Full text of DOI:10.1002/j.2161-4296.2000.tb00215.x

Extended Theory of Early-Late Code
Tracking for a Bandlimited GPS Receiver
JOHN W. BETZ and KEVIN R. KOLODZIEJSKI
The MITRE Corporation, Bedford, Massachusetts
Received September 1999; Revised May 2000
ABSTRACT: Code-tracking accuracy, an important attribute of GPS receivers, depends both on characteristics of
the signal being tracked and on the design of the receiver. A simple expression has been available to predict
code-tracking accuracy for early-late processing of signals with sinc-squared spectra in white noise for an infinite
front-end bandwidth receiver. However, the literature has not indicated when this approximation holds. This
paper provides a more exact analysis of code-tracking accuracy for early-late discriminators processing conven-
tional binary phase shift keyed signals in white noise. New analytical expressions apply for various front-end
bandwidths, discriminator spacings, and code-tracking loop bandwidths, while using linear models based on a
small-error assumption. A theoretical lower bound is also supplied, indicating inherent limits on accuracy for
given conditions. While evaluation of the exact expressions requires numerical integrations, new algebraic
approximations are also provided. Numerical results compare the various expressions and approximations.
INTRODUCTION
The code-tracking performance of a receiver with
a bandlimited front end is of increasing importance.
Advances in receiver technology enable wider band-
widths and narrower early-late spacing. The emerg-
ing question is what values are best, rather than
what values can be built. Very wide-bandwidth
CrA-code receivers with narrow early-late spacing
are being built; while the primary motivation is
discrimination against multipath, it is important to
understand the consequences for code-tracking
accuracy in white noise. In addition, the new
Theory has long been available for predicting the
code-tracking accuracy of conventional biphase sig-
nals with pseudonoise spreading of rectangular
chips for code-tracking loops using early-late dis-
criminators and receivers having infinite front-end
bandwidth. However, since all receiver front ends
are bandlimited, it is necessary to know the condi-
tions under which the infinite bandwidth approxi-
mation applies, and to characterize code-tracking
errors when this approximation does not hold. Ref-
Ž
.
civil signal design at L5 1176.45 MHz em-
w x
1 presents this theory in the context of
erence
w x
ploys 10.23 MHz chip rates 3 ; the infinite band-
GPS, providing an algebraic approximation for
infinite front-end bandwidth and outlining an
approach for numerical calculations when the
front-end bandwidth is finite.
width theory does not fully apply for this important
situation.
Ž
This paper presents new expressions that as
long as the standard assumptions apply accurately
.
w x
Although the algebraic approximation in 1 has
describe code-tracking accuracy in white noise with
any degree of bandlimiting, along with a new
closed-form approximation and the conditions un-
der which this approximation applies. All of the
discussion in this paper assumes that the early-late
spacing in the discriminator is less than or equal to
a chip period, and that the receiver front-end band-
width is wide enough to pass at least half the main
lobe of the signal’s sinc-squared spectrum. We show
been applied to cases in which the front-end band-
width is finite, later work has attempted to develop
explicit theory that accounts for bandlimiting at the
w x
receiver front end 2 . This extension does not com-
pletely avoid assumptions of infinite bandwidth,
and thus yields misleading results. In particular, it
does not account for the effects of noise correlation
due to front-end bandlimiting.
w x
that the infinite bandwidth approximation 1 ap-
N
AVIGATION: Journal of The Institute of Navigation
plies when the receiver is spacing-limited: when the
Vol. 47, No. 3, Fall 2000
Printed in the U.S.A.
211

Ž
product of the front-end complex bandwidth in
Furthermore, over relatively small ranges of
front-end bandwidths and early-late spacings, the
.
Ž
.
Ž
hertz and the two-sided early-late spacing in
.
w x
Ž
.
seconds is greater than ␲. As has been shown 1 ,
the approximate normalized code-tracking error in
the spacing-limited case depends on the early-late
spacing, and not the receiver front-end bandwidth.
The infinite bandwidth approximation does not
hold, and a new approximation is supplied, when
the receiver is bandwidth-limited: the product of
root-mean-squared RMS code-tracking error oscil-
lates over a peak-to-peak range more than 25 per-
cent. Choosing appropriately matched bandwidth
and spacing can significantly lower the noise-
related contribution to code-tracking error.
Finally, we also show that, for any fixed front-end
bandwidth and high enough signal power, the
code-tracking error converges, for vanishing early-
Ž
.
.
the front-end complex bandwidth in hertz and the
two-sided early-late spacing in seconds is less
Ž
.
Ž
Ž
.
late spacing, to a lower bound LB computed for
than unity. In the bandwidth-limited case, the ap-
proximate normalized code-tracking error depends
on the receiver front-end bandwidth, and not the
the bandlimited signal. This result is in contrast to
previous claims 2 .
w x
This paper draws on a more general theory of
w x
w
x
early-late spacing. The previous expression 2 is
code-tracking accuracy developed in 4, 5 . While
the more general theory applies to signals with
arbitrary spectra and for additive Gaussian noise
having arbitrary spectra, this paper focuses on the
case of bandlimited white noise and signals having
sinc-squared spectra, such as the conventional
not accurate for the bandwidth-limited case. A new
approximation is also supplied for the transition
region between the spacing-limited case and the
bandwidth-limited case.
