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Coherent integration time limit of a mobilereceiver for indoor GNSS applications
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CoherentintegrationtimelimitofamobilereceiverforindoorGNSSapplications
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TheUniversityofCalgary
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1
Coherent Integration Time Limit of a Mobile
Receiver for Indoor GNSS Applications
Ali Broumandan†, John Nielsen, and Gérard Lachapelle
Schulich School of Engineering
Position Location and Navigation (PLAN) Group
http://plan.geomatics.ucalgary.ca/
University of Calgary
2500 University Drive, N.W.,Calgary, Alberta, Canada, T2N 1N4 †Corresponding author: [email protected], +1 (403) 993 1358 (Tel)
Abstract— There is an emerging requirement for processing Global Navigation Satellite System
(GNSS) signals indoor where the signal is very weak and subjected to spatial fading. Typically
longer coherent integration intervals provide the additional processing gain required for the
detection and processing of such weak signals. However, the arbitrary physical motion of the
handset imputed by the user limits the effectiveness of longer coherent integration intervals due
to the spatial decorrelation of the multipath faded GNSS signal. In this paper, limits of coherent
integration due to spatial decorrelation are derived and corroborated with experimental
verification. A general result is that the processing gain resulting from direct coherent integration
saturates after the antenna has moved through a certain distance, which for typical indoor
propagation, is about half a carrier wavelength. However, a refined Doppler search coupled with
a prolonged coherent integration interval extends this limit, which is effectively a manifestation
of selective diversity.
Keywords: Coherent integration, temporal/spatial processing, channel decorrelation,
multipath, fading, selective diversity.
2
I. INTRODUCTION
A microwave propagation channel from a satellite source to an indoor receiver is characterized
by a channel gain factor that typically randomly varies with the spatial location and orientation
of the receiver (Parsons 2000). When the receiver antenna is located in a dense multipath
scattering environment with no well-defined line-of-sight (LOS) signal components, fading
conforms approximately to Rayleigh statistics with spatial coherency intervals of typically less
than the carrier wavelength of the propagated signal (Rensburg and Friedlander 2004; Kim
2004). There is a requirement for advanced processing for GNSS handsets such that they can
aptly support location based services in a NLOS multipath environment. Such processing is
typically based on extending the coherent integration time as in (Pany et al. 2009) where the
coherent integration time reaches several seconds. The receiver processing used in Pany et al
(2009) incorporates a high stability receiver clock, MEMS IMU, magnetometer, barometer, WiFi
power reading, ZigBEE-based radio navigation system and sophisticated GNSS/INS integration
to compensate for the receiver nonlinear motion. The ancillary sensors and processing facilitate
extended coherent integration interval by providing a better matched replica signal for the GNSS
signal demodulation. Unfortunately there are limits to such processing enhancements in diffuse
multipath environments as the channel gain is spatially and temporally random. This limits the
performance gains typically afforded by longer integration intervals when the antenna is moving.
Issues of receiver clock instability further exacerbate the processing of direct coherent
integration (Watson 2006). In this paper, it is shown theoretically and experimentally that, for a
moving antenna in a multipath faded environment, the processing gain necessarily saturates as
the coherent interval is increased indefinitely. As such, beyond a certain integration interval,
there is no further processing gain that can be extracted directly from coherent integration.
3
However, it will also be shown that through a refined Doppler search, which occurs naturally for
longer coherent integration intervals, that there is an additional incremental gain due to the innate
selective combining diversity that becomes part of the overall receiver processing.
Typically, GNSS receivers take a burst sampling of the signal which will be referred to
here as a “signal snapshot”. This is correlated with the receiver synthesized replica signal
encompassing the Pseudo Random Noise (PRN) code of the desired GNSS signal appropriately
modified by the estimated Doppler and code phase (Kaplan and Hegarty 2006). The duration of
the signal snapshot, and therefore the coherent integration time, will be denoted by T. Provided
that the replica signal is perfectly correlated with the desired signal component of the signal
snapshot then the processing gain will be proportional to T. This assumes that the associated
independent additive channel noise is spectrally white relative to the bandwidth of the GNSS
signal. However, due to the spatial and temporal decorrrelation of the received signal as well as
the residual instability of the receiver clock, the synthesized replica and snapshot signals become
more decorrelated as T is increased. As such the processing gain saturates as T is increased.
