On Second Order Statistics of the Satellite-to ... - Semantic Scholar
On Second Order Statistics of the Satellite-to-Indoor Channel Based on Field Measurements Abdelmonaem Lakhzouri, Elena Simona Lohan, Ilkka Saastamoinen , and Markku Renfors Institute of Communications Engineering, Tampere University of Technology P.O. Box 553, FIN-33101, Finland, {abdelmonaem.lakhzouri, elena-simona.lohan, markku.renfors}@tut.fi u-Nav Microelectronics, Hermiankatu 6-8 D, FIN-33720, Tampere Finland. {[email protected]}
Abstract— Wireless positioning has received increased attention during the past few years, where several wireless applications have been envisaged and promoted such as the E-911/E-112 regulations. The positioning needs to be carried out in all the environments covered by the wireless communication services, including the most constraining areas such as dense urban areas and obstructed indoor environments. The most known positioning system is the Global Navigation Satellite System (GNSS), which demonstrated quite reliable positioning capabilities when the receiver is in direct view with the sky. However, in indoor environments the signal characteristics are not well understood yet and positioning capabilities are quite poor. Therefore, understanding the multipath propagation indoors and fading characteristics are quite important to make GNSS works indoors. In this paper, we describe measurement-based modeling results of the level-crossing rate (LCR) and average duration of fades (ADF) with the purpose of giving further insight on the real satellite-to-indoor channel characteristics.
I. I NTRODUCTION The global positioning system (GPS) is a positioning and timing system based on 24 satellites in operation. Each satellite orbits the earth in approximately 12 hours [1]. With the development of the new European navigation system, Galileo, the number of satellites will increase to approximately 60 satellites. This will offer a variety of navigation services that can be classified into Open Services providing global positioning and timing services free of charge, Safety-of-Life Services providing integrity services with a defined time-to-alert limit, Public Regulated Services, and Commercial Services [2]. These applications are mostly considered in outdoor navigation where the receiver is in direct view with the sky. However with the growth of indoor navigation applications, the use of same concepts as in outdoor environments based on GNSS is gaining more attention [3], [4], [5]. This will introduce new challenges in the receiver designs, such as signal acquisition and tracking, as well as Line-of-sight occurrence and availability. Therefore, understanding the satellite-to-indoor propagation and fading characteristics is of utmost importance. In the literature, very few studies have attempted to develop channel models specifically for the GPS-indoor channel. The first attempt was back to 1997 [6], where the authors used strong reference data to augment indoor processing capabilities
and conduct coherent integrations of up to 160 ms. The existence of deep fades and their impact on indoor signals was observed. In [7], [8], the authors have evaluated the Karasawa model and Urban Three-State Fade Model (UTSFM) as applied to GPS signals in difficult environments using standard and high-sensitivity GPS receivers. The most recent study was found in [9], where the author analyzed highbandwidth raw GPS data with high-sensitivity techniques to characterize fading and multipath characteristics indoors This paper presents an analysis of experimental data for the satellite-to-indoor channel obtained in different measurement campaigns carried out by the authors based on GPS signals. We remark that the authors expect to find similar results once the Galileo signals from the satellites will be available, with the main difference that more resolution in modeling the channel delays is expected. The aim of this work is to analyze the Level-Crossing Rate (LCR) and Average Duration of Fades (ADF) to give further insight on the real satellite-to-indoor channel characteristics. Analytical expressions of these quantities have been derived in the context of Rayleigh [10], Rice [11], and Nakagami [12] distributions. All these stochastic processes play an important role for modeling mobile fading channels. However, the validation of these models for the current application through real measurement data is not often encountered in the literature. These statistics are important for indoor mobile applications, since they characterize the rate and duration of fade occurrences. The sequel of this paper is organized as follows. In Section 2, the experiment settings and measurement environments are described. Data processing and navigation data detection are described in Section 3. Section 4 is dedicated to the analysis of the second order statistics, and Section 5 gives the conclusions. II. M EASUREMENT C AMPAIGNS A study related to satellite-to-indoor channel modeling was undertaken by Tampere University of Technology and u-Nav microelectronics, Finland. During this study, 2 measurement campaigns were carried out in April and September, 2004 in typical office environments. The transmitters are the different GPS satellites available in view during the measurements dates
and the receiver is an integrated GPS receiver with sampling rate of 16.36 MHz.
in the indoor environment. Both receivers are shown in Fig. 2.
A. Measurement Environments During the measurement campaigns, different indoor environments were tested. The first scenario, denoted as Room scenario, corresponds to a small room without any windows (about 5 m2 ), where in the front there is small corridor with large windows. Here the Line-of-sight (LOS) signal is more likely to be absent. The second scenario, denoted by Corridor scenario, corresponds to a long corridor with windows and doors open. The movement inside the environments is in all x, y, and z directions (The GPS antenna is placed in hand, and the movement is in all directions). The movement is random and it is at the walking speed of about 2-3 km/h (see Fig. 1). For the Room scenario, the movement is performed in and out of the room. All the measurements were taken for a duration of 1 to 3 minutes for reliable statistics.
