I read your feature about ‘Digital Problems, Practical Solutions’ (www.soundonsound.com/sos/feb08/articles/digitalaudio.htm),
which said that digital audio can capture and recreate analogue signals
accurately, and that the ‘steps’ on most teaching diagrams are
misleading. Does that mean that the graph should really show lines, or
plot ‘x’s, instead of looking like a standard bar-graph?
Remi Johnson via email
SOS
Technical Editor Hugh Robjohns replies:
Good question! The graphs in
that article are accurate as far as they go, but offer a very simplified
view of only one part of the whole, much more complex, process.
When
an analogue signal (the red line on Graph 1: Sample & Hold) is
sampled, an electronic circuit detects the signal voltage at a specific
moment in time (the sampling instant) and then holds that voltage as
constant as it can until the next sampling instant. During that holding
period the quantising circuitry works out which binary number represents
the measured sample voltage. This, not surprisingly, is called
a ‘sample and hold’ process, and that’s what that diagram is trying to
illustrate.
Graph 1: Sample & Hold
So the sampling moment is, theoretically, an
instant in time, best represented on the graph as a thin vertical line
at the sample intervals (the blue lines in the picture Graph 1: Sample
& Hold), but the actual output of the sample and hold process is the
grey bar extending to the right of the blue line.
However,
the key to understanding sampling is understanding the maths behind
that theoretical sampling ‘instant’, and that means delving into the
maths of ‘sinc’ (sin(x)/x) functions, which is the time-domain response
of a band-limited signal sample. At this point most musicians’ eyes
glaze over…
As we know, the measured amplitude
of each sample from an analogue waveform is represented by a binary
number in the digital audio system. When reconstructing the analogue
waveform that number determines the height of the sinc function.
The
important point is that we are not just creating a simple ‘pulse’ of
audio at the sample point, because the sinc signal actually comprises
a main sinusoidal peak at the sampling instant (and of the required
amplitude), plus decaying sine wave ‘ripples’ that extend (theoretically
for ever) both before and after that central pulse. The reconstructed
analogue waveform is the sum of all the sinc functions for all
the samples.
The clever bit is that the points
where those decaying sinc ripples cross the zero line always occur at
the adjacent sampling instants. This is shown in the next diagram (Graph
2: Two Sinc Functions) where, for simplicity, just two sample sinc
functions are shown for samples 23 (red) and 27 (blue). You can see that
at the intermediate sample points (26, 25, 24 and so on) the sinc
functions are always zero.
Graph 2: Two Sinc Functions
That means that the ripples don’t contribute to
the amplitude of any other sample, but they do contribute to the
amplitude of the reconstructed signal in between the samples, with the
adjacent sample sinc functions having the greatest influence, and lesser
contributions from the more distant samples. This is shown in the next
diagram (Graph 3: 3kHz Sinc Addition), in which the sinc functions of
a number of adjacent samples are shown, and when summed together produce
the dotted line that is a sampled 3kHz sine waveform
Graph 3: 3kHz Sinc Addition
These last two diagrams have been borrowed from
a superb paper by Dan Lavry (of Lavry Engineering), which explains
sampling theory extremely well, and can be found here: www.lavryengineering.com/documents/Sampling_Theory.pdf.
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