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Knife Edge Diffraction Model
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This model was developed by Scientists Fresnel and Kirchhoff. Knife Edge diffraction model is used for a pure diffractive environment. To understand this model, it is necessary to know the concept of Fresnel zone.

Diffraction: Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle. Diffraction allows radio signals to propagate around the curved surface of the earth, beyond the horizon, and to propagate behind obstructions. Although the received field strength decreases rapidly as a receiver moves deeper into the obstructed (shadowed) region, the diffraction field still exists and often has sufficient strength to produce a useful signal.

The phenomenon of diffraction can be explained by Huygen's principle, which states that all points on a wavefront can be considered as point sources for the production of secondary wavelets, and that these wavelets combine to produce a new wavefront in the direction of propagation. When the signal bends around the obstruction a loss is seen . This loss is called diffraction loss. The concept of diffraction loss as a function of the path difference around an obstruction is explained by Fresnel zones.

Fresnels Zones: Transmitted electromagnetic waves can follow slightly different paths before reaching a receiver, especially if there are obstructions or reflecting objects between the two. The waves can arrive at different times and will be slightly out of phase due to the different path lengths. The strong signal gets divided into multi-path components and reaches the receiver through multiple zones as shown in Figure 6. d1 refers to the distance between Tx and obstruction. d2 refers to the distance between Rx and obstruction. n is the path loss exponent of the environment. Figure 6: Fresnel zones

Fresnel zones represent successive regions where secondary waves have a path length from the transmitter to receiver which are n. $\lambda/2$ greater than the total path length of a line-of-sight path. Some important points that the reader should note are:

1. Since the components are traversing through different path lengths, they can either add up or try to nullify the strong LOS component.

2. Also, signal components traversing through the nth Fresnel zone travel a distance of $\lambda/2$ greater than components traversing in the preceding i:e (n-1)th Fresnel zone.

3. Whether a component is constructive or destructive in nature, will depend on the Fresnel zone number. The successive Fresnel zones have the effect of alternately providing constructive and destructive interference to the total received signal. Signals passing through even number of Fresnel zones are constructive in nature. Signals passing through odd number of zones are destructive in nature.

4. Any diffraction in the path will increase the number of Fresnel zones.

5. Radius of the $n^{th}$ Fresnel zone can be calculated as follows i.e. $r_{n}$=$\sqrt{\frac{n \lambda d_{1} d_{2}}{d_{1}+d_{2}}}$

The model can be explained as follows: Figure 7: The physical model for knife edge.

Consider a transmitter and receiver separated in free space as shown in Figure 7. Let an obstructing screen of effective height h with infinite width (going into and out of the paper) be placed between them at a distance d, from the transmitter and d: from the receiver. It is apparent that the wave propagating from the transmitter to the receiver via the top of the screen travels a longer distance than if a direct line-of-sight path (through the screen) existed. Assuming h $\lt d$ and $h \gt \gt d1,d2$, then the difference between the direct path and the diffracted path, called the excess path length $\Delta$ can be found as:

• To find path difference $\Delta$: The simplified model is given as, Figure 8: Simplified model to find path difference

Path difference

$\triangle = (p_1 + p_2) – (d_1 +d_2)$

$p_1 = \sqrt{d_1^2 + h^2}$ $\quad$ $p_2 = \sqrt{d_2^2 + h^2}$

$p_1 = (d_1^2 + h^2)^{1/2}$

$p_1 = d_1 [ 1 + \frac{h^2}{d_1^2}]^{1/2}$

= $d_1 + \frac{h^2}{2d_1}$

Similarly,

$p_2 = d_2 + \frac{h^2}{2d^2}$

$\triangle = (p_1 + p_2) – (d_1 + d_2)$

= $d_1 + \frac{h^2}{2d_1} + d_2 + \frac{h^2}{2d_2} – d_1 – d_2$

= $\frac{h^2}{2d_1} + \frac{h^2}{2d_2}$

$\triangle = \frac{h^2}{2} ( \frac{1}{d_1} + \frac{1}{d_2})$

$\theta_{ \triangle} = \frac{2 \pi \triangle }{\lambda}$

$\theta_{ \triangle} = \frac{ \pi h^2}{ \lambda} ( \frac{d_1 + d_2}{ d_1 d_2})$

• To find diffraction angle $\alpha$:

Referring to knife edge diffraction geometry, (Figure 7)

By exterior angle property,

$\alpha = \beta + \gamma$

$\beta = tan^{-1} (\frac{h}{d_1}) \quad \gamma = \ tan^{-1} (\frac{h}{d_2})$

$\alpha = \ tan^{-1} ( \frac{h}{d_1}) + \ tan^{-1} \ (\frac{h}{d_2})$

$\beta$ and $\gamma$ are very small.

$\alpha = \frac{h}{d_1} + \frac{h}{d_2} = h \ ( \frac{d_1 + d_2}{ d_1 d_2})$

Fresnel –Kirchoff parameter: This is an important parameter in the model which helps to predict path loss. The electric field strength, Ed. of a knife-edge diffracted wave is given by

$\frac{E_{d}}{E_{o}}=F(v)=\frac{(1+j)}{2} \int_{v}^{\infty} \exp \left(\left(-j \pi t^{2}\right) / 2\right) d t$

where $E_o$is the free space field strength in the absence of both the ground and the knife edge, and F (v) is the complex Fresnel integral. The Fresnel integral, F(v),is a function of the Fresnel Kirchoff diffraction parameter v defined as

$v=h \sqrt{\frac{2\left(d_{1}+d_{2}\right)}{\lambda d_{1} d_{2}}}=\alpha \sqrt{\frac{2 d_{1} d_{2}}{\lambda\left(d_{1}+d_{2}\right)}}$

The equation of electric field intensity which is a complex function of v is difficult to calculate. Hence empirical list of formulae is proposed for the same which is as follows:

$\begin{array}{ll}{G_{d}(d B)=0} & {v \leq-1} \\ {G_{d}(d B)=20 \log (0.5-0.62 v)} & {-1 \leq v \leq 0}\end{array}$

$\begin{array}{ll}{G_{d}(d B)=20 \log (0.5 \exp (-0.95 v))} & {0 \leq v \leq 1} \\ {G_{d}(d B)=20 \log (0.4-\sqrt{0.1184-(0.38-0.1 v)^{2}})} & {1 \leq v \leq 2.4} \\ {G_{d}(d B)=20 \log \left(\frac{0.225}{v}\right)} & {v\gt2.4}\end{array}$

In real time, the path contains multiple diffractive objects.Many models that are mathematically less complicated have been developed to estimate the diffraction losses due to multiple obstructions which is beyond the scope of this book.