Fig. 2. The two GPS receivers synchronized to a common clock, operating in parallel.
III. M EASUREMENT DATA PROCESSING The signal at the reference GPS receiver can be written as [1]:
Fig. 1. Measurement environments: Corridor and Room scenarios, random movements.
B. Measurement Settings The indoor signal is expected to be very weak and embedded in noise. Therefore long coherent and non-coherent integrations are required. For the outdoor signal, usually the coherent integration intervals are around 1, 4, 5 msec [1] (a divisor of the data bit duration T b , which is 20 msec). However, in indoor reception for channel measurement purposes, the coherent integration must be much longer than 20 msec in order to compensate for the increase in the noise level. Therefore, an estimation and removal of the navigation data must be undertaken before any coherent integration. The blind estimation of the navigation data from the indoor signal alone, which is expected to be very noisy, can not guarantee good performance. Besides, we need to estimate the code-phase and Doppler frequency. For these reasons, two GPS receivers synchronized to a common clock, operating in parallel are used. The first one acquires the signal from an outdoor antenna placed on the roof of the building. This signal is expected to be strong, and it will be used as reference signal for code-phase and Doppler frequency acquisition, as well as for frequency drift estimation and correction. The second antenna is moving
r(t) =
N sv
ds (t − τs )αs (t − τs )e+j2πfDs t Cs (t − τs )+
s=1
(1)
η(t),
where t is the GPS time, ds (t), αs (t), τs , and fDs are the data bit, complex channel coefficient, instantaneous delay, and Doppler frequency from satellite s at time t corresponding to the first arriving path, respectively. Here C s (t) is the transmitted code and η(t) is an additive noise incorporating the multipath effect. We point out here that equation (1) models the radio channel as a single tap channel, this assumption is reasonable since in outdoor reception, it is more likely that we have strong single path [13]. The extension to multipath model is straightforward The output of the correlator corresponding to the satellite v, to the tentative delay τ , and Doppler frequency f D is: yv (τ, fD ) = T1 r(t)e−j2πfD t Cv(r) (t − τ )dt, (2) T
(r)
where T is the integration time and C v (·) is the replica of the pseudorandom (PRN) code. In the outdoor reception, it is possible to estimate the carrier phase θˆv (t) arg dv (t)αv (t) at the peak of the correlation function (τ = τˆv and fD = fˆDv ). The data modulation represents an additional phase uncertainty at the boundaries of the data intervals. Therefore, in order to detect the possible transition at the end of data bit
Therefore, the measurement data processing is done as the following. From the reference signal, we estimate first the code phase, the Doppler frequency, and the navigation data. Then, once all these parameters are stored in memory, we process the indoor data as shown in the block diagram of Fig. 3. First, we remove the Doppler shift and we apply the integrate and dump (I&D) operation over 1 msec (time domain correlation), where the navigation data is removed. Then, via Doppler frequency multiplication, we estimate the Doppler shift error. The resolution in the Doppler frequency is related to the bandwidth after 1 msec I&D and to the coherent integration time. Here, the bandwidth after 1 msec I&D is 1 kHz, therefore the frequencies will span the range from −500 Hz to +500 Hz, and the number of frequency bins is directly related to the coherent integration length N c . For example, for N c = 5 msec, each frequency bin has the width of 100 Hz, and the total number of bins is M=11. The resulting signal, which is a 2-D mesh, is then coherently integrated over N c blocks. In LOS situations, the output 2-D signal has a peak at zero frequency/ zero code phase if the estimates obtained from the reference signal were accurate. IV. S ECOND ORDER STATISTICS : LCR AND ADF The Level Crossing Rate (LCR) and the Average Duration of Fades (ADF) are the second order statistics of the received signal. LCR, which is the rate at which the envelope crosses a certain threshold in a predetermined direction (either positive or negative), and ADF, which is the length of the time that the envelope stays below a given threshold, are two important second-order statistics of fading channels, which, give further insight on the fading characteristics . The LCR is defined as: ∞ a˙ p(a, a)d ˙ a, ˙ (4) NLCR (a) = 0
where a˙ is the time derivative of the signal amplitude a = |α v | and p(a, a) ˙ is the joint PDF of the received amplitude and its derivative taken at the same time t. In [13], the authors showed that the measured fading statistics matched with the Nakagami-m distribution with quite low m factor. Therefore, the expression of LCR satisfies the following equation [12], [14]: σ (5) NLCR (a) = √ W (a, m, µ), 2π were W (a, m, µ) is the Nakagami distribution (with parameters m and µ) at the envelope level a, and σ is the standard deviation of a. ˙ The parameter m of the theoretical distributions can be computed via [12]: {E(a2 )}2 m= Var(a2 )
(6)
NLCR (crossing/sec)
(3)
Indoor, svindex= 03, Cohinteg = 200 msec
2000 1500 1000
LCR
if |θˆv (t + 1) − θˆv (t)| > π/2 if |θˆv (t + 1) − θˆv (t)| < π/2.
