Circuits and Systems, 2011, 2, 139-144
doi:10.4236/cs.2011.23021 Published Online July 2011 (http://www.SciRP.org/journal/cs)
Linearized Phase Detector Zero Crossing DPLL
Performance Evaluation in Faded Mobile Channels
Qassim Nasir1, Saleh Al-Araji2
1Department of Electrical and Computer Engineering, University of Sharjah, Sharjah, UAE
2Communication Engineering Department, Khalifa University of Science, Technology and Research, Sharjah, UAE
Received February 13, 2011; revised April 15, 2011; accepted April 22, 2011
Abstract
Zero Crossing Digital Phase Locked Loop with Arc Sine block (AS-ZCDPLL) is used to linearize the phase
difference detection, and enhance the loop performance. The loop has faster acquisition, less steady state
phase error, and wider locking range compared to the conventional ZCDPLL. This work presents a Zero
Crossing Digital Phase Locked Loop with Arc Sine block (ZCDPLL-AS). The performance of the loop is
analyzed under mobile faded channel conditions. The mobile channel is assumed to be two path fading
channel corrupted by additive white Gaussian noise (AWGM). It is shown that for a constant filter gain, the
frequency spread has no effect on the steady state phase error variance when the loop is subjected to a phase
step. For a frequency step and under the same conditions, the effect on phase error is minimal.
Keywords: Non-uniform Sampling, Digital Phase Locked Loops, Zero Crossing DPLL, Mobile Faded
Channels
1. Introduction
Phase Lock Loops (PLLs) are used in a wider range of
communication applications such as carrier recovery
synchronization, and demodulation [1]. A PLL is a clo-
sed loop system in which the phase output tracks the
phase of the input signal. It consists of a phase detector,
filter, and voltage controlled oscillator. Digital Phase
locked Loops (DPLLs) were introduced to minimize
some of the problems associated with the analogue
counter part such as sensitivity to DC drift and the need
for periodic adjustments [1,2]. Conventional Zero Cros-
sing DPLL (ZCDPLL) is the most widely used due to its
simplicity in modeling and implementation [3,4].
In this paper an Arc-Sine ZCDPLL is analyzed under
mobile faded channel. The purpose of including the Arc-
-Sine in the loop is to linearize the phase difference de-
tection. The peak detector guarantees the input amplitude
to the Arc-Sine block to remain between –1 and +1. It
has been shown that the AS-ZCDPLL loop offers im-
proved performance in the lock range and acquisition
with reduced steady state phase error [5]. The proposed
ZCDPLL-AS can be characterized by a linear difference
equation in module (π/2) sense.
The mobile radio channel is characterized by fast Ray-
leigh fading and random phase distribution. This consid-
erably degrades the tracking performance and increase
the jitter of the loop. In this paper, the performance of
ZCDPLL-AS with phase and frequency step inputs in the
mobile radio environment is studied. The ZCDPLL-AS,
in this work is considered as part of a mobile receiver.
The mobile channel is assumed to be a two path fading
channel corrupted by additive white Gaussian noise
(AWGN). The fading in each path of the channel follows
Rayleigh distribution and has power spectral density as
given by Jakes [6].

2
2
π1
m
m
Sf
f
ff





where fm = vfc/c is the Doppler frequency that depends
on the speed of the vehicle v and carrier frequency fc.
The performance of the proposed algorithm will be
evaluated for Doppler frequencies of 6 Hz, 100 Hz and
222 Hz, corresponding to a pedestrian (3.5 km/hr) and
vehicular channels with speeds of 54 km/hr and 120 km/hr
respectively.
The stochastic difference equation describing the ZC-
DPLL-AS loop operation is derived in Section 2. Finally,

140
1k
Q. NASIR ET AL.
the probability density function (pdf) of the steadys-
tate phase error is derived and calculated numerically in
Section 3. Experimental simulation results are presented
in Section 4 and finally conclusion are given, in Section
5.
2. ZCDPLL-AS System Operation in Mobile
Faded Channels
The ZCDPLL-AS is composed of a sampler as a phase
detector, inverse sine block, a digital loop filter, and a
Digital Controlled Oscillator (DCO) as shown in Figure
1 [5]. The input signal to the loop is taken as x(t) = s1(t) +
n(t) , where s1(t) is the noise free input signal to the loop
after passing through the mobile channel. If s(t) =
Asin(ω0t + θi(t)), n(t) is Additive white Gaussian Noise
(AWGN); θi (t) = θ0+Ω0t , from which the signal dynam-
ics are modeled; θ0 is the initial phase which we will
assume to be zero; Ω0 is the frequency offset from the
nominal value ω0. Then s1(t) = r(t)sin(ω0t + θi(t) + φch(t)),
r(t) is Rayleigh faded envelope and φch(t), is a uniform
distribution channel phase.
The input signal is sampled at time instances tk deter-
mined by the Digital Controlled Oscillator (DCO). The
DCO period control algorithm as given by [7-10] is
01kkk
TTc tt

 (1)
where

0

2πT0

 is the nominal period, 1k c

is the
output of the loop digital filter D(z). The sample value of
the incoming signal x(t) at is
k
t
 
1kkk
x
tstnt (2)
or
kkk
x
sn (3)
where k
 
0
sin
kki

s
tt   , The sequence k
x
is
passed through the Arc-Sine block with output
. The output is passed through a digital
filter D(z) whose output is used to control the period
of the DCO.

