Cover of track Heisenberg phase modulation function by Inavon
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Heisenberg phase modulation function

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Here I demonstrate how much modulator modulates the carrier in the Heisenberg's phase modulation. The purpose is so that you know how much you're modulating based on modulation values.

When the modulator's value is 0 (the value is in the range [-1,1]), no phase shift of the carrier occurs. The amount of phase shift is a positive linear function of the value of the modulator when the amount of modulation is constant.

I set out to find the phase shift as a function of amount of modulation and value of modulator. I used the sawtooth down as the modulator because it is linear and starts at 1 and ends at -1, and a sine wave at 0 Hz as the carrier so that the phase is always (0 + phase shift). Note that there is an imperfection at the jump of the sawtooth wave because it's band limited.

At modulation amount = 0, nothing is heard (no phase shift).

At modulation amount = 50, a sine wave is heard at the modulator's frequency. This means that for every wave of the modulator, a wave of carrier is played. The carrier is shifted 1/2 wave at modulator = 1 and -1/2 wave at modulator = -1, or range [-π,π].

When the amount of modulation is 100, a sine wave is heard at 4 times the modulator's frequency (carrier shift is 2 waves at modulator = 1, -2 waves at modulator = -1, or range [-4π,4π]).

I extrapolated from these two points that at a modulation amount of sqrt(3)*100/2≈86.603, the carrier would be shifted 3/2 wave in each direction (range [-3π,3π]), and at sqrt(2)*100/2≈70.711, it would be shifted 1 wave in each direction (range [-2π,2π]. I was correct so I could be reasonably certain that the phase shift is

s = modulator*π*(2x/100)^2

where "s" is the phase shift in radians, "x" is the amount of modulation, and "modulator" is value of the modulator in [-1,1]. Alternatively,

λ = (modulator*(2x/100)^2)/2

where λ is the number of wavelengths of the carrier.

If you modulate the carrier using two modulators, then the phase shifts add.

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