Symbolic Dynamics of Squave Waves
Symbolic Dynamics of Squave Waves
Demonstration 1: Symbol Definitions
The instaneous values of pairs of pulse waveforms may be used to define symbols which change over type. The resulting sequence of symbols and their generation may be viewed as a simple symbolic dynamics system. Properties of the symbolic dynamics symbol sequence can be used to implement patent pending new approaches to frequency comparators and other related functions.
The novel frequency comparator implementations may be realized in either hardwave or software, and have several surprising advantages over known classical approaches. These advantages include only feedback-free structures (as only original signals and feed-forward paths are involved), operation over a very wide frequency range, no need for quadrature input signals, and expanded to include more than two signal inputs and detections of attributes of asymmetric pulse waveforms. Additionally, implementations may employ either state or transition analysis and may be event-driven or periodically sampled.
States Dynamics of a Square Wave Pair
A 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
This is illustrated in the animation below. Here the relative frequencies are in a ratio of 5/6: in particular the red curve has frequency 3.0 Hz and the blue curve has frequency 2.5 Hz.
The state dynamics of the pair of pulse waveforms may also be represented using a positive/negative relative frequency viewpoint. The red curve is thus represented as having "positive frequency" relative to average frequency of (2.5 Hz + 3.0 Hz)/2 = 2.75 Hz, while the blue curve is thus represented as having "negative frequency" relative to average frequency of 2.75 Hz. The beat frequency, which may be viewed as the "wave-to-wave separation rate" in the positive/negative relative frequency model is the difference in frequency | 2.5 Hz + 3.0 Hz | = 0.5 Hz. The separation rate from the center is half the frequency difference 0.5 Hz. = 0.25 Hz. All numbers scale linearly, for example Hz may be replaced with KHz, MHz, etc.
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].
REFERENCES
[1] Susan G. Williams, "Introduction to Symbolic Dynamics," in Symbolic Dynamics and Applications, American Mathematics Society, Providence, 2004, pp. 1-11, ISBN 0-8218-3157-7.
[2] H. Furstenberg and E. Glasner, "Robert Ellis and the Algebra of Dynamical Systems," in Topological Dynamics and Applications, American Mathematics Society, Providence, 1998, pp. 3-12, ISBN 0-8218-0608-4.