Percezione uditiva (1)

Corso di Principi e Modelli della Percezione

Prof. Giuseppe Boccignone

Dipartimento di InformaticaUniversità di Milano

boccignone@di.unimi.ithttp://boccignone.di.unimi.it/PMP_2017.html

Cos’è il suono?

• I suoni sono creati dalle vibrazioni degli oggetti• Le vibrazioni di oggetti producono vibrazioni nelle molecole in prossimità degli

oggetti stessi.• Questo causa differenze di pressione nell’aria che si propagano in ogni direzione

(Es. sasso nello stagno)

Cos’è il suono?

• Le onde sonore viaggiano con una certa velocità di propagazione• Questa dipende dal mezzo di trasmissione. • Esempio: La velocità del suono attraverso l’aria è di circa 340 metri al secondo,

ma nell’acqua essa cresce sino a 1500 metri al secondo

Velocita’ 340 m/sec

1200 km/ora

• Caratteristiche fisiche delle onde sonore– Ampiezza: Grandezza del profilo di variazione

di pressione dell’onda sonora– Intensità: Quantità di energia di un suono che

cade su una unità di area

– Frequenza: Per i suoni è il numero di cicli (in termini di variazione di pressione) che si ripetono in un secondo

• Caratteristiche psicologiche delle onde sonore– Volume (loudness): L’aspetto psicologico del

suono correlato alla intensità percepita– Altezza (pitch): L’aspetto psicologico del suono

correlato alla frequenza percepita

Cos’è il suono?

Livello psicologico

Livello fisico

Pitch Loudness

Frequenza Ampiezza / Intensità

dipende da un insieme di proprietà spettrali

(armoniche, ecc)

dipende da frequenza, rumore, ambiente acustico

Caratteristiche del suono

Cos’è il suono? //Ampiezza e frequenza

• L’udito degli umani è sensibile ad un ampio range di intensità• Il rapporto fra il volume più basso e quello più alto di un suono che risulta percepibile

è quasi di uno su un milione• Al fine di descrivere differenze in ampiezza, i livelli del suono sono misurati su una

scala logaritmica le cui unità sono i decibels (dB)• Cambiamenti relativamente piccoli in decibel possono corrispondere a cambiamenti

fisici molto consistenti (un incremento di 6 decibel corrisponde circa ad un raddoppio della pressione del suono)

Cos’è il suono? //Ampiezza e frequenza

dB = 20 log(p/po)

per p = po, dB = 0

Cos’è il suono? //Ampiezza e frequenza• La frequenza è associata con l’altezza (pitch) di un suono

• Suoni a basse frequenze corrispondono a suoni “bassi” (e.g., i suonati da una tuba)

• Suoni ad alta frequenze corrispondono a suoni “acuti” (e.g., i suonati da un clarino)

Cos’è il suono? //Ampiezza e frequenza• La frequenza è associata con l’altezza

di un suono• L’udito degli umani è sensibile ad un

ampio range di frequenze: da circa 20 a 20,000 Hz

Cos’è il suono? //Ampiezza e frequenza: rumori ambientali

• Uno dei più semplici tipi di suoni: onde sinusoidali = toni puri• Onde sinusoidali: Onde per cui le variazioni in funzione del tempo sono descritte

da un’ onda sinusoidale• Il tempo per un ciclo completo dell’onda sinusoidale è definito periodo• Ci sono 360 gradi di fase in un intero periodo

Cos’è il suono? //Sinusoidi e toni puri

Diapason

Cos’è il suono? //Sinusoidi e toni puri

Cos’è il suono? //Sinusoidi e toni puriPeriodo:

Numero d'onda:

Pulsazione:

Frequenza:

Equazione di un'onda progressiva :

Equazione di un'onda retrograda :

Velocità:

Cos’è il suono? //Funzioni trigonometriche e suono

Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude and frequency, and the brain interprets those signals as sound. In this paper, we focus on how sound is generated and imagine generating sounds using a computer with a speaker.

