RMS (root mean square)

Definition

The RMS (root mean square) is the square root of the mean of the squares of the samples in a window of N samples:

x_rms = √( (1/N) · Σ x[n]² ), with n from 0 to N − 1.

In continuous form: x_rms = √( (1/T) · ∫ x(t)² dt ). Squaring makes all values positive and weights energy: the RMS is proportional to the square root of the signal's average power. In dBFS: L_rms = 20 · log10(x_rms / FS).

There are two reference conventions:

• mathematical: a full-scale sine wave has RMS = 1/√2 = −3.01 dBFS;
• AES17: 3.01 dB is added, so that the full-scale sine wave reads 0 dBFS RMS. The two values always differ by 3.01 dB.

RMS values of waveforms with peak amplitude A: sine A/√2 (−3.01 dB), square wave A (0 dB), triangle and sawtooth waves A/√3 (−4.77 dB). RMS measured continuously uses a moving window (rectangular, of duration T) or an exponential integrator: y[n] = α · y[n−1] + (1 − α) · x[n]², with α = e^(−1/(τ · fs)). For a stereo file the overall RMS is √((ΣL² + ΣR²) / 2N). The RMS is not frequency-weighted: low frequencies, which carry most of the energy, dominate the measurement.

Parameters and units

• Unit: dBFS RMS (specifying the convention) or dBu/dBV in the analog domain.
• Integration window: 300 ms (similar to the VU), 50–100 ms (fast meters), 3 s or the whole song (summary measurements).
• Analog/digital alignment: 0 VU = +4 dBu; −18 dBFS = 0 dBu (EBU R68) or −20 dBFS = +4 dBu (SMPTE RP155).
• Indicative values (mathematical convention, whole song): single track −24/−18 dBFS; pre-mastering mix −18/−14 dBFS; modern pop master −10/−7 dBFS.
• Derived quantities: crest factor = peak − RMS (dB); RMS detection in compressors.

Musical function

The RMS estimates the average power of the signal and is therefore closer to the perception of volume than the peak is. It serves as a level reference in gain staging (−18 dBFS RMS as the nominal track level), for calculating the crest factor, for setting compressors with RMS detection, which react to energy rather than to individual transients, and for comparing tracks with a similar spectrum. For comparing different songs, RMS is less suitable than LUFS loudness, because it does not take into account the ear's sensitivity to different frequencies.

Practical example

• 1 kHz sine wave with a peak of −6.0 dBFS: RMS = −9.01 dBFS (mathematical convention) or −6.0 dBFS (AES17 convention).
• Pre-mastering mix: peak −3.2 dBFS, RMS of the whole file −15.8 dBFS → crest factor 12.6 dB.
• Two basses with the same RMS of −18 dBFS: the first has its energy concentrated at 45 Hz, the other at 110 Hz with harmonics up to 1 kHz. The latter sounds 3–5 dB louder because the ear is less sensitive below 60 Hz; LUFS loudness registers the difference, RMS does not.
• A song with 20 s of silence at the end out of 3 minutes: the RMS of the whole file drops by 10 · log10(180/160) = 0.5 dB without the musical material having changed.

Common mistakes

• Comparing RMS values from different plugins without knowing whether they use the mathematical or the AES17 convention (a 3 dB gap).
• Using windows that are too short (under 10 ms): the RMS follows the low-frequency waveform and fluctuates.
• Including silences and long tails in the whole-file measurement: the RMS drops and the crest factor rises.
• Comparing songs with different spectra by RMS instead of integrated loudness.
• Reading a high RMS as a guarantee of power: an excess of sub can raise the RMS without increasing the perceived loudness.

How to check it

• App analysis: the RMS row is the RMS value of the whole file over both channels, in the mathematical convention (a full-scale sine wave reads −3.01 dBFS). The Crest factor (peak − RMS) row is the difference between the sample peak and the RMS.
• Meters: Wave Candy (FL Studio) in meter mode shows peak and RMS; Voxengo SPAN shows the RMS with an adjustable window; a VU meter plugin shows the RMS with 300 ms integration.
• Convention test: generate a sine wave at 0 dBFS and read the meter's RMS: −3.0 dBFS indicates the mathematical convention, 0.0 dBFS the AES17 convention.
• Trim the silences before analyzing a file if you want to compare the RMS of the musical material alone.