Canceling and Selecting Partials from Musical Tones Using Fractional-Delay Filters

This article discusses canceling and extracting harmonics from a musical signal using digital filters. This is an old technique that has been proposed in different forms by Moorer in the 1970s for pitch detection of speech signals (Moorer 1974) and for analyzing music data for additive synthesis (Moorer 1977). The basic idea is to use a multiple- notch filter to extract individual harmonic components as signals. The filter structure can be obtained as the inverse transfer function of a comb filter (i.e., a delay line in a feedback loop).

In this article, we expand on a recently proposed idea that the delay line in the inverse comb filter (ICF) can be replaced with a high- order fractional-delay filter to obtain very accurate cancellation of neighboring harmonics to select a single harmonic or to extract the residual signal by canceling all harmonics (Välimäki, Ilmoniemi, and Huotilainen 2004; Välimäki, Lehtonen, and Laakso 2007). The proposed signal analysis method is useful for many practical cases. Many musical instruments, including all woodwind, brass, and bowed string instruments, produce a sound signal that is inherently harmonic, that is, the spectral components are integral multiples of a fundamental frequency. This follows from the sound- production mechanism of these self- excited systems, which involves mode locking in the time domain (Fletcher and Rossing 1991). It forces the sustained tones of such instruments to be periodic. There is often a noise component in these musical tones, however, making them pseudo- periodic in practice.

Another method for this kind of signal decomposition is sinusoidal modeling (McAulay and Quatieri 1986; Serra 1989; Serra and Smith 1990). In this method, the signal is analyzed using the windowed Fast Fourier transform (FFT), and the frequency and amplitude tracks are obtained by connecting data in the neighboring analysis frames. This approach has its roots in the phase vocoder technique and its efficient transform- domain implementation. For periodic or pseudo- periodic musical tones it is unnecessary to resort to an overly generic analysis method, because the frequencies of the harmonic components are known after the estimation of the fundamental frequency. Advantages of the proposed filter- based analysis method (compared with the more general FFT- based techniques) are simplicity, which follows mainly from the small number of parameters, and the possibility of designing filter coefficients in closed form. Additionally, the resulting decomposition is obtained directly as a set of time- domain signals, and no separate synthesis stage is required.

Other signal processing methods proposed for analyzing the harmonic structure of musical signals include wavelets (Evangelista 1993) and high-resolution tracking methods ({Badeau, David, and Richard 2006}). These methods provide excellent frequency accuracy at the expense of a complicated algorithm and a high computational cost. The method proposed in this article can also provide amplitude and frequency accuracy that is sufficient for musical signal analysis, but at the same time, the analysis method remains easy to apply.

In this article, we first discuss the theory behind canceling and selecting partials using fractional delay filters. Then, we present three test cases to demonstrate the power of this approach in musical signal analysis. Sound samples corresponding to these examples are provided on the forthcoming Computer Music Journal DVD (to be released with the Winter 2008 issue).

Figure 1.

(a) Conventional ICF and (b) a fractional-delay filter based ICF (after Välimäki, Ilmoniemi, and Huotilainen 2004). Two other ICF structures are presented in (c) and (d) that allow the use of a fractional- delay filter Hfd(z) of arbitrary order for signals with both (c) low and (d) high fundamental frequencies.

Fractional- Delay Inverse Comb Filters

The inverse comb filter (ICF) is a finite impulse response (FIR) filter in which input signal is delayed by L samples and is then subtracted from the original input signal (see Figure 1a). Following the convention of Steiglitz (1996), the term "inverse comb filter" is used for the feedforward system with a delay line. The "comb filter," in contrast, has a delay line...