Accurate Sinusoidal Model Analysis and Parameter Reduction by Fusion of Components

A method is described, with which two stable sinusoids can be represented with a single sinusoid with timevarying parameters and in some conditions approximated with a stable sinusoid. The method is util ized in an iterative sinusoidal analysis algorithm, which combines the components obtained in different iteration steps using described the method. The proposed algorithm improves the quali ty of the analysis at the expense of an increased number of components. INTRODUCTION Sinusoidal modeling is a powerful parametric representation for audio signals. It represents the periodic components of a signal with sinusoids with time-varying frequencies, amplitudes, and phases. The parameters are updated from frame to frame, and sinusoidal analysis algorithms are usually frame-based, too. In polyphonic, real-world signals the density of sinusoidal components can be very high. Also the sinusoids are usually not stable, which makes it difficult to estimate their parameters accurately. There are complex algorithms, which do the analysis in only one pass, and iterative methods that try to get a better estimation of the parameters in each iteration, for example [1]. Because of errors and inaccuracies in the sinusoidal analysis, there might be some harmonic components left in the residual. One approach to correct this phenomenom is to detect sinusoids iteratively from the residual. There are algorithms, which detect only one sinusoid at time, synthesize it, and then remove from the residual, for example [2]. Our system detects several sinusoids at each pass, therefore requiring only two or three iterations. ITERATIVE ANALYSIS The sinusoids that are not detected are left in the residual. If the parameters of the detected sinusoids are inaccurate, there remain sinusoids in the residual, the frequencies of which are close to the original ones. A natural approach to remove the sinusoids from the residual is to analyze the residual iteratively with the same analysis algorithms. If the sinusoids obtained from the residual are combined with the trajectories obtained from the original signal, a sinusoid which parameters were inaccurate becomes presented with two or more sinusoids. Normally, this is an undesirable situation. The proposed method combines the sinusoids obtained in different iterations, therefore reducing the total number of the parameters. The block diagram of the system is illustrated in Figure 1. In the first iteration, the input signal is analyzed using a conventional sinusoidal analysis system. This block can itself be very complex, but basically any sinusoidal analysis system can be used. In our experiments, sinusoidal li keness measure was used to detect the meaningful sinusoidal peaks [3]. The frequency resolution was improved using quadratic interpolation [4]. The ampli tudes and phases are obtained using non-iteratively the least-squares solution proposed in [1]. The peaks are tracked into trajectories by VIRTANEN ACCURATE SINUSOIDAL ANALYSIS AES 110 CONVENTION, AMSTERDAM, NETHERLANDS, 2001 MAY 12–15 2 synthesizing the possible continuations and comparing them to the original signal. The trajectories are filtered using the methods presented in [5]. The obtained trajectories are then synthesized and subtracted from the original signal in time domain to obtain the residual. In the following iterations, the residuals are analyzed with the same sinusoidal analysis algorithms. The parameters of the analysis, for example the sensitivity in the peak detection, can be varied from iteration to iteration. The sinusoidal trajectories obtained in different iterations are fused together using the methods proposed in the next section. Using the trajectories obtained in the first iteration and the remaining errors obtained in the following iterations, the parameters of the underlying sinusoids can be estimated. Again, the combined sinusoids are synthesized and the iteration continues. The iterative procedure can be repeated as long as desired. For example, the iteration can be stopped if no significant harmonic components are found from the residual. In our analysis system, two iterations was found to be quite enough. The iterative algorithm is computationally expensive, since each iteration requires one pass of a conventional analysis, and synthesis of the sinusoids, too. Compared to the analysis and syntesis, the fusion of sinusoids is computationally cheap. FUSION OF TWO SINUSOIDS Representation of Two Sinusoids with a Single Sinusoid and Time-varying Parameters Let us have two sinusoids, the ampli tudes, frequencies, and phases of which are a1, a2, ω1, ω2, φ1, and φ2, respectively. The sum of the sinusoids at time t is denoted by x(t): ) sin( ) sin( ) ( 2 2 2 1 1 1 φ + ω + φ + ω = t a t a t x (1) Using the basic trigonometric formulas this can be converted into a form where the two terms have equal frequencies and time-varying amplitudes: