Homework: Please number questions as numbered on assignment, and turn in solution pages in order.

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1 ECE 5325/6325: Wireless Communiation Systems Leture Notes, Spring 2010 Leture 6 Today: (1) Refletion (2) Two-ray model (3) Cellular Large Sale Path Loss Models Reading for today s leture: 4.5, 4.6, For Tue: Haykin/Moher handout ( ), get from WebCT. Homework: Please number questions as numbered on assignment, and turn in solution pages in order. Exam 1 is Tue, Feb 9. Please be prepared. 1 Refletion and Transmission There are eletri and magneti waves that serve to propagate radio energy. The eletri waves an be represented as a sum of two orthogonal polarization omponents, for example, vertial and horizontal, or left-hand and right-hand irular. What happens when these two omponents of the eletri field hit the boundary between two different dieletri media? We talk about the plane of inidene, that is, the plane ontaining the diretion of travel of the waves (inident, refleted, and transmitted), and perpendiular to the surfae (plane where the two media meet). See Figure 4.4 on page 115, reprodued in Fig 1. Figure 1: Figure 4.4 from Rappaport. Notes about the notation from Rappaport: Parallel refers to the E-field having diretion parallel to the plane of inidene (as in Figure 4.4(a)); perpendiular means perpendiular (normal) to the plane of inidene (as in Figure 4.4(b)).

2 ECE 5325/6325 Spring Use subsripts i, r, and t to refer to the inident, refleted, and tranmitted field. ǫ 1, ǫ 2, is the permittivity of medium 1 and 2. (units Farads/meter) (Note: F = se / Ω) µ 1, µ 2 is the permeability of medium 1 and 2. (units Henries/meter) (Note: H = Ω se ) σ 1, σ 2 is the ondutane of medium 1 and 2 (units Siemens/meter). (Note: S = 1 / Ω) The omplex dialetri onstant is ǫ = ǫ 0 ǫ r jǫ where ǫ 0 = F/m is free spae permittivity, j = 1, and ǫ = σ 2πf, and ǫ r is the relative permittivity. Don t get onfused by the subsripts in this equation. A material is a good ondutor when σ k > fǫ k. The intrinsi impedane of medium k is µ k /ǫ k. Then the refletion oeffiients are given by Γ E r E i = η 2 sin θ t η 1 sin θ i η 2 sin θ t + η 1 sin θ i Γ E r E i = η 2 sin θ i η 1 sin θ t η 2 sin θ t + η 1 sin θ i (1) where θ t is determined by Snell s Law: µ1 ǫ 1 sin(90 o θ i ) = µ 2 ǫ 2 sin(90 o θ t ) (2) Also, the angle of inidene is equal to the angle of refletion: θ i = θ r Finally, the refleted and transmitted field strengths are: E r E t = ΓE i = (1 + Γ)E i where you hose Γ based on the polarization of the inident E-field, i.e., use either Γ or Γ. There is a speial ase of (1) when the first medium is free spae (or approximately, air) and µ 1 = µ 2. These two onditions are the ase for most of the things we are about. In this ase you an show (good HW problem!) that Γ = ǫ r sin θ i + ǫ r os 2 θ i ǫ r sin θ i + ǫ r os 2 θ i Γ = sin θ i ǫ r os 2 θ i sin θ i + ǫ r os 2 θ i (3)

3 ECE 5325/6325 Spring See Figure 4.6 on page 118 of Rappaport. At some angle θ i, there is no refletion of the parallel E-field from (3). This angle is alled the Brewster angle, whih is given by ǫ1 sin θ B = ǫ 1 + ǫ 2 When medium 1 is free spae, and ǫ 2 = ǫ 0 ǫ r, sin θ B = ǫr This is the same as Equation 4.28 in Rappaport. Note that as θ i 0, Γ 1 and Γ 1. Also, for perfet ondutors (as desribed in Setion 4.5.3), we also have Γ = 1 and Γ = 1. Example: Refletion from ground Find the refletion oeffiients for typial ground at an inident angle of 15 degrees at 100 MHz. Solution: Assume free spae is medium 1 and that typial ground has ǫ r = 15. Note sin 15 o = 0.259, and os 15 o = 0.933, so from (3), Γ = 15(0.259) (0.259) = Γ = = Two-Ray (Ground Refletion) Model Setion 4.6 in Rappaport develops a theoretial model for propagation slightly better than the free spae assumption. This model inludes not just one path, but also another path that reflets off of the ground. The World Is Flat, if you will. The model isn t hard to develop, and provides an important theoretial underpinning to the multiple breakpoint model we overed in leture 5. Remember, powers of multipath DON T add together. Only voltages or field strength of multipath atually add together. The voltage on the antenna is proportional to the eletri field at the antenna position. So let s talk about adding eletri fields. See Figure 4.7 on page 121 of Rappaport. 2.1 Diret Path Reall that the eletri field magnitude deays as 1/d in free spae. So, similar to how we wrote the reeived power with a referene distane, we write the E-field strength as the E-field strength

