# dBFS or dBdig ?

Michael Arnao mikea at ensoniq.com
Wed Mar 3 19:25:20 EST 1999

```OK, this is even better: AES17-1991, "AES standard method
for digital audio engineering--Measurement of digital audio
equipment," which appears in JAES Vol. 39 No. 12.

Section 2.5 defines decibels, full scale, as "20 times the
common logarithm of the amplitude over the full-scale
amplitude."

Section 2.3 in turn defines the full-scale amplitude
as the "amplitude of a 997-Hz sine wave whose positive
peak value reaches the positive digital full scale, leaving
the negative maximum code unused. In 2's-complement
representation, the negative peak will be 1 LSB away from
the negative maximum code."

As this standard is in reference to taking measurements on
audio equipment, I would assume that a sine wave is used
simply because most audio equipment is AC-coupled, and that the
frequency of 997 Hz was chosen because it's not a factor of the
common sampling rates and thus a full-scale digital level
corresponding to the peak amplitude of the sine wave will be
reached during some cycles of the waveform.

Section 2.3.1 states that amplitude measurements relative
to this value shall be reported in decibels, full scale
(dB FS), or percent, full scale (%FS).

In Section 2.6, there is a note that states "Because the
definition of full scale is based on a sine wave, it will
be possible with square-wave test signals to read as much
as +3.01 dB FS. Square-wave signals at this level are
not recommended since tilt or overshoot introduced by
any filtering operations will cause clipping of the signal."

(Notice that the "+3.01 dB FS" in the note in Section 2.6
corresponds to the peak amplitude reading taken on the
ANALOG square wave relative to the peak amplitude of the
ANALOG sine wave.)

In light of Section 2.3, I would revise the equation I'd
previously given for converting a linear value to dB FS

y dB FS = 20 * log10 (x / (2**(N-1) - 1)), 0 <= x < 2**(N-1)

(Assuming dB FS are appropriate units for DIGITAL signals)
so that 0x7fff => 0 dB FS for a 16-bit word length,
0x7fffff => 0 dB FS for a 24-bit word length, etc.

But there is still nothing here that states that dB FS
has anything in particular to do with rms measurements!

In light of what is stated in Sections 2.3 and 2.5, one
can conclude that dB FS are appropriate as units of
peak amplitude.

A clue as to where the rms confusion might initially have
arisen is found in Section 3, "Input characteristics,"
specifically in Section 3.4, "Input for full-scale
amplitude," which describes the procedure for measuring
clipping occurs. It is stated that this voltage should
be specified as the "rms voltage of a 997-Hz sine wave
that shall be applied to the input to obtain a digital
signal whose positive peak value reaches the positive
digital full scale." But this is simply a way of reporting
equipment at which digital clipping occurs, and is
not a definition of dB FS.

The question that I now have is whether dB FS are
appropriately used as units of DIGITAL amplitude, as the
definition is relative to the amplitude of an analog
sine wave.

--Mike

Michael A. Arnao
Sr. DSP Engineer
Emu <> Ensoniq Music and Audio Systems
155 Great Valley Parkway
Malvern, PA 19355
email: mikea at ensoniq.com
Voice:(610)647-3930  Fax:(610)647-8908

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```