Sound Listener s perception

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1 Inversion of Loudspeaker Dynamics by Polynomial LQ Feedforward Control Mikael Sternad, Mathias Johansson and Jonas Rutstrom Abstract- Loudspeakers always introduce linear and nonlinear distortions in a stereo system. It is however possible to partially cancel these distortions by digitally pre-inverting the signal. We here discuss one of several possible solutions to this problem, namely the use of a linear inverse obtained from a measured ARX loudspeaker model. The inverse is obtained by well known polynomial methods as a H2-optimal feedforward lter. Due to numerical reasons, sound signals sampled at 44kHz will have to be separated into 3-6 sub-bands, to reduce the complexity of the model in each band to less than 7 poles and 7 zeros. An inverse is then implemented in each band and with this implementation, the method works well. The resulting performance is competitive to that obtained with direct adaptation of FIR pre-compensators by the idealized ltered-x algorithms, and a far shorter computation time is required for the lter tuning. ters pose a challenge to the numerical performance of polynomial tools for Wiener lter design [3], such as solvers for spectral factorizations and Diophantine equations. We have had the opportunity to investigate these issues when developing and teaching a new projectoriented course in Adaptive Signal Processing within the Engineering Physics program at Uppsala University [4, 5, 6]. In one of the projects during 999, the aim has been to use system identication to model loudspeakers of various quality. We then compute an LQG feedforward lter and pre-compensate the digital audio stream by using this lter before the DA-converter and amplier, see Figure. The aim of the projects for the year will be to use these designs in a larger system, which also compensates for the low-frequency room acoustics at one specic location in the room, as outlined by Figure 2. Introduction Inverse system Hi-fi system The increasing computing capacity of dedicated digital signal processors as well as general purpose processors are enabling the use of advanced signal processing on line for CD-quality audio signals, sampled at 44kHz. Possible applications include systems for multimedia computers, to produce sound eects localized in direction and distance. We can also implement advanced equalizers for use in conventional stereo systems. Adaptive methods for the compensation of stereophonic sound have been pioneered by Nelson, Elliot and co-workers, see e.g. [, 2], but these solutions have so far been implemented in restricted frequency bands. When the frequency range is increased, the required complexity of models and lters increases and the available computation time per sample is reduced. The high complexity of models and l- Signals and Systems, Uppsala University, PO Box 528, SE-75, Uppsala, Sweden. ms,majo,joru@signal.uu.se Figure : The principle of audio pre-compensation. 2 Models of Audio Channels Loudspeakers are characterized by rather long impulse responses when sampled at 44kHz. See Figure 3 for an example. Depending on the quality of the equipment, we may also experience nonlinear distortions. However, for most equipment we can for reasonable signal amplitudes obtain good models of the loudspeaker dynamics by using linear ARX models. This requires the use of suciently long times series, obtained by microphones of good quality and a broadband input. The modelling is improved if the data is collected in an anechoic chamber, to reduce background noise and echos. Loudspeaker models will have numerous zeros close to the unit circle, both inside and outside jzj =.

2 Sound source Inverse of room Inverse of stereo Stereo Room Listener s perception Figure 2: A block diagram of the total system. The listeners perception should equal the original sound source. The room acoustics, on the other hand, is more appropriately characterized as a sparse impulse response, see Figure 4. It varies from point to point in space, and the spatial covariance decreases with increasing frequency [7, 8] Impulse Response Time Figure 3: Example of loudspeaker impulse response. 3 Dynamic Pre-compensation of Loudspeakers Models of the loudspeakers can be used for computing an inverse in two ways.. In a direct adaptive approach the models are used to obtain sensitivity derivatives with respect to the parameters of an inverse lter, such as a FIR lter. The sensitivity derivatives are then used by a gradient or Newton algorithm to optimize the inverse. When using FIR lters and LMS adaptation, this is commonly referred to as a Under normal conditions, the diameter of the sphere where more than db reverberation reduction can be achieved is approximately r = =7 where is the wavelength of the signal [8]. The equalization of acoustic echos therefore becomes increasingly dicult for wider frequency bands, and equalization over a meaningful spatial volume will be possible for frequencies up to khz only. Figure 4: Example of room acoustic impulse response with direct wave and rst echo. ltered-x LMS algorithm [9]. If a known covariance matrix of the model output is used, we obtain a Newton algorithm, the Idealized ltered-x LMS algorithm. In a general framework, the approach can be regarded as self-tuning control via the method of Explicit Criterion Minimization []. 2. In an indirect approach, we analytically perform a model-based design of the pre-compensator, by minimizing an appropriate criterion. We will here discuss the indirect approach, using polynomial LQG feedforward compensator design. Without restrictions, the loudspeakers can be modeled and compensated individually. Based on a scalar (mono loudspeaker) model y(t) = q?k B(q? ) u(t) () A(q? ) where u(t) is the input to the system, q? is the backward shift operator, q?k is the time delay and y(t) is the measured sound, our objective is to design a regulator, or an IIR pre-compensator lter, given by u(t) = Q(q? ) w(t) (2) P (q? ) 2

