Symbolic Dynamics of Squave Waves

    Symbolic Dynamics of Squave Waves

    Demonstration 4: Implementation of Pulse Waveform Symbolic Dynamics Frequency Comparators

    As described in an earlier demonstration page, time-dependent state may be associated with pairs of pulse waveforms by representing the instantaneous values (0 or 1) of the waveforms as a two-component vector. For a first pulse wave signal A and second pulse wave signal B, one may use vector notation Sx = A B with indexing formula S(2a+b) (where "a" and "b" are the instantaneous values of A and B, respectively), the four resulting states are:

      S0 = 0 0
      S1 = 0 1
      S2 = 1 0
      S3 = 1 1

    Application of Theory to Frequency Comparsion

    By detecting simple patterns in sequences of the symbols 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 (termed "enveloping events") where the wave of higher frequency will make two consecutive transitions among 0 and 1 while the wave of lower frequency makes no such transition. The envelope events create a symbolic signature that characterizes relative frequency and duty cycle relationships among the square wave signals. 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

    It can be seen from the figure above that it is impossible for two square waves of the same frequency to have any of these eight symmetric symbol sequences. The adaptation of these ideas to produce hardware and algorithms for frequency comparators and realted extensions are covered in a pending patent [1] and associated theory [2] and implementation [3] whitepapers.

    Frequency Comparitor Implementations

    Hardware and software implementations are described in a companion whitepaper [3], and these have several advantages over some 50 prolific years of prior approaches (see, for example [4]-[22]). The advantages include:

    • Implementations are feedback-free as only original signals and feed-forward are involved;
    • Implementations may operate over a very wide frequency range;
    • No input signal needs to be in quadrature form;
    • Implementations may employ either state or transition analysis;
    • Implementations may be event-driven or periodically sampled;
    • Software implementations are small and suitable for microcode;
    • Hardware implementations are of small complexity and may be implemented as a dedicated integrated cicuit, a SoIC IP core, or with a modest amount of inexpensive commonly available components.

    Any of the implementation approaches may be further expanded to include more than two signal inputs and asymmetric pulse waveforms. Additionally, the theory and associated implementations imply considerable promise for further development in a number of theoretical and applications areas currently under study.

    State-Based Implementations

    The examplary state-based implementation provided in the pending patent application [1] and associated whitepaper [3] employs the use of retained history of the last two symmetry events. The real-time behavior for the symmetric square wave case is effectively equivalent to symmetry event reporting illustrated in the symbol, torus, and tiling animations of earlier demonstration pages. Because of this, no additional demonstration animation of state-based implementation is provided here, and the remaining attention is directed to the transition-based implementation.

    Transition-Based Implementations

    The examplary transition-based implementation provided in the pending patent application [1] and associated whitepaper [3] employs a different approach for detecting enveloping events. The real-time behavior for the symmetric square wave case employs a different process than the symmetry event reporting illustrated in the symbol, torus, and tiling animations of earlier demonstration pages. For this reason, a demonstration animation of a transistion-based implementation is provided below. Diagrams and theory of the implementations are detailed in the pending patent [1] and associated implementation whitepaper [3].

    Four flip-flops are set and reset according to the primitive actions of the rising and falling of pulse waveform edges. The states of these four flip-flops are submitted to combinational logic designed to indicate conditions that directly correspond to enveloping events. In the case of symmetric square waves, this provides direct indication as to which waveform is instaneously determined to be faster or if there is no current distinguishing instantaenous information ("ambiguity signal")..

     

    Latches and additional gates can be added to further process the above results so as to indicate which waveform was most recently determined to be faster, and provide the aforemention ambiguity siganl if that information is either instaneously current or pending an update.

    Exemplary Research, Demonstration, and Prototyping Environment

    The photograph below shows a hardware environment for research, demonstration and prototyping the ideas presented in these demonstration pages, the pending patent [1] and associated whitepapers [2]-[3].

    • The board on the left is an expanded state-based implementation, about one-third of which is relevant to the basic squarewave-driven state-based implementation. Some additional chips and LEDs are used to indicate state information and detailed symmetry event reporting, and the remaining chips, LEDs, switches, and surface patch wiring are used to support further study of asymmetric pulse waveform dynamics.
    • The sparsely populated and sparely wired board on the right is the transition-based implementation. (Unlike the state-based board, no additional facilities are included for the additional detailed study of asymmetric pulse waveform dynamics.) The small replaceable daughter board is one of several types used to provide pulse-transition detection. The central cluster of LEDs match functions of those depicted in the animation above. The two additional chips and associated LEDs on the far right are used to provide memory of which waveform was last detected as faster and other types of additional output reporting.
    • The clear plexiglass panel and associated circuit board provide a rich-featured dual reference-oscillator waveform source. Both squarewaves and asymmetric pulses of adjustable (individually or tracking) duty-cycle pulse waveforms can be produced analog-adjustable stable frequencies. The oscillators can also be replaced with pushbuttons for careful step-by-step study. Polarity reversal switches, signal indication LEDs, and buffered replicated outputs for parallel connections to accurate test equipment (dual-trace oscilloscope, a pair of frequency meters, and pair of duty-cycle meters) are also provided.
    • The black box includes a power switch permitting either or both units to be active as is useful in directing attention to detail during demonstrations. It also includes a power supply connector and external signal input jacks which override the dual dual reference-oscillator waveform source.

