# PowerPoint Presentation - Fundamentals of Digital Audio by dyr60218

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```									Fundamentals of Digital Audio
The Central Problem
   Sound waves consist of air pressure changes
   This is what we see in an oscilloscope view:
changes in air pressure over time
The Central Problem
   Waves in nature, including sound waves,
are continuous:
Between any two
points on the curve,
no matter how close
together they are,
there are an infinite
number of points
The Central Problem
   Analog audio (vinyl, tape, analog synths, etc.)
involves the creation or imitation of a continuous
wave.
   Computers cannot represent continuity (or
infinity).
   Computers can only deal with discrete values.
   Digital technology is based on converting
continuous values to discrete values.
Digital Conversion
   The instantaneous amplitude of a continuous wave is
measured (sampled) regularly. The measurement values,
samples, may be stored in a digital system.
Digital Conversion
   The instantaneous amplitude of a continuous wave is
measured (sampled) regularly. The measurement values,
samples, may be stored in a digital system.

0.9998 1.0 0.9998
0.9993             0.9993
0.9986                      0.9986
0.9975                            0.9975
0.9961                                 0.9961

0.9945                                              0.9945

0.9925                                                    0.9925
Digital Conversion
    The amplitude of a continuous wave is measured
(sampled) regularly. The measurement values, samples,
may be stored in a digital system.

[ 0.9925, 0.9945, 0.9961, 0.9975, 0.9986, 0.9993, 0.9998, 1.0, 0.9998, 0.9993, 0.9986, 0.9975, 0.9961, 0.9945, 0.9925 ]
Digital Audio
   Digital representation of audio is analogous to
cinema representation of motion.
   We know that “moving pictures” are not really
moving; cinema is simply a series of pictures of
motion, sampled and projected fast enough that
the effect is that of apparent motion.
   With digital audio, if a sound is sampled often
enough, the effect is apparent continuity when the
samples are played back.
Digital Audio
   Con:
–   It is, at best, only an approximation of the wave
   Pros:
–   Significantly lower background noise levels
–   Sounds are more reliably stored and duplicated
–   Sounds are easier to manipulate:
Rather than worry about how to change the shape of a wave,
engineers need only perform appropriate numerical operations.
e.g., changing the volume level of a digital audio file is simply a
matter of multiplication: each sample value is multiplied by a value
that raises or lowers it by a certain percentage.
Digital Audio
 The theory behind digital representation has
existed since the 1920s.
 It wasn’t until the 1950s that technology
caught up to the theory, and it was possible to
implement digital audio.
Digital Audio
   Bell Labs produced the first digital audio synthesis in
the 1950s.
   For computer synthesis, a series of samples was
calculated and stored in a wavetable.
   The wavetable described, in connect-the-dots fashion,
the shape of a wave (i.e., its timbre).
   Reading through the wavetable at different rates
(skipping every n samples, the sampling increment)
allowed different pitches to be created.
   Audio was produced by feeding the samples that were
to be audified through a digital to analog converter
(DAC).
Digital Audio
   Contemporary computer sound cards often contain a
set of wavetable sounds.
   The function is the same: a library of samples
describing different waveforms.
   They are triggered by MIDI commands. (These will
be covered fully in a few weeks.) For example, a
given note number will translate to the table being
read at a certain sampling increment to produce the
desired pitch.
Digital Audio
 Digital recording became possible in the
1970s.
 Voltage input from a microphone is fed to an
analog to digital converter (ADC), which
stores the signal as a series of samples.
 The samples can then be sent through a DAC
for playback.
Digital Audio
 Thus, the ADC produces a “dehydrated”
version of the audio.
 The DAC then “rehydrates” the audio for
playback.
(Gareth Loy, Musimathics v. 2)
Characteristics of Digital Audio
   With digital audio, we are concerned with
two measurements:
–   Sampling rate
–   Quantization
   With these measurements, we can describe
how well a digitized audio file represents
the analog original.
Sampling Rate
   This number tells us how often an audio signal is sampled,
the number of samples per second.
   The more often an audio signal is sampled, the better it is
represented in discrete form:
Sampling Rate
   This number tells us how often an audio signal is sampled,
the number of samples per second.
   The more often an audio signal is sampled, the better it is
represented in discrete form:
Sampling Rate
   This number tells us how often an audio signal is sampled,
the number of samples per second.
   The more often an audio signal is sampled, the better it is
represented in discrete form:

