D/A and A/D | Digital Show and Tell (Monty Montgomery @ xiph.org)

Original Video: http://xiph.org/video/vid2.shtml

More videos in this series: http://xiph.org/video/

Get the software - https://wiki.xiph.org/Videos/Digital_Show_and_Tell

Monty at Xiph presents a well thought out and explained, real-time demonstrations of sampling, quantization, bit-depth, and dither on real audio equipment using both modern digital analysis and vintage analog bench equipment.

This video has been reproduced with permission of Monty @ xipg.org and in accordance with Attribution-ShareAlike 3.0 http://creativecommons.org/licenses/by-sa/3.0/

Further reading: Audio Myths & DAW Wars - http://www.image-line.com/support/FLHelp/html/app_audio.htm

This is a video about the digital vs analog audio quality debate. It explains, with examples, why analog audio within the accepted limits of human hearing (20 Hz to 20 kHz) can be reproduced with perfect fidelity using a 44.1 kHz 16 Bit digital signal.

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Closed Caption:

hi I'm Monty Montgomery from red hat and xiph.org
a few months ago i wrote
an article on digital audio and why 24
bit 192 kilohertz music downloads don't
make sense. in the article i mentioned
almost in passing that a digital wave
form is not a stair-step and it was
certainly don't get a stair-step when
you convert from digital back to analog
of everything in the entire article that
was the number one thing people wrote
about in fact more than half the mail I
got those questions and comments about
basic digital signal behavior since
there is interest
let's take a little time to play with
some simple digital signals
pretend for a moment that we have no
idea how digital signals really behave
in that case it doesn't make sense for
us to use digital test equipment either
fortunately for this exercise there's
still plenty working along lab equipment
out there
first up we need a signal generator to
provide us with analog input signals in
this case and HP 3325 from 1970
it's still a pretty good generator so if
you don't mind the size the weight the
power consumption than the noisy fan you
can find them on ebay occasionally for
only slightly more than you'll pay for
shipping
next we'll observe our analog wave forms
on analog oscilloscopes like this
tektronix 2246 from the mid-nineties one
of the last and very best analog scopes
ever made everyone laugh should help
and finally inspect the frequency
spectrum of our signals using an analog
spectrum analyzer this HP 3585 from the
same product line is the signal general
here like the other equipment here
it has a rudimentary and hilariously
large microcontroller but the signal
path from input to what you see on the
screen is completely analog all of this
equipment is vintage
but aside from its raw tonnage this
picture still quite good at the moment
we have our signal generator set to
output a nice one will hurt sine wave at
12 volts rms we see the sine wave on the
oscilloscope can verify that it is
indeed one kilohertz at 10 volts rms
which is two point eight volts peak to
peak and that matches the measurement on
the spectrum analyzer as well
the analyzer also shows some low-level
white noise and just a bit of harmonic
distortion with the highest peak are all
70 decibels were so below the fund
now this doesn't matter at all in our
demos but i wanted to point it out now
just in case you didn't notice it till
later
now we drop digital sampling in the
middle for the conversion will use a
boring consumer grade ii magic USB one
audio device
it's also more than 10 years old at this
point it's getting obsolete
a recent converter can easily have an
order of magnitude better specs flatness
linearity jitter noise Behavior
everything
you may not have noticed just because we
can measure an improvement
doesn't mean we can hear it and even
these all the consumer grade boxes were
already at the edge of ideal
transparency the e magic connects to my
thinkpad which displays a digital wave
form and spectrum for comparison
then the thinkpad since the digital
signal right back out to the e magic for
reconversion to analog and observation
on the outlet scopes input to output
left to right
okay it's go time
we begin by converting an analog signal
to digital and that right back analog
again with no other steps
the signal generator is set to produce a
1 kilohertz sine wave just like before
we can see our analog sine wave on our
input side oscilloscope we digitize our
signal to 16-bit PCM at 44.1 kilohertz
same as on a CD the spectrum of the
digitized signal matches what we saw
earlier and what we see now on the
analog spectrum analyzer
aside from its high impedance input
being just a smidge noisy for now the
waveform display shows are digitized
sine wave as a stair step pattern one
step for each sample and when we look at
the output signal that's been converted
from digital back to analog we see it's
exactly like the original sine wave
no stair steps ok 1 kilohertz is still a
fairly low frequency
maybe the stair steps are just hard to
see or they're being smooth away
fair enough let's choose a higher
frequency something close to my question
- 8:15 collects
now the sine wave is represented by less
than three samples per cycle in the
digital wave form looks pretty awful
well looks can be to see the analog
output is still a perfect sine wave
exactly like the original
let's keep going up let's see if I can
do this without blocking any cameras 16
killers 17 color it's 18 killers it's
19 killers 20 kilohertz
welcome to the upper limits of human
hearing the output waveform is still
perfect
no jagged edges no drop-off no stair
steps
so where the stair steps go don't answer
it's a trick question
they were never there drawing a digital
wave form as a stair-step was wrong to
begin
why a stair-step is a continuous time
function its jagged and it's piecewise
but it has a defined value with every
point in time a sample signal is
entirely different
it's discrete time it's only got a value
right at each instantaneous sample . and
it's undefined
there is no value at all everywhere in
between a discrete-time signal is
properly drawn as a lollipop draft
the continuous analog counterpart of a
digital signal passes smoothly through
each sample . and that's just as true
