Analysis of Fading Statistics based on Angle of Arrival Measurements

Zia M. Loni, Rahat Ullah COMSATS Institute of Information Technology

Abbotabad 22060, Pakistan zialoni@ciit.net.pk, rahatzz@yahoo.com

Noor M. Khan

Muhammad Ali Jinnah University Islamabad 44000, Pakistan

noor@ieee.org

Abstract—In this paper, we analyze various spatial channel measurements using their second order statistics like level crossing rate, average fade duration, auto covariance and coherence distance. Occurrence rates of the angle of arrival in considering macro, micro and Pico cellular environment are used for the analysis. Effect of directional antennas employed at the base station and mobile station on second order statistics is also analyzed. We also present the quantative analysis of the measurements using multipath shape factors i.e. angle spread, the angular constriction and direction of maximum fading. Effect of increasing Doppler spread on the LCR and AFD is also elaborated in detail.

Index Terms— Angle of arrival, multipath shape factors, spatial measurements, Level crossing rates, Average fade durations.

I. INTRODUCTION In recent years adaptive antenna arrays have become a part of MIMO communication systems. Topologies incorporated in antenna arrays are strongly related to azimuth dispersion of mobile radio channel. For small scale fading, understanding of the channel statistics like angle spread is required [1]. Recent measurement and models have shown that fading statistics depend on the direction of receiver travel relative to arriving multipath. An omni directional channel model does not accurately describe fading statistics if directional antennas are employed [2]. In [4], Spencer et al. studied the temporal and spatial characteristics of indoor environment. The authors observed the clustering of the AOA both in time and space. Based on data the authors proposed a laplacian distribution for modeling the spatial channel. In [5], Jalden et al. observed after field measurements that the angle spread measurement both at BS and MS in outdoor environment. The angle spread at BS depended on height. The highest elevated BS had the lowest angle spread. The angle spread at MS was larger than that at BS due to the closely-located scatterers. Pedersen et al. studied a typical urban environment [6]. The authors found that power azimuth spectrum (PAS) and Power azimuth delay spectrum (PDS) could be modeled by a laplacian function and a one sided exponential decay function respectively. They declared that in an environment having high angle spread would result in high delay spreads. In [7], Zhang et al. observed the effect of building architecture on AOA distribution. The authors observed both elevation and azimuth

AOA distribution and concluded that azimuth AOA followed a laplacian distribution and elevation AOA follows a Gaussian distribution in indoor environments. In [1], Khan et al. studied the effect of AOA distribution truncation on angle spread. The authors observed that standard deviation of the angular energy distribution depended on angular span of the angular data. As the angular span of the angular data decreased, the standard deviation of the full span Gaussian, did not represented the standard deviation , �� , of the actual angular energy distribution anymore. The authors in [1], after analyzing the measurement result of [4], concluded that in addition to the bell shaped Gaussian function, there is always some additional part, which distorts the Gaussian function. As far as, the analysis of the AOA is concerned, comparatively little work has been done exploiting its second order statistics. In this paper we find multipath shape factors the angle spread, the angular constriction and direction of maximum fading from the AOA measurements and utilize them to analyze the second order statistics including level crossing rates, average fade durations, auto covariance and coherence distance. We than compare the results for their fitness to macro, micro and Pico cellular environments.

II. AVALABLE MEASUREMENTS IN LITERATURE Many measurement campaigns were executed to avail the actual angle of arrival distribution of the multipath signals in azimuth in the last decade. Out of these campaigns, we have chosen the following notable measurements for our analysis.

� Q. H. Spencer et al. took measurement in an indoor environment [4]. Measurements were taken in two buildings made of steel and concrete. A dish antenna of 60cm diameter operating at 7GHz was used for data collection. Antenna was placed in separate rooms during experiment. Doors were made of wood and closed during measurement. People were also moving in the rooms during measurement.

