Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 30

A Tutorial on Kalman Filter-Based Techniques
filter, in charge of filtering out noise and driving the error to zero;
and finally, the NCO is used to generate the local replica (i.e., a
complex exponential using the predicted or tracked CP). The basic
PLL architecture is sketched in Fig. 1.
The role of the phase discriminator is to produce an output that
is proportional to the phase estimation error. In the presence of data
modulating the phase of the input signal (and thus producing phase
jumps), noncoherent discriminators (usually known as Costas-type
discriminators) should be adopted. The two quadrant arctangent
discriminator atan (I(yk)/ℜ(yk)) is usually the preferred noncoherent
choice [3], although other discriminators may result from different optimization criteria and signal-to-noise ratio (SNℜ) regimes
[41], [42]. Otherwise, if phase jumps produced by data have been
removed (either because they belong to a training sequence known
at the receiver or because they have been estimated) or pilot (dataless) signals are available, coherent discriminators may be used
[39]. In the absence of data, the optimal maximum likelihood
(ML) estimator is the four-quadrant arctangent discriminator atan2
(I(yk)/ℜ(yk)). In addition to the phase discriminators, one may be
interested in directly tracking the input signal frequency by using
a frequency-locked loop (FLL). ℜefer to [43] for detailed analysis
and an updated overview on carrier tracking techniques.
The order of the PLL refers to the overall closed loop (used
throughout the article to unify the notation) and not only to the
filter loop, which may lead to confusion because the latter is always one order lower than the entire loop due to the NCO. The
order of the PLL basically determines the input signal dynamics
that the filter is able to track. In other words, a second-order PLL is
able to track a constant frequency mismatch, whereas a third-order
PLLs properly tracks frequency drifts. The loop coefficients are
usually optimized to minimize the mean-squared error (MSE) [3],
but other approaches may be adopted as well [44].
The dynamics of a PLL are heavily dependent on the type and
response of the loop filters. A PLL with nth order filter is of (n +
1)th order. In the general case, a PLL of order n has a closed-loop
transfer function that can be expressed as [45]
(1)

H ( s ) = H PI ( s ) L( s ),

K
K
where HPI(s) = K1 + 2 + 23 + ... is the transfer function of the
s
s
proportional plus integrator part, Ki is the gain of the loop of order
i, and L(s) is the transfer function of the loop filter.

ANALOG VERSUS DIGITAL

where a, b, and c are the loop filter coefficients, and ωn (radian
per second) is the so-called natural frequency of the loop filter.
The PLL loop is closed with a voltage-controlled oscillator (VCO),
with transfer function V(s) = 1/s (unity VCO gain). The closedloop transfer functions for these second- and third-order PLLs are
H 2 (s) =

aωn s + ωn2
,
s + aωn s + ωn2

(3)

H 3 (s) =

cωn s 2 + bωn2 s + ωn3
.
s + cωn s 2 + bωn2 s + ωn3

(4)

2

3

The single-sided loop noise bandwidth Bn in hertz (defined as the
bandwidth of a perfect rectangular filter that produces the same
integrated noise power as that of the actual filter) is obtained from
the frequency response (setting s = j2π f) of the closed-loop system
as
∞

2

Bn ,i =  H i ( j 2π f ) df ,

(5)

0

where i is the loop order. This bandwidth can be used to compute the
receiver's noise floor (kTBn) and its sensitivity (the minimum input
signal power required to produce a specified SNℜ at the receiver's
output). Equation (5) can be analytically computed from the loop
parameters and the natural frequency [3]. Again, considering the
second- and third-order loops, the noise bandwidth can be written as

(

 bc 2 + b 2 − c
Bn ,3 = 
 4(bc − 1)


 a2 + 1 
Bn ,2 = 
ω ;
2  n
 4a 

)  ω .



(6)

n

Therefore, the design of the analog PLL is completely specified
by the desired noise bandwidth and the filter parameters, which
are usually designed to minimize the estimation's MSE. Typical
values are [3] a = 2 , b = 1.1, and c = 2.4, which is the setup used
in the literature to compute the PLL parameters from a specified
noise bandwidth.

