Spectral Analysis - Fourier Decomposition

frequencies –

Makes the sound less

shrill or bright

Sampling

If sampled every

period then the

entire wave is lost

The shorter the sampling

spacing, the better the wave

is measured --- more high

frequency information

More on sampling

Two sample rates A. Low sample rate that distorts the original sound

wave. B. High sample rate that perfectly reproduces the original sound

wave. Image from Adobe Audition Help.

Guideline for sampling rate

•  Turning a sound wave into digital data: you must measure

the voltage (pressure) as a function of time. But at what

times?

•  Sampling rate (in seconds) should be a few times faster

than the period (in seconds) of the fastest frequency you

would like to be able to measure

•  To capture the sharp bends in the signal you need short

sampling spacing

•  What is the relation between frequency and period?

Guideline for choosing a digital

sampling rate

Period is 1/frequency

Sampling rate should be a few times shorter than

1/(maximum frequency) you would like to measure

For example. If you want to measure up to 10k Hz.

The period of this is 1/104 seconds or 0.1ms.

You would want to sample at a rate a few times less

than this or at ~0.02ms.

Recording in Audition

The most common sample

rates for digital audio

editing are as follows:

•  11,025 Hz Poor AM Radio

Quality/Speech (low-end

multimedia)

•  22,050 Hz Near FM Radio

Quality (high-end

multimedia)

•  32,000 Hz Better than FM

Radio Quality (standard

broadcast rate)

•  44,100 Hz CD Quality

•  48,000 Hz DAT Quality

•  96,000 Hz DVD Quality

Demo –degrading sampling and

resolution

•  Clip of song by Lynda Williams sampling is

48kHz resolution 16 bit

•  48kHz sampling , 8 bit

•  11kHz sampling, 16bit

Bits of measurement

8 bit binary number

00000000b = 0d

00000001b = 1d

00000010b = 2d

00000011b = 3d

00000100b = 4d

…

11111111b = 511d

can describe 2^8 = 512 different levels

sampling

Bit of

precision

Error in amplitude of

signal

loudness error

error in recording the

strength of signal

Bits of measurement

A signal that goes between 0Volt and 1Volt

•  8 bits of information

•  You can measure 1V/512 = 0.002V = 2mV

accuracy

•  16bits of information 2^16 = 65536

•  1V/65536= 0.000015V = 0.015mV =

15micro Volt accuracy

Creating a triangle wave with Matlab

using a Fourier series

dt = 0.0001; % sampling

time = 0:dt:0.01; % from 0 to 0.01 seconds total

with sampling interval dt

% Here my sample interval is 0.0001sec or a

frequency of 10^4Hz

frequency1 = 440.0; % This should be the note A

% harmonics of this odd ones only

frequency2 = frequency1*3.0;

frequency3 = frequency1*5.0;

frequency4 = frequency1*7.0;

% here are some amplitudes

a1 = 1.0;

a2 = 1.0/9.0;

a3 = 1.0/25.0;

a4 = 1.0/49.0;

% here are some sine waves

y1 = sin(2.0pifrequency1*time);

y2 = sin(2.0pifrequency2*time);

y3 = sin(2.0pifrequency3*time);

y4 = sin(2.0pifrequency4*time);

% now let's add some together

y = a1y1 - a2y2 + a3y3 - a4y4;

plot(time, y); % plot it out

Playing the sound

%Modify the file so the second line has

time = 0:dt:2; %(2 seconds)

%Last line: play it:

sound(y, 1/dt)

Save it as a .wav file for later

wavwrite(0.8*y,1/dt,'triangle.wav')

Phase

Up to this point we

have only discussed

amplitude and

frequency

x = 0:pi/100:2*pi;

y = sin(x);

y2 = sin(x-.25);

y3 = sin(x-.5);

plot(x,y,x,y2,x,y3)

Sine wave

period

e

d

u

t

i

l

p

m

a

phase

What happens if we vary the phase

of the components we used to make

the triangle wave?

y1 = sin(2.0pifrequency1*time);

y2 = sin(2.0pifrequency2*time - 1.6);

y3 = sin(2.0pifrequency3*time - 0.1);

y4 = sin(2.0pifrequency4*time +1.3);

y = a1y1 + a2y2 + a3y3 + a4y4;

Shape of wave is changed even

though frequency spectrum is

the same

Is there a difference in the sound?

