THE HELMHOLTZ RESONATOR TREE

Transcription

1 THE HELMHOLTZ RESONATOR TREE Rafael C. D. Paiva and Vesa Välimäki Deartment of Signal Processing and Acoustics Aalto University, School of Electrical Engineering Esoo, Finland ABSTRACT The Helmholtz resonator is a rototye of a single acoustic resonance, which can be modeled with a digital resonator. This aer extends this concet by couling several Helmholtz resonators. The resulting structure is called a Helmholtz resonator tree. The height of the tree is defined by the number of resonator layers that are interconnected. The overall number of resonance frequencies of a Helmholtz resonator tree is the same as its height. A Helmholtz resonator tree can be modeled using wave digital filters (WDF), when electro-acoustic analogies are alied. A WDF tool for imlementing Helmholtz resonator trees has been develoed in C++. A VST lugin and an Android mobile alication were created, which can run short Helmholtz resonator trees in real time. Helmholtz resonator trees can be used for the real-time synthesis of ercussive sounds and for realizing novel filtering which can be tuned using intuitive hysical arameters. 1. INTRODUCTION Musical acoustics have many alications in understanding musical instruments and building comutational tools. These comutational tools may be used to create new digital musical instruments as well as to build digital effects based on intuitive hysical henomena. Some interesting henomena derive from Helmholtz resonances [1]. These resonances are related to the body of acoustic string instruments, such as the guitar or violin, wind instruments, such as the flute or a simle ocarina, and the cavity of ercussion instruments, such as the hang [2]. The hysical descrition of musical henomena use a variety of techniques [3]. Digital waveguides are used to model vibrating henomena like strings and airflow in wind instruments [4]. The mass-sring models are used to create vibrating structures in an intuitive way [5, 3]. The modes of a system may be reresented using couled mode synthesis [6], modal synthesis [7], or the functional transformation method [8]. Various ercussive musical instruments have been successfully modeled using a digital waveguide mesh [9, 1, 11], modal synthesis [12], and finite differences [13]. Additionally, there are a variety of models of electric circuits. These include state-sace methods [14, 15], the K method [16] and wave digital filters (WDF) [17, 18] among others. WDF alications include the model of a iano hammer [19], models of vacuum tube amlifiers [], and a model of an audio transformer in vacuum tube amlifiers [21]. This aer rooses the combination of several Helmholtz resonators in a tree-like structure, which is called a Helmholtz resonator tree. Such structures can be modeled using WDFs when elec- This work was suorted by the Nokia Foundation. tro-acoustic analogies are alied. This leads to real-time hysical modeling of comlex resonant structures, where the inut and outut oints can be located at any of the interconnected resonators. The Helmholtz resonator tree yields an intuitive way of creating hysically-insired resonating structures for synthesis. In this tye of structure several resonators influence each other, and their arameters can be modified using acoustical arameters, which are intuitive even for non-technical eole, although the exact resonance frequencies are unknown. Additionally, it rovides the ossibility of exciting the system at different oints, resulting in timbre variations of the same musical instrument. This aroach differs from the imlementation of cascaded second order filters, since the resonators interact with each other, changing their resonance frequencies. This aer is organized as follows. Section 2 resents a review of the acoustic-electric analogy for obtaining an equivalent model a Helmholtz resonator and extending it to a Helmholtz resonator tree structure. Section 3 reviews basic WDF concets. Section 4 introduces the Helmholtz resonator tree tool with a VST lugin and an Android mobile alication as imlementation examles. Section 5 shows simulation results to evaluate the effect of the Helmholtz resonator tree structure and comares the comutational cost of the imlemented system with a commercial circuit simulator software. Section 6 concludes the aer Basic concet 2. HELMHOLTZ RESONATOR The Helmholtz resonator, which is a rototye of a simle acoustic resonant system, can be thought to be a hollow shell, like a bottle, enclosing a volume connected to the external environment through an oen ie, or neck, as