- Email: [email protected]
Doppler spread for Gaussian scatter density environments employing smart antennas
Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636 www.elsevier.de/aeue
LETTER
Doppler spread for Gaussian scatter density environments employing smart antennas Mario Castrellona , David Muñoza,∗ , Cesar Vargasa , Claudio Lopezb , David Covarrubiasc a Tecnologico de Monterrey, Campus Monterrey, Ave. Eugenio Garza Sada 2501, Monterrey N.L. C.P. 64849, Mexico b University of Sonora, Rosales y Transversal, Hermosillo, Son., Mexico c Electronics and Communications, CICESE Research Center Km. 107 Tijuana-Ensenada, Ensenada B.C., C.P. 22860, Mexico
Received 8 June 2006; accepted 19 December 2006
Abstract In this paper, the effects of actual smart antennas on the Power Doppler Spectrum (PDS) are studied when smart antennas are deployed at the base station (BS), employing a statistical Gaussian Scatter Density Model (GSDM). This proposed approach is general and it can be applied to omnidirectional and sectorized antennas as well, for beam patterns of linear and circular arrays, the results obtained are compared with those for omnidirectional and sectorized antennas. The characterization of PDS is presented for channel scenarios including different motion direction, scatterer spread, and a variety of beam patterns. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: Power Doppler Spectrum; Smart antennas; Gaussian Scatter Density Model
1. Introduction The tremendous growth in wireless communications has led to a crowding of the authorized radio spectrum. To provide high-quality services and to increase capacity in wireless systems, improved methods are introduced, such as the deployment of smart antennas. Among the benefits for smart antenna implementation are the possibility for increased spectrum efficiency of cellular systems, interference cancellation, and the introduction of higher bit-rate data services for third generation systems [1]. However, for an uneven antenna gain, the different propagation rays are affected differently, so that the Doppler characteristics depart from those ∗ Corresponding author. Tel./fax: +52 81 83 59 72 11.
E-mail addresses: [email protected] (M. Castrellon), [email protected] (D. Muñoz), [email protected] (C. Vargas), [email protected] (C. Lopez), [email protected] (D. Covarrubias). 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2006.12.010
obtained with omni and sectorized antennas. In this paper, we analyze the Doppler shift for both actual circular and linear arrays [2], which is characterized through the PDS that describes the way the power is spread around a reference frequency. This has been studied in the literature when the transmitter and receiver antennas are omnidirectional and sectorized [3–5]. The PDS is used to calculate the level crossing rate (LCR), to determine if a signal is attenuated and the coherent time of the communication channel, which represents the interval of time when the characteristics of the channel stay without change [6]. In this paper, we analyze the PDS for a Gaussian Scatter Density environment by employing actual intelligent beam patterns at the BS, which allow for an increase in the system performance through spatial filtering. The statistical analysis will be conducted for macrocell environments, for a variety of scenarios, by relating the scattering region width (srw) [7], the subscriber motion direction, and the antenna’s beam characteristics.
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M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
The Gaussian Scatter Density Model (GSDM) [7,8], assumes that the effective scattering points are Gaussian distributed with the center at the mobile station (MS). The model assumes that the energy received by the path passing through several scattering points can be neglected. Therefore, only one bounce in the path from the transmitter to the receiver is considered. Other additional assumptions commonly adopted in the model are: • The signals received at the BS are plane waves propagating along the horizon. • Scatterers are treated as omnidirectional re-radiating elements. • The scatterers have identical scattering coefficients. • The phases associated with each scatterer are independent. Thus, for a MS located D units from the BS, as seen in Fig. 1, the scattering location points (xm , ym ) can be distributed as 2 2 1 xm + y m (1) exp − fxm ,ym (xm , ym ) = 22 22 when the mobile location is taken as the coordinate reference. If the origin is placed at the BS, the scattering points are denoted (xb , yb ) and they will be distributed according to the probability density function (pdf) (xb − D)2 + yb2 1 fxb ,yb (xb , yb ) = , (2) exp − 22 22 where the parameter is used to characterize the spread of the scatterers refered as srw. Alternatively, a polar representation can be adopted. The corresponding polar representation using (rb , b ) and (rm , m ) for instance, can be obtained by using the Jacobian yb
Probability of being in the window |90 °|
2. Gaussian Scatter Density Model 1 0.95 0.9 0.85 0.8 0.75
0.7
0
0.2
0.4
0.6
0.8 srw/D
1.0
1.2
1.4
Fig. 2. Probability the AOA at the BS is within the window ±/2.