In (Rappaport 2002) the limit of the processing gain of a receiver with a moving antenna in
a Rayleigh multipath environment was theoretically determined based on the assumptions that
multipath was characterized by the ring of scatterers model. In this paper, the limit of coherent
processing is derived based on a generalized multipath scattering model with experimental
verification. In Section II, the description of the moving receiver system is given along with the
statistics of the received signal. In Section III, the coherent integration limit due to antenna
motion is theoretically analyzed for a general scattering model. The optimum trajectory of the
4
antenna that provides maximum processing gain is also presented. Experimental results are given
in Section IV with overall conclusions provided in Section V.
II. MOVING ANTENNA SYSTEM MODEL
Consider a moving antenna scenario as shown in Figure 1 where ( )tp denotes the position vector
of the moving antenna as a function of time, t, relative to a rectangular coordinate system. The
complex baseband signal representation of the antenna output received signal is denoted as ( )r t
which is despread with a locally generated replica signal and coherently integrated over a
snapshot interval of [ ]0,t T∈ .
The signal component of ( )r t , originating at a GNSS satellite is denoted ( )( ),s t tp , which
is a function of t and the changing antenna position ( )tp . As the GNSS signal is of sufficiently
narrow bandwidth, ( )( ),s t tp can be approximated as ( ) ( ), ( ) ( ) ( )os t t A t s t≈p p where ( )o
s t is
the deterministic complex baseband component of the received signal that is known to the
receiver and ( )( )A tp is the complex channel response to the incident signal at the antenna
position of ( )tp which is a function of the multipath channel and the antenna response. Implicit
in this simplification is that the maximum extent of the trajectory over the snapshot interval T is
much shorter than the inverse of the bandwidth of ( )o
s t (Van Trees 2002).
The signal is written as
( )( ) ( ) ( ) j 2 ftos t D t c t e
π ψτ τ ∆ += − − (1)
5
where ( )D t is the navigation data modulation, ( )c t is the PRN code, τ is the code phase, f∆
represents the carrier frequency offset, ψ is the initial phase offset. ( )os t is assumed to be
completely known to the receiver with the navigation data, code phase and carrier frequency
determined through an assumed acquisition process.
In diffuse Rayleigh faded multipath environments, as assumed herein, ( )A p can be
modeled by a Complex Normal (CN) random variable (Friedlander and Scherzer 2004), such that
( ) ( )20, AA CN σp ∼ where ∼ denotes the PDF of the left-hand side variable and ( )2,CN µ σ
signifies a complex normal Probability Density Function (PDF) of mean µ and variance 2σ . It
is also assumed that during signal snapshots, the channel at a specified location is temporally
static such that ( )A p only changes when p changes. Based on the assumed diffuse multipath
model, ( )A p is a stationary random process relative to the position vector p . This motivates the
definition of the normalized spatial correlation of the channel gain as
( ) ( ) ( )*
2
1( ) ( ) ( ) ( )
A
g t t t E A t t A tσ
+ ∆ − = + ∆ p p p p (2)
where 2
Aσ is the variance of ( )( )A tp , ( )( ) ( )g t t t+ ∆ −p p is the normalized correlation and t∆
is a time shift. If the antenna trajectory is spatially smooth then a further simplification can be
made as
( ) ( )( ) ( )g t t t g tν+ ∆ − ≈ ∆p p (3)
where ν is defined as the magnitude of the antenna velocity in terms of carrier wavelengths per
second. There is some loss of generality by imposing the constraint of a smooth trajectory.
6
However, the advantage is that closed form expressions for g emerge that are of practical use
which are sufficiently accurate for typical trajectories.
