# Should calculated time domain RMS and frequency domain RMS be approximately similar?

I have an acceleration measurement for a full day with sampling rate 10.24 Hz. I have divided the signal into 30 minute intervals because I want to compare the amplitude spectra for each 30 minute interval to other data I have for 30 minute intervals of the day.

I have not used any windowing, overlap or derating of the time signal before applying DFT by the fft functions from numpy (Python).

As a quality check, I want to calculate RMS for the time domain and RMS for the frequency domain, because my understanding is that they should be approximately equal. So questions are:

1. Should time signal RMS and amplitude spectra RMS be approximately equal?

2. If so, will amplitude spectra RMS always be smaller than time signal RMS?

I have calculated RMS for both, and found that there are larger variation in time (RMS decreasing/increasing) for the amplitude spectra RMS compared to the time signal RMS.

1. Assuming 1 is correct, should the two RMS calculations have similar trends (increase/decrease), or will frequency domain RMS vary more depending on the what my DFT/FFT can detect with my sampling rate of 10.24 Hz?

• From Pareseval's theorem, the energies in the time and frequency domains will be equal.
– jojeck
Oct 27, 2022 at 8:02
• @birki Are you using complex or real FFT? If real: is your input even/odd length, and are you doing anything special for bin 0 and Nyquist when you calculate your RMS? Oct 27, 2022 at 8:24
• @jojek strictly speaking, and also very practical, because not all FFTs are unitary, proportional, not identical but yes. Oct 27, 2022 at 8:36
• @cloudfeet real FFT, one-sided. Even, length 18432. Not doing anything for the 0 bin and nyquist. The amplitude spectra RMS is smaller, so I assume something is wrong. The 0 bin amplitude and nyquist amplitude should be halfed in the RMS calculation, or does that depend on if the length of the input is even or odd? Oct 28, 2022 at 8:57
• @jojek Is this the case for random vibration signals as well? Or signals with noise? WIll they be exactly equal? Oct 28, 2022 at 9:20

$$\sum |x[n]|^2 = \sum |X[k]|^2$$
This equality answers your other questions I think. The scaling you need to deploy is $$\frac{1}{\sqrt{N}}$$ for both the forward and inverse transforms