# Problem

I have two USRP antennas close to each other. I know that signal strength on the receiving end has to be at least -80 dBm. Both antennas have 2 dBi of gain. I want to compute what is maximum propagation loss so that this system still works. Signal is being transmitted with power of 10 mW.

# Attempt to solve

Since i want to know the maximum amount of power loss during transmission to one antenna to another. I can form equation as. $$p_0=$$ power of transmitted signal, $$p_{m}=$$ minimum acceptable signal strength, $$x=$$ maximum amount of signal loss during transmission. $$p_0 \cdot x = p_{\text{ml}}$$ $$x = \frac{p_0}{p_{ml}}$$

My solution is ratio of powers so I can use dB units, by converting them with base-10 logarithm.

$$x_{dB}=10\cdot \log_{10}\left(\frac{p_0}{p_{ml}}\right)$$

i need to change -80 dBm to millliWatts with $$-80\text{ dBm}=10^{-80/10}\text{ mW}$$. After this I can sum antenna gain to my answer since we are using logarithms.

$$x_{dB}=10\log_{10}\left(\frac{10^{-80/10}\text{ mW}}{10\text{ mW}}\right)+2\text{ dBi} +2\text{ dBi}$$

$$x_{dB}=-86\text{ dB}$$

However my answer seems to be incorrect ?

There is simple error on second line: it should be like this $$p_0\cdot x = p_{ml}$$ $$x = \frac{p_{ml}}{p_0}$$ the fraction was wrong. Then we have: $$x_{dB}= 10 \log_{10}(\frac{10\text{mW}}{10^{-80/10}\text{mW}})+2\text{dBi}+2 \text{dBi}= 94 \text{dB}$$