A qualitative description of the problem being
analyzed and an intuitive explanation of the results
help motivate the remainder of this paper. It has
been established that analysis of code-tracking ac-
curacy, under conventional assumptions of station-
ary noise, no dynamics, and moderate or high sig-
nal-to-noise ratio, can proceed by modeling a nonco-
herent early-late code-tracking loop in linearized
form: the difference in estimated power between
early and late correlator taps provides an error
signal, which is scaled by the discriminator gain
and smoothed in the code-tracking loop.
Bandlimiting of the signal smoothes its correla-
tion function, tending to reduce the discriminator
gain and increase code-tracking error. This effect
has been accounted for in previous attempts to
analyze the effects of limited front-end bandwidth.
But a further effect occurs that has not previously
been taken into account: bandlimiting of the noise
introduces additional correlation between the error
at the early tap and the error at the late tap,
beyond the correlation introduced at smaller chip
spacings from correlating the noise against the early
and late reference signals. The consequence is that
the code-tracking error of bandlimited receivers can
differ from what has been predicted previously.
One consequence of this new and more accurate
theory involves application of the discriminator
gain. While it has long been recognized that larger
discriminator gain implies improved code-tracking
accuracy for infinite bandwidth receivers, this rela-
tionship does not necessarily hold except in the
spacing-limited case. If the front-end bandwidth is
not extremely large, there is a point of diminishing
returns in the reduction of code-tracking error once
the early-late spacing becomes smaller than the
reciprocal of the receiver bandwidth, even though
the discriminator gain may continue to increase.
Ž .
signals used for CrA-code and P Y -code and the
new civil signal proposed for L5. In contrast with
w
x
the more detailed development in 4, 5 , this
paper emphasizes results and their application and
interpretation.
The following analysis applies to conditions that
satisfy a set of fundamental assumptions. It is as-
sumed that the integration time used in the dis-
criminator is longᎏmuch longer than the recipro-
cal of the receiver front-end bandwidth. The signal
Ž
.
is assumed to be a known to the receiver segment
of pseudorandomly modulated direct sequence
spread-spectrum chips. Any effects from short peri-
Ž
w x.
ods of the spreading code e.g., for CrA-code 6 are
deemed to be negligible in white noise. The noise is
statistically stationary, white, and Gaussian. The
resulting errors are assumed to be small, so that
the linearized analysis applies.
The paper models the code-tracking loop as an
Ž
.
estimator of the time of arrival TOA of the re-
ceived signal. The next section defines the problem
being analyzed, introduces the notation and as-
sumptions that are employed, and shows how the
tracking loop affects errors from TOA estimation.
The following section presents exact analytical re-
Ž
.
sults within the conventional assumptions used
that predict code-tracking accuracy in bandlimited
white noise, derives new approximations to these
exact expressions, and compares the new approxi-
mations and previous expressions. This section also
presents a lower bound on code-tracking accuracy
that applies for any discriminator with a band-
limited receiver. A set of numerical results is
then presented to explore and compare the differ-
ent expressions and approximations. The last sec-
tion summarizes the results and discusses their
implications.
212
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PROBLEM FORMULATION AND
MATHEMATICAL MODELS
is defined as a power spectral density normalized to
unit power over infinite bandwidth
This section formulates the problem being ana-
lyzed, introducing notation and describing the over-
all processing flow under consideration. It provides
mathematical models and assumptions, and defines
some common quantities that are employed in sub-
sequent sections.
The analysis uses complex baseband representa-
tions of signals and noise, and lowpass equivalent
models of filtering and processing. Thus, frequency
values of zero in the baseband representations cor-
respond to the carrier frequency of the radio fre-
quency signal.
ϱ
Ž .
Gs f df s 1
ϱ
Ž .
2
H
y
and Cs is commonly referred to as the signal car-
rier power, also defined over an infinite bandwidth.
It is assumed that this power spectral density is
symmetric about f s 0. The noise is white with
power spectral density N0 , so that the signal has a
carrier power-to-noise density ratio of CsrN0 Hz.
Signals of interest here are phase shift keyed
Ž
.
Ž
PSK signals with rectangular chip shapes i.e.,
.
non-return-to-zero chips . For conciseness, we refer
to these modulations in this paper as conventional
PSK signals, in contrast, for example, with the
subcarrier signals described in 7 . For the conven-
tional PSK signals, if the chip period is Tc , the
normalized power spectrum of the signal is
Ž
.
Ž .
Assume a complex-valued signal s t that is
Ž
.
known except for unknown delay and carrier phase
at the receiver. It is assumed that the frequency of
arrival is known perfectly. However, as long as the
frequency of arrival is known to within a fraction of
the reciprocal of the coherent integration time, this
approximation is valid. The received data is then
w x
Ž .
Gs f s Tc sinc ␲fTc
2 Ž
.
Ž .
3
Ž .
the sum of the signal and white noise, n t . The
where we neglect any spectral lines introduced by
received data is observed over a long time interval
i␪
short spreading codes.
Ž .
Ž
.
Ž .
Tobs: x t s e s t y t0 q n t , 0 F t F Tobs, where
the unknown delay, or time of arrival, t0 , is defined
relative to an arbitrary origin, and ␪ is the carrier
phase. Consistent with most analyses of code-
tracking accuracy, it is assumed that t0 either is
fixed or has time variation that is being tracked
perfectly by the receiver over the time period of
interest, so no dynamics need be considered explic-
itly. However, the results hold as long as any change
in t0 not tracked by the receiver is somewhat slower
than the reciprocal bandwidth of the code-tracking
loop.
