How is channel frequency response/channel state information related to free space path loss?

Question

As we know, the path loss can be described by the equation as $$ \frac{P_{r}}{P_{t}}=D_{t} D_{r}\left(\frac{\lambda}{4 \pi d}\right)^{2} $$ In the OFDM system, such as Wi-Fi, there are infinite paths exist, assume it is in free space, and only one path exist, for the specific path (LoS path), the channel frequency response(CFR)/channel state information(CSI) describes how is the channel affect the transmitted signal, it calculated by: $$ H = \frac{Y}{X} $$ where Y is the received signal at receiver and X is the transmitted signal at transmitter. the signal power is square of the amplitude of the signal, that is $$ P_{t}=|Y|^2 $$ $$ P_{r}=|X|^2 $$ I have two questions:

1. Is the following equation holds? $$ \left|H_{}\right|^{2} = \frac{P_{r}}{P_{t}}=D_{t} D_{r}\left(\frac{\lambda}{4 \pi d}\right)^{2} $$
2. According to the first equation, is the following equation holds? $$ \frac{X}{Y}=\sqrt{D_{t} D_{r}\left(\frac{\lambda}{4 \pi d}\right)^{2}} $$

Thanks in advance.

Answer 1

In the OFDM system, such as Wi-Fi, there are infinite paths exist, assume it is in free space, and only one path exist,

Exactly. So if you use OFDM in a scenario where FSPL describes your transmission sufficiently, then you're doing something wrong, because you don't need OFDM, as you have a flat channel.

We can still describe large scale fading in multipath channels using models similar to your FSPL formula, but neither of your equations hold! Then, we're describing distance-dependent stochastic properties of the random variables that make up your CSI.

You need to cross over from your deterministic "this channel has a defined attenuation" model to a stochastic "and here's how we describe the properties of random variable, whose realizations are channels".