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PLL Tracking Performance in the Presence of Oscillator Phase Noise

Article in GPS Solutions · April 2002

DOI: 10.1007/PL00012911

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 PLL Tracking Performance

in the Presence of Oscillator Phase Noise

Number of text pages: 22

Number of figures: 12

Number of tables: 3

Markus Irsigler Bernd Eissfeller

Institute of Geodesy and Navigation Institute of Geodesy and Navigation
University FAF Munich University FAF Munich
D-85577 Neubiberg, Germany D-85577 Neubiberg, Germany
Phone: +49 89 6004 3552 Phone: +49 89 6004 3017
Fax: +49 89 6004 3019 Fax: +49 89 6004 3019
Email: markus.irsigler@unibw-muenchen.de Email: bernd.eissfeller@unibw-muenchen.de
ABSTRACT

The tracking performance of a Phase Lock Loop (PLL) is affected by the influence of several

error sources. In addition to thermal noise and dynamic stress error, oscillator phase noise can

cause significant phase jitter which degrades the tracking performance. Oscillator phase noise

is usually caused by two different effects: Allan deviation phase noise is primarily caused by

frequency instabilities of the receiver’s reference oscillator. It can be termed as system-

inherent phase noise and is relevant for both static and dynamic applications. “External”

phase noise, however, is caused by vibration and is a major problem for dynamic applications.

In the context of this paper, both types of phase noise will be modeled and the resulting

integrals will be evaluated for PLLs up to the third order. Besides, phase jitter induced by

thermal noise and signal dynamics will also be discussed, thus providing all necessary

formulas for analyzing the performance of a phase lock loop in case of different forms of

stress. Since the main focus is centered on the effects of oscillator phase noise, the overall

PLL performance is graphically illustrated with and without consideration of oscillator phase

noise.

INTRODUCTION

Different approaches are used to ensure robust and continuous signal tracking inside a

satellite receiver. The most important ones are the Delay Lock Loop (DLL) and the Phase

Lock Loop (PLL). The DLL aligns the code contained in the received signal with an identical

code generated within the receiver (pseudorange determination), whereas the PLL aligns the

phase of the incoming satellite signal with respect to the receiver-generated reference signal

(carrier phase determination). Satellite signal tracking can also be performed by using a

Frequency Lock Loop (FLL). In this case, the frequency of the received satellite signal is

aligned with that of the receiver-generated reference signal. However, signal alignment is not

a straightforward process because of occurring signal dynamics resulting from permanent

satellite and possible receiver motion. Therefore, the general functionality of a tracking loop

is first described below.

A generic tracking loop block diagram is illustrated in Figure 1. Principal constituents are a

numerical controlled oscillator (NCO), an integrator 1/s and a loop filter of bandwidth BL.

According to (Van Trees, 1971) and (Cugny, 1989), a loop filter of order n can be described

mathematically by its Laplace transform F(s):

n −1 α k ,n
F ( s) = ∑ ; α0 = 1 (1)
k =0 sk

The variable s can be interpreted as a complex frequency, so that equation (1) can be deemed

to be a low pass filter of order n. Loop filters up to 3rd order are usually used in today’s GPS

receivers. The objective of the loop filter is to reduce noise, so that an accurate estimate of the

original signal can be ensured. The filter order and noise bandwidth not only determine the

effects of thermal noise but also the loop filter’s response to signal dynamics. As it can be

derived from Figure 1, the loop filter’s output signal is subtracted from the original signal to

produce an error signal. This error signal is fed back into the filter’s input with the objective

to adjust the frequency of the NCO in such a way that the error signal can be minimized

(closed loop process).

Figure 1. Generic tracking loop block diagram. Potential error sources are represented

by blue-colored arrows.
It should be noted that oscillator phase noise does not significantly enter directly at the NCO

but rather at the receiver reference oscillator. Since the operating frequency of the NCO is

derived from that of the (noisy) reference oscillator, the NCO will also be affected by the

influences of phase noise. Figure 2 illustrates generic block diagrams of 1st, 2nd and 3rd order

loop filters (a similar illustration can be found in (Kaplan,1996)). The integrators are

represented by 1/s, the Laplace transform of the time domain integration function. The input

signal is multiplied by the filter coefficients α2,α3 and β3 and generally processed as shown in

Figure 2. Some GPS receivers, in particular high dynamics receivers, have an additional phase

feedback term. In this case, the final integrators are not implemented. The filter coefficients

α2,α3,β3 and the actual number of integrators completely determine the loop filter’s

characteristics. The most important characteristics for 1st, 2nd and 3rd order loop filters are

listed in Table 1.