transformation
fxb ,yb (xb , yb ) frb ,b (rb , b ) = |J (xb , yb )| xb =rb cos(b )
(3)
yb =rb sin(b )
integration on the entire rb range allows us to obtain the marginal pdf of the angle of arrival (AOA) at the BS as D cos b 1 D2 exp − 2 + √ fb (b ) = 2 2 2 2 D 2 sin2 b D cos b × exp − erfc − √ 22 2 (4) − b , the erfc is known as the complementary error function [9]. Integrating (4) we obtain the probability of scatterers in an angular window. This proportion depends on srw relative to the MS–BS distance, that is the srw/D ratio. It is a fact that for (srw/D) 3/4, P {b ± /2}0.90. This probability increases as the MS–BS separation grows or as the srw diminishes see Fig. 2. Therefore, most of the presented results are obtained for the b = ±/2 window.
ym
3. Power Doppler spectrum
rm θm
rb
θb D
MS
BS
Fig. 1. Gaussian scattering scenario for a circular array.
xb xm
A scatterer will produce a Doppler shift induced by the apparent transmitter/receiver motion. For a mobile traveling in the direction observed v , with respect to the straight line passing through the BS and the MS [3,6], the Doppler frequency is given by fi =fm cos(m −v ), |fi | < fm , where fm = v/ is the maximum Doppler shift, v is the mobile speed, is the carrier wavelength, and (m − v ) is the angle subtended by connecting the MS to scatterer line. For the purposes of this paper, we adopt the common reference speed of 54 km/h and the carrier frequency of 2 GHz thus, the maximum Doppler shift will be 100 Hz.
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
Previous works have studied the PDS through the Macrocell Circular Scattering Model (MCSM) and GSDM for sectorized cases [5,10]. For omnidirectional cases, the PDS is given by the well known Clarke’s model regarding uniform distribution of the AOA at the MS [3]. In general, the PDS of the received signal can be expressed as [6]
Li2
ri2 Li1
A2 S(f ) = × [fm (v + |cos−1 (f/fm )|) √ 0 fm 1 − (f/fm ) + fm (v + |cos−1 (f/fm )|)], (5) where A0 is a constant and f is the Doppler frequency. Doppler spread performance for actual smart antennas will be characterized in the following section.
4. Effect of smart antennas on the AOA pdf in the MS Typical smart antenna beam patterns are expected to have a main lobe pointing in the expected direction of the MS. However, there are also lateral lobes that impact the Doppler spectrum. In order to assess the use of smart antennas, the beamwidth is quantized into K =/ sectors of angular width . In this form the gain G(b ) in any directions b can be expressed as K G(b ) ≈ G(bi )rect b − bi − , 2 i=1 (6) |b | ± . 2 This formulation allows us to take advantage of the fact that the PDS has been characterized for sectorized antenna [5]. However, the Doppler contribution depends not only on the gains, but also on the probability of the scatterers being in the concerned angular window and angular scatterer location as seen from MS site m . This angular location can be described, after using a Jacobian transformation on (1) and after integrating over all possible scatterer–MS distance in the considered sector, thus r2 fm (m ) = frm ,m (rm , m ) drm r1 2 −ri1 1 = exp 2 cos2 (m − i1 )
2 −ri2 − exp , cos2 (m − i2 ) bi < |m |, where r1 and r2 are ri1 , r1 = cos(m − i1 ) ri2 r2 = cos(m − i2 )
(7)
633
ri1
r2 r1 θbi BS
D
θi1 θi2
θm
x MS
Fig. 3. Scattering boundary region.
and ri1 , ri2 are distances from the MS location to the lines Li1 and Li2 that define the sector [bi − (/2), bi + (/2)] and i1 , i2 are the subtended angles by the line orthogonal to Li1 and Li2 , respectively, see Fig. 3. This is ri1 = D sin bi − , (10) 2 , (11) ri2 = D sin bi + 2 i1 = + bi − , (12) 2 2 (13) i2 = + bi + 2 2 note that (7) is applied for the values greater than bi , since the imaginary line formed by m must reach the sector studied; and it is zero for the rest [11]. The joint pdf of (rm , m ) is given by 2 rm rm frm ,m (rm , m ) = . (14) exp − 22 22 In this way, we relate the angular observations at the MS m and the viewed angles bi from the BS, and we conduct the same procedure for each of the antenna’s subsectors so that the whole PDS will be the weighted addition to the individual PDS contribution Si (f ) multiplied by the power gain of the corresponding antenna sector. Thus, the total PDS will be approximately S(f ) ≈
K
Si (f )[G(bi )].
(15)
i=1
(8) (9)
Formulations in (7)–(15) are general and they can also be used for an omnidirectional antenna case (i.e., G(bi ) = 1; ∀i) as well as for a sectorized case centered in the direction bi (i.e., G(bi ) = Gij ; where ij is a Kronecker delta).