The received signal is corrupted with additive channel noise which has an equivalent
complex baseband representation denoted by ( )w t , which is assumed to be spectrally white
relative to the bandwidth of ( )r t with a power spectral density o
N . Furthermore, ( )w t is
assumed to be a complex normal random process. The representation of ( )r t can then be
expressed as
( ) ( ( )) ( ) ( )o
r t A t s t w t= +p (4)
The signal snapshot of ( )r t is collected by the receiver and despread by the locally
generated replica of ( )o
s t and coherently integrated resulting in an output variable denoted by x
expressed as
( )
( )
0
0 0
1( )
1 1( ) ( ) ( ) ( ) ( )
T
o
T T
o o o
x r t s t dtT
A t s t s t dt w t s t dtT T
∗
∗ ∗
=
= +
∫
∫ ∫p
(5)
The channel noise term of Eq. 5 is a zero mean CN random variable with variance
2
0 02
0 0
00 02
0 0
0
1( ) ( ') ( ) ( ') '
( ) ( ') '
T T
w
T T
E w t w t s t s t dtdtT
Ns t s t dtdt
T
N
T
σ ∗ ∗
∗
=
=
≈
∫ ∫
∫ ∫ (6)
7
where it is assumed that the GNSS signal is nominally of unit power and has a bandwidth that is
much larger than 1 T such that
0 0
0 0
( ) ( ') '
T T
s t s t dtdt T∗ ≈∫ ∫ (7)
The signal component of x in Eq. 5 is also a zero mean CN random variable but with
variance
( ) ( )2 *
2
0 0
1( ) ( ') '
T T
sE A t A t dtdt
Tσ = ∫ ∫ p p (8)
where the approximation of 2
( ) 1o
s t ≈ was used. Based on Eq. 2 and Eq. 3, Eq. 8 can be
rewritten as
( )2
2
2
0 0
( ') '
T T
As
g v t t dtdtT
σσ = −∫ ∫ (9)
which can be simplified by introducing a change of variable of
't tq
T
−= (10)
Eq. 9 is then written as
( ) ( )
( )
1
2 2
0
2
2 1s A
A
q g vTq dq
vT
σ σ
σ
= −
= Ω
∫ (11)
where the function Ω is defined as
8
( ) ( ) ( )1
0
2 1vT q g vTq dqΩ = −∫ (12)
( )vTΩ is a fundamental parameter in regards to this paper which relates the signal coherent
integration losses to the type of diffuse multipath with arguments of the antenna velocity and the
snapshot interval. Finally it is convenient to normalize the signal x by the standard deviation of
the additive channel noise such that ( )( )0, 1Tx CN Tρ νΩ +∼ where
2
AT
o
TN
σρ = (13)
is the averaged signal to noise ratio (SNR) with respect to the snapshot interval of T.
A common approximate model for describing the multipath spatial correlation is based on a
ring of scatterers with density function of ( )S ϕ where ϕ is the azimuth angle results in (Van
Trees 2002; Fulghum et al. 2002):
( )( )( ) exp 2 ( )sin
( ') '
S j v t d
g v t
S d
π
ππ
π
ϕ π ϕ ϕ
ϕ ϕ
−
−
∆
∆ =∫
∫ (14)
Specifically for a uniform ring of scatterers, ( )g v t∆ becomes
( ) ( )0 2g v t J v tπ∆ = ∆ (15)
where 0J is the zero-order Bessel function of the first kind. Substituting into (12) gives
( ) ( ) ( )1
0
0
2 1 2vT q J vTq dqπΩ = −∫ (16)
9
For the sphere of scatterers diffuse multipath model, ( )g v t∆ is given as
( ) ( )sin 2 2g v t v t v tπ π∆ = ∆ ∆ (17)
such that (Van Trees 2002).
( ) ( )( )1
0
sin 22 1
2
TqT q dq
Tq
πνν
πνΩ = −∫ (18)
Note that if the antenna during the signal snapshot is stationary, then 0ν = and ( )0 1g =
such that ( ) ( )0 1TνΩ = Ω = . The signal component of the coherently integrated despread signal
will have a variance of 2
s Tσ ρ= . Increasing ν from 0 to represent the antenna being moved
along a smooth trajectory during the snapshot interval results in a reduction of ( )TνΩ from a
maximum value of 1.
It is convenient to define a pair of parameters that will be used throughout this paper. The
ratio of the variance of the signal component of the moving antenna relative to that of the
stationary antenna is denoted as the coherent integration Gain Degradation (GD), expressed as
( )( )T
T
vTGD vT
ρ
ρ
Ω= = Ω (19)
Note that GD is completely characterized by ( )vTΩ . The second parameter is the Normalized
Integration Gain (NIG) metric which is the ratio of the variance of the signal component of the
moving antenna for an arbitrary snapshot interval of T to the variance of the signal component of
the stationary antenna specifically for one second integration period, expressed as
10
( )1
T
T
NIG vTρ
ρ =
= Ω (20)
III. COHERENT INTEGRATION LIMIT DUE TO ANTENNA MOTION
Consider the scenario in Figure 2 where the receiver is moving with a constant speed ν along
the x axis for the snapshot interval of [ ]0,t T∈ . A sectored ring of scatterers with
( )1
2
0
sm
sS
otherwise
ϕϕ ϕ
ϕϕ
− ≤
=
(21)
where s
ϕ is the angular extent or spread of the scattering and mϕ is the mean angle and assumed
to represent the diffuse multipath. Figure 3a shows GD values versus v for 1T = , 90mϕ = and
different angle spread values of 0s
ϕ = , 45, 90, 135, and 360 degrees. Since T=1, the horizontal
axis in Figure 3a may be considered as the antenna normalized aperture during the signal
snapshot. As an example, there is no integration gain loss in the case of 0s
ϕ = regardless of the
antenna velocity. However, in the case of 360s
ϕ = , which represents a complete uniform ring of
scatterers, moving the antenna by half a wavelength (v=0.5) results in a GD value being reduced
to 0.7 implying a 30 percent loss relative to the integration gain of a stationary antenna. It should
be noted that the spatial channel correlation coefficients and therefore GD values are also
function of mϕ . However, in NLOS multipath environments where signals impinge on the
antenna from all directions the correlation coefficient values become independent of mϕ .
