# Power spectral density of a pulse

A little background: I am simulating the response of an antenna to a pulse. The pulse has a wide frequency range, but the antenna only responds to a fairly narrow frequency. I can extract voltage and current from the antenna: they look like sinusoidal signals quickly decaying in time.

I now want to calculate the power spectral density (PSD) of that received signal. If I calculate the FFT of my voltage and current signals I see a nice isolated peak at the resonance frequency, so I would expect the power to also be concentrated around that frequency.

My first attempt was simply to multiply the voltage and current in time domain and then calculating the PSD of that. It did not work, the peak of the PSD happens at a frequency roughly double the expected one and at the desired frequency there is actually a "notch". I suspect that it is because by simply multiplying current and voltage I have effectively modulated one onto the other, therefore mixing their frequencies.

I can get a PSD that seem to make sense if instead I multiply the voltage and current in frequency domain and then calculate the PSD from the resulting spectrum.

So my questions are:

• Why can't I just calculate power as v*i in time domain?
• Does multiplying voltage and current spectra to then calculate the PSD make sense?
• so let's answer your second question first: no. While the PSD is indeed a description of the power over frequency, it is not calculated from the power signal, but from the signal itself. No big deal, if you observe the current, the voltage contains zero additional info (and vice versa); your power is U·I consideration comes from a world where you'd tell apparent from real power, but that concept doesn't apply to RF. – Marcus Müller Mar 3 '19 at 20:06
• Pulses don't have power, they have energy. Power spectral densities exist for pulse trains but not for individual pulses. – Dilip Sarwate Mar 3 '19 at 20:50
• @MarcusMüller: I understand that if I observe the current the voltage contains zero additional info. Yet, if I want to calculate how much power goes into my antenna, as if I were to measure it in reality, I would need to know the current and multiply the two, no? – DrEarlGray Mar 4 '19 at 17:59
• @DilipSarwate: pulses have energy in the sense that, being contained in time, they allow me to calculate the total energy they contain. This is not possible for an infinitely long signal because that total would go to infinity. But they obviously still have power: my pulse will physically do work and the energy that the pulse have is the integration of that power over time. Or not? – DrEarlGray Mar 4 '19 at 18:04
• @DrEarlGray as I said, no. – Marcus Müller Mar 4 '19 at 20:10