Optimization Techniques for Parametric Modeling of Acoustic Systems and Materials

Linear and time-invariant signal and system models are useful in compact characterization of acoustic transfer functions. In addition to compact representations of responses, such models are efficient in simulating acoustic systems for sound synthesis, artificial reverberation, etc. In this paper we propose parametric modeling techniques for room impulse responses (RIRs), insitu acoustic material measurements, and musical instrument modeling, based on ARMA models — including Kautz filter models — which require nonlinear optimization of the parameters. Low order models are applied to surface impedance modeling, and high-order models are used for complex responses such as RIRs and musical instruments. INTRODUCTION Physical modeling of acoustic systems is in practice based on numerical simulation, for example by solving partial difference equations. Often the problem is to model the response from a point to another so that the spatial distribution of waves is not of primary interest. This leads to using transfer functions and signal modeling. The signal processing approach has the advantage of being computationally highly efficient, which is important especially when real-time simulation is needed. This is the case for example in model-based sound synthesis of musical instruments, audio effects such as artificial reverberators, auralization of room or concert hall acoustics in acoustics design software, or equalization of loudspeaker-room responses in audio reproduction. In all these cases the system to be modeled can be considered being linear and time-invariant (LTI), and in practice also stable and causal. Further cases where the signal processing methodology is highly useful can be found in acoustic and audio measurements. For LTI acoustic systems the impulse response or corresponding frequency response can be measured and the task is often to find a compact parametric model to capture the essential features of the target response. Digital filters, in the form of finite impulse response (FIR) and infinite impulse response (IIR) filters, are powerful means of modeling target responses, both from analysis and synthesis points of view. In analysis applications we may obtain essential information about the measured process, such as (eigen)mode parameters of a room or a musical instrument body. In synthesis applications a compact parametric model in the form of a digital filter makes it possible to simulate a given system in real time. Although LTI models are linear in input-output relationships, the solving for optimal model parameters is not necessarily so. There are a number of parameter estimation techniques for LTI systems [1,2]. MA (moving average) modeling leads to an FIR type filter model with ak coefficients being zero in its z-transform, in Eq.(1) below. While MA models are easily estimated, for reverberant and resonating systems they are not compact, computationally efficient, or analytically interesting. AR (autoregressive) models, with only b0 being non-zero in the numerator of z-transform in Eq.(1), are able to describe recursive feedback in a system, and are also relatively straightforward due to linearity of normal equations used to estimate filter parameters for example in the linear predictive (LP) autocorrelation method [3]. AR models are however, due to their minimum-phase property, limited in modeling capabilities, e.g., in modeling responses exhibiting inherent latencies in their frequency components. ARMA (autoregressive moving average) modeling is the most general form of LTI modeling with both ak and bk non-zero coefficients in Eq.(1), but there is no way to solve the optimal model parameters in a closed form, and thus iterative techniques of nonlinear optimization are needed, which brings potential problems in convergence, e.g., trapping to a local minumum not close to the global optimum.