Advanced wireless networks 4g technologies phần 2 pot

JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 WIRELESS MIMO LAN ENVIRONMENTS (5.2 GHz) 67 additional correction by visual inspection of the Scree Graph, showing that the eigenvalue is an option used for generating the results presented in the sequel. After estimation of the parameters τ i , we can determine the corresponding ‘steering’ matrix, A τ . Subsequent beamforming with its Moore–Penrose pseudoinverse [34,53–56] A + τ gives the vector of delay-weights for all x R ,x T h τ ( x T , x R ) = A + τ T f ( x T , x R ) (3.36) where T f is the vector of transfer coeffcients at the 192 frequency sub-bands sounded. This gives us the transfer coeffcients from all positions x T to all positions x R separately for each delay τ i . Thus, one dimension, namely the frequency, has been replaced by the parameterized version of its dual, the delays. For the estimation of the direction of arrivals (DOA) in each of the two-dimensional transfer functions, ESPRIT estimation and beamforming by the pseudo-inverse are used h ϕR ( τ i , x T ) = A + ϕR h xR ( τ i , x T ) (3.37) Finally for the direction of departure (DOD) we have h ϕT  τ i ,ϕ R,i, j  = A + ϕT h xT  τ i ,ϕ R,i, j  (3.38) Figure 3.12 illustrates these steps. The procedure gives us the number and parameters of the MPCs, i.e. the number and values of delays, which DOA can be observed at these delays and which DOD corresponds to each DOA at a specific delay. Furthermore, we also obtain the powers of the multi- path channels (MPCs). One important point in the application of the sequential estimation procedure is the sequence in which the evaluation is performed. Roughly speaking, the number of MPCs that can be estimated is the number of samples we have at our disposal. Figure 3.12 Sequential estimation of the parametric channel response in the different do- mains: alternating estimation and beamforming. (Reproduced by permission of IEEE [52].) JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 68 CHANNEL MODELING FOR 4G 3.6.2 Capacity computation In a fading channel, the capacity is a random variable, depending on the local (or instan- taneous) channel realization. In order to determine the cdf of the capacity, and thus the outage capacity, we would have to perform a large number of measurements either with slightly displaced arrays, or with temporally varying scatterer arrangement. Since each single measurement requires a huge effort, such a procedure is highly undesirable. Toimprove thissituation,an evaluationtechniquethat requiresonlya singlemeasurement of the channel is used. This technique relies on the fact that we can generate different realizations of the transfer function by changing the phases of the multipath components. It is a well-established fact in mobile radio that the phases are uniformly distributed random variables, whose different realizations occur as transmitter, receiver or scatterers move [27]. We can thus generate different realizations of the transfer function from the mth transmit to the kth receive antenna as h k,m ( f ) =  i a i exp  −j 2π λ d  k sin  φ R,i  + m sin  φ T,i  ×exp ( −j2π f τ i ) exp ( jα i ) (3.39) where α i is a uniformly distributed random phase, which can take on different values for the different MPCs