Through-Zero Pulse-Width Modulations

    Through-Zero Pulse-Width Modulations

    Demonstration 3: Comparative Constructions of Through-Zero Pulse-Width Processes

    The notion of "through-zero pulse-width modulation" was introduced in [1]. Here a brief treatment is provided, not comprehensively but with some new results and commentary. A more comprehensive treatment is to be provided in a forthcoming publication. The principal purpose here is in relation to the study of the cross-products of square-waves employed in the signal processing patent U.S. 6,849,795 [2], the original multiple-octave square-wave cross-product article [3], and products derived from the latter (see for example [4]).

    As discussed earlier, the Fourier series for a periodic pulse waveform having a pulse-width to waveform-period ratio d and frequency f is given by [5]:

     

    In this construction, a value of d = 0 is the limiting pathological case of a pulse of zero-width and a value of d = 1 is the limiting pathological case of a pulse of 100%-width. Neither of these limiting pathological cases have any time-oscillatory terms since for all integers n the amplitude terms are zero

    sin(np) = 0
    (d = 1 case)
    sin(0) = 0
    (d = 0 case)

     

    However, sin(x) is a periodic function so mathematically there is no need for d to be confined to the interval 0 < d< 1. One can thus constructively consider the situation where d is allowed to exceed the 0 < d < 1 limitation, particularly with respect to the time-oscillatory terms

    .

    As both the pathological cases of a pulse of zero-width with d = 0 and pulse of 100% with d = 1 involve zero amplitude time-oscillatory terms, (this being the case in fact because the sine term in the above Fourier series is zero), then from the viewpoint of time-oscillatory terms the cases where d = 0, d = 1, and in fact d passing through the value of any integer, may be called "through-zero pulse-width modulation." Here several types and representations of through-zero pulse-width modulation will be presented and explored.

    Sawtooth-Difference Representations

    An interesting way to begin such study is to simply let d linearly increase with time, at least over some interval. This might be realized, for example, by modulating a pulse-width with a low-frequency ramp waveform (e.g., a low-frequency upward or downward sawtooth). Wherein at least for a duration d increases linearly with time, one may set d = st , where s denotes the slope, yielding

    .

    Using one of the trigonometric sum and product relations ("Prosthaphaeresis formulas") [5]-[6])

     

    one obtains

    ,

    so the Fourier series becomes

    .

    This is immediately recognizable as the sum of the Fourier series for two increasing sawtooth waves [5], one at frequency 2f + s and the other frequency 2f - s . This consistent with Hutchin's decomposition or construction of pulse width modulation as the sum of three sawtooth waveforms [7]. As a brief presentation of this, consider the exemplary case of an upward sawtooth waveform of one frequency being compared to another upward sawtooth waveform of 1/8 that frequency, producing pulse output with pulse-width modulation behavior as depicted below:

    Now consider the three sawtooth waveforms in the next figure (below). Waveform A is again the original upward sawtooth at the higher frequency, and waveform C is the negative of the second upward sawtooth waveform of 1/8 that higher frequency, resulting in a downward sawtooth at 1/8 of the higher frequency. Introduced is the sawtooth waveform B at the difference frequency of 7/8 of the higher frequency, again inverted. Adding waveforms A and B together give waveform D, which looks like the pulse output of the figure above perched on a rising ramp. Adding descending waveform C cancels out the rising ramp, producing the same pulse output of the figure above.

    Another way to think of this involves swapping the roles of waveforms B and C, since waveform C may equally be thought of as having frequency equal to the difference frequency of waveforms A and B: the difference between two sawtooth waveforms whose ratio of frequencies is the ratio of integers produces

    • a third sawtooth waveform with a frequency of the difference of the two frequencies,
    • a modulated-width pulse waveform with a frequency of the higher of the two frequencies,
    • a modulation pattern that is periodic with a "macro-cycle" rate equal to the difference frequency.

    (This in fact can be explicitly calculated in this general case by adding and subtracting Fourier series of the sawtooth and pulse waveforms.)

    This is also not surprising as the Hutchin's decomposition/construction can be shown to follow from the use of a properly-phased variable pulse-width waveform to shift the phase of a sawtooth waveform [8] when the phase shift itself is a linear ramp. This "algebraic sawtooth phase-shifting principle" is illustrated in the pictorial below. Here an upward sawtooth waveform R of unit amplitude is compared to a reference threshold c with 0>c>1 to produce a pulse waveform P. When these are added waveform S results, which is a phase-shifted version of the original waveform R perched on a constant level amounting to the value of the threshold level c itself. Subtracting this value c gives a properly positioned waveform T which is a phase-shifted version of waveform R. (A circuit realization of this may be found in [8]). The phase-shift varies over the full period of the waveform R linearly as the threshold value c varies from 0 to 1. If one lets R be the mid-frequency sawtooth waveform B in the previous figure and varies threshold c as the lowest frequency sawtooth waveform C, the resultant waveform T below becomes the highest frequency waveform A.

    Returning to waveform E of the previous figure, which is simply the sum of the three sawtooth waveforms A, B, and C, one sees a pattern that periodically evolves from full-width to zero-width and begins again. In this sense there is a type of "through-zero pulse-width modulation." Thus second representation of at least one type of "through-zero pulse-width modulation" is as the sum and difference of sawtooth waves of three sawtooth waves whose frequencies are such that two of the frequencies add to the value of the third and are all multiplicatively related by ratios of integers.