� Niklas Jalden et al. conducted measurements in an urban environment with six to eight storey stone buildings [5]. MS consisted of a 4-element box antenna transmitter and BS had a three 4-element uniform linear array (ULA), with a antenna spacing

314978-1-4244-9134-6/11/$26.00 ©2011 IEEE

of 0.56. Measurements are taken both at Base station and Mobile station.

� K. I. Pedersen et al. conducted measurement campaign in an urban environment [6]. Measurement area had buildings ranging from four to six floors and an irregular street grid. BS antenna was placed on two different locations 20 and 30m high. BS antenna had an eight element uniform linear array (ULA) and a MS with an omni directional car-mounted antenna. There was N-LOS communication between MS and BS.

� Yongwei Zhang et al. in [7] conducted experiments in two rooms "office 1" and "office 2". The sizes of the room were 6m x 6m and 5.7m x 4.8m respectively. A probe antenna was used operating at a frequency of 3.1 to 10.6GHz. A receiver array consisted of 100 x 100 elements with a uniform spacing of d = 0.01m.

� Cramer et al. measurement campaign in [8] consisted of a fixed transmitter in an office building. Reading were taken in 14 different rooms and hallways of the office building, transmitting a pulse train of duty cycle 0.2% and a period of 500ns

III. PROPOSED METHODOLOGY AND SYSTEM MODEL Our proposed methodology is based on Fourier coefficients of the PDF of the angle of arrival, )(�p . In [3], Durgin et al. used exponential moments for calculating the statistics of directional data, while the same were obtained by using the trigonometric moments in [1]. The above two methods work on the same basis and hence yield the same results. We adopt the latter method for our analysis of the directional data, because it is advantageous in manipulating the directional data. Let nnn SjCR �� be defined as the nth complex trigonometric moments of the angular energy distribution,

)(�p [1]. The trigonometric parameters, nC and nS for the angular energy distribution, )(�p are defined as

� ��

�N

iiin nf

FC

10

cos1 � (1)

and

� ��

�N

iiin nf

FS

10

sin1 � (2)

where �

�N

iifF

10

for Ni ,,1 � and if is the number of

occurrences for the AOA, i� , in the ith bin of histogram.

nR can also be written as nj

nn eRR �� (3)

where 22nnn SCR �� is the mean resultant of the nth

trigonometric moment and ���

���

�� �

n

nn

CS1tan� is its direction.

1� , gives the mean angle of distribution, � ��p . 1R is the

magnitude of the first trigonometric moment. If 1R is closer to

1, it means the AOA are tightly clustered about the mean position, 1� .If AOA are widely spread than 1R will be small

and closer to 0. 1R is related to the shape factor , � , and

standard deviation ,��, in [1].

211 R��� (4)

and

� �1ln2 R���� (5)

where �� is the standard deviation of the angular energy distribution in radians. The shape factors,� , and ��,are also inter-related as

� �21ln ������ (6)

2

1 ������ e (7) Angle spread is often measured interms of the total span occupied by the non-zero values of occurrences [1]. It is denoted by �span and defined as,

�span = �max — �min Two more shape factors, i.e. the angular constriction,� , and orientation parameter, MF� , have also been proposed.� is the measure of how multipath concentrates about two azimuthal directions and MF� provides the azimuth direction of maximum fading [1]. These two shape factors are defined as [1]-[3];

21

212

1 R

RR

�

��� (8)

and

� �21221 RRMF �� phase� (9)

In [2]-[3], Durgin et al. related the multipath shape factors to level crossing rate , RN , average fade duration , � , auto-covariance ,p(r,�), and Coherence distance , cD , which are

315

Environ-

ment

Measurements �span �

Standard Deviation

��

Direction of Max Fading

MF�

Angular Spread

�

Angular Constriction

�

Indoor

Cramer et al. [8] 360o -2.6o 38.68o 76.16o 0.6012 0.1802 Zhang 1 et al. [7] 135o -53.6o 34.79o 72.75o 0.5553 0.8093 Spencer et al. [4] 350o 11.7o 36.85o -77.22o 0.5821 0.1827 Zhang 2 et al. [7] 320o 10.2o 34.14o 89.49o 0.5468 0.0933

