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

  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
to read

 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
the analog input signal level at which digital
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
the analog input level into a piece of audio
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.


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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