1
sin
kk
x

k
c
y
The time instances can be rewritten as
k
t
1
0
10
,1,2,3,
kk
ki i
ii
tTkTck


 
  (4)
Thus
1
00 ,
0
sin
k
kki kchk k
i
x
rwkT c 





 


n

i
i
(5)
The phase error is defined to be [5]
1
,0
0
k
kkchk
i
wc 


  (6)
Also
11,10
0
k
kkchk
i
wc 
 
 
 (7)
D(z)
DCO
c
k−1
t
k
x
k−1
y
k−1
x
k
S’(t)
x(t)
n(t)
S(t)
S
/
H
Rayleigh
faded
Channel
Peak
Detector
Register
I
ARC-SINE
Bloc
k
Figure 1. Block diagram of the ZCDPLL-AS.
Taking the difference of (7) and (8) results in
11,1,kkk kchkchk
wc
0k



 

 (8)
The Arc-Sine (1
sin

) block has been added to lin-
earize the equation and avoid the nonlinear behaviour of
the systems [5]. The output of the Arc-Sine block can be
expressed as


1
kk
xsinyk



, and 11
k
x 
π2yπ2
k

 . The z transform of the output of the
digital filter is


Cz DzYz

(9)
where


Yz is the z transform of

y
t. The order of
the loop is determined by the type of the digital filter.
For first order, the digital filter is simply a gain block


Dz 1 G

, where is the block gain. However, for
second order loop,
1
G



1
12
1z

DzG G
k
.
Let us consider a first order AS-ZCDPLL loop, then
the digital filter output which controls the DCO is given
by
1k
cGy

(10)
Then the stochastic difference equation describing the
loop behaviour is given by

11,1,01kkkkchkchk kkk
wrGn  
 
 (11)
For phase step input where 1kk


 for , (11)
becomes
0k

1,1,01kkchkchk kkk
wrGn 

 

k
(12)
And for frequency step and for
, (12) becomes

0k
t  
0k

1,1,
10
kkchkchk
kkk
wr GnT

 





(13)
In practical mobile communication systems and in the
800 MHz band, an IF frequency of 10.7 MHz is usually
used; therefore, the sampling period T is on the order of
0.1 ps. The maximum Doppler frequency shift is on the
order of 100 Hz (at vehicle velocity, about 60 mph). In
other words, the ,1chk


and ,ch k

are equal, then (12)
and (13) are reduced to

Q. NASIR ET AL.
141
1k1010kk kk k
wrG wrGn


 (14)

11kkkkk
wr GnT  
  0
k
(15)
For both cases, the probability density function of
steady state phase error became a function of two inde-
pendent random variables r(k) and n(k).
3. Phase Error Probability Density Function
(pdf)
3.1. Phase Step without Noise
In steady state 1k


, (14) can be rewritten as

101
1
kk
rG

k



 (16)
If the expected value of 01k is 1, then the ex-
pected value of k
wGr

is zero for all values of k. This will
lead to rapid convergence of the steady state. Since the
probability density function of is Rayleigh then
k
r

2
2
2
2e,
s
k
r
r
s
r
Pr r



0 (17)
which has an average of π2
s

. Therefore, the opti-
mum value of the gain is

0
12π
opt s
G  . Let b =
, where , n is integer. Then the tran-
sition pdf can be shown as to be

01
sinGz  πzn

2
22
k+1
2
22
e.
s
k
uz
b
s
uz
u
P
zb










 uz
(18)
3.2. Phase Step plus Noise
Let 01k yZ Gn

 be a Gaussian random variable
with a mean of z and variance 222
01 n
G

 ,where 2
n

is
the variance of the noise n(t). Then the pdf of y is given
by


2
222
01
2
01
1e
2π
n
yz
G
y
n
Py G





 (19)
So

10
,sin
kk 1
y
brbw Gz 
  (20)
When 1
0, k z



π,k zn
will be zero mean Gaussian. Also
when 1


 will be Gaussian with mean of z.
The transition pdf can be rewritten as [4]