Figure 2. Generic block diagrams of first-, second- and third-order tracking loops
As already stated, the design of the loop filter determines the loop filter’s response to thermal

noise and signal dynamics. Furthermore, additional error sources, such as oscillator phase

noise, have to be taken into account. Induced by the influence of such error sources, the

tracking performance of either a DLL, PLL or FLL is affected in a negative manner (DLLs,

however, are less affected than PLLs or FLLs). If the total sum of these error sources exceeds

a certain threshold, the tracking loop loses lock and the satellite signal cannot be tracked

anymore. Since we intend to focus on the PLL tracking performance, other types of tracking

loops (DLL, FLL) will not be discussed further below.

According to (Kaplan, 1996), the most important error sources affecting the PLL tracking

performance are thermal noise σT, dynamic stress error e(t) (induced by signal dynamics),

vibration-induced oscillator phase noise σV and Allan deviation oscillator phase noise σA (see

Figure 1). Considering all these error sources, the total PLL jitter and its corresponding rule-

of-thumb tracking threshold can be computed by using the following expressions (Kaplan,

1996):

e(t ) e(t ) λ
σ PLL = σ T2 + σ A2 + σ V2 + ≤ 15 o or σ PLL = σ T2 + σ A2 + σ V2 + ≤ [m] (2)
3 3 24

Equation (2) indicates that, due to the above mentioned error sources, the total PLL jitter

should not exceed 15° or 1/24 of the carrier wavelength λ. Evaluation of equation (2b) for the

GPS L1 and L2 signals (λL1 = 0.19m, λL2 = 0.24m) leads to (metric) tracking thresholds of

7.9mm and 10.2mm, respectively. In case that the total PLL phase jitter exceeds this tracking

threshold, the PLL does not reliably maintain lock and the satellite signal cannot be tracked

any longer. Otherwise, if the total phase jitter is less than this threshold, the PLL can be

considered as stable. Note, however, that loss of lock is a non-deterministic, statistical effect

and that PLL tracking performance is generally a non-linear problem. Within the scope of the

following analysis, PLL tracking performance will be treated as a linear problem. Major

problems arise from the fact that the error sources mentioned above cannot be accurately
modeled. In particular Allan deviation oscillator phase noise and vibration-induced oscillator

phase noise are complex analysis problems. This article will therefore focus on these error

sources. Nevertheless, phase jitter induced by thermal noise and dynamic stress error will also

be discussed.

As a matter of fact, a lot of work has been done in this area in the past (e.g., Lindsay (1972),

Spilker (1977) or Viterby (1966)). Thus, the general approaches to calculate the influences of

the error sorces mentioned above as well as the models of all relevant error sources are

principally known. Therefore, one intention of this article is to summarize the results of

previous work and to provide the necessary formulas for analyzing the performance of a PLL

in case of different forms of stress. Furthermore, the influences of all relevant error sources as

well as the overall PLL performance will be graphically illustrated.

MODELING OF ERROR SOURCES

Vibration Induced Phase Noise

Since the PLL of the satellite receiver tracks the phase of the incoming satellite signal with

respect to the receiver’s reference oscillator, the tracking performance will be degraded if

either the satellite’s or the receiver’s oscillator show significant phase noise. Phase noise can

be either natural phase noise caused by the oscillator itself or “external” phase noise caused

by vibration. The PLL tracking error caused by oscillator phase noise can be expressed as

follows (Parkinson & Spilker, 1996):

∞
1
σ φ2 = ∫ Gφ (ω ) 1 − H (ω ) dω
2
(3)
2π 0

The integration variable ω=2πf represents the radian frequency of vibration; Gφ(ω) is the

single-sideband oscillator phase noise spectral density. 1−Η(ω)  depends on the loop order

n and is, according to (Parkinson & Spilker, 1996), given by

 ω 2n
1 − H (ω ) =
2
. (4)
ω L2 n + ω 2 n

Note that ωL also depends on the loop order and that it is a function of the loop noise

bandwidth BL (expressed in Hz). As it can be derived from Table 1, ωL can be computed by

using ωL,1 = 4BL, ωL,2 = 1.885BL and ωL,3 = 1.2BL for 1st, 2nd and 3rd order PLLs, respectively

(Parkinson & Spilker, 1996). In case of phase noise caused by vibration, Gφ(ω) can be

expressed by

G g (ω )
Gφ (ω ) = (2πf 0 ) 2 k g2 (ω ) . (5)
ω2

There, Gg(ω) is the single-sided vibration spectral density [g²/Hz], kg(ω) is the oscillator’s g-

sensitivity in parts-per-g [1/g] and f0 the carrier frequency (Parkinson & Spilker, 1996). Note

that, strictly speaking, the oscillator’s g-sensitivity depends on the vibration frequency ω.

