SBR Method - Revision history

Kazem Sabet: /* Wedge Diffraction from Edges */

2016-10-02T16:28:04Z

‎Wedge Diffraction from Edges

Kazem Sabet at 13:55, 13 September 2016

2016-09-13T13:55:42Z

Kazem Sabet at 13:55, 13 September 2016

2016-09-13T13:55:24Z

Kazem Sabet at 13:54, 13 September 2016

2016-09-13T13:54:49Z

Kazem Sabet at 13:54, 13 September 2016

2016-09-13T13:54:28Z

Kazem Sabet at 13:54, 13 September 2016

2016-09-13T13:54:06Z

Kazem Sabet: /* Wedge Diffraction from Edges */

2016-09-13T13:53:25Z

‎Wedge Diffraction from Edges

Kazem Sabet at 13:51, 13 September 2016

2016-09-13T13:51:54Z

Kazem Sabet at 13:50, 13 September 2016

2016-09-13T13:50:54Z

Kazem Sabet at 13:46, 13 September 2016

2016-09-13T13:46:08Z

@@ Line 158: / Line 158: @@
 :<math> \mathbf{ \hat{u}_b' = \hat{k}' \times \hat{u}_f' } </math>
 The diffraction coefficients are calculated in the following way:
@@ Line 166: / Line 165: @@
 :<math> D_h = \frac{-e^{-j\pi/4}}{2n \sqrt{2\pi k} \sin\beta_0'} \left\lbrace \begin{align} & \cot \left(\frac{\pi + (\phi-\phi')}{2n}\right) F[kLa^+(\phi-\phi')] +  \cot \left(\frac{\pi - (\phi-\phi')}{2n}\right) F[kLa^-(\phi-\phi')] + \\ & R_{0 \|} \cot \left(\frac{\pi - (\phi+\phi')}{2n}\right) F[kLa^-(\phi+\phi')] +  R_{n \|} \cot \left(\frac{\pi + (\phi+\phi')}{2n}\right) F[kLa^+(\phi+\phi')] \end{align} \right\rbrace </math>
 where ''F(x)'' is the Fresnel Transition function:
 :<math> F(x) = 2j \sqrt{x} e^{jx} \int_{\sqrt{x}}^{\infty} e^{-j\tau^2} \, d\tau </math>
 In the above equations, we have
@@ Line 183: / Line 180: @@
 where <math>N^{\pm}</math> are the integers which most closely satisfy the equations <math> 2n\pi N^{\pm} - \nu = \pm \pi </math>.
-<p>&nbsp;</p>
+<br />
 [[Image:Top_icon.png|48px]] '''[[#Free-Space Wave Propagation | Back to the Top of the Page]]'''

← Older revision		Revision as of 13:55, 13 September 2016
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	[[Image:Maxwell1.png\|right\|800px]]		[[Image:Maxwell1.png\|right\|800px]]
−	~~<p> </p>~~
	== Free-Space Wave Propagation ==		== Free-Space Wave Propagation ==

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 [[Image:Maxwell1.png|right|800px]]
 <p>&nbsp;</p>
 == Free-Space Wave Propagation ==

@@ Line 1: / Line 1: @@
 [[Image:Maxwell1.png|right|800px]]
 <p>&nbsp;</p>
 <p>&nbsp;</p>
 <p>&nbsp;</p>
 In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e<sup>j(&omega;</sup><sup>t-k<sub>0</sub>R)</sup>/R, where R is the distance between the transmitter and receiver, <math>\omega = 2\pi f</math>, f is the signal frequency, <math>k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda}</math>, c is the speed of light, and &lambda;<sub>0</sub> is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R<sup>2</sup>, and additionally, it experiences a phase shift of <math>\frac{2\pi R}{\lambda_0}</math>, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P<sub>R</sub>) to the transmitted signal power (P<sub>T</sub>). Assuming isotropic transmitting and receiving radiators (<i>i.e.</i> radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:

@@ Line 1: / Line 1: @@
 [[Image:Maxwell1.png|right|800px]]
 <p>&nbsp;</p>
 <p>&nbsp;</p>
 <p>&nbsp;</p>
 In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e<sup>j(&omega;</sup><sup>t-k<sub>0</sub>R)</sup>/R, where R is the distance between the transmitter and receiver, <math>\omega = 2\pi f</math>, f is the signal frequency, <math>k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda}</math>, c is the speed of light, and &lambda;<sub>0</sub> is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R<sup>2</sup>, and additionally, it experiences a phase shift of <math>\frac{2\pi R}{\lambda_0}</math>, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P<sub>R</sub>) to the transmitted signal power (P<sub>T</sub>). Assuming isotropic transmitting and receiving radiators (<i>i.e.</i> radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:

@@ Line 180: / Line 180: @@
 :<math>a^{\pm}(\nu) = 2\cos^2 \left( \frac{2n\pi N^{\pm} - \nu}{2} \right), \quad \nu = \phi \pm \phi' </math>
 where <math>N^{\pm}</math> are the integers which most closely satisfy the equations <math> 2n\pi N^{\pm} - \nu = \pm \pi </math>.
 <p>&nbsp;</p>
 [[Image:Back_icon.png|40px]] '''[[EM.Terrano | Back to EM.Terrano Main Page]]'''

@@ Line 98: / Line 98: @@
 == Penetration through Thin Walls or Surfaces ==
 In &quot;Thin Wall Approximation&quot;, we assume that an incident ray gives rise to two rays, one is reflected at the specular point, and the other is transmitted almost in the same direction as the incident ray. The reflected ray is assumed to originate from a virtual image source point. Similar to the case of reflection and transmission at the interface between two dielectric media, here too we have three triplets of unit vectors, which all form orthonormal basis systems.
@@ Line 129: / Line 127: @@
 == Wedge Diffraction from Edges ==
 For the purpose of calculation of diffraction from building edges, we define a &quot;Wedge&quot; as having two faces, the 0-face and the ''n''-face. The wedge angle is a = (2-''n'')p, where the parameter ''n'' is required for the calculation of diffraction coefficients. All the diffracted rays lie on a cone with its vertex at the diffraction point and a wedge angle equal to the angle of incidence in the opposite direction. A diffracted ray is assumed to originate from a virtual image source point. Three triplets of unit vectors are defined as follows:
@@ Line 185: / Line 181: @@
 :<math>a^{\pm}(\nu) = 2\cos^2 \left( \frac{2n\pi N^{\pm} - \nu}{2} \right), \quad \nu = \phi \pm \phi' </math>
-<!--
 [[File:frml10.png]]
 [[File:frml13.png]]
 where <math>N^{\pm}</math> are the integers which most closely satisfy the equations <math> 2n\pi N^{\pm} - \nu = \pm \pi </math>.

