1. Fundamentals of Multimedia
2nd Edition 2014
Ze-Nian Li
Mark S. Drew
Jiangchuan Liu
Chapter 6 :
Basics of Digital Audio

2. Sound Sound comprises the spoken word, voices, music and even noise.
It is a complex relationship involving a vibrating object (sound source), a transmission medium (usually air), a receiver (ear) and a perceptor (brain). Example banging drum.
As the sound vibrates it bumps into molecules of the surrounding medium causing pressure waves to travel away from the source in all directions

3. Waveforms
Sound waves are manifest as waveforms
A waveform that repeats itself at regular intervals is called a periodic waveform
Waveforms that do not exhibit regularity are called noise
The unit of regularity is called a cycle
This is known as Hertz (or Hz) after Heinrich Hertz
One cycle = 1 Hz
Sometimes written as kHz or kiloHertz (1 kHz = 1000 Hz)

4. Waveforms
Time for one cycle
distance
along wave
Cycle

5. The characteristics of sound waves
Sound is described in terms of two characteristics:
Frequency
Amplitude (or loudness)
the rate at which sound is measured
Number of cycles per second or Hertz (Hz)
Determines the pitch of the sound as heard by our ears
The higher frequency, the clearer and sharper the soundthe higher pitch of sound

6. The characteristics of sound waves
Amplitude
Sound’s intensity or loudness
The louder the sound, the larger amplitude.
In addition, all sounds have a duration and successive musical sounds is called rhythm

7. The characteristics of sound waves
Time for one cycle
Amplitude
pitch
distance
along wave
Cycle

8. Example waveforms
Piano
flute
drum

9. Capture and playback of digital audio
Air pressure
variations
DAC
Converts back into voltage
Digital to
Analogue
Converter
Captured via
microphone
Air pressure
variations
ADC
Signal is
converted into
binary (discrete form)
Analogue
to Digital
Converter

10. 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

11. 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

12. 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.

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

14. 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

15. Digital Conversion
The amplitude of a continuous wave is measured (sampled) regularly. The measurement values, samples, may be stored in a digital system.
[ , , , , , , , 1.0, , , , , , , ]

16. 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.

17. Digital Audio Con: Pros:
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.

18. 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.

19. 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).

20. 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. For example, a given note number will translate to the table being read at a certain sampling increment to produce the desired pitch.

21. 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.

22. 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)

23. The Analogue to Digital Converter (ADC)
An ADC is a device that converts analogue signals into digital signals
An analogue signal is a continuous value
It can have any single value on an infinite scale
A digital signal is a discrete value
It has a finite value (usually an integer)
An ADC is synchronised to some clock

24. 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.

25. 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:

26. 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:

27. 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.

28. 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.

29. 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:

30. 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.

31. 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.

32. Sampling Rate
For most frequencies, we will oversample (the audio frequency is below the Nyquist frequency):

33. Sampling Rate
For most frequencies, we will oversample (the audio frequency is below the Nyquist frequency):

34. 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.

35. 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:

36. 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.

37. 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
Suppose the Sampling Frequency is 1.5 times the True Frequency. What is the Alias Frequency?

38. 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 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

39. 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.