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1Once cycle of sin(2π t)

Time t in seconds

sin(2

π t)

Frequency of oscillation We need to describe oscillations that occur many times per second. The graph of a sine function that oscillates through one cycle in a second looks like: A function that oscillates 440 times per second will look more like this. Note that the time axis only runs to 1/20 of a second. We say that the oscillation is 440 Hertz, or 440 cycles per second. Generating sound with a computer speaker A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to oscillate back and forth. If you make a speaker cone oscillate 440 times per second, it will sound like a pure A note. Click here to listen. If you make a speaker cone oscillate 880 times per second, it will sound like an A, but one octave higher. Click here to listen. We’ll call this A2. On a later page, there are graphs of the location of the speaker cone as a function of time, each for one twentieth of a second. Raising the frequency to 1760 Hertz raises the pitch another octave to A3. Changing the amplitude of oscillation, that is, how high and how low the graph goes, or how far forward and backward the speaker cone goes, changes the volume of the sound. The middle graph on the next page shows the amplitude of an oscillation of 1760 Hertz rising from 0 to 1. Click here to listen to this increasing volume A3 four times.

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1Once cycle of sin(2π 440 t)

Time t in seconds

sin(2

π 4

40 t)

Chromatic and major scales The chromatic scale increases the frequency of oscillation by 12 steps from one octave to another. Starting at A 440, the frequencies of the chromatic scale would be

8802440,2440,2440,2440 1212

122

121

0 =⋅⋅⋅⋅ K Each of the notes in between has a name of its own; they are: A, A#, B, C, C#, D, D#, E, F, F#, G, G#, A2. The A major scale starts at 440 Hertz, then increases over 8 steps to A2, along the notes A (0/12), B (2/12), C# (4/12), D (5/12), E (7/12), F# (9/12), G# (11/12) and A2 (12/12). A graph of how the speaker cone moves in an A-major scale can be found on a later page. Click here to listen to the A-major scale.

Una “cassa” è fatta di - una componente che vibra e vibrando

genera onde di pressione - un elettromagnete che genera

l’oscillazione

Inviando una corrente elettrica variabile all’elettromagnete, questo fa oscillare la “membrana”

Se l’oscillazione è di 440 cicli per secondo (Hz), viene generata la nota LA

Sound Waves

• Loud speakers produce sound by– The diaphragm of the speaker moves out, pushing air molecules

together– The diaphragm also moves in, pulling the air molecules apart– The cycle of this process creates alternating high- and low-pressure

regions that travel through the air

Cos’è il suono? //Funzioni trigonometriche e suono

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−4−2

024

A (440 Hz)

Spea

ker l

ocat

ion

Time (seconds)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−4−2

024

A (880 Hz)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.1

0

0.1A (1760 Hz) getting louder

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−4−2

024

A−C#−E chord (440, 554, 659 Hz)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−5

0

5A−C#−E−A2 chord (440, 554, 659, 880 Hz)

Se l’oscillazione è di 880 Hz viene generata la nota LA ma di un’ottava superiore (LA2)

A 1760 Hz ottengo LA3

Cos’è il suono? //Funzioni trigonometriche e suonoEsempio: scala cromatica

Aumenta la frequenza di oscillazione per passi di 1/12

Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude and frequency, and the brain interprets those signals as sound. In this paper, we focus on how sound is generated and imagine generating sounds using a computer with a speaker.