Ž
Ž
..
w x
It is well known see equation B-8 in 8 , for
.
example that when the receiver front end has com-
Ž
plex bandwidth ␤r produced with ‘‘ideal filtering’’
ᎏa rectangular bandpass filter with no phase re-
sponse , correlation of the received signal plus noise
with the reference signal produces a maximum out-
put signal-to-noise ratio SNR
.
Ž
.
TCs
N0
TCs
N0
␤
rr2
br2
Ž .
Gs f df s
␳o (
sinc2 ␲f df
Ž .
H
H
y
␤rr2
ybr2
Ž .
4
The integration time in the discriminator is de-
Ž
.
noted T, while the one-sided equivalent rectangu-
where the approximation applies for output SNR
somewhat greater than unity, the normalized re-
ceiver front-end bandwidth is denoted b s Tc␤r , and
the integral indicates the fraction of signal power
passed by the receiver front end.
This paper evaluates different code-tracking loop
approaches that all satisfy the canonical form illus-
trated in Figure 1. The received signal plus noise
enters a discriminator, which effectively is an esti-
mator of the signal’s TOA. A previous estimate of
Ž
lar bandwidth also known as the noise equivalent
bandwidth of the code-tracking loop is denoted BL .
.
Furthermore, it is assumed that Tobs is much
greater than 1rBL , and that the analysis is not
concerned with the transient start-up phase, so the
time frame of interest is 1rBL < t F Tobs
.
Since the signal is modeled as known and not a
random process at the receiver, a power spectral
density is found as follows. The Fourier trans-
form of the signal observed over the time interval
Ž
.
Ž .
k y 1 T F t - kT is denoted SkT f :
kT
e
yi2 ␲ ft dt
Ž .
SkT f s
Ž
.
Ž .
1
s t y t0
.
H
Ž
ky1 T
<
Ž .<2
f rT,
Define a power spectral density as SkT
and recognize that, as long as T includes many chip
<
Ž .<2
f rT is approximately the same for
periods, SkT
any value of k. The signal’s power spectral density
Ž . Ž .
Ž .<2
<
is then denoted by CsGs f s SkT rT, where Gs f
f
Fig. 1 ᎐ Representation of Code-Tracking Loop
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
213

the signal’s TOA is also provided to this TOA esti-
mator, which uses an integration time of T seconds
to produce an unsmoothed TOA estimate. This esti-
mate is the update to the previous TOA estimate
based on the received signal plus noise. Un-
smoothed TOA estimates are processed by a TOA
smoothing filter, producing smoothed estimates that
are provided to the TOA estimator as previous
estimates.
This relationship between the variance of un-
smoothed TOA estimate and variance of smoothed
TOA estimate is independent of the specific pro-
cessing used to obtain the unsmoothed TOA esti-
mate, as long as the estimation process conforms to
the canonical form shown in Figure 1.
Ž .
Observe that equation 5 implies that the code-
tracking loop bandwidth is constant in the expres-
sion for variance of the code-tracking loop output.
In practice, this may not be the case, since chang-
ing the early-late spacing changes the loop gain and
Different versions of the TOA estimator are con-
sidered in this analysis. When the TOA is esti-
mated from early-late processing of the cross-corre-
lation between the received signal plus noise and a
reference signal, noncoherent early-late processing
Ž
w
x
.
hence the loop bandwidth see 10 , for example .
Numerical calculations performed by the authors
have shown that for limited front-end bandwidths,
the discriminator gain diminishes as the early-late
spacing becomes small, increasing the loop band-
Ž
.
NELP involves the magnitude-squared of this
cross-correlation, implying no knowledge of align-
ment between the phase of the reference signal and
that of the received signal. Coherent early-late pro-
Ž
Ž ..
width and thus the output variance equation 5 .
Attempts using computer simulations or hard-
ware to explore the behavior described in this paper
must either account for this change in loop band-
width or keep the loop gain hence the loop band-
width constant as the early-late spacing is modified.
Ž
.
cessing CELP uses only the in-phase part of the
cross-correlation function, implying perfect align-
ment between the phase of the reference signal and
that of the reference signal. Finally, an LB on
code-tracking error is obtained based on the
Ž
.
Cramer-Rao lower bound on TOA estimation, given
´
the integration time used in the TOA estimator.
Although the code-tracking loop representation
in Figure 1 differs from those traditionally shown
ANALYTICAL RESULTS
Early-late processing is one of the more common
approaches used for TOA estimation in the code-
tracking process portrayed in Figure 1, or equiva-
lently as a discriminator. Noncoherent processing,
which does not rely on knowledge or estimates of
the carrier phase, is of interest, as well as coherent
processing. The LB calculated for a bandlimited
signal provides a bound on processing performance.
