Phase distortion synthesis

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Figure A depicting the saw wave phase distortion transform

Phase distortion synthesis is a synthesis method introduced in 1984 by Casio in its CZ range of synths, and similar to Frequency modulation synthesis in the sense that they are both built on phase modulation. Basically a sine wave is played, but by modifying the phase angle, the sine wave is bent out of shape.

Figure A shows how a sine wave gradually turns into a saw wave as the readout phase angle gets more and more distorted. Depending on whether a frequency counter x, is above or below the current distortion point d (the sharp knee in the phase angle insert) one of two equations must be solved:

 if(x > d) 
   return sin( x * d / pi);
 else  
   return sin((x-d) * pi / (1-d)  +  pi);

Figure B depicting the square wave phase distortion transform

Figure B is similar to A, only showing the development of a square wave instead. For x values below the first knee in the phase angle insert, the y values will vary rapidly between -3.14 and 0 resulting in the initial sharp rise of the square. Values between the first knee and the center knee all equals 0.0 resulting in the squares flat ceiling in the first half wave. At this point there is again a sudden rise in the phase angle up to 3.14, resulting in the fall of the square wave down to its flat floor at -1.0 where it stays until the frequency counter wraps around and repeats the process.

Other phase distortion patterns that are supported on the CZ range are impulse, half-sine and double impulse. The CZ synths also generate synchronised and windowed sine waves in order to emulate resonant filter sweeps.

Internally, phase distortion synthesis works by reading a sine wave table stored in memory. The most significant bits of an initial linear frequency counter is under envelope control transformed into a secondary phase angle signal which is then used to read the sine wave. Depending on the wiring, the transform will change and a wide variety of waveshapes can be produced.

Figure 19 from the uspto CZ-series patent application depicting how to eliminate the sudden jumps in the variable resonance circuitry (here showing the second harmonic coming into view.)

The phase transforms are all assembled from piecewise linear functions under binary logic control and shows characteristic sharp knees (and for some transforms, even sudden jumps) as they move from minimum to maximum, where the frequency counters accumulator wraps around and starts over. The sharp knees are smoothened out by the roundness of the modulated sine wave and not too noticeable in the resulting signal.

[edit] Simulating a resonant filter

Figure 19 from the 1985 CZ-series patent shows how to emulate the variable resonance found in analogue voltage controlled filters:

(a) The base frequency counter, wrapping around every period.

(b) The resonance frequency counter at a slightly higher frequency, being reset (or "synced") when the base counter wraps around.

(d) The inverted base frequency counter.

(e) Multiplying c by d. The sudden jump in c is now leveled out.

Casio made five different synthesizers using this method of phase distortion synthesis.

The VZ-1's synthesis method ("Interactive phase distortion") includes Frequency modulation synthesis as well as an improved version of phase distortion.