# How to compute the channel gain from the path loss model of a wireless channel

I did my research but I did not get a satisfying answer.
We have a transmitter and a receiver over a wireless communication channel. The channel's path loss is modeled as
$$128 + 37 \log_{10} (d)$$, with a shadowing standard deviation of $$10$$ dBs.

I am assuming that this path loss equation is expressed with dBs (nothing explicitly says that).

I have read that the combined path loss and the shadowing model is just the addition of the path loss model and a normally distributed random variable with mean $$0$$ and std. $$10$$ dBs.

• Is this addition of path loss and shadowing is the same as the channel gain ?
• If not, how to calculate the channel gain $$\vert h \vert$$ from the path loss model, combined with the shadowing ?
• Do you know how to calculate channel gain based on path loss? – ddddd Mar 29 at 4:58

## 2 Answers

To answer the literal questions you asked:

• yes
• see above

To Frame Challenge your question and give you what you were probably looking for:

The path loss equation you cited: $$128+37\log_{10}(d) ,$$ is equivalent to: $$\frac{P_r}{P_t} = \frac{10^{-12.8}}{d^{3.7}},$$ where $$P_r$$ is the received power in Watts, and $$P_t$$ is the transmit power in Watts. From this equation it can be seen that the path loss exponent is 3.7.

Under log-normal shadowing, the random shadowing component is modeled as a zero-mean Gaussian random variable (in dB) added to the path loss.

The answers are

• No.
• You cannot.

Small scale (channel gain) and large scale (path loss + shadowing) fading effects cannot be deduced from each others. They have independent characterizations. The channel gain characteristics must be given. Usually in rich multi-path fading with no line-of-sight (LOS), Rayleigh fading is assumed, while if there is a strong LOS, Rician fading is assumed.