Sub-Octave Waveform Cross-Products
- The red waveforms are peer-cross products;
- The green waveforms involve a one-octave difference;
- The cyan waveforms involve a two-octave difference;
- The violet waveforms involve a three-octave difference.
- As with the square wave cross-product case, the waveforms above the peer-cross product diagonal are time-reversals of the waveforms below the diagonal, with mirror-symmetry.
- As with the square wave cross-product case, moving along the diagonal simply increases or decreases the rate of similarly colored waveforms by factors of two.
- Unlike the square wave cross-product case, no waveform instaneously vansishes; rather each continuously morph through configurations of piece-wise parabolic curve segments. In some intervals a parabolic shape is quite evident, while in others segments appear almost linear.
- Unlike the square wave cross-product case, there does not appear a well-defined time-reversals nor mirror symmetry of waveforms below the diagonal compared with waveforms above the diagonal;
- As with the square wave cross-product case, moving along the diagonal simply increases or decreases the rate of similarly colored waveforms by factors of two.
- Unlike the square wave cross-product case, no waveform instaneously vansishes; rather each continuously morph through configurations of piece-wise linear line segments, some with discontinous jumps.
- Above the diagonal, all waveforms are piece-wise continuous;
- Below the diagonal, all waveforms include discontinous jumps;
- On the diagonal, the waveforms periodically morph between piece-wise continuous triangle waves and discontinous sawtooth waves. The periodic morph includes both inverted and non-inverted versions of these triangle and sawtooth waves. The sawtooth waves when fully formed have no constant ("D.C.") term, while the triangle waves when fully formed exhibit a constant ("D.C.") term of half its amplitude. The triangle waves when they are fully formed have half the amplitude of the sawtooth waves when they are fully formed.
Sub-Octave Waveform Cross-Products
Demonstration 2: Sub-Octave Cross-Products of Triangle Waves
Sub-octave square-wave cross-product signal processing technology is introduced in [1], productized in [2], developed further in U.S. Patent 6,849,795 [3], and explained further in the associated whitepaper [4] and a future publication. In this webpage, extensions to the case of triangle waves is considered.
The figure below depicts a general implementation framework for multiple-octave cross-products of arbitrary waveforms. Two audio sources (for example, oscillators) provide signals of different audio frequencies. Typically these frequencies are nearly but strategically not quite at a consonant musical interval (i.e., related by a ratio of small integers). Frequency dividers, frequency multipliers or pitch shifters are used to produce a range of octaves from the original signal of each of the two audio sources. The corresponding signals are pair-wise multiplied together and mixed with the original audio signals and their corresponding octave signals. The resulting mixes may be subjected to signal processing and/or used to create a stereo or other multiple-channel sound field. If the waveforms produced by the audio sources and octave frequency dividers, frequency multipliers, or pitch shifters are anti-symmetric over the waveform period (as is the sine wave, square wave, and triangle wave), the spectrum (for example, Fourier series) of each waveform will comprise only odd harmonics. In this case, the multiples of powers of two in frequency produced by the octave frequency dividers, frequency multipliers, or pitch shifters multiply the frequencies of the odd harmonics. The result is a spectral partition so that the octave frequency reducing (or increasing) chain separates the overall collection of frequencies into non-overlapping groups. This, together with the multiplying cross products which produce low-frequency modulation (beat-frequency) effects and other harmonic enrichment, provide a rich pallet for further parallel signal processing and spatial distribution over a stereo or other multiple-channel sound field.
Triangle waves of a given frequency can be readily transformed into triangle waves of frequencies that are octaves higher or lower though the use of simple electronic circuits or algorithms. One exemplary approach for frequency doubling of triangle waves is to use the operation:
If [ x<1] then output is 2x, else output is 2(1-x)
on a triangle wave centered at zero with amplitude range [-1,1]. Absolute value operations can also be used in various configurations. These transformations may be used repeatedly to create a frequency doubling chain. One exemplary approach for frequency halving of triangle waves is to multiply the triangle wave by +/- 1 depending on the sign of its time derivative. This transformations may be used repeatedly to create a frequency reducing chain. A number of other techniques are possible and offered in a forthcoming publication.
The following animation illustrates sub-octave cross-products of triangle waves.
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In the animation, a positive and negative relative frequency representation is used wherein the waveform dynamics are depicted with respect to the average frequency of the two original sources. The black waveforms along the left side and top represent the source and derived octave waveforms of various octaves. The row at the top represent signals from the lower frequency source and appear to have negative relative frequency with respect to the average of the two frequencies. The column at the far left represent signals from the higher frequency waveforms appear to have positive relative frequency with respect to the average of the two frequencies.
Note several properties:
The following animation illustrate the interesting case of the sub-octave cross-products of triangle waves with square waves (each with no contanst ("D.C.") offsetting term).
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Note several properties:
REFERENCES
[1] L. Ludwig, "A Square-Wave Frequency-Division Sub-Octave Cross-Product Module," Electronotes, Vol. 11, No.98, February 1980, pp. 9-15.
[2] Synthesis Technology, "MOTM-120 Sub-Octave Multiplexer Module," product description, http://www.synthtech.com/motm120.html.
[3] U.S. Patent 6,849,795 "Controllable Frequency-Reducing Cross-Product Chain," February 11, 2005.
[4] NRI Whitepaper, "Controllable Frequency-Reducing Cross-Product Chain," available at XXXXXXXXXXXXXXX, October 2005.