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M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
5. Results
-16
-16 Lineal array Circular array Sectorized Omnidirectional
-18
PSD (Bd)
-20 -22 -24 -26 -28 -30 -32 -100
-50
0 50 Doppler Frequency(Hz)
Fig. 4. Macrocell PDS with v = 180◦ and BW = 15◦ .
100
Lineal array Circular array Sectorized Omnidirectional
-18
PSD (Bd)
-20
-22
-24
-26
-28 -100
-50
50 0 Doppler Frequency (Hz)
100
Fig. 5. Macrocell PDS with v = 90◦ and BW = 15◦ .
-16 Omnidirectional
0° 45° 90° 135° 180°
-18 -20
PSD (Bd)
In this section, we report results for linear and circular arrays. We take as reference the macrocell environment with a base to mobile separation of D = 1000 m and srw = 100 m as in [5,7]. The Doppler effect depends on a variety of parameters such as the mobile travel direction, but it also depends on the receiving antenna’s characteristics (e.g., beam pattern). In the case of an omnidirectional pattern (Clarke’s Model), all the scatterers will contribute to the PDS, while for the sectorized antenna the scatterers outside the antenna’s view are cut off. Since those scatterers outside the sector window are neglected, this implies lower receiving power. In the case of smart antenna, the scatterers contribution to the PDS will be affected by the antenna’s gain in the corresponding direction. Fig. 4 shows the PDS for an omnidirectional and 15◦ sectorized receiving pattern as well as for linear and circular antenna arrays for the same beamwidth (BW). Fig. 4 also shows a slight asymmetry that favors the low frequencies. This is because the number of scatterers to the right side of the BS (producing a negative shift) is larger than the number to the left side (producing positive shift). For a very narrow BW and a mobile travel direction of v = 90◦ , the Doppler shift tend to be small. This shows that the PDS is concentrated around zero. However, as the BW increases the Doppler shift spreads. See Fig. 5. We also show that for this direction there is no asymmetry. It can be observed that PDSs for circular and linear arrays are close. However, small differences are seen, because the lateral lobes for the linear array are approximately 4 dB lower than for the omnidirectional arrays. Frequency spread strongly depends on the motion direction, this is illustrated in Fig. 6 where the PDS is shown using a linear array at different mobile moving directions v . It can be observed that the PDS for 0◦ and 180◦ are mirrored images. The same effect occurs for 45◦ and 135◦ .
-22 -24 -26 -28 -30 -32 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 6. Macrocell PDS with variable v and BW = 15◦ .
Although the above results were obtained for a BW of 15◦ , we point out that the calculations involved the whole antenna pattern. Note, the BW was taken as the angular spacing of the two nulls around the main gain direction. Figs. 7 and 8 show the effect of the BW on the PDS, for travel directions v of 180◦ and 90◦ , respectively, when linear arrays are used. It can be observed that as the BW becomes wider the PDS approaches that of the Clarke’s Model. Similar results can be obtained for circular arrays. A common convention has been to take the straight line passing through the BS and MS as a coordinate axis with its origin at the BS (Fig. 1). This convention is useful as omni and directional antenna, as well as circular arrays present a near symmetric receiving pattern with respect to the MS direction. It has also been customary to consider only those scatterers in an angular window (o − 2 , o + 2 ) where o denotes the mobile azimuth location. However, in the case
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636 -25
-16 -18 -20
Omnidirectional
No inclination
60° 45° 30° 15°
45° 30°
-30
-22 PDS (dB)
PSD (Bd)
635
-24 -26
-35
-28 -30 -32 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 7. Macrocell PDS with variable BW and v = 180◦ .
-40 -100
-50
0 Doppler frequency (Hz)
50
100
Fig. 10. Macrocell PDS for different angles in the main beam BW = 15◦ and v = 90◦ .
-16 Omnidirectional
60° -18
45° 30° 15°
PSD (Bd)
-20
-22
-24
-26
-28 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 8. Macrocell PDS with variable BW and v = 90◦ . MS scatterer antenna axis
ξ Ψ °
Specular beam
Main beam
Fig. 9. Gaussian scattering scenario for a linear array with the main beam at 45◦ .
of linear arrays, the beam presents a specular pattern with respect to the antenna’s axis see Fig. 9. Consequently, the Doppler contributions are not symmetric. The performance will depend on the angle formed by the BS–MS line and the antenna’s axis. This behavior is apparent in Fig. 10, where Doppler spectra for angles =45◦ , =30◦ , and where there is no inclination, are presented. Using D = 1000 m, srw = 500 m, v for 90◦ and a BW = 15◦ , an asymmetry can be observed due to the scatterers that are illuminated by the specular beam. The asymmetry is more blatant with the inclination of = 30◦ , because this beam reaches more scatterers.
