP) is increased. The ratio of sampling rate to the PLL closed-loop unity-gain frequency will be defined in this memorandum as 
in which Fs is the sampling rate in Hz, and ωn is the PLL’s natural frequency in radians/s. Although this definition was not formally presented in , it is a very convenient definition because closed-loop unity gain frequency is precisely given by the denominator in (1) for the type-2 CTPLL, and this relationship is independent of the damping factor ζ. For a second-order, type-2 CTPLL, the damping factor and phase margin are closely related
by the approximation1
It is of course no surprise that system stability margin and the time-domain behavior are interconnected, but this is a useful approximation for the discussions that are presented later. We would otherwise lack a simple means to relate a discrete-time system’s characteristics back to the CTPLL damping factor equivalent.
The main purposes of this memorandum are:
- To provide a mathematical basis and means to convert the classical second-order type-2 CTPLL into a discrete-time system
- To investigate different metrics of equivalence between CTPLL and DTPLL designs
- To provide the mathematical framework to assist in analyzing other arbitrary designs that may be of interest.
The PLL satisfies a cornucopia of different engineering needs in today’s world, and as such it is necessary to at least attempt to capture some of these different perspectives in the discussions that follow. To that end, we will consider the extracted DTPLL designs from several different perspectives including: (i) timedomain response, (ii) frequency-domain closed-loop gain, (iii) equivalent noise bandwidth.
There are many graphical plots contained in this memorandum. This was purposely done in order to enhance readability but it does make this a very lengthy monologue. Conclusions are provided at the end of each major section where appropriate in boxed regions in order that key points are not overlooked or lost.
Continuous-Time and Discrete-Time System Important Relationships
Two equations from  ( equations 1 and 8 ) form much of the foundational basis for the technical discussions that follow. The first of these two equations makes use of the Poisson Sum formula to permit the discrete-time and continuous-time transform descriptions of a system to be translated as
In (3), the left-hand side is mathematically the z-transform (scaled by Ts) of the time-series given by hk = h( k Ts ) whereas H( f ) is the continuous-time Fourier transform of h( t ). The sampling rate is Fs with Fs= Ts-1. The second equation of interest makes it possible to use the continuous-time Fourier transform of the open-loop gain function to compute the closed-loop frequency-domain description of the sampled control system and is given by
These equations receive substantial consideration in  should the interested reader wish to follow the underlying details more rigorously. One of the most attractive features of (4) is that it permits the exact inclusion of sampling in an otherwise continuous-time system without the need of first computing the z-transforms involved. Although we will not exploit this perspective in this memorandum, but it is nonetheless worthy of special note.
Since the CTPLL and DTPLL are not precisely the same particularly for small OSR values, the notion of an “equivalent” DSP redesign of a CTPLL must receive additional definition. The question, “Equivalent how?” must be asked. Different systems will naturally require a different definition of equivalence such as equivalence in:
- Equivalent Noise Bandwidth
- Bandwidth at -3 dB
- Natural Frequency and Damping Factor
- Stability Margin (Gain Peaking)
- Impulse Response
Ultimately, we must decide how the Laplace transform operator s and the z-transform operator z are to be related in our redesign methodology. Equivalently, we must decide how differentiation and integration in the time domain will be handled between the continuous-time and discrete-time systems. A glimpse of this issue is offered in Figure 1 where the forward and backward Euler integration methods are compared for the continuous time domain systems which has an impulse response given as
Comparison Between Forward and Backward Euler Integration Methods
for t ≥ 0. The resulting responses are substantially different even for appreciable OSR values as evidenced here. More information is provided in the next section concerning such integration methods.
Finally, a word about system stability is in order. As is true with any numerical simulation work, stability is crucial for achieving any meaningful results. In the context of the material presented in this memorandum, two different forms of stability must be considered. First of all, the underlying numerical integration formulas that are adopted must be in a parameter range where they are themselves stable as discussed in Section 2. Secondly, the design of the DTPLL must itself be stable. The first stability type is easily achieved by using an adequate OSR parameter whereas the second depends upon proper pole-zero placement for the DTPLL design itself. Stability issues need to be considered in context since there are different types of stability that must be considered.
- “Equivalence” between CTPLL and DTPLL is a subjective term that must be precisely quantified.
- Overall stability requires both (i) stability of the underlying numerical integration model and (ii) the system barring any imperfections in the numerical integration.
s- and z-Operators in Terms of Integration Formulas
It is very convenient to think about the Laplace transform s as a mathematical operator that is to be replaced by a discrete time operator in terms of z which is the unit-time element of z-transform theory. It is desirable that the adopted discrete time operator be both simple and accurate. The accuracy issue is particularly important if the OSR parameter is small.
In this section, we will introduce 4 different discrete-time methods for approximating the continuous-time Laplace transform operator s. These methods will subsequently be used to redesign the CTPLL into a discrete-time system implementation. A fifth technique is developed at the beginning of Section 3 which is based on an impulse-invariant perspective.
All of the integration formulas that we will consider in this memorandum are documented in numerical method textbooks. Two references that provide extensive treatment of this subject are  and .
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