    Taking stock so far:

    • A first type of "through-zero pulse-width modulation" involves modulating the pulse-width of a pulse waveform with a linearly-increasing modulation signal and ignoring the non-sinusoidal term proportional to pulse-width.
    • A second type of "through-zero pulse-width modulation" is as the sum and difference of three sawtooth waves of specially selected frequencies.
    • In fact, the specially selected frequencies of the second case are actually such that the lowest-frequency sawtooth discontinuity is "snuck in" as the sine term in the pulse waveform Fourier series
      •  

      cannot distinguish d = 0 from d = 1 .

    • Additionally, the Hutchins decomposition/construction only depicts one macro-cycle period. One could imagine a through-zero pulse-width modulation process that would continue indefinitely. If one focuses on only the audible frequencies, the Fourier series term proportional to pulse-width can be dropped. This dropped term corresponds to the lowest-frequency sawtooth in the Hutchins decomposition/construction.

    Thus these two representation cases are even more closely related. They may be united in that both involve summing two sawtooth waveforms and, in one way or another, rendering mute the Fourier series term proportional to pulse-width. Going forward, these will collectively be called the "sawtooth-difference representation."

    In taking care to study what occurs as the pulse width in the sawtooth-difference representation exceeds the 0%-100% range, one sees that the expanding pulse-width uniformly repeats a pattern of monotonic change from one extreme limit of pulse width to the other extreme limit, doing this without a sign reversal.

    Triangle-Wave Centered-Modulation Representation

    Another way to construct a through-zero pulse-width modulation process is to modulated the width of a centered pulse-width modulation process between 0% and 100% with a low-frequency triangle waveform. In an earlier demonstration page, it was shown that a centered pulse-width modulation process could be realized by applying a symmetric triangle waveform and the pulse-width reference to a comparator, for example as shown in the configuration below:

    In the above, which shows two periods of the pulse waveform, the pulse-width is modulated symmetrically around the dashed green vertical center line of each period. If the pulse-width is modulated by a triangle so that its extremes are exactly that giving 0% and 100% pulse-width, the edges of the pulses meet as the pulses merge and vanish, as illustrated in the anim ation below.

    Tracing any particular pulse edge as the pulses merge and vanish, the edges of the pulses can be interpreted as smoothly moving through the waveform period boundary into the adjacent pulse. This may be seen in the animation below where the cyan pulse edge may be interpreted as smoothly moving through pulses merge and vanish events:

    Summarizing the properties of this through-zero pulse-width modulation process,

    In taking care to study what occurs as the pulse width in the sawtooth-difference representation exceeds the 0%-100% range, one sees that the pulse-width follows a symmetric oscillatory expanding and contracting pattern, doing this without a sign reversal.

    Square-Wave Amplitude Modulation (Cross-Product) Representation

    Through-zero pulse-width modulation processes also occur in the sub-octave square-wave cross-product signal processing technology discussed in U.S. Patent 6,849,795 [2] and [3]. In this technology, two square-waves of different frequencies 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 corresponds to a gating function, with one square wave periodically turning the transmission of the other square wave on and off. It turns out that "Exclusive-OR" corresponds to classical amplitude modulation. It also turns out, as will be shown, 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.

    To see how these logical operations on square waves produce through-zero pulse-width modulation as the the pulse edges of the two square waves migrate past one another in time, all that is required is a simple accounting of the specific pulse edges and square wave values maintained between these. The following animation provides directional arrows specifically indicating the up-transitions and down-transitions of the two source waveforms, and subsequently tracks these through the two logical operation processes of "AND" and "XOR."

    Note in logical "AND" through-zero modulation only every other intermingled pair of source waveform transitions give rise to a pulse, while in "Exclusive-OR" through-zero modulation all intermingled pairs of source waveform transitions give rise to a pulse. Since one of four input conditions give a logical "1" in an "AND" operation, while two of four input conditions give a logical "1" in an "Exclusive-OR" operation, the "Exclusive-OR" process has twice the number of pulses as the "AND" process.

    To begin further comparative study of these through-zero pulse-width modulation processes, first note that these logical "AND" or "Exclusive-OR" operations may naturally be viewed as multiplications, in the following senses, applied to the two square waves:

    • If the square waves are interpreted as oscillating between logical "0" and "1", then the "AND" operation can be interpreted as the multiplication of the instantaneous values of the two square waves since the numerical multiplication A*B =1 only if both A=1 and B=1 (and A*B =0 if either A=0 or B=0). Note that a square wave oscillating between "0" and "1" has an average of 0.5, and thus an offset ("D.C.") term is implicitly present.
    • If the square waves are interpreted as oscillating between numerical values of "-1" and "+1" then the average is zero and no offset ("D.C.") term is implicitly present. In this case, however, the numerical multiplication a*b=1 only if a=b (specifically a=b=-1 or a=b=+1, with a*b = -1 if either a=+1 with b=-1 or a=-1 with b=+1). This behaves like a logical "Exclusive-OR" operation.

    The "AND" case looks like a gating function and is attractive from that conceptual level of understanding. The "Exclusive-OR" case, however, corresponds to amplitude modulation without offset terms. Since, in terms of the definitions A, B, a, b used above:

      A = (a+1)/2
      B = (b+1)/2

    the logical "AND" operation is

      A*B = [(a+1)/2] * [(b+1)/2] = (a*b + a + b + 1)/4

    which amounts to an attenuated mix of the "Exclusive-OR" operation plus the oscillatory portions of each square waves plus an offset constant. Thus, when 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.

    The animations below show the logical "AND" and "Exclusive-OR" through-zero modulation effects. In all cases shown, the two waveforms are represented utilizing a positive/negative relative frequency viewpoint, where the faster waveform is portrayed as its positive frequency difference from the average of the two frequencies and the slower waveform is portrayed as its negative frequency difference from the average of the two frequencies