numberedi. Note, however, that α i stays unchanged as we consider different antenna elements k and m. To simplify discussion, we for now consider only the flat-fading case, i.e. τ i = 0. We can thus generate different realizations of the channel matrix H H = ⎛ ⎜ ⎜ ⎝ h 11 h 12 ··· h 1N T h 21 h 22 ··· h 2N T ··· ··· ··· ··· h N R1 h N R2 ··· h N R N T ⎞ ⎟ ⎟ ⎠ (3.40) by the following two steps: (1) From a single measurement, i.e. a single snapshot of the channel matrix, determine the DOAs, and DODs of the MPCs as described earlier in the section. (2) Compute synthetically the impulse responses at the positions of the antenna ele- ments, and at different frequencies. Create different realizations of one ensemble by adding random phase factors (uniformly distributed between 0 and 2π) to each MPC. For each channel realization, we can compute the capacity from [ 97] C = log 2 det  I + ρ N T H H H  (3.41) where ρ denotes theSNR.I istheidentity matrixandsuperscript HmeansHermitian transposition. For the frequency-selective case, we have to evaluate the capacity by integrating over all frequencies C =  log 2 det  I + ρ N T H H ( f ) H ( f )  d f. (3.42) Here, H( f ) is the frequency-dependent transfer matrix. The integration range is the bandwidth of interest. JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 WIRELESS MIMO LAN ENVIRONMENTS (5.2 GHz) 69 3.6.3 Measurement environments As an example the following scenarios are evaluated with the procedure described above [52]: r ScenarioI–acourtyard with dimensions 26 × 27m, open on one side. The RX-array broadside points into the center of the yard; the transmitter is located on the positioning device8mawayinLOS. r Scenario II – closed backyard of size 34 ×40 m with inclined rectangular extension. The RX-array is situated in one rectangular corner with the array broadside of the linear array pointing under 45 ◦ inclination directly to the middle of the yard. The LOS connection between TX and RX measures 28 m. Many metallic objects are distributed irregularly along the building walls (power transformers, air-condition fans, etc.). This environment looks very much like the backyard of a factory (Figure 3.13). r Scenario III – same closed backyard as in Scenario II but with artificially obstructed LOS path. It is expected that the metallic objects generate serious multipath and higher-order scattering that can only be observed within the dynamic range of the device if the LOS path is obstructed. r Scenario IV – same as scenario III but with different TX position and LOS obstructed. The TX is situated nearer to the walls. More details about the senarios can be found in Steinbauer et al. [57]. Some of the measurements results for these scenarios are presented in Figure 3.14 TX RX 0 1020304050 60 X-Coordinate (m) Y-Coordinate (m) -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Figure 3.13 Geometry of the environment of scenarios II–IV (backyard) in top view. Su- perimposed are the extracted DOAs and DODs for scenario III. (Reproduced by permission of IEEE [52].) JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 70 CHANNEL MODELING FOR 4G II I III IV 0 10 25 0 0.1 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Capacity (bits/s/Hz) ideal cdf (Capacity) 20 Scenario I Scenario II Scenario III Scenario IV Ideal Figure 3.14 The