Borio D., Lo Presti L.

English

PORTO Publications Open Repository TOrino

22      InsideGNSS   march/april 2008	 www.insidegnss
   GNSS 
Solutions: 
What	does	
“geometry-based”	
and	“geometry-
free”	mean	in	the	
context	of	GNSS?	
T
he geometry-based model is 
the usual mathematical model 
for solving high-precision posi-
tions from phase (and code) data 
using a relative GNSS receiver setup. 
The relative receiver-satellite geom-
etry plays a crucial role and has a sig-
nificant effect on the precision of the 
parameters to be determined, which 
are, among others, the carrier phase 
ambiguities and the relative receiver 
position. 
The geometry-free model dispenses 
this receiver-satellite geometry: it 
solves for the receiver-satellite ranges 
instead of positions. However, ambigu-
ity resolution is still possible using this 
model.
To clarify this, recall the general 
observation equations for GNSS phase 
and code data:
Here   denotes the phase observ-
able (in distance units) from satellite 
s to receiver r, tracked on frequency 
j and     the corresponding code 
observable. The receiver-satellite range 
is denoted as   and the receiver clock 
errors for phase and code as   and 
, respectively. Satellite clock errors 
for phase and code are denoted as   
and  , respectively, while   denotes 
the tropospheric and   the ionospher-
ic errors. 
The initial phase of the signal at 
time of transmission is denoted as 
, and   denotes the initial 
phase of the signal generated in the 
receiver. The integer phase ambiguity is 
denoted as   (in cycles), whereas the 
wavelength corresponding to frequen-
cy j is denoted as  . In this context, we 
should mention that the term in square 
brackets is often referred to as non-
integer (undifferenced) ambiguity. 
Finally, all other remaining errors, 
such as multipath and receiver noise, 
are denoted as   for the phase observ-
able and as   for the code observable. 
The unknown receiver position is of 
course “hidden” in the receiver-satellite 
range:
where xr = (Xr, Yr, Zr)T denotes the 
receiver coordinates and xs = (Xs, Ys, 
Zs)T the satellite coordinates. To solve 
Mathematical	
Models	
and	GNSS	
Interference
“GNSS Solutions” is a 
regular column featuring 
questions and answers 
about technical aspects of 
GNSS. Readers are invited 
to send their questions to 
the columnists, professors 
Gérard lachapelle and mark 
petovello, Department of 
Geomatics Engineering, 
University of Calgary, 
who will find experts to 
answer them. Their e-mail 
addresses can be found with 
their biographies at the 
conclusion of the column.
FIGURE 1  Unit direction or line of sight (LOS 
vector
FIGURE 2  Double difference geometry
s
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r
2 1
1 2
x12
l
1
1
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1
2
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1
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2
2www.insidegnss.com    march/april 2008	 InsideGNSS  23
the receiver position, the phase and code observation equa-
tions need to be linearized  as follows: 
where    denotes the unit direction 
vector between receiver and satellite (line-of-sight vector), 
see Figure 1. Due to the linearization all observables and 
unknown parameters are denoted as incremental (delta) 
quantities. 
In principle, for all these variables we need approximate 
values, which are denoted using the zero superscript. For 
example, in the linearization the receiver position is decom-
posed as  . It is assumed that the satellite positions 
are known (computed using either broadcast ephemeris or 
precise orbits), implying that  .
To exploit the high precision of the phase data, the ambi-
guities need to be resolved to their integer values. This can 
be accomplished by a relative measurement setup in which 
GNSS data are simultaneously tracked by at least two receiv-
ers and two satellites in order to eliminate common errors. 
See Figure 2. 
The resulting model of linearized observation equations 
is the geometry-based model, in which the position of the 
first receiver, the reference, should be known, and the relative 
position (baseline) of the second receiver is parameterized. 
This model is then usually based on double differences of the 
observables and parameters. 
For example, the geometry-based model for a single 
epoch of dual-frequency phase and code data of m satellites 
tracked by two receivers reads, ignoring the differential iono-
spheric errors (this is allowed for short baselines): 
 