Of course, this
staircase-shaped
wave needs to be
smoothed.
This process will be
covered during the
discussion on
filtering.
Sampling Rate
 So we want to sample an audio wave every
so often.
The question is: how “often” is “often
enough”?
 Harry Nyquist of Bell Labs addressed this
question in a 1925 paper concerning
telegraph signals.
Sampling Rate
   Given that a wave will be smoothed by a
subsequent filtering process, it is sufficient
to sample both its peak and its trough:
Sampling Rate

   Thus, we have the sampling theorem
(also called the Nyquist theorem):
To represent digitally a signal containing frequency
components up to X Hz, it is necessary to use a
sampling rate of at least 2X samples per second.

 Conversely, the maximum frequency
contained in a signal sampled at a rate of SR
is SR/2 Hz.
 The frequency SR/2 is also termed the
Nyquist frequency.
Sampling Rate

   In theory, since the maximum audible
frequency is 20 kHz, a sampling rate of
40 kHz would be sufficient to re-create a
signal containing all audible frequencies.
Sampling Rate
   For most frequencies, we will oversample
(the audio frequency is below the Nyquist
frequency):
Sampling Rate
   For most frequencies, we will oversample
(the audio frequency is below the Nyquist
frequency):
Sampling Rate
   If we sample at precisely the Nyquist frequency,
our critically sampled signal runs the risk of
missing peaks and troughs:

or

   This problem is also addressed by filtering.
Sampling Rate
   More serious is the problem of undersampling a
frequency greater than the Nyquist frequency:
Audio                                    RESULT:
signal at
30 kHz,
sampled
at 40 kHz
Sampling Rate
   More serious is the problem of undersampling a
frequency greater than the Nyquist frequency:
Audio                                           RESULT:
signal at                                       The frequency is
30 kHz,                                         misrepresented
sampled                                         at 10 kHz, at
at 40 kHz                                       reverse phase

Misrepresented frequencies are termed aliases.
Sampling Rate
   In general, if a frequency, F, sampled at a
sampling rate of SR, exceeds the Nyquist
frequency, that frequency will alias to a
frequency of:
- (SR - F)

Sampling Rate
   It is useful to illustrate sampled frequencies on a polar diagram,
with 0 Hz at 3:00 and the Nyquist frequency at 9:00:

f     The upper half of the circle
represents frequencies from 0 Hz
to the Nyquist frequency

Nyquist                     0 Hz

The lower half of the circle represents
-f    negative frequencies from 0 Hz to the
Nyquist frequency (there is no distinction
Any audio frequency above the
in a digital audio system between ±NF)
Nyquist frequency will alias to a
frequency shown on the bottom
half of the circle, a negative
frequency between 0 Hz and the                  Frequencies above the
Nyquist frequency.                              Nyquist frequency do not
exist in a digital audio system
Sampling Rate
   In the recording process, filters are used to remove
all frequencies above the Nyquist frequency
before the audio signal is sampled.
   This step is critical since aliases cannot be
removed later.
   Provided these frequencies are not in the sampled
signal, the signal may be sampled and later
reconverted to audio with no loss of frequency
information.
Sampling Rate
   The sampling rate for audio CDs is 44.1 kHz.
   The origin of this rate lies in video formats.
   When digital audio recording began, audio tape was
not capable of handling the density of digital signals.
   The first digital masters were stored on video as a
psuedo video signal, in which binary values of 1 and
0 were stored as video levels of black and white.
Sampling Rate
Video is drawn left to right,               O
starting from the top of the                E
O
screen and moving down.                     E
O
First the odd numbered                      E
O
lines are drawn, then the                   E
O
even numbered lines.                        E

