Journal of Global Positioning Systems (2004)
Vol. 3, No. 1-2: 45-48
Aperture Jitter Effects in Software Radio GNSS Receivers
Andrew Dempster
School of Surveying and Spatial Information Systems, University of New South Wales, Sydney 2052
e-mail: a.dempster@unsw.edu.au Tel: +61 2 93854182; Fax: +61 2 93137493
Received: 15 Nov 2004 / Accepted: 3 Feb 2005
Abstract. Increasingly, software radio techniques are
being used in the implementation of communications
receivers in general, and GNSS receivers in particular. In
such a receiver, the received signal is sampled as close to
the receive antenna as possible, and all subsequent
processing uses digital signal processing (DSP)
techniques. The sampling clock will suffer from phase
noise instabilities, leading to a phenomenon known as
aperture jitter. This paper examines the effects of aperture
jitter for a number of “typical” software radio GNSS
receivers. A jitter specification is derived which restricts
the noisy effects due to jitter to 10dB below thermal
noise. It transpires that regardless of the new signals that
are selected to accompany it, it is the L1 signal that drives
this jitter specification.
Key words: Software radio, GNSS receiver, aperture
jitter.
1 Introduction
1.1 Bandpass Sampling
With the increase in the variety of signals required to be
processed by radio receivers, versatile receiver designs
are gaining in popularity. Software radio (SWR) receivers
(which perform the “radio” functions in a processor) and
software-defined radio (SDR) receivers (where these
procedures are also performed digitally but using
hardware controlled and configured in software) are two
approaches that deliver this versatility. In both these
approaches, the ideal is to convert the received signal to a
sampled digital signal as closely as possible to the receive
antenna. With the advent of new GPS signals, and a range
of new Galileo signals, GNSS is an application where
SWR and SDR are likely to have an impact.
Sampling the received signal using low-pass sampling
(LPS), the more usual interpretation of Nyquist’s
sampling theorem, requires a sampling rate of twice the
maximum frequency of interest. For GNSS, this is of the
order of 3.2Gs/s. Analogue-to-digital converters (ADCs)
do exist that will operate at these frequencies (Lee, 2003,
Poulton, 2003), but they are expensive and consume a
great deal of power. A cheaper, more efficient method of
conversion in a SWR receiver is “bandpass sampling”
(BPS) (Vaugan, 1991), where, consistent with Nyquist’s
sampling theorem, it is possible to sample an RF signal at
twice the bandwidth of that signal. That signal is the
“aliased” into the baseband used for that sampling rate.
For bandpass sampling to be successful, out of band
signals must be attenuated to reduce aliasing of unwanted
signals and noise, requiring a high-Q bandpass filter. This
is a more demanding design requirement than the low
pass anti-aliasing filter required for LPS at the same
sampling frequency. When several bands are required for
downconversion, the minimum sampling rate is twice the
sum of the bandwidths. However, there are constraints
that mean that the sampling rate is usually higher than
that minimum.
Software receivers have been designed for the GPS L1
signal (Akos, 1996, Ledvina, 2003, Lin, 2001) and for the
L5 signal (Ries, 2002). Bandpass sampling dual band
receivers have also been designed, for L1 GPS and
Glonass (Akos, 1999), and for GPS L1 and L3 (Thor,
2003).