11
In Figure 3b, NIG is plotted for a variable T with the antenna velocity held constant at 1v =
for angular spreads of 0s
ϕ = , 45, 90, 135, and 360 degrees. In the range of 0,0.5T ∈ the
integration gain increases approximately linearly with T for all the cases of s
ϕ shown. However,
increasing T beyond 0.5 for the case of 360s
ϕ = and 135s
ϕ = results in losses in NIG due to
signal decorrelation. Further, as observed in Figure 3b, NIG saturates for T larger than about 1
for 360s
ϕ = . Note that in the case of 0s
ϕ = , the slope of NIG does not decrease as T is
increased implying that the processing gain is proportional to T for this case. Figure 4 shows
mesh plots of GD and NIG as a function of s
ϕ , ν and T revealing similar observations.
It is of interest to determine the asymptotic value of NIG for different scatterer models as a
function of v when T → ∞ . Consequently
( )
( )
01
0
lim lim ,
2 lim
T
T TT
T
T
NIG v T
T g vTq dq
ρ
ρ→∞ →=
→∞
= Ω
= ∫ (22)
In the ring and sphere of scatterers cases the NIG limit become
1lim
1lim
2
ringT
sphereT
NIGv
NIGv
π→∞
→∞
=
=
(23)
Hence, the higher the velocity, the smaller the asymptotic value of NIG for large integration
intervals.
12
IV. EXPERIMENTAL RESULTS
In the previous sections, the limit of coherent integration, based on commonly used diffuse
multipath fading models, was analyzed. In this section, experimental results for indoor fading
environments will be shown to corroborate the theoretical model assumptions and findings.
a) Data collection setup
The experimental measurements are based on the indoor reception of GPS L1 C/A signals.
A commercial dual-channel front-end was used to down-convert GPS raw data to the baseband
frequency. A temperature controlled crystal oscillator (TCXO) was utilized to provide the
reference frequencies required for down-conversion and digitization. To account for the
navigation data modulation, reference signals for each GPS signal were obtained from a
stationary antenna located on the rooftop directly above the indoor measurement area. The roof
top location provided line of sight propagation with low multipath to each of the desired GPS
satellite transmitters. The estimated navigation data bit and Doppler frequency from the
reference channel were used to form a coherent replica signal for the indoor receiver. GNSRx-
rr™, a modified version of the GNSRx™ software receiver (O’Driscoll 2009) was utilized to
estimate the navigation data bits and extract signal samples to facilitate the long coherent
integration times required. The justification for the reference channel based replica signal is that
the focus of the measurements is the reduction of GD resulting from the antenna motion in an
indoor diffuse multipath environment. As such the complexities of compensating for the
oscillator instability, atmospheric decorrelation effects and so fourth as well as providing
navigation data stripping for the C/A code was conveniently provided by the reference channel.
13
Data collection was performed in a large size laboratory located on the ground floor of a
four-story commercial office building. During the data collection, sources of scattering
distributed throughout the indoor measurement environment were stationary. The sky plot of
available GPS satellites during the data collection, measurements setup and data collection
locations are shown in Figure 5. An active RHCP antenna mounted on a linear moving table was
utilized to realize a moving receiver scenario with a constant velocity. During the data collection
process the antenna was moved back and forth on the east-west direction with a constant speed
of 0.5 m/s on a 2 m linear trajectory.
b) Experimental verification
The overall objective was to experimentally determine the reduction of processing gain due
to the motion of the receiver antenna in a diffuse faded propagation environment. As such
several verification steps were required to ensure the validity of the theoretical assumptions and
measurement setup. The first verification test was to ensure that the replica signal synthesized
from the reference antenna was sufficiently correlated with the signal received on the indoor
antenna to enable relatively long coherent integration time intervals when the indoor antenna was
stationary. This was provided by computing NIG for the indoor stationary antenna and insuring
that it increased proportionally to T. The second verification test was that the indoor fading
channel was indeed approximately Rayleigh. The third verification test was a statistical
measurement of the spatial decorrelation to determine an adequate model for ( )g Tν . These tests
are described below:
14
a. Coherency test of a stationary antenna
As discussed, the NIG was measured for the indoor stationary antenna with the replica
signal generated from the rooftop reference antenna. Figure 6 shows time series of the SNR
values over a 200 second measurement window. As shown the SNR values of the roof top
signals remains constant during this measurement time as required. However, the SNR plots of
indoor antenna change slowly with time due to the constant slewing of the GNSS satellite
bearing which slowly modulates the multipath response as observed by the stationary indoor
antenna. While this modulation is a potential limitation, the longest coherent integration time is
1.6T = seconds, well within the coherency time of the channel which from Figure 6b is on the
order of 10’s of seconds.
