Electronic – Calculate power of an LTE Signal

The easy solution is to get an RF wattmeter. Those will measure transmit power directly.

Alternately, you can transmit into a \$50\Omega\$ dummy load, measure the RMS voltage, and calculate power as \$P = V^2/50\Omega\$.

This will give you total power. To calculate the spectral density, divide this by the bandwidth of your signal, which you should know since you are making it. The power in this spectrum isn't flat: some smaller areas will have greater spectral density, some will have less. I'm no expert on FCC regulations so I can't say precisely what their rules are. I also can't say precisely how they define "bandwidth".

To get an idea of the spectral density of smaller slices of spectrum within your signal, take the FFT of your transmitted signal, and each bin will give you a relative measure of power in the frequency range covered by that bin. Divide your measured total power by the sum of these bin powers, and you have a scaling factor that relates the unitless power given to the FFT to power in watts.

If the USRP and your software has fixed gain, then this scaling factor will be the same for any transmitted signal. It might be easiest to transmit a simple carrier and calculate the scaling factor that way, then apply it to more complex signals.

Note that your software probably displays the FFT in decibel units; you will want to convert these to linear units to do the math as described.

Your choice of windowing function will affect how the power spreads out between bins. See How does the energy of non-resolved spectral lines get distributed in an FFT?

While "-50dBm/Hz" can be interpreted as "with 1 Hz RBW, no point exceeds -50dBm", it is generally expected that your signal power is somewhat evenly distributed over the entire bandwidth. Thus, it means that your transmit power maximum is this value times the transmission bandwidth, i.e. for 1 MHz bandwidth, you get +10dBm.

However, that assumes ideal distribution of power, which you hardly ever get, so that is a purely theoretical value. What is practically possible depends to a large extent on the scrambling algorithm you use.

The regulations should also specify how compliance is tested -- typically with an RBW of a few kHz, VBW > RBW, peak detector and a fairly long sweep time.

Measurements with an RBW of 1 Hz are not really useful in compliance tests, because they take ages and see only little of the actual signal power.