Insertion of equations (4) and (5) into equation (3) leads to the following expressions for

vibration-induced phase noise:

∞
k g2 (ω )G g (ω )
σ = 2πf ∫ dω
st 2 2
1 order PLL: φ (6)
ω L2 + ω 2
0
0

∞
k g2 (ω )G g (ω )ω 2
σ = 2πf ∫ dω
nd 2 2
2 order PLL: φ (7)
ω L4 + ω 4
0
0

∞
k g2 (ω )G g (ω )ω 4
σ = 2πf ∫ dω
rd 2 2
3 order PLL: φ (8)
ω L6 + ω 6
0
0

Assuming constant values for g-sensitivity (kg(ω)=kg) and power spectral density (PSD) of

vibration (Gg(ω)=Gg), equations (6)-(8) can easily be integrated. In case of constant spectral

density, all occurring vibrations have identical intensities. The integration limits depend on

the actual shape of the vibration spectrum. Assuming that all occurring vibrations are equally

distributed across the entire frequency range (white vibrational noise), the integration limits

can be set to ω=[0 ∞] and the resulting vibration spectrum is represented by Figure 3
(schematic representation). In practice, however, the occurring vibrations will not be equally

distributed across the entire frequency range. Therefore, the vibration spectrum will rather

look like Figure 4 and the integration interval can be limited to ω=[ω1 ω2]. If the PSD of

vibration cannot be assumed to be constant, the vibration spectrum may look like Figure 5 and

equations (6)-(8) must be solved by piecewise integration.

Figure 3. Constant PSD of vibration (f=[0 ∞])

Figure 4. Constant PSD of vibration (band-limited, f=[f1 f2])

Figure 5. Typical aircraft random vibration envelope

In dependency on the chosen integration limits and in case of the above mentioned

simplifications (constant g-sensitivity and constant spectral density of vibration), integration

of equations (6)-(8) leads to the following expressions for vibration-induced phase jitter:

1st order PLL, ω=[0 ∞]:

∞
dω π 2 f 02 k g2 G g
σ φ2 = 2πf 02 k g2 G g ∫ = [rad] (9)
0 ωL + ω
2 2
ωL
1st order PLL, ω=[ω1 ω2]:

ω2
dω
σ = 2πf k G g ∫ω
2 2 2
φ
+ω2
0 g 2
ω1 L
(10)
2πf k G g 
2
 ω2
2
 ω 
σ φ2 = arctan  − arctan 1  [rad]
0 g

ωL  ωL  ωL 

2nd order PLL, ω=[0 ∞]:

ω 2 dω π 2 f 02 k g2 G g
∞
σ = 2πf k G g ∫ 4
2
φ
2 2
= [rad] (11)
0 ωL + ω
0 g
2ω L
4

2nd order PLL, ω=[ω1 ω2]:

ω2
ω 2 dω
σ = 2πf k G g ∫ 4 4
2 2 2
φ
ω1 ω L + ω
0 g

πf 02 k g2 G g   ω2 2  ω 2 
σ =  arctan + 1 + arctan 2 − 1
2
φ
2ω L   ω L   ω L 
(12)
ω 2  ω 2 
− arctan 1 + 1 − arctan 1 − 1
 ωL   ωL 
1  (ω12 + ω Lω1 2 + ω L2 )(ω 22 − ω Lω 2 2 + ω L2 )  
+ ln  2   [rad]
2  (ω1 − ω Lω1 2 + ω L2 )(ω 22 + ω Lω 2 2 + ω L2 )  

3rd order PLL, ω=[0 ∞]:

∞
ω 4 dω 2πf 02 k g2 G g
σ φ2 = 2πf 02 k g2 G g ∫ = [rad] (13)
0 ωL +ω
6 6
3ω L
3rd order PLL, ω=[ω1 ω2]:

ω2
ω 4 dω
σ = 2πf k G g ∫ 6 6
2 2 2
φ
ω1 ω L + ω
0 g

2πf 02 k g2 G g  1   ω2  ω 
σ φ2 =  arctan  − arctan 1 
ωL  3  ωL  ωL 
1  − 3ω L + 2ω 2   3ω L + 2ω 2 
+ arctan  + arctan
 

 (14)
6   ω L   ω L 
 − 3ω L + 2ω1   3ω L + 2ω1 
− arctan  − arctan
 


 ω L   ω L 
1  (ω L2 − ω Lω 2 3 + ω 22 )(ω L2 + ω Lω1 3 + ω12 )  
+ ln    [rad]
4 3  (ω L2 + ω Lω 2 3 + ω 22 )(ω L2 + ω Lω1 3 + ω12 )  

Equations (9)-(14) can be converted to [deg] by multiplying the obtained phase jitter σφ with

(180°/π). When expressed in [rad] or [deg], the resulting vibration-induced phase jitter is

proportional to the carrier frequency and thus becomes larger with increasing carrier

frequency.

Allan Deviation Phase Noise

A perfect oscillator can be described mathematically by a sinusoidal waveform y(t)=cos(ω0t).