CDFs of the MIMO channel capacity encountered in scenarios I–IV, and the cdf for an ideal channel. The SNR is 20 dB, and 4 × 4 antenna elements were used. 3.7 INDOOR WLAN CHANNEL (17 GHz) In this section we discuss the indoor radio propagation channel at 17 GHz. The presentation is based on results reported in Rubio et al. [58]. Wideband parameters, such as coherence bandwidth or rms delay spread, and coverage are analyzed for the design of an OFDM-based broadband WLAN.The methodusedto obtainthechannel parametersisbased onasimulator described in Rubio et al. [58]. This simulator is a site-specific propagation model based on three-dimensional (3-D) ray-tracing techniques, which has been specifically developed for simulating radio coverage and channel performance in enclosed spaces such as buildings, and for urban microcell and picocell calculations. The simulator requires the input of the geometric structure and the electromagnetic properties of the propagation environment, and is based on a full 3-D implementation of geometric optics and the uniform theory of diffraction(GO/UTD). Examples ofthemeasurement environmentsare giveninFigure 3.15. The results for coherence bandwidth B c = 1/ατ rms are given in Table 3.13 and Figure 3.16. A further requirement related to the correct and efficient channel estimation process by the receiver is the selection of a number of subcarriers in OFDM satisfying the condition of being separated between approximately B c /5 and B c /10. Results for delay spread are shown in Figure 3.17 and Tables 3.14 – 3.17. The results for the path loss exponent and k factor are given in Figure 3.18 and Table 3.18 and Table 3.19. For channel modeling purposes, the mean power of the received signal will be represented as P RX | dB = P TX | dB + G TX | dB + G RX | dB − L fs | dB + 10 · log   ∞ 0 PDP ( t ) dt  (3.43) where T TX is the mean power at the transmitting antenna input, G TX is the transmitting antenna gain while G RX is the receiving antenna gain. L fs is free space propagation losses, JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 Figure 3.15 (a) ETSIIT hall (49 ×26 m); (b) DICOM, floors 2 and 3 (34 ×20 m); (c) office building (72 ×38 m); 3-D representations 63. (Reproduced by permission of IEEE [58].) Table 3.13 B c at 17 GHz Coherence bandwidth (MHz) Place Mean Standard deviation Hall 24.85 12.35 Floors 14.44 9.85 Building 22.86 10.24 Total 20.72 11.56 71 JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 72 CHANNEL MODELING FOR 4G 0 5 10 15 20 25 30 35 40 45 B c (MHz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF Figure 3.16 B c CDF at 17 GHz. (Reproduced by permission of IEEE [58].) 0 102030405060 RDS (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF (a) 40 60 80 100 120 140 160 180 200 T max (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) CDF 20 40 60 80 100 120 140 160 T max (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF (c) 0123456 Alpha 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF (d) Figure 3.17 (a) The RMS delay spread CDF (B c = 1/ατ rms ). (b) Maximum delay CDF, 30 dB criterion. (c) Maximum delay CDF, 20 dB criterion. (d) Alpha CDF. JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 INDOOR WLAN CHANNEL (17 GHz) 73 Table 3.14 The RMS delay spread CDF. (Reproduced by permis- sion