shown in Fig. 1. This structure is secified by the cavity volume V, the neck length l, the neck cross-sectional area S, and arameters of the surrounding environment [22], namely the seed of sound c and the density ρ of the surrounding gas where the resonator is located. These arameters are tyically c = 345 m/s and ρ = 1.2 kg/m 3 at room temerature. In order to derive the Helmholtz resonator model, the imedance of each acoustic art needs to be determined. The general imedance of an acoustic system is given by Z = U = Su, (1) where is the gas ressure, U is the volume velocity, u the article velocity, and S the cross sectional area. By using the analogy ressure voltage and volume flow current, it is ossible to DAFX-1

2 V, l S U 1, 1 U 1 R L 1 C U V, Figure 1: Helmholtz resonator and its equivalent circuit (adated from [1]). U V, U 1 V 1, 1 U 21 V 21, 21 model an acoustic imedance as a circuit and use circuit simulation techniques to model the entire system. The equivalent circuit of a Helmholtz resonator is given by an RLC circuit as Fig. 1. In this model, the imedance of an oen tube is reresented by the inductor L given by [1] U 11 V 11, 11 R L L = ρl S. (2) R 1 L 1 U C Additionally, the cavity is modeled by the caacitor C, which is given by [1] U R L U 1 C 1 1 R 21 L 21 C = ρc2 V. (3) Finally, the resistance is related to the cross-sectional area of the ie aerture and the hysical characteristics of the medium as [23] C R 11 U 11 L 11 C U 21 C R = ρc S. (4) Equations 2, 3 and 4 show that a Helmholtz resonator can be reresented by an equivalent circuit as in Fig. 1. In this circuit, the external ressure is reresented by the inut voltage 1, while the cavity ressure is reresented by the caacitor voltage. Additionally, the volume velocity in the neck is reresented by the current U 1 flowing through the resistor and inductor. Z K Z K-1,1 Z K-1,N (c) 2.2. Helmholtz resonator tree The concet of the Helmholtz resonator can be exanded to build more comlex structures as illustrated in Fig. 2. This structure is named the Helmholtz resonator tree. In Fig. 2, five Helmholtz resonators are connected through their necks. Hence, the ressure at each cavity will be the result of the integration of the volume flows from the tubes connected to this cavity. As a result, each cavity of Fig. 2 can be reresented as a caacitor in Fig. 2, and the connections between cavities through the tubes are reresented by circuit connections with resistors and inductors corresonding to the imedance of each tube. The acoustic imedance of a Helmholtz resonator tree can be derived in an iterative way. The acoustic imedance of a single resonator is given by Z 1(ω) = (jω)2 LC + jωrc + 1, (5) jωc where ω is the frequency in rad/s and j = 1 is the imaginary unit. The imedance of a Helmholtz resonator tree of height two and with N leaves can be determined as Z 2(ω) = jωl + R + 1 jωc +, (6) N 1 n=1 Z 1,n Figure 2: Examle of a Helmholtz resonator tree; its equivalent circuit; and (c) imedance calculation. where Z 1,n(ω) is the imedance of the n th leaf, given by Eq. 5. The same rocedure erformed for a tree of height two can be generalized for a tree of height K, where the imedance of this tree Z K(ω) is calculated based on the imedances of the subtrees Z K 1,n(ω) connected to it, as illustrated in Fig. 2 (c). This rocedure results in the general form of the Helmholtz resonator tree imedance Z K(ω) = ω2 LC + jωrc + 1 jωc +. (7) N 1 n=1 Z K 1,n Finally, the ressure-to-volume flow transfer function, can be obtained with the imedance Z K(ω) as H K(ω) = u(ω) (ω) = Z 1 K (ω). (8) This results in a comlex resonating system where the resonances are given by the oles of H K(ω) in Eq. 8. The oles of H K(ω) are deendent on all the resonator arameters, and thus no simle DAFX-2

3 closed-formula solution can be found for the resonating frequencies of this tye of system. A demonstrative video illustrates the effect of changing a single resonator at htt:// hut.fi/go/dafx12-helmholtztree Helmholtz resonator tree structure effect The arameters of the Helmholtz resonator tree were evaluated with Eq. 8 in two sets of results. In the first, the effect of the tree height and number of branch divisions is evaluated, while in the second set the effect of individual hysical resonator arameters is evaluated. Figure 3 shows the results for the evaluation of the Helmholtz resonator tree size arameters. In all the simulations, the curves are labeled K B, where K indicates the tree height and B indicates the number of branch divisions. Additionally, the arameters for individual