Normalized N
dˆv (t + 1) = −dˆv (t) dˆv (t + 1) = dˆv (t)
The empirical LCR was compared to the theoretical LCR computed from eq. (5). The LCR of all tests was computed for amplitudes between the maximum and the minimum levels received indoors with a step δ = (max level − minlevel )/50 dBHz. The results of this comparison are shown in Fig. 4 for Sat 03 in Room Scenario and in Fig. 5 for Sat 13 in Corridor Scenario.
500 0 10
15
20
25
30
35
Level in dB−Hz
0.08
N meas. LCR N theo.
0.06
LCR
0.04 0.02 0 10
15
20
25
30
35
Level in dB−Hz
0.08
Normalized NLCR
boundaries and assuming that −π ≤ θˆv (t) ≤ π, we may apply the following equations:
NLCR meas. N theo., m=6.1 LCR N theo., m=20 LCR
0.06 0.04 0.02 0 10
15
20
25
Level in dB−Hz
30
35
Fig. 4. LCR of the indoor propagation, Meas. 06/04/2004, Satindex = 03, Room Scenario, Nc = 200 msec. Upper curve: LCR values in crossing per second, lower curves: LCR fitting between measurements and theory (middle curve: m-factor computed according (6)).
The maximum LCR was observed to be around the global mean level of the received signal, which means around 20 dB-Hz in the Room Scenario and around 23 dB-Hz in the Corridor Scenario. In the later case, the average value of the maximum LCR of the indoor signal is around 2100 crossings per second and with standard deviation of 57.7 crossings per second. The poor fit observed in the Room Scenario can be explained by that the estimation of the Nakagami-m parameters from the measurement data according to equation (6) is not very accurate since the noise is quite strong. But still the fitting between measurement and theory holds rather well for different m-factor as shown in Fig. 4, lower curve. Those m-factors were found empirically. Another measure of statistics that helps to characterize the fading channel is the ADF below a specified level. We denote by τi the duration of fade below an arbitrary signal level a. Then the ADF below a can be written as: i τi i τi = . (7) ADF(a) number of fades T × NLCR (a) where T is the observation interval. Under the assumption of Nakagami-distributed envelope, it can be shown that the ADF satisfies [14]:
I Storage: reference signal Q
I&D (1ms)
1. Code phase 2. Doppler frequency 3. Navigation data Navigation data removal
Doppler frequency removal
Storage: rover signal
I
Doppler Frequency Multiplication (M points)
I&D (1ms)
Q
Fig. 3.
Indoor, svindex= 13, Cohinteg = 200 msec, m
Therefore, by substituting t =
= 8.1
(crossing/sec) LCR
N
2000 1500
0
1000
20
25
30
35
a
m 2 µr ,
1 W (r, m, µ)dr = Γ(m)
it can be shown that
m 2 µ a
0
tm−1 e−t dt
(10)
The integration in (10) is the incomplete gamma function γ(·). Therefore, the expression of ADF can be given by: 2 √ γ m, m µa ADF(a) = 2π , (11) Γ(m)σW (a, m, µ)
500
15
Time/Frequency mesh
Measurement data processing block diagram
2500
0 10
Nc
40
Level in dB−Hz
Normalized N
LCR
0.06
NLCR meas. NLCR theo.
0.05 0.04 0.03 0.02 0.01 0 10
15
20
25
30
35
40
Level in dB−Hz
Fig. 5. LCR of the indoor propagation, Meas. 30/09/2004, Satindex = 13, Corridor Scenario, Nc = 200 msec. Upper curve: LCR values in crossing per second, lower curve: LCR fitting between measurements and theory, mfactor computed according (6).
ADF(a) =
0
a
V. C ONCLUSIONS W (r, m, µ)dr NLCR (a)
.