1.2 Aperture Jitter
Aperture jitter is defined as the sample-to-sample
variation in time between the effective points at which the
samples are actually taken (Sheingold, 1986). The effects
of aperture jitter can be modelled as noise (we will use
the terminology of (Sun, 2004). If the signal into the
analogue-to-digital converter (ADC) is y(t), then the jitter
noise can be modelled as:
46 Journal of Global Positioning Systems
()()( )
nnn tytyn
+=
τ
ε
τ
(1)
where tn is the ideal sampling instant and
τ
n is the offset
to that instant due to jitter. The power of this jitter noise
signal is:
()
[]
()
[
]
2
2nEnEN
τττ
εε
−= (2)
Previous work has identified two formulae for this jitter
noise power. Both assume a sinusoidal signal and zero-
mean gaussian jitter around fixed “perfect” sampling
points. This latter assumption has been found
experimentally to be reasonable (Shinagawa, 1990),
although the spectrum is not white if a phase-locked loop
is used to generate the clock (Da Dalt, 2002), or if the
jitter is accumulative(Awad, 1998). The first formula
assumes that 12 <<
nc
f
τπ
, where fc is the carrier
frequency of the sampled signal, i.e. that the jitter offset
is much less than a period of the carrier (Shinagawa,
1990):
2222
2AfN c
ττ
σπ
= (3)
where
σ
τ
2 is the variance of
τ
n and A is the amplitude of
the sinusoid. The second formula makes no assumption
about the size of
τ
n (Awad, 1998):
(
)
222
2
21
τ
σπ
τ
c
f
eAN
−= (4)
Other simple non-sinusoidal signals such as square and
triangular waves have also been examined with regard to
their jitter noise power (Kobayashi, 1999). Comparisons
have been made between the noise generated by jitter and
thermal (Shinagawa, 1990) and quantisation (Kobayashi,
1999) noise.
2 Example GNSS Receivers
2.1 Frequency Bands
Sampling requirements for a number of GNSS example
systems are examined, restricting interest to GPS and
Galileo signals that are available to commercial users.
Commercial satellite navigation receivers have until
recently largely been restricted to using the GPS signal at
L1 (carrier 1575.42MHz, chipping rate 1.023Mcps) (ICD,
2003). Soon GPS will provide a similar signal on L2
(carrier 1227.6 MHz) (ICD, 2003, Fontana, 2001) and a
new signal on L5 (carrier 1176.45 MHz, chipping rate
10.23Mcps) (ICD, 2002, RTCA, 2000, van Dierendonck
2000).
Galileo will have several signals available to commercial
users. Open services (OS) are available on E5 (1164-
1215MHz, nominal carrier 1189MHz) and E2-L1-E1,
known for convenience as L1 (1559-1592Mz).
Commercial services (CS), for which a fee is required but
are not restricted to security services, are available on E6
(1215-1300MHz, carrier 1278.75MHz) and L1. At the
time of writing, the above frequency allocations were the
latest to have been formally released (Hein, 2002, Hein,
2003), although they will change because of agreements
between the Galileo and GPS teams (GPSW, 2004,
2004A). For instance, the OS L1 signal bandwidth has
been reduced by a factor of 2 so that it has a 1.013MHz
chipping rate, and a binary offset code of 1.013MHz, a
so-called BOC(1,1) code.
Combinations of these signals were selected that are
likely to be common within a “GNSS” receiver. Example
1 is the most expensive receiver processing GPS L1, L2,
L5 and Galileo OS and CS, using L1, E5 and E6.
Example 2 uses only the free-to-air GPS L1 and L2 and
Galileo OS, on L1 and E5. Example 3 uses L1 and E5
(ignoring L2) and example 4 uses L1 and L2 (i.e. the
GPS bands that are “currently” available, and
incorporating the Galileo L1). Example 5 is cheapest
arrangement, simply using “L1 only” and incorporating
the new Galileo signal.
Table 1. Frequency bands for example receiver designs
Band E5
(L5)
L2 E6 L1
Fmin 1164 1217 1260 1573
fmax 1214 1238 1300 1577
Ex. 1 X X X X
Ex. 2 X X X
Ex. 3 X X
Ex. 4 X X
Ex. 5 X
The frequencies required for the four examples are shown
in Table 1. It can be seen that because the frequency
bands are contiguous, in all cases, only two bands are
required. Example 1, for instance, must receive in the
range 1164-1300MHz and 1573-1577MHz.
2.2 Sampling Rates
For simplicity, as the type of results sought need only to
be indicative, bandpass sampling is assumed, but only the
minimum sampling rate required by Nyquist’s theorem is
considered (i.e. the full set of constraints identified in
(Vaugan, 1991) are not all accounted for). These
minimum sampling rates are shown in Table 2.
2.3 Data Bandwidths
As we have seen, aperture jitter tends to be modelled as
noise. This noise can be considered to be additive with
the thermal noise present in the receiver. For the new
signals, which have data in one channel and a dataless
Dempster: Aperture Jitter Effects in Software Radio GNSS Receivers 47
pilot signal in quadrature, the effects of noise are greater
in the data channel, so we will consider the bandwidth of
the data channel as being the relevant bandwidth in which
thermal noise appears. The data bandwidths of the GNSS
signals are shown in Table 3.
