Digital Sound Synthesis by Physical ?· Digital Sound Synthesis by Physical Modelling ... Subtractive…

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<ul><li><p>Digital Sound Synthesis by Physical Modelling </p><p>Rudolf Rabenstein and Lutz Trautmann Telecommunications Laboratory University of Erlangen-Nurnberg D-91058 Erlangen, Cauerstr. 7 </p><p>{rabe, traut} ( </p><p>Abstract </p><p>After recent advances in coding of natural speech and audio signals, also the synthetic creation of musical sounds is gaining importance. Various methods for waveform syn- thesis are currently used in digital instruments and software synthesizers. A family of new synthesis methods is based on physical models of vibrating structures (string, drum, etc.) rather than on descriptions of the resulting waveforms. This article describes various approaches to digital sound syn- thesis in general and discusses physical modelling methods in particular: Physical models in the form of partial differ- ential equations are presented. Then it is shown, how to derive discrete-time models which are suitable for real-time DSP implementation. Applications to computer music are given as examples. </p><p>1. Introduction </p><p>The last 150 years have seen tremendous advances in electrical, electronical, and digital information transmission and processing. From the very beginning, the available technology has not only been used to send written or spoken messages but also for more entertainig purposes: to make music! An early example is the Musical Telegraph of Elisha Gray in 1876, based on the telephone technology of that time. Later examples used vacuum tube oscilators through- out the first half of last century, transistorized analog syn- thesizers in the 1960s, and the first digital instruments in the 1970s. By the end of last century, digital soundcards with various methods for sound reproduction and genera- tion were commonplace in any personal computer. </p><p>give desktop computers the functionality of stereo equip- ment or sound studios. An example are new coding schemes for high quality audio. Together with rising bitrates for , file transmission on the internet, they have made digital music recordings freely avaiable on the world wide web. Another example is the combination of high performance sound cards, high capacity and fast access hard disks, and sophisticated software for audio recording, processing and mixing. A high-end personal computer equipped with these. components and programs provides the full functionality for a small home recording studio. </p><p>While more powerful hard- and software turn a single computer into a music machine, advances in standardiza- tion pave the way to networked solutions. The benefits of audio coding standards has already been mentioned. But the new MPEG-4 video and audio coding standard does not only provide natural but also synthetic audio coding. This means, that not only compressed samples of recorded music can be transmitted, but also digital scores similar to MIDI in addition to algorithms for the sound generation. Finally the concept of Structured Audio allows to break down an acoustic scene into their components and to transmit and manipulate them independently. </p><p>While natural audio coding is a well researched subject with widespread applications, the creation of synthetic high quality music is a topic of active development. For some time, applications have been confined to the refinement of digital musical instruments and software synthesizers. Re- cently, digital sound synthesis finds its way into the h4PEG- 4 video and audio coding standard. The most recent and maybe most interesting family of synthesis algorithms is based on physical models of vibrating structures. </p><p>The development is rapidly going on. One driving force is certainly the availablity of ever more powerful hardware. Cheap memory allows to store sound samples in high qual- ity and astonishing variety. The increase in processing power makes it possible to "compute" sounds in real time. </p><p>But also new algorithms and more powerful software </p><p>This article will higlight some of the methods for digi- tal sound synthesis with special emphasis on physical mod- elling. Section 2 presents a survey of synthesis methods. Two algorithms for physical modelling are described in sec- tion 3. Applications to computer music are given in sec- tion 4. </p><p>12 </p><p></p></li><li><p>2. Digital Sound Synthesis </p><p>2.1. Overview </p><p>Four methods for the synthesis of musical sounds are be presented in ascending order of modelling complexity [5]. The first method, wavetable synthesis, is based on samples of recorded sounds with little consideration of their phys- ical nature. Spectral synthesis creates sounds from mod- els of their time-frequency behaviour. The parameters of these models are derived from descriptions of the desired waveforms. Nonlinear synthesis allows to create spectrally rich sounds with very modest complexity of the synthesis algorithms. In contrast to spectral synthesis the parameters of these nonlinear models are not related to the produced waveforms in a straightforward way. The