An actual oscillator, however, will exhibit both an amplitude noise modulation and a phase

noise modulation. Both modulation processes have random character. Phase noise modulation

results from oscillator frequency instabilities and has negative influences on the PLL tracking

performance. These frequency instabilities can in turn be described by the Allan deviation, a

measure of the oscillator’s short-term stability. In contrast to the vibration-induced phase

noise discussed in the previous section, this form of phase noise can be characterized as

natural phase noise. The approach to model the PLL jitter due to Allan deviation phase noise

is the same as for vibration induced phase noise. Equation (3) can therefore be rewritten as

∞
1
σ = ∫ S φ (ω ) 1 − H (ω ) dω .
2 2
φ (15)
2π 0
There, Sφ(ω) is the single-sideband oscillator phase noise spectral density resulting from

frequency instabilities. 1−Η(ω) again depends on the loop order n and has already been

defined by equation (4):

ω 2n
1 − H (ω ) =
2
(16)
ω L2 n + ω 2 n

As it is the case for the computation of vibration-induced phase jitter, ωL depends on the loop

order n and is a function of the loop noise bandwidth BL (expressed in Hz). Values for ωL can

again be derived from Table 1. The oscillator phase noise spectral density S φ(ω) resulting

from frequency instabilities can be expressed in the same manner as equation (5):

S y (ω )
S φ (ω ) = (2πf 0 ) 2 (17)
ω2

f0 is the carrier frequency; Sy(ω) is the clock’s power spectrum which can be expressed,

according to (NovAtel Inc., 1998), as a function of the clock parameters h -2, h-1 and h0:

h− 2 h−1 4π 2 h− 2 2πh−1
Sy ( f ) = + + h0 ⇔ S y (ω ) = + + h0 (18)
f2 f ω2 ω

The clock parameters h-2 [s-1], h-1 (dimensionless) and h0 [s] are related to the Allan deviation

σy by the expression σy2(T)= h0/(2T)+2⋅ln2⋅h-1+(2⋅π2⋅T⋅h-2)/3 with T being the averaging time

(Allan, 1966). As it is the case for vibration-induced phase jitter, we assume a single-sided

power spectrum, so that equation (18) can be rewritten as

2π 2 h− 2 πh−1 h0
S y (ω ) = + + (19)
ω2 ω 2

and Sφ(ω) can finally be expressed as follows:

 2π 2 h− 2 πh−1 h 
Sφ (ω ) = (2πf 0 ) 2  + 3 + 02  (20)
 ω ω 2ω 
4

This leads to the following expressions for the Allan deviation induced phase jitter:
 ∞
 2π 2 h− 2 πh−1 h0  ω 2
st
1 order PLL: σ A2 = 2πf 02 ∫  + + 2 
dω (21)
0  ω 4
ω 3
2ω ω
 L
2
+ ω 2

∞
 2π 2 h− 2 πh−1 h0  ω 4
2nd order PLL: σ A2 = 2πf 02 ∫  + + 2 
dω (22)
0  ω 4
ω 3
2ω ω
 L
4
+ ω 4

∞
 2π 2 h− 2 πh−1 h0  ω 6
3rd order PLL: σ A2 = 2πf 02 ∫  + + 2 
dω (23)
0  ω 4
ω 3
2ω ω
 L
6
+ ω 6

Equations (21)-(23) can now be integrated. Since the integrals for the 1st order PLL do not

converge, results are only given for the 2nd and 3rd order PLL:

2nd order PLL:

 ∞
dω
∞
ω dω h0 ∞ ω 2 dω 
σ A2 = 2πf 02 2π 2 h− 2 ∫ 4 −1 ∫
2 ∫0 ω L4 + ω 4 
+ πh + 
0 ωL +ω 0 ωL +ω
4 4 4

(24)
 π 2 h− 2 πh−1 h0 
σ A2 = 2π 2 f 02  + +  [rad]
 2ω L 4ω L 4 2ω L 
3 2

3rd order PLL:

 ∞
ω 2 dω
∞
ω 3 dω h0 ∞ ω 4 dω 
σ A2 = 2πf 02 2π 2 h− 2 ∫ 6 −1 ∫
2 ∫0 ω L6 + ω 6 
+ πh + 
0 ωL +ω 0 ωL + ω
6 6 6

(25)
π 2 h− 2 πh−1 h0 
σ A2 = 2π 2 f 02  + +  [rad]
 3ω L 3 3ω L2 6ω L 
3

Equations (24) and (25) can again be converted to [deg] by multiplying the obtained phase

jitter σφ with (180°/π). As it is the case for vibration-induced phase jitter, the resulting Allan

deviation phase jitter is proportional to the carrier frequency and thus becomes larger with

increasing carrier frequency.

Thermal Noise

The PLL tracking loop jitter due to thermal noise is independent of loop order, but depends on

the actual PLL implementation. In case of noncoherent carrier tracking (e.g. Costas tracking),

the resulting thermal noise jitter can be expressed as

BL  1 
σT = 1 +  [rad] . (26)
c n0  2T ⋅ c n0 

There, c/n0 is the carrier to noise power expressed as a ratio10C/10No (C/N0 expressed in [dB-

Hz]), T is the predetection integration time [s] and BL the PLL loop noise bandwidth [Hz].

