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Page 00000001 On Searching the Model Parameters Of Digital Waveguide Filters By Using Error Backpropagation Methods Alvin W.Y. Su Department of Computer Science Chung-Hwa Polytechnic Institute Of Technology, Hsin-Chu, Taiwan, R.O.C. email: email@example.com Abstract Digital Waveguide Filter technology has been widely used in the model-based electronic music synthesis. However, it is sometimes difficult to obtain the appropriate model parameters so that the synthetic results can sound like tones produced by any particular instruments. In this paper, we employ an algorithm called "Error Backpropagation" which is used for the supervised training for Neural Networks to search for the model parameters for plucked acoustic strings. Before the training process, we build a measurement device for string vibrations and the measured data is used as the training vector. Experiments show that the synthetic outputs produced by the resultant Digital Waveguide Filter are quite close to the measured data. Future works are also suggested in this paper. 1 Introduction The Digital Waveguide Filter (DWF) have been the most popular physical modeling synthesis technique . Synthetic tones of wind instruments and plucked-string instruments are successfully produced by using DWF technologies   . However, to find those suitable model parameters for DWF such that the synthetic sounds can sound like those generated by a specific acoustic instrument usually requires lots of effort  . In this paper, we present a technique borrowed from the supervised training for Artificial Neural Networks to help solve this problem. There are a few important procedures to carry out such that the modeling work can succeed. First of all, we need to design the DWF whose structure is physically close to that of the corresponding instrument to be modeled. Secondly, we have to measure and record the timedomain response of the instrument under excitation such that we can use the measured data as the training vectors. Finally, a good training algorithm based on the DWF structure has to be developed. In this paper, we use the modeling for the dynamics of a plucked string as an example to demonstrate the proposed method. Being an universal approximator, Artificial Neural Networks (ANN) can be trained to simulate any linear and nonlinear functions . The DWF can be mapped into an ANN structure and Error backpropagation method used in the supervised training of ANNs can be employed to derived the correspondent model parameters of the DWF . In section 2, we build a DWF such that it can be used to model a plucked string. In section 3, we introduce our string vibration measurement system. In section 4, we propose a training algorithm to obtain the model parameters for the "Jargar" cello A-string we use in our experiment. In section 5, The experimental result for the modeling work is shown. Conclusions and suggestions for future works are given in section 6. 2 DWF Model For A Plucked String Based on the DWF modeling of an acoustic string, the dynamics of an ideal uniform string with simple passive loss can be described by the following discrete-time equation . y ( xm,t) = gmf (n-m)+ g-mf, (n+m) =(, ( Xm tn)+ q, ( m tn ) = (p + (p, (1) where t, = n - T and xm = m. c - T. T is the sampling period used by the system and c is the speed of the traveling wave for the string. (x, t, ) will be dropped for notional simplicity. If the string is not uniform, there are scattering junctions on the string due to the variation of the acoustic impedance. If the acoustic impedances belong to the two adjacent segments of a scattering junction are Z1 and Z2,
Page 00000002 respectively, the reflection coefficient of the junction can be obtained as follows. P 2-z (2) Z1 + Z2 The system of equations of a scattering junction can be expressed as follows. " =(l-p) +(1+p))2 (3) and f= -p(P ( +p))2 J r2 (1- P)(Pq + P(P/= P/ where y J is the junction displacement. (p1 and (p2 represent the arrival traveling waves into the junction for the first and the second segments, respectively. ft' and f,2 represent the departure traveling waves. Figure.1 shows the DWF for the modeling of acoustic strings. In this DWF, we assume that the string is terminated by two fixed ends. Therefore, the displacements of the two terminals are zero and the reflection coefficients for these two positions are -1. 3 Measurement System The measurement system consists of a string setup device which is shown in Figure. 2, an analog multitrack mixer, and a TASCAM DA88. There are seven electromagnetic pickups which are widely used in electrical guitars and are placed under the string. The mixer provides the necessary gain to the DA-88. The acoustic string we use is a Jargar cello A-string. The string is plucked and the vibrations picked up by the seven sensors are sampled and stored in the DA-88. The digital data is then transferred to an IBM PC and used as the training vectors. The electrical signals from the electromagnetic pickups are certainly not what we hear from an acoustic instrument most of the time and the string should be fixed on a cello instead of the device shown in Figure. 2. However, we intend to show that it is possible to reproduce these measured tones by the proposed synthesis technique. Even though the string is set up on a cello, it is also possible to synthesize the tones through proper modeling. 4 Training Algorithm The target of the proposed technique is to let the synthetic time-domain outputs be as close to the measured signals as possible. Therefore, we define the error function as follows. de (di(t)-y(t), if ieA (t) S0, otherwise (5) where d, (t) is the measured and desired signal from the i-th electromagnetic pickup at time t and y, (t) is the synthetic output produced by the DWF shown in Figure. 