of IEEE [58]) CDF value RDS value 0.2 12.1 ns 0.4 14.3 ns 0.6 17.5 ns 0.8 34.3 ns 1 58.3 ns RDS, root delay spread. Table 3.15 Maximum delay CDF, 30 dB criterion. (Reproduced by permission of IEEE [58]) CDF value T max value 0.2 62 ns 0.4 76 ns 0.6 101 ns 0.8 122 ns 1 197 ns Table 3.16 Maximum delay CDF, 20 dB criterion. (Reproduced by permission of IEEE [58]) CDF value T max value 0.2 51 ns 0.4 56 ns 0.6 69 ns 0.8 94 ns 1 156 ns Table 3.17 Alpha CDF, B c = 1/ατ rms CDF value Alpha value 0.2 2.17 0.4 2.67 0.6 3.75 0.8 4.44 1 5.78 JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 74 CHANNEL MODELING FOR 4G 123 44.55 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF LOS OLOS NLOS 1.5 2.5 3.5 n 1 Figure 3.18 CDF of path loss exponent n. Table 3.18 Mean values of n Type of path LOS OLOS NLOS n Mean value 1.68 2.14 2.61 Table 3.19 Fading statistic over distance, LOS case Radius (m) K factor 417 510 69 78 86 95 10 1 given by L fs | dB = 32.45 dB +20 ·log 10 ( d km + f MHz ) and PDP(t) the modeled power delay profile. Once thePDFismodeled, to obtain the discrete channel impulse response, h i , we only have to add a random phase to the square root of each delay bin amplitude, as follows: h i = √ p i e jφ i φ i r.υ. unif [ 0, 2π ] (3.44) where h i is the ith bin of the modeled channel impulse response and p i , the module of the ith bin of the modeled power delay profile. JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 INDOOR WLAN CHANNEL (17 GHz) 75 It can be assumed that phases of different components of the same channel impulse response are uncorrelated at the frequency of interest (17 GHz), because their relative range is higher than a wavelength, even for high-resolution models [59]. As the total bandwidth assigned to the communication is 50 MHz, a selection of 10 ns for the bin size must be made. Using 99 % of the total power criterion for the maximum duration of the PDF, the former bin size selection leads to a total of nine taps for the LOS case and 17 for the NLOS case. The statistical variability of the bin amplitudes has been modeled following different probability density functions. Taking into account the fact that the area of service of future applications (SOHO – small office, home office) has small ranges, the variability has been analyzed considering a medium-scale, that is, the environment is divided in to the LOS area and the NLOS one. In the LOS case, a Frechet PDF [60] is chosen for the first bin and exponential PDFs for the rest. A continuous random variable X has a Frechet distribution if its PDF has the form f ( x; σ ;λ ) = λ σ  σ x  λ+1 exp  −  σ x  λ  ; x ≥ 0; σ, λ > 0 (3.45) A Frechet variable X has the CDF F ( x; σ ;λ ) = exp  −  σ x  λ  (3.46) This model has a scale structure, with σ a scale parameter and λ a shape parameter.A continuous random variable X has an exponential distribution if its PDF has the form f ( x; μ ) = 1 σ exp  −  x − μ σ  ; x ≥ 0; μ, σ > 0 (3.47) This PDF has location-scale structure, with a location parameter, μ, and a scale one, σ . The CDF of the exponential variable X is F ( x; μ ) = 1 − exp  −  x − μ σ  (3.48) These PDFs were considered the most suitable after a fitting process. The NLOS case needs a combination of exponential and Weibull PDFs for the first bin and exponential PDFs for the others. A