Functional Link Expansions for Nonlinear Modeling of Audio and Speech Signals Danilo Comminiello, Simone Scardapane, Michele Scarpiniti, Raffaele Parisi, and Aurelio Uncini Abstract-Nonlinear distortions pose a serious problem for the quality preservation of audio and speech signals. To ad dress this problem, such signals are processed by nonlinear models. Functional link adaptive iter (FLAF) is a Iinear-in-the parameter nonlinear model, whose nonlinear transformation of the input is characterized by a basis function expansion, satisfy ing the universal approximation properties. Since the expansion type affects the nonlinear modeling according to the nature of the input signal, in this paper we investigate the FLAF modeling performance involving the most popular functional expansions when audio and speech signals are processed. A comprehensive analysis is conducted to provide the best suitable solution for the processing of nonlinear signals. Experimental results are assessed also in terms of signal quality and intelligibility. I. INTRODUCT ION One of the most challenging problems to be addressed in the modeling of audio signals is the presence of nonlinear distortions. Nonlinearities may be introduced in audio signals by loudspeakers during large signal peaks, by vibrations of plastic loudspeaker shells, by DI A converters and power ampliiers, or also by some signal preprocessing. Such distor tions afect audio and speech signals and impair their quality and intelligibility. This problem is particularly frequent in several digital audio applications involving the use of a loudspeaker, from high-quality listening to hands-ree speech conunications [1]-[3]. In order to compensate or reduce distortions affecting audio signals, nonlinear models are oten adopted. In the literature, Volterra series expansions have been widely used for the modeling of loudspeaker distortions [4], [5]. Such solution has been successfully used for a large range of nonlinearities afecting audio and speech signals, but its high computational cost limits its use in real-time applications. Other kind of nonlinear transformation have been proposed to model different kinds of nonlinearities. Based on Fourier expansions, even miror Fourier nonlinear (EMFN) ilter is presented in [6], showing superior convergence properties than Volterra ilters. In [7], a raised-cosine function is used to model both sot-clipping and hard-clipping nonlinearities. A lexible solution is proposed in [8], [9] based on spline functions, that are smooth parametric curves deined by interpolation of properly control points collected in a look up table. F unctional link-based models have been recently used for active noise control and nonlinear acoustic echo cancellation [3], [10]. Authors are with the Department of Information Engineering, Elec tronics and Telecommunications (DIET), "Sapienza" University of Rome, Via Eudossiana 18, 00184 Rome, Italy (Corresponding author's email: danilo.comminiello@uniromal.it. The work of Danilo ComminieUo was partly funded by bdSound. 978-1-4799-1959-8/15/$31.00 @2015 IEEE In this paper, we focus on this last class of Iinear-in the-parameter nonlinear model. The functional link adaptive ilter (FLAF) [3] is characterized by a nonlinear expansion of the input signal and a subsequent linear adaptation in the higher dimensional space. The nonlinear transformation in the FLAF is performed by the functional expansion block (FEB), which contains the functional links, i.e. a series of linearly independent functions, which might be a subset of a complete set of orthonormal basis functions, satisfying universal approximation constraints. The use of functional links for the nonlinear expansion has been quite popular over the years for the identiication of nonlinear systems, e.g. [10], [11], and recently it has been proposed also for echo cancellers [3], [12]. One of the main advantages of FLAF-based architectures lies in the lexibility, since the setting of several parameters is allowed in order to it the model to a speciic application. In this regard, an important choice in the FLAF design concerns the expansion type, i.e. the basis functions, or a subset of it, to be assigned for the functional link expansion. This choice mostly depends on the application and on the signals involved in the processing. Basis functions must satisfy universal approximation constraints and may be a subset of orthogonal polynomials, such as Chebyshev, Legendre, Laguerre and trigonometric polynomials, or just approximating functions, such as sigmoid and Gaussian functions. Among such func tional expansions, the one based on Chebyshev polynomials has been widely used for several applications, in partic ular for nonlinear dynamic system identiication [l3] and channel equalization [14], showing a strong effectiveness. Legendre expansion was used as an alternative to Chebyshev polynomials in channel equalization [15] and, recently, even in real nonlinear system identiication [16]. Trigonometric polynomials represent one of the most popular expansions [11], [17], besides being the only series to be used for applications involving audio and speech input signals [3], [10], although no other comparison with different expansion types is present in the literature. One of the irst expansions used for functional links is the random vector (RV) [17] [20]. Recently, similar models have been popularized under the name of extreme leaning machines [21], [22], and used in the audio context when dealing with static music classiication tasks [23], [24]. In this work, we want to provide a comprehensive analysis about the effect of diferent functional expansion types on the nonlinear modeling of distorted audio signals. Analyses are conducted by considering both error-based criteria and signal quality measures. Computational costs are also discussed. In particular, we assess the efects of different functional link expansions in some of the most representative digital audio applications involving nonlinear signals, such