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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 ?

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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: $$ x_{dB}= 10 \log_{10}(\frac{10\text{mW}}{10^{-80/10}\text{mW}})+2\text{dBi}+2 \text{dBi}= 94 \text{dB} $$

This answer seems to be correct since it corresponds to physical measurements.

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