with   the identity matrix of dimension m-1 and the fol-
lowing vectors defined as:24      InsideGNSS   march/april 2008	 www.insidegnss
Thus the phase and code observables (plus their error 
components) are in double difference form, and the integer 
ambiguities as well. As a result of the double differencing 
all clock errors and initial phases are eliminated from the 
unknown parameter set; thus the ambiguities are now truly 
integer in nature. 
As for the (differential) tropospheric errors, in the geom-
etry-based model they are mapped to one delay in the local 
zenith, i.e.:  , where   denotes the mapping coeffi-
cient. The mapping coefficients of all m satellites enter matrix 
G, together with the line-of-sight vectors, as follows:
This matrix G is referred to as the (receiver-satellite) 
geometry matrix. We should point out that the geometry-
based model is also solvable without the code data, but in 
that case at least two epochs of phase data should be includ-
ed. At least five satellites are needed to solve the geometry-
based model in which a tropospheric zenith delay is param-
eterized; in the absence of this parameter one needs at least 
four satellites.
Without linearization, the observation equations (1) 
themselves can still be used for ambiguity resolution. Instead 
of the receiver coordinates, one then solves for the receiver-
satellite ranges. The resulting model is referred to as the 
geometry-free model. 
For example, the geometry-free model for a single epoch 
of dual-frequency phase and code data of m satellites reads
where vector   represents the troposphere-biased receiver-
satellite ranges, i.e.  . 
Only these combined troposphere-range parameters are 
estimable, because the range and troposphere error cannot 
be distinguished from each other. This lumping can be seen 
as an advantage compared to the geometry-based model, 
because possible residual errors created by modeling the tro-
pospheric delay do not affect ambiguity resolution.
Other differences exist between the geometry-free and 
geometry-based models. Because the geometry-free model  
 
