RMS question

[ulysses]

0 (volts? dBSPL?) RMS would be no signal (silence) and the sky is the theoretical limit. It's okay that squares and roots behave the way they do for values below one because you're performing both operations, so they "cancel." The whole point of root-mean-squared calculations are to compensate for the fact that (approximately) half of the signal has a negative polarity. If you just took the average value of a sine wave it would be zero. So you square all the values first, take the average, then take the square root of that. This gives you an RMS level (for sine waves only) that's equal to the peak value divided by the square root of two. A square wave has an RMS value equal to the peak levrel. Music is less predictable.

Of course it's impossible to calculate the RMS, or any average value, of a continuous waveform pecause there are an infinite number of points along the curve (any two points will have an infinite number of points in-between). So we don't calculate this stuff by hand, instead we use electronic devices with integrators and capacitive time constants to do the "math" for us. the calculus would get crazy for any waveform that wasn't perfectly periodic and repeating.

And no, it's not the same as any kind of "logarithmic conversion" (such as converting the ratios of volts or millibars into decibels). A voltage or SPL level can be expressed in terms of RMS. A ratio between two RMS levels can be expressed in decibels, which is a logarithmic calculation.

[jbuntz]

So could RMS be replaced in DSP with if sample value < 0 then sample value = sample value * -1 and then do the mean calculation?

[SteveHalko]

That wouldn't be the same as RMS. For a pure sine wave, if you take all the negative values and make them positive, you get a full-wave rectified waveform. Thats a different waveform than if you squared all the values. Since its a different waveform, then the mean would also be different.

For the full-wave rectified waveform, the mean is .637 times the peak value, whereas the RMS value is .707 times the peak.

[volki]

Originally posted by ulysses
A voltage or SPL level can be expressed in terms of RMS. A ratio between two RMS levels can be expressed in decibels, which is a logarithmic calculation.

which is where all those different db-expressions (and misunderstandings) come in...


logarithmic conversion basically goes like

L = A*lg(X/Y)

where X,Y - compared rms values; A - scaling factor.

when inserting spl (sound pressure level), A equals 10.
when inserting volts, X equals 20 (proof of which would be too confusing here i guess)

for relative comparison of two measured values, insert them for X & Y, and get dBr (r - relative).

for absolute levels, you want to take one measured value and relate it to a commonly known base value. so it's like

Lp = 10*lg(P/Po) or Lu = 20*lg(U/Uo) respectively.

Po is 2*10^-5 Pa (pascal) and relates to the human hearing threshold @ 1kHz. this gives you dB SPL.

Uo can either be 1V (for dBV) or .775V (for dBu or dBm). dBV is easier to handle without a calculator and is found more in the consumer field i believe. the others are standard in pro audio, the odd value being due to historical reasons. the conversion is pretty easy though, it gets pretty close to dBV - 2,2dB = dBu.

there's a ton of other dB-expressions, like e.g. dB(A), which is a frequency-weighted (i.e. filtered) curve, approximating the frequency-dependent human ear. or, in the digital world, you'll come across dBfs (full scale). where 0dBfs = maximum, any positive value will mean overload.

hope this helps