Each video frame has two
fields: the odd field and
the even field.                 Frame n,   Frame n,   Frame n+1, Frame n+1, Frame n+2,
odd        even       odd        even       odd

each other on the video
tape.
Sampling Rate
   There are two video formats:
–   525 lines, 30 frames per second (USA)
Minus 35 blank lines, leaving 490 lines per frame
60 fields per second, 245 lines per field
–   625 lines, 25 frames per second (European)
Minus 37 blank lines, leaving 588 lines per frame
50 fields per second, 294 lines per field
   Three samples could be stored on each line, allowing:
60 x 245 x 3 = 44,100 samples per second
or
50 x 294 x 3 = 44,100 samples per second
   44.1 kHz remains the standard sampling rate for CD
audio.
Quantization
   This has a few names:
–   Sample size
–   Bit depth
–   Word size
   The term “quantization” takes its origin from
quantum physics:
–   Electrons orbit an atom’s nucleus in one of a number of
well-defined layers;
–   An electron may be knocked from one layer to another,
but it can never stay between one of the layers.
Quantization
   In the discussion of sampling rate, we only considered how often
the amplitude of the wave was measured.
   We did not discuss how accurate these measurements were.
   The effectiveness of any measurement depends on the precision
of our ruler. (Measuring the thickness of something with many
small indentations with a ruler only marking feet will probably
not give a very accurate measurement; we have to estimate many
measurements.)
   Just as there are limits to how often we can sample, there are
limits to the resolution of our ruler.
Quantization
   Like all numbers stored in computers, the amplitude values are stored as
binary numbers.
   The value that gets stored is the closest available binary number - akin to
the nearest marking on a ruler.
   The accuracy of our measurement depends on how many bits we have to
represent these values.
   Clearly, the more bits we have, the finer the resolution of our ruler.

2 bits

Each change of bit
represents a change in
voltage level
Quantization
   Like all numbers stored in computers, the amplitude values are stored as
binary numbers.
   The value that gets stored is the closest available binary number - akin to
the nearest marking on a ruler.
   The accuracy of our measurement depends on how many bits we have to
represent these values.
   Clearly, the more bits we have, the finer the resolution of our ruler.

3 bits

Each change of bit
represents a change in
voltage level
Quantization
   Like all numbers stored in computers, the amplitude values are stored as
binary numbers.
   The value that gets stored is the closest available binary number - akin to the
nearest marking on a ruler.
   The accuracy of our measurement depends on how many bits we have to
represent these values.
   Clearly, the more bits we have, the finer the resolution of our ruler.

4 bits

Each change of bit
represents a change in
voltage level
Quantization
   CD audio uses 16-bit quantization.
Quantization
 While aliasing is eliminated if our signal
contains no frequencies above the Nyquist
frequency, quantization error can never be
completely eliminated.
 Every sample is within a margin of error
that is half the quantization level (the
voltage change represented by the least
significant bit).
Quantization
   For a sine wave signal represented with n
bits, the signal to error ratio is:
S/E (dB) = 6.02n + 1.76
   The problem is that low-level signals do not
use all available bits, and therefore the error
level is greater.
Quantization
   While quantization error may be masked at high
audio levels, it can become audible at low levels:

Worst case: a sine
wave fluctuating
within one
quantization
increment is stored
as a square wave

Thus, unlike the constant hissing noise of analog recordings,
quantization error is correlated with the signal, and is thus a
type of distortion, rather than noise.
Quantization
   The problem of quantization distortion is
   Dither is low-level noise added to the audio signal
before it is sampled.

Low level
audio signal
with dither
Quantization
 Dither adds random errors to the signal,
therefore the quantization results in added
noise, rather than distortion.
 The noise is a constant factor, not correlated
with the signal like quantization distortion.
 The result is a noisy signal, rather than a
signal broken up by distortion.
Quantization
 The auditory system averages the signal at
all times. We do not hear individual
samples.
 With dither, this averaging alows the
musical signal to co-exist with the noise,
rather than be temporarily eliminated due to
distortion.
Quantization
   Dither allows resolution below the least significant
quantization bit.
   Without dither, digital recordings would be far less
satisfactory than analog recordings - a plucked guitar
string, for example, fades into something close to a
sine tone. Without dither, a guitar sound would
gradually turn into the sound of a square wave.
   With dither, there is significantly less noise in digital
recordings than in analog recordings.
Quantization and Sampling Rate
 The sampling rate determines the signal’s
frequency content.
 The number of quantization bits determines
the amount of quantization error.
Size of Audio Files

44,100       x 2         x 2           x 60          ≈ 10 MB/minute
samples     bytes per    channels       seconds
per second    sample     (for stereo    per minute
(16 bits)     audio)

```
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