Phase-aligned formant (PAF) generator

Combining the two-cosine carrier signal with the waveshaping pulse generator gives the phase-aligned formant generator, usually called by its acronym, PAF. The combined formula is,

$$x[n] = { {\underbrace {g \left ( a \sin (\omega n/2) \ri... ...) + q \cos( (k+1) \omega n)} \right ] } _\mathrm{carrier}} }$$

Here the function $g$ may be either the Gaussian or Cauchy waveshaping functions, $\omega$ is the fundamental frequency, $a$ is a modulation index controlling bandwidth, and $k$, $p$, and $q$ control the formant center frequency.

Figure 6.8 shows the PAF as a block diagram. The block diagram is separated into a phase generation step, a carrier, and a modulator. The phase generation step outputs a sawtooth signal at the fundamental frequency.

Figure 6.8: The PAF generator as a block diagram.

The modulator is done by standard waveshaping, with a slight twist added. The formula for the modulator signals in the PAF call for an incoming sinusoid at half the fundamental frequency, i.e.,

$$\sin({\omega \over 2})$$

and this nominally would require us to use a phasor tuned to half the fundamental frequency. However, since the waveshaping function is even, we may substitute the absolute value of the sinusoid:

$$\left \vert \sin({\omega \over 2}) \right \vert$$

which repeats at the frequency $\omega$ (the first half cycle is the same as the second one.) We can compute this simply by using a half-cycle sinusoid as a wavetable lookup function (with phase running from $-\pi/2$ to $\pi /2$), and it is this rectified sinusoid that we pass to the waveshaping function.

Although we picture the wavetable function over both negative and positive values (reaching from -10 to 10), in actuality we're only using the positive side for lookup, ranging from 0 to $b$, the index of modulation. If the index of modulation exceeds the input range of the table (here set to stop at 10 as an example), the table lookup address should be clipped. The table should extend far enough into the tail of the waveshaping function so that the effect of clipping is inaudible.

The carrier signal is a weighted sum of two cosines, whose frequencies are increased by multiplication (by $k$ and $k+1$, respectively) and wrapping. In this way all the lookup phases are controlled by the same sawtooth oscillator.

The quantities $k$, $q$, and the wavetable index $b$ (labeled ``index" in the figure), are calculated as shown in Figure 6.9. They are functions of the specified fundamental frequency, the formant center frequency, and the bandwidth, which are the original parameters of the algorithm. The quantity $p$, not shown in the figure, is just $1-q$.

Figure 6.9: Calculation of the time-varying parameters $a$ (the waveshaping index), k, and q for use in the block diagram of Figure 6.8.

As described in the previous section, the quantities $k$, $p$, and $q$ may only change at phase wraparound points, that is to say, at periods of $2 \pi / \omega$. Since the calculation of $k$, etc., depends on the value of the parameter $\omega$, it follows that $\omega$ itself may only be updated when the phase is a multiple of $2\pi$; otherwise, a change in $\omega$ could send the center frequency $(k+q)\omega$ to an incorrect value for a (very audible) fraction of a period. In effect, all the parameter calculations should be synchronized to the phase of the original oscillator.

Having the oscillator's phase control the updating of its own frequency is the first example we have seen of feedback, which in general means using any process's output as one of its inputs. When processing digital audio signals at a fixed sample rate (as we're doing), it is never possible to have the process's current output as an input, since at the time we would need it we haven't yet calculated it. The best we can hope for is to use the previous sample of output--in effect, adding one sample of delay. In block environments (such as Max, Pd, and Csound) the situation becomes more complicated, but we will delay discussing that until the next chapter (and avoid dealing with the problem in the examples at the end of this one).

The amplitude of the central peak in the spectrum of the PAF generator is roughly $1/(1+b)$; in other words, close to unity when the index $b$ is smaller than one, and falling off inversely with larger values of $b$. For values of $b$ less than about ten, the loudness of the output does not vary greatly, since the introduction of other partials, even at lower amplitudes, offsets the decrease of the center partial's amplitude. However, if using the PAF to generate formants with specified peak amplitudes, the output should be multiplied by $1+b$ (or even, if necessary, a better approximation of the correction factor, whose exact value depends on the waveshaping function). This amplitude correction should be ramped, not sampled-and-held.

Since the expansion of the waveshaping (modulator) signal consists of all cosine terms (i.e., since they all have initial phase zero), as do the two components of the carrier, it follows from the cosine product formula that the components of the result are all cosines as well. This means that any number of PAF generators, if they are made to share the same oscillator for phase generation, will all be in phase and combining them gives the sum of the individual spectra. So we can make a multiple-formant version as shown in figure 6.10.

Figure 6.10: Block diagram for making a spectrum with two formants using the PAF generator.

Figure 6.11 shows a possible output of a pair of formants generated this way; the first formant is centered halfway between partials 3 and 4, and the second at partial 12, with lower amplitude and bandwidth. The Cauchy waveshaping function was used, which generates linearly sloped spectra (viewed in dB). The two superpose additively, so that the spectral envelope curves smoothly from one formant to the other. The lower formant also adds to its own reflection about the vertical axis, so that it appars slightly curved upward there.

Figure 6.11: Spectrum from a two-formant PAF generator.

The PAF generator can be altered if desired to make inharmonic spectra by sliding the partials upward or downward in frequency. To do this, add a second oscillator to the phase of both carrier cosines, but not to the phase of the modulation portion of the diagram, nor to the controlling phase of the sample-and-hold units. It turns out that the sample-and-hold strategy for smooth parameter updates still works; and furthermore, mutiple PAF generators sharing the same phase generation portion will still be in phase with each other.

Our technique for superposing spectra does not work as predictably for phase modulation as it does for the PAF generator; the partials of the phase modulation output have complicated phase relationships and they seem difficult to combine coherently. In general, phase modulation will give more complicated patterns of spectral evolution, whereas PAF is easier to predict and turn to specific desired effects.