Subtractive Synthesis

In prior chapters we found that spectral shape is important to our perception of sounds, such as vowel/consonant distinctions, the different timbres of the vowels eee and ahh, etc. We also discovered that sinusoids are not the only way to look at modeling the spectra of sounds (or soimd components), and that sometimes just capturing the spectral shape is the most important thing in parametric sound modeling. Chapters 5 and 6 both centered on the notion of additive synthesis, where sinusoids and other components are added to form a final wave that exhibits the desired spectral properties. In this chapter we will develop and refine the notion of subtractive synthesis and discuss techniques and tools for calibrating the parameters of subtractive synthesis to real sounds. The main technique we will use is called Linear Predictive Coding (LPC), which will allow us to automatically fit a low-order resonant filter to the spectral shape of a sound. 

The basic idea of subtractive synthesis is to use a complex soimd soiuce to excite resonant filters. Certain types of spectrally rich sounds are simple to 

Periodic impulse train has equal-amplitude harmonic spectrum 

From a signal processing point of view, the resonator acts as a filter (or bank of filters) applied to a non-linear (i.e. noisy) excitation signal. Emulation of various sound spectra is achieved by combining the production of suitable source signals with the specification of appropriate coefficients for the filters. 

Subtractive synthesis has been used to model percussion-like instruments and the human voice mechanism, but the results are somewhat inferior to those obtained with other source 

In general, a filter is any device that performs some sort of transformation on the spectrum of a signal. For simplicity, however, in this section we refer only to filters which cut off or favour the resonance of specific components of the spectrum. In this case, there are four types of filters, namely low-pass (LPF), high-pass (HPF), band-pass (BPF) and band-reject (BRF). 

The BRF amplitude response is the inverse of a BPF. It attenuates a single band of frequencies and discounts all others. Like a BPF, it is characterised by a central frequency and a bandwidth but another important parameter is the amount of attenuation in the centre of the stopband. 

Under special conditions a BPF may also be used as an LPF or an HPF. An LPF can be simulated by setting the BPF s centre frequency to zero. In this case, however, the cut-off frequency of the resulting low-pass is 70.7 per cent of the specified bandwidth, and not 50 per cent. For example, if the desired LPF cut-off frequency is to be 500 Hz, the bandwidth value for the BPF must be 1 kHz multiplied by 1.414 (that is, 1000 x 1.414). This is because the BPF is a two-pole filter at its cut-off frequency (where output is 50 per cent) the output power of a true LPF of the same cut-off would be in fact 70.7 per cent. 

---

Source-filter Synthesis (Subtractive synthesis, Karplus-Strong Plucked string algorithm [Karplus and Strong, 1983])... 

Speech spectra have important features called formants which are the three five gross peaks in the spectral shape located between 200 and 4000 Hz. These correspond to the resonances of the acoustic tube of the vocal tract. We will discuss formants further in Chapter 8 (Subtractive Synthesis). For now, we will note that the formant locations for the ahh vowel are radically different from those for the eee vowel, even though the harmonic spacing (and thus, the perceived pitch) is the same for the two vowels. We know this, because a singer can sing the same pitch on many vowels (different spectral shapes, but same harmonic spacings), or the same vowel on many pitches (same spectral shape and formant locations, but different harmonic spacings). 

In Chapter 4, we developed the notion that individual resonant filters can be used to model each vibrational mode of a system excited by an impulse. Thus, modal synthesis is a form of subtractive synthesis. For modeling the gross peaks in a spectrum, which could correspond to resonances (although these resonances are weaker than the sinusoidal modes we talked about in Chapter 4), we can exploit the resonance-factored form of a filter to perform our subtractive synthesis. The benefits of this are that we can control the resonances (and thus, spectral shape) independently. The filter can be implemented in series or cascade (chain of convolutions) as shown in Figure 8.4. The filter can also be implemented in parallel (separate subband sections of the spectrum added together), as shown in Figure 8.5. 

But how do we know how to set the filter parameters (coefficients and gains) for subtractive synthesis If we have control over the input and access... 

Subtractive synthesis uses a complex source wave—such as an impulse, a periodic train of impulses, or white noise—to excite a spectral-shaping filter. Linear prediction, or linear predictive coding (LPC), gives us a mathematical technique for automatically decomposing a sound into a source and a filter. For low order LPC (6-20 poles or so), the filter is fit to the coarse spectral... 

In prior chapters we looked at subtractive synthesis techniques, such as modal synthesis (Chapter 4) and linear predictive coding (Chapter 8). In these methods a complex source is used to excite resonant fQters. The source usually has a flat spectnun, or exhibits a simple roll-off pattern like f or ip (6 dB or 12 dB per octave). The filters, possibly time-varying, shape the spectrum to model the desired sound. 

From our experience with subtractive synthesis and LPC (Chapter 8), it might occur to us to fit a filter to the resonant sound and decompose the sound into a source/filter system. Figure 13.11 shows the results of deconvolving the resonance from the maraca sounds of Figure 13.10. 

The resynthesis process results from two simultaneous synthesis processes one for sinusoidal components and the other for the noisy components of the sound (Figure 3.15). The sinusoidal components are produced by generating sinewaves dictated by the amplitude and frequency trajectories of the harmonic analysis, as with additive resynthesis. Similarly, the stochastic components are produced by filtering a white noise signal, according to the envelope produced by the formant analysis, as with subtractive synthesis. Some implementations, such as the SMS system discussed below, generate artificial magnitude and phase information in order to use the Fourier analysis reversion technique to resynthesise the stochastic part. 

Figure 4.3 The basic building block of subtractive synthesis is the band-pass filter (BPF)...

There have been several attempts to minimise the limitations of subtractive synthesis, ranging from better filtering strategies to the use of alternative source modules. However successful these attempts have been, none of them matches the realism of a physical model. 

Synthesising human vocal sounds using subtractive synthesis... 

Figure 6.3 The subtractive synthesis of formants is determined by two main components excitation source and resonator. The former produces a raw signai that is shaped by the iatter...

In Chapter 4 we introduced the physiology of the human vocal mechanism from a physical modelling point of view. Here we will study the vocal tract from a subtractive synthesis standpoint. 

The slots must be loaded with synthesis parameter values in order to instantiate a certain sound. For each different sound produced by the instrument there is a corresponding instantiation. For example, considering the case of a subtractive synthesis instrument which simulates the vocal tract mechanism (see Chapter 6) an instantiation would correspond to the set of parameters for the simulation of a specific position of the vocal tract when producing a certain sound. 

---