(8)
The denominator of (8) has already been evaluated in (5). The numerator of (8) is the integration of the Nakagami distribution that yields the incomplete gamma function [12]: 0
a
W (r, m, µ)dr =
The empirical values of ADF were compared to the theoretical ones computed from eq. (11). The ADF of all tests was computed for levels between the maximum and the minimum levels received indoors with a step δ = (max level − minlevel )/50 dB-Hz. The results of this comparison are shown in Fig. 6 for Sat 03 in Room Scenario and in Fig. 7 for Sat 13 in Corridor Scenario. With most satellite signals tested (around 7 satellites in view), we found that the theoretical ADF follows the empirical ADF. However, some deviations were observed when there is an abrupt change in the indoor signal level (for example moving from inside the room to the outside corridor) or when moving from a clear Line of Sight condition to a deep NonLOS situation.
m a m 2 m 2 r2m−1 e− µ r dr (9) Γ(m) µ 0
In this paper,we presented an analysis of the high-order statistics of the satellite-to-indoor channel propagation based on real measurement data. The presented results are based on two measurement campaigns carried out in typical office environments. Here,we focused on the level crossing rate and the average duration of fades observed in indoor propagation. First, we described the theoretical models of these quantities based on the assumption that the envelope is Nakagami-m distributed. Then, we studied their matches to the measured quantities. In some cases, we found clear fitting between theory and measurements. However, in some scenarios, we
noticed that the estimated distribution parameters are not accurate enough and the fitting is not perfect. ACKNOWLEDGMENT
Indoor, svindex= 03, Cohinteg = 200 msec 60
This work was carried out in the project ”Advanced Techniques for Mobile Positioning” funded by the National Technology Agency of Finland (Tekes).
50
ADF (sec)
40 30
R EFERENCES
20 10 0 10
15
20
25
30
35
30
35
Level in dB−Hz 0.25
ADF
0.2
ADF meas. ADF theo.
0.15 0.1 0.05 0 10
15
20
25
Level in dB−Hz
Fig. 6. ADF of the indoor propagation, Meas. 06/04/2004, Satindex = 03, Room Scenario, Nc = 200 msec. Upper curve: ADF values in second, lower curve: ADF fitting between measurements and theory, using the mfactor given by equation (6)
Indoor, svindex= 13, Cohinteg = 200 msec 60
ADF (sec)
50 40 30 20 10 0 18
20
22
24
26
28
30
32
34
36
30
32
34
36
Level in dB−Hz 0.2
ADF
0.15
ADF meas. ADF theo.
0.1
0.05
0 18
20
22
24
26
28
Level in dB−Hz
Fig. 7. LCR of the indoor propagation, Meas. 30/09/2004, Satindex = 13, Corridor Scenario, Nc = 200 msec. Upper curve: ADF values in second, lower curve: ADF fitting between measurements and theory. , using the mfactor given by equation (6)
[1] E. Kaplan, Understanding GPS, Principles and applications. Boston: Artech House, 1996. [2] L. Ruis and E. Breeuwer, “Galileo mission and services,,” in European Navigation Conference - GNSS, 2004. [3] H.S. Cobb, GPS Pseudolites: theory, design and applications. PhD thesis, Stanford University,, Palo Alto CA, USA, 1997. [4] C. Kee, H. Jun, B. Parkinson, T. Lengastein, and S. Pullen, “Precise calibration of pseudolite positions in indoor navigation system.,” in Proc. of ION GPS Conference, pp. 1499–1507, 1999. [5] J. Hill and I. Progri, “Techniques for reducing the near-far problem in indoor geolocation systems.,” in Proc. of ION NTM, pp. 860–865., 2001. [6] B. Peterson, D. Bruckner, and S. Heye, “Measuring GPS signals indoors,” in Proc. of ION GPS, pp. 615–624, Sep. 1997. [7] C. Ma, G. Jee, G. MacGougan, G. Lachapelle, S. Bloebaum, G. Cox, L. Garin, and J. Shewfelt, “GPS signal degradation modeling,” in Proc. of ION GPS, pp. 882–893, Sep. 2001. [8] R. Klukas, G. Lachapelle, C. Ma, and G. Jee, “GPS signal fading model for urban centres,” IEE Proceedings on Microwaves, Antennas and Propagation, pp. 245–252, 2003. [9] J. R. Watson, “High-sensitivity GPS L1 signal analysis for indoor channel modelling,” Master’s thesis, Department of Geomatics Engineering, University of Calgary, Alberta, Canada, April 2005. [10] W. C. Jakes, Microwave Mobile Communications. New York: IEEE Press, 1993. [11] S. O. Rice, “Distribution of the duration of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech., vol. 37, pp. 581–635, May 1958. [12] N. Youssef, T. Munakata, and M. Takeda, “Fade statistics in nakagami fading environments,” in IEEE International Symposium on Spread Spectrum Techniques & Applications,, pp. 1244–1247., Sep 1996. [13] A. Lakhzouri, E. S. Lohan, I. Saastamoinen, and M. Renfors, “Measurement and characterization of satellite-to-indoor radio wave propagation channel.” To appear in Proc. of The European Navigation Conference (ENC-GNSS05), Jul 2005. [14] A. Abouraddy and S. M. Elnoubi, “Statistical modeling of the indoor radio channel at 10 GHz through propagation measurements. part i: Narrow-band measurements and modeling,” IEEE Transaction on Vehicular Technology, vol. 49, pp. 1491–1507, September 2000.