Outdoor

Pedersen et al.[6] 50o 0.1o 7.08o -89.96o 0.1232 0.9760 Jalden et al. at MS[5] 100o 70.6o 19.22o -26.36o 0.3263 0.9145

Jalden et al. at BS [5] 32o 8.41o 4.89o -78.81o 0.0852 0.9908 Table 1: Numerical values of first and second order statistics of the measurement campaign for indoor and outdoor environments.

useful for studying fading statistics in non-omni directional multipath propagation channel. The level crossing rate, RN , is the measure of rapidity of fading and can be defined as [2]-[3].

� �� � � �2exp2cos12 �����

������

�� MFRN (10)

where � is the wavelength of the radiation. The average fade duration , � , is the time that the signal spends below the threshold level and can be formulated as

� �� �� �� �MF������

������

���

2cos121exp 2

(11)

The auto covariance ,p(r,�), is envelope correlation between two points in space separated by a distance r, in an azimuth direction� . p (r,�) can be defined as

� � � �� �� �� ����

�

!

"��

��������

22 2cos123exp,

����� rrp MF

(12)

The coherence distance, Dc , is the distance in space over which the channel response remains constant. Dc can be formulated as

� �� �� �� �MF

cD���

����

�21232

cosln (13)

IV. RESULTS AND DISCUSSION

The mathematical findings discussed in the previous section empower us to analyze the chosen measurement data on the basis of second order statistics. Table 1 enlists the first and second order spatial statistics of the multipath signal observed in the measurements. Table 1 shows the numerical results obtained from simulating equations (1) to (11). From Table 1, it can be

easily observed that the �span has greater values for indoor measurements as compared to outdoor measurements. It is in accordance to the fact that in indoor or pico cellular environment, scatterers are usually present in the vicinity of the base station which is in contrast to the outdoor where BS is usually free from scattering objects. This is why �span for the measurements at BS in indoor is greater than that for outdoor environment. However, �span have always higher values if AOAs are recorded at MS. This shows that in indoor environment the multipath power is arriving over a wide range of angles. However in outdoor environment these multipaths are restricted over a small angular area. The angle spread shape factor ,�, of the measurements in Table- 1 follows the same trend as observed in [9]. For outdoor receivers the angle spread falls within the range of lower values 0-0.3. For indoor receivers, the angle spread, �, falls in the range 0.54-1.00. This is due to the fact that the density of scatterers (doors, walls, windows, shelves etc) is higher in the vicinity of BS as well as MS. Due to these scattering objects, receivers located in indoor environment get multipath signals over a wide range of angles that results in increased angle spread. The angular constriction shape factor, �, is high for almost all channels indicating that the

Fig. 1. Level crossing rate of measurement data

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Fig. 2. Average fade duration of the measurement data multipath power is not distributed uniformly instead there is greater probability of multipath power arriving from several discrete directions. The value of standard deviation (SD), ��, is used in determining spacing between antenna elements. Table 1 shows high values of SD ,��, for indoor environment. This means that BS receivers operating in indoor environment have to produce wider beam width in comparison to the BS located in outdoor environment. Fig. 1 and Fig. 2 illustrate the behavior of the fading statistics in different propagation environments. Fig. 1 and Fig. 2 show the plots of the normalized level crossing rate (LCR), i.e.