Dynamic Stress Error

Due to the permanent motion of the satellites and due to possible receiver motion, the PLL

has to track the resulting signal dynamics. Signal dynamics is a major problem for non-static

applications and principally degrades the PLL tracking performance because it causes phase

jitter. The PLL tracking loop dynamic stress error can be expressed by

dR n
en (t ) = dt n [ m] (27)
ω Ln

and is thus dependent on the loop order n. It is furthermore dependent on the loop noise

bandwidth BL. Values for ωL can again be derived from Table 1. This leads to the following

equations for the dynamic stress errors (expressions are valid for 1st, 2nd and 3rd order loops):

x& (t ) &x&(t ) &x&&(t )

e1 (t ) = [ m] e2 (t ) = [ m] e3 (t ) = [ m] (28)
4 ⋅ BL (1.885 ⋅ B L ) 2 (1.2 ⋅ B L ) 3

x& (t ) is the line-of-sight velocity [m/s], &x&(t ) the line-of-sight acceleration [m/s2] and &x&&(t ) the

line-of-sight jerk stress [m/s3]. Thus, 1st order loops are sensitive to velocity stress, whereas

2nd and 3rd order loops are sensitive to acceleration and jerk stress, respectively. The dynamic

stress errors can be converted to [rad] or [deg] by multiplying with (2π/λ) or (360°/λ),
respectively, where λ is the carrier wavelength. When expressed in [rad] or [deg], the

resulting dynamic stress errors become larger with increasing carrier frequency.

RESULTING PHASE JITTER

Random Vibration

As shown in the previous section, the influence of random vibration depends on the carrier

frequency, the oscillator’s g-sensitivity, the vibration intensity (PSD of vibration) and the

frequency range of vibrations. In order to illustrate the influences of random vibration, the

resulting phase jitter was plotted for 1st, 2nd and 3rd order PLLs as a function of the loop noise

bandwidth BL, assuming the GPS L1 signal (f0=1575.42MHz), an oscillator’s g-sensitivity of

kg = 1·10-9 [1/g] and a constant PSD of vibrations of Gg = 0.05 [g²/Hz] (typical value for

aircraft applications). Assuming furthermore that the vibrations are equally distributed across

the entire frequency range (ω=[0 ∞]), the vibration spectrum is correspondingly represented

in Figure 3. The resulting phase jitter was determined by evaluation of equations (9), (11) and

(13) and is illustrated in Figure 6. Under these conditions, vibration-induced phase jitter

increases with smaller loop noise bandwidth and increasing loop order (see Figure 6).

Figure 6. Influence of random vibration as a function of the loop noise bandwidth BL.
The assumption, that the occurring vibrations are equally distributed across the entire

frequency range (ω=[0 ∞]), leads to maximum values for the vibration-induced phase jitter.

We already mentioned that, in practice, however, the occurring vibrations will not be equally

distributed across the entire frequency range. Therefore, we now assume that the vibration

spectrum either does not contain very low vibration frequencies or that low-frequency

vibrations can be neglected. Under these conditions, the constant PSD of vibration has the

shape illustrated in Figure 4, resulting in significantly reduced vibration-induced phase jitter

(see Figure 7). Since it can be shown that variation of the upper vibration limit f2 has only

marginal effects on the resulting phase jitter (high frequency vibrations barely affect the PLL

tracking performance), it has been set to the constant value of f2=2500Hz; only the lower limit

f1 has been varied to analyze the resulting phase jitter. The following diagrams were

computed for 2nd and 3rd order PLLs only (Phase Lock Loops are never implemented as 1st

order PLLs).

Figure 7. Reduction of vibration-induced phase jitter by neglecting low-frequency

vibrations.

It is obvious that vibration-induced phase jitter can be significantly reduced if low frequency

vibrations are not present or can be neglected. Another outcome of ignoring low frequency

vibrations is that the resulting phase jitter becomes independent of loop order and loop noise

bandwidth.
Another promising approach to reduce the negative influence of random vibration is the

minimization of the oscillator’s g-sensitivity. Figure 8 illustrates the effects if the g-sensitivity

is significantly reduced by a factor 5 and is thus set to k g=2·10-10 [1/g] instead of kg=1·10-9

[1/g]. Since vibration-induced phase jitter is, according to (9)-(14), proportional to the g-

sensitivity of the receiver clock, the resulting phase jitter can be reduced to one fifth of its

original value. All illustrations are based on a carrier frequency of f0=1575.42MHz (GPS L1)

and a constant PSD of vibration of Gg=0.05 [g²/Hz]. The vibration frequency ranges have

been set to fvib = [0Hz 2500Hz] and fvib = [25Hz 2500Hz], respectively. It is obvious that

vibration-induced phase jitter can be significantly reduced if the clock’s g-sensitivity is

minimized.

fvib = [0Hz 2500Hz]

fvib = [25Hz 2500Hz]

Figure 8. Reduction of vibration-induced phase jitter by enhancing the oscillator’s g-

sensitivity for second- and third-order PLLs.

