Abstract

The need frequently arises to design a discretetime digital signal processing (DSP) algorithm that closely mimics a conventional continuoustime phaselocked 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 type2 PLL. It also provides a plethora of detailed design formulas for implementing CTPLL designs with finitedifference equations for digital implementation.
For readers who prefer to skip the exhaustive material and jump right to the preferred redesign method, go to Section 7 which envelops the bilinear transform method for redesign directly.

Date:

2 April 2005 
Author:

J.A. Crawford 
Ref. No.

U11700 
Introduction to Digital ReDesign of Analog PhaseLocked Loops
Discretetime and continuoustime phaselocked loop (DTPLL and CTPLL) implementations are fundamentally different. The behavioral differences between these different systems becomes almost indistinguishable however, as the oversampling rate (OSR) in the digital signal processing (DSP) is increased. The ratio of sampling rate to the PLL closedloop unitygain frequency will be defined in this memorandum as [1]
in which F_{s} is the sampling rate in Hz, and ω_{n} is the PLL’s natural frequency in radians/s. Although this definition was not formally presented in [1], it is a very convenient definition because closedloop unity gain frequency is precisely given by the denominator in (1) for the type2 CTPLL, and this relationship is independent of the damping factor ζ. For a secondorder, type2 CTPLL, the damping factor and phase margin are closely related
by the approximation^{1}
It is of course no surprise that system stability margin and the timedomain 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 discretetime 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 secondorder type2 CTPLL into a discretetime 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) frequencydomain closedloop 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.
ContinuousTime and DiscreteTime System Important Relationships
Two equations from [1] ( 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 discretetime and continuoustime transform descriptions of a system to be translated as
In (3), the lefthand side is mathematically the ztransform (scaled by T_{s}) of the timeseries given by h_{k} = h( k T_{s} ) whereas H( f ) is the continuoustime Fourier transform of h( t ). The sampling rate is F_{s} with F_{s}= T_{s}^{1}. The second equation of interest makes it possible to use the continuoustime Fourier transform of the openloop gain function to compute the closedloop frequencydomain description of the sampled control system and is given by
These equations receive substantial consideration in [2] 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 continuoustime system without the need of first computing the ztransforms 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 ztransform 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 continuoustime and discretetime 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 polezero 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.
Key Points
 “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 zOperators 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 unittime element of ztransform 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 discretetime methods for approximating the continuoustime Laplace transform operator s. These methods will subsequently be used to redesign the CTPLL into a discretetime system implementation. A fifth technique is developed at the beginning of Section 3 which is based on an impulseinvariant 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 [2] and [3].
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