Digital Signal Levels

For analog signals, we looked at the voltage amplitudes through the analog signal path in a previous post.  How do we ‘measure’ the signal amplitude level in the digital signal path?   We know the highest allowed amplitude voltage in the quantization process is assigned the binary number whose digits are all “1’s” .  We can define the amplitudes of the allowed voltage levels with reference to this “full scale” level as follows,

In our example of a bit depth  n = 4,  the highest level is  i = 2^3 = 8.   So this “full scale” digital signal level is specified by  0 dBFS, which is read  “zero dB full scale”.    The lowest (non-zero) level is i = 1.  So this lowest digital signal level is specified by  -18.1 dBFS.
 
For a more typical n = 16 bit depth digital system, we would have the following digital signal levels,

It should be kept in mind that dBFS is a logarithmic unit. In linear voltage units, the interval (step) between the huge number of allowed voltage levels is very small and uniform across the full dynamic range.  In your digital audio workstation, you often see the signal input/output level meters expressed in  dBFS units.  An example of an input signal level meter in dBFS units is shown below.  

Input Signal Level Meter dBFS

 
Gain Staging

An interesting question arises --  in the analog-to-digital converters used in the audio interfaces,  what is the maximum voltage amplitude |Vmax| that is chosen to correspond to the digital full scale  0 dBFS ?  There seems to be some guidance in this regard from various industry standards that align the digital signals levels (dBFS) to analog signal levels (dBu), such as in the following alignment,

This implies that the designers of the A/D converter use a value of  
|Vmax| =  +24dBu  = 12.28 V.  This may be true in high-end professional A/D units.  But it’s probably not the case in my budget audio interface that is powered via the USB bus.  So different manufacturers may be designing to different |Vmax| values.  The choice affects the high side of the dynamic range. On the low side of the dynamic range, the noise floor may be dictated by the existing noise present with the signal from the analog circuitry, and not by the theoretical noise floor from quantization error.  So the actual dynamic range available for recording music is certainly much less than the theoretical values for the different bit depths used in the digitization process, e.g., theoretical dynamic range = 144 dB for n = 24 bit depth.  Actual dynamic range for bit depth   n = 24  varies from  105 dB  in inexpensive audio interfaces to 125 dB in top professional units.  In the figure above, the dynamic range is shown to be 119 dB.  The bottom line is that we would like to have a dynamic range available that well exceeds the dynamic range of comfortable human hearing, which is roughly 85-90 dB.
 
Managing the sound levels through the recording, mixing, and mastering processes is called "gain staging" .  The ultimate goal is to capture and recreate the full dynamic range needed for the type of music being recorded.  There is A LOT written about this topic, some of which is just plain wrong. The article on gain staging from the folks at Sound On Sound provides an excellent overview.  The key to setting proper input recording levels is maintaining sufficient “headroom” – the difference between the clipping point (0 dBFS) and the average sound level.  Headroom is necessary to accommodate peaks in sound level as well as changes in level that will occur during signal processing and channel mixing, and ultimately in mastering the audio file.  The current best practice is to allow for 20 dB of headroom.  As seen in the figure above, using 20 dB of headroom still provides more than enough dynamic range for recording very quiet sounds. So when setting input levels, we should adjust the gain of the audio interface pre-amplifiers to yield average levels around  -20 dBFS and keep anticipated peak levels around -12 dBFS.

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The digital data created by the A/D converter in the audio interface is now transferred through the USB-C bus to the computer, the next major piece of equipment in the home music studio.  We’ll look at the computer hardware setup in the next post.