6. Conclusion In this paper, a new methodology to evaluate the effects of the use of smart antennas at the BS on the PDS has been presented. Omnidirectional and sectorized antennas are particular cases of the presented formulation. The methodology is assessed in a GSDM macrocell environment using both linear and circular arrays. Also, we have calculated the scatterer probability of every sector in the studied scenario and, by the gain power G(b ) weighted, a close approximation of the behavior of the PDS is obtained when smart antennas are deployed, in the case of omni and sectorized antennas the presented formulation leads to results consistent to those reported in literature for the same scenarios. The comparison between PDS tells us that Clarke’s model and sectorized antennas give more power to the lateral Doppler frequencies than smart antennas. The impact of the BW of the main lobe and the direction of motion v of the MS were also included. For linear arrays the PDS is compared when the MS location is not orthogonal to the antenna array axis.
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M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
References [1] Anderson S, Millnert M, Viberg M, Wahlberg B. An adaptive array for mobile communications systems. IEEE Trans Veh Technol 1991;40:230–6. [2] Litva J, Kwong-Yeung T. Digital beamforming in wireless communications. Boston, MA: Artech House; 1996. [3] Clarke RH. A statistical theory of mobile-radio reception. Bell Syst Tech J 1968;1:957–1000. [4] Jakes WC. Microwave mobile communications, IEEE Communications Society, 1974. [5] Lopez CA, Covarrubias DH, Muñoz D, Panduro M. Statistical cellular gaussian scatter density channel model employing a directional antenna for mobile environments. Int J Electron Commun 2005;59:195–9. [6] Liberti JJ, Rappaport TS. Smart antennas for wireless communications: Is-95 and third generation cdma applications. New York: Prentice-Hall; 1999. [7] Lotter MP, Van Rooyen P. Modeling spatial aspects of cellular cdma/sdma systems. IEEE Commun Lett 1999;3:128–31. [8] Crohn I, Bonek E. Modeling of intersymbol – interference in a rayleigh fast fading channel with typical delay profile. IEEE Trans Veh Technol 1992; 438–47. [9] Janaswamy R. Angle and time of arrival statistics for the gaussian scatter density model. IEEE Trans Wireless Commun 2002;1:488–97. [10] Petrus P, Reed J, Rappaport TS. Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans Commun 2002;50:495–502. [11] Castrellon M. Statistical cellular scatterer density model employing smart antenas for macrocell environments. MSc in Electronic Engineering Thesis. Tec de Monterrey, Campus Monterrey, N.L. Mexico; 2005. Mario Castrellon was born in Durango, Mexico 1980. He received his B.S. and M.S. degrees in electrical engineering from the Instituto Tecnologico de Durango, Durango, Mexico and Instituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico, in 2002 and 2005, respectively.
David Muñoz received his B.S., M.S., and Ph.D. degrees in electrical engineering from the Universidad de Guadalajara, Guadalajara, Mexico, Cinvestav Mexico City, México, and University of Essex, Colchester, England, in 1972, 1976 and 1979, respectively. He has been Chairman of the Communication Department and Electrical Engineering Department at Cinvestav,
IPN. In 1992, he joined the Centro de Electronica y Telecomunicaciones, Instituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico, where he is the Director. His research interests include transmission and personal communication systems. Cesar Vargas received his M.S., and Ph.D. degrees in electrical engineering from Louisiana State University, Baton Rouge, Louisiana, USA, in 1992, and 1996, respectively. In 1996, he joined the Centro de Electronica y Telecomunicaciones, Tec de Monterrey, Campus Monterrey, Mexico, where he is the Telecommunications Graduate Program Coordinator. His research interests include reconfigurable networks and routing. Claudio A. Lopez-Mirinda received the Bachelor’s degree in Mathematics from the University of Sonora, Hermosillo, Sonora, Mexico, in 1987, and his Master’s degree in Operation Research from the Universidad Nacional Autónoma de Mexico, in 1995. He received his Ph.D. in Electronics and Telecommunications at the CICESE Research Center, Ensenada, B.C., Mexico. His research interests are radio channel modeling and evolutionary computation applied to angle of arrival estimations in smart antennas. David H. Covarrubias-Rosales received his Ph.D. in Telecommunications (Cum Laude) from the Universitat Politécnica de Catalunya, Barcelona, Spain; his Master of Science in Telecommunications from CICESE, México, and his BS in Electronics and Communications Engineering from the Universidad Nacional Autónoma de México, UNAM, Mexico City. He is a faculty member of the Electronics and Telecommunications Department, CICESE Research Center, Ensenada, B.C., México, from June 1984 to date. His research areas are 3G mobile communications systems and smart antennas.