Digital PLLs
Digital PLLs are usually derived from their analog counterpart.
Using s = (1 − z−1)/Ts, with Ts the sampling period, one can obtain
the PLL loop transfer function and the overall closed-loop transfer
function. In this case, the VCO is replaced by a NCO with transfer
function

To start from scratch and see how digital PLLs are derived from
their analog design, some results on analog PLLs are given together with their digital counterpart.

N ( z) =

Analog PLLs

Considering the following loop filter transfer functions [15]

The Laplace transform of the continuous time domain transfer
functions for the second- and third-order PLL loop filters are, respectively,

L2 ( z ) = α1 +

aω s + ωn2
;
L2 ( s ) = n
s

30

cω s 2 + bωn2 s + ωn3
,
L3 ( s ) = n
s2

(2)

z −1
.
1 − z −1

L3 ( z ) = α1 +

IEEE A&E SYSTEMS MAGAZINE

α2

1 − z −1

α2

1 − z −1

(7)

,

+

(8)

α3

(1 − z )
−1

2

,

(9)
JULY 2017, Part II of II



Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017 Tutorial XI

No label
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - No label
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - Cover2
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 1
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 2
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 3
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 4
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 5
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 6
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 7
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 8
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 9
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 10
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 11
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 12
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 13
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 14
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 15
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 16
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 17
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 18
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 19
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 20
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 21
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 22
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 23
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 24
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 25
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 26
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 27
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 28
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 29
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 30
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 31
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 32
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 33
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 34
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 35
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 36
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 37
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 38
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 39
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 40
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 41
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 42
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 43
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 44
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 45
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 46
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 47
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 48
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 49
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 50
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 51
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 52
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 53
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 54
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 55
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 56
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 57
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 58
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 59
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 60
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 61
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 62
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 63
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 64
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 65
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 66
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 67
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 68
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 69
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 70
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 71
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 72
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - Cover3
Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - Cover4
http://www.brightcopy.net/allen/aesm/34-2s
http://www.brightcopy.net/allen/aesm/34-2
http://www.brightcopy.net/allen/aesm/34-1
http://www.brightcopy.net/allen/aesm/33-12
http://www.brightcopy.net/allen/aesm/33-11
http://www.brightcopy.net/allen/aesm/33-10
http://www.brightcopy.net/allen/aesm/33-09
http://www.brightcopy.net/allen/aesm/33-8
http://www.brightcopy.net/allen/aesm/33-7
http://www.brightcopy.net/allen/aesm/33-5
http://www.brightcopy.net/allen/aesm/33-4
http://www.brightcopy.net/allen/aesm/33-3
http://www.brightcopy.net/allen/aesm/33-2
http://www.brightcopy.net/allen/aesm/33-1
http://www.brightcopy.net/allen/aesm/32-10
http://www.brightcopy.net/allen/aesm/32-12
http://www.brightcopy.net/allen/aesm/32-9
http://www.brightcopy.net/allen/aesm/32-11
http://www.brightcopy.net/allen/aesm/32-8
http://www.brightcopy.net/allen/aesm/32-7s
http://www.brightcopy.net/allen/aesm/32-7
http://www.brightcopy.net/allen/aesm/32-6
http://www.brightcopy.net/allen/aesm/32-5
http://www.brightcopy.net/allen/aesm/32-4
http://www.brightcopy.net/allen/aesm/32-3
http://www.brightcopy.net/allen/aesm/32-2
http://www.brightcopy.net/allen/aesm/32-1
http://www.brightcopy.net/allen/aesm/31-12
http://www.brightcopy.net/allen/aesm/31-11s
http://www.brightcopy.net/allen/aesm/31-11
http://www.brightcopy.net/allen/aesm/31-10
http://www.brightcopy.net/allen/aesm/31-9
http://www.brightcopy.net/allen/aesm/31-8
http://www.brightcopy.net/allen/aesm/31-7
https://www.nxtbookmedia.com