3 so that the criterion J = E[jy(t)? w(t? d)j 2 + ju(t)j 2 ] (3) is minimized, see Figure 5. Assume the system to be stable, i.e. all roots of A(z? ) to be inside the unit circle and assume w(t) to be white with zero mean. The MMSE-optimal compensator is then given by u(t) = Q(q? ) P (q? ) w(t) = Q (q? )A(q? ) (q? ) w(t) (4) where (q? ) is the stable solution to the spectral factorization r(q? ) (q) = B(q? )B (q) + A(q? )A (q) (5) with r >. Together with a polynomial L (q), the polynomial Q (q? ) is the unique solution to the Diophantine equation q k?d B (q) = r (q)q (q? ) + ql (q) : (6) Here, the input penalty plays the role of a regularization parameter, giving ideal inversion for = and d!. See e.g. [] for a derivation of a generalization of (4)-(6) that also takes model errors into account. The Diophantine equation (6) is numerically wellbehaved, since the known factors on the right-hand side, (q) and q, do not have zeros close to each other. When >, the spectral factorization (5) will always be solvable: since A is assumed stable, the right-hand side can then have no zeros on the unit circle. For =, (5) will have a stable solution if B(z? ) has no zeros on the unit circle. While there will be numerous zeros close to the unit circle, models of loudspeaker dynamics will in general not have zeros precisely on the unit circle. Still, the numerical properties of spectral factorizations when applied to these types of models needs to be evaluated. The problem ()-(3) and solution (4)-(6) can be generalized in dierent directions, such as multivariable and robust designs [, 2]. Its use is exemplied below. 4 Example We here use the LQG method to compensate one loudspeaker. Test measurements with a broadband input and a sound-to-noise ratio of db were performed in an anechoic chamber. A broadband parametric ARX models was identied using the least squares method for the whole 5Hz-9kHz frequency range. Six sub-band models were also identied for the frequency ranges 5-7, 7-4, 4-9, 9-, -5 and 5-9 Hz, respectively. The broadband model required model orders na = 9; nb = 25 and k = 2, while the subband models were considerably less complex. relative phase POWER PLOT (db) RELATIVE PHASE PLOT (rad) Figure 6: Measured Bode plot of the uncompensated system. The average linear phase shift, corresponding to an average delay, has been subtracted from the phase plot. Numerical problems were encountered when trying to solve the design problem for the total broadband model. For the subband models, no such problems were encountered. Pre-compensators were thus calculated with = based on the six subband models. After bandpass ltering, the outputs of the six pre-compensators are added together and sent to the loudspeaker. The measured Bode plots before and after compensation are displayed in Figure 6 and Figure 7, respectively. It was possible to reduce the variations of the frequency response noticeably. We also applied a direct adaptive ltered-x LMS algorithm. It was found to converge towards a FIR pre-compensator with similar performance as that of the IIR solution after about 3 iterations. An interesting display of the performance of the compensated system versus that of the original system is presented in Figure 8. The top diagram shows an excerpt from Eric Clapton's Tears In Heaven taken from CD. The plot shows a distinct amplitude variation caused by a stroke on the guitar by Clapton. In the diagram below, the music has been sent through the broadband model of the speaker system. It is now somewhat distorted. In the bottom dia- x 4 x 4 3

