Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6...
Transcript of Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6...
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COMP 546
Lecture 19
Sound 2:frequency analysis
Tues. March 27, 2018
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Speed of Sound
Sound travels at about 340 m/s, or 34 cm/ ms.
(This depends on temperature and other factors)
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Wave equation
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𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝐼𝑎𝑡𝑚 + 𝐼(𝑋, 𝑌, 𝑍, 𝑡)
𝐼(𝑋, 𝑌, 𝑍, 𝑡) is not an arbitrary function.
Rather:
𝜕2
𝜕𝑋2+𝜕2
𝜕𝑌2+𝜕2
𝜕𝑍2𝐼 𝑋, 𝑌, 𝑍, 𝑡 =
1
𝑣2𝜕2
𝜕𝑡2𝐼 𝑋, 𝑌, 𝑍, 𝑡
𝑣 = 340 m/s
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The wave equation + boundary conditions give complicated shadow and reflection effects.
What happens when sound enters the ear ?
4plane wave + single slit sea waves + islands
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Musical sounds
(brief introduction)
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Modes are sin(𝜋
𝐿𝑗𝑥) where 𝐿 is the length of the string, 𝑗 is an integer.
Write one string displacement at t = 0 as sum of sines.
Example: guitar
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Physics says:
where constant 𝑐 depends on physical properties of string(mass density, tension)
𝜔 =𝑐
𝐿
𝐿
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Modes of a vibrating string each have fixed points which reduce the effective length.
𝜔 =𝑐
𝐿
2𝑐
𝐿
3𝑐
𝐿
4𝑐
𝐿
𝐿
2
𝐿
3
𝐿
4𝐿
Physics says:
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𝜔 =𝑐
𝐿
2𝑐
𝐿
3𝑐
𝐿
4𝑐
𝐿
𝜔0
“fundamental” “overtones”(1st harmonic)
The temporal frequency 𝑚 𝜔0 is called the 𝑚-th harmonic.
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For stringed instruments, most of the sound is produced by vibrations of the instrument body (neck, front and back plates). http://www.acs.psu.edu/drussell/guitars/hummingbird.html
The lines in the sketches below are the nodal points. They don't move.
These are vibration modes, not harmonics. The guitar sound is a sum of these modes.
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Difference of two frequencies 𝜔1 and 𝜔2 :
𝑙𝑜𝑔2𝜔2
𝜔1octaves.
e.g. 1 octave is a doubling of frequency.
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(Western) Musical Notes
Each “octave” ABCDEFGA is divided into 12 “semitones”, separated into 1/12 octave.
C-D, D-E, F-G, G-A, A-B are two semitones each
E-F, B-C are one semitone each.
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Q: How many semi-tones are there from 𝜔0 to 𝜔 ?
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Q: How many semi-tones are there from 𝜔0 to 𝜔 ?
A: 12 𝑙𝑜𝑔2𝜔
𝜔0
𝜔0
Fundamental frequency of note
𝜔
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88 fundamental frequencies (Hz) on a keyboard
The fundamental frequencies of successive notes define a geometric progression.
This is different from the harmonics of a vibrating string which define an arithmetic progression.
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Speech Sounds
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What determines speech sounds?
• voiced vs. unvoiced
‘zzzz’ vs. ‘ssss’, ‘vvvv’ vs. ‘ffff’
• articulators (jaw, tongue, lips)
‘aaaa’, ‘eeee’, ‘oooo’, …
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Voiced sounds are produced by “glottal pulses”.
𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑔 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
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Exercise 16 Q7.
𝑔 𝑡 − 𝑡0 = 𝑔 𝑡 ∗ 𝛿(𝑡 − 𝑡0)
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Voiced sounds are produced by “glottal pulses”.
𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑔 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 = 𝑔 𝑡 ∗ 𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝛿 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
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𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑔 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
decrease 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 by increasing tension in vocal cords
increase frequency of pulses≡
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𝐼 𝑡 = 𝑎 𝑡 ∗ 𝑔 𝑡 ∗ 𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝛿 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
Let 𝑎 𝑡 be the impulse response function of the articulators.
(jaw, tongue,lips)
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𝑛𝑝𝑢𝑙𝑠𝑒 −1
𝑛𝑝𝑢𝑙𝑠𝑒 −1
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𝑛𝑝𝑢𝑙𝑠𝑒 −1
𝑛𝑝𝑢𝑙𝑠𝑒 −1
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Oral and nasal cavity have resonant modes of vibration, like air cavity in guitar does.
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Time domain Temporal frequency domain
Peaks are called “formants”
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𝑇𝑔 is the period of the glottal pulse train.
The pulse train has 𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙 pulses in 𝑇 time steps, i.e.
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙 = 𝑇.
Assume that the Fourier transform is taken over 𝑇 samples.
𝐅
𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝛿 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 = ?
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29𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
Assignment 3: Show
𝐅
𝑗=0
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙−1
𝛿 𝑡 − 𝑗 𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 = 𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑚=0
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 −1
𝛿 𝜔 −𝑚 𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙
𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙
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Units of temporal frequency 𝜔
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𝑇𝑔𝑙𝑜𝑡𝑡𝑎𝑙 is the period of the glottal pulse train.
𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙 pulses in 𝑇 time samples.
To convert 𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙 to ‘pulses per second’, we divide 𝑇 (to get
pulses per sample) and then multiply by ‘time samples per second’. High quality audio uses 44,100 samples per second.
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𝑛𝑔𝑙𝑜𝑡𝑡𝑎𝑙 is the fundamental frequency of the voiced sound.
It determines the "pitch".
Adult males : 100-150 Adult females : 150-250 HzChildren: over 250 Hz
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𝜔0 = 100 𝐻𝑧
𝜔0 = 200 𝐻𝑧
glottal pulse spectrum formant spectrum sound spectrum
glottal pulse spectrum “formants” sound spectrum
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Voiced vowel sounds
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Unvoiced sounds noise instead of glottal pulses
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Unvoiced sounds noise instead of glottal pulses
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Flat amplitude spectrum on average ( ‘white noise’)
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Restrict flow of air by moving tongue, lips into contact with the teeth & palate.
Fricatives
- voiced z, v, zh, th (the)
- unvoiced ?
Stops
- voiced b, d, g
- unvoiced ?
Nasals (closed mouth)
- m, n, ng
Consonants
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Restrict flow of air by moving tongue, lips into contact with the teeth & palate.
Fricatives
- voiced z, v, zh, th (the)
- unvoiced s, f, sh, th (theta)
Stops
- voiced b, d, g
- unvoiced p, t, k
Nasals (closed mouth)
- m, n, ng
Consonants
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I did not have time to cover the following slides properly.
I will present them again in lecture 22.
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Spectrogram
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Partition a sound signal into 𝐵 blocks of 𝑇 samples each (i.e. the sound has 𝐵𝑇 samples in total).
Take the Fourier transform of each block.
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Spectrogram
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Partition a sound signal into 𝐵 blocks of 𝑇 samples each (i.e. the sound has 𝐵𝑇 samples in total).
Take the Fourier transform of each block.
Let 𝑏 be the block number, and 𝜔 units be cycles per block.
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Cycles per second (Hz)
Time (samples)
𝜔0 =
![Page 42: Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6 Modes are sin ... Write one string displacement at t = 0 as sum of sines. Example:](https://reader034.fdocuments.net/reader034/viewer/2022050404/5f81f225492e726c4c29fec5/html5/thumbnails/42.jpg)
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e.g. T = 512 samples (12 ms), 𝜔0 = 86 Hz T = 2048 samples (48 ms) 𝜔0 = 21 Hz
![Page 43: Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6 Modes are sin ... Write one string displacement at t = 0 as sum of sines. Example:](https://reader034.fdocuments.net/reader034/viewer/2022050404/5f81f225492e726c4c29fec5/html5/thumbnails/43.jpg)
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e.g. T = 512 samples (12 ms), 𝜔0 = 86 Hz T = 2048 samples (48 ms), 𝜔0 = 21 Hz
You cannot simultaneously localize the frequency and the time. This is a fundamental tradeoff. We have seen it before (recall the Gaussian).
![Page 44: Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6 Modes are sin ... Write one string displacement at t = 0 as sum of sines. Example:](https://reader034.fdocuments.net/reader034/viewer/2022050404/5f81f225492e726c4c29fec5/html5/thumbnails/44.jpg)
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Narrowband
(good frequency resolution, poor temporal resolution … ~50ms)
Wideband
(poor frequency resolution,good temporal resolution)
![Page 45: Sound 2: frequency analysiscim.mcgill.ca/~langer/546/19-sound2-slides.pdf · (brief introduction) 6 Modes are sin ... Write one string displacement at t = 0 as sum of sines. Example:](https://reader034.fdocuments.net/reader034/viewer/2022050404/5f81f225492e726c4c29fec5/html5/thumbnails/45.jpg)
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Examples: Spectrograms of 10 vowel sounds