Table 1 tabulates the mean values of the measured SNRs for different T values for the static
roof and indoor antennas. As shown in Table 1, the SNR at the output of the process linearly
increases with processing time T. Figure 7 compares the NIG values of the static roof and indoor
antennas with the ideal case where the signal remains coherent during the signal snapshot. This
clearly demonstrates ample coherency of the despreading replica signal synthesized from the
reference receiver at least for T=1.6 seconds.
Table 1: SNR output of coherent integration for roof top and indoor static scenarios
T SNR of static roof
antenna (PRN 20)
SNR of static indoor
antenna (PRN 31)
100 ms 38 16.3
200 ms 41 19.2
400 ms 44 22.1
800 ms 47 25.1
1000 ms 48 26.1
1600 ms 50 28.1
15
Considering the results of Table 1 and Figure 7, one may notice that in the static case, full
processing gains are achieved during maximum 1600 ms snapshot periods and the signal
decorrelation effects during the observation time can be neglected. Hence, during the maximum
integration time, with the equipment utilized and at the data collection time and location, other
decorrelation factors such as oscillator instability and atmospheric activities are not significant.
b. Verification that the channel is Rayleigh faded
The theoretical limit of the coherent integration gain described in Section III is based on the
Rayleigh fading assumption. Hence, firstly the validity of this assumption is examined by
measuring the statistics of the received signal. Figure 8 shows the measured Probability Density
Function (PDF) of the received signal amplitude. A Rayleigh fit to the measured signal
amplitude PDF is also overlaid. As observed the experimentally measured amplitude PDF agrees
fairly well with the theoretical one verifying that the Rayleigh fading assumption for the indoor
channel is valid.
c. Spatial correlation model
Spatial correlation measurements based on the setup in Figure 5 using the linear motion
table was performed in this section. The correlation coefficient of the received signals in
different spatial/temporal locations can be quantified as
*
* *
( ( )) ( ( ))
( ( )) ( ( )) ( ( )) ( ( ))
i j
ij
i i j j
E x P t x P t
E x P t x P t E x P t x P tρ
=
(24)
16
where * is the conjugate operator. The measured correlation coefficient of the PRN 20 signal is
shown in Figure 9 which agrees fairly well with the sphere of scatterers model.
d. Experimental measurements of coherent integration limit
Based on these three verification tests in the previous section, it is possible to contrast the
measurements of coherent integration limit in a multipath environment with the theoretical
findings. In the experimental part, the antenna was moved at a speed of 0.5 m/s corresponding to
2.5ν ≈ wavelengths per second of the GPS signal. Hence, the antenna motion can cause up to a
±2.6 Hz frequency offset in addition to the Doppler frequency resulting from the satellite motion
and clock bias. Hence, during the despreading process, a frequency search by the maximum
Doppler frequency centered at the reference antenna frequency was adopted and the peak of the
correlation function selected as the test statistics. As for the static antenna test, different coherent
integration times were considered to evaluate integration performance. Figure 10 shows the
experimentally determined PDFs of post SNR as obtained from various T values. These
measurements are based on the processing of 30 minutes of data. The measured SNR mean
values for each T shown in Figure 10 are tabulated in Table 2.
Table 2: Mean value of measured SNR for different T values
T (ms) 100 200 400 800 1600
Mean SNR (dB) 13.2 15.9 18.4 20.3 21.7
17
The NIG values based on measurements results for different T are shown in Figure 11.
Figure 11 also shows the theoretical NIG values for the static and the moving antenna with a
sphere of scatterers model.
Considering the results shown in Table 2 and Figure 11, it is apparent that NIG does not
reach an asymptotic limit but rather increases with T. From Figure 11, NIG for the sphere of
scatterers model has an asymptotic value much less than the measured values. According to the
correlation coefficient results of Figure 9, the channel statistically decorrelates when the antenna
displacement of Tν exceeds about half a wavelength. As 2.5ν ≈ , this occurs when 0.2T >
seconds.