    REFERENCES

    [1] Pending NRI U.S. Patent Application "Frequency Comparator Utilizing Enveloping-Event Detection Via Symbolic Dynamics Of Fixed Or Modulated Waveforms," Pre-grant publication to be provided by the US PTO in 2006 (PDF link then will be provided).

    [2] NRI Technology Whitepaper, "A Frequency Comparator for Fixed and Modulated Waveforms Utilizing Enveloping-Event Detection: A Theoretical Background Employing a Symbolic Dynamics Framework," November 2005, PDF available at www.newrenaissanceinstitute.com/Whitepaper-PDF/Symbolic.Dynamics.Theory.pdf.

    [3] NRI Technology Whitepaper, "A Frequency Comparator for Fixed and Modulated Waveforms Utilizing Enveloping-Event Detection: Implementations and Applications," November 2005, PDF available at www.newrenaissanceinstitute.com/Whitepaper-PDF/Symbolic.Dynamics.Implement.pdf.

    [4] Blaine Quentin Geddes, U.S. Patent 5,939,901, "Synthesizable Flip-Flop Based Phase-Frequency Comparator for Phase-Locked Loops," August 17, 1999.

    [5] Walter L. Zweig, U.S. Patent 4,527,080, "Digital Phase And Frequency comparator Circuit," July 18, 1983.

    [6] Ryo Tamaki; Tastuya Kubo, U.S. Patent 6,218,907, "Frequency Comparator And Pll Circuit Using The Same," April 19, 1999.

    [7] James B. Rhode, U.S. Patent 4,278,898, "Frequency Comparator For Electronic Clocks," August 13, 1979.

    [8] Raymond Huguenin; Hubert Mattey; Jean Engdahl, U.S. Patent 4,608,171, "Frequency Comparator," Setpember 9, 1976.

    [9] Jean-Jacques Thiebaut, U.S. Patent 3,947,775, "Phase And Frequency Comparator," April 30, 1974.

    [10] Henri Butin, U.S. Patent 4,322,686, "Frequency Comparator Circuit," Febuary 27, 1980.

    [11] Dicy D. Davis, U.S. Patent 3,958,269, "Color Subcarrier Frequency Comparator," August 20, 1974.

    [12] Klye L. Burson; Scott O. Campbell; Apparajan Ganesan; Ronald A. Morrison, U.S. Patent 4,677,322, "Frequency Comparator Circuits," August 16, 1984.

    [13] Maurizio, U.S. Patent 4,772,852, "Phase-Frequency Comparator for Phase-Locked Loops," September 20, 1988.

    [14] Koyo Kegasa, U.S. Patent 4,940,952, "Phase And Frequency Comparator Circuit For Phase Locked Loop," July 10, 1990.

    [15] Jean-Claude Abblate; Carl Cederbaum, U.S. Patent 6,563,346 B2, "Phase Independent Frequency Comparator ," May 13, 2003.

    [16] Masato Takeyabu; Akira Kikuchi; Toshiyuki Sakai, U.S. Patent 6,392,494 B2, "Frequency Comparator And Colck Regenerating Device Using The Same," May 21, 2002.

    [17] Tsuguhiro Okada; Akira Endo, U.S. Patent 3,987,365, "Digital Frequency Comparator Circuit," October 19, 1976.

    [18] Yin-Shang Liu; Kuo-sheng Huang; Hung-chih Liu, U.S. Patent 6,6501,46 B2, "Digital Frequency Comparator," November 18, 2003.

    [19] Dalius Baranauskas, "Frequency Comparator draws 8 A," EDN Access, www.e-insite.net/ednmag/archives/1997/030397/05DI_05.htm, March 03,1997.

    [20] Edward Foster, "The Phase/ Frequency Comparator Simplified," Electronics World, February 1997 (available at http://ecforster.netfirms.com/phfrqdet/Doc1.html).

    [21] W. Dijkstra, "Frequency Window Comparator has Hysteresis, " EDN Access,
    www.e-insite.net/ednamag/archives/1996/110796/23DI_05.htm, November 7,1996.

    [22] (Author unknown), "Frequency Comparator, " IEC, www.geocities.com/IECMaster/circuits_msr/cir_msr011.html, February 15, 2002.