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1Once cycle of sin(2π t)

Time t in seconds

sin(2

π t)

Frequency of oscillation We need to describe oscillations that occur many times per second. The graph of a sine function that oscillates through one cycle in a second looks like: A function that oscillates 440 times per second will look more like this. Note that the time axis only runs to 1/20 of a second. We say that the oscillation is 440 Hertz, or 440 cycles per second. Generating sound with a computer speaker A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to oscillate back and forth. If you make a speaker cone oscillate 440 times per second, it will sound like a pure A note. Click here to listen. If you make a speaker cone oscillate 880 times per second, it will sound like an A, but one octave higher. Click here to listen. We’ll call this A2. On a later page, there are graphs of the location of the speaker cone as a function of time, each for one twentieth of a second. Raising the frequency to 1760 Hertz raises the pitch another octave to A3. Changing the amplitude of oscillation, that is, how high and how low the graph goes, or how far forward and backward the speaker cone goes, changes the volume of the sound. The middle graph on the next page shows the amplitude of an oscillation of 1760 Hertz rising from 0 to 1. Click here to listen to this increasing volume A3 four times.

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1Once cycle of sin(2π 440 t)

Time t in seconds

sin(2

π 4

40 t)

Chromatic and major scales The chromatic scale increases the frequency of oscillation by 12 steps from one octave to another. Starting at A 440, the frequencies of the chromatic scale would be

8802440,2440,2440,2440 1212

122

121

0 =⋅⋅⋅⋅ K Each of the notes in between has a name of its own; they are: A, A#, B, C, C#, D, D#, E, F, F#, G, G#, A2. The A major scale starts at 440 Hertz, then increases over 8 steps to A2, along the notes A (0/12), B (2/12), C# (4/12), D (5/12), E (7/12), F# (9/12), G# (11/12) and A2 (12/12). A graph of how the speaker cone moves in an A-major scale can be found on a later page. Click here to listen to the A-major scale.

LA LA# SI DO, DO#,RE,RE#,MI,FA,FA#,SOL,SOL# LA2

Cos’è il suono? //Funzioni trigonometriche e suono

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.

2−1

−0.5

0

0.5

1

A−major scaleSp

eake

r pos

ition

Time

A 440 Hz B 493.88 Hz

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4−1

−0.5

0

0.5

1 C# 554.37 Hz D 587.33 Hz

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6−1

−0.5

0

0.5

1 E 659.26 Hz F# 739.99 Hz

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8−1

−0.5

0

0.5

1 G# 830.61 Hz A 880 Hz

Cos’è il suono? //accordi e sovrapposizione di suoniUn accordo è ottenuto suonando più note contemporaneamente

Esempio: accordo di LA maggiore

LA DO# MI

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

Cos’è il suono? //accordi e sovrapposizione di suoni

Possiamo: - generare contemporaneamente le tre note su tre speaker diversi - sommare le tre note e generare l’onda ottenuta, su un unico speaker

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

Sono possibilità equivalenti: al nostro orecchio comunque arrivano sovrapposte

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

TimeSp

eake

r con

e lo

catio

n

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

Cos’è il suono? //spettro di frequenza

Notiamo come l’onda del tono puro LA sia molto più “semplice” dell’onda che corrisponde all’accordo

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

Time

Spea

ker c

one

loca

tion

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

frequenza frequenza frequenza frequenza

Cos’è il suono? //spettro di frequenza

In generale, per suoni arbitrariamente complessi, scriviamo Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

TimeSp

eake

r con

e lo

catio

n

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

frequenza (udibile)

frequenza frequenza frequenza

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

Cos’è il suono? //spettro di frequenza

In generale, per suoni arbitrariamente complessi, scriviamo Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

Time

Spea

ker c

one

loca

tion

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

frequenza (udibile)

frequenza frequenza frequenza

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

ampiezza per la frequenza k

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

Cos’è il suono? //spettro di frequenza

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

Time

Spea

ker c

one

loca

tion

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speakers side by side and making each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to make a more complicated oscillatory function and make your speaker cone oscillate according to that function. For example, if you want to play an A – C# - E chord, you can separately make three speakers oscillate according to the functions:

)22440sin(),22440sin(),2440sin( 127

124

ttt πππ ⋅⋅⋅⋅⋅ If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaker will make pressure waves in the air, and these pressure waves from different speakers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaker they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and make the speaker cone of a single speaker oscillate following the sum:

)22440sin()22440sin()2440sin( 127

124

ttt πππ ⋅⋅+⋅⋅+⋅ Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Click here to hear an A – C# - E chord. Click here to hear an A – C# - E – A2 chord. The graphs on the preceding page show what happens when you add the three functions to make the A – C# - E chord and the A – C# - E – A2 chord. Frequency spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn’t entirely clear that sine functions have anything to do with them. However, it can be shown that any “continuous” sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaker is playing a continuous sound by making the speaker cone follow some function L(t) over the time interval from 0 to 1 second, then we can write L(t) as a sum of sine functions this way:

∑=

⋅=000,20

20)2sin()(

kk tkatL π

The number k is the frequency, and for sounds that humans can hear, we should use frequencies from 20 Hertz to about 20,000 Hertz. As we age, we can’t hear sounds at 20,000 Hertz very well anymore. Click here for: 20 Hz, 60 Hz, (the sound of mechanical devices that hum because of alternating current being 60 Hz) 100 Hz, 10000 Hz, 16000 Hz, 20000Hz. Be aware that some of these may be outside the range of your speakers or your ears! The numbers are called the amplitudes for frequency k. It is easy to find the values of using integrals:

ka

ka

.)()2sin(1

0∫ ⋅= dttLtkak π

(Actually, this isn’t the whole story – for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close

frequenza frequenza frequenza frequenza0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

Pure A (440 Hz)

Time

Spea

ker c

one

loca

tion

0 0.01 0.02 0.03 0.04 0.05−1

−0.5

0

0.5

1

A−C#−E chord (440, 554, 659 Hz)

TimeSp

eake

r con

e lo

catio

n

0 200 400 600 800 10000

1

2

3

4

5Pure A

Stre

ngth

Frequency

Frequency spectrum for pure A

0 200 400 600 800 10000

0.5

1

1.5

A−C#−E chord

Stre

ngth

Frequency

Frequency spectrum for A−C#−E chord

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Stre

ngth

Frequency

Frequency spectrum for spoken long A

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken long E

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Stre

ngth

Frequency

Frequency spectrum for spoken long O

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

Stre

ngth

Frequency

Frequency spectrum for spoken OO

spettro di frequenza

• Se il segnale non è periodico

Cos’è il suono? //Sinusoidi e toni puri

• Non sono molto comuni fra i suoni che sentiamo tutti i giorni perché poche vibrazioni sono così pure• I suoni più comuni nel mondo sono suoni complessi (e.g., voci umane, di uccelli,

suoni di macchine etc.)

• Però tutti i suoni complessi possono essere descritti come combinazioni di onde sinusoidali (teorema di Fourier).

Cos’è il suono? //Suoni complessi

input

• Generalizziamo la rappresentazione del segnale acustico

Introduzione ai sistemi lineari 3

• Il comportamento di un sistema lineare spazio-invariante e completamente caratterizzato dalla suarisposta all’impulso luminoso �(x, y), cioe dalla PSF h(x, y) = S(�(x, y))

• Data un’immagine in ingresso f , un sistema lineare spazio-invariante caratterizzato dalla PSF h,produce un’immagine in uscita g e�ettuando la convoluzione

g = f ⇥ h =� +⇤

�⇤

� +⇤

�⇤f(x� x⇥, y � y⇥)h(x⇥, y⇥)dx⇥dy⇥ (16)

In ultima analisi, conoscendo h conosciamo perfettamente il sistema.