rN and normalized average fade duration (AFD), i.e. � versus increasing threshold level ,�, respectively. The factors affecting LCR and AFD are density of scatterers, Doppler spread and physical propagation channel (indoor or outdoor). LCR is related to the fading behavior of the physical channel. Severe fading environment results in high LCR and vise versa. Results show that indoor measurement campaigns of Spencer et al. [4], Zhang et al. [7], and Cramer et al. [8], show high LCR. This seems to be due to the fact that in indoor environment the elevation of base station and receiver terminals is low. Thus in indoor environment, the received signal strength is greatly affected by body shadowing and hence a walking pedestrian can also block the LOS path. Due to the blockage of the LOS path, receivers operating in indoor environment are dependent on multipath arrivals due to scatterers like (doors, walls and shelves). Due to these multipaths, �span for indoor environment is larger than that of outdoor environment. We can say that the density of scatterers encircling MS and BS in indoor environment is greater. These factors cause the indoor measurements of Spencer et al. [4], Zhang et al. [7], and Cramer et al. [8], to show high LCR and lower AFD as shown in Fig. 1 and Fig. 2 respectively. Outdoor environment can be further subdivided to macro and micro cellular environments [1]. In macro cellular environment the BS height is usually kept higher and the scatterers around BS are almost negligible. In micro cellular environment the

Fig. 3. Auto covariance of the measurement data

Fig. 4. Coherence distance, cD , calculated from measurement data

Fig. 5. Level crossing rate versus increasing Doppler frequency

317

Fig. 6. Average fade duration versus increasing Doppler frequency BS height is kept lower, so the high rise buildings near the BS also cause angular dispersion of signals. Fig. 1 and Fig. 2 show that the measurement taken by Jalden et al. [5], and Pedersen et al.[6], show low LCR than those got in indoor measurement campaign. Measurement campaign of Jalden et al. [5], at MS shows high LCR than measurement campaign of Pedersen et al. [6]. This is due to the fact that Pedersen et al. [6] has taken measurements in an urban environment having four to six-storey buildings while measurement campaign of Jalden et al. [5] is in an environment having much higher buildings. Measurement campaign Jalden et al. [5] at BS show lowest LCR proving the scattering free environment around BS. One other factor responsible for low LCR at BS is the use of antenna array. BS is usually equipped with directional antennas. Due to this directionality, scatterers falling out of the beam-range of antenna do not cause significant signal fading. LCR of a BS with directional antenna is lower than that of omni directional antenna. We can conclude from the results shown in Fig. 1 and Fig. 2 that the LCR at BS working in an indoor environment is higher than that working in outdoor environment. Moreover, in an outdoor urban environment, the LCR at BS will be higher than that in sub urban environment. Fig. 3 shows the auto correlation behavior of the spatial measurements plotted versus r/�. High correlation behavior shows that that AOA distribution of that particular measurement is narrower. Fig. 3 reveals that outdoor measurements show high correlation than indoor measurements. These results can be validated from the numerical results listed in Table 1, where low values of SD ,��, and angle spread, �, confirms such behavior for outdoor environment. This means that in outdoor environment the multipath components arrive at the receiver array over small angular area which results in high correlation while in indoor environment the multipath components arrive at broadside to receiver array which results in low correlations for indoor measurements. In Table 1, measurement campaign of Cramer et al. [8], has the largest SD ,��, which results in lowest correlation in Fig. 3 while the SD ,��, of

Fig. 7. Level crossing rate versus SIR threshold for different values of fD

considering Pedersen's measurements

Fig. 8. Average fade duration versus SIR threshold for different values of

fD considering Pedersen's measurements

Fig. 9. Level crossing rate versus SIR threshold for different values of fD

considering Spencer's measurements

318

Fig. 10. Average fade duration versus SIR threshold for different values of

fD considering Spencer's measurements the measurements of Jalden et al. [5] at BS is the lowest exhibiting the highest correlation in Fig. 3. Fig. 4 shows the plot of coherence distance ,Dc, versus �. Coherence distance shows the degree of stability of a particular environment. Durgin et al. in [3], claimed that as � decreases, the coherence distance increases. This claim can be validated from Table 1 and Fig. 4. Table 1 shows lower values of � for outdoor environment and Fig. 4 shows large coherence distances for such cases. This proves that the channel in outdoor environment is more stable than that in indoor environment. Earlier, we have already discussed that the measurement data of outdoor environment show high AFD than that of indoor environment. Large coherence distances and high AFDs have a disadvantage of channel instability. A receiver traveling in outdoor environment if comes under deep fade it will take more time to come out of that fade. Fig. 5 and Fig. 6 show the plots of LCR and AFD for different values of maximum fD. Results show that LCR increases linearly with an increase in Doppler frequency. These results follow the same trend as shown in [10]. Indoor measurements show high LCR with increasing Doppler frequency than outdoor measurements and a converse behavior is observed for AFD. This proves that indoor environment is more sensitive to Doppler frequency. We further take one measurement as example in each outdoor and indoor case to observe the effect of Doppler spread. Fig. 7 and Fig. 8 show such an effect for LCR and AFD, respectively, in outdoor environment taking the measurement data of Pedersen et al. [6]. Fig. 9 and Fig. 10 present the above results for indoor environment taking the measurement data of Spencer et al. [4].