resonators is ket constant, V =.1 m 3, L = 1 m, S = 1 m 2, ρ = 1.2 and c = m/s, which results in a resonance frequency of Hz. In Fig. 3, the height of the tree is modified, while the number of branch divisions at every height ste is ket constant. Figure 3 shows that for a tree of height two (2 3) only two resonances are observed, while for a height four (4 3) four resonances are observed. Additionally, even though the individual resonators have the same resonance frequency, the osition of the magnitude resonse eaks is different when a different tree size is chosen. The evaluation of the effect of the number of branch division is evaluated in Fig. 3. In this case, one notices that the number of resonances is the same, indeendently of the number of branch divisions. Additionally, as the number of branch division gets larger, the frequency difference between the resonant eaks increases x1 tree 2x3 tree 3x3 tree 4x3 tree x1 tree 4x2 tree 4x3 tree 4x4 tree Figure 3: Magnitude resonse showing the effect of changing the Helmholtz resonator tree height and number of branch division. same resonator arameters are evaluated: V =.1 m 3, L = 1 m, S = 1 m 2, ρ = 1.2 and c = m/s. The evaluated structures are shown in Fig. 4, and (c), with tye, tye 1, and tye 2 resonator trees, resectively. Figure 4(c) resents the frequency resonse for each tree tye, where one sees that the tye and 1 trees have the resonances at Hz, 89.1 Hz and Hz, and Hz, Hz and Hz, resectively. On the other hand, the tye 2 tree has four resonances, at Hz, Hz, Hz, and Hz. H H21 H22 H H H21 H12 H22 H H21 (c) Tye Tye 1 Tye Figure 4: Evaluation of the tree structure tye. Schematic view of a tye tree, tye 1 tree, and (c) tye 2 tree. (d) The magnitude resonse shows the effect of the tree structure. (d) 3. WAVE DIGITAL FILTERS Wave digital filter (WDF) is an efficient technique for imlementing circuit models [17]. By using this technique, each circuit element is reresented by one block with one ort through which the voltages and currents are maed onto the wave domain, where the comutation of the iteration between circuit comonents is simlified. Hence, a WDF rovides a way to simulate circuit models without matrix inversions and iterative rocesses for circuits with one nonlinear element. The main WDF aradigm is maing the Kirchhoff variables, i.e. voltages and currents, onto the wave variables used in the comutation. Each WDF ort has an incoming wave A and an outgoing wave B, which are related to the ort voltage V and current I by A = k (V + R PI) (9) B = k (V R PI), (1) H12 H22 Figure 4 shows the effect of different tree structures. Two different tree structures with the same number of resonators and the where R P is the ort imedance and k is a real-valued scaling factor [17, 3]. On the other hand, the wave variables can be easily DAFX-3

4 converted again into Kirchhoff variables as V = A + B 2k (11) I = A B 2kR P. (12) It is imortant to notice that the ort imedance R P is not necessarily related to the imedance of the element being simulated. In most of the cases, R P is adjusted in order to rovide reflection-free elements, but this is not necessarily mandatory for all elements. Reflection-free orts constitute an imortant issue in WDF. Elements using reflection-free orts are also called adated elements. When this kind of ort is used, the outut wave of an element has no instantaneous deendency on the incoming wave. This means that the comutation of the outgoing wave for that element will imly in no recursive method for the comutation of the circuit resonse. Tyically, multiort elements can only have one reflection-free ort. One examle of how this is reresented in the drawings of this work is resented in Fig. 5, where a generic tree-ort element X is connected to other elements through orts with imedance R, R 1 and R 2. In this reresentation, the dash at the beginning of the line reresenting the ort connection R indicates that this ort is adated. and it may be connected to several other resonators in its cavity. In this system, each resonator can be excited with an external source of volume flow U, which simulates something hitting that resonator. The equivalent circuit of the resonator is shown in Fig. 6, which includes a caacitor simulating the resonator cavity, the neck model with a resistor and an inductor, and a current source simulates the volume flow disturbance at the cavity. The WDF model for this resonator is shown in Fig. 6(c). This model is rendered generic for the connection of other resonators at the neck and cavity. Additionally, the model combines a