Korzh R.A.

Krivoy Rog national university, Ukraine

SOUND WAVE MODEL CONSTRUCTION IN CONTEXT OF ROBUST SIGNAL GENERATION

 

Introduction

Sound wave model construction allows to describe sound parameter modification in the time domain used in various sound synthesizers. While a real musical instrument is playing its loudness changes through the time. Each musical instrument has its own changes of loudness features. For example, organ plays the notes with the permanent loudness, but guitar plays the sounds with maximal loudness only in the attack moment of the string after that it smoothly fades out. Wind musical instruments are characterized by reaching the maximum loudness not in the attack moment but after some period of time.

 

ADSR-envelope

ADSR-envelope realizes such changes by using a small collection of parameters. This is the simplest and the most widespread sound dynamic model which helps to describe the progress laws in time for most of the sounds. According to mathematics, the loudness level modulation of a signal comes to the simple multiplication of the signal form (timbre) and ADSR-form, as a result the output signal form is limited by the form of ADSR-oscillation.

ADSR-envelope is an important sound characteristic of the musical instruments and a one of the main criteria of musical instrument identification. The envelope consists of four main sections (stages) (fig. 1):

1.     Attack (A) – the period of initial signal loudness increasing;

2.     Decay (D) – the period of signal decaying after the initial increasing;

3.     Sustain (S) – the period of constant signal power;

4.     Release (R) – the period of final signal fading.

1 – attack, 2 – decay, 3 – sustain, 4 – release

Figure 1 – ADSR-envelope

 

Not all of the stages could be represented in ADSR-envelope: it depends on a specific musical instrument. For example, piano has all of these stages, but flute can be viewed only in sustain stage, the others may be ignored. Because of nonlinearity of the initial and final sound stages for the robust digitization it is necessary to have a sample rate at least in five times over the sound oscillation frequency [44].

Attack time determines a period of time to reach the maximum loudness level (key pressing event).

Decay time determines a period of time to come from the maximum loudness level to sustain level.

Sustain time describes a sound level which is playing while key is holding (key holding event).

Release time determines a period of time to the final fading of the note sound level (key releasing event).

 

Sound wave mathematical model

Now, it is necessary to implement and describe a mathematical model of the sound wave, following by ADSR-scheme.

The main advantage of this model is that it is not mathematically complicated and fits for most sound simulation. But for the parameter completeness and of course for the reliability improving another one period is needed – hold time which could describe the maximal loudness sound level conservation law.

Our received model of the sound oscillations will be called AHDSR (Attack-Hold-Decay-Sustain-Release) and shows it in more detail (fig. 2).

 

AHDSR-model description

Let the individual musical note is describing as (fig. 2):

 

,

(1)

where

t – time ();

tn – onset time ();

 – duration time (sustain time) ();

 – relative attack time ();

 – relative hold time ();

 – relative decay time ();

 – relative release time ();

1 – attack, 2 – hold, 3 – decay, 4 – sustain, 5 – release

Figure 2 – AHDSR-envelope

 

a0 – sustain loudness (a0 > 0);

f0 – pitch (f0 > 0);

 – coefficient determing the maximal value of sounding duration (sustain time) ():

 

if , then  and  don't change,

if , then , but ;

 

 – relative frequencies of the partial harmonics (fi > 0);

 – percentage of the partial harmonics ();

 – oscillation phases of the partial harmonics;

 – timbre function:

 

,

(2)

where

 – oscillation phase of the fundamental harmonic ();

m – number of partial harmonics ();

ah – hold loudness coefficient ();

ad – decay loudness coefficient ();

ar – release loudness coefficient ();

 

Parameters ah, ad, a0 and ar are choosing in the way to satisfy the following inequality:

 

;

(3)

ga(t) – attack function of time t;

gh(t) – hold function of time t;

gd(t) – decay function of time t;

gs(t) – sustain function of time t;

gr(t) – release function of time t.

 

Let add some more model parameters for convenience and system completeness (4) – (12):

 

;

(4)

where

 – coefficient determing instantaneous () oscillation process duration of frequency up to 1 Hz and intensity up to 1 dB;

 

 

;

(5)

 

;

(6)

 

;

(7)

 

;

(8)

 

;

(9)

 

;

(10)

 

;

(11)

 

;

(12)

 

In terms of above-listed formula, let set the values of real sound wave times (13) – (18):

 

;

(13)

 

;

(14)

 

;

(15)

 

;

(16)

 

;

(17)

 

;

(18)

 

Including the following conditions are to be satisfied (19) – (21):

 

;

(19)

 

;

(20)

 

;

(21)

From formula (1) – (21) we receive the equation (22) for the inividual sound wave:

 

(22)

The equation (22) is true in ideal cases of the practical modelling techniques, as it doesn't contain any deviations and noise. Thus, it is necessary to add a noise component  which would distort the source sound oscillation by k times:

, in case of additive noise model,

, in case of multiplicative noise model;

The function  returns random values in the range (-1, 1), which cause more (if ) or less (if ) effect on the oscillation system:

 

;

(23)

In addition to that, it is necessary to add an external noise component  which would be the same relation as  but applied to the whole sound signal (would effect on all of the sound objects of the system which are taking part in the oscillation process).

 

Conclusions

Therefore, a musical piece is a N-set (24) – a set of the elementary sound objects ni(t) ():

 

(24)

where

m – number of the notes in a musical piece.

 

If (24) is written more detailed then:

 

(25)

where

p – external noise level.

 

Thereby, formula (22) specifies the equation of the sound wave according to the AHDSR-scheme, including its harmonic, amplitude, temporal and functional characteristics.

 

References:

1.     Белобородов А. Ю. Распознавание аудиообразов с применением обертонового ряда. — Инженерия программного обеспечения. — №3. — Киев, 2010.

2.     Голд, Б. Цифровая обработка сигналов (под ред. Трахтмана А. М.) / Б. Голд, Ч. Рэйдер. — М.: Советское радио. — 1973.

3.     Попов, О. Б. Цифровая обработка сигналов в трактах звукового вещания: учебное пособие / О. Б. Попов, С. Г. Рихтер. — "Горячая линия - Телеком". — М. — 2007.

4.     Френкс Л. Теория сигналов (под ред. Вакмана Д. Е.). — М.: Советское радио. — 1974.