for high frequencies as it is for low
now the interesting and not at all
obvious videos
there's only one and limited signal that
passes exactly through each sample .
it's a unique solution so if you sample
a band limited signal and then convert
it back the original input is also the
only possible output and before you say
oh I can draw a different signal that
passes through those points
well yes you can but if it differs even
minutely from the original it contains
frequency content at or beyond my quest
breaks the band limiting requirement and
isn't a valid solution
so how did everyone get confused and
start thinking of digital signals and
stair steps i can think of two good
reasons first
it's easy enough to convert a sampled
signal to a true stair-step just extend
each sample value forward until the next
sample . this is called a zero-order
hole and it's an important part of how
some digital-to-analog converters work
especially the simplest ones
so anyone who looks up digital to analog
converter or digital-to-analog
conversion is probably going to see a
diagram of a stair-step waveform some
way but that's not a finished conversion
and it's not the signal that comes out
second and this is probably the more
likely reason engineers who supposedly
no better like me
process steps even though they're
technically wrong it's sort of like a
one dimensional version of families in
an image pixels are squares either their
samples of a two dimensional function
space and so they're all so conceptually
infinitely small points
practically it's a real pain in the ass
to see or manipulate infinitely small
anything so big squares it is digital
stair-step drawings are exactly the same
thing
it's just a convenient drawing the stair
steps aren't really there
when we convert a digital signal back to
analog the result is also smooth
regardless the bit depth 24 bits or 16
bits rapists
it doesn't matter so does that mean that
the digital bit depth makes no
difference at all
of course them channel 2 here is the
same sine wave input but we quantize
with either down 28 bids on the scope we
still see a nice smooth sine wave on
channel two look very close and you'll
also see a bit more noise that's a clue
if we look at the spectrum of the signal
haha r sine wave is still there
unaffected but the noise level of the
8-bit signal on the second channel
he is much higher and that's the
difference the number of bits makes
that's it
when we digitize the signal first we
sampling the sampling step is perfectly
loses nothing but then we quantize it
and quantization ads noise
the number of bits determines how much
noise and so the level of the noise
floor
what does this dinner quantization noise
sound like let's listen to our a bit
sign
that may have been hard to hear anything
but the tone
let's listen to just the noise after we
watch out the side wave and then bring
the gain up a bit because the noise is
quite those of you who have used analog
recording equipment may have just got to
yourselves
my goodness that sounds like tape hiss
well it doesn't just sound like take
this it acts like it too
and if we use a Gaussian did that it's
mathematically equivalent in every way
it is tape it's intuitively that means
that we can measure tape this in dusty
noise for magnetic audio tape in fits
instead of decibels in order to put
things in a digital perspective
compact cassettes for those of you who
are old enough to remember them could
reach as deep as 9 bits in perfect
conditions though 526 was more typical
especially if it was a recording made on
the tape deck
that's right your mix tapes were only
about six bits deep
if you're lucky the very best
professional open reel tape used in
studios could barely hit any guesses
13 bits with advanced noise reduction
and that's why I seeing
ddd on a compact disc used to be such a
big behind deal
I keep saying that I'm quantizing with
dinner
so what is doing exactly and more
importantly what does it do
the simple way to quantize the signal is
to choose the digital amplitude value
closest to the original analog a pretty
obvious right
unfortunately the exact noise you get
from this simple quantization scheme
depends somewhat on the input signal so
we make it noise that's inconsistent or
causes distortion or is undesirable in
some other way to there is specially
constructed noise that substitutes for
the noise produced by simple
quantization if it doesn't drown out or
mask quantization noise
it actually replaces it with noise
characteristics of our choosing that
aren't influenced by the important
let's watch what better dance the signal
generator has too much noise for this
test so will produce a mathematically
perfect sine wave on the thinkpad and
quantize it to eight bits with every we
see a nice sine wave on the waveform
display and output scope and once the
analog spectrum analyzer catches up a
clean frequency peak with a uniform
noise floor
on both special displays just like
before
again this is with it now
I turned everything off the quantization
noise that did it spread out into a nice
flat noise floor piles up into harmonic
distortion makes the noise floor is
lower but the level of distortion
becomes non zero and the distortion
peaks higher than the differing noise
gate at eight bits
this effect is exaggerating at 16 bits
even without dinner
harmonic distortion is going to be so
low as to be completely normal
still we can use together to eliminate
it completely
if we so choose turning the dinner off
again for a moment you'll notice that
the absolute level of distortion from
under quantization stays approximately
constant regardless of the input
amplitude but when the signal level
drops below half a bit
everything quantizes 20 in a sense
everything quantizing 20 is just one
hundred percent distortion either
eliminate this distortion - we really
able to gather and there's our signal
back at one-quarter bit with our nice
flat noise the noise floor doesn't have
to be flat
there is noise of our choosing so let's
choose a noise as in offensive and
difficult to notice as possible are
hearing is most sensitive in the mid
range from two kilohertz 24 colors so
that's where background noise is going
to be the most obvious we can shake
differing noise away from sensitive
frequencies
- we're hearing is less sensitive
usually the highest frequencies 16-bit
