Modal Expansion

A special case of transfer-function modeling is known as modal synthesis [9,10,143,380]. It can also be seen as the basis for formant synthesis and the source-filter synthesis model (resonators excited by various means). In this technique, the physical system is characterized in terms of its resonant modes. It finds extensive application in industry for determining parametric frequency responses from measured vibration data. Each mode is typically described in terms of its resonant frequency, bandwidth (or damping), and gain (and perhaps phase).

Since the modal parameters specify the spectral formants of the system, modal synthesis can be regarded as including formant synthesis so often applied to the synthesis of voice [220,262,390,40,83,500,499,318,367,298]. Since the importance of spectral formants in sound synthesis has more to do with the way we hear than with the physical parameters of a system, formant synthesis is best viewed as spectral modeling synthesis techniques as opposed to a true physical modeling technique [440]. An exception to this rule may occur when the physical system truly consists of a parallel bank of second-order resonators, such as an array of tuning forks or Helmholtz resonators. In that case, the mode parameters correspond to to physically independent objects. However, this is rare in practice.

Since only the modes in the range of human hearing need be retained, and since ``uncontrollable'' or ``unobservable'' modes can be left out, the modal representation is generally more efficient than an explicit mass-spring-dashpot digitization such as obtained using wave digital filters. On the other hand, since the modal representation normally sacrifices a physical description, it may be difficult or impossible to add nonlinearities that behave naturally.

Given any order $n$ transfer function $H(s)$ describing the input-output behavior of a physical system, a modal description can be obtained immediately from the partial fraction expansion:

$$H(s) \isdef \frac{B(s)}{A(s)} \isdef \frac{b_m s^m + \cdots b_1 s... ...od_{i=1}^m (s-\xi_i) }{\prod_{i=1}^n (s-p_i) } = \sum_{i=1}^n \frac{r_i}{s-p_i}$$ (Q.4)

where $p_i$ denotes the $i$th pole, $\xi_i$ is the $i$th zero, and $r_i$ is called the residue of the $i$th pole [455]. (For simplicity of notation, (Q.4) is written for the case $m<n$ and $p_i$ distinct.) Since the system is real, complex poles will occur in complex conjugate pairs. Thus, for each complex term $r_i/(s-p_i)$ there is also the term $\overline{r}_i/(s-\overline{p}_i)$, where $\overline{z}$ denotes the complex conjugate of $z$, and first-order complex terms can be combined to provide half as many real second-order terms:

$$\frac{r_i}{s-p_i} + \frac{\overline{r}_i}{s-\overline{p}_i} = \fr... ...\vert p_i\right\vert^2} \isdef \frac{b_{i1} s + b_{i0}}{s^2 - a_{i1}s + a_{i0}}$$ (Q.5)

Let $n_c$ denote the number of complex pole pairs, and $n_r = n-2n_c$ be the number of real poles. Then the modal expansion can be written as

$$H(s) = \sum_{i=1}^{n_r} \frac{r_i}{s-p_i} + \sum{i=1}^{n_c} \frac{b_{i1} s + b_{i0}}{s^2 - a_{i1}s + a_{i0}}$$ (Q.6)

Note that each component second-order mode consists of one zero and two poles. The zero is necessary to adjust the phase of the resonant mode to obtain all possible responses at frequencies between resonances (from smoothly varying to the appearance of a deep notch between them). Also, note that a general modal expansion requires provision for real poles in addition to resonances.

A typical implementation of the modal representation is by means of second-order filter, or biquad (biquadratic transfer function), which is easily converted from continuous- to discrete-time form. The final discrete-time model may thus consist of a parallel bank of second-order filters

$$H_d(z) = \sum_{i=1}^N G_i \frac{1 + b_i z^{-1}}{1 + a_{i1} z^{-1} + a_{i2} z^{-2}}$$ (Q.7)

where $N$ denotes the appropriate order. (Note that real poles can also be paired into second-order sections, and a second-order section can degenerate to provide a single real pole.) The filter coefficients $b_i$ and $a_{ij}$ are now digital second-order section coefficients which can be computed from the continuous-time filter coefficients if they are known, or they can be directly estimated from discrete-time data.

Expressing complex poles of $H_d(z)$ in polar form as $p_i \isdef R_i e^{j\theta_i}$, (where now we assume $p_i$ denotes a pole in the $z$ plane), we obtain the classical relations

$$a_{i1} = p_i + \overline{p}_i = 2 R_i \cos(\theta_i) \protect$$ (Q.8)

$$a_{i2} = \left\vert p_i\right\vert^2 = R_i^2. \protect$$ (Q.9)

Given for each mode its resonant frequency $F_i$ Hz, bandwidth $B_i$, gain $G_i$, and phase $\phi_i$, we can compute the modal parameters as [455]

$$\begin{eqnarray*} \theta_i & = & 2\pi f_i / f_s\\ R_i & = & e^{-\pi B_i/f_s} \\ b_i & = & \lim_{z\to p_i} (1-p_iz^{-1})H_d(z) \end{eqnarray*}$$

and these values are substituted into Equations (Q.8-Q.7) to obtain the final representation in terms of second-order digital filter sections. Sections implementing two real poles, of course, are not conveniently specified in terms of resonant frequency and bandwidth. In such cases, it is common to work with the filter coefficients $b_i$ and $a_i$ directly, as they are often computed by system-identification software.

Given a model in terms of elemantary modes, numerous techniques exist for measuring the modal parameters. For example, resonant modes can be estimated from the magnitude, phase, width, and location of peaks in the Fourier transform of a recorded (or estimated) impulse response (see Appendix R). Another well-known technique is Prony's method [455, Section G.4.4], [298], and there are more advanced methods for obtaining parametric fits to exponentially decaying sinusoids [280].