LETTER
Doppler spread for Gaussian scatter density environments employing smart antennas Mario Castrellona , David Muñoza,∗ , Cesar Vargasa , Claudio Lopezb , David Covarrubiasc a Tecnologico de Monterrey, Campus Monterrey, Ave. Eugenio Garza Sada 2501, Monterrey N.L. C.P. 64849, Mexico b University of Sonora, Rosales y Transversal, Hermosillo, Son., Mexico c Electronics and Communications, CICESE Research Center Km. 107 Tijuana-Ensenada, Ensenada B.C., C.P. 22860, Mexico
Received 8 June 2006; accepted 19 December 2006
Abstract In this paper, the effects of actual smart antennas on the Power Doppler Spectrum (PDS) are studied when smart antennas are deployed at the base station (BS), employing a statistical Gaussian Scatter Density Model (GSDM). This proposed approach is general and it can be applied to omnidirectional and sectorized antennas as well, for beam patterns of linear and circular arrays, the results obtained are compared with those for omnidirectional and sectorized antennas. The characterization of PDS is presented for channel scenarios including different motion direction, scatterer spread, and a variety of beam patterns. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: Power Doppler Spectrum; Smart antennas; Gaussian Scatter Density Model
1. Introduction The tremendous growth in wireless communications has led to a crowding of the authorized radio spectrum. To provide high-quality services and to increase capacity in wireless systems, improved methods are introduced, such as the deployment of smart antennas. Among the benefits for smart antenna implementation are the possibility for increased spectrum efficiency of cellular systems, interference cancellation, and the introduction of higher bit-rate data services for third generation systems [1]. However, for an uneven antenna gain, the different propagation rays are affected differently, so that the Doppler characteristics depart from those ∗ Corresponding author. Tel./fax: +52 81 83 59 72 11.
E-mail addresses: [email protected] (M. Castrellon), [email protected] (D. Muñoz), [email protected] (C. Vargas), [email protected] (C. Lopez), [email protected] (D. Covarrubias). 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2006.12.010
obtained with omni and sectorized antennas. In this paper, we analyze the Doppler shift for both actual circular and linear arrays [2], which is characterized through the PDS that describes the way the power is spread around a reference frequency. This has been studied in the literature when the transmitter and receiver antennas are omnidirectional and sectorized [3–5]. The PDS is used to calculate the level crossing rate (LCR), to determine if a signal is attenuated and the coherent time of the communication channel, which represents the interval of time when the characteristics of the channel stay without change [6]. In this paper, we analyze the PDS for a Gaussian Scatter Density environment by employing actual intelligent beam patterns at the BS, which allow for an increase in the system performance through spatial filtering. The statistical analysis will be conducted for macrocell environments, for a variety of scenarios, by relating the scattering region width (srw) [7], the subscriber motion direction, and the antenna’s beam characteristics.
632
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
The Gaussian Scatter Density Model (GSDM) [7,8], assumes that the effective scattering points are Gaussian distributed with the center at the mobile station (MS). The model assumes that the energy received by the path passing through several scattering points can be neglected. Therefore, only one bounce in the path from the transmitter to the receiver is considered. Other additional assumptions commonly adopted in the model are: • The signals received at the BS are plane waves propagating along the horizon. • Scatterers are treated as omnidirectional re-radiating elements. • The scatterers have identical scattering coefficients. • The phases associated with each scatterer are independent. Thus, for a MS located D units from the BS, as seen in Fig. 1, the scattering location points (xm , ym ) can be distributed as 2 2 1 xm + y m (1) exp − fxm ,ym (xm , ym ) = 22 22 when the mobile location is taken as the coordinate reference. If the origin is placed at the BS, the scattering points are denoted (xb , yb ) and they will be distributed according to the probability density function (pdf) (xb − D)2 + yb2 1 fxb ,yb (xb , yb ) = , (2) exp − 22 22 where the parameter is used to characterize the spread of the scatterers refered as srw. Alternatively, a polar representation can be adopted. The corresponding polar representation using (rb , b ) and (rm , m ) for instance, can be obtained by using the Jacobian yb
Probability of being in the window |90 °|
2. Gaussian Scatter Density Model 1 0.95 0.9 0.85 0.8 0.75
0.7
0
0.2
0.4
0.6
0.8 srw/D
1.0
1.2
1.4
Fig. 2. Probability the AOA at the BS is within the window ±/2.