CONCLUSIONS

The spatial channel measurements have been analyzed quantatively on the basis of multipath shape factors. We have drawn the following conclusions from this analysis.

1) The angular spread values for indoor environment are higher than those for outdoor environments.

2) The angular constriction values are higher for indoor as well as outdoor showing that multipath power is arriving from several discrete directions.

3) The standard deviation is large for indoor environment confirming that antennas operating in indoor have high beam width as compared to outdoor environments.

4) Level crossing rates are higher for indoor than those for outdoor environment and depend on the density of the scatterers surrounding BS or MS.

5) Higher values of average fade duration for outdoor, especially in macro cellular environment have a disadvantage of channel unstability. A receiver traveling in outdoor environment if comes under deep fade it will take more time to come out of the fade.

6) Level crossing rate increases with increasing Doppler spread and average fade duration reduces with increasing Doppler spread.

REFERENCES

[1] Noor M. Khan, “Modeling and characterization of multipath fading channels in cellular mobile communication system,” PhD Thesis Dissertation, School of Electrical Engineering and Telecommunication, University of New South Wales (UNSW), Australia, March 2006.

[2] G. D. Durgin and T. S. Rappaport, “Basic relationship between multipath angular spread and narrow band fading in wireless channels,” IEE Electron. Lett., vol. 24, no. 25, pp 2431-2432, December 1998.

[3] G. D. Durgin and T. S. Rappaport, “Theory of Multipath shape factors for small-scale fading wireless channels,” IEEE Trans. Antennas Propagat., vol. 48, no. 5, pp.682 – 693, May 2000.

[4] Q. H. Spencer, B. D. Jeffs, M. A. Jensen, and A. L. Swindlehurst, “Modeling the statistical time and angle of arrival characteristics of an indoor multipath channel,” IEEE J.Select. Areas Commun., vol.18, no. 3, pp. 347-360, 2000

[5] Niklas Jalden, Per Zetterberg, Bjorn Ottersten and Laura Garcia," Inter- and intrasite correlations of large-scale parameters from microcellular measurements at 1800 MHz, “EURASIP journal on wireless comm and networking, vol 2007

[6] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury,” A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environment,” IEEE Trans. Veh. Technol., vol. 49, no. 2, pp. 437-447 March 2000.

[7] Yongwei Zhang, Anthony K. Brown, Wasim Q. Malik, and David J.Edwardes," High resolution 3-D angle of arrival determination for indoor UWB multipath propagation," IEEE Trans. Wireless commun., vol. 7, no. 8, Aug 2008

[8] R. J.-M. Cramer, M. Z. Win, R. A. Scholtz,” Impulse radio multipath characteristics and diversity reception,” IEEE conference on comm (ICC) 98, vol. 3, pp.1650-1654, June 1998.

[9] G. D. Durgin, Vikas kukshya, and T. S. Rappaport,” Wideband Measurement of Angle and Delay Dispersion for outdoor and indoor peer-to-peer radio channels at 1920 MHz,” IEEE Trans. Antennas propagate., vol. 51, no. 5, May 2003.

[10] Paul Petrus, Jeffrey H. Reed, “Geometrical-based statistical macro cell channel model for mobile environments," IEEE Trans. Commun., vol. 50, no.3, March 2000.

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