2
222
01
1
π
2
01
1e
π2π
n
kk
un
G
n
u
PnG











 (21)
Given kz

, then
 
111 0
0
2π
sin ,
kk k
zGrzGn  

  (22)
Define a random variable Y as

10
0
2π
k
Yz Gn 

  (23)
Y will be Gaussian with mean
 
0
2π1z  
and variance 22
1n
G

 . Therefore

2
0
222
1
2π1
2
1
1e
2
n
yz
G
Y
n
Py G




 
 






 (24)

1
', 'sin
kk
YbrbGz 
1
(25)
Since k

is a discrete time continuous variable Mar-
kov process, its conditioned on an initial condition error
0

satisfies Chapman-Kolmogrov equation, then
11
0
d
kkkk
z
PPP
z




 


 

 
 z
(26)
Equation (23) is valid whether k and 1k rare mutu-
ally independent or not. This is solved numerically as
was done in [4]. The transition pdf
r

P
z  is stored in
a matrix starting with

 
000
zPz

  ,


10 k
P



is calculated from


0
k
P


with k = 1,2,···, until the
values of successive k differ by a prescribed small
amount.
4. Simulation Results
The performance of the loop was evaluated in simulation
by subjecting it to phase as well as frequency steps. The
input signal s(t) = sin(2000_t) is considered as modula-
tion free and the DCO center frequency is 1000 Hz. In
the simulation process, the Signal to Noise Ratio is de-
fined as SNRdb = 10


2
log 1n

, where 2
n

represents
noise variance. The loop is studied under phase step in
the presence of noise. It is noticed from Figure 2, and as
derived in section (2), that the steady state phase error
variance depends on the value of the filter gain, as shown
in Figure 3. The increase in gain causes the phase error
to increase sharply which results in degradation in sys-
tem’s performance. The effect of SNR on the phase error
variance is shown in Figure 4. This variance is directly
proportional to SNR as shown in Figure 5. As shown
from the figure, the loop performance due to phase jitter
improves as SNR increases. The frequency spread has no
direct effect on the steady state phase error variance if
the filter gain is kept constant, as shown in Figure 6.
However, if the loop input signal is subjected to a fre-
quency step, then the loop jitter is slightly affected by the

142 Q. NASIR ET AL.
step size if the filter gain is kept constant. The Doppler
spread will increase the jitter if the spread is increased as
shown in Figure 6. The loop probability density function
of the phase performance when subjected to a frequency
step is shown in Figure 7 for different frequency offsets,
while Figure 8 is for different wireless channel Doppler
spreads. It is seen from the figures that the impact of
frequency offset and channel speed of variations (Dop-
pler spread) on the system performance is minimal. The
loop performance, when a frequency step is applied to
the loop, is also affected by the channel SNR as shown in
Figure 9. The variance of timing error in the loop is in-
creased as the loop gain G1 is increased and this primary-
Figure 2. Probability Density Function (pdf) of DCO Period
when SNR = 10 dB and when Phase step is applied with
different values of filter gain G1.
Figure 3. Variance of DCO period against filter gain G1.
Figure 4. Probability Density Function (pdf) of DCO period
for SNR = 10, 20 dB and when phase step is applied.
Figure 5. Variance of DCO period versus input signal SNR.
Figure 6. Probability Density Function (pdf) of DCO period
for SNR = 20 dB with phase step with different doppler
spreads.

Q. NASIR ET AL.
143
Figure 8. Probability Density Function (pdf) of DCO period
for SNR = 20 dB with frequency step with doppler spread of
6 and 100 Hz.
Figure 7. Probability Density Function (pdf) of DCO period
for SNR = 20 dB with Frequency step with different fre-
quency spre a ds.
Figure 9. Probability Density Function(pdf) of DCO period
for SNR = 10 and 20 dB with frequency step of frequency
offset of 0.01.
Figure 10. Variance of DCO period versus the loop gain G1
for different frequency offsets.
ly depends on the value frequency step input as shown in
Figure 10.
5. Conclusions
The ZCDPLL-AS loop is studied under phase and fre-
quency steps in the presence of noise. It is shown that the
frequency spread, under phase step condition, has no
direct effect on the steady phase error variance if the
filter gain is kept constant. For frequency step, the error
is slightly affected under the same conditions. From the
results, it has been shown that the variance of the DCO
period increases with the Doppler spread. The system
was tested with Doppler spreads of 6 Hz, 100 Hz, and
222 Hz. ZCDPLL-AS loop has been tested and has
shown to give improved locking and acquisition per-
formance.
6. References
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Crossing Digital Phase Locked Loop using Multi-Sam-
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cember 2003, pp. 719-722.
[3] Q. Nasir, “Digital Phase Locked Loop with Broad Lock

144 Q. NASIR ET AL.
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