Problem #$2(75\%)$: Channel gain variation with distance

You are walking away from a WiFi access point $($transmitter$).$ Your receiver Rx is located $x$ meters away from the transmitter Tx$.$ A perfectly reflective wall is located $x_{\text{w}\text{a}\text{l}\text{l}}$ meters from the transmitter. The signal you receive at $Rx$ is the sum of the line$-$of$-$sight signal and the reflected signal from the wall.

Hint: denote the transmitted signal as ${}_{tx}(t)\frac{?}{b}=ar(y)(t)e^{j2f_{c}t}.$

The speed of light is $c=3{10}^8\frac{m}{s}.$

i$.$ Write an expression for the received signal ${}_{rx}(t)$ in terms of ${}_{tx}(t)$ and the path delays ${}_1$ and ${}_2($Hint: find the delays $$ by calculating the time$-$of$-$flight of the signal from each path$)$

ii$.$ Assume ${}_1,{}_2{T}_b,$ so $\frac{?}{\text{b}\text{a}\text{r}}(y)(t-)$~$\frac{\text{~}}{b}ar(y)(t).$ Write an expression for $\frac{?}{\text{b}\text{a}\text{r}}(y{)}_{rx}(t).$

iii. Suppose $x_{\text{w}\text{a}\text{l}\text{l}}=50m$ and $f_c=902MHz.$ Plot $($in dB$)$ the channel gain $P_h$ vs$.x($i$.$e$.,$ Rx distance from Tx $)$ as $x$ ranges from $25$ to $30$ meters.

iv$.$ From your plot in iii., how far apart $($in meters$)$ are the nulls of the channel?