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Calculating Shape Factors for Laplacian Azimuth Spectrum in Class Project, Study Guides, Projects, Research of Guiding Electromagnetic Systems

Georgia Institute of Technology - Main CampusGuiding Electromagnetic Systems

Hints for calculating the shape factors, including angular spread (λ), angular constriction (γ), and direction of maximum fading (θmax), for the laplacian azimuth spectrum used in a class project. The laplacian azimuth spectrum is given by the equation with constants a, θ0, and θ1. The nth fourier coefficient is calculated using the given formula with the provided values for θ0 and θ1. The results should be numbers between 0 and 1, with larger angular spread and smaller angular constriction for larger θ1.

Typology: Study Guides, Projects, Research

Pre 2010

Uploaded on 08/05/2009

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Hints for Shape Factors on Class Project
Due Date: 22 April 2004 (Thursday)
1UsefulHint
Several students requested some help in calculating the shape factors for the Laplacian azimuth
spectrum used in the project. Recall that the Laplacian azimuth spectrum is given by
p(θ)=Aexp −
θ−θ0
θ1
The constant Ais arbitrary (it will only change the average received power, not the shape of the
distribution or the statistics of the fading.) The value θ0is the azimuth direction of peak arrival
and, in general, points in the direction of the base station transmitter. The value θ1is related to
the thickness of the distribution.
The nth Fourier coefficient of this spectrum is given by
Fn=2Aθ1exp(jnθ0)
n2θ2
1+1 1−(−1)nθ1exp −
π
θ1
where θ0and θ1are in radians. Just plug in the values for your θ1= 120◦and θ1=3
◦models and
solve for angular spread (Λ), angular constriction (γ), and direction of maximum fading (θmax).
A few hints about what your results should look like:
•The angular spread should always be a number between 0 and 1. The angular spread for the
θ1= 120◦case will be much larger than the angular spread for the θ1=3
◦case.
•The angular constriction should also always be a number between 0 and 1. The angular
constriction for the θ1= 120◦case will be smaller than the angular constriction for the θ1=3
◦
case.
•The direction of maximum fading for both cases should be 90◦away from the peak of the
azimuth distribution (θ0).

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Hints for Shape Factors on Class Project

Due Date: 22 April 2004 (Thursday)

1 Useful Hint

Several students requested some help in calculating the shape factors for the Laplacian azimuth spectrum used in the project. Recall that the Laplacian azimuth spectrum is given by

p(θ) = A exp

θ − θ 0 θ 1

The constant A is arbitrary (it will only change the average received power, not the shape of the distribution or the statistics of the fading.) The value θ 0 is the azimuth direction of peak arrival and, in general, points in the direction of the base station transmitter. The value θ 1 is related to the thickness of the distribution. The nth Fourier coefficient of this spectrum is given by

Fn =

2 Aθ 1 exp(jnθ 0 ) n^2 θ^21 + 1

[

1 − (−1)nθ 1 exp

π θ 1

)]

where θ 0 and θ 1 are in radians. Just plug in the values for your θ 1 = 120◦^ and θ 1 = 3◦^ models and solve for angular spread (Λ), angular constriction (γ), and direction of maximum fading (θmax ). A few hints about what your results should look like:

  • The angular spread should always be a number between 0 and 1. The angular spread for the θ 1 = 120◦^ case will be much larger than the angular spread for the θ 1 = 3◦^ case.
  • The angular constriction should also always be a number between 0 and 1. The angular constriction for the θ 1 = 120◦^ case will be smaller than the angular constriction for the θ 1 = 3◦ case.
  • The direction of maximum fading for both cases should be 90◦^ away from the peak of the azimuth distribution (θ 0 ).