These two are sums with the same amplitude

sine waves components, however the phases of

the sine waves differ.

Another example

This sound file has varying phases of its

frequencies.

Do we hear any difference in time?

Sound file from

http://webphysics.davidson.edu/faculty/dmb/py115/MusTechS05.htm

Spectrum of this sound

Waveform views at different times

Do we hear phase?

Helmholtz and Ohm argued that our ear and brain

are only sensitive to the frequencies of sounds.

Timbre is a result of frequency mix.

There are exceptions to this (e.g., low frequencies)

Two major psycho-acoustic models

1) Place theory – each spot in basal membrane is

sensitive to a different frequency

2) Timing – rate of firing of neurons is important and

gives us phase information

What is the role of each in how our ear and brains

process information? Open questions remain on this.

Cutting and pasting audio

High frequencies

introduced

Phase

shift

Sharp changes in wave form

Demo with a cut and paste in Audition/Audacity of a

generated sine. Note: the effect in spectral view depends on

the length of the FFT used, also you need to be fairly zoomed

out horizontal to see the noise.

Transform and

inverse

transform

e

d

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p

m

a

Time

f 3f 5f 7f

frequency

I have shown how to go this way

How we will talk about how to take a

signal and estimate the strength of its

frequency components

Multiplying two cosines with different

frequencies

Multiplying two cosines with the

same frequency

The

average is

not zero.

The

average is

1/2

Multiplying two cosines with different

frequencies

What if your

window fits

here?

Windowing and errors

Calculating the amplitude of each

Fourier component

What is the average of

Over a long interval this averages to zero

unless

f=g

Sine/Cosine functions are “orthogonal”

Calculating the amplitude of each

Fourier component

•  Procedure: multiply the waveform f(t) by a

cosine or sine and take the average.

•  Multiply by 2. This gives you the

coefficient Am or Bm.

Predicting the spectrum of a

plucked string

•  Can one predict the amplitude of each mode

(overtone/harmonic?) following plucking?

•  Which pluck will contain only odd harmonics?

•  Which pluck has stronger higher harmonics?

Odd vs Even Harmonics and

Symmetry

•  Sines are Anti-

symmetric about

mid-point

•  If you mirror

around the

middle you get

the same shape

but upside down

More on

Symmetry

•  Sines are anti-

symmetric

•  Cosines are

symmetric

n=1

odd

Symmetry

n=3

odd

n=2

even

•  Additional symmetry

of odd sines if you

consider reflection at

the black line.

•  About this line, Odd

harmonics are

symmetric but even

ones are anti-

symmetric

Symmetry of the triangle wave

Obeys same

symmetry as

the odd

harmonics so

cannot

contain even

harmonic

components

Odd Fourier components

Both triangle waves and square waves contain odd

Fourier components.

e

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t

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p

m

a

f 3f 5f 7f

frequency

Sawtooth

•  What overtones are present in this wave?

Use its symmetry to guess the answer.

Spectrum of sawtooth

All integer harmonics are present. The

additional symmetry about the ¼ wave that both

triangle and square wave have is not present in

the sawtooth.

Generated tones

Order of 440Hz tones:

Sine, Triangle, Sawtooth, Square, Rectangular

with 10%/90%

Sawtooth

Triangle

Symmetry as a compositional

element

•  From Larry Solomon’s “Symmetry as a compositional

element” -- last phrase of Bartok’s Music for Strings,

Percussion and Celesta, movement I

•  Reflection symmetry in tones --- axis of symmetry is an A

•  microcosmos vol 6 141 Free variations

Reflection in

time

Axis of

symmetry is a

time

(Example from

Larry Solomon)

Anton Webern,

Opus 27

Predicting the spectrum of a

plucked string

•  Can one predict the amplitude of each mode

(overtone/harmonic?) following plucking?

•  Using the procedure to measure the Fourier

coefficients it is possible to predict the amplitude

of each harmonic tone.

Predicting the

spectrum

of a plucked

string

• You know the shape just before it is plucked.

