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The Simulation of the Movement of Synthesized Sounds with the Aid of Small Computers
'Klaus Buhlert
Klaus Buhlert
1. INTRODUCTION It is a daily experience when hearing environmental sounds and also music, that we try to localize the sound source in our surroundings. For musical (and broadcasting) applications it is often desireable to simulate this localization of different sound sources by different channels, i. e. we must apply models describing both the sound characteristics of the source and the volume envelope in space as well as the frequence differences with the approach or the removal of the source from the listener.
For the localization of real sound sources in the space a listener needs two kinds of information:
(1) Distance between source and listener with the following parameters becoming important: - Share of the directly incoming energy compared to the indirect share, with the direct energy share decreasing more rapidly with distance than the intensity of the reflected share within closed spaces.
- Loss of low energy spectral parts of a source with increasing distance to the listener (change in timbre).
(2) Direction (angle) of the source relatively to the listener: Here the important parameters are: - Signal time-lag in signal perception by the ears (unless the source is not exactly in front or in back of the listener).
- Energy differences in the high frequences reaching the ears due to the shading of the head.
By the means of analog combined analog/digital-techniques, in the past the first attempts to control tape recorders and VCA-units have been made and such to simulate a movement of an in-reality stationary sound source in a multi-channel system (3), (4).
Chowning (1) presented a calculation model using a computer system for measuring the parameters for the simulation of synthesized sources. The disadvantage of his system was the long calculation time for the synthesis of sources and for the calculation of the space curve.
The present work shall present a model for the simulation of sources and movements on a digital synthesizer/computer system in real-time mode. Such a software-module means much shorter processing times and a direct access to the change of parameters until the final result is reached; on the other hand this system reduces the possibilities of the description of complex sound sources and complicated movements.
In spite of the limited possibilities of applications relative to sound synthesis, choice of the space curve or consideration of reverberation rates compared to Chowning's model, very interesting results were reached after the development of appropriate software for the "SYNCLAVIER II".2. SYNTHESIS MODELS FOR THE DESCRIPTION OF THE SOURCE In an analyzation-based synthesis we try to analyze the parameters decisive for the sound characteristics and to use them as input for the synthesis-model.
2.1. Additive Synthesis
Additive Synthesis (also Fourier-synthesis) is based on the fact that even the most complex oscillations can be described as the sums of sinus and cosinus functions (fig. 1). Fourier-analyses of sounds are possible via filter gates or by using the Fast-Fourier-Transformation (FFT) Algorithm on a computer system.
Thus we get the descriptive parameters for the synthesis:- Frequences of the prime tone and the overtones as well as their characteristics in terms of time.
- Amplitudes of the different sinus-components and their characteristics in terms of time. (6)
The additive synthesis technique is an important and multiply useable technique in digital sound synthesis. A change in the input parameters often leads to new sounds out of the ordinary. The technique also allows the interpolation of two and more sounds and so for instance transfussing the sound of a cello into that of a trumpet and reaching very interesting intermediary stages.
The disadvantage of this system lies in the gigantic amount of data to be processed.
For the analysis or synthesis of a signal with an upper frequence limit of 15 kHz (and a scanning frequence of 30 kHz), having an amplitude resolution of 12 bit, we have a data-bulk of 360,000 bits per second. Now, it is possible to create an approximation to the curves by line segments in order to reduce the number of data, but the real-time construction of complex spectrums on digital systems is additionally limited (and expensive) through the great number of necessary oscillators.
These considerations lead to the construction of synthesis models in the audio sector, based on digital systems in AM, FM or non-linear function synthesis, etc. modes. The advantage of these techniques is that complex structures can be created although a much smaller number of parameters has to be controlled. This is why these techniques are also applicable on smaller DP-systems.
As a disadvantage, however, we find that there are no analysis-techniques for the generation of parameters for copying natural sounds. There are various (convincing) imitations of instruments, but they have been made after a long series of "trial and error" experiments.
The hardware configuration of the "Synclavier II", which was at disposal for the experiments described here, allowed two methods of synthesis: - a strongly limited additive synthesis in a so called Real-Time-Performance-Program. (No frequency-envelopes and only a limited number of amplitude envelopes of the ADSR-type for overtone-construction but complex time functions calculable for digital oscillators from 24 weighted harmonic tones).