continuous random variable X has a Weibull distribution if its PDF has the form f ( x; σ ;λ ) = λ σ  x σ  λ−1 exp  −  x σ  λ  ; x ≥ 0; σ, λ > 0 (3.49) While the CDF is F ( x; σ ;λ ) = 1 − exp  −  x σ  λ  (3.50) This model has a scale structure, that is, σ is a scale parameter, while λ is a shape parameter. Tables 3.20 and 3.21 show the probability density functions employed for LOS and NLOS channel models [58]. For both tables, the units of σ parameters are Hz (s −1 ), while λ has no units. These units have no physical correlation but make the last term of Equation (3.43) nondimensional, as it represents a factor scale between the free space behavior and the real one. The mean JWBK083-03 JWBK083-Glisic March 6, 2006 11:18 Char Count= 0 76 CHANNEL MODELING FOR 4G Table 3.20 Wind-flex channel model PDFs, LOS case. (Reproduced by permission of IEEE [58]) Bin 1 Frechet (σ = 2.66 ×10 8 ,λ= 7) Bin 4 exp (σ = 1.45 ×10 7 ) Bin 7 exp (σ = 0.41 ×10 7 ) Bin 2 exp (σ = 5.44 ×10 7 ) Bin 5 exp (σ = 1.03 ×10 7 ) Bin 8 exp (σ = 0.27 ×10 7 ) Bin 3 exp (σ = 2.51 ×10 7 ) Bin 6 exp (σ = 0.79 ×10 7 ) Bin 9 exp (σ = 0.71 ×10 7 ) Table 3.21 Wind-flex channel model PDFs, NLOS case. (Reproduced by permission of IEEE [58]) Bin 1 0.5 * [exp (σ = 4.378 ×10 6 )+ Weibull(σ = 4.207 ×10 7 ,λ= 5) Bin 7 exp (σ = 1.88 ×10 5 ) Bin 13 exp (σ = 9.21 ×10 4 ) Bin 2 exp (σ = 3.04 ×10 6 ) Bin 8 exp (σ = 2.51 ×10 5 ) Bin 14 exp (σ = 1.27 ×10 5 ) Bin 3 exp (σ = 2.47 ×10 6 ) Bin 9 exp (σ = 5.69 ×10 5 ) Bin 15 exp (σ = 2.76 ×10 4 ) Bin 4 exp (σ = 2.14 ×10 6 ) Bin 10 exp (σ = 1.53 ×10 5 ) Bin 16 exp (σ = 6.71 ×10 4 ) Bin 5 exp (σ = 1.1 ×10 6 ) Bin 11 exp (σ = 3.29 ×10 5 ) Bin 17 exp (σ = 6.42 ×10 4 ) Bin 6 exp (σ = 3.71 ×10 5 ) Bin 12 exp (σ = 2.67 ×10 5 ) [...]... 1.5 1.6 1.7 2. 1 2. 2 2. 3 2. 4 2. 5 3.1 3 .2 4.1 4 .2 4.3 4.4 5.1 5 .2 5.3 5.4 6.1 6 .2 8.1 LOC no 5 10 20 30 40 50 60 5 10 20 30 40 4 .2 3.3 7.1 3.8 5 .2 4 .2 2.4 2. 4 2. 4 2. 4 3 3 2 TR 1 .20 6.16 32. 61 15.50 27 .60 46. 42 6.38 2. 22 2.78 2. 3 22 . 02 77.3 0.74 0. 92 2.74 2. 4 12. 88 21 .3 0.83 2. 46 0.71 1.16 10.67 14. 82 7.63 τ ¯ 6.95 5.88 47 .25 31.15 37.04 28 .17 22 .57 6 .24 6.48 4.56 33.87 45.07 4.85 4.95 4. 72 4.98 31.10... 5.19 14. 72 21.78 24 .59 στ 6.33 5.06 32. 89 10.16 25 .89 36.70 5.99 7. 52 8 .24 7.81 13.17 105.04 6 .20 5.97 11.16 11.11 26 .36 31.5 2. 41 2. 61 1.30 1.85 23 .07 34.30 10 .24 δτ ¯ 1.91 1 .20 8.43 3.43 8.81 8.10 1. 82 2.38 2. 61 2. 55 4.60 34.41 1.88 1.87 3.08 3.17 6.86 7.4 0.69 0.84 0.41 0.61 6. 62 8.57 2. 66 τ ¯ 1 .20 6.16 32. 61 15.50 27 .60 46. 42 6.38 2. 22 2.78 2. 30 22 . 02 77.30 0.74 0. 92 2.47 2. 40 12. 88 21 .3 0.83 2. 46... 1.1 1 .2 1.3 1.4 1.5 1.6 1.7 2. 1 2. 2 2. 3 2. 4 2. 5 3.1 3 .2 4.1 4 .2 4.3 4.4 5.1 5 .2 5.3 5.4 6.1 7.1 7 .2 8.1 LOS, hallway Durham Hall LOS, outdoor Room to room LOS, outdoor parking lot Hallway toroom LOS, room Whittemore LOS, room Durham Hall LOS, hallway Whittemore No Site 5 10 20 30 40 50 60 5 10 20 30 40 4 .2 3.3 7.1 3.8 5 .2 4 .2 2.4 2. 4 2. 4 2. 4 3 1.9 1.9 2 TR 80.0 52. 0 85.9 116.6 84.9 52. 1 53 .2 51.0 62. 1... 0.84 0. 72 0.13 0.44 0.40 0 .27 0.40 0.97 0.94 0.76 γ Max AOA −4.0 4.0 8.0 5.0 5.0 10.0 2. 0 5.0 21 .0 4.0 10.0 1.0 0.0 5.0 −60.0 −1.0 49.0 −49.0 0.0 5.0 0.0 5.0 52. 0 2. 0 20 .0 3.0 θmax −80.7 −86.6 −61.9 −66.4 4.3 8 .2 4.0 −73.5 − 72. 