@@ Line 2: / Line 2: @@
 == Free-Space Wave Propagation ==
 In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e<sup>j(&omega;</sup><sup>t-k<sub>0</sub>R)</sup>/R, where R is the distance between the transmitter and receiver, <math>\omega = 2\pi f</math>, f is the signal frequency, <math>k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda}</math>, c is the speed of light, and &lambda;<sub>0</sub> is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R<sup>2</sup>, and additionally, it experiences a phase shift of <math>\frac{2\pi R}{\lambda_0}</math>, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P<sub>R</sub>) to the transmitted signal power (P<sub>T</sub>). Assuming isotropic transmitting and receiving radiators (<i>i.e.</i> radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:
@@ Line 11: / Line 10: @@
 where '''u<sub>T</sub>''' and '''u<sub>R</sub>''' are the unit polarization vectors of the transmitting and receiving antennas, and G<sub>T</sub> and G<sub>R</sub> are their gains, respectively.
 == Basic Wave Interaction Mechanisms ==
@@ Line 21: / Line 28: @@
 == Ray Reflection & Transmission ==
 The incident, reflected and transmitted rays are each characterized by a triplet of unit vectors:
@@ Line 29: / Line 34: @@
 * <math>( \mathbf{ \hat{u}_{\|}^{\prime}, \hat{u}_{\perp}', \hat{k}' } )</math> representing the reflected parallel polarization vector, reflected perpendicular polarization vector and reflected propagation vector, respectively.
 * <math>( \mathbf{ \hat{u}_{\|}^{\prime\prime}, \hat{u}_{\perp}^{\prime\prime}, \hat{k}^{\prime\prime} } )</math> representing the transmitted parallel polarization vector, transmitted perpendicular polarization vector and transmitted propagation vector, respectively.
 The reflected ray is assumed to originate from a virtual image source point. The three triplets constitute three orthonormal basis systems. Below, it is assumed that the two dielectric media have permittivities &epsilon;<sub>1</sub> and &epsilon;<sub>2</sub>, and permeabilities &mu;<sub>1</sub> and &mu;<sub>2</sub>, respectively. A lossy medium with a conductivity &sigma; can be modeled by a complex permittivity &epsilon;<sub>r</sub> = &epsilon;'<sub>r</sub> –j&sigma;/&epsilon;<sub>0</sub>. Assuming '''n''' to be the unit normal to the interface plane between the two media, and Z<sub>0</sub> = 120&Omega; , the incident polarization vectors as well as all the reflected and transmitted vectors are found as:
@@ Line 89: / Line 102: @@
 In &quot;Thin Wall Approximation&quot;, we assume that an incident ray gives rise to two rays, one is reflected at the specular point, and the other is transmitted almost in the same direction as the incident ray. The reflected ray is assumed to originate from a virtual image source point. Similar to the case of reflection and transmission at the interface between two dielectric media, here too we have three triplets of unit vectors, which all form orthonormal basis systems.
 The transmission coefficients are calculated for the two parallel and perpendicular polarizations as:
@@ Line 116: / Line 137: @@
 * <math>\mathbf{(\hat{u}_f, \hat{u}_b, \hat{t})}</math> representing the incident forward polarization vector, incident backward polarization vector and incident propagation vector, respectively.
 * <math>\mathbf{(\hat{u}_f', \hat{u}_b', \hat{t}')}</math> representing the diffracted forward polarization vector, diffracted backward polarization vector and diffracted propagation vector, respectively.
 The three triplets constitute three orthonormal basis systems. The propagation vector '''k'''' of the diffracted ray has to be constructed based on the diffraction cone as follows:

← Older revision		Revision as of 13:46, 13 September 2016
Line 1:		Line 1:
	[[Image:Maxwell1.png\|right\|800px]]		[[Image:Maxwell1.png\|right\|800px]]
		+	== Free-Space Wave Propagation ==
		+
		+	[[Image:los.png\|thumb\|600px\|A Line-of-Sight (LOS) Propagation Scenario.]]
		+	In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e<sup>j(ω</sup><sup>t-k<sub>0</sub>R)</sup>/R, where R is the distance between the transmitter and receiver, <math>\omega = 2\pi f</math>, f is the signal frequency, <math>k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda}</math>, c is the speed of light, and λ<sub>0</sub> is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R<sup>2</sup>, and additionally, it experiences a phase shift of <math>\frac{2\pi R}{\lambda_0}</math>, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P<sub>R</sub>) to the transmitted signal power (P<sub>T</sub>). Assuming isotropic transmitting and receiving radiators (<i>i.e.</i> radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:
		+
		+	:<math> \frac{P_R}{P_T} = \left( \frac{\lambda_0}{4\pi R} \right)^2 </math>
		+	The above formula assumes that the receiving antenna is polarization-matched. Normally, there is a polarization mismatch between the transmitting and receiving antennas. In the case of directional transmitting and receiving antennas, Friis’ formula takes the following form:
		+
		+	:<math> P_R = P_T \, G_T G_R \left( \frac{\lambda_0}{4\pi R} \right)^2 \left\| \mathbf{ \hat{u}_T \cdot \hat{u}_R } \right\|^2 </math>
		+
		+	where '''u<sub>T</sub>''' and '''u<sub>R</sub>''' are the unit polarization vectors of the transmitting and receiving antennas, and G<sub>T</sub> and G<sub>R</sub> are their gains, respectively.
		+
	== Basic Wave Interaction Mechanisms ==		== Basic Wave Interaction Mechanisms ==