- a complex frequence-modulation with up to 24 modulation pairs per synthesizer channel: (As to the choice of envelopes in the Real-Time-Performance-Program we are subject to the same limitations as with additive synthesis enlarged by our own software).
2.2. Synthesis by Frequence Modulation
The mathematical relations in FM mode have been known for a considerable period. Therefore we shall only give a short outline of this technique in a new field of application, in the synthesis of sounds.
In general, the FM method can be described by the following formula for sinusoidal carrier waves and modulator:
y(n) = A(n) sin [2p nDt fT + l(n) sin(2p nDt fM)] (1) wherein:
y(n) = basic value of the modulated signal
A(n) = amplitude value
fT = carrier frequence
fM = modulation frequence
Dt = difference between two scanning values. By a transformation the above expression can be described as a sum of Bessel-functions of nth order Jn.
With
a = 2p n Dt fT
b = 2p n Dt fM
cn = l(n), we get:
y(n) = A(n)
J0(cn)sin a
+ J1(cn)[sin (a + b) – sin (a – b)]
+ J2(cn)[sin (a + 2b) – sin (a – 2b)]
+ J3(cn)[sin (a + 3b) – sin (a – 3b)]
+ … The value of the Bessel-function is weighted for the respective frequence shares. So, in the modulation index cn = l(n) = 0, there exists a pure sinus-function A(n)sina, da J0(0) = 1 and all Bessel-functions of the first order are zero at the origin of coordinates. If cn increases, more and more energy goes to the side bands (Fig. 3. and 4).
The negative spectral shares are reflected into the positive areas of frequency with a change in the algebraic sign (phase is turned at an angle of 180°) and added to the other spectral components.
For the sound synthesis the above mathematical context leads to the following conclusions:
Harmonic spectrums (characteristics of most of the musical instruments) develop when the ratio of carrier frequences to modulation frequences fM/fT or its reciprocal value can be expressed by integers.
Example A:
fM/fT = 1
The spectrum encludes all harmonic overtones of the carrier frequence.
Example B:
fM/fT = 2
The spectrum includes only the odd harmonic overtones of the carrier frequence. - Inharmonic spectrums (bells, percussion instruments etc.) are generated, if the ratio fM/fT or its reciprocal value can be expressed by irrational numbers, e.g. fT/fM = 1/Ö2.
- Dynamic spectres (facsimiles of transient processes of instruments) may be described by the change of the modulation index over time. Here the modulation index determinates the real range of the signal. The energy changes of the different spectral shares are given by the value of the relative Bessel-functions (fig. 5).
2.3. Outline of other methods
The modulation of the amplitude of a signal many also lead to complex spectres and is a widely used technique of synthesis. A large-scale explanation of this technique of synthesis seems to have little sense here. The following techniques, therefore, are presented only as a short survey: - Synthesis by non-linear functions:
Normally, a sinus cycle (with its amplitude controlled by an envelope generator) is used as input for a non-linear processor. The transmission function of this process controls the spectrum of the output signal. For the calculation of this function (depending on time) from the desired spectrum, the Tchebysheff-polynomes may be used profitably.
- Granular Synthesis
Within a fixed or variable time-window-area a function f(t) is generated (mostly a Gauss-curve or a sin2wt-function) wherein the window range, the amplitude of the signal and frequency can be influenced (5). The stimulation function can change between purely sinus-shaped and a band-limited sequence of pulses.
3. MOTION MODEL In the description of moving point-shaped sources we have to simulate the most important parameters for the localization of sources by the listener with sufficient exactness.
3.1. Change in intensity through change in distance
An observer will notice a change in intensity when approaching (or going away from) a sound source. By sound yield considerations at the source, we get the relation that the sound yield p at the place of measurement is inversely proportional to the distance. r:
p ~ 1/r
(6dB diminution of sound pressure level when doubling the distance)
In accordance with this relation we get a volume envelope in the place of survey with a linear movement of the source (fig. 6) as in fig. 7 (distance A towards zero).