Allan Deviation

As shown in the previous section, PLL jitter due to Allan deviation phase noise depends on

the carrier frequency f0, the loop noise bandwidth BL and the clock parameters h-2, h-1 and h0

(equations (24) and (25)). These clock parameters represent the frequency (in)stability of a

certain oscillator and can be used to compute the Allan deviation for a certain period of time.

Table 2 lists the clock parameters h-2, h-1 and h0 for different types of oscillators, namely for a

temperature compensated crystal oscillator (TCXO), an ovenized crystal oscillator (OCXO)

and for a rubidium and cesium clock (NovAtel Inc., 1998). These clock parameters were used

to illustrate the influences of oscillator frequency instabilities on PLL tracking performance

by means of equations (24) and (25). The resulting phase jitter is plotted for both 2nd and 3rd

order PLLs as a function of the loop noise bandwidth BL.

Figure 9. PLL phase jitter due to frequency instabilities of the receiver clock.

It is obvious that PLL phase jitter due to Allan deviation phase noise is dependent on the loop

noise bandwidth BL. Especially at small noise bandwidths, the resulting PLL phase jitter

increases significantly. Figure 9 reveals two potential approaches to reduce the negative

influence of Allan deviation phase noise and in return to enhance the PLL tracking

performance. Firstly, the negative influence of oscillator frequency instabilities can

principally be reduced by increasing the loop noise bandwidth BL. This approach, however,

results in increased thermal noise which in turn decreases the closed loop C/N0. As a result,
the receiver becomes more prone to loss of lock. Therefore, increasing the loop noise

bandwidth is a suitable approach only at high C/N0. Secondly, additional enhancements can

be achieved by choice of an adequate and suitable oscillator (e.g. an OCXO). Note that the

phase jitter for the 3rd order PLL is slightly higher than for the 2nd order PLL. Table 3 lists the

resulting PLL jitter for a 2nd and 3rd order PLL, assuming a loop noise bandwidth of

BL=18Hz. The computation is based on the clock parameters listed in Table 2.

Thermal Noise

According to equation (26), the signal characteristics that affect the PLL performance are the

predetection integration time, the carrier-to-noise ratio and the loop noise bandwidth. When

expressed in [rad] or [deg], the thermal noise performance does not depend on the carrier

frequency. Figure 10 illustrates the influences of thermal noise for the present GPS signals.

The predetection integration time is chosen to be the inverse navigation data rate (50 bps).

This assumption leads to the following illustration of PLL thermal noise tracking error:

Figure 10. PLL jitter due to thermal noise for the GPS signals.

The PLL jitter due to thermal noise is plotted as a function of the carrier-to-noise ratio C/N0

and the PLL loop noise bandwidth BL. It is obvious that the influence of thermal noise

increases both with increasing loop noise bandwidth and with degradation of C/N0.
Dynamic Stress Error

The PLL dynamic stress error is a function of the signal dynamics and the PLL loop noise

bandwidth BL. When expressed in [rad] or [deg], it also depends on the carrier frequency f0.

Depending on its implementation as 1st, 2nd or 3rd order loop, the PLL is sensitive to velocity

stress, acceleration stress or jerk stress. The resulting dynamic stress errors are computed by

evaluation of equations (28) and are illustrated in Figure 11.

Figure 11. Dynamic stress errors for first-, second- and third-order PLLs.

The dynamic stress errors are to a great extent dependent on the loop noise bandwidth B L.

Narrow loop bandwidths result in large tracking errors whereas the tracking errors can be

significantly reduced by using larger bandwidths. The dynamic stress errors are also

dependent on the line-of-sight signal dynamics. Higher signal dynamics will result in larger

dynamic stress errors, especially in case of narrow loop bandwidths.

TOTAL PLL JITTER

In order to assess the actual tracking performance of a Phase Lock Loop, the total sum of

error sources discussed above has to be taken into account. The total sum of all occurring

tracking errors is termed as total PLL jitter and can be computed by means of equation (2).