Abstract— Wireless positioning has received increased attention during the past few years, where several wireless applications have been envisaged and promoted such as the E-911/E-112 regulations. The positioning needs to be carried out in all the environments covered by the wireless communication services, including the most constraining areas such as dense urban areas and obstructed indoor environments. The most known positioning system is the Global Navigation Satellite System (GNSS), which demonstrated quite reliable positioning capabilities when the receiver is in direct view with the sky. However, in indoor environments the signal characteristics are not well understood yet and positioning capabilities are quite poor. Therefore, understanding the multipath propagation indoors and fading characteristics are quite important to make GNSS works indoors. In this paper, we describe measurement-based modeling results of the level-crossing rate (LCR) and average duration of fades (ADF) with the purpose of giving further insight on the real satellite-to-indoor channel characteristics.
I. I NTRODUCTION The global positioning system (GPS) is a positioning and timing system based on 24 satellites in operation. Each satellite orbits the earth in approximately 12 hours [1]. With the development of the new European navigation system, Galileo, the number of satellites will increase to approximately 60 satellites. This will offer a variety of navigation services that can be classified into Open Services providing global positioning and timing services free of charge, Safety-of-Life Services providing integrity services with a defined time-to-alert limit, Public Regulated Services, and Commercial Services [2]. These applications are mostly considered in outdoor navigation where the receiver is in direct view with the sky. However with the growth of indoor navigation applications, the use of same concepts as in outdoor environments based on GNSS is gaining more attention [3], [4], [5]. This will introduce new challenges in the receiver designs, such as signal acquisition and tracking, as well as Line-of-sight occurrence and availability. Therefore, understanding the satellite-to-indoor propagation and fading characteristics is of utmost importance. In the literature, very few studies have attempted to develop channel models specifically for the GPS-indoor channel. The first attempt was back to 1997 [6], where the authors used strong reference data to augment indoor processing capabilities
and conduct coherent integrations of up to 160 ms. The existence of deep fades and their impact on indoor signals was observed. In [7], [8], the authors have evaluated the Karasawa model and Urban Three-State Fade Model (UTSFM) as applied to GPS signals in difficult environments using standard and high-sensitivity GPS receivers. The most recent study was found in [9], where the author analyzed highbandwidth raw GPS data with high-sensitivity techniques to characterize fading and multipath characteristics indoors This paper presents an analysis of experimental data for the satellite-to-indoor channel obtained in different measurement campaigns carried out by the authors based on GPS signals. We remark that the authors expect to find similar results once the Galileo signals from the satellites will be available, with the main difference that more resolution in modeling the channel delays is expected. The aim of this work is to analyze the Level-Crossing Rate (LCR) and Average Duration of Fades (ADF) to give further insight on the real satellite-to-indoor channel characteristics. Analytical expressions of these quantities have been derived in the context of Rayleigh [10], Rice [11], and Nakagami [12] distributions. All these stochastic processes play an important role for modeling mobile fading channels. However, the validation of these models for the current application through real measurement data is not often encountered in the literature. These statistics are important for indoor mobile applications, since they characterize the rate and duration of fade occurrences. The sequel of this paper is organized as follows. In Section 2, the experiment settings and measurement environments are described. Data processing and navigation data detection are described in Section 3. Section 4 is dedicated to the analysis of the second order statistics, and Section 5 gives the conclusions. II. M EASUREMENT C AMPAIGNS A study related to satellite-to-indoor channel modeling was undertaken by Tampere University of Technology and u-Nav microelectronics, Finland. During this study, 2 measurement campaigns were carried out in April and September, 2004 in typical office environments. The transmitters are the different GPS satellites available in view during the measurements dates
and the receiver is an integrated GPS receiver with sampling rate of 16.36 MHz.
in the indoor environment. Both receivers are shown in Fig. 2.
A. Measurement Environments During the measurement campaigns, different indoor environments were tested. The first scenario, denoted as Room scenario, corresponds to a small room without any windows (about 5 m2 ), where in the front there is small corridor with large windows. Here the Line-of-sight (LOS) signal is more likely to be absent. The second scenario, denoted by Corridor scenario, corresponds to a long corridor with windows and doors open. The movement inside the environments is in all x, y, and z directions (The GPS antenna is placed in hand, and the movement is in all directions). The movement is random and it is at the walking speed of about 2-3 km/h (see Fig. 1). For the Room scenario, the movement is performed in and out of the room. All the measurements were taken for a duration of 1 to 3 minutes for reliable statistics.
