# Does the RMS value for a non-periodic signal exist?

Does the concept of RMS VALUE applicz to a non periodic continuous time signal? If yes how to calculate it? If no, why is it not defined?

My doubt is : What value does an ac voltmeter display for an input of non periodic square pulse of say 5V peak?

• If it's non-periodic how long a period do you want to calculate the RMS over? Oct 1 '15 at 13:23
• Gen an oscilloscope (with memory).
– Fizz
Oct 1 '15 at 13:25

Does the concept of RMS VALUE applicz (sic) to a non periodic continuous time signal? If yes how to calculate it?

RMS is rigorously defined for an infinite (not necessarily periodic) time-signal as $$\lim_{T\to\infty}\sqrt{\frac{1}{2T}\int_{-T}^T (f(t))^2dt}$$

or for signals starting at $$\t=0\$$ ($$\f(t)=0\$$ for $$\t<0\$$) as$$\lim_{T\to\infty}\sqrt{\frac{1}{T}\int_{0}^T (f(t))^2dt}$$

so RMS is quite applicable to nonperiodic signals. When the signal is periodic though, the calculation can be done over a single period (since it's time-averaged) and the above equation reduces to $$\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2}(f(t))^2dt}$$ and even for specific signals we can make further simplifications, such as that for sinusoidal signals the RMS is just the peak voltage divided by $$\\sqrt{2}\$$.

What value does an ac voltmeter display for an input of non periodic square pulse of say 5V peak?

Your voltmeter won't be doing anything as analytic as the first equation though, since it's only sampling a signal and thus can't know it's behavior into infinity. Instead they are designed to do the time-averaged approach under the assumption that you will only ever want RMS for a periodic signal. If the signal you feed is nonperiodic, the voltmeter readout will be next to meaningless in most cases as it's probably attempting to guess the interval ($$\T_2-T_1\$$) over which to average. This is why you see the value fluctuate on the readout.

• That depends on the actual voltmeter and its measuring technique. Some take the ac coupled rectified mean and multiply, some do some analogue integration, and some do digital sampling at a few hundred kHz and sum that up. Depending on the actual signal, one of these types may or may not make sense. Oct 1 '15 at 14:27
• @PlasmaHH: Yes, the really expensive ones are nearly oscilloscope sans scope display, can give rough shape (square, sine) and peak info, plot values over time, etc. But I think that's 0.01% of the market. The OP almost certainly asked about some cheap one.
– Fizz
Oct 1 '15 at 14:50
• (continued) If you have something like this you could probably abuse it to measure non-periodic square pulses (on DC/autorange) assuming the pulse on-time isn't too short for the meter to sample.
– Fizz
Oct 1 '15 at 14:56
• Could you share the source or book from which you got those formulas? It reminds of energy and power signals, and of the derivation of Fourier's transform from Fourier's series. Jul 5 '20 at 19:08
• @AlejandroNava, I had gotten them from Wikipedia's RMS article at the time but the book by Lathi begins with a treatment in the context of signal power. Jul 6 '20 at 20:34

The definition is simple : the RMS voltage is the voltage that would produce the average (i.e. mean) power. There is nothing here that depends on periodicity of the signal.

Practically, for an aperiodic signal, the issue comes down to: how do you measure its average value? One approach is to average it over some defined period, and state the measurement conditions.

Sure, it's applicable, but you have to decide what "mean" means for a non-periodic siganl. With a periodic signal, the mean for a single period is the same as the mean for any number of whole periods, so it's the obvious value to use for the averaging period. For a non-periodic signal, you'll have to explicitly specify the interval over which you're averaging.