most advanced method, physical modelling, is based on models of the phys- ical properties of the vibrating structure which produces the sound. Rather than imitating a waveform, they simulate the physical behaviour of a string, drum, etc. Such simulations are numerically demanding, but modem hardware allows real-time implementations under practical conditions. </p><p>2.2. Wavetable Synthesis </p><p>The most widespread method for sound generation in digital musical instruments today is wavetable synthesis, also simply called sampling. Here, the term wavetable syn- thesis will be used, since sampling strictly denotes time dis- cretization of continuous signals in the sense of signal the- ory. </p><p>In wavetable synthesis recorded or synthesized musical events are stored in the internal memory and are played back on demand. Therefore wavetable synthesis does not require a parameterized sound source model. It only consists of a database of digitized musical events (the wavetable) and a set of playback tools. The musical events are typically </p><p>. temporal parts of single notes recorded from various in- struments and at various frequencies. The musical events must be long enough to capture the attack of the real sounds as well as a portion of the sustain. Capturing the attack is necessary to reproduce the typical sound of an instrument. Recording a sufficiently long sustain period avoids a strict periodicity during playback. </p><p>The playback tools consist of various techniques for sound variation during reproduction. The most important components of this toolset are pitch shifting, looping, en- veloping, and filtering. They are discussed here only briefly. See [3, chapter 81 and [5] for a more detailed treatment. </p><p>Pitch shifting allows to play a wavetable at different pitches. Recording notes at all possible frequencies for all instruments of interest would require excessive mem- ory. To avoid this situation only a subset of the frequency </p><p>range is recorded. Missing keys are reconstructed from the closest recorded frequency by pitch variation during play- back. Pitch shifting is accomplished by sample rate con- version techniques. Pitch variation is only possible within the range of a few semitones without noticeable alteration of the sound characteristics (Micky-Mouse effect). </p><p>Looping stands for recursive read out of the wavetable during playback. It is applied due to memory limitations as well as length variations of the played notes. As mentioned above, only a certain period is recorded, long enough to cap- ture the richness of the sound. This period is extended by looping to produce the required duration of the tone. Care has to be taken to avoid discontinuities at the loop bound- aries. </p><p>Enveloping denotes the application of a time varying gain function on the looped wavetable. Since the typical attack-decay-sustain-release (ADSR) envelope of an instru- ment is destroyed by looping, i t can be reconstructed or modified by enveloping. </p><p>Filtering modifies the time dependent spectral content of a note as enveloping changes its amplitude. Usually recur- sive digital filters of low order with adjustable coefficients are used. This allows not only a better sound-variability than present in the originally recorded wavetables but also time-varying effects which are not possible with acoustic instruments. </p><p>Despite these playback tools for sound alteration (and others not mentioned here), the sound variation of wavetable synthesis is limited by the recorded material. However, with the availability of cheap memory, wavetable synthesis has become popular for two reasons: Low com- putational cost and ease of operation. More advanced syn- thesis techniques need more processing power and require more skill of the performing musician to fully exploit their advantages. </p><p>2.3. Spectral Synthesis </p><p>While wavetable synthesis is based on sampled wave- forms in the time domain, spectral synthesis produces sounds from frequency domain models. There is a variety of methods based on a common generic signal representation: the superposition of basis functions $(t) with time-varying amplitudes Fl ( t ) </p><p>Only a short description of the main approaches is given here, based'on [3, chapter 91, [5] , and [2 ] . Practical imple- mentations often consist of combinations of these methods. </p><p>13 </p></li><li><p>Additive Synthesis the superposition of sinusoids </p><p>In additive synthesis, (1) describes </p><p>f ( t ) = xfi(t) sin(Ql(t)) + 4 t h (2) </p><p>Sometimes a noise source n(t) is added to account for the stochastic character which is not modelled well by sinu- soids. In the simplest case, each frequency component 81 ( t ) is given by a constant frequency and phase term &amp;(t) = wl t + + l . In practical synthesis, the time signals in (2) are represented 'by samples and the synthesized sound is pro- cessed in subsequent frames. The time variation of the am- plitude and the frequency of the sinusoids are considered by changing the values of 4, wl, and possibly 41 from frame to frame. </p><p>1 </p><p>Subtractive Synthesis Subtractive synthesis shapes sig- nals