Wireless signal propagation (two ray model) Thanasis Korakis University of Thessaly [email protected] Two-ray model ● Used when a single ground reflection dominates the multipath effect ● The received signal consists of ○ the LOS ray (the transmitted signal) and ○ a reflected ray, which is reflected off the ground, and may interfere constructively or destructively Why is this model so important? Free-space path loss model does not consider interference. Consider the problem of delivering wireless coverage to an island using a wireless backhaul from mainland (this problem is based on a real scenario) Low tide Constructive interference High-quality signal High tide Destructive interference Very poor signal Two-ray interference model ● Explains deep fades caused by multipath interference ○ where they will occur, and how deep they will be ● Validated by measurements on many different wireless systems Two ray interference model Two ray interference model Distance traveled by direct ray: Distance traveled by reflected ray: Two ray interference model Phase difference between rays: Reflection coefficient at ground (depends on material): Two ray interference model The received signal is the superposition of the two rays. Using the free space path loss model: the power of the two rays adds to: Two ray interference model Assume that delay spread is small compared to signal bandwidth (narrowband assumption): Then we can express the received power as: What does the power look like over distance? Example on next slide based on real measurements Two ray model, large d If we assume d becomes large, equal gains, perfect reflection: Two ray model, large d Then the phase difference becomes (using Taylor expansion): Two ray model, large d and the expression for received power can be simplified as: Two ray model, as multi-slope model ● We saw that for large d received power Pr decays according to Pr = K P t / d 4 ● For small d, two-ray model decays like freespace model, according to Pr = K P t / d 2 (but two-ray model also include multipath) Two ray model, as multi-slope model Two ray model, as multi-slope model What is considered large d? ● Power decays with d-4 when d > dc dc = 4πhthr / λ this is known as the critical distance, and comes from the expression for phase difference ● Power decays with d-2 if d < dc Simplified path loss models ● More generally: the main characteristics of path loss (without multipath interference) are captured by the expression: Pr = K Pt [d0 / d]n 2≤n≤6 Includes free-space, two-ray, log-distance models K factor (in detail) ● In free space & two ray models respectively: K(free space) = Gt Gr λ2 / (4π)2 L K(two ray) = Gt Gr ht2 hr2 / L ● Where ○ ○ ○ ○ Gt and Gr are the Tx and Rx antenna gains ht and hr are the Tx and Rx antenna heights λ is the wavelength L >=1 is the system loss factor (if not specified, L=1) Log-distance PL model PL = PL0 + 10nlog10( d / d0 ) Where: ○ PL0 is the Path loss in an indicative distance d0 ○ n varies with propagation environments Example 1 ● Calculate free space & two ray path loss, given L=1, if : ○ frequency of signal is 2GHz ○ Tx and Rx antenna gains are 9dBi and 30dBi ○ Tx and Rx antenna heights are 15m and 1.5m ○ the distance Tx - Rx is 21Km ● 9dBi → 7.94 and 30dBi → 1000 ○ since G(dBi)= 10 log10(G) (further details later) ● λ = c / f = 3x108 / 2x109 m = 0.15m Example 1 (K factors) ● K(free space) = Gt Gr λ2 / (4π)2 L = 7.94 x 1000 x (0.15m)2 / (4π)2 x 1 = 1.13m2 ● K(two ray) = Gt Gr ht2 hr2 / L = 7.94 x 1000 x (15m)2 x (1.5m)2 / 1 = 4019625m4 Example 1 (Path losses PL and dc) ● PL(free space) = 10log10(d2/K(free space) ) = 10log10((21x103) 2 /1.13) = 10log10(345.4x106) = 85.38dB ● PL(two ray) = 10log10(d4/K(two ray))= 10log10((21x103)4/4019625) = 10log10(0.048x1012) = 106.85 dB ● dc = 4π ht hr / λ = 4π x 15m x 1.5m / 0.15m = 1884m ○ d > d , so the two ray model predicts better Multipath reception ● The received signal contains: ○ the direct LOS radio wave, but also ○ a large number of reflected radio waves (not only one like in two ray reflection model) ● In urban centers, LOS is often blocked by obstacles and a collection of differently delayed waves is all what is received Rayleigh Fading Assumes that a received multipath signal consists of a (theoretically infinitely) large number of reflected waves This model has played a major role in our understanding of mobile propagation, first proposed by Lord Rayleigh in 1889 ○ describing the resulting signal if many violinists in an orchestra play in unison... References 1. T. S. Rappaport, Wireless Communications: Principle and Practice, 2nd ed. Singapore: Pearson Education, Inc., 2002. 2. S. Haykin and M. Moher, Modern Wireless Communications. Singapore: Pearson Education, Inc., 2002.

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