Fig. 2. The two GPS receivers synchronized to a common clock, operating in parallel.
III. M EASUREMENT DATA PROCESSING The signal at the reference GPS receiver can be written as [1]:
Fig. 1. Measurement environments: Corridor and Room scenarios, random movements.
B. Measurement Settings The indoor signal is expected to be very weak and embedded in noise. Therefore long coherent and non-coherent integrations are required. For the outdoor signal, usually the coherent integration intervals are around 1, 4, 5 msec [1] (a divisor of the data bit duration T b , which is 20 msec). However, in indoor reception for channel measurement purposes, the coherent integration must be much longer than 20 msec in order to compensate for the increase in the noise level. Therefore, an estimation and removal of the navigation data must be undertaken before any coherent integration. The blind estimation of the navigation data from the indoor signal alone, which is expected to be very noisy, can not guarantee good performance. Besides, we need to estimate the code-phase and Doppler frequency. For these reasons, two GPS receivers synchronized to a common clock, operating in parallel are used. The first one acquires the signal from an outdoor antenna placed on the roof of the building. This signal is expected to be strong, and it will be used as reference signal for code-phase and Doppler frequency acquisition, as well as for frequency drift estimation and correction. The second antenna is moving
r(t) =
N sv
ds (t − τs )αs (t − τs )e+j2πfDs t Cs (t − τs )+
s=1
(1)
η(t),
where t is the GPS time, ds (t), αs (t), τs , and fDs are the data bit, complex channel coefficient, instantaneous delay, and Doppler frequency from satellite s at time t corresponding to the first arriving path, respectively. Here C s (t) is the transmitted code and η(t) is an additive noise incorporating the multipath effect. We point out here that equation (1) models the radio channel as a single tap channel, this assumption is reasonable since in outdoor reception, it is more likely that we have strong single path [13]. The extension to multipath model is straightforward The output of the correlator corresponding to the satellite v, to the tentative delay τ , and Doppler frequency f D is: yv (τ, fD ) = T1 r(t)e−j2πfD t Cv(r) (t − τ )dt, (2) T
(r)
where T is the integration time and C v (·) is the replica of the pseudorandom (PRN) code. In the outdoor reception, it is possible to estimate the carrier phase θˆv (t) arg dv (t)αv (t) at the peak of the correlation function (τ = τˆv and fD = fˆDv ). The data modulation represents an additional phase uncertainty at the boundaries of the data intervals. Therefore, in order to detect the possible transition at the end of data bit
Therefore, the measurement data processing is done as the following. From the reference signal, we estimate first the code phase, the Doppler frequency, and the navigation data. Then, once all these parameters are stored in memory, we process the indoor data as shown in the block diagram of Fig. 3. First, we remove the Doppler shift and we apply the integrate and dump (I&D) operation over 1 msec (time domain correlation), where the navigation data is removed. Then, via Doppler frequency multiplication, we estimate the Doppler shift error. The resolution in the Doppler frequency is related to the bandwidth after 1 msec I&D and to the coherent integration time. Here, the bandwidth after 1 msec I&D is 1 kHz, therefore the frequencies will span the range from −500 Hz to +500 Hz, and the number of frequency bins is directly related to the coherent integration length N c . For example, for N c = 5 msec, each frequency bin has the width of 100 Hz, and the total number of bins is M=11. The resulting signal, which is a 2-D mesh, is then coherently integrated over N c blocks. In LOS situations, the output 2-D signal has a peak at zero frequency/ zero code phase if the estimates obtained from the reference signal were accurate. IV. S ECOND ORDER STATISTICS : LCR AND ADF The Level Crossing Rate (LCR) and the Average Duration of Fades (ADF) are the second order statistics of the received signal. LCR, which is the rate at which the envelope crosses a certain threshold in a predetermined direction (either positive or negative), and ADF, which is the length of the time that the envelope stays below a given threshold, are two important second-order statistics of fading channels, which, give further insight on the fading characteristics . The LCR is defined as: ∞ a˙ p(a, a)d ˙ a, ˙ (4) NLCR (a) = 0
where a˙ is the time derivative of the signal amplitude a = |α v | and p(a, a) ˙ is the joint PDF of the received amplitude and its derivative taken at the same time t. In [13], the authors showed that the measured fading statistics matched with the Nakagami-m distribution with quite low m factor. Therefore, the expression of LCR satisfies the following equation [12], [14]: σ (5) NLCR (a) = √ W (a, m, µ), 2π were W (a, m, µ) is the Nakagami distribution (with parameters m and µ) at the envelope level a, and σ is the standard deviation of a. ˙ The parameter m of the theoretical distributions can be computed via [12]: {E(a2 )}2 m= Var(a2 )
(6)
NLCR (crossing/sec)
(3)
Indoor, svindex= 03, Cohinteg = 200 msec
2000 1500 1000
LCR
if |θˆv (t + 1) − θˆv (t)| > π/2 if |θˆv (t + 1) − θˆv (t)| < π/2.