• You know that each mode moves at its own

frequency

• The shape when released

• We rewrite this as

Predicting the motion of a plucked

string (continued)

Each harmonic has its own frequency of

oscillation, the m-th harmonic moves at a

frequency or m times that of the

fundamental mode.

Moving string in general

Does this make sense? Some checks:

Are left and right boundaries fixed?

Is the string not moving at t=0?

Sum of forward

+ backwards

travelling

waves

Initial condition given above, and the velocity every

where is zero.

This is equal to the sum of two traveling waves

Shape of wave form can be predicted at future times by

considering each traveling wave and how it reflects off

of the boundaries

Violin and stick slip motion

Figure and animation from http://

www.phys.unsw.edu.au/jw/Bows.html

Iphone films

•  http://www.wired.com/gadgetlab/2011/07/

iphones-rolling-shutter-captures-amazing-

slo-mo-guitar-string-vibrations/

Each line scanned at a different time.

The “rolling shutter”

Between 24 and 30fps.

1280 x 720 pixels

At fastest 0.033s per frame If I

divide by 1/1000 then ~ 30

microseconds delay between lines

Guitar string

•  Length of string, L, is about a meter, frequency of lowest

string is 82Hz, P=0.012s

•  Speed on the string

v/(2L) = f ! v = 2Lf ~160 m/s

•  The delay between lines is 30 microseconds corresponding

to a distance of 160m/s x 30 microseconds = 5e-3m =

0.5cm

•  Number of lines to get there and back travel times

0.012/33e-6=400 (half the picture) as expected

•  Maybe could do this calculation more efficiently by

considering what fraction of wavelength fits in view of

camera, giving phase information

Clarinet spectrum

172 Hz

506 Hz

333 Hz

172*2=344

172*3=516

Why is the third harmonic stronger than the second?

Piano spectrum

347Hz 697Hz

1094Hz

1396Hz

347*2=694

347*3=1041

347*4=1399

Even harmonics are the same size

Piano spectrum

•  C4 piano on left,

sawtooth at same

frequency on

right.

•  High overtones

are higher in

piano.

•  Why?

Are these frequency shifts

important?

Butler (example 2.4).

a) Piano playing C4

b) Piano playing C4 but the partials have been

lowered by digital processing so that their

frequencies are exact integer multiples of

the fundamental.

Pair of tones repeated 3 times.

Synthesized voicing

•  Voice and many instruments make a nearly

periodic signal

•  Overtones are all integer multiples of each other

•  Frequencies are fixed

•  However if a tone is synthesized to have exact

integer overtones and fixed frequencies it sounds

electronic

•  How do you synthesize more realistic tones?

Irregularities are important

•  Slight frequency shifts

•  Slight timing differences in the periodic waveform

Timing differences

from turbulence in

throat and other

sources.

If there is no

irregularity then

the tones are

unnatural and dull.

Synthesized singer

Cookdemo70

a. No vibrato

b. Random and periodic vibrato and singer

scooping slightly upward at beginning of each

note

Nearly Periodic Waveforms

•  Voice, guitar, flute, horn, didgeridu, piano: all have ladder spectrum

Why nearly periodic signals?

•  Stringed instruments. Modes of vibration have frequencies that are

integer multiples of a fundamental tone. All modes are excited by

plucking. Harmonics are modes.

•  Wind instruments. Mode frequencies are close to integer multiples of

a fundamental. Excitation builds on one mode. Excitation (mouth) is

nearly periodic. Resulting sound contains harmonics. The harmonics

may not be modes. Sometimes other modes can be seen in the sound

spectrum that are not harmonics.

•  Voice. Excitation is nearly periodic. Tract resonances give formants,

but not key toward driving sound. Emerging sound since nearly

periodic contains harmonics.

Not all musical sounds are nearly periodic in nature

Some history

Images and information from http://physics.kenyon.edu/

EarlyApparatus/Rudolf_Koenig_Apparatus/

Helmholtz_Resonator/Helmholtz_Resonator.html

Earliest sound spectra taken by Helmholtz ~1860 who used

glass spheres or cylinders, each with a difference size and

hole diameter setting its resonant frequency. The opposite

side would have a slender opening that could be held in the

ear. The enclosed volume of air acts as a spring connected to

the mass of the slug of air, and vibrates in an adiabatic

fashion at a frequency dependent on the density and volume

of the air, its molecular composition, and the mass of the slug

of air in the neck.