series inductor and resistor, and a arallel current source and a caacitor in order to reduce the comutational comlexity of the system. It is imortant to notice that, for avoiding delay-free loos in the WDF imlementation, the neck can contain only one neighbor. Connections to cavity neighbors U V, l S U 1, 1 Connection to neck neighbor R Cavity neighbors Z l Z r Neck neighbor X U U 1 Z c 1 R 1 R 2 Connection to neck neighbor R, L Figure 5: Drawing nomenclature used in this aer. A generic multi-ort element with one reflection-free ort and tree ort series and (c) arallel adators with one reflection-free ort. (c) Connections to cavity neighbors (c) C, U Figure 6: Helmholtz resonator circuit elements: general resonator with neck and cavity connections, equivalent circuit and (c) WDF imlementation. 4. HELMHOLTZ RESONATOR TREE TOOL In order to encasulate the behavior of Helmholtz resonators, a C++ tool for simulating these resonators was imlemented. This tool is built using WDF classes to simulate circuit elements and uses its own classes for generating Helmholtz resonator tree structures and managing the connections between resonators. Figure 6 shows the basic elements of a Helmholtz resonator imlemented in the C++ tool. The acoustic reresentation of a generic Helmholtz resonator is resented in Fig. 6, which may be connected to the cavity of another resonator through its neck, The comlexity of imlementing the WDF resonator of Fig. 6(c) is given as follows. The series combination of a resistor R and inductor L is imlemented with two multilications, one sum, and one memory element. The arallel combination of a caacitor C and a current source U has one multilication, one sum, and one memory element. The three-ort series and arallel adators imlies one multilication and four additions [24]. This results in four multilications and six additions for each resonator. If an additional voltage/current source is connected to the root element of the Helmholtz resonator tree, it may be imlemented using one multilication and one addition [17, 3]. DAFX-4

5 4.1. VST Plugin In order to evaluate the erformance of the Helmholtz resonator tree, a Virtual Studio Technology (VST) lugin was created. VST lugins are based on a technology develoed by Steinberg, which allows the creation of lugins that are easily used in a audio host rogram [25]. The lugin creates a Helmholtz resonator tree of variable size. The size of the tree is controlled by two arameters, the height of the tree and the number of branch divisions at each height ste. Additionally, the lugin enables modification of the hysical arameters of the resonators where all the resonators have the same hysical dimensions. In this lugin, the inut signal is fed as an inut ressure at the root resonator and the outut signal is the volume flow at this resonator Mobile imlementation A mobile imlementation of the Helmholtz resonator tree was develoed for the Android tablet. The main interface used was develoed using Java in the Android Software Develoment Kit. The interface with the existing C++ Helmholtz Tool was develoed using Android s Native Develoment Kit framework, which uses the Java Native Interface. Two aroaches were tested for imlementing the audio interface. The first used the AudioTrack library. Although this is a straightforward method, when tested with the Android 3.1 OS in a Samsung Galaxy tablet, the maximum allowed frame size was 18 ms. This created a minimum algorithm delay, which is not desirable in real-time alications. In the second aroach, the audio interface used the OenSL ES library to access audio outut. When using this library no restrictions on the minimum frame size were observed. Since it was verified that OenSL ES has reduced algorithm latency comared to AudioTrack, this library was chosen to imlement the final mobile alication. Additionally, the samling rate for the alication could be modified to between 8 khz and 44.1 khz, with audible artifacts being observed only when using the 44.1-kHz samling rate. As an examle imlementation, a seven-element Helmholtz resonator tree was develoed, where the arameters of individual resonators can be modified in the user interface. The resonator tree structure is resented in Fig. 7. In this structure, the user may feed a constant ressure signal into the first resonator s neck or hit any of the resonators alying a short noise burst of volume flow in the cavity of the resonator. U U U U U U U H H21 H22 H23 Figure 7: Helmholtz resonator tree structure used in the mobile alication. 