Basic Acoustics + Digital Signal Processing

1. Basic Acoustics + Digital Signal Processing January 14, 2014

2. Just so you know... • Some ideas for finding consultants: • Kijiji and “couch surfing” • For today: • Part 1: Go through a review of the basics of (analog) acoustics. • Part 2: Converting sound from analog to digital format. • Any questions so far?

3. Part 1: An Acoustic Dichotomy • Acoustically speaking, there are two basic kinds of sounds: • Periodic • = an acoustic pattern which repeats, over time • The “period” is the length of time it takes for the pattern to repeat • Periodic speech sounds = voiced segments + trills 2. Aperiodic • Continuous acoustic energy which does not exhibit a repeating pattern • Aperiodic speech sounds = fricatives

4. The Third Wheel • There are also acoustic transients. • = aperiodic speech sounds which are not continuous • i.e., they are usually very brief • Transient speech sounds: • stop release bursts • clicks • also (potentially) individual pulses in a trill • Let’s look at the acoustic properties of each type of sound in turn…

5. Acoustics: Basics • How is a periodic sound transmitted through the air? • Consider a bilabial trill: Fad Pin Fad

6. What does sound look like? • Air consists of floating air molecules • Normally, the molecules are suspended and evenly spaced apart from each other • What happens when we push on one molecule?

7. What does sound look like? • The force knocks that molecule against its neighbor • The neighbor, in turn, gets knocked against its neighbor • The first molecule bounces back past its initial rest position initial rest position

8. What does sound look like? • The initial force gets transferred on down the line rest position #1 rest position #2 • The first two molecules swing back to meet up with each other again, in between their initial rest positions • Think: bucket brigade

9. Compression Wave • A wave of force travels down the line of molecules • Ultimately: individual molecules vibrate back and forth, around an equilibrium point • The transfer of force sets up what is called a compression wave. • What gets “compressed” is the space between molecules • Check out what happens when we blow something up!

10. Compression Wave area of high pressure (compression) area of low pressure (rarefaction) • Compression waves consist of alternating areas of high and low pressure

11. Pressure Level Meters • Microphones • Have diaphragms, which move back and forth with air pressure variations • Pressure variations are converted into electrical voltage • Ears • Eardrums move back and forth with pressure variations • Amplified by components of middle ear • Eventually converted into neurochemical signals • We experience fluctuations in air pressure as sound

12. Measuring Sound • What if we set up a pressure level meter at one point in the wave? Time pressure level meter

13. Sine Waves • The reading on the pressure level meter will fluctuate between high and low pressure values • In the simplest case, the variations in pressure level will look like a sine wave. pressure time

14. Other Basic Sinewave concepts • Sinewaves are periodic; i.e., they recur over time. • The period is the amount of time it takes for the pattern to repeat itself. • A cycle is one repetition of the acoustic pattern. • The frequency is the number of times, within a given timeframe, that the pattern repeats itself. • Frequency = 1 / period • usually measured in cycles per second, or Hertz • The peakamplitude is the the maximum amount of vertical displacement in the wave • = maximum (or minimum) amount of pressure

15. Waveforms • A waveform plots air pressure on the y axis against time on the x axis.

16. Phase Shift • Even if two sinewaves have the same period and amplitude, they may differ in phase. • Phase essentially describes where in the sinewave cycle the wave begins. • This doesn’t affect the way that we hear the waveform. • Check out: sine waves vs. cosine waves!

17. Complex Waves • It is possible to combine more than one sinewave together into a complex wave. • At any given time, each wave will have some amplitude value. • A1(t1) := Amplitude value of sinewave 1 at time 1 • A2(t1) := Amplitude value of sinewave 2 at time 1 • The amplitude value of the complex wave is the sum of these values. • Ac(t1) = A1 (t1) + A2 (t1)

18. Complex Wave Example • Take waveform 1: • high amplitude • low frequency + • Add waveform 2: • low amplitude • high frequency = • The sum is this complex waveform:

19. A Real-Life Example • 480 Hz tone • 620 Hz tone • the combo = ?

20. Spectra • One way to represent complex waves is with waveforms: • y-axis: air pressure • x-axis: time • Another way to represent a complex wave is with a power spectrum (or spectrum, for short). • Remember, each sinewave has two parameters: • amplitude • frequency • A power spectrum shows: • amplitude on the y-axis • frequency on the x-axis

21. One Way to Look At It • Combining 100 Hz and 1000 Hz sinewaves results in the following complex waveform: amplitude time

22. The Other Way • The same combination of 100 Hz and 1000 Hz sinewaves results in the following power spectrum: amplitude frequency

23. The Third Way • A spectrogram shows how the spectrum of a complex sound changes over time. time frequency 1000 Hz 100 Hz • intensity (related to amplitude) is represented by shading in the z-dimension.

24. Fundamental Frequency • One last point about periodic sounds: • Every complex wave has a fundamental frequency (F0). • = the frequency at which the complex wave pattern repeats itself. • This frequency happens to be the greatest common denominator of the frequencies of the component waves. • Example: greatest common denominator of 100 and 1000 is 100. • GCD of 480 and 620 Hz is 20. • GCD of 600 and 800 Hz is 200, etc.

25. Aperiodic sounds • Not all sounds are periodic • Aperiodic sounds are noisy • Their pressure values vary randomly over time “white noise” • Interestingly: • White noise sounds the same, no matter how fast or slow you play it.

26. Fricatives • Fricatives are aperiodic speech sounds [s] [f]

27. Aperiodic Spectra • The power spectrum of white noise has component frequencies of random amplitude across the board:

28. Aperiodic Spectrogram • In an aperiodic sound, the values of the component frequencies also change randomly over time.

29. Transients • A transient is: • “a sudden pressure fluctuation that is not sustained or repeated over time.” • An ideal transient waveform:

30. A Transient Spectrum • An ideal transient spectrum is perfectly flat:

31. As a matter of fact • Note: white noise and a pure transient are idealizations • We can create them electronically… • But they are not found in pure form in nature. • Transient-like natural sounds include: • Hand clapping • Finger snapping • Drum beats • Tongue clicking

32. Click Waveform some periodic reverberation initial impulse

33. Click Spectrum • Reverberation emphasizes some frequencies more than others

34. Click Spectrogram some periodic reverberation initial impulse

35. Part 2: Analog and Digital • In “reality”, sound is analog. • variations in air pressure are continuous • = it has an amplitude value at all points in time. • and there are an infinite number of possible air pressure values. analog clock • Back in the bad old days, acoustic phonetics was strictly an analog endeavor.

36. Part 2: Analog and Digital • In the good new days, we can represent sound digitally in a computer. •  In a computer, sounds must be discrete. • everything = 1 or 0 digital clock • Computers represent sounds as sequences of discrete pressure values at separate points in time. • Finite number of pressure values. • Finite number of points in time.

37. Analog-to-Digital Conversion • Recording sounds onto a computer requires an analog-to-digital conversion (A-to-D) • When computers record sound, they need to digitize analog readings in two dimensions: X: Time (this is called sampling) Y: Amplitude (this is called quantization) quantization sampling

38. Thanks to Chilin Shih for making these materials available. Sampling Example

39. Sampling Example

40. Sampling Rate • Sampling rate = frequency at which samples are taken. • What’s a good sampling rate for speech? • Typical options include: • 22050 Hz, 44100 Hz, 48000 Hz • sometimes even 96000 Hz and 192000 Hz • Higher sampling rate preserves sound quality. • Lower sampling rate saves disk space. • (which is no longer much of an issue) • Young, healthy human ears are sensitive to sounds from 20 Hz to 20,000 Hz

41. One Consideration • The Nyquist Frequency • = highest frequency component that can be captured with a given sampling rate • = one-half the sampling rate Harry Nyquist (1889-1976) • Problematic Example: • 100 Hz sound • 100 Hz sampling rate samples 1 2 3

42. Nyquist’s Implication • An adequate sampling rate has to be… • at least twice as much as any frequency components in the signal that you’d like to capture. • 100 Hz sound • 200 Hz sampling rate samples 1 2 3 4 5 6