transformation
fxb ,yb (xb , yb ) frb ,b (rb , b ) = |J (xb , yb )| xb =rb cos(b )
(3)
yb =rb sin(b )
integration on the entire rb range allows us to obtain the marginal pdf of the angle of arrival (AOA) at the BS as D cos b 1 D2 exp − 2 + √ fb (b ) = 2 2 2 2 D 2 sin2 b D cos b × exp − erfc − √ 22 2 (4) − b , the erfc is known as the complementary error function [9]. Integrating (4) we obtain the probability of scatterers in an angular window. This proportion depends on srw relative to the MS–BS distance, that is the srw/D ratio. It is a fact that for (srw/D) 3/4, P {b ± /2}0.90. This probability increases as the MS–BS separation grows or as the srw diminishes see Fig. 2. Therefore, most of the presented results are obtained for the b = ±/2 window.
ym
3. Power Doppler spectrum
rm θm
rb
θb D
MS
BS
Fig. 1. Gaussian scattering scenario for a circular array.
xb xm
A scatterer will produce a Doppler shift induced by the apparent transmitter/receiver motion. For a mobile traveling in the direction observed v , with respect to the straight line passing through the BS and the MS [3,6], the Doppler frequency is given by fi =fm cos(m −v ), |fi | < fm , where fm = v/ is the maximum Doppler shift, v is the mobile speed, is the carrier wavelength, and (m − v ) is the angle subtended by connecting the MS to scatterer line. For the purposes of this paper, we adopt the common reference speed of 54 km/h and the carrier frequency of 2 GHz thus, the maximum Doppler shift will be 100 Hz.
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
Previous works have studied the PDS through the Macrocell Circular Scattering Model (MCSM) and GSDM for sectorized cases [5,10]. For omnidirectional cases, the PDS is given by the well known Clarke’s model regarding uniform distribution of the AOA at the MS [3]. In general, the PDS of the received signal can be expressed as [6]
Li2
ri2 Li1
A2 S(f ) = × [fm (v + |cos−1 (f/fm )|) √ 0 fm 1 − (f/fm ) + fm (v + |cos−1 (f/fm )|)], (5) where A0 is a constant and f is the Doppler frequency. Doppler spread performance for actual smart antennas will be characterized in the following section.
4. Effect of smart antennas on the AOA pdf in the MS Typical smart antenna beam patterns are expected to have a main lobe pointing in the expected direction of the MS. However, there are also lateral lobes that impact the Doppler spectrum. In order to assess the use of smart antennas, the beamwidth is quantized into K =/ sectors of angular width . In this form the gain G(b ) in any directions b can be expressed as K G(b ) ≈ G(bi )rect b − bi − , 2 i=1 (6) |b | ± . 2 This formulation allows us to take advantage of the fact that the PDS has been characterized for sectorized antenna [5]. However, the Doppler contribution depends not only on the gains, but also on the probability of the scatterers being in the concerned angular window and angular scatterer location as seen from MS site m . This angular location can be described, after using a Jacobian transformation on (1) and after integrating over all possible scatterer–MS distance in the considered sector, thus r2 fm (m ) = frm ,m (rm , m ) drm r1 2 −ri1 1 = exp 2 cos2 (m − i1 )
2 −ri2 − exp , cos2 (m − i2 ) bi < |m |, where r1 and r2 are ri1 , r1 = cos(m − i1 ) ri2 r2 = cos(m − i2 )
(7)
633
ri1
r2 r1 θbi BS
D
θi1 θi2
θm
x MS
Fig. 3. Scattering boundary region.
and ri1 , ri2 are distances from the MS location to the lines Li1 and Li2 that define the sector [bi − (/2), bi + (/2)] and i1 , i2 are the subtended angles by the line orthogonal to Li1 and Li2 , respectively, see Fig. 3. This is ri1 = D sin bi − , (10) 2 , (11) ri2 = D sin bi + 2 i1 = + bi − , (12) 2 2 (13) i2 = + bi + 2 2 note that (7) is applied for the values greater than bi , since the imaginary line formed by m must reach the sector studied; and it is zero for the rest [11]. The joint pdf of (rm , m ) is given by 2 rm rm frm ,m (rm , m ) = . (14) exp − 22 22 In this way, we relate the angular observations at the MS m and the viewed angles bi from the BS, and we conduct the same procedure for each of the antenna’s subsectors so that the whole PDS will be the weighted addition to the individual PDS contribution Si (f ) multiplied by the power gain of the corresponding antenna sector. Thus, the total PDS will be approximately S(f ) ≈
K
Si (f )[G(bi )].
(15)
i=1
(8) (9)
Formulations in (7)–(15) are general and they can also be used for an omnidirectional antenna case (i.e., G(bi ) = 1; ∀i) as well as for a sectorized case centered in the direction bi (i.e., G(bi ) = Gij ; where ij is a Kronecker delta).
634
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636
5. Results
-16
-16 Lineal array Circular array Sectorized Omnidirectional
-18
PSD (Bd)
-20 -22 -24 -26 -28 -30 -32 -100
-50
0 50 Doppler Frequency(Hz)
Fig. 4. Macrocell PDS with v = 180◦ and BW = 15◦ .
