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## Basic Acoustics + Digital Signal Processing

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Basic Acoustics + Digital Signal Processing

January 14, 2014

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?

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

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…

Acoustics: Basics

- How is a periodic sound transmitted through the air?
- Consider a bilabial trill:

Fad

Pin

Fad

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?

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

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

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!

Compression Wave

area of high pressure

(compression)

area of low pressure

(rarefaction)

- Compression waves consist of alternating areas of high and low pressure

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

Measuring Sound

- What if we set up a pressure level meter at one point in the wave?

Time

pressure level meter

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

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

Waveforms

- A waveform plots air pressure on the y axis against time on the x axis.

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!

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)

Complex Wave Example

- Take waveform 1:
- high amplitude
- low frequency

+

- Add waveform 2:
- low amplitude
- high frequency

=

- The sum is this complex waveform:

A Real-Life Example

- 480 Hz tone
- 620 Hz tone
- the combo = ?

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

One Way to Look At It

- Combining 100 Hz and 1000 Hz sinewaves results in the following complex waveform:

amplitude

time

The Other Way

- The same combination of 100 Hz and 1000 Hz sinewaves results in the following power spectrum:

amplitude

frequency

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.

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.

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.

Aperiodic Spectra

- The power spectrum of white noise has component frequencies of random amplitude across the board:

Aperiodic Spectrogram

- In an aperiodic sound, the values of the component frequencies also change randomly over time.

Transients

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

A Transient Spectrum

- An ideal transient spectrum is perfectly flat:

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

Click Spectrum

- Reverberation emphasizes some frequencies more than others

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.

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.

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

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

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

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

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