5. RESULTS 5.1. Comutational comlexity evaluation The comutational comlexity of the Helmholtz resonator tree imlementation was evaluated by comaring the simulation time of the VST lugin against the commercial circuit simulation software LTSice. For this urose, a tree with a height of four stes and two branch divisions er ste was built using the Helmholtz resonator tree tool using a 48-kHz samling frequency. The same circuit was simulated in LTSice with the maximum simulation ste size set to 1/48 in order to enable a fair comarison. Additionally, a stereo signal was simulated, meaning that the same network was simulated twice, using both aroaches and the inut signal consisted of imulses saced at 1 s with a 1-s long file. The comuter used in this test is equied with a Intel R Core TM 2 Quad CPU of 3 GHz, 8 GB RAM and running the Windows 7 oerating system. The host for the VST lugin was Audacity 1.3 Beta. The simulation time was measured for both cases and the CPU usage was monitored while the simulation was running. A first evaluation was erformed using a 1-s random noise inut signal. When using LTSice, the measured simulation time was 3 min 56 s (1856 s) with 75% CPU usage. On the other hand, when using the imlemented VST lugin, the simulation time was 1 min 52 s (112 s) with 25% CPU usage. In order to evaluate the effect of different inut signals in LTSice, a second test was conducted using a 5-Hz sine wave as inut. In this case, the simulation time using LTSice decreased to 6 min and 42 s (2 s). This shows that for the tested comuter configuration LTsice takes 3.35 to 15.5 s to simulate each second of a four stes high Helmholtz resonator tree with two divisions er branch while using 75% of the rocessing ower. For the same circuit, the Helmholtz resonator tree lugin took.93 s to simulate each second of the same Helmholtz resonator tree while using only 25% of the rocessing ower. For the case of 1% of CPU ower, the time to rocess 1 s with LTSice would be between 2.5 and 11.6 s, while for the Helmholtz resonator tree tool it would be.21 s. This indicates that the Helmholtz resonator tree imlemented as a VST lugin is 1 to 55 times faster than LTSice for the circuit under test Accuracy evaluation The accuracy of the Helmholtz resonator tree tool was evaluated comaring the VST lugin s results with the ones obtained with LTSice. Figure 8 shows the comarison of the frequency resonse obtained with a Helmholtz resonator tree with a height of 4 stes and 2 branch divisions er ste. Additionally, the Helmholtz resonator was built with a cavity V =.1 m 3, L = 1 m, S = 1 m 2, ρ = 1.2, and c = m/s. In this result, both the Helmholtz resonator tree tool and the LTSice results have three resonances at the same frequencies, although some amlitude deviation is observed for the high frequency eaks and the amlitude between these eaks. Overall, this result shows good agreement between the LTSice reference simulation and the Helmholtz resonator tree tool Mobile alication results The frequency reresentation of recorded examles using the mobile alication is shown in Figs. 9 and 1. These examles were DAFX-5

6 LTsice Helmholtz tree VST Figure 8: Helmholtz resonator tree tool and LTSice frequency resonse comarison for a four stes high Helmholtz resonator tree with two branch divisions er height ste. collected by recording the outut of a Samsung Galaxy tablet, oerating at a 24-kHz samling frequency and with a frame size of 1 ms. Two tyes of excitation were used. The first one was constant noise ressure alied at the root element of the tree. This is modeled as a voltage source alied at the neck connection in the model of Fig. 6 (c). The second one is a windowed white noise burst of 1 ms simulating the volume flow disturbance caused by hitting a resonator. Figure 9 shows the results for the default arameters of the alication. These arameters include ρ = 1.2, V =.1 m 3, A = 1 m 2 and L = 1 m for all the resonators. Figure 9 shows four main resonances at Hz, Hz, Hz and Hz. For one excitation signal at, Fig. 9 shows that most of the energy is concentrated at the higher frequency resonances and small variations when exciting the resonating structure at different ositions. Noise Figure 9: Magnitude resonse for different excitations of the same Helmholtz resonator structure. Noise ressure excitation at and volume flow disturbance at and volume flow disturbance at and H. The results for the second configuration are shown in Fig. 1. In this configuration the air density was set to ρ = 398, while the individual resonator arameters were with V =.575 m 3, A = 87 m 2 and L =.8 m; with V =.131 