Normalized N
dˆv (t + 1) = −dˆv (t) dˆv (t + 1) = dˆv (t)
The empirical LCR was compared to the theoretical LCR computed from eq. (5). The LCR of all tests was computed for amplitudes between the maximum and the minimum levels received indoors with a step δ = (max level − minlevel )/50 dBHz. The results of this comparison are shown in Fig. 4 for Sat 03 in Room Scenario and in Fig. 5 for Sat 13 in Corridor Scenario.
500 0 10
15
20
25
30
35
Level in dB−Hz
0.08
N meas. LCR N theo.
0.06
LCR
0.04 0.02 0 10
15
20
25
30
35
Level in dB−Hz
0.08
Normalized NLCR
boundaries and assuming that −π ≤ θˆv (t) ≤ π, we may apply the following equations:
NLCR meas. N theo., m=6.1 LCR N theo., m=20 LCR
0.06 0.04 0.02 0 10
15
20
25
Level in dB−Hz
30
35
Fig. 4. LCR of the indoor propagation, Meas. 06/04/2004, Satindex = 03, Room Scenario, Nc = 200 msec. Upper curve: LCR values in crossing per second, lower curves: LCR fitting between measurements and theory (middle curve: m-factor computed according (6)).
The maximum LCR was observed to be around the global mean level of the received signal, which means around 20 dB-Hz in the Room Scenario and around 23 dB-Hz in the Corridor Scenario. In the later case, the average value of the maximum LCR of the indoor signal is around 2100 crossings per second and with standard deviation of 57.7 crossings per second. The poor fit observed in the Room Scenario can be explained by that the estimation of the Nakagami-m parameters from the measurement data according to equation (6) is not very accurate since the noise is quite strong. But still the fitting between measurement and theory holds rather well for different m-factor as shown in Fig. 4, lower curve. Those m-factors were found empirically. Another measure of statistics that helps to characterize the fading channel is the ADF below a specified level. We denote by τi the duration of fade below an arbitrary signal level a. Then the ADF below a can be written as: i τi i τi = . (7) ADF(a) number of fades T × NLCR (a) where T is the observation interval. Under the assumption of Nakagami-distributed envelope, it can be shown that the ADF satisfies [14]:
I Storage: reference signal Q
I&D (1ms)
1. Code phase 2. Doppler frequency 3. Navigation data Navigation data removal
Doppler frequency removal
Storage: rover signal
I
Doppler Frequency Multiplication (M points)
I&D (1ms)
Q
Fig. 3.
Indoor, svindex= 13, Cohinteg = 200 msec, m
Therefore, by substituting t =
= 8.1
(crossing/sec) LCR
N
2000 1500
0
1000
20
25
30
35
a
m 2 µr ,
1 W (r, m, µ)dr = Γ(m)
it can be shown that
m 2 µ a
0
tm−1 e−t dt
(10)
The integration in (10) is the incomplete gamma function γ(·). Therefore, the expression of ADF can be given by: 2 √ γ m, m µa ADF(a) = 2π , (11) Γ(m)σW (a, m, µ)
500
15
Time/Frequency mesh
Measurement data processing block diagram
2500
0 10
Nc
40
Level in dB−Hz
Normalized N
LCR
0.06
NLCR meas. NLCR theo.
0.05 0.04 0.03 0.02 0.01 0 10
15
20
25
30
35
40
Level in dB−Hz
Fig. 5. LCR of the indoor propagation, Meas. 30/09/2004, Satindex = 13, Corridor Scenario, Nc = 200 msec. Upper curve: LCR values in crossing per second, lower curve: LCR fitting between measurements and theory, mfactor computed according (6).
ADF(a) =
0
a
V. C ONCLUSIONS W (r, m, µ)dr NLCR (a)
.
(8)
The denominator of (8) has already been evaluated in (5). The numerator of (8) is the integration of the Nakagami distribution that yields the incomplete gamma function [12]: 0
a
W (r, m, µ)dr =
The empirical values of ADF were compared to the theoretical ones computed from eq. (11). The ADF of all tests was computed for levels between the maximum and the minimum levels received indoors with a step δ = (max level − minlevel )/50 dB-Hz. The results of this comparison are shown in Fig. 6 for Sat 03 in Room Scenario and in Fig. 7 for Sat 13 in Corridor Scenario. With most satellite signals tested (around 7 satellites in view), we found that the theoretical ADF follows the empirical ADF. However, some deviations were observed when there is an abrupt change in the indoor signal level (for example moving from inside the room to the outside corridor) or when moving from a clear Line of Sight condition to a deep NonLOS situation.