is linear, in this model the original observations should be 
used, while in the geometry-based model these should be 
the “observed-minus-computed” observations. Because of 
this linearity, approximate values are not needed to solve the 
geometry-free model; also the model does not need a priori 
values for the satellite positions or the position of the refer-
ence receiver. 
Another difference is that the geometry-free model can 
be solved using only two satellites, which means that the 
ambiguities corresponding to a single double difference can 
be resolved independently. Increasing the number of satel-
lites does significantly benefit ambiguity resolution for the 
geometry-based model, but for the geometry-free model this 
improvement is only marginal. 
One can easily demonstrate that the geometry-free model 
is weaker than its geometry-based counterpart, because in 
the latter model all ranges are coupled to the same three 
baseline components. Hence, the redundancy of the geom-
etry-based model is higher and therefore its ability to detect 
outliers in the data is improved. 
Finally we should mention that the geometry-free model 
cannot be solved without code data, because the absence of 
code data would make the design matrix, which consists of 
the columns of the two matrices in Equation (6), rank defi-
cient.
In practice, the use of the geometry-free model is usu-
ally seen in combination with the geometry-based model for 
the purpose of ambiguity resolution. As a final remark one 
may argue whether the term “geometry-free” is completely 
justified here, as the receiver-satellite ranges are still pres-
ent in the geometry-free model (a better name would be the 
“coordinate-free” model). In this context the well-known 
geometry-free linear combination of phase data is really 
geometry-free, because the receiver-satellite ranges are elimi-
nated by taking the difference of the phase observables at two 
frequencies.
Dennis ODijk
Dr. Dennis Odijk	is	a	
researcher	at	Delft	
University	of	Technology.	
His	interests	are	focused	on,	
among	others,	GNSS	
ambiguity	resolution	and	
ionosphere	modeling.	
GNSS	SOLUTIONSwww.insidegnss.com    march/april 2008	 InsideGNSS  25
What	are	the	main	classes	of	
interference	that	can	degrade	
the	GNSS	signals?	What	are	the	
possible	countermeasures?	
T
he extremely low received power of a GNSS signal 
makes it vulnerable to different kinds of interference, 
such as spurious and out-of-band emissions. These 
emissions can originate from telecommunication 
and electronic systems, either operating in adjacent bands or 
working at frequencies relatively far from the GNSS bands. 
Interference can be intentional or unintentional. 
Intentional interference can be generated easily by GNSS 
jammers that adopt different strategies for jamming 
GNSS receivers, such as the emission of strong continuous 
waves (CW) or strong Gaussian noise occupying the same 
frequencies as the GNSS signal. Unintentional interference 
can be generated by a variety of sources, because every 
electronic device potentiallys emit strong electromagnetic 
signals that can jam the GNSS receiver. 
Examples of unintentional interference are UHF and 
VHF television, VOR (VHF omnidirectional radio-range), 
and ILS (instrument landing system) harmonics and 
spurious signals caused by power amplifiers working in their 
non-linear region or by oscillators present in many electronic 
devices. Regardless of their source, interfering signals can 
generally be classified as pulsed, continuous wave, or swept. 
Pulsed interference is characterized by intermittent 
bursts of signals, such as arise, for instance, from DME 
(distance measuring equipment) and the TACAN (TACtical 
Air Navigation) system. The former is a transponder-based 
radionavigation technology that measures distances by 
timing the propagation delay of VHF/UHF radio signals, 
whereas the latter is a navigation system used by military 
aircraft. 
Both DMEs and TACAN emit series of pulse-pairs 
modulated at the frequencies in the range 962–1213 MHz. 
These frequencies include the Galileo E5 and GPS L5 bands; 
thus, DME/TACAN can represent a serious threat for GNSS 
receivers operating at these frequencies. The issue of DME/
TACAN signals has been deeply analyzed in the literature 
and some details can be found in the article, “GNSS Album: 
Images and spectral signatures of the new GNSS signals,” 
appearing in the June 2006 issue of Inside GNSS. 
The class of continuous wave (CW) interference includes 
all those narrowband signals that can be reasonably 26      InsideGNSS   march/april 2008	 www.insidegnss
represented as pure sinusoids with respect to the Galileo/GPS 
bands. Many electronic devices, such as personal computers 
and mobile phones, are equipped with local oscillators that 
can produce harmonics in the GNSS bands: these harmonics 
are examples of CW signals. 
In Figure 1 the power spectral density (PSD) of a GPS 
signal acquired by a GPS receiver front-end in the presence 
of a CW interferer is reported. As can be seen, the spectrum 
at the output of the front-end filter contains two peaks, each 
being a CW interference source.
Swept interferences are signals characterized by a 
narrow instantaneous band at a central frequency that 
changes over time. These interferences can be harmonics 
of frequency modulated (FM) signals and can be produced 
by telecommunication systems such as television and radio 
broadcasting.  
An example of swept interference is depicted in Figure 2, 
which represents the spectrogram of a GPS signal acquired 
