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Currently, I'm trying to understand the Two-Ray Ground-Reflection Model. During my research, I came across the following simplification expressing the received power as follows (taken from Wikipedia): enter image description here

The part that is not clear to me is $e^{-j\Delta\phi}$: How is that derived from $e^{-j2\pi l/\lambda}$ and $e^{-j2\pi (x + x\prime)/\lambda}$?

I know that $\phi$ can be calculated with $\phi = 2\pi d/\lambda$, where $d$ is the distance between the transmitter and the receiver.

In the above equation, the ground-reflected path is expressed as $(x + x\prime)$ and $l$ is the direct line-of-sight path.

Moreover, the approximation only holds if the signal is "narrow band relative to the inverse delay spread $1/\tau$, so that $s ( t ) \approx s ( t − τ )$".

Why is that important?

Thank you for your answers in advance!

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The part that is not clear to me is e−jΔϕ: How is that derived from e−j2πl/λ and e−j2π(x+x′)/λ?

Attempting to take $e^{-j2\pi l/\lambda}$ as a common factor we get $$ \left|e^{-j2\pi l/\lambda} \right|^2 \left| \frac{\sqrt{G}}{l} +\Gamma(\theta)\frac{e^{-j2\pi (x+x\prime)/\lambda}}{(x+x\prime) (e^{-j2\pi l/\lambda})} \right|^2 $$

On the above, you might have to use the simplifications $\frac{e^a}{e^b} = e^{a-b}$ and $|e^{j\ c}|=1$.

Moreover, the approximation only holds if the signal is "narrow band relative to the inverse delay spread 1/τ, so that s(t)≈s(t−τ)". Why is that important?

They might be interested in finding out the locations or angles for which the signal and its reflected version destructively interfere and result in low signal strength. If a signal $s(t)$ and its time delayed version $s(t-\xi)$ needs to cancel out, for all values of $t$, then the assumption above ($s(t)\approx \color{red}{-}^{\dagger}s(t-\xi)$) is required. The word narrow band is simply used to indicate that the signal needs to looks like a sinusoidal wave (which has the required property of periodicity mentioned above).

$\dagger$ The minus sign may be provided by the act of reflection. I am not sure.

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