Table 2. For the 5 examples, the lower bandwidth, the upper bandwidth,
the total bandwidth, and the minimum sampling rate, considering only
Nyquist’s theorem. All figures in MHz
BW
1
BW
2
Total
BW
fsmin
Ex. 1 136 4 140 280
Ex. 2 74 4 78 156
Ex. 3 50 4 54 108
Ex. 4 21 4 25 50
Ex. 5 - 4 4 8
Table 3 GNSS data bandwidths
GPS band
[ICD (2003), van
Dierendonck (2000)]
Data bandwidth (Hz)
L1 100
L2 100
L5 2000
Galileo band [Hein
(2002), Hein (2003)]
L1 500
E5 500
E6 2000
2.4 Signal Strengths
We will be comparing SNR due to noise to SNR due to
jitter. In order to do this, we will need to know the signal
levels for each of the signals we are examining. These are
shown in Table 4.
Table 4 GNSS signal strengths
GPS signal
[ICD (2003), van
Dierendonck (2000)]
Signal strength
(dBW)
L1 -160
L2 -160
L5 -154
Galileo signal [Hein
(2002), Hein (2003)]
L1 -155
E5 -152
E6 -155
3 Derivation of Acceptable Aperture Jitter
It is assumed that the jitter is relatively small with respect
to the carrier frequency, and hence equation (3) rather
than (4) is used to evaluate the contribution to the noise
power made by aperture jitter. The SNR due to jitter
arising from (3) is:
()
22222
2
2
1
2
2
τ
τ
τ
σπ
σπ
c
c
A
jf
Af
N
S
SNR === (5)
In order to keep this contribution to an insignificant size,
the jitter noise power is restricted to be 10dB down from
the thermal noise power, as in equation (6).
th
jN
S
SNR 10 (6)
where S is the signal power from Table 4, and
kTBNth
=
is the thermal noise power, with the
bandwidths of interest B being taken from Table 3. Thus,
using equation (6), it is possible to evaluate the required
jitter standard deviation allowable for each GNSS band of
interest, by using
S
N
f
th
c102
1
π
στ
(7)
Evaluating this expression for the values selected in the
previous sections gives the jitter requirements in Table 5.
Table 5 Jitter requirements (standard deviation) for each of the signal
bands
GPS
signal
Jitter
requirement
(ps)
L1 2.05
L2 2.63
L5 6.15
Galileo
signal
L1 2.58
E5 2.42
E6 6.35
These jitter requirements do not vary very much with the
loosest (6.35ps) being only 3.1 times the tightest (2.05ps).
The tightest of these jitter requirements is for the GPS L1
band, which is probably not surprising as it is the oldest
of the signals. However, it has been included in all of our
examples, so the 2ps jitter requirement is the one that
drives the specification for all of the examples. This
requirement can be turned into a phase noise standard
deviation requirement of the sampling clock:
48 Journal of Global Positioning Systems
τ
τ
θ
σ
π
σ
s
f2= (8)
where fs is the bandpass sampling frequency.
Interestingly, as the sampling frequency increases, the
phase noise requirement gets looser. Therefore, in phase
noise terms, the most difficult receiver to design is what
in other ways is the easiest receiver to design, the “L1-
only” example, giving a phase noise requirement of
50µrads.
4 Conclusion
The analysis evaluated the jitter requirement such that
noise due to sampling jitter at the carrier frequencies of
GNSS signals was 10dB less than the thermal noise. For
all GNSS bands, this requirement was of the order of
picoseconds. Several combinations of bands were
examined but they all used the L1 band, which has the
tightest requirement for jitter, a standard deviation less
than 2ps. However, that requirement is not much less than
for L2 and E5. The jitter analysis assumed that
12 <<
nc
f
τπ
, and for the tightest specification
02.02 =
τ
σπ
c
f, which satisfies that assumption.