by means of a receiver front-end (thanks to L. Camoriano 
and T. C. G. Corsini for their support during data collection 
performed in Torino near the installation of different 
television emitters). The spectrogram represents the temporal 
evolution of the frequency content of the received signal: 
— the faint, noisy band between 2 and 5 MHz represents 
the filtered and down-converted GPS signal the spectral 
characteristics of which are essentially stationary. The more 
prominent (red) line shows the sinusoidal evolution of the 
frequency content of the swept interference.
For each kind of interference, an appropriate 
countermeasure exists that exploits the time-frequency 
characteristics of the disturbing signal. One such method 
is based on spatial diversity, which effectively mitigates 
interference but is not discussed in detail here.  Briefly, 
however, these techniques exploit the correlation between 
the received signals at different spatially separated antennas 
in order to minimize the interference impact. A short 
introduction to this topic can be found in “GNSS Solutions” 
of April 2006.
Aside from spatial diversity, interference mitigation 
techniques can be divided into time, frequency, and time-
frequency algorithms according to the specific domain in 
which they operate.
Time domain mitigation techniques essentially consist 
of excision and filtering. Time excision is usually employed 
GNSS	SOLUTIONS
FIGURE 1  Power spectral density of a GPS signal corrupted by two CWs. FIGURE 2  Spectrogram of a GPS signal corrupted by swept interference
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Frequency (MHz)
Power Spectral Density Estimate via Welch
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Time (ms)www.insidegnss.com    march/april 2008	 InsideGNSS  27
for removing pulsed interference because pulsed signals are 
concentrated in the time domain.  The few samples of the 
received GNSS signal that are affected by the interference 
are thus excised together with the interference without 
significantly affecting overall receiver performance.
Adaptive time filtering is generally used for removing 
CW and swept interference: an adaptive notch filter is, for 
instance, able to track the frequency variations of a swept 
signal and to efficiently mitigate its effects by placing a 
notch at the frequencies where the interference is most 
concentrated. 
Adaptive filters are efficient devices requiring a low 
computational load and thus can be easily implemented in 
GNSS receivers; however, their use is often limited to mono-
component signals. Special solutions have to be adopted if 
more than one CW or swept signal is present.
In frequency domain mitigation algorithms, the 
GNSS signal is FFT (fast Fourier transform)-transformed, 
processed, and inverse FFT-transformed back into the time 
domain. The FFT allows the CW and pulsed interference to 
be concentrated in the frequency domain, where they can be 
easily excised. These techniques can effectively handle multi-
component signals and thus can cope with more than one 
interfering signal.
A drawback of frequency domain algorithms appears 
in the fact that a swept interference source, when FFT 
transformed, can occupy a large portion of the digital 
frequency range.  This makes efficient excision more difficult, 
because the quality of the GNSS signal can be seriously 
compromised if too many frequency bins are removed.
A solution can be found in time-frequency mitigation 
techniques: the received signal is first expanded into the 
time-frequency plane by means of an appropriate time-
frequency transformation such as the spectrogram. In this 
domain the interference is well localized, as in Figure 2, and 
thus can be removed with a little degradation of the GNSS 
signal. Once the interference is excised, the time domain 
signal is reconstructed. 
The use of time-frequency mitigation techniques is 
limited by the high computational load required to transform 
the signal into the time-frequency plane. However, as the 
technology continuously evolves, such techniques may be 
adopted in future GNSS receivers. Because they are able to 
cope with a large variety of interfering signals, they combine 
the advantages of both time and frequency mitigation 
algorithms.
Manufacturers
The data in figures 1 and 2 was derived from a NordNav-R30 
receiver from NordNav (now a part of CSR, Cambridge, 
England). 
Daniele BOriO anD leTizia lO PresTi
Daniele Borio	(daniele.
borio@polito.it)	is	a	
Ph.D	student	at	the	
Politecnico	di	Torino	in	
Turin,	Italy.	His	research	
activity	is	mainly	
focused	on	GNSS	signal	
acquisition.	letizia lo presti	(letizia.lopresti@polito.it)	is	full	professor	
at	the	Politecnico	di	Torino.	Her	fields	of	expertise	include	digital	signal	
processing,	communications,	and	navigation.	 
Mark	Petovello	is	an	assistant	professor	in	the	Department	
of	Geomatics		Engineering	at	the	University	of	Calgary.	He	
has	been	actively	involved	in	many	aspects	of	positioning	
and	navigation	since	1997	including	GNSS	algorithm	
development,	inertial	navigation,	sensor	integration,	and	
software	development.	
Email:	mpetovello@geomatics.ucalgary.ca
Professor	Gérard	Lachapelle	holds	a	CRC/iCORE	Chair	
in	Wireless	Location	in	the	Department	of	Geomatics	
Engineering	at	the	University	of	Calgary.	He	has	been	
involved	with	GNSS	since	1980	and	has	received	numerous	
awards	for	his	contributions	in	the	area	of	differential	
kinematic	GPS	and	indoor	location.	
Email:	lachapel@geomatics.ucalgary.ca

Mathematical models and GNSS interference

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