Friis law and the power delay profile Assume we have communication system between transmit and receive over wireless channel.

The Friis path loss model is given by the following equation 

$$P_r = P_t G_t G_r (\frac{\lambda}{4\pi D})^2$$

Where $P_{r(t)}$ is the received (transmit) power, $G_{t(r)}$ is the gain of the antennas at transmitter (receive) and $\lambda $ is the signal wavelength while $D$ is the distance between transmit receive ends.

In theoretical (academia) and simulation scenarios (for example MATLAB)I have come across examples wireless channel where a power delay profile is defined, that is time of arrival (TOA) of rays versus amplitude of those rays. For example, assume I have five rays arriving and delay spread $40 ns$ with amplitude 

$$channel= [ 0.5+i0.5, 0.3+i0.3, 0.2+i0.2, 0.1+i 0.1, 0.3 ] $$

$$TOA= [0, 10*10^{-9} , 20*10^{-9}, 30*10^{-9} , 40*10^{-9}]$$

My simple question is: How are the two concepts related??

Thanks
Basically, multipath propagation causes time dispersion (hence causing a delay spread ($\sigma_{\tau}$)) and that dispersion influences the fading characteristics of the wireless channel.So delay spread is related to the fading characteristics of a channel and not really to the path loss. Particularly, if $\sigma_{\tau}$ is greater than the symbol period ($T_s$) then the wireless channel is said to be frequency selective, meaning that a signal will interfere with itself causing additional fading.

You should also note that the rms delay spread is not the maximum delay that your signal experiences as I think you may asssume, $\sigma_{\tau}$ can be calculated as follows:

$\sigma_{\tau} = \sqrt{\overline{\tau ^2} - \overline{\tau} ^2}$

where

$$\overline{\tau} =  \frac{\Sigma P(\tau _i) \tau _i}{\Sigma P(\tau _i)},$$
$$\overline{\tau ^2} =  \frac{\Sigma P(\tau _i) \tau _i ^2}{\Sigma P(\tau _i)}$$
and $P(\tau _i)$ is the power of the component with delay $\tau _i$