Tag Archives: Electromagnetic

Knife Edge Diffraction Model

Diffraction is a phenomenon where electromagnetic waves (such as light waves) bend around corners to reach places which are otherwise not reachable i.e. not in the line of sight. In technical jargon such regions are also called shadowed regions (the term again drawn from the physics of light). This phenomenon can be explained by Huygen’s principle which states that as a plane wave propagates in a particular direction each new point along the wavefront is a source of secondary waves. This can be understood by looking at the following figure. However one peculiarity of this principle is that it is unable to explain why the new point source transmits only in the forward direction.

Image result for diffraction

The electromagnetic field in the shadowed region can be calculated by combining vectorially the contributions of all of these secondary sources, which is not an easy task. Secondly, the geometry is usually much more complicated than shown in the above figure. For example consider a telecom tower transmitting electromagnetic waves from a rooftop and a pedestrian using a mobile phone at street level. The EM waves usually reach the receiver at street level after more than one diffraction (not to mention multiple reflections).  However, an approximation that works well in most cases is called knife edge diffraction, which assumes a single sharp edge (an edge with a thickness much smaller than the wavelength) separates the transmitter and receiver.

The path loss due to diffraction in the knife edge model is controlled by the Fresnel Diffraction Parameter which measures how deep the receiver is within the shadowed region. A negative value for the parameter shows that the obstruction is below the line of sight and if the value is below -1 there is hardly any loss. A value of 0 (zero) means that the transmitter, receiver and tip of the obstruction are all in line and the Electric Field Strength is reduced by half or the power is reduced to one fourth of the value without the obstruction i.e. a loss of 6dB.  As the value of the Fresnel Diffraction Parameter increases on the positive side the path loss rapidly increases reaching a value of 27 dB for a parameter value of 5. Sometimes the exact calculation is not needed and only an approximate calculation, as proposed by Lee in 1985, is sufficient.

Fresnel Diffraction Parameter (v) is defined as:

v=h*sqrt(2*(d1+d2)/(lambda*d1*d2))

where

d1 is the distance between the transmitter and the obstruction along the line of sight

d2 is the distance between the receiver and the obstruction along the line of sight

h is the height of the obstruction above the line of sight

and lambda is the wavelength

Diffraction Loss Using Knife-Edge Model

The MATLAB codes used to generate the above plots are given below (approximate method followed by the exact method). Feel free to use them in your simulations and if you have a question drop us a comment.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculation of the path loss based on the value of
% Fresnel Diffraction Parameter as proposed by Lee
% Lee W C Y Mobile Communications Engineering 1985
% Copyright www.raymaps.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
 
v=-5:0.01:5;
 
for n=1:length(v)
 
    if v(n) <= -1
        G(n)=0;
    elseif v(n) <= 0
        G(n)=20*log10(0.5-0.62*v(n));
    elseif v(n) <= 1
        G(n)=20*log10(0.5*exp(-0.95*v(n)));
    elseif v(n) <= 2.4
        G(n)=20*log10(0.4-sqrt(0.1184-(0.38-0.1*v(n))^2));
    else
        G(n)=20*log10(0.225/v(n));
    end
 
end
 
plot(v, G, 'b')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exact calculation of the path loss (in dB)
% based on Fresnel Diffraction Parameter (v)
% T S Rappaport Wireless Communications P&P
% Copyright www.raymaps.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
 
v=-5:0.01:5;
 
for n=1:length(v)
 
    v_vector=v(n):0.01:v(n)+100;
 
    F(n)=((1+1i)/2)*sum(exp((-1i*pi*(v_vector).^2)/2));
 
end
 
F=abs(F)/(abs(F(1)));
 
plot(v, 20*log10(F),'r')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')

An interesting video explaining the phenomenon of diffraction.

How to Find Point of Intersection of Two Lines

Finding the point of intersection of two lines has many important application such as in Ray-Tracing Simulation.  Two lines always intersect at some point unless they are absolutely parallel, like the rails of a railway track. We start with writing the equations of the two lines in slope-intercept form.

y1=b1+m1*x1

y2=b2+m2*x2

straight-lines

Here m1 and m2 are the slopes of the two lines and b1 and b2 are their y-intercepts. At the point of intesection y1=y2, so we have.

b1+m1*x1=b2+m2*x2

But at  the point of intersection x1=x2 as well, so replacing x1 and x2 with x we have.

b1+m1*x=b2+m2*x

or

b1-b2=-x*(m1-m2)

or

x=-(b1-b2)/(m1-m2)

Once the x-component of the point of intersection is found we can easily find the y-component by substituting x in any of the two line equations above.

y=b1+m1*x

In future posts we would like to discuss the cases of intersection of two surfaces and the intersection of two volumes.