SECOND YEAR COURSE AUTUMN
PERCEPTION
Hearing Lecture notes (1): Introductory Hearing
1. What is hearing for ?
* (i) Indicate direction of sound sources (better than eyes since omni-directional,
no eye-lids; but poorer resolution of direction).
* (ii) Recognise the identity and content of a sound source (such as speech
or music or a car).
* (iii) Give information on the nature of the environment via echoes, reverberation
(normal room, cathedral, open field).
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2. Waveforms and Frequency Analysis
Sound is a change in the pressure of the air. The waveform of any
sound shows how the pressure changes over time. The eardrum moves
in response to changes in pressure.
Any waveform shape can be produced by adding together sine waves
of appropriate frequencies and amplitudes. The amplitudes (and phases)
of the sine waves give the spectrum of the sound. The spectrum of
a sine wave is a single point at the frequency of the sine wave. The spectrum
of white noise is a line covering all frequencies.
The cochlea breaks the waveform at the ear down into its component
sine waves - frequency analysis. Hair cells in the cochlea
respond to these component frequencies. This process of frequency analysis
is impaired in sensori-neural hearing loss. It cannot be compensated
for by a conventional hearing aid.
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3. Why does the auditory system analyse sound
by frequency ?
Some animals do not analyse sound by frequency, but simply transmit the
pressure waveform at the ear directly. We could do this by having hair cells
on the eardrum. But instead we have an elaborate system to analyse sound
into its frequency components. We do this because, since almost all sounds
are structured in frequency, we can detect them, especially in the presence
of other sounds, more easily by "looking" at the spectrum than
at the waveform.
In the six panels below, the left-hand column shows plots of the waveform
of a sound - the way that pressure changes over time. The right-hand column
shows the spectrum of the sound - how much of each sine-wave you have to
add together in order to make that particular waveform.
The upper panel is a sine wave tone with a frequency of 1000 Hz. A sine
wave has energy at just one frequency, so the spectrum is just one point.
waveform ----------------------------------spectrum
The middle panel is white noise (like the sound of a waterfall). White noise
has equal energy at all frequencies, so the spectrum is a horizontal
line.
The lower panel is the sine tone added to the noise. The spectrum of the
sum is just the sum of the spectra of the two components.
Notice that you can see the tone very easily in the spectrum, but it is
completely obscured by the noise in the waveform.
Click on the icon to hear Noise then Noise+Tone twice 
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4. Sine waves
A sine wave has three properties which appear in the basic equation:
p(t) = a* sin(2 pi ft +phase)
(i) frequency (f) - measured in Hertz (Hz), cycles per second.
Click on the icon to hear a 500 Hz , a 1000 Hz and a 4000 Hz sine wave
(ii) amplitude (a) - is a measure of the pressure change of
a sound. It is usually measured in decibels (dB) relative to another sound;
the dB scale is a logarithmic scale : if we have two sounds p1 and
p2, then p1 is 20*log10(p1/p2) dB greater than p2. Doubling pressure (amplitude)
gives on increase of 6dB: 20 * log10(2/1) = 20 * 0.3 = 6.
Amplitude squared is proportional to the energy, or level, or intensity
(I) of a sound. The decibel difference between two sounds can also be expressed
in terms of intensity changes: 10*log10(I1/I2). Doubling intensity gives
an increase of 3dB (10 * 0.3). The just noticeable difference (jnd)
in intensity between two sounds is about 1dB.
(iii) phase (phase) - measured in degrees or radians, indicates
the relative time of a wave.
The sine wave shown above has an amplitude of 1, a frequency of 1000 Hz,
and it starts in zero sine phase, phase = 0.
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5. Complex periodic sounds
A sound which has more than one (sine-wave) frequency component is a complex
sound. A periodic sound is one which repeats itself at regular intervals.