dithering noise is normally much too
quiet to hear it all right let's listen
to our noise shaping example again with
the game brought way up
yeah
lastly different quantization noise is
higher power over all been under the
quantization noise even when it sounds
quieter and you can see that on the vu
meter during passages of your silence
but there isn't only an on or off choice
we can reduce the dinners power to
balance less noise against a bit of
distortion to minimize the overall
effect will also modulate the input
signal like this to show how a very
important effects the quantization noise
at full during power the noises uniform
constant and featureless just like we
expect as we reduce the dinners power
the input increasingly affects the
amplitude and the character of the
quantization noise
shaped dinner behaves similarly but
noise shaping lens one more nice
advantage to make a long story short it
can use a somewhat lower dinner power
before the input has as much effect on
the output
despite all the time i just spent on
dinner
we're talking about differences that
start a hundred decibels and more below
full scale
maybe if the CD had been 14 bits as
originally designed there might be more
important made at 16 bits really it's
mostly a wash
you can think of dinner as an insurance
policy that gives several extra decibels
of dynamic range
just in case the simple fact is though
no one ever ruined a great recording by
not during the final master
we've been using sideways through the
obvious choice when what we want to see
is a systems behavior at a given
isolated frequency
now let's look at something a bit more
complex what should we expect to happen
when i change the input to a square way
the input scope confirms our 1 kilohertz
square wave the output scope shows
exactly what it should
what is a square wave really well we can
say it's a waveform that's some positive
value for half a cycle and then
transitions instantaneously to a
negative value for the other half but
that doesn't really tell us anything
useful about how this input becomes this
output then we remember that any
waveform is also the sum of discrete
frequencies and a square wave is a
particularly simple some fundamental and
an infinite series of odd harmonics some
of them all up
you get a square wave at first glance
that doesn't seem very useful either you
have to sum up an infinite number of
harmonics to be the answer but we don't
have an infinite number of harmonics
we're using a quite sharp anti-aliasing
filter that cuts off right above 20
kilohertz
so our signal is band limited which
means we get this
and that's exactly what we see on the
output scope the rippling you see around
sharp edges in a band limited signal is
called the Gibbs effect
it happens whenever you slice off part
of the frequency domain in the middle of
non zero energy
the usual rule of thumb you'll hear is
sharper the cut off the stronger the
rippling which is approximately true but
we have to be careful how we think about
it
for example what would you expect are
quite sharp and the aliasing filter to
do if i run our signal through a second
time
yeah
aside from adding a few fractional
cycles of the lack the answer is nothing
at all
the signal is already been limited and
limiting it again
doesn't do anything a second pass .
remove frequencies that we already
removed and that's important
people tend to think of the ripples as a
kind of artifact that's added by
anti-aliasing and anti imaging filters
implying that the ripples get worse each
time the signal passes through
we can see in this case that didn't
happen so was it really the filter that
added the rebels the first time through
no not really it's a subtle distinction
but gives effect ripples aren't added by
filters
they're just part of our band limited
signal is even if we synthetically
construct what looks like a perfect
digital square wave
it's still limited to the channel
bandwidth remember the stair step
representation is misleading
what we really have here are
instantaneous sample points and only one
band limited signal fits those points
all we did when we do our apparently
perfect square wave was line up the
sample points just right so it appeared
that there were no ripples if we played
connect the dots but the original band
limited signal
complete with ripples was still there
and that leads us to one more important
. you've probably heard that the timing
precision of digital signal is limited
by its sample rate
put another way that digital signals
can't represent anything that falls
between the samples implying that
impulses or fast attacks have to align
exactly with a sample or the time and
gets mangled or they just disappeared
at this point we can easily see why
that's well again our input signals are
badly and digital signals are samples
not Stairsteps not connect the dots
we most certainly can for example with
the rising edge of our band limited
square wave anywhere we want
between samples it's represented
perfectly
and it's reconstructed perfectly
yeah
just like in the previous episode we've
covered a broad range of topics and yet
barely scratched the surface of each one
if anything my sins of omission or
greater this time around but this is a
good stopping point
or maybe a good starting point dig
deeper experiment
I chose my demos very carefully to be
simple and give clear results you can
reproduce every one of them on your own
if you like but let's face it sometimes
we learn the most about a 50 toy by
breaking it open and studying all the
pieces that fall out and that's okay
where engineers play with the demo
parameters pack up the code set up
alternate experiments the source code
for everything including a little push
button demo application is up at xiph.org. in
the course of experimentation
you're likely to run into something that
you didn't expect and can't explain
don't worry. like earlier snark aside
wikipedia is fantastic for exactly this
kind of casual research and if you're
really serious about understanding
signals, several universities have
advanced materials online such as the
six double O 3 and six double 07 signals
and systems modules at MIT
OpenCourseWare and of course there's
always the community here at xiph.org
digging deeper or not I am out of
coffee, so until next time, happy hacking