m 3, A = 1 m 2 and L = 5.75 m; with V =.31 m 3, A = 758 m 2 and L = 1 m; H with V =.912 m 3, A = 1 m 2 and L = 1 m; H21 with V = 1 m 3, A = 1 m 2 and L = 83.1 m; H22 with V =.173 m 3, A = 1 m 2 and L = 1 m; and H23 with V =.1 m 3, A = 87 m 2 and L = 1 m. 8 Noise H H (c) H22 H (d) Figure 1: Magnitude resonse for different excitations of the same Helmholtz resonator structure. Noise ressure excitation at and volume flow disturbance at ; volume flow disturbance at and ; (c) volume flow disturbance at H and H21; and (d) volume flow disturbance at H22 and H23. Figure 1 shows that, indeendently of how the Helmholtz resonator tree is excited, seven distinct resonances are visible at 159 Hz, Hz, Hz, 742 Hz, 1426 Hz, 2493 Hz and DAFX-6

7 Hz. Moreover, the outut energy is observed to be concentrated at some resonances deending on which resonator is excited. When exciting the resonator H, the energy is concentrated at Hz, whereas when exciting H21 energy concentrates at 159 Hz. This results in a distinct timbre every time a different art of the resonator tree is excited, which can be interreted as hitting different arts of the same musical structure. When comaring the results of Figs. 1 and 9, the resonators with different arameters are seen to yield larger differences when exciting the different resonators. Additionally, the number of resonances increases when setting different arameters for individual resonators. 6. CONCLUSIONS This aer has introduced the new idea of a Helmholtz resonator tree and has shown how the new structure can be imlemented. The equivalent circuit analogy for acoustic systems was reviewed, and the model of a Helmholtz resonator was extended to include connections of several resonators. Even when all the resonators of a tree have the same arameters, a comlex resonator with many resonance frequencies is obtained. In this case, the height of the Helmholtz resonator tree determines the number of resonances observed in the frequency resonse of the tree. Furthermore, the number of branch divisions er layer influences the sacing between the frequency resonse eaks. WDFs can be used to grow Helmholtz resonator trees. A C++ tool called HelmTree was imlemented for real-time emulation of Helmholtz resonator tree structures. This tool consists of a WDF block library extended with features to simulate acoustic henomena in Helmholtz resonator trees. The tool was used to create a VST lugin and a mobile alication using the Android oerating system. These ieces of software can emulate Helmholtz resonator trees in real time. The Helmholtz resonator tree tool was comared against a commercial circuit simulator LTSice. The comarison showed that the simulation results obtained with the Helmholtz resonator tree are in line with results obtained with standard circuit simulators, because no aroximations are done. The VST lugin using the Helmholtz resonator tree tool was 1 to 55 times faster than LTSice. Moreover, the comuting times indicate that the Helmholtz resonator tree tool is suitable for real-time simulation of short Helmholtz resonator trees. Since the Helmholtz resonator tree creates resonant filters related to hysical objects, it is useful for synthesizing ercussive sounds. Testing with the mobile alication revealed that exciting different resonators of the tree leads to a different timbre. This can be directly alied to modeling of a musical instrument whose timbre deends on the osition where it is excited. The comlex frequency resonses obtained with these resonators also aear suitable for filtering musical signals. The Helmholtz resonator tree introduced in this aer can serve as a method for real-time synthesis and effects rocessing. One advantage of having arameters related to Helmholtz resonators to build filters is the close relation to understandable hysical roerties. These arameters are intuitive for users with no training, since it is easy to learn what haens to the sound when the volume of a bottle or its neck length is changed. Sulementary material to this aer, including sound examles, is available at htt:// 7. ACKNOWLEDGMENTS The authors would like to thank Nokia Foundation for funding, and Dr. Jyri Pakarinen, Dr. Henri Penttinen, Dr. Antti Jylhä, and Dr. Cumhur Erkut for their helful comments. 8. REFERENCES [1] N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, Sringer-Verlag, [2] A. B. Morrison and T. D. Rossing, The extraordinary sound of the hang, Physics Today, vol. 62, no. 3, , Mar. 9. [3] V. Välimäki, J. Pakarinen, C. Erkut, and M. 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