4 ECE 5325/6325 Spring at a referene distane, multiplied by /d, for a path (distane of travel for waves) length d. Also, assume the signal is a simple sinusoid at the arrier frequeny, f. So E(d,t) = E 0 d os (2πf ( t d For the LOS path, given a distane along the ground of L, an- heights h t and h r at the TX and RX, respetively, the d = tenna L 2 + (h t h r ) 2. So ( )) L E LOS = E 0 L 2 + (h t h r ) os 2πf (t 2 + (h t h r ) 2 2 (5) 2.2 Refleted Path Let s assume that L is very long ompared to the antenna heights. So, the angle of inidene is approximately 0. In this ase the refletion oeffiient (assume perpendiular polarization) is -1. The refleted path travels longer than the diret path, for total length L 2 + (h t + h r ) 2 (one an use the method of images to show this). Then ( E g = E 0 L 2 + (h t + h r ) os 2πf (t Total Two-Ray E-Field )) (4) )) L 2 + (h t + h r ) 2 We are interested in the magnitude of the total E-field, E TOT = E LOS +E g, that is, the quantity that multiplies the os(2πf t) term. Using trig identities, and this approximation: = L 2 + (h t + h r ) 2 L 2 + (h t h r ) 2 2h th r L we an show that ( E TOT 2E 0 d sin 2πf ) 2h t h r L But at large L, the argument of the sin is approximately 0, and the sin x x. This is when x < 0.3 radians, whih in our equation, means that for L > 20h th r f we an use this approximation (also noting that L d): E TOT 2E 0 d ( 2πf 2h t h r d ) = onst d 2 (6)

5 ECE 5325/6325 Spring This means that the power deays as 1/d 4! See Figure 2. In summary, when there are two paths, one diret and one ground refletion, the theoretial models show behavior that has two different path loss exponents, 1/d 2 for d less than a threshold, and 1/d 4 for d above the threshold. This mathes what we ve observed from measurements and presented as the empirial multiple breakpoint model. Figure 2: Reeived power as a funtion of log distane in two-ray model, Figure 2.5 from Andrea Goldsmith, Wireless Communiations, Cambridge University Press, However, a note: this is just a theoretial model. Typial ellular or indoor hannels do not have just two paths. One of the 6325 assignments is to onsider T-ray model, for T > 2. For example, if you had a eiling refletion as a 3rd path. Or a eiling-floor twoboune path as a 4th ray. As T goes up, you don t see the 1/d 4 behavior. 3 Cellular Large Sale Path Loss Models Now, let s onsider some empirial models that are used in ellular systems. These types of models are better than free spae or the logdistane model, and have been designed to better fit measured data. Note that there are many modifiations that people ontinually add to these models to better fit their environments of interest. 3.1 Okumura-Hata Models The median (50th perentile) propagation loss at frequeny f, distane d, and TX and RX antenna heights h te and h re, is given in the Okumura model as, L 50 (db) = L F (f,d) + A mu (f,d) G(h te ) G(h re ) G AREA

6 ECE 5325/6325 Spring where all terms are in db even though Rappaport does not expliitly denote thenm as suh, and where L F (f,d) is the free spae path loss in db at distane d and frequeny f, A mu is the median loss ompared to free spae (found from Table 4.23), G AREA is a fator due to the type of environment (open, quasi open, or suburban, as given in Figure 4.24), and h te G(h te ) = 20log m, for 30m < h te < 1000m { h 10log10 re G(h re ) = 3m, 0m < h re < 3m h 20log re 10 3m, 3m < h re < 10m Example: Rappaport Example 4.10, page 153 The Hata model is largely funtion-fitting, expressing a formula that aptures most of the results of Okumura. Again, these are based on large sets of measurements. There are many adjustments to this model, partiularly for PCS (the COST- 231 model for 1900 MHz), for miroells, et. In general, these models have a standard deviation of about 10 or more db when ompared to the atual measurements. So, it is typially useful to obtain more aurate means of prediting reeived powers, in partiular, software that predits based on the geometry of a ity or area, and measurements whih give feedbak to the model to allow it to (to some extent) orret its errors). Disussion What are some of the assumptions made in these models that we have not talked about, or in partiular, that may be inorret?