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CoherentintegrationtimelimitofamobilereceiverforindoorGNSSapplications

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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: abrouman@ucalgary.ca, +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