m a m 2 m 2 r2m−1 e− µ r dr (9) Γ(m) µ 0
In this paper,we presented an analysis of the high-order statistics of the satellite-to-indoor channel propagation based on real measurement data. The presented results are based on two measurement campaigns carried out in typical office environments. Here,we focused on the level crossing rate and the average duration of fades observed in indoor propagation. First, we described the theoretical models of these quantities based on the assumption that the envelope is Nakagami-m distributed. Then, we studied their matches to the measured quantities. In some cases, we found clear fitting between theory and measurements. However, in some scenarios, we
noticed that the estimated distribution parameters are not accurate enough and the fitting is not perfect. ACKNOWLEDGMENT
Indoor, svindex= 03, Cohinteg = 200 msec 60
This work was carried out in the project ”Advanced Techniques for Mobile Positioning” funded by the National Technology Agency of Finland (Tekes).
50
ADF (sec)
40 30
R EFERENCES
20 10 0 10
15
20
25
30
35
30
35
Level in dB−Hz 0.25
ADF
0.2
ADF meas. ADF theo.
0.15 0.1 0.05 0 10
15
20
25
Level in dB−Hz
Fig. 6. ADF of the indoor propagation, Meas. 06/04/2004, Satindex = 03, Room Scenario, Nc = 200 msec. Upper curve: ADF values in second, lower curve: ADF fitting between measurements and theory, using the mfactor given by equation (6)
Indoor, svindex= 13, Cohinteg = 200 msec 60
ADF (sec)
50 40 30 20 10 0 18
20
22
24
26
28
30
32
34
36
30
32
34
36
Level in dB−Hz 0.2
ADF
0.15
ADF meas. ADF theo.
0.1
0.05
0 18
20
22
24
26
28
Level in dB−Hz
Fig. 7. LCR of the indoor propagation, Meas. 30/09/2004, Satindex = 13, Corridor Scenario, Nc = 200 msec. Upper curve: ADF values in second, lower curve: ADF fitting between measurements and theory. , using the mfactor given by equation (6)
[1] E. Kaplan, Understanding GPS, Principles and applications. Boston: Artech House, 1996. [2] L. Ruis and E. Breeuwer, “Galileo mission and services,,” in European Navigation Conference - GNSS, 2004. [3] H.S. Cobb, GPS Pseudolites: theory, design and applications. PhD thesis, Stanford University,, Palo Alto CA, USA, 1997. [4] C. Kee, H. Jun, B. Parkinson, T. Lengastein, and S. Pullen, “Precise calibration of pseudolite positions in indoor navigation system.,” in Proc. of ION GPS Conference, pp. 1499–1507, 1999. [5] J. Hill and I. Progri, “Techniques for reducing the near-far problem in indoor geolocation systems.,” in Proc. of ION NTM, pp. 860–865., 2001. [6] B. Peterson, D. Bruckner, and S. Heye, “Measuring GPS signals indoors,” in Proc. of ION GPS, pp. 615–624, Sep. 1997. [7] C. Ma, G. Jee, G. MacGougan, G. Lachapelle, S. Bloebaum, G. Cox, L. Garin, and J. Shewfelt, “GPS signal degradation modeling,” in Proc. of ION GPS, pp. 882–893, Sep. 2001. [8] R. Klukas, G. Lachapelle, C. Ma, and G. Jee, “GPS signal fading model for urban centres,” IEE Proceedings on Microwaves, Antennas and Propagation, pp. 245–252, 2003. [9] J. R. Watson, “High-sensitivity GPS L1 signal analysis for indoor channel modelling,” Master’s thesis, Department of Geomatics Engineering, University of Calgary, Alberta, Canada, April 2005. [10] W. C. Jakes, Microwave Mobile Communications. New York: IEEE Press, 1993. [11] S. O. Rice, “Distribution of the duration of fades in radio transmission: Gaussian noise model,” Bell Syst. Tech., vol. 37, pp. 581–635, May 1958. [12] N. Youssef, T. Munakata, and M. Takeda, “Fade statistics in nakagami fading environments,” in IEEE International Symposium on Spread Spectrum Techniques & Applications,, pp. 1244–1247., Sep 1996. [13] A. Lakhzouri, E. S. Lohan, I. Saastamoinen, and M. Renfors, “Measurement and characterization of satellite-to-indoor radio wave propagation channel.” To appear in Proc. of The European Navigation Conference (ENC-GNSS05), Jul 2005. [14] A. Abouraddy and S. M. Elnoubi, “Statistical modeling of the indoor radio channel at 10 GHz through propagation measurements. part i: Narrow-band measurements and modeling,” IEEE Transaction on Vehicular Technology, vol. 49, pp. 1491–1507, September 2000.