References
Akos, D M, and J B Y Tsui (1996), Design and
Implementation of a Digital Digitization GPS Receiver
Front End, IEEE Trans Microwave Theory and
Techniques, vol 44, no 12, Dec 1996
Akos, Dennis M, et al (1999), Direct Bandpass Sampling of
Multiple Distinct RF Signals, IEEE Trans
Communications, vol 47, n0 7, July 1999, pp983-988
Awad, Selim Saad (1998), Analysis of Accumulated Timing-
Jitter in the Time Domain, IEEE Trans Instrumentation
and Measurement, vol 47, no 1, Feb 1998, pp69-73
Da Dalt, Nicola, et al (2002), On the Jitter Requirements of the
Sampling Clock for Analogue-to-Digital Converters,
IEEE Trans Circuits and Systems I, vol 49, no 9, Sep 2002,
pp1354-1360
Fontana, R D, et al (2001), The Modernized L2 Civil Signal,
GPS World, Sep 1, 2001
GPSW (2004), GPS and Galileo Reach Signal Agreement,
Global View, GPS World, Mar 2004, pp12-13
GPSW (2004A), EC: GPS-Galileo Deal Has Mutual Benefits,
Global View, GPS World, Apr 2004, p 13
Hein, G, et al (2002), Status of Galileo Frequency and Signal
Design, Proc ION-GPS 2002
Hein, G, et al (2003), Galileo Frequency and Signal Design,
GPS World, June 2003, pp30-37
ICD-GPS-705 (2002), Interface Control Document: Navstar
GPS Space Segment/ Navigation L5 User Interfaces, US
DOD, 29 Mar 2002
ICD-GPS-200C (2003), Interface Control Document: Navstar
GPS Space Segment/ Navigation User Interfaces, US
DOD, IRN-200C-005R1, 14 Jan 2003
Kobayashi, Haruo, et al (1999), Aperture Jitter Effects in
Wideband ADC Systems, Proc ICECS '99, vol 3 ,5-8 Sept.
1999 pp1705-1708
Ledvina, B.M., M.L. Psiaki (2003), S.P. Powell, and P.M.
Kintner, A 12-Channel Real-Time GPS L1 Software
Receiver, Proc. ION NTM, Jan 22-24, 2003. Anaheim, CA
Lee, J, et al (2003), 10 Gsample/s optoelectronic A/D
converter, Electronics Letters, vol 39, no 23, 13 Nov 2003
Lin, D M, and J B Y Tsui (2001), A Software GPS Receiver for
Weak Signals, Microwave Symposium Digest, 2001 IEEE
MTT-S International , vol 3, 20-25 May 2001, pp2139 –
2142
Poulton, K, et al (2003), A 20GS/s 8b ADC with a 1MB
Memory in 0.18
µ
m CMOS, Proc Int Solid-State Circ Conf,
2003, IEEE
Ries, L, et al (2002), A Software Receiver for GPS-IIF L5
Signal, Proc ION-GPS 2002, Sep 2002, pp1540-1553
RTCA SC-159 (2000), DO-261 Navstar GPS L5 Signal
Specification, 14 Dec 2000,
Sheingold, Daniel H, (ed) (1986), “Analog-Digital Conversion
Handbook, 3rd ed., Prentice Hall, 1986
Shinagawa, Mitsuru, et al (1990), Jitter Analysis of High-Speed
Sampling Systems, IEEE J of Solid State Circuits, vol 25,
no 1, Feb 1990, pp220-224
Sun, Yi-Ran, and Svante Signell (2004), Effects of Noise and
Jitter on algorithms for Bandpass Sampling in Radio
Receivers, Proc. ISCAS 2004, IEEE, Vancouver, Canada,
2004, pp I-761 – I-764
Thor, Jonas, and Dennis M Akos (2002), A Direct RF
Sampling Multifrequency GPS Receiver, Proc Position
Location and Navigation Symposium, 2002 IEEE, 15-18
April 2002 pp44-51
Van Dierendonck, A J, (2000), The New L5 Civil GPS Signal,
GPS World, Sept 2000, pp64-71
Vaugan, Rodney G et al (1991), The Theory of Bandpass
Sampling, IEEE Trans Signal Processing, vol 39, no 9,
Sept 1991, pp1973-1984