100
Lineal array Circular array Sectorized Omnidirectional
-18
PSD (Bd)
-20
-22
-24
-26
-28 -100
-50
50 0 Doppler Frequency (Hz)
100
Fig. 5. Macrocell PDS with v = 90◦ and BW = 15◦ .
-16 Omnidirectional
0° 45° 90° 135° 180°
-18 -20
PSD (Bd)
In this section, we report results for linear and circular arrays. We take as reference the macrocell environment with a base to mobile separation of D = 1000 m and srw = 100 m as in [5,7]. The Doppler effect depends on a variety of parameters such as the mobile travel direction, but it also depends on the receiving antenna’s characteristics (e.g., beam pattern). In the case of an omnidirectional pattern (Clarke’s Model), all the scatterers will contribute to the PDS, while for the sectorized antenna the scatterers outside the antenna’s view are cut off. Since those scatterers outside the sector window are neglected, this implies lower receiving power. In the case of smart antenna, the scatterers contribution to the PDS will be affected by the antenna’s gain in the corresponding direction. Fig. 4 shows the PDS for an omnidirectional and 15◦ sectorized receiving pattern as well as for linear and circular antenna arrays for the same beamwidth (BW). Fig. 4 also shows a slight asymmetry that favors the low frequencies. This is because the number of scatterers to the right side of the BS (producing a negative shift) is larger than the number to the left side (producing positive shift). For a very narrow BW and a mobile travel direction of v = 90◦ , the Doppler shift tend to be small. This shows that the PDS is concentrated around zero. However, as the BW increases the Doppler shift spreads. See Fig. 5. We also show that for this direction there is no asymmetry. It can be observed that PDSs for circular and linear arrays are close. However, small differences are seen, because the lateral lobes for the linear array are approximately 4 dB lower than for the omnidirectional arrays. Frequency spread strongly depends on the motion direction, this is illustrated in Fig. 6 where the PDS is shown using a linear array at different mobile moving directions v . It can be observed that the PDS for 0◦ and 180◦ are mirrored images. The same effect occurs for 45◦ and 135◦ .
-22 -24 -26 -28 -30 -32 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 6. Macrocell PDS with variable v and BW = 15◦ .
Although the above results were obtained for a BW of 15◦ , we point out that the calculations involved the whole antenna pattern. Note, the BW was taken as the angular spacing of the two nulls around the main gain direction. Figs. 7 and 8 show the effect of the BW on the PDS, for travel directions v of 180◦ and 90◦ , respectively, when linear arrays are used. It can be observed that as the BW becomes wider the PDS approaches that of the Clarke’s Model. Similar results can be obtained for circular arrays. A common convention has been to take the straight line passing through the BS and MS as a coordinate axis with its origin at the BS (Fig. 1). This convention is useful as omni and directional antenna, as well as circular arrays present a near symmetric receiving pattern with respect to the MS direction. It has also been customary to consider only those scatterers in an angular window (o − 2 , o + 2 ) where o denotes the mobile azimuth location. However, in the case
M. Castrellon et al. / Int. J. Electron. Commun. (AEÜ) 61 (2007) 631 – 636 -25
-16 -18 -20
Omnidirectional
No inclination
60° 45° 30° 15°
45° 30°
-30
-22 PDS (dB)
PSD (Bd)
635
-24 -26
-35
-28 -30 -32 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 7. Macrocell PDS with variable BW and v = 180◦ .
-40 -100
-50
0 Doppler frequency (Hz)
50
100
Fig. 10. Macrocell PDS for different angles in the main beam BW = 15◦ and v = 90◦ .
-16 Omnidirectional
60° -18
45° 30° 15°
PSD (Bd)
-20
-22
-24
-26
-28 -100
-50
0 50 Doppler Frequency (Hz)
100
Fig. 8. Macrocell PDS with variable BW and v = 90◦ . MS scatterer antenna axis
ξ Ψ °
Specular beam
Main beam
Fig. 9. Gaussian scattering scenario for a linear array with the main beam at 45◦ .
of linear arrays, the beam presents a specular pattern with respect to the antenna’s axis see Fig. 9. Consequently, the Doppler contributions are not symmetric. The performance will depend on the angle formed by the BS–MS line and the antenna’s axis. This behavior is apparent in Fig. 10, where Doppler spectra for angles =45◦ , =30◦ , and where there is no inclination, are presented. Using D = 1000 m, srw = 500 m, v for 90◦ and a BW = 15◦ , an asymmetry can be observed due to the scatterers that are illuminated by the specular beam. The asymmetry is more blatant with the inclination of = 30◦ , because this beam reaches more scatterers.