While analytical results for various discriminator
processing approaches have been developed by the
authors, this paper concentrates for simplicity on
early-late processing, carrying along the LB for
reference. General expressions for code-tracking ac-
curacy using early-late processing are based on the
Ž
w x
.
in 9 , for example , it is mathematically equiva-
lent under the conventional linearizing assump-
tions used to analyze code-tracking error. This
representation was selected to show explicitly the
distinction and the relationship between the TOA
estimator and the TOA smoothing filter. Of course,
the smoothed TOA estimate is the conventional
output of the code-tracking loopᎏthe current esti-
mate of the timing of the received code.
w
x
As described in 4, 5 , the TOA estimator in
Figure 1 is modeled as performing block processing,
processing the received signal plus noise over each
T seconds, and using the smoothed TOA estimate
s
w x
derivations in 4 . While these expressions are ex-
␶
as the previous TOA estimate to provide an
uknsmoothed TOA estimate that persists for the
subsequent T seconds until it is replaced by the
next unsmoothed TOA estimate. The unsmoothed
TOA estimate produced at kT seconds is denoted
␶uk. Consistent with most analyses of code-tracking
accuracy, it is assumed that the unsmoothed TOA
estimates are unbiased and that the errors are
act under the assumptions given, they are compli-
cated, do not provide much insight, and require
numerical integrations in their evaluation. Exact
expressions are presented for the special case of
conventional PSK signals in bandlimited white
noise, which are much simpler but still require
numerical evaluation. We also present simpler ap-
proximations, derived from the exact expressions,
that can be evaluated using algebra, as well as the
conditions under which these approximations hold.
General expressions for code-tracking accuracy in
white noise with a bandlimited receiver front end
and early-late discriminator are obtained from those
in the appendix, which are not restricted to white
noise. The following results hold for conventional
PSK signals with chip period Tc seconds in white
noise when the integration time used to form a
u
u
Ä
small. Hence, the conditional mean of ␶ is E ␶k N
s 4
.
Ä
s 4 k
ÄŽ
u
u
␶
s t0 , and its variance is Var ␶k N ␶ s E ␶k y
t0k 2 N ␶ s ␴u .
k
2 4
2
k
w
x
It is shown in 4, 5 that for 0 - TBL - 0.5, the
variance of the code-tracking loop output is approx-
imately related to the variance of the unsmoothed
TOA estimate by
2
␶
2
Ä
s 4
Ž
.
Ž .
5
␴ s Var ␶ ( ␴u 2BLT 1 y 0.5BLT
k
214
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correlation estimate in the discriminator is T sec-
Using the observation that for 0 - b - 1, 1 y
Ž
.
Ž
.
< <
Ž
.
onds assumed to be much longer than Tc ; the
bandlimiting at the receiver is represented by a
rectangular bandpass filter with complex band-
sinc ␲b ( b, while for 1 F b , 1 y sinc ␲b ( 1,
Ž .
equation 6 can be approximated as
Ž
.
.
¡
~
BL 1 y 0.5BLT
Ž
.
width ␤r Hz assumed to be much larger than 1rT ,
,
b F 1
b ) 1
Cs
␤r s brTc; BL is the one-sided code-tracking loop
2
b2
Ž
.
bandwidth in hertz assumed to be less than 0.5rT ;
2
␶
N0
␴
Ž
.
⌬ s DTc is the two-sided early-late spacing in
Ž .
7
(
2
ž T /
Ž
BL 1 y 0.5BLT
c
Ž .
Ž
.
seconds; Gs f is the normalized unit area power
spectral density of the signal in 1rHz; Cs is the
signal carrier power over infinite bandwidth in
watts; and N0 is the power spectral density of white
Gaussian noise in wattsrhertz.
LB
,
C
2
s b
¢
N0
Note that the LB does not depend on the early-late
spacing, but instead relates how well any discrimi-
nator can perform with a given front-end band-
width.
Figure 2 illustrates the definition of the normal-
ized front-end complex bandwidth, b. Figure 3 illus-
trates the definition of the normalized two-sided
discriminator spacing D.
Ž .
Ž
.
Substituting Gw f s N0 into equation 12 and
performing some manipulations yields the error
variance of a code-tracking loop using NELP, in
normalized units of chip periods squared:
An LB on code-tracking error variance, denoted
2
␶
␴ in units of seconds squared, for a given front-end
filter bandwidth and given code-tracking loop is
Ž .
obtained by substituting Gw f s N0 into equation
2
␶
␴
Ž
.
14 , yielding the normalized code-tracking error
2
variance.
ž T /
c
NELP
2
Ž .
BL 1 y 0.5BLT
b
r2
␴
BL 1 y 0.5B T
sinc2 ␲f sin ␲fD df
Ž
.
Ž
.
2 Ž
.
␶
H
s
s
s
L
2
Cs
y
br2
ž T /
␤
rr2
2
c
2␲ Tc3
f 2 sinc2 ␲fTc df
Ž .
LB
s
Ž
.
H
2
Cs
N0
H r2
b
y
␤rr2
2
f sinc2 ␲f sin ␲fD df
Ž . Ž .
Ž
.
2␲
ž
/
N0
y
br2
Ž
Cs
.
BL 1 y 0.5BLT
H r2
b
sinc2 ␲f cos ␲fD df
Ž
.
2 Ž
.
w
Ž
.x
2␤rT
1 y sinc ␲␤rTc
c N0
y
br2
= 1 q
2
Cs
b
H r2
Ž
Cs
.
T
sinc2 ␲f cos ␲fD df
Ž . Ž .
BL 1 y 0.5BLT
Ž .
6
ž
/
ž N /
y
br2
0
w
Ž
.x
2b
1 y sinc ␲b
Ž .
8
N0
Fig. 2 ᎐ Signal Power Spectral Density and Two Different Values of Normalized Front-End
( )
Bandwidth b
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
215

Fig. 3 ᎐ Ideal Signal Autocorrelation Function and Two Different Values of Normalized Early-Late
(
)
Spacing D
Ž .