To justify this phenomenon consider a 5-second time series of measured SNRs for different
T values for the static and the moving antenna as shown in Figure 12, where during this
observation time the moving antenna was displaced by about 13 wavelengths.
As shown in Figure 12a for the static antenna case, SNR for different T values change
proportional to T and all plots vary with the same mean slope. Note that, as shown in Figure 6b,
the channel remains essentially static over the maximum integration time of T=1.6 seconds.
Thus, sampling the fading channel with T of 100 ms or 1600 ms will result in the same signal
variation with different SNR values which is proportional to T. Now, consider the moving
antenna case where the SNR plots are shown in Figure 12b. As shown in Figure 12b the T of 100
ms samples the spatial field with higher resolution (every quarter of carrier wavelength) and the
SNR values varies rapidly as a function of time and it seems there is low correlation between
successive samples. However, other T values have almost a smooth SNR variation with time
which is due to the spatial filtering posited by the longer T. An interesting observation in Figure
18
12b is that the SNR values do not vary proportionally to T. This phenomenon is not similar to the
SNR variations shown in Figure 12a.
As discussed previously, in a dense multipath environment the NIG values theoretically
saturates beyond 0.5Tν > . This is contrary to the results of Table 2 and Figure 11 where there is
no sign of saturation even up to the experimental range of antenna displacement of 4Tν = . To
justify this phenomenon, consider the first 1.6 seconds of SNR values shown in Figure 12b, the
area of which is shown by a dashed rectangle. The plots with T values of 100, 400, 800, and
1600 ms have 16, 4, 2, and 1 sample(s) each. If one takes the maximum values of each T case
during this observation time (1.6 s) it can be seen that they all have almost the same maximum
SNR values, which is about 18 dB. This phenomenon cannot be seen in the SNR variation of the
static antenna case shown in Figure 12a. Thus, for the given observation time T, the coherent
integration gain for T seconds yields the same processing gain results if the observation time is
divided into k= 2, 4 and 16 sections and the receiver uses the section which provides the
maximum SNR value. This procedure selects the maximum value of a portion of data set in each
observation time which has maximum energy for a given correlated subset of data. This is
similar to a selective diversity system described in detail in Parsons (2000) and Rappaport
(2002). Other diversity schemes such as equal gain combining or maximum ratio combining
methods can be implemented to improve the signal detection and parameter estimation
performance which is beyond the scope of this paper.
To investigate the validity of the selective diversity process explained earlier the PDF of
measured SNRs in different scenarios have been considered. Figure 13 shows the measured PDF
of SNR values for different T values. In Figure 13 Max (τ, k) expresses the PDF of a scenario for
19
which the observation time is T= τk ms and the maximum SNR values at the output of k
successive τ ms coherent integration is considered to yield the Max (τ, k) plot. Table 3 tabulates
the mean values of the different scenarios shown in Figure 13.
Table 3: Comparing coherent integration gain vs. selective diversity for different
observation times
Observation
time (ms)
200 400 800 1600
T 200 ms Max(100,2) 400 ms Max(200,2) 800 ms Max(400,2) 1600
ms
Max(400,4) Max(800,2)
Mean SNR
(dB)
16 14.5 18.4 17.5 20.3 20.1 21.7 21.4 21.7
As shown in Figure 13a and Table 3, there is some processing gain when coherently
integrating for 200 ms as opposed to taking the maximum value of two successive samples of
100 ms each. This is because during a 200 ms integration time, the antenna is displaced by half a
wavelength and according to Figure 9, the signal remains partially correlated during this period.
As shown in Figure 13a the measured PDF of T=200 ms does not match with that of Max (100,
2). Now consider observation times of 400 ms, 800 ms, and 1600 ms the results of which were
shown in Figure 13b, Figure 13c, Figure 13d and Table 3. As shown, the SNR statistics of
coherent integration for T seconds are almost the same as those for the case where the
observation time is divided into some successive segments and the maximum SNR value of these
segments in each observation time is chosen. As an example, in the case of a 1600 ms
observation time (Figure 13d), coherent integration for such a period gives exactly the same SNR
profile of Max (400,4) and Max (800,2). Considering the results of Table 3 one may notice that
increasing observation time increases the overall SNR value. It should be noted that this gain is
20
not due to the coherent integration gain but to selecting the most powerful correlated signal
subinterval or implementing a selective diversity system. As mentioned before, this is not the
optimum way to process data and other processing techniques can be implemented to improve
the processing performance further.