Equation (2) also fixes the tracking threshold (15° or 1/24 of the carrier wavelength) which -

in terms of this linear analysis - should not be exceeded to ensure robust and continuous

signal tracking. In order to estimate the PLL tracking performance subject to different

parameters, the total PLL jitter can be plotted as a function of both the carrier-to-noise ratio

C/N0 and the line-of-sight signal dynamics. The latter parameter depends on the PLL

implementation. In the following, we want to examine the tracking performance of a 2 nd order

PLL, so that the total PLL jitter depends on the line-of-sight acceleration. Since we intend to

point out the influences of oscillator phase noise, the total PLL jitter is first computed without

considering the negative influences of Allan deviation phase noise and vibration-induced

phase noise. In a second step, oscillator phase noise is added to the illustration, so that the

influences of oscillator phase noise on PLL tracking performance become evident. The result

of this analysis is illustrated Figure 12. Both diagrams base on the following parameters:

Carrier wave: f = 1575.42 MHz (GPS L1), λL1 = 0.19m

Tracking threshold: s = 15°

Loop Order: 2

Loop noise bandwidth: BL = 18 Hz

The left diagram illustrates the PLL tracking performance when oscillator phase noise is

neglected, i.e. thermal noise and dynamic stress error are the only existing error sources. The

right diagram illustrates the tracking performance under consideration of all important error

sources (thermal noise, dynamic stress error, oscillator phase noise). The computation of

oscillator phase noise is based on the following parameters:

Receiver clock: TCXO (clock parameters according to Table 2)

Constant PSD of vibration: G = 0.05 g²/Hz (typical for aircraft applications)

Frequency range of vibration: fvib = [5Hz 2500Hz]

G-Sensitivity (receiver clock): k = 1·10-9 [1/g]

Figure 12. Tracking performance of a second-order PLL. The left diagram illustrates

the influences of thermal noise and dynamic stress error, whereas the right diagram

illustrates the tracking performance in the presence of additional oscillator phase noise.

The total PLL jitter is represented by the curved red surface and strongly depends on the

carrier-to-noise ratio and the line-of-sight acceleration. The total PLL jitter increases with

degradation of C/N0 (due to the resulting increase of thermal noise) and increasing

acceleration. The tracking threshold is represented by the gray plane having a constant height

of 15°. Robust and continuous signal tracking can be ensured if the total PLL jitter is less than

the tracking threshold, i.e. the curved red surface has to be situated below the gray plane

representing this threshold. It can be derived from the left diagram in Figure 12 that the signal

cannot be tracked even for static applications if the C/N0 falls below 25dB-Hz. Even if we

assume a low-noise signal with a carrier-to-noise ratio of 50dB-Hz, the PLL loses lock if the

line-of-sight acceleration exceeds approximately 2.6g. The situation gets even worse if

additional oscillator phase noise as defined above is taken into account (right diagram). In this
case, the required minimum C/N0 is 26dB-Hz and the PLL already loses lock if the line-of-

sight acceleration exceeds approximately 1.2g (for C/N0=50dB-Hz). As a result, the presence

of oscillator phase noise primarily minimizes the dynamic range of the PLL.

As already discussed in the previous sections, there are several approaches to enhance the

PLL tracking performance, namely the increase of loop noise bandwidth and the use of a high

quality oscillator (low Allan deviation phase noise and low g-sensitivity). Both approaches

primarily enlarge the dynamic range of the PLL, i.e. higher accelerations may occur for a

given C/N0 before the PLL loses lock. Note, however, that increase of the loop noise

bandwidth results in increased thermal noise.

SUMMARY

In addition to thermal noise and dynamic stress error, the performance of a Phase Lock Loop

is affected by the influence of oscillator phase noise. Oscillator phase noise can be divided

into “external” and natural phase noise. External phase noise is caused by vibration and is a

major problem for dynamic applications whereas natural phase noise is a result of the

oscillator’s frequency instabilities and is relevant for both static and dynamic applications.

Both types of phase noise have been modeled and the resulting integrals have been evaluated

for PLLs up to the 3rd order.

Thermal noise is independent of loop order and increases when the loop noise bandwidth is

widened. Therefore, the negative influence of thermal noise can principally be reduced by

narrowing the noise bandwidth. The opposite is true for dynamic stress error; its negative

influence on PLL tracking performance decreases with increasing loop noise bandwidth. The

negative effects on tracking performance can therefore be reduced by increasing the loop

noise bandwidth. As a result, the loop noise bandwidth has to be chosen in such a way that it

is wide enough to ensure robust signal tracking even in high dynamics environments but is, on

the other hand, small enough to avoid excessive thermal noise.

It has been shown that the negative influences of Allan deviation phase noise can be reduced

by either increasing the loop noise bandwidth or by choice of a suitable oscillator (e.g. an

OCXO). Increase of the loop noise bandwidth, however, will also result in increased thermal

noise which may in turn degrade the PLL performance. Therefore, the choice of a suitable

oscillator seems to be the most promising approach.

The influences of random vibration mainly depend on the oscillator’s g-sensitivity, the

vibration intensity (PSD of vibration) and the frequency range of vibrations. Assuming a

band-limited vibration spectrum, the resulting phase jitter is independent of loop order and

loop noise bandwidth. One feasible approach to reduce vibration-induced phase jitter is to

improve (minimize) of the oscillator’s g-sensitivity. Another option is suitable oscillator

mounting. Such physical mounting techniques can absorb the occurring vibrations to some

extent.

