Symbolic Dynamics of Squave Waves
Symbolic Dynamics of Squave Waves
Demonstration 2: Torus Representation
States Dynamics of a Square Wave Pair
As described in the previous demonstration page, time-dependent state may be associated with pairs of pulse waveforms by representing the instantaneous measured values of the two waveforms as a two-component vector. For example, a first pulse wave signal A and second pulse wave signal B may each have values of 0 or 1 at any particular time (ignoring noise, transition, and transient phenomena). There would be four resulting states, for example named by symbols S0, S1, S2, S3 defined using vector notation Sx = A B and indexing formula S(2a+b) where "a" is the instantaneous value of A and "b" is the instantaneous value of B:
S0 = 0 0
S1 = 0 1
S2 = 1 0
S3 = 1 1
The collection of all states of a system is called the "state space" of the system, and paths tracing the sequence of states over time through the state space are called "trajectories." The collection of all four above states is a four-element set and may be represented in various ways, such as a four-point lattice. However, as the pulse waveforms themselves are produced from a parent dynamical process (for example, an underlying continuous oscillator or other periodic phenomena), it is useful to embed the the pulse-wave state space into a larger state space representing the parent dynamical process. This embeding may be achieved by quantizing the larger representing parent dynamical process state space into four subspaces. This is in keeping with the origins of symbolic dynamics study as proposed and refined by Hadamard, Poincaré, Birkhoff, Morse, Hedlund, and others [1,2].
The figure below illustrates a torus representation for the state-space of two continuous-valued oscillators. Here the periodic oscillations of one oscillator (oscillator A) are represented as periodic motion in a vertical circle, while the periodic oscillations of the second oscillator (oscillator B) are represented as periodic motion in a horizontal circle.
The combined actions of the two circular patterns sweep out the donut/bagel shape of a torus. If one associates the timekeeping within each period of each oscillator with a point moving in a circle, and keeps track of this moving point for the combined pair of oscillators, the center of one circle is located along the circumference of the other circle in a manner similar to adding the components of orthogonal vectors. As the state of each oscillator evolves, a trajectory is traced out along the surface of the torus as illustrated in the figure above.
If one of the oscillators is faster than the other, the trajectory will wrap around the surface of the torus faster in its rotation direction than the other oscillator will wrap around the surface in its rotation direction. Note that all that is needed of the torus is its surface, and its interior is, for present purposes, hollow. In more formal terms, this torus surface represents a continuous-valued state, continuous-valued time dual oscillator manifold, often used to describe or study uncoupled and coupled linear and nonlinear differential equations and other systems.
The figure below illustrates how the torus manifold may be symmetrically quantized into regions associated with the symbols defined above. In particular, the symmetric quantization corresponding to the case where the input signals are symmetric square waves is depicted. The square wave model is naturally obtained by quantizing the continuous-valued continuous-time oscillator torus manifold into four regions corresponding to the symbols {S0, S1, S2, S3} as shown.
Portions of trajectory paths on the torus surface (manifold) may then be characterized according to which of the quantized sections they lie in, each corresponding to the symbols {S0, S1, S2, S3}. Referring to the left side of the torus above, oscillations occurring only in horizontally-oriented loops (corresponding to, for example, oscillator A) alternate between symbols S0 and S2 while oscillations occurring only in horizontally-oriented loops on the right side of the torus alternate between symbols S1 and S3. Similarly, oscillations occurring in vertically oriented loops (corresponding to, for example, oscillator B) alternate between symbols S0 and S1 in the lower portion of the torus and between symbols S2 and S3 in the upper portion of the torus. With both oscillators A and B active, the trajectories may then cross through all four sections generating an event-driven symbol sequence comprising the symbols {S0, S1, S2, S3}. As a trajectory progresses from section to section, it may be thought of as generating an event-driven symbol sequence comprising the symbols {S0, S1, S2, S3}.
The following animation illustrates a trajectories where the horizontally-oriented oscillator oscillates 2 times (blue curve) and 2 times (red curve) as fast as the vertically-oriented oscillator. The current state in each trajectory is indicated with the moving dot of corresponding color. The symbol associated with the current state (represented by location on the torus manifold) is displayed in the corresponding color.
Asymmetric Pulses
The dynamic behavior of pairs of asymmetric pulse waveforms can be rendered in terms of modified versions of the symbolic dynamics models introduced for the symmetric case. The figure below illustrates a variation of the symmetrically quantized torus considered above adapted for use with asymmetric pulse waveforms. Here, the region boundaries determining the quantization thresholds on the continuous-valued, continuous-time oscillator torus surface may be adjusted (as indicated by the arrows) so as to obtain different duty cycles. Clearly, this affects the resulting event-driven symbol sequence. For example, a rising trajectory rotating clockwise and having current state S0 may next attain any of states S0 , S1 , or S2 depending on the location of the variable movable boundary for oscillator A.
Application of Theory to Frequency Comparsion
By detecting simple patterns in sequences of symbols generated in the manner described above, it is possible to determine which waveform has the higher frequency. As the relation between transitions in the two non-phase-locked pulse waveforms fluctuates over time, there will be intervals over which the wave of higher frequency will make two consecutive transitions between its higher and lower amplitudes while the wave of lower frequency makes no such transition. This creates a symbolic signature that characterizes relative frequency and duty cycle relationships among the square wave signals. The approach, discussed here for symmetric pulse waves (square-waves), may also be extended to explain the dynamics of asymmetric pulse waves.
Intervals over which the wave of higher frequency will make two consecutive transitions between its higher and lower amplitudes while the wave of lower frequency makes no such transition may be called "enveloping events." It may be seen from the figure below that there are eight types of enveloping events.
Comparable enveloping events may occur wherein either of the square waves having a given or opposite polarity, giving four types of events. Either square wave may be the "faster" (higher frequency) one, giving two cases for these four types, or eight cases altogether. These eight cases may be organized using the state symbols S0, S1, S2, and S3 as follows:
Cases where B is faster than A:
S2 S3 S2
S3 S2 S3
S0 S1 S0
S1 S0 S1
Cases where A is faster than B:
S1 S3 S1
S3 S1 S3
S0 S2 S0
S2 S0 S2
Noting the symmetric form of each of these, a determanation "signature" that one square wave has a faster rate (higher frequency) than another is the existence of "symmetry events" of the form
wpq = Sp Sq Sp
since it is impossible for two square waves of the same frequency to have any of these eight symmetric symbol sequences.
The animation below shows trajectories on the torus together with the detection of symmetry events.
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