Sub-Octave Waveform 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.
- The waveforms above the peer-cross product diagonal are time-reversals of the waveforms below the diagonal, with mirror-symmetry.
- Moving along the diagonal simply increases or decreases the rate of similarly colored waveforms by factors of two.
Sub-Octave Waveform Cross-Products
Demonstration 1: Sub-Octave Cross-Products of Square Waves
The sub-octave square-wave cross-product signal processing technology introduced in [1] and productized in [2] is developed further in U.S. Patent 6,849,795 [3] and explained further in the associated whitepaper [4] and a future publication. The simple operations involved may be readily realized in inexpensive logic circuitry or dramatically modest line-count software.
The figure below depicts a general implementation framework for 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 (single or multiple channel output) 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.
In the square wave version of this technology, two square waves of different frequencies and their derived sub-octaves are submitted to a logical "AND" or "Exclusive-OR" ("XOR") operations. As the pulse edges migrate past one another in time, the logical "AND" or "Exclusive-OR" ("XOR") operations create pulses of varying widths. The "AND" operation may be viewed as a "mutual gating" type of multiplying function with either square wave periodically turning the transmission of the other square wave on and off. The following animation illustrate the sub-octave cross-products of square waves utilizing logical "AND" operations as the multiplying function.
|
||||||
|
 |
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 (and hence make left to right, as negative frequency can be interpreted as time reversal by changing the association of the minus sign). 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. The red waveforms are peer-cross products, and exhibit through-zero pulse width modulation. The other colored waveforms represent non-peer, or off-diagonal cross-products:
Note the generic structural properties:
It also turns out that in listening to the "AND" of two audio frequency square waves one hears the two square waves themselves mixed equally with the "Exclusive-OR" of the two square waves. It can be shown that "Exclusive-OR" corresponds to classical amplitude modulation. Thus the case where a logical "Exclusive-OR" operation is used as the multiplying function is also of interest. This is shown in the animation below.
|
||||||
|
 |
Both the logical "AND" or "Exclusive-OR" ("XOR") operations produce through-zero pulse-width modulation processes, initially described in [5]. More specifically, these operations on the square wave pair produce the type of through-zero pulse-width modulation which resembles the symmetric 100% pulse-width modulation of an audio frequency triangle wave by a low-frequency triangle wave. The resemblance can be mathematically formalized wherein the audio frequency triangle wave has the average frequency of the two square waves while the pulse-width modulating low-frequency triangle wave has a frequency proportional to the difference in the frequency of the two square waves.
The peer-cross products for the XOR-products produce two varying with pulses which are centered, vanish and reappear in the same location. The peer cross-products for the AND operations jump between these same two locations, alternately vanishing at one and reappearing at the other, giving this version of the process (equivalent to a mix of the XOR case with the source waveforms) a rich character.
The through-zero pulse-width modulation processes, together with spectral partitioning invoked by sub-octave square-wave frequency division described above, give the sub-octave square-wave cross-product signal processing technology its characteristic sounds and variational mixing 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.
[5] L. Ludwig and B. Hutchins, "A New Look at Pulse-Width Modulation -- Part 3," Electronotes, Vol. 12, No.118, October 1980, pp. 3-18.