by taking away frequency components from a spec- trally rich excitation signal. This is achieved by exciting time-varying filters with noise. This approach is closely re- lated to filtering in wavetable synthesis. However, in sub- tractive synthesis, the filter input is a synthetic signal rather than a wavetable. Since harmonic tones cannot be well ap- proximated by filtered noise, subtractive synthesis is mostly used in conjunction with other synthesis methods. </p><p>Granular Synthesis In granular synthesis the basis func- tions $l(t) in (1) are chosen to be concentrated in time and frequency. These basis functions are called atoms or grains here. Building sounds from such grains is called granular synthesis. Sound grains can be obtained by various means: from windowed sine segments, from wavetables, from Ga- bor expansions, or with wavelet techniques. </p><p>2.4. Nonlinear Synthesis </p><p>In the previous sections linear sound synthesis methods have been described. They varied from the computational cheap wavetable synthesis with low variability to the com- putational expensive additive synthesis where arbitrary ac- cess on the basics of a sound is possible. </p><p>Using nonlinear models for sound synthesis leads to computational cheap methods with rich spectra. The dis- advantage of these methods is that the resulting time func- tions or spectra cannot be calculated analytically in most cases. Also the effect of parameter changes on the tim- bre of the sound cannot be predicted except for very simple schemes. Nevertheless nonlinear synthesis provides com- putational low-cost synthetic sounds with a wide variety of time functions and spectra. </p><p>The simplest case of nonlinear synthesis is discussed here. Making the phase term in the sine function time de- </p><p>- </p><p>I I I I </p><p>Figure 1. Frequency Modulation </p><p>pendent leads to the frequency modulation (FM) method. In its simplest form, the time function f ( t ) is given by </p><p>f ( t ) = F ( t ) sin(wot + $(t)) . (3) The implementation consists of at least two coupled oscil- lators. In (3) the carrier sin(w0t) is modulated by the time- dependent modulator 4(t) such that the frequency becomes time-dependent with w ( t ) = WO + (a/&amp;)q5(t). If the mod- ulator is also sinusoidal with $(t) = q sin(wmt) as shown in Fig. 1 the resulting spectrum consists of the carrier fre- quency WO and side frequencies at WO f nwm, n E N. The relations between the amplitudes of the discrete frequen- cies can be varied with the modulation index q. They are given by the values of the Bessel functions of order n with argument q. Four different FM spectra for WO = 1 kHz and different modulator frequencies and different modula- tion indices q are shown in Fig. 2. The spectrum for q = 1 has a simple rational relation between WO and wm resulting in a harmonic spectrum. Increasing the modulation index to q = 2 preserves the distance of the frequency lines but increases their number (top right). A slight decrease of wm moves the frequency components closer together and pro- duces a non-harmonic spectrum (bottom left). Spectrally very rich sounds can be produced by combining small val- ues of the modulation frequency wm with high modulation indices, as shown for q = 8. However, due to the depen- dence on only a few parameters, arbitrary spectra as in ad- ditive synthesis cannot be produced. Therefore this method fails to reproduce natural instruments. Nevertheless FM is frequently used in synthesizers and in sound cards for per- sonal computers, often with more than just two oscillators in a variety of different connections. </p><p>2.5. Physical Modelling </p><p>Wavetable synthesis, spectral synthesis as well as the nonlinear synthesis are based on sound descriptions in the time and frequency domain. A family of methods called physical modelling goes one step further by modelling directly the sound production mechanism instead of the sound. Invoking the laws of acoustics and elasticity the- ory results in the physical description of the main vibrat- ing structures of musical instruments by partial differential equations. Most methods are based on the wave equation which describes wave propagation in solids and in air ([17]). </p><p>14 </p></li><li><p>"0 1 2 f i n kHz </p><p>f i n kHz </p><p>11 . I </p><p>f i n kHz </p><p>Figure 2. Typical FM spectra </p><p>Finite Difference Methods The most direct approach is the discretization of the wave equation by finite difference approximations of the partial derivatives with respect to time and space. However, a faithful reproduction of the har- monic spectrum of an instrument requires small step sizes in time and space. The resulting numerical expense is consid- erably. The application of this aproach to piano strings has for example been shown by [I]. A physical motivation of the space discretization is given by the mass-spring-models described in [4]. </p><p>Modal Synthesis Vibrating structures can also be de- scribed in terms of their characteristic frequencies or modes and the associated decay rates. This approach allows the formulation...</p></li></ul>


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