To quantify the selective diversity gain analytically, it has been shown in Parsons (2000)
that if the average SNR of each diversity branch (here the portion of data which remains
correlated) is denoted by ρ the average SNR at the output of the K branch selective diversity,
denoted by selective
ρ statistically becomes
1
1K
selective
k kρ ρ
=
= ∑ (25)
This can be easily verified by comparing the results of Table 3 with the theoretical SNR
values of Eq. 25. This is shown in Table 4 where Ms and Th indicate the measured and
theoretical values, respectively.
Table 4: Average SNR comparison based on selective diversity principle
T (ms) 200 Max
(200,2)
400 Max
(400,2)
Max
(400,4)
800 Max
(800,2)
Ms Th Ms Th Ms Th Ms Th
Mean SNR
(dB)
16 17.5 17.7 18.4 20.1 20.2 21.4 21.4 20.3 21.7 22
Average SNR in the case of Max (τ, k) represents the selective diversity method with k
branches, with each branch coherently integrating for T ms. As an example the theoretical
average SNR in the case of Max (200,2) with two diversity branches using Eq. 25 becomes
16+10log(1.5)=17.7, which is in close agreement with the measured value. Results of Table 4
21
confirm that the additional processing observed when the antenna displacement is increased
beyond 0.5Tν = is primarily a result of selective diversity.
To justify the measured NIG anomaly shown in Figure 11, one can utilize Eq. 25 and
subtract the SNR gain due to the selective diversity. In Broumandan et al. (2010) it is shown that
considering a quarter of a wavelength antenna displacements (here each 100 ms) as a diversity
branch provides a practical trade off between diversity gain and coherency loss. Hence, the NIG
measurements for different T values have adjusted accordingly. The modified NIG curve where
the SNR values were compensated for the diversity gain is shown in Figure 11 as well. As shown
the compensated measured NIG value for selective diversity gain agrees fairly well with that of
the theoretical values of the sphere of scatterers model.
V. CONCLUSIONS
The coherent integration performance of GNSS signals has been considered for a moving
antenna in a diffuse fading environment. Of specific interest was to determine the coherent
integration limit for different fading scenarios and antenna motion. From the theoretical
perspective it was shown that the coherent integration gain of a moving antenna is limited due to
spatial decorrelation such that the coherent integration time should be commensurate with an
antenna displacement. It was also shown that in a case where the angular density of the multipath
scattering is uniformly distributed, knowledge of the antenna trajectory is of no benefit to
enhancing the processing gain. Experimental measurements were performed to verify the
assumption of the Rayleigh fading and the diffuse scattering model. However, the experimental
results seemingly contradicted the theoretical asymptotic limit of the processing gain available
by increasing integration time. As shown by increasing processing time, SNR values surpassed
22
the theoretical asymptotic limit due to the hitherto unaccounted selective diversity. The effect of
selective diversity on the overall processing gain was analyzed theoretically and experimentally
and the measurement results were found to agree well with the theoretical findings.
23
References
Broumandan A (2009), Enhanced Narrowband Signal Detection and Estimation with a Synthetic
Antenna Array for Location Applications, PhD Thesis, University of Calgary.
Broumandan A, Nielsen J, and Lachapelle G (2010) Enhanced Detection Performance of Indoor
GNSS Signals based on Synthetic Aperture. IEEE Transaction on Vehicular Technology,
VOL. 59, NO.6, July.
Friedlander B, Scherzer S (2004) Beamforming Versus Transmit Diversity in the Downlink of a
Cellular Communications Systems. IEEE Trans. Vehicular Tech., vol.53, no. 4, July.
Fulghum TL, Molnar KJ, Duel-Hallen A. (2002) The Jakes Fading Model for Antenna Arrays
Incorporating Azimuth Spread. IEEE Trans. Vehicular Tech., vol. 51, no. 5, Sept.
Kaplan ED, Hegarty C (2006) Understanding GPS Principles and Applications, 2nd ed., Artech
House.
Kim S (2004) Acquisition Performance of CDMA Systems with Multiple Antennas. IEEE Trans.
Vehicular Tech., VOL. 53, No. 5, September.
O’Driscoll C, Borio D, Petovello MG, Williams Lachapelle T (2009) The Soft Approach: A
Recipe for a Multi-System, Multi-Frequency GNSS Receiver. Inside GNSS Magazine,
Volume 4, Number 5, pp. 46-51.
Pany T, Riedl B, Winkel J, et al (2009) Coherent Integration Time: The Longer, the Better.
InsideGNSS November/December 2009.
Parsons JD (2000) The Mobile Radio Propagation Channel. John Wiley & Sons LTD, Second
ed.