E’ facile, dalla definizione, dimostrare che la convoluzione gode delle proprieta commutativa e associativa

In definitiva:

f ⇥ h = h⇥ f (17)(f ⇥ h1)⇥ h2 = f ⇥ (h1 ⇥ h2) (18)

2 Frequenze

Supponiamo un segnale f(t) = A cos ⇤t oppure f(t) = A sin⇤t. L’output di un sistema LSI sara un segnaletraslato e scalato

g(t) = A⇥ cos(⇤t + ⇥) oppure g(t) = A⇥ sin(⇤t + ⇥)

Generalizzando f(t) = Aeiwt g(t) = A⇥eiwt

3 Rilevamento di bordi

Il problema della modellazione di un processo di rilevamento dei bordi puo’ essere riformulato secondo lametodologia di Marr come illustrato nella figura Con riferimento alla figura ,

marr.png

Figura 1: Rilevamento dei bordi e schema di Marr

I bordi sono caratterizzati da una brusca transizione di luminanza nel dominio dello spazio (x, y) e come altefrequenze nel dominio trasformato (u, v).

Da un punto di vista teorico (modello computazionale), possiamo pensare di caratterizzare l’operatore didetezione come un derivatore dell’immagine f(x, y), ovvero un operatore definito mediante le operazioni diderivazione parziale

�f(x,y)�x , �f(x,y)

�y ,

Formule di Eulero

Cos’è il suono? //Sinusoidi e toni puri

rappresentazione in campo complesso

Cos’è il suono? //Sinusoidi e toni puri

• Se il segnale non è periodico

Cos’è il suono? //Sinusoidi e toni puri

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.1

−0.05

0

0.05

0.1Snippet of a piece of music

Time in seconds

Spea

ker p

ositi

on

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.1

−0.05

0

0.05

0.1Digital samples are taken 44100 times per second

Time in seconds

Spea

ker p

ositi

on

This handout is available at http://www-math.bgsu.edu/z/sound

• Se il segnale non è periodico

Cos’è il suono? //Sinusoidi e toni puri

spettro di frequenza (Trasformata di Fourier)

rappresentazione del segnale (Antitrasformata di Fourier)

• Esempio:

Cos’è il suono? //Sinusoidi e toni puri

which makes questions of convergence of the integral that much more delicate. Furthermore,applying the Fourier inversion formula

f(t) =

Z 1

�1e

2⇡ist

f(s) ds

involves knowing first that e

2⇡ist

f(s) can be integrated and that the integration leads backto f(t).

As a cautionary example, the trig functions sine and cosine – the building blocks of periodicphenomena! – do not have simple Fourier transforms. More general theories of integrationnotwithstanding, there is no way to make sense of

Z 1

�1e

�2⇡ist cos 2⇡t dt or

Z 1

�1e

�2⇡ist sin 2⇡t dt

as any kind of classical functions.Plainly there’s a lot going on. Why then the certainty in employing these Secrets of the

Universe – and believe me, they’re fully employed. Several reasons. First, by now thereis a quite sophisticated mathematical understanding of the Fourier transform and Fourierinversion, including definitions that go beyond the formulas via integrals. This culminateswith the theory of ‘distributions,’ which is the rigorous treatment of ‘delta functions’ andthe like.2 Second, nature seems to take care of the existence of the integrals quite nicely,thank you, not to mention Fourier inversion. Recall that your inner ear finds the frequencycontent of a given signal (signals which generally are not periodic) and that your brain mustthen take the inverse transform to get the signal back. In the field of optics one discoversthat lenses and prisms have the e↵ect of taking a Fourier transform. Finally, for modern,computational uses of Fourier analysis, which are ubiquitous, one works with discrete datanot continuous signals. It’s a discrete form of the Fourier transform and the inverse transformthat are used, with finite sums replacing integrals. There are no questions of convergence orexistence for such finite sums.3 There are other questions, but not the same questions.

An example. Before doing anything else grand and sweeping, let’s do something modestand confined. Let’s compute an example.