A sine wave is a simple periodic sound. Musical instruments or the voice
produce complex periodic sounds. They have a spectrum consisting of a series
of harmonics. Each harmonic is a sine wave that has a frequency that
is an integer multiple of the fundamental frequency.
For example, the note 'A' played by the oboe to tune the orchestra has a
fundamental frequency of 440 Hz, giving harmonics at 440, 880, 1320, 1760,
2200, 2640, etc. If the oboe played a higher pitch, the fundamental
frequency (and so all the harmonic frequencies of the note would be higher.
The period of a complex sound is 1/fundamental frequency (in this
case 1/440 = 0.0023s = 2.3ms). A different instrument, with a different
timbre, playing the same pitch as the oboe, would have harmonics
at the same frequencies, but the harmonics would have different relative
amplitudes. The overall timbre of a natural instrument is partly deptermined
by the relative amplitudes of the harmonics, but the attack of the note
is also important. Different harmonics start at different times in different
instruments, and the rate at which they start also differs markedly across
instruments. Cheap synthesisers cannot imitate the attack, and so they do
not make very lifelike sounds. Expensive synthesisers (like Yamaha's Clavinova)
store the whole note including the attack and so sound very realistic.
Here is one second of the waveform and also the spectrum of a complex periodic
sound consisting of the first four harmonics of a fundamental of 100 Hz.
Notice that there are 100 cycles of the waveform in 1s, and all the frequency
components are integer multiples of 100 Hz.
Here is a sound with the same period, but a different timbre. Notice that
the waveform has a different shape, but the same period. The change in timbre
is produced by making the higher harmonics lower in amplitude.
We can also change the shape of the waveform by changing the relative phase
of the different frequencies. In this example four components were all in
sine phase, in the next example the odd harmonics are in sine phase and
the even in cosine phase. This change produces very little change in timbre.
Click on the icon to hear these three sounds in order
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6. Linearity
Most studies of the auditory system have used sine waves. If we know how
a system responds to sine waves, then we can predict exactly how it will
behave to complex waves (which are made up of sine waves), provided that
the system is linear.
* The output of a linear system to the sum of two inputs, is equal to the
sum of its outputs to the two inputs separately.
* Equivalently, if you double the input to a linear system, then you double
the output.
* A linear system can only output frequencies that are present on the input,
non-linear systems always add extra frequency components.
The filters we describe below are linear. The auditory system is only linear
to a first approximation.
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7. Filters
A filter lets through some frequencies but not others. A treble control
acts as a low-pass filter, letting less of the high frequencies through
as you turn the treble down. A bass control acts as a high-pass filter,
letting less of the low frequencies through as you turn the bass down. A
band-pass filter only lets through frequencies that fall within some
range. A slider on a graphic equalizer controls the output level of a band-pass
filter. In analysing sound into its frequency components, the ear acts like
a set of band-pass filters.
We can represent the action of a filter with a diagram like a spectrum which
shows by how much each frequency is attenuated (or reduced in amplitude)
when it passes through the filter.
Input sound
Filter
Output sound
Click on the icon to hear the unfiltered and the filtered sounds
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8. Resonance
A resonant system acts like a band-pass filter, responding to a narrow range
of frequencies. Examples are: a tuning fork, a string of a harp or piano,
a swing. Helmholtz was almost right in thinking that the ear consisted of
a series of resonators - like a grand-piano with the sustaining pedal held
down. Here is what happens when a complex sound is passed through a sharply-
tuned band-pass filter. Notice that a complex wave goes in, but a sine wave
comes out. Each part of the basilar membrane acts like a band-pass filter
tuned to a different frequency.
Input sound
Filter
Output sound
Click on the icon to hear the unfiltered and the filtered sounds twice
What you should know.
- You should understand the meaning of all the terms shown in italics.
- You should also be able to explain all the diagrams in this handout.
If you do not understand any of the terms or diagrams, first try asking
someone else in the class who you think might.
If you still don't, then ask me either in a lecture, after a lecture or
in my office.
Chris Darwin
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