Considerable manipulation of equation 8 , using
a variety of trigonometric identities and series
approximations, yields approximations for three
cases of particular interest under the constraint
that b ) 1, which is ensured when the front-end
filter is wide enough to pass the center half of the
signal spectrum’s main lobe. The resulting approxi-
mation is
¡
Ž
.
.
.
BL 1 y 0.5BLT
2
D 1 q
,
␲ F Db
Cs
Cs
N0
Ž
.
2 y D
2
T
N0
2
Ž
BL 1 y 0.5BLT
1
b
1
2
2
␶
␴
q
b
D y
1 q
,
1 - Db - ␲
Db F 1
~
Ž .
9
(
␲ y 1 ž /
Cs
Cs
N0
b
2
ž T /
Ž
.
2 y D
2
T
c
NELP
N0
Ž
BL 1 y 0.5BLT
1
1
Cs
1 q
,
ž b /
Cs
2
T
¢
N0
N0
that for CELP, the error variance is
The condition ⌬␤r s Db G ␲ is referred to as spac-
ing-limited, since the error depends primarily on
the early-late spacing and not the front-end band-
width. The condition ⌬␤r s Db - 1 is referred to as
bandwidth-limited, since the error depends primar-
ily on the front-end bandwidth and not the early-
late spacing. The condition 1 - Db - ␲ indicates a
transition region between the two distinct limiting
cases. These three different regions are illustrated
2
␶
␴
2
ž T /
c
CELP
b
r2
sinc2 ␲f sin ␲fD df
Ž
.
Ž
.
2 Ž
.
BL 1 y 0.5B T
H
L
y
br2
s
2
Cs
H r2
b
2
f sinc2 ␲f sin ␲fD df
Ž . Ž .
Ž
.
2␲
ž
/
N0
y
br2
in Figure 4.
Ž
.
10
w x
Given the above, it is straightforward to show 4
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Fig. 4 ᎐ Region Boundaries in Discriminator Parameter Space Where Different Approximations
Apply
which can be approximated, again using a variety
of trigonometric identities and series approxima-
tions, as
Ž
.
.
.
¡
~
BL 1 y 0.5BLT
D,
␲ F Db
Cs
2
N0
2
Ž
BL 1 y 0.5BLT
1
b
b
1
2
␶
␴
q
D y
,
1 - Db - ␲
Db F 1
␲ y 1 ž /
Cs
Ž .
11
(
b
2
ž T /
2
c
CELP
N0
Ž
BL 1 y 0.5BLT
1
,
ž b /
Cs
2
¢
N0
The result for coherent early-late processing is a
lower bound on noncoherent early-late processing.
The two diverge significantly only when the ideal
postcorrelation SNR, TCsrN0 , is small.
Note that while the code-tracking error, ex-
pressed in fractional chip periods, is independent of
the chip period for the spacing-limited case when
the front-end bandwidth is large, the error variance
is inversely proportional to the normalized band-
width and independent of the early-late spacing for
the bandwidth-limited case. Also, the expression in
the reciprocal of the front-end bandwidth, early-late
processing provides optimal performance, and that
Ž
there is no benefit in terms of code-tracking accu-
.
racy in white noise to further reducing the early-
late spacing.
NUMERICAL RESULTS FOR NONCOHERENT
EARLY-LATE PROCESSING
The results in this section compare numerical
Ž
calculations that use the expression for LB equa-
Ž
.
Ž ..
equation 11 for the bandwidth-limited case is
tion 6 , the expression for noncoherent early-late
Ž .
w x
identical to the lower bound in equation 7 , indicat-
processing with infinite bandwidth from 1 , the
ing that once the discriminator spacing is less than
previous expression for finite front-end bandwidths
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
217

w x
from 2 , the exact expression for noncoherent
early-late processing equation 8 , and the new
approximation for noncoherent early-late process-
smaller for larger bandwidths and smaller discrimi-
nator spacings, with an oscillatory structure super-
posed for larger values of b and D.
Ž
Ž ..
Ž
Ž ..
ing equation 9 . The results are calculated using
a coherent integration time of T s 0.02 s and a
Figure 6 shows, in contrast, the normalized
code-tracking error predicted using the infinite
Ž
w x
code-tracking loop bandwidth one-sided equivalent
rectangular bandwidth of 1 Hz.
bandwidth approximation from 1 , which is equiva-
.
lent to using the expression for the spacing-limited
Ž
. Ž .
Figure 5 shows the normalized code-tracking er-
Db G ␲ case in equation 9 over all values of b
Ž .
ror predicted by the exact expression of equation 8
and D. Unlike the exact expression in Figure 5,
Figure 6 shows no dependence on front-end band-
width, shows the error approaching zero for dimin-
over a range of normalized front-end bandwidths
and discriminator spacings. The error tends to be
(
( ))
Fig. 5 ᎐ Exact Expression equation 8 Evaluated at CsrN0 of 30 dB-Hz
Fig. 6 ᎐ Infinite Bandwidth Approximation Evaluated at CsrN0 of 30 dB-Hz
218
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ishing early-late spacing, and does not display any
of the oscillatory structure.
The approximation error becomes very large for
small values of D, corresponding to the bandwidth-
limited case.
Figure 8 shows the normalized code-tracking er-
ror predicted using the previous expression from
w x
2 , which, like the exact expression shown in Fig-
ure 5, must be evaluated numerically. Like the
exact expression in Figure 5, Figure 8 shows depen-
dence on front-end bandwidth and displays some
oscillatory structure. However, it predicts that the
Figure 7 shows the absolute value of the frac-
tional error between the infinite bandwidth approx-
imation shown in Figure 6 and the exact expression
shown in Figure 5. While the error in the approxi-
mation is quite small for large bandwidth and large
early-late spacing, it increases in an arc whose
location in the D-b plane corresponds to the loca-
tion of the transition region portrayed in Figure 4.