2 at the associated i-th physical position. A (t) is the set of positions where the vibrations can be measured. Because the computation required by the training is very large, it is impossible to use the whole duration of signals. If the duration of the measured data used in the training process is T, the total error function can be defined as follows . Etotal (tt,o ) e(t) t =too+1 ieA ( t ) (6) where T = t - to. In order to minimize the total error function, we need to modify the model parameters along the opposite direction of the gradient of the total error function. In the DWF for plucked-string modeling, the model parameters consist of the loss factors and the reflection coefficients. We have assumed that the string is lossy and non-uniform at every sampling positions. The model parameters are iteratively modified during the process by the following equations. Wt^l, =Wt Awt^ ';,,=,+l, A i+l, i Wt+l =Wt Awt i-, i i-1, i +p - w p+ = p + -Apt (7) (8) (9) Aw;, i Aw;.,, E total (to 0 t) E tot aw-1, A aE t ' ( to I t, _p E = -f r)t ap; (10) where fl is the learning constant and should be a small number. The superscript t denotes the iteration number. 5 Experiment We use the measurement system described in section 3 with seven pickups. Each is placed under the string equally separated from each other. The sampling rate is 32 KHz. The length of the DWF is 100 which means that the fundamental frequency of the signals is 160 Hz. 5000 samples of the measured signal from each pickup are shown in Figure. 3. The first 2000 samples are used as the training data. The next 300 samples are used to
Page 00000003 be compared with the synthetic output of the DWF after the model parameters are obtained. The learning constant is 0.0002. Let the initial DWF be an ideal string. Therefore, every loss factor is initiated to be unity and each reflection coefficient is set to be zero. The training is stopped after 15000 iterations. We use the DWF with the model parameters obtained in the training process to generate the synthetic tones. The result is shown in Figure. 4. Due to the limited number of pages allowed, only the result of one sampled position is shown. By comparing the first 2000 samples of the two figures, we find that they are very close. This is due to the fact that these samples are used as the training vectors for the total error function represented by Eq.(6). The corresponding Short-TimeFourier-Transforms (STFT) are shown in Figure. 5 and Figure. 6. The DWF generates the next 3000 samples with the same set of model parameters. We can see that the difference between the synthetic signal and the measured signal is still small. However, the difference becomes larger and larger after 7000 samples. This is due to the fact that the acoustic characteristic of the string changes over time. To have a more accurate synthesis result, it may be desired to update the model parameters during the synthesis process. There is also one important advantage of the proposed technique. Unlike some model-based synthesis methods which require a segment of tones taken from acoustic instruments , we can use inputs such as triangular and trapezoid waveforms which are similar to the normal plucks on acoustic strings and the sound is still close to that of its acoustic counterpart. This can significantly reduce the required memory space. 6 Conclusions and Future Works In this paper, a systematic method for searching the model parameters of the Digital Waveguide Filter for plucked strings is presented. The experimental result shows that the Error Backpropagation method used in the supervised training of Artificial Neural Networks can be used to obtain these model parameters as long as the mapping between the DWF and the ANN exists. Though the result shown in this paper is encouraging, we have to point out some important issues about the proposed technique. First of all, it is sometimes very difficult to obtain the measurement to be used as the training vectors. Secondly, there may not exist an appropriate DWF for certain instruments. Reeds widely used in woodwinds are the most immediate examples. Thirdly, though the training can be performed off-line, the computation required by the re-synthesis part is still too large for real-time implementations. To reduce the computational complexity of the re-synthesis part of the proposed method is necessary. Finally, the proposed training algorithm has to extended such that the model parameters of the multi-dimensional DWFs can also be found. 8 Acknowledgments The author would like to thanks Dr. Julius O. Smith for the precious discussions and suggestions. This works was initiated when the author was a visiting scholar in CCRMA, Stanford University in 1994. The research continued after the author joined CHPI in 1995. References  Smith,J.O., "Music Application of Digital Waveguide." CCRMA Technical Report STANM-67. Stanford University.  Smith,J.O., "Physical Modeling Using Digital Waveguides", Computer Music Journal, Vol. 8, No.3, p495-493.  Smith,J.O " Efficient Synthesis of String Musical Instruments", ICMC, 1993.  Julius O. Smith, "Efficient simulation of the reedbore and bow-string mechanisms," in Proceedings of the 1986 International Computer Music Conference, The Hague, San Francisco, 1986, pp. 275-280, Computer Music Association.  Matti Karjalainen et. al., "Physical Modeling of Plucked String Instruments with Application to Real-Time Sound Synthesis", JAES, Vol.44, No.5, p331-pp353, May, 1996.  Smith J.O., Techniques For Digital Filter Designs and System Identification With Applications To Violin", Ph.D. dissertation, Stanford University, 1983.  Cichocki A. and Unbehauen R., "Neural Networks For Optimization and Signal Processing", John Wiley, New York, 1993.  McClelland J. and Rumelhart D., "Parallel Distributed Processing", MIT Press, 1987.  Ronald J. Williams, and David Zipser, "A Learning Algorithm for Continually Running Fully Recurrent Neural Networks", Neural Computation 1, p270-280, 1989.
Page 00000004 1-pw1 1-p2 w 1-PN -Pi P 2 N -p -1 --1 -p +p 1,2 NN-1 +PN Figure 1 Digital Waveguide Filter for string modeling Figure.2 The string setup and measurement device 0.5 0.5 00 0 j -0.5 -0.5 0 1000 2000 3000 4000 5000 time Figure. 3 The measured string vibration 0 1000 2000 3000 4000 5000 time Figure. 4 The synthetic vibration output of DWF _ -10 0-20 E - 10 0-20 - E -30 --------------l;lr;' ~----- --------- -_ý- ------------ 0 -5 -- --------- 2000 1000 -50 400 600 1000 time steps Figure. 5 STFT of signal in Figure 3. 40 50 00200 400 time steps Figure. 6 STFT of signal in Figure.4