BIOGRAPHIES

Markus Irsigler is research associate at the Institute of Geodesy and Navigation at the

University of the Federal Armed Forces Munich. He received his diploma in Geodesy and

Geomatics from the University of Stuttgart, Germany. His scientific research work focuses -

among other topics - on GNSS receiver design and performance.

Bernd Eissfeller is Full Professor and Vice-Director of the Institute of Geodesy and

Navigation at the University of the Federal Armed Forces Munich. He is responsible for

teaching and research in the field of GPS/GLONASS and inertial technology. Till the end of

1993 he worked in industry as a project manager on the development of GPS/INS navigation

systems. From 1994-2000 he was head of the GNSS Laboratory of the Institute of Geodesy

and Navigation.
REFERENCES

Allan, D. (1966): Statistics of Atomic Frequency Standards, Proceedings of the IEEE,

Vol. 54, No. 2, 221-230, New York

Cugny, B. (1989): Techniques de localisation et de navigation: Mise en oeuvre de

principes, Système Spatiaux de Localisation et de Navigation, 89-131, CNES,

Toulouse

Kaplan, E.D. (1996): Understanding GPS, Principles and Applications, Mobile

Communications Series, Artech House, Norwood, 1996

Lindsey, W. C. (1972): Synchronisation Systems in Communication and Control,

Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1972

NovAtel Inc. (1998): Millenium GPSCardTM Command Descriptions Manual,

Calgary, Alberta, Canada, 1998

Parkinson, B. W., Spilker, J. J. (1996): Global Positioning System: Theory and

Applications, Volume I, Progress in Astronautics and Aeronautics, Volume 163,

American Institute of Aeronautics and Astronautics, Inc., Washington, 1996

Spilker, J. J. (1977): Digital Communication by Satellite, Prentice-Hall Inc.,

Englewood Cliffs, New Jersey, 1977

Viterby, A. J. (1966): Principles of Coherent Communication, Part I, McGraw-Hill,

New York, 1966

Van Trees, H. L. (1971): Detection, Estimation and Modulation Theory, Part II, John

Wiley & Sons, New York

FIGURES AND ILLUSTRATIONS

Figure 1: Generic tracking loop block diagram. Potential error sources are represented by

blue-colored arrows.

Figure 2: Generic block diagrams of 1st, 2nd and 3rd order tracking loops

Figure 3: Constant PSD of vibration (f=[0 ∞])

Figure 4: Constant PSD of vibration (band-limited, f=[f1 f2])

Figure 5: Typical Aircraft Random Vibration Envelope

Figure 6: Influence of random vibration as a function of the loop noise bandwidth BL.

Figure 7: Reduction of vibration-induced phase jitter by neglecting low frequency

vibrations.

Figure 8: Reduction of vibration-induced phase jitter by enhancing the oscillator’s g-

sensitivity for 2nd and 3rd order PLLs.

Figure 9: PLL phase jitter due to frequency instabilities of the receiver clock

Figure 10: PLL jitter due to thermal noise for the GPS signals.

Figure 11: Dynamic stress errors for 1st, 2nd and 3rd order PLLs.

Figure 12: Tracking performance of a 2nd order PLL. The left diagram illustrates the

influences of thermal noise and dynamic stress error, whereas the right diagram

illustrates the tracking performance in the presence of additional oscillator phase

noise.
 TABLES

Loop
Laplace Transform F(s) Typical Filter Values Remarks
Order
BL = 0.2500·ωL
1st 1 Sensitive to velocity stress
ωL = 4.00·BL
α 1,1 α2 BL = 0.5305·ωL
2nd 1+ = 1+ Sensitive to acceleration stress
s s ωL = 1.885·BL
α 1,3 α 2,3 α β BL = 0.8333·ωL
3rd 1+ + = 1 + 3 + 23 Sensitive to jerk stress
s s2 s s ωL = 1.200·BL
Table 1: Loop Filter Characteristics

Oscillator h0 [s] h-1 [-] h-2 [1/s]

TCXO 1.00*10-21 1.00*10-20 2.00*10-20
OCXO 2.51*10-26 2.51*10-23 2.51*10-22
Rubidium 1.00*10-23 1.00*10-22 1.30*10-26
Cesium 2.00*10-20 7.00*10-23 4.00*10-29
Table 2: Clock parameters of different oscillators

Resulting PLL Jitter Due to Allan Deviation Phase Noise

Oscillator
2nd order PLL 3rd order PLL
TCXO 1.58° 0.84 mm 2.09° 1.11 mm
OCXO 0.10° 0.05 mm 0.14° 0.07 mm
Rubidium 0.14° 0.07 mm 0.18° 0.10 mm
Cesium 4.09° 2.16 mm 4.98° 2.63 mm
Table 3: Allan deviation induced PLL jitter (expressed in [deg] and [mm]) for 2 nd and

3rd order PLLs when using different types of oscillators. The computations were carried

out for the GPS L1 signal.

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