3 −73.8 −64.8 5.0 −79 .2 −79.1 −88.0 −89.6 −35 .2 −38 .2 −76.3 −89.6 −88.1 72. 3 25 .3 −81 .2 −66.7 −66.3 12. 3 12. 0 14.5 14.7 13.9 13.3 13 .2 12. 5 11.4 12. 9 13.8 13 .2 12. 5 13.1 12. 3 13.1... 0.74 0. 92 2.47 2. 40 12. 88 21 .3 0.83 2. 46 0.71 1.16 10.67 14. 82 7.63 δστ 0 .29 1.73 9. 02 5.69 9.76 10.73 1.57 0.73 0. 82 0.55 6.30 25 .86 0 .20 0 .23 0.36 0.47 2. 95 5.43 0. 32 0.94 0 .25 0.36 1.30 3.37 1.75 στ −13.7 20 .3 −36.6 −31 .2 −40.5 − 42. 8 −41.5 −16.7 24 .4 − 32. 86 −34.7 −36.3 − 12. 1 − 12. 9 29 .7 24 .2 −56 .2 −57.9 −5.5 −14.3 −6.7 −9.1 − 12. 8 −48.3 2. 4 Pr Corner Center Corner Center Corner, ⊥ to TX Corner,... 51.0 62. 1 90.7 41 .2 83.7 42. 6 47.7 46.6 64.3 66.3 77.8 49.1 41.6 95.8 80.3 42. 7 41.3 56.6 24 .4 τ ¯ 14.7 18.8 40.1 38.7 60.0 26 .1 30.3 20 .7 29 .4 14.6 12. 3 53.8 16 .2 17.5 13.0 13.3 17.7 13.3 21 .4 18.1 14.6 16.0 16.6 17.4 16.1 7.7 στ 0.46 0.44 0.56 0. 42 0.69 0.66 0.78 0.48 0.66 0.36 0.41 0. 72 0.86 0.81 0.84 0. 62 0.73 0.78 0.81 0.74 0.63 0.68 0.80 0. 12 0.49 0 .26 0.83 0.74 0 .28 0 .22 0 .25 0 .26 0.36 0.88 0.79... 14 .2 12. 0 10.3 12. 1 11.9 11.5 13.9 8.5 13.9 Peak/avg −14.9 −18 .2 28 .8 28 .3 −38 .2 −38 .2 −40.8 −13 21 .7 29 .8 −31.7 −36.0 −11.8 − 12. 1 26 .8 25 .6 −30.4 28 .6 −6.0 −14.1 −5.6 −8.9 −36.4 −15.0 29 .9 −10.1 Pmax Corner Center Corner Center Corner, ⊥ to TX Corner, ⊥ to TX LOS Through wall LOS Through glass Through wall TX pattern RX pattern Near Durham Hall Intersection Open area Comments Table 3 .22 Spin... Distance (feet) -PLTOTAL (dB) (a) 3 0 -3 -6 -9 - 12 -15 -18 -21 -24 -27 -30 -33 -36 -39 - 42 -45 -48 -51 -54 σ N = 3.55 dB Free space N = 2. 1 data 0 1 10 2 10 10 Distance (feet) -PLPEAK+RAKE (dB) (b) 3 0 -3 -6 -9 - 12 -15 -18 -21 -24 -27 -30 -33 -36 -39 - 42 -45 -48 -51 -54 σN = 4.04 dB Free space N = 2. 5 data 0 10 1 10 2 10 Distance (feet) (c) Figure 3 .26 (a) Peak PL vs distance; (b) total PL vs distance;... for 5, 10, 15, 20 and 30 dB threshold levels 50 % NLOS Threshold 5 dB 10 dB 15 dB 20 dB 30 dB 90 % NLOS Power % L τm (ns) τRMS (ns) Power % L τm (ns) τRMS (ns) 46.8 89 .2 97.3 99.4 99.97 7 27 39 48 60 1.95 7.1 8.6 9.87 10.83 1. 52 5.77 7.48 8.14 8.43 46.9 86.5 96 99.5 99.96 8 31 48 69 82 2 .2 8.1 10.3 12. 2 12. 4 1.65 6.7 9.3 11 11.5 REFERENCES [1] S Glisic, Advanced Wireless Communications, 4G Technology... path loss model for in-home UWB channels, 20 02 IEEE Conf Ultra Wideband Systems and Technologies, Digest of Papers, 21 23 May 20 02, pp 59–64 [101] W Turin, R Jana, S.S Ghassemzadeh, C.W Rice and T Tarokh, Autoregressive modelling of an indoor UWB channel, IEEE Conference on Ultra Wideband Systems and Technologies, Digest of Papers, 21 23 May 20 02, pp 71–74 [1 02] S.S Ghassemzadeh and V Tarokh, UWB path . 1. 82 6.38 1.57 −41.5 LOS, hallway Whittemore 2. 1 5 2. 22 6 .24 7. 52 2.38 2. 22 0.73 −16.7 2. 2 10 2. 78 6.48 8 .24 2. 61 2. 78 0. 82 24 .4 Intersection 2. 3 20 2. 3 4.56 7.81 2. 55 2. 30 0.55 − 32. 86 2. 4 30 22 . 02. 0. 92 0 .23 − 12. 9 Center LOS, room Whittemore 4.1 7.1 2. 74 4. 72 11.16 3.08 2. 47 0.36 29 .7 Corner 4 .2 3.8 2. 4 4.98 11.11 3.17 2. 40 0.47 24 .2 Center 4.3 5 .2 12. 88 31.10 26 .36 6.86 12. 88 2. 95 −56 .2. 4.0 2. 0 13 .2 −40.8 LOS, hallway Whittemore 2. 1 5 51.0 20 .7 0.48 0.88 −73.5 5.0 12. 5 −13 2. 2 10 62. 1 29 .4 0.66 0.79 − 72. 3 21 .0 11.4 21 .7 Intersection 2. 3 20 90.7 14.6 0.36 0.43 −73.8 4.0 12. 9 29 .8 2. 4