Take

⇧(t) =

(1, �1

2 t 12

0, otherwise

This is called, variously, the ‘top hat’ function, the ‘rect’ function (for rectangle), the ‘in-dicator’ function for the interval [�1/2, 1/2], or the ‘characteristic’ function for the interval[�1/2, 1/2]. One can debate whether to define the function at the endpoints as I have. Irefuse to be drawn into the debate – for our purposes it will never matter. The graph is:

2Though delta functions are often associated with P. Dirac and his treatment of quantum mechanics,their operational use was realized much earlier by O. Heaviside in his elucidation of Maxwell’s theory ofelectromagnetism.

3The discrete form of the Fourier transform is essentially a Riemann sum approximation to the integral,or at least it can be developed from that point of view. This, the DFT (discrete Fourier transform), is thefundamental operator in applications. It is not to be confused with the FFT (Fast Fourier Transform). TheFFT is an algorithm for the e�cient calculation of the DFT. Keep this straight. It’s easy to detect whensomebody doesn’t know what they’re talking about when they casually refer to the FFT when they mean,or should mean the DFT.

4

The function is zero outside of [�1/2, 1/2] so the limits on the integral defining b⇧ only gofrom �1/2 to 1/2. The integration is straightforward:

b⇧(s) =

Z 1

�1⇧(x)e�2⇡isx

dx =

Z 1/2

1/2

e

�2⇡isx

dx

=1

�2⇡is

e

�2⇡isx

�1/2

�1/2

=1

�2⇡is

(e�⇡is � e

⇡is)

=1

2⇡is

2i sin ⇡s

=sin ⇡s

⇡s

Note the special values

sinc s =

(1, s = 0,

0, s an integer other than 0.

You may have thought that sin x/x only comes up as an example of 0/0 in calculus classesand could have no possible interest beyond that. No, no! It’s in your CD player, for example,for reasons we’ll explain. In fact, the function comes up so often it’s given a name, the sinc

function, and denoted

sinc s =sin ⇡s

⇡s

.

Here’s the graph.

5

Engineers see this curve when they sleep, and when they shop:

6

The function is zero outside of [�1/2, 1/2] so the limits on the integral defining b⇧ only gofrom �1/2 to 1/2. The integration is straightforward:

b⇧(s) =

Z 1

�1⇧(x)e�2⇡isx

dx =

Z 1/2

1/2

e

�2⇡isx

dx

=1

�2⇡is

e

�2⇡isx

�1/2

�1/2

=1

�2⇡is

(e�⇡is � e

⇡is)

=1

2⇡is

2i sin ⇡s

=sin ⇡s

⇡s

Note the special values

sinc s =

(1, s = 0,

0, s an integer other than 0.

You may have thought that sin x/x only comes up as an example of 0/0 in calculus classesand could have no possible interest beyond that. No, no! It’s in your CD player, for example,for reasons we’ll explain. In fact, the function comes up so often it’s given a name, the sinc

function, and denoted

sinc s =sin ⇡s

⇡s

.

Here’s the graph.

5

• Se il segnale non è periodico

Cos’è il suono? //Sinusoidi e toni puri

• Spettro armonico: causato da una semplice fonte di vibrazioni (pe. corda di chitarra o un sassofono)

• Prima armonica: la componente fondamentale più bassa del suono

• Suoni con la stessa armonica fondamentale possono essere percepiti come diversi

Cos’è il suono? //Spettro armonico

• Il timbro è quella particolare qualità del suono che permette di giudicare diversi due suoni con uguale intensità e altezza.

• Rappresenta quell'attributo della sensazione uditiva che consente all'ascoltatore d'identificare la fonte sonora, rendendola distinguibile da ogni altra.

• Suggerisce numerose analogie con il colore: il timbro viene designato come colore del suono tanto in inglese (tone-colour) quanto in tedesco (klangfarbe)

• Concezione classica, basata sulla teoria del suono di Helmholtz:

• il timbro viene determinato sulla base della sola composizione spettrale del suono, ossia in base alla distribuzione dell'energia delle diverse componenti di frequenza che compongono il suono.

Cos’è il suono? //il timbro