Not-Linear Convolution: A New Approach For The Auralization Of Distorting Systems
Enrico Armelloni, A. Bellini, Angelo Farina · Journal of the Audio Engineering Society · 2001
This work defines a new method for processing audio signals, with the aim to recreate an audible simulation (auralization) of the modification imposed on the original signal by a complex system. The new method is the extension of the classic auralization process based on the linear convolution of the original signal with the impulse response of the system. The extension allows for the emulation of non-linear systems, characterized in terms of harmonic distortion at several orders. The work first presents the mathematical framework of the proposed implementation, then it is shown how a not linear system can be experimentally characterized by a new measurement method of multiple impulse responses at various harmonic orders, and finally it is shown how these impulse responses can be employed in a multiple convolution process: an experimental demonstration is given of the similarity of the numerically processed sound with the live recording coming from a highly distorting device. INTRODUCTION The traditional auralization process is in common use since some years [1]. The method is usually employed for adding to dry music or speech recordings a set of information related to an acoustic space (and optionally to the sound system installed in it) such as reverberation and frequency response. Usually the system is modeled as a linear, time invariant process, and thus it is completely characterized by its impulse response. Several measurement techniques have been developed for measuring the impulse response of a system [2,3,4] or for predicting it in the case of large rooms [5,6,7], small rooms [8,9] and taking into account the directivity of sources and receivers [10]. Furthermore, efficient implementations are available for realizing the convolution process in frequency domain, for example with the traditional selectand-save algorithm [11], or with partitioned-block frequency filtering methods, pioneered by Soo and Pang [12] for the case of an impulse response subdivided in equally-sized blocks, and further refined by Gardner and McGrath [13,14] with the introduction of the partition of the impulse response in different-size blocks, which allows for very little Input/Output delay. In all those cases, anyway, the two constraints imposed on the system (linearity and time invariance) must be closely respected. In fact, even minor deviations from linearity or time invariance can disrupt completely the measurement of the impulse response [15,16], and even if these problems are circumvented with more advanced measurement techniques [15], the auralization obtained by the linear convolution process cannot represent faithfully the non-linear effects, which instead are often present in real-world systems, and which happen to be subjectively well noticeable. This situation is responsible for the fact that the sounds obtained with the traditional auralization method usually are perceived as slightly unrealistic, artificial, or unnaturally “clean”, due to the complete absence of the “natural” artifacts that are usually present in the real systems. FARINA ET AL. NOT LINEAR CONVOLUTION AES 110 CONVENTION, AMSTERDAM, NETHERLANDS, 2001 MAY 12–15 2 These effects are particularly important when the auralization method is employed for subjective comparison of different electro acoustic reproduction systems, as it is common in the car-audio field of application. The non-linear effects are an important part of the evaluation of the perceived sound quality, and their complete removal causes a harmful bias of the subjective responses. Recently the authors developed a novel measurement method [15], which allows for the complete characterization of the linear and not linear behavior of a complex system with a single measurement. The result of this measurement procedure is a set of impulse responses, the first being the traditional linear response, and the other the responses at several harmonic orders. From these measurement results, all the traditional metrics for describing the distortion of a reproduction system can be derived easily. Here it is proposed to employ this set of impulse responses in a multiple convolution process, capable of reconstructing the complete modification that happens to a signal passing through the complex system. The theory behind the new processing method is briefly recalled, then the results of actual measurement on distorting systems are shown, and finally it is demonstrated (also with audible examples) how the proposed method can accurately reproduce the distortion effects produced by non-linear reproduction systems THEORY The following picture describes the flow diagram of a system obtained by a distorting transducer (memory-less distortion) driving a subsequent linear system with memory: Not-linear system K[x(t)] Noise n(t) input x(t) + output y(t) linear system w(t)⊗h(t) distorted signal w(t) Figure 1. Flow diagram of the complex system Neglecting the noise, the transfer function of this system can be described, in general, by means of a Volterra series expansion: ( ) ( ) ( ) ( ) ( ) ∑ ∑ ∑ −