3.2. Multi channel movement simulation
In a stereo- or quadro-reproduction the exact position of the listener is mostly not known, thus the parameters time-lag and headposition shall be neglected.
The angle the information is given to the listener by the change in energy between the different loudspeaker-pairs of the quadrant in question depends on a factor of attenuation kn, which depends on the following relations:
k1 = 1 – a/amax
k2 = a/amax,
with amax = 90°.
The same applies for the other pairs of loudspeakers in a multichannel system.
3.3. Change of frequence by Dopplershift
An important piece of information in the simulation of the movement of sound sources is the so-called Dopplershift, i.e., the subjective perception of a frequency change when the source is in a movement towards or away from the listener.
If the listener be fixed in one place and the source is in movement, the perceived frequency fR in dependence on the relative speed vR is:
fR = f0(1± vR/c),
wherein f0 is the real frequency emitted by the source and c is the speed of sound in the medium (air = 340 m/s).
The quotient is added when the source is approaching, and subtracted when leaving the listener.
3.4. Timbre changes
A simulation of the changes in sound colour (as to the influence of different parameters such as humidity, temperature, deviation and refraction of the soundwaves) is almost impossible and shall be neglected here, also because of the limited possibilities of application.
4. REALIZATION The above described model for the simulation of the movement of synthetic sound sources was realized as a software-module for a "SYNCLAVIER II" Computer Music System.
The programming languages were XPL/4, a language similar to PASCAL and developed for this system, as well as MAX, a subset of the former. By integrating the MAX library, synthesis and control programs for the control of necessary parameters may be described on a high level of language, i. e. with such real-time processes in the audio-area no additional assembler language is necessary.
The three parts of the MAX library include: - Sub-programs for the control of the synthesizer charts (input of frequence, wave shape, amplitude envelope etc.),
- Sub-programs for the control of the input and output procedures at the control unit for the user-keyboard (including among others 8´16 control knobs, A/D-transformer, D/A-transformer, Display, etc.)
- "Time Sharing Algorithm" for the control of a quasi simultaneous synthesis of different synthesizer channels (complex or polyphonic sounds).
Part one of the MAX sub-programs was used for the description of the sound synthesis of the source in question, but served also for the realization of the frequence-shifts under consideration of the Dopplershifts.
Part two was used for the input of control parameters (for the sound synthesis, choice of motion functions etc.) and for the output of control functions (multichannel envelopes for the control of VCA-units).
Although there are limits to such a Computer Music System (monophonic audio output channel, little working memory etc.), a user program for the synthesis of the sound source as well as for the calculation of the motion function and of the Dopplershift could be developed.
The calculated volume envelopes for the different loudspeakers (8-bit resolution) may be put out via maximally 8 D/A-transformers for the control of external VCA-units. The Dopplershift, calculated according to the motion function and to speed, was considered internally as a frequence parameter during the synthesis.
The user's advantage is in the short calculation time and the immediate feed-back of changes applied in the description of the source characteristics with respect to the motion parameters.
LITERATURE:
(1)
Chowning, J. M.: The Simulation of Moving Sound Sources. J. Audio Eng. Soc. Bd. 13 (1971), S. 2-6.  back
(2)
Chowning, J. M.: The Synthesis of Complex Audio Spectra by Means of Frequency Modulation. J. Audio Eng. Soc. Bd. 21 (1973) No. 7. [Note is not in the text]
(3)
Schulz, W.: Steuereinrichtung für Magnetbandgeräte mit elektronischer Drehzahlregelung zur Simulation bewegter Schallquellen aus Aufzeichnungen stationärer Schallquellen. Diplomarbeit Institut für Hochfrequenztechnik der TU Berlin, 1976. back
(4)
Haller, H. P.: Heinrich-Strobel-Stiftung des Südwestfunks e.V. Broschüre des Experimentalstudios Freiburg im Breisgau, 1981. back
(5)
Buhlert, K: Synthesemodelle auf Computersystemen. Script zur Lehrveranstaltung, TU Berlin, 1982. back
(6)
Moorer, J. K; Grey, J.; Strawn, J.: Lexicon of Analyzed Tones. Computer Music Journal Bd. 1. (1977), No. 3. back
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