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Who's work on the spread of heat energy through solid objects help scientists understand complex sounds?
Joseph Fourier

What did Fourier propose and why is this is this useful in real life?
 Any signal, no matter how complex, could be described as the sum of a family of simple sine waves.
 Sine waves are cyclical and many physical events in the world depend on periodic, repeating processes.
 We can thus decompose complex signals into a number of parts that depend only on 3 parameters.

Why is the Fourier analysis so useful? (2)
 1. it provides compact representation of complex signal  like an alphabet, sine waves encode properties of the environment using compact representation so that parameters of that representation can be sent off to brain rather than transmitting the whole signal.
 2. we have opportunity understand individual processes underlying complex behaviour (the more components we have, the more accurate the model)
 F Analysis provides good description of perceptual processes because our sensory apparatus evolved to encode statistical properties of the world around us.

Amplitude/frequency __ represents a __ __. A pure tone is represented on an __ __ by a __ __ that plots the __ of that sine for that single __.
 spectra
 complex tone
 amplitude spectrum
 single line
 amplitude
 frequency

Give a classic example of how Fourier decomposition works to give a good approximation of a wave.
 Square wave
 Special because it is composed of family of sine components where sines have odd harmonics (3F, 5F, 7F)
 and amplitude of each component decreases exponentially
 While amplitude spectrum continues to infinity, we can have reasonable approximation of square wave signal with a small number of components.

Remember that most sounds do not have __ Fourier series. They generally plot a changing function of amplitude across multiple frequency values. Eg white noise in radio is just a flat line on a spectrum.
What can we call this?
 discrete
 Continuous spectra

What are the limitations of strict Fourier analysis as a biologically realistic model? MAKE SURE YOU ASK.
 Basically, sine waves are very simple models.
 Sensory systems process signals are understood to use space and time limited frequencey bands, where a range of frequency signals are considered within each band, rather than being limited to a single discrete frequency value. (By analogy, Fourier's alphabets are replaced by a dictionary of words that provide better descriptions of how real signals vary). So F Analysis is useful as a scaffold for understanding how sensory systems work but it is not a biologically realistic model.
 Difficult to present a genuinely pure tone > amplitude energy is 'splattered' across frequency space
 A simple click sound needs a large number of sine waves that add together in destructive interference.
 ASK!!! MAKE SURE YOU ARE CLEAR WITH WHAT THE HANDOUT IS SAYING!

Briefly outline what the linear systems approach is.
 Many real systems are (quasi) linear
 Easy to characterise all possible inputoutput values based on limited measurements
 Do this by measuring the impulse response function (measure the output to a single impulse input and characterise the behaviour of the whole system)
 So, can understand human hearing by measuring auditory system reponsds to pure tones of range of frequencies
 Useful to think of the system as a set of linear systems (filters)

What are the 3 rules for a system to be linear?
 1. Homogeneity: increase input (x2) will lead to corresponding increase in output (x2)
 2. Additivity: response of 2 inputs together is sum of each input presented alone
 3. Shift invariance: response to an input at one point in time is same as at another point in time

What is another important property of linear systems that is helpful for undestanding human auditory systems?
 Input is sine then output is a sine with the same frequency
 the only thing that can change are amplitude and phase
 So, for our case, we are looking at how auditory system, as a set of linear systems, change the amplitude of the input.
 THEREFORE: can characterise auditory system by measuring amplitude of sine wave inputs of a range of different frequencies

Using this linear systems approach, we can think of the auditory system as a set of __ __ and therefore composed of a number of __ __ that....
 linear systems
 individual filters
 that only let through particular frequencies of sounds, thus determining the amplitude of the output.

Filters which only let through a particular range of sounds is called a what?
 Band pass filters
 only let through paricular range of frequencies relative tot he centre frequency of the filter.

What is one basic way we can understand how linear system will change the amplitude of hearing?
 Measure the minimum audible pressure (MAP) at ear frum for a range of frequencies.
 Detectable response amplitude changes with frequency  traces out a contour which shows we are good at 15kHz but bad at lower or higher freq.

Describe a classic experiment which attempted to find out the filtering properties of the human auditory system.
 Fletcher (1940)
 Used technique of masking.
 Measured the frequency range over which masking signal (white noise) interfered with detection of a pure tone.
 He used band pass filtered noise as a masker.
 Measured detection perfroamcne when he varied the frequency bandwidth of the noise.
 Result: as the badnwidth of the noise increased, performance got worse, but after a certain bandwidth (critical band), adding more noise didn't affect performance.
 From this he was able to estimate width of perceptual auditory filter.

How has Fletcher's approach been expanded?
 Psychophysical tuning curves
 eg. Vogten, 1974
 to estimate width of a range of different auditory frequency channels
 masking stimulus filtered to cover only a small frequency range
 this is done for several frequencies
 resulting in an inverted shape of auditory filters, made up of smaller filters at different channels/frequencies.

What is a limitation of psychophysical tuning curves?
 It assumes that the listener's performance depends on using a single bandpass frequency channel at all times.
 In reality, off frequency listening happens whereby there will be overlap between perceptual channels so they may have used neighbouring channels to perform the task.

What are the practical applications of this linear systems inspired audio psychophyics?
 Assessing hearing function: minimum audible pressure levels measured at different frequencies  compared to performance of normal individuals (Hearing theshold level  dB HL). Hearing aids can be controlled so that it only amplifies frequenceis where individual has deficit.
 Compression for music: masking properties help guide approaches such as mp3 compression. mp3 compression throws away frequencies that are outside hearing range, as well as those which are inaudible because of masking by higher amplitude nearby frequencies.

Protecting the ears. Loud sounds can...
 rupture ear drum
 break ossicles
 tear basilar membrane
 shear hairs off hair cells