4 Compensator Model of loudspeaker w(t) Q P u(t) q -k B A y(t) + Delay + - q -d e(t) Error to minimize Figure 5: Schematic view of loudspeaker inversion with an IIR compensator lter POWER PLOT (db) Orginal music source.5 relative phase RELATIVE PHASE PLOT (rad) Figure 7: Measured Bode plot for the compensated system consisting of six sub-band pre-compensators acting in parallel, before the amplier and loudspeaker Original music sent through model of stereo x Compensated music sent through model of stereo x Figure 8: A comparison of an actual music source, the source sent through the broadband model of stereo and nally the compensated source sent through the model. x 4 gram we see the same piece of music, compensated and thereafter sent through the broadband model. The stroke is now much more similar to the original excerpt. 5 Implementation and numerics The system has been implemented on a 4MHz Dual Pentium T M II PC, with one of the two processors, running compiled C++ code, dedicated to the pre- ltering. We could implement lters with 6 taps in single-precision oating point arithmetic. The other processor performs monitoring and possibly adaptation using Matlab T M and communicates with the real-time ltering processor via shared memory. We have in Matlab solved the Diophantine equation as a linear system of equations and solved the spectral factorization by calculating the roots of the right-hand side of (5). In all our experiments, this has been found to work reliably on sub-band models of loudspeakers of orders up to na = 7 and nb = 7. Numerical diculties, mainly in the solution of the spectral factorization, have been encountered if the orders are increased signicantly above 7. The use of a nonzero regularization parameter will not improve the situation appreciably, since the pattern of zeros in A is very similar to that in B. Thus the average spacing between zeros on the right-hand side of (5) is not increased by using >. The polynomial Wiener lter design tools such as polynomial spectral factorization have thus been 4

5 found to work well with discrete-time models of acoustic elements of rather high order. A main reason is that, although the sub-band model numerators B have zeros close to the unit circle, these zeros are well spread out along the unit circle. [2] Ohrn K., A. Ahlen and M. Sternad, A probabilistic approach to multivariable robust ltering and openloop control, IEEE Trans. on Automatic Control, Vol. 4, pp , 995. References [] Nelson, P. A., H. Hamada and S.J. Elliott, Adaptive Inverse Filters for Stereophonic Sound Reproduction, IEEE Trans. on Signal Processing, Vol. 4, No. 7, 992. [2] Nelson, P. A., F. Orduna-Bustamante and H. Hamada, Inverse lter design and equalization zones in multi-channel sound reproduction IEEE Trans. on Speech and Audio Processing, Vol. 3, No. 3, pp , 997. [3] Ahlen, A. and M. Sternad, Derivation and Design of Wiener Filters using Polynomial Equations, Digital Signal Processing Techniques and Applications, Academic Press, 994. [4] Adaptive Signal Processing at Uppsala university: Results and sound examples can be found on the course home page /sigproject.html [5] Botella, D., Inversion of Loudspeaker Dynamics and Room Acoustics, Master Thesis, Uppsala University, Report UPTEC F 99, February 999. [6] Johansson M. and J. Rutstrom, Inversion of audio systems by direct and indirect adaptive methods, Master Thesis, Uppsala University, Report UPTEC IT 2 99, July 999. [7] Haneda, Y., S. Makino and Y. Kaneda, Multiplepoint equalization of room transfer functions by using common acoustical poles, IEEE Trans. on Speech and Audio Processing, Vol. 5, No. 4, 997. [8] Radlovic, B. D., R. C. Williamson and R. A. Kennedy, On the poor robustness of sound equalization in reverberant environments, ICASSP, Phoenix, Arizona, USA, March 999. [9] Widrow, B. and S.D.Stearns, Adaptive Signal Processing, Prentice-Hall, 985. [] Trulsson E. and L. Ljung, Adaptive control based on explicit criterion minimization, Automatica, vol 2, , 985. [] Sternad M. and A. Ahlen, Robust ltering and feedforward control based on probabilistic descriptions of model errors, Automatica, vol 29, pp ,