•  Sets of these were built and ordered by universities to

allow spectra of sounds to be measured in the lab

•  This very large set of twenty two Helmholtz resonators is in the Garland

Collection of Classic Physics Apparatus at Vanderbilt University. These were

bought by Chancellor Garland to outfit the Vanderbilt physics department for

the opening of the university in 1875. Garland had previously gone to visit

Koenig in Paris to discuss his order. in 1889 a set of nineteen resonators cost

170 francs.

Tunable

resonators

•  a cylindrical resonator

permits the volume of the

resonator to be changed by

sliding the tubes in and out.

The notes (and hence the

resonant frequencies) are

engraved on the side of the

apparatus. This is one of a

number of tunable

Helmholtz resonators at the

University of Vermont.

Tunable resonators

Ocarinas and

whistling

Unlike with flutes

the pitch is not set

by the effective

length of the

instrument

Whistle

•  To do : film a whistle of across an octave

Fourier analysis

in 1890

•

•

"Manometric Flame Analyser for the timbre of

sounds, with 14 universal resonators --- originally

650 francs" ($130). The adjustable Helmholtz

resonators are tuned to the fundamental frequency

of the sound to be analyzed, plus its harmonics.

The holes on the other side of the resonators are

connected by the rubber tubes to manometric

flame capsules, and the variation in the height of

the flames observed in the rotating mirror. The

variation is proportional to the strength of the

Fourier component of the sound.

The picture at the left, below, shows the

manometric capsules and the jets where the flames

are produced. Note the black background to made

the flames more visible.

•  BTW nice display at U Toronto!

Now how is the frequency analysis

computed?

•  The fast Fourier transform (FFT) is a discrete Fourier

transform algorithm which reduces the number of

computations needed for N points from 2N2 to 2Nlog2N

computations

•  Discrete: works on data points rather than a function.

•  A nice, space efficient algorithm exists for the number of

points N equal to a power of 2.

•  When you do a frequency analysis in Adobe Audition one

of the parameters you can choose is N (and you will notice

that the menu only allows powers of 2).

The FFT algorithm

•  A nice, memory efficient algorithm exists if

the number of points is a power of 2

•  Each component can be written as a sum of

components from a transform of the interval

divided in half.

•  It maybe makes sense that the number of

steps depends on log N

Taking an FFT

P

•  Total interval P

•  Number of points N

•  Sampling dt

•  P=N*dt

•  Windowing function –

entire interval is

multiplied by a function

dT

Output of FFT

•  Frequencies are computed at frequencies

•

f, 2f, 3f, 4f, ……Nf where 1/f=P is the length of the

interval used to compute the FFT and N is the number

of points

•  Difference between frequencies measured is set by

the length of the whole interval P.

•  If P (or number of points N) is too small then

precision of FFT is less.

Accuracy of FFT

•  To get better frequency measurements you

need a larger interval to measure in

•  You can’t make extremely fine frequency

measurements over extremely small time

intervals

•  Similar to a Heisenberg uncertainty relation

Sum of two sine waves with

frequencies very close together

Frequency f and 1.02f and their sum

The closer the two frequencies, the longer it takes until

they start to cancel

If I measure a fixed frequency over a small window then I don’t

know whether I have a single frequency of a sum of nearby

frequencies. The longer the window I measure a pure sine wave,

the more exactly I know the frequency of the sine wave.

Effect of window length on FFT

precision

•  Demo in Audition or Audacity different

FFT lengths and windows on a sine wave

green 1024 sample window

red 16834 sample window

piccolo sound

Window length and precision

green 1024 sample window

red 16834 sample window

digi low frequency sound

Effect of Window function on

FFT

red = Blackman-Harris

blue = triangle

n=2408 on digi sound

Terminology

•  Fourier decomposition

•  Spectrum

•  Spectral analysis

•  Sampling rate

•  Phase

•  FFT (Fast Fourier Transform)

Good/Bad physics -- Animusic

Good/Bad Physics

•  Donald Duck in Mathemagic land

Recommended Reading

•  Berg and Stork Chap 4