# How to understand EIRP, PSD for a given bandwidth?

I have troubles to understanding what is the meaning of parameters for transmitting power of RF-signal.

So for RF signal in the same frequency band and same application, on one hand I see the max. allowed mean power is e.g 50dBm. On the other hand I see another max. allowed power spectral density is e.g -3dBm/MHz. This really confuses me.

So the questions are:

1. What is the relation between mean power and power spectral density? How to convert one to the other ?

2. For e.g. if my device uses a 2GHz bandwidth, how do I know how much power the device is allowed to transmit ? How to calculate it ?

Thanks very much.

BL

• In practice it's hard to calculate ERP outside a lab since it is measured as field strength in the air some distance away from the antenna. Don't confuse ERP with the maximum power you are allowed the have out of the antenna connector. It's rather output power plus antenna gain minus losses in cables/connectors minus losses in the air. Aug 11, 2021 at 14:30
• Hi @Lundin, thanks for the hint.
– BL_
Aug 12, 2021 at 9:04

If your Power Spectral Density is -0.3 dBm / MHz that's a real power of 0.5 milliwatts per MHz hence, if your bandwidth is 2000 MHz, that's a total permissible power of 1 watt across the whole width of the spectrum you are allowed to use.

If you are allowed to use a signal of 50 dBm (100 watts) AND, you must simultaneously ensure that it doesn't exceed the PSD of -0.3 dBm / MHz, you have to consider the signal bandwidth. If the bandwidth is 5 Hz, that's 100 watts in 5 Hz and, equivalent to 0.5 milliwatt spread over 1 MHz.

• Thanks @Andy. The example is extremely helpful to understand. So just to confirm my understand: the PSD is the bottle neck when having a wide Bandwitdh (>1MHz) and the mean power is the bottle neck when having a narrow bandwidth (<1MHz) ? Is it correct ?
– BL_
Aug 12, 2021 at 9:01

What is the relation between mean power and power spectral density? How to convert one to the other ?

The integral over the PSD is the power. That's it.

For e.g. if my device uses a 2GHz bandwidth, how do I know how much power the device is allowed to transmit ? How to calculate it ?

2 GHz · -3 dBm/MHz = 33 dB + -3 dBm = 30 dBm = 1 W.

It's forbidden, because you mustn't transmit more than -50 dBm on average. So, you're bound by both limits, and either one might kick in.

You might, however, transmit a 1 ns pulse with a power of 1W, then be silent for -50 dBm - 30 dBm = -80 dB of a nanosecond, so -10 dBs = 0.1s

More realistically, your PSD won't be flat – you're doing anything but a pulsed radar, after all. In that case, assigning of power might be implicit e.g. through transmitter symbol pulse shaping, or explicit, e.g. through subcarrier power allocations according to subcarrier SNR, for example employing the waterfilling algorithm.

• I think you meant to use -3dBm/MHz Aug 11, 2021 at 17:17
• @TonyStewartEE75 yes, you're right! Fixing this right away, thank you! Aug 11, 2021 at 17:28
• thanks @Marcus. The silent trick does help to utilize ave transmit power optimally.
– BL_
Aug 12, 2021 at 8:57
• @bienle that really depends! Many things would be better off with a low average transmit power, long spreading sequences / low-rate channel coding and appropriate processing in the receiver. Aug 12, 2021 at 9:57