Fundamentals of Digital Audio

1
Fundamentals of Digital Audio
2
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

3
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
4
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.

5
Digital Conversion

• The instantaneous amplitude of a continuous wave
  is measured (sampled) regularly. The measurement
  values, samples, may be stored in a digital
  system.

6
Digital Conversion

• The instantaneous amplitude of a continuous wave
  is measured (sampled) regularly. The measurement
  values, samples, may be stored in a digital
  system.

1.0
0.9998
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
7
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
8
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.

9
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 manipulateRather 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.

10
Digital Audio

• The theory behind digital representation has
  existed since the 1920s.
• It wasnt until the 1950s that technology caught
  up to the theory, and it was possible to
  implement digital audio.

11
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).

12
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.

13
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.

14
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)

15
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.

16
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

17
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

18
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.
19
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.

20
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

21
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.

22
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.

23
Sampling Rate

• For most frequencies, we will oversample (the
  audio frequency is below the Nyquist frequency)

24
Sampling Rate

• For most frequencies, we will oversample (the
  audio frequency is below the Nyquist frequency)

25
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.

26
Sampling Rate

• More serious is the problem of undersampling a
  frequency greater than the Nyquist frequency

Audio signal at 30 kHz, sampled at 40 kHz
RESULT
27
Sampling Rate

• More serious is the problem of undersampling a
  frequency greater than the Nyquist frequency

Audio signal at 30 kHz, sampled at 40 kHz
RESULT
The frequency is misrepresented at 10 kHz, at
reverse phase
Misrepresented frequencies are termed aliases.
28
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)

The minus sign indicates that the frequency is in
opposite phase
29
Sampling Rate

• It is useful to illustrate sampled frequencies on
  a polar diagram, with 0 Hz at 300 and the
  Nyquist frequency at 900

f
The upper half of the circle represents
frequencies from 0 Hz to the Nyquist frequency
0 Hz
Nyquist
The lower half of the circle represents negative
frequencies from 0 Hz to the Nyquist frequency
(there is no distinction in a digital audio
system between NF)
-f
Any audio frequency above the Nyquist frequency
will alias to a frequency shown on the bottom
half of the circle, a negative frequency between
0 Hz and the Nyquist frequency.
Frequencies above the Nyquist frequency do not
exist in a digital audio system
30
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.

31
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.

32
Sampling Rate
Video is drawn left to right, starting from the
top of the screen and moving down.
First the odd numbered lines are drawn, then the
even numbered lines.
Each video frame has two fields the odd field
and the even field. The fields are adjacent to
each other on the video tape.
Frame n1, even
Frame n2, odd
Frame n, even
Frame n1, odd
Frame n, odd
33
Sampling Rate

• There are two video formats
• 525 lines, 30 frames per second (USA)Minus 35
  blank lines, leaving 490 lines per frame60
  fields per second, 245 lines per field
• 625 lines, 25 frames per second (European)Minus
  37 blank lines, leaving 588 lines per frame50
  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.

34
Quantization

• This has a few names
• Sample size
• Bit depth
• Word size
• The term quantization takes its origin from
  quantum physics
• Electrons orbit an atoms 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.

35
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.

36
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
37
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
38
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
39
Quantization

• CD audio uses 16-bit quantization.

40
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).

41
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.

42
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.
43
Quantization

• The problem of quantization distortion is
  addressed by dither.
• Dither is low-level noise added to the audio
  signal before it is sampled.

Low level audio signal with dither added
44
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.

45
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.

46
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.

47
Quantization and Sampling Rate

• The sampling rate determines the signals
  frequency content.
• The number of quantization bits determines the
  amount of quantization error.

48
Size of Audio Files
x 2
x 2
x 60
10 MB/minute
44,100
samples per second
bytes per sample(16 bits)
channels(for stereo audio)
secondsper minute