Rappaport TS (2002) Wireless Communications: Principles and Practice. Prentice Hall PTR, 2nd
Edition.
24
Rensburg C, Friedlander B (2004) Transmit Diversity for Arrays in Correlated Rayleigh Fading.
IEEE Trans. Vehicular Tech., vol.53, no. 6, Nov.
Van Trees HL (2002) Optimum Array Processing, part IV, Detection, Estimation, and
Modulation Theory. John Wiley & Sons, Inc., New York.
Watson R, Lachapelle G, Klukas R (2006) Testing Oscillator Stability as a Limiting Factor in
Extreme High-Sensitivity GPS Applications. ENC, Manchester, U.K., 8-10 May.
25
Figures:
Antenna
trajectory
Spatial
coordinates
Antenna
T second
p(0) p(t)
p(T)
multipath
scattered
signal field
Figure 1: Moving antenna trajectory during a signal snapshot interval of duration T
Antenna
trajectory
x
y
multipath scattered
signal field
sϕ
mϕ
Figure 2: Moving antenna scenario and scatterers topology
26
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
v (λλλλ /s)
GD
(a), T=1 s
0 1 2 3 4 50
0.5
1
1.5
2
2.5(b), v=1 λλλλ/s
T (s)
NIG
φφφφs = 0
φφφφs = 45
φφφφs = 90
φφφφs = 135
φφφφs = 360
φφφφs = 0
φφφφs = 45
φφφφs = 90
φφφφs = 135
φφφφs = 360
Figure 3: a) GD value of a moving receiver versus receiver velocity for T=1 s and different
scatterer angle spread, b) NIG value versus T for constant 1v = and different scatterer
angle spread
Figure 4: GD and NIG variations as a function of antenna velocity and angle spread
27
(a)
Antenna
Linear Motion
Table Receiver
(b)
Figure 5: a) Sky plot of available satellites b) Measurements setup in a large size
laboratory
0 20 40 60 80 100 120 140 160 180 20030
40
50
60
70
Time (s)
SN
R (
dB
)
(a) Roof top antenna, PRN 20
T = 100 ms
T =1600 ms
20 40 60 80 100 120 140 160 180 200
10
20
30
40
50
Time (s)
SN
R (
dB
)
(b) Indoor static antenna, PRN 31
T = 100 ms
T = 1000 ms
mean = 49.94 dB
mean = 37.94 dB
mean = 16.3 dB
mean = 26 dB
Figure 6: Static SNR variation for a) roof antenna, b) indoor antenna
28
200 400 600 800 1000 1200 1400 16000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
T (ms)
NIG
Static roof
Static indoor
Ideal
Figure 7: NIG values for roof top and indoor static receivers
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized amplitude
Den
sity
Measured pdf
Rayleigh fit
Figure 8: Theoretical and measured PDF of received signal amplitude
29
0 0.5 1 1.5 2-0.5
0
0.5
1
Antenna displacement (λλλλ)
Co
rrela
tio
n c
oeff
icie
nt
Measured
Ring model
Sphere model
Figure 9: Correlation coefficient of PRN 20
-15 -10 -5 0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2PRN 20
SNR (dB)
Den
sity
T = 100 ms
T = 200 ms
T = 400 ms
T = 800 ms
T = 1600 ms
Figure 10: SNR PDFs for different T values
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
T (s)
NIG
Static
Moving, sphere model
Measured
Compensated for diversity
Figure 11: Measured and theoretical NIG values
0 1 2 3 4 50
5
10
15
20
25
Time (s)
SN
R (
dB
)
(b) Moving
0 1 2 3 4 515
20
25
30
35(a) Static
Time (s)
SN
R (
dB
)
T = 100 T = 400 T = 800 T = 1600
Figure 12: Time series of measured SNR for a) static antenna, b) moving antenna
31
0 10 20 300
0.05
0.1
SNR (dB)
De
nsi
ty
(b)
0 10 20 300
0.05
0.1
0.15
SNR (dB)
De
nsi
ty
(c)
0 10 20 300
0.05
0.1
0.15
0.2
SNR (dB)
De
nsi
ty
(d)
0 10 20 300
0.02
0.04
0.06
0.08
0.1
SNR (dB)
De
nsi
ty
(a)
T =100 ms
T = 200 ms
Max(100,2)
T = 200 ms
T = 400 ms
Max(200,2)
T = 400 ms
T = 800 ms
Max(400,2)
T = 1600 ms
Max(400,4)
Max(800,2)
Figure 13: SNR PDFs for different observation times