Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 31

Vilà-Valls et al.
the discrete-time closed-loop transfer function for the second- and
third- order PLLs are

H 2 ( z) =

H3 ( z) =

(α1 + α 2 ) z − α1

( z − 1) 2 + α1 ( z − 1) + α 2 z

,

α2

( z − 1)3 + α1 ( z − 1) 2 + α 2 z ( z − 1) + α 3 z 2

(10)

,

(11)

where the loop filter coefficients can be directly computed from
the analog loop filter parameters, the desired noise bandwidth and
the sampling period as
C

second order: α1 = aωnTs and α2 ωn2Ts2.

C

third order: α1 = cωnTs, α2 = bωn2Ts2 and α3 = ωn3Ts3.

For the digital PLL, the equivalent noise bandwidth can be directly
computed from H(z) (results given in hertz) [15], leading to
Bn ,2 =

2α12 + 2α 2 + α1α 2
,
2Tsα1 ( 4 − 2α1 − α 2 )

(12)

Bn ,3 =

γ 3,1
,
2Tsγ 3,2γ 3,3

(13)

with γ3,1 = 4α12α 2 − 4α1α3 + 4α 22 + 2α1α 22 + 4α12α 3 + 4α2α3 + 3α1α2α3
+ α 32 + α1α 32, γ3,2 = α1α2 − α3 + α1α3 and γ3,3 = 8 − 4α1 − 2α2 − α3.
These results verify that digital PLL parameters computed
from the analog coefficients are equivalent to the parameters directly derived in the discrete-time domain, because the resulting
equivalent noise bandwidths coincide.

STANDARD AND ADVANCED PLLS
A key step for the practitioner may be how to interpret (8) or (9)
and the way to turn them into a useful architecture. The block diagram of a third-order PLL loop filter is sketched in Fig. 2. The
standard PLL-based architectures are somehow limited because
of the noise reduction versus dynamic range trade-off, which may
lead the filters to lose lock.
This trade-off is mainly driven by the bandwidth and order of
the loop. A small bandwidth is needed to filter out as much noise
as possible to be able to operate at low SNℜ, whereas a large one
is required for coping with fast variations of the parameters of
interest. Moreover, the loop's order also plays an important role
in such scenarios. For instance, the second-order PLL is unconditionally stable at all noise bandwidths, but it is not suitable to
deal with complex dynamics. The third-order PLL, while being
more flexible in front of high dynamics, only remains stable for
bandwidths below 18 Hz [3]. Another issue is the PLL constant
bandwidth, a priori fixed by the designer. A time-varying bandJULY 2017, Part II of II

Figure 2.

Block diagram of a third-order digital PLL loop filter.

width would seem to be more suitable in practice. These two key
points have led to propose a plethora of advanced PLL-based
techniques.
One possible solution to provide robustness and extra flexibility to the stand-alone PLLs is to consider cooperative loops
architectures, where several loops interact to counteract its individual limitations. The most basic solution is the so-called
switching architecture, where a PLL is used under nominal operation but the system switches to a FLL in harsh conditions to
not lose lock [46]. The FLL is, in general, more robust than the
PLL because the variability of the incoming signal frequency
is orders of magnitude lower than the phase variability. However, the solution usually adopted to overcome the problems of
standard architectures in dynamic or harsh conditions is the use
of a hybrid approach in which the FLL permanently assists the
PLL (F-PLL) [13], [47], which is capable to maintain lock in
situations in which the PLL diverges. The second concern that
typically arises from standard architectures is the constant bandwidth operation, which may limit its applicability to rather constant propagation conditions. A possible solution is to directly
use the input working conditions to automatically adjust the loop
bandwidth, what is usually known as adaptive bandwidth PLL
(A-PLL) [48]. Several contributions appeared in the literature
using the same concept [15], [49], [50].
The following section presents a systematic, unified approach
to design digital phase tracking filters that is based on the technically sound Bayesian filtering theory.

STANDARD KF-BASED CARRIER TRACKING
OPTIMAL FILTERING BACKGROUND AND KF GENERAL
FORMULATION
The optimal filtering problem involves the recursive (i.e., online)
estimation of time-varying unknown states of a system by using
the incoming flow of information (observations) from the system,
along some prior statistical knowledge about the variations of such
states. The general dynamic state-space model (assuming additive
noises) can be expressed as
x k = f k −1 ( x k −1 ) + νk  ,

(14)

y k = hk ( xk ) + nk ,

(15)

IEEE A&E SYSTEMS MAGAZINE

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