6. Conclusion In this paper, a new methodology to evaluate the effects of the use of smart antennas at the BS on the PDS has been presented. Omnidirectional and sectorized antennas are particular cases of the presented formulation. The methodology is assessed in a GSDM macrocell environment using both linear and circular arrays. Also, we have calculated the scatterer probability of every sector in the studied scenario and, by the gain power G(b ) weighted, a close approximation of the behavior of the PDS is obtained when smart antennas are deployed, in the case of omni and sectorized antennas the presented formulation leads to results consistent to those reported in literature for the same scenarios. The comparison between PDS tells us that Clarke’s model and sectorized antennas give more power to the lateral Doppler frequencies than smart antennas. The impact of the BW of the main lobe and the direction of motion v of the MS were also included. For linear arrays the PDS is compared when the MS location is not orthogonal to the antenna array axis.
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References [1] Anderson S, Millnert M, Viberg M, Wahlberg B. An adaptive array for mobile communications systems. IEEE Trans Veh Technol 1991;40:230–6. [2] Litva J, Kwong-Yeung T. Digital beamforming in wireless communications. Boston, MA: Artech House; 1996. [3] Clarke RH. A statistical theory of mobile-radio reception. Bell Syst Tech J 1968;1:957–1000. [4] Jakes WC. Microwave mobile communications, IEEE Communications Society, 1974. [5] Lopez CA, Covarrubias DH, Muñoz D, Panduro M. Statistical cellular gaussian scatter density channel model employing a directional antenna for mobile environments. Int J Electron Commun 2005;59:195–9. [6] Liberti JJ, Rappaport TS. Smart antennas for wireless communications: Is-95 and third generation cdma applications. New York: Prentice-Hall; 1999. [7] Lotter MP, Van Rooyen P. Modeling spatial aspects of cellular cdma/sdma systems. IEEE Commun Lett 1999;3:128–31. [8] Crohn I, Bonek E. Modeling of intersymbol – interference in a rayleigh fast fading channel with typical delay profile. IEEE Trans Veh Technol 1992; 438–47. [9] Janaswamy R. Angle and time of arrival statistics for the gaussian scatter density model. IEEE Trans Wireless Commun 2002;1:488–97. [10] Petrus P, Reed J, Rappaport TS. Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans Commun 2002;50:495–502. [11] Castrellon M. Statistical cellular scatterer density model employing smart antenas for macrocell environments. MSc in Electronic Engineering Thesis. Tec de Monterrey, Campus Monterrey, N.L. Mexico; 2005. Mario Castrellon was born in Durango, Mexico 1980. He received his B.S. and M.S. degrees in electrical engineering from the Instituto Tecnologico de Durango, Durango, Mexico and Instituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico, in 2002 and 2005, respectively.
David Muñoz received his B.S., M.S., and Ph.D. degrees in electrical engineering from the Universidad de Guadalajara, Guadalajara, Mexico, Cinvestav Mexico City, México, and University of Essex, Colchester, England, in 1972, 1976 and 1979, respectively. He has been Chairman of the Communication Department and Electrical Engineering Department at Cinvestav,
IPN. In 1992, he joined the Centro de Electronica y Telecomunicaciones, Instituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico, where he is the Director. His research interests include transmission and personal communication systems. Cesar Vargas received his M.S., and Ph.D. degrees in electrical engineering from Louisiana State University, Baton Rouge, Louisiana, USA, in 1992, and 1996, respectively. In 1996, he joined the Centro de Electronica y Telecomunicaciones, Tec de Monterrey, Campus Monterrey, Mexico, where he is the Telecommunications Graduate Program Coordinator. His research interests include reconfigurable networks and routing. Claudio A. Lopez-Mirinda received the Bachelor’s degree in Mathematics from the University of Sonora, Hermosillo, Sonora, Mexico, in 1987, and his Master’s degree in Operation Research from the Universidad Nacional Autónoma de Mexico, in 1995. He received his Ph.D. in Electronics and Telecommunications at the CICESE Research Center, Ensenada, B.C., Mexico. His research interests are radio channel modeling and evolutionary computation applied to angle of arrival estimations in smart antennas. David H. Covarrubias-Rosales received his Ph.D. in Telecommunications (Cum Laude) from the Universitat Politécnica de Catalunya, Barcelona, Spain; his Master of Science in Telecommunications from CICESE, México, and his BS in Electronics and Communications Engineering from the Universidad Nacional Autónoma de México, UNAM, Mexico City. He is a faculty member of the Electronics and Telecommunications Department, CICESE Research Center, Ensenada, B.C., México, from June 1984 to date. His research areas are 3G mobile communications systems and smart antennas.