Fig. 7 ᎐ Absolute Value of Fractional Error Between Infinite Bandwidth Approximation and Exact
Expression, with Larger Absolute Error Indicated by Lighter Shading
Fig. 8 ᎐ Previous Expression Evaluated at Cs rN0 of 30 dB-Hz
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
219

error increases for small bandwidth or for small
spacing, and the plot is truncated near the origin.
Figure 9 shows the absolute value of the frac-
tional error between the previous expression shown
in Figure 8 and the exact expression shown in
Figure 5. While the error is quite small over much
corresponds roughly to the location of the band-
width-limited region portrayed in Figure 4. The
approximation error becomes very large for small
values of D, corresponding to the bandwidth-limited
case.
Figure 10 shows the normalized code-tracking
Ž
Ž
of the plane note that it benefits from numerical
error predicted using the new approximation equa-
Ž ..
tion 9 . While the surface does not display the
.
integration rather than algebraic approximation , it
Ž
oscillatory structure in the exact expression allow-
increases in an arc whose location in the D-b plane
Fig. 9 ᎐ Absolute Value of Fractional Error Between Previous Expression and Exact Expression,
with Larger Absolute Error Indicated by Lighter Shading
Fig. 10 ᎐ New Approximation Evaluated at CsrN0 of 30 dB-Hz
220
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.
ing the algebraic expression to remain simple , it
end bandwidths that are not significantly larger
than the chip rate.
correctly shows the increased code-tracking error
with diminished front-end bandwidth, and also
shows that the error approaches a nonzero asymp-
tote with diminishing early-late spacing.
Figure 11 shows the absolute value of the frac-
tional error between the new approximation shown
in Figure 10 and the exact expression shown in
Figure 5. The error in the approximation is consis-
tently small over the entire range of parametersᎏa
substantial improvement over the previous expres-
sion for cases with small early-late spacing or front-
While the surfaces shown in the preceding fig-
ures indicate qualitatively the behavior of the exact
expression and the approximations, more quantita-
tive results are obtained by considering slices of
these surfaces. Figure 12 shows predicted code-
tracking error for CsrN0 of 30 dB-Hz and normal-
ized front-end bandwidth of b s 10ᎏa wide front-
end bandwidth. Except for the oscillatory behavior
predicted by the exact expression, the infinite band-
width approximation is accurate for wider discrimi-
Fig. 11 ᎐ Absolute Value of Fractional Error Between New Approximation and Exact Expression
Fig. 12 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Front-End
Bandwidth of 10
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
221

nator spacings. However, as the spacing becomes
narrower, the infinite bandwidth approximation
predicts too large an error, then predicts errors
smaller than the LB indicates are achievable. The
previous expression matches the exact expression
well in the spacing-limited region and the transi-
tion region, but has large error in the bandwidth-
limited region. In contrast, the new approximation
tracks the exact expression much better for the
smaller discriminator spacings. Both the exact ex-
pression and the new approximation show that with
small discriminator spacing, the code-tracking
error approaches the lower bound.
20 dB-Hz. The results are similar except that with
small discriminator spacing, the error approaches
an asymptote somewhat greater than the LB. This
behavior is due to the squaring loss incurred with
noncoherent processing. Since this behavior at lower
signal powers is well known, the remaining presen-
tation of results uses the higher CsrN0 of 30 dB-Hz.
Figure 14 shows predicted code-tracking error for
CsrN0 of 30 dB-Hz and normalized front-end band-
width of b s 3ᎏa moderate front-end bandwidth.
The infinite bandwidth approximation exhibits dif-
ferences from the exact expression for most dis-
criminator spacings, overestimating the error for
larger spacings and underestimating it for smaller
spacings. The previous expression deviates from the
Figure 13 shows predicted code-tracking error for
the same conditions as Figure 12, except CsrN0 of
Fig. 13 ᎐ Predicted Code-Tracking Error for CsrN0 of 20 dB-Hz and Normalized Front-End
Bandwidth of 10
Fig. 14 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Front-End
Bandwidth of 3
222
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Ž
exact expression for normalized discriminator spac-
ings less than 0.5, and incorrectly predicts increas-
ing error for smaller spacings. In contrast, the new
approximation tracks the exact expression accu-
rately over the range of discriminator spacings.
Figure 15 shows predicted code-tracking error for
CsrN0 of 30 dB-Hz and normalized front-end band-
width of b s 1.5ᎏa narrow front-end bandwidth
that may be of interest in designing discriminators
for the new civil signal at 1176 MHz. While the
infinite bandwidth approximation predicts continu-
ing reduced error from smaller discriminator spac-
ings, the previous expression predicts increasing
error from smaller discriminator spacings. In con-
trast, both the new approximation and the exact
expression indicate that there is little benefit in
.
terms of code-tracking accuracy in white noise to
using a discriminator spacing smaller than 1 chip
period, but neither does the error increase. This is
why the parameter set Db - 1 is called bandwidth-
limited: the small bandwidth imposes an inherent
limitation on code-tracking accuracy that cannot be
overcome with smaller discriminator spacing.
Figure 16 shows predicted code-tracking error for
CsrN0 of 30 dB-Hz and normalized discriminator
spacing of D s 1ᎏa wide spacing. The infinite
bandwidth approximation applies well over the
wider bandwidths, but does not predict the reduc-
tion in code-tracking error obtained at normalized
front-end bandwidths near 2. However, the new
Fig. 15 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Front-End
Bandwidth of 1.5
Fig. 16 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Discriminator
Spacing of 1
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
223

approximation and previous expression both per-
form well over the entire range of bandwidths,
since the bandwidth-limited region does not occur
with wide discriminator spacing. The performance
is much worse than the LB, indicating the opportu-
nity to reduce error with smaller discriminator
spacing.
Figure 17 shows predicted code-tracking error for
CsrN0 of 30 dB-Hz and normalized discriminator
spacing of D s 0.5ᎏa moderate spacing. The por-
tion of the graph left of the transition region may
be of particular interest in designing receivers for
the new civil signal at 1176 MHz. The infinite
bandwidth approximation applies well over the
wider bandwidths, but does not predict the reduc-
tion in code-tracking error obtained at normalized
front-end bandwidths below 6 or the increase in
error for normalized front-end bandwidths smaller
than 2. The error predicted by the previous expres-
sion is too large for smaller bandwidths. Although
the error is smaller than in Figure 16, the perfor-
mance is still much worse than the LB, indicating
the opportunity to reduce error further with even
smaller discriminator spacing.
Figure 18 shows predicted code-tracking error for
CsrN0 of 30 dB-Hz and normalized discriminator
spacing of D s 0.1ᎏa narrow spacing. The infinite
bandwidth approximation applies well over the
wider bandwidths, but does not predict the larger
code-tracking error for normalized front-end band-
widths less than 10. In contrast, the previous ex-
pression predicts excessively large code-tracking er-
Fig. 17 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Discriminator
Spacing of 0.5
Fig. 18 ᎐ Predicted Code-Tracking Error for CsrN0 of 30 dB-Hz and Normalized Discriminator
Spacing of 0.1
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rors for the bandwidth-limited region. Performance
is very close to the LB, and is clearly bandwidth-
limited for normalized front-end bandwidths less
than 10.
widths and discriminator spacings, and can be eval-
uated readily since they involve only algebraic ex-
pressions. The results show that there is no perfor-
Ž
mance penalty in the sense of increased code-
.
The oscillatory behavior of the exact expression
for code-tracking error in the spacing-limited region
of parameter space can be observed in Figures 10,
11, 14, and 15. In each of these cases, when the
receiver design is spacing-limited so that Db G ␲
there are local minima in code-tracking error for
Db s 2, 6, 10, . . . . Conversely, there are local max-
ima in code-tracking error for Db s 4, 8, . . . . Conse-
quently, there may be advantages in selecting
paired values of front-end bandwidth and discrimi-
nator spacing that avoid these local maxima and
seek the local minima.
tracking error for very small discriminator spac-
w
x
ings. Recent work 10 has also verified the purely
theoretical results in this paper with experimental
results.
Discriminator performance using parameter val-
ues associated with the bandwidth-limited and
transition regions is likely to take on increasing
importance with the higher chip rate signals pro-
posed for the new civil signal on L5, and with
continuing interest in narrow correlator spacing for
multipath processing as well as improved code-
tracking accuracy. Either the exact expression or
the new approximation presented in this paper
should be employed.
CONCLUSIONS
This paper has presented more accurate analyti-
cal expressions for the code-tracking accuracy of
conventional binary phase shift keyed signals in
white noise for limited front-end bandwidth and a
code-tracking loop using an early-late discrimina-
tor. While evaluation of the exact expressions re-
quires numerical integration, new approximations
have been presented that involve only algebraic
expressions. An LB on code-tracking error for a
bandlimited receiver has also been presented. The
new approximation shows that there are three re-
gions of receiver parameter space that must be
considered separately. These regions are defined in
terms of the normalized front-end bandwidth b,
which is the complex bandwidth of the receiver
front end multiplied by the signal chip period, and
the normalized early-late discriminator spacing D,
ACKNOWLEDGMENTS
This work was sponsored by the U.S. Air Force
under contract number F19628-99-C-0001. The au-
thors express their appreciation to Dr. Christopher
Hegarty for his suggestions on the presentation of
this work.
APPENDIX
GENERAL EXPRESSIONS FOR CODE-TRACKING
ERROR IN GAUSSIAN NON-WHITE NOISE PLUS
INTERFERENCE
This appendix presents general analytical results
w
x
for code-tracking accuracy, as derived in 4, 5 .
Special cases of these expressions for bandlimited
white noise and conventional binary phase shift
keyed signals have been developed in the main
body of this paper.
Ž
.
which is the two-sided spacing of the discrimina-
tor expressed as a fraction of the signal chip period.
The region defined by ␲ F Db is referred to as
spacing-limited, since the error depends primarily
on the early-late spacing and not the front-end
bandwidth. The infinite bandwidth approximation
When the integration time used to form a correla-
tion estimate in the discriminator is T seconds as-
Ž
sumed to be much longer than the signal’s spread-
w x
1 applies for this set of receiver parameters, but
.
ing code period , the bandlimiting at the receiver is
does not apply for other parameter values. The
region defined by Db - 1 is referred to as band-
width-limited, since the error depends primarily on
the front-end bandwidth and not the early-late
spacing. The condition 1 - Db - ␲ indicates a
transition region between the two distinct limiting
cases. The previous expression for the bandlimited
case is incorrect for the bandwidth-limited case
since it does not account for the effect of bandlimit-
ing on the noise correlationᎏan important effect
for narrow discriminator spacings.
represented by a rectangular bandpass filter with
complex bandwidth ␤r Hz assumed to be much
Ž
.
larger than 1rT ; BL is the one-sided code-tracking
Ž
loop bandwidth, in hertz assumed to be less than
0.5rT ; ⌬ is the two-sided early-late spacing in
.
Ž
.
Ž .
Ž
.
seconds; Gs f is the normalized unit area power
spectral density of the signal in 1rhertz, assumed
to be symmetric; Cs is the signal carrier power over
Ž .
infinite bandwidth in watts; Gw
f is the power
spectral density of Gaussian noise plus interference
in wattsrhertz; and a code-tracking loop using non-
Ž
.
The new approximations presented in this paper
are accurate over the range of front-end band-
coherent early-late processing NELP has output
variance, in units of seconds squared, of
Vol. 47, No. 3
Betz et al.: Extended Theory of Early-Late Code Tracking
225

␤
rr2
Ž . Ž . 2 Ž
Gw f Gs f sin ␲f⌬ df
.
H
y
␤rr2
2
␶
Ž
.
Ž
.
␴
NELP s BL 1 y 0.5BLT
2
␤
rr2
2␲ 2Cs
fGs f sin ␲f⌬ df
Ž
.
Ž .
Ž
.
H
ž
/
y
␤rr2
2
2
␤
rr2
␤rr2
Gw f Gs f ei2 ␲ f⌬ df
Ž . Ž .
Gw f Gs f df
Ž . Ž .
y
H
H
ž
/
y
␤rr2
y␤rr2
Ž
.
12
q
2
␤
rr2
␤rr2
2
4 2␲ TCs2
fG f sin ␲f⌬ df
Gs f cos ␲f⌬ df
Ž
.
Ž .
Ž
.
Ž .
Ž
.
H
H
s
ž
/
y
␤rr2
y␤rr2
2. Holmes, J. K., Noncoherent Late Minus Early Power
Code Tracking Loop Performance with Front End
Filtering, Proceedings of ION GPS-97, The Institute
of Navigation, September 1997.
3. Spilker, J. J., Proposed New Civil GPS Signal at
1176.45 MHz, Proceedings of ION GPS-99, The Insti-
tute of Navigation, September 1999.
4. Kolodziejski, K. R. and J. W. Betz, Effect of Non-White
Gaussian Interference on GPS Code-Tracking Accu-
racy, Technical Report MTR99B21R1, The MITRE
Corporation, McLean, VA, June 1999.
5. Betz, J. W., and K. R. Kolodziejski, Generalized The-
ory of GPS Code-Tracking Accuracy with an Early-
Late Discriminator, Submitted to IEEE Transactions
on Aerospace and Electronic Systems, July 2000.
In contrast, a code-tracking loop using coherent
early-late processing CELP has output variance,
also in units of seconds squared, of
Ž
.
2
␶
Ž
.
Ž
.
␴
CELP s BL 1 y 0.5BLT
␤
rr2
Ž . Ž . 2 Ž
Gw f Gs f sin ␲f⌬ df
.
H
y
␤rr2
=
2
␤
rr2
2␲ 2Cs
fGs f sin ␲f⌬ df
Ž
.
Ž .
Ž
.
H
ž
/
.
y
␤rr2
Ž
13
Ž
.
A lower bound LB on code-tracking loop output
Ž
.
6. Parkinson, B. W. and J. J. Spilker, Jr. eds. , Global
Positioning System: Theory and Applications, Volume
I, Progress in Astronautics and Aeronautics, Volume
163, 1996.
variance is obtained by computing the Cramer-Rao
´
lower bound on time-of-arrival estimation for inte-
gration time T, then assuming the estimates are
smoothed with the code-tracking loop, yielding
the LB
7. Betz, J. W., The Offset Carrier Modulation for GPS
Modernization, Proceedings of The Institute of Navi-
gation’s 1999 National Technical Meeting, January
1999.
2
␶
Ž
.
Ž
.
␴
LB s BL 1 y 0.5BLT
8. Betz, J. W., Effect of Narrowband Interference on
GPS Code-Tracking Accuracy, Proceedings of ION-
2000, The Institute of Navigation, January 2000.
9. Holmes, J. K., Coherent Spread Spectrum Systems,
Krieger Publishing, 1990.
10. Stephens, S. A. and J. B. Thomas, Controlled-Root
Formulation for Digital Phase-Locked Loops, IEEE
Transactions on Aerospace and Electronic Systems,
Vol. 31, No. 1, January 1995, pp. 78᎐95.
11. Betz, J. W., Effect of Narrow Correlator Spacing on
Code-Tracking Accuracy, Proceedings of The Insti-
tute of Navigation’s 1999 National Technical Meet-
ing, January 2000.
1
Ž
.
14
=
Ž .
Gs f
␤
rr2
2␲ 2C
f 2
df
Ž
.
sH
Ž .
Gw
f
y
␤rr2
REFERENCES
1. Van Dierendonck, A. J., P. Fenton, and T. Ford,
Theory and Performance of Narrow Correlator Spac-
ing in a GPS Receiver, NAVIGATION, Journal of The
Institute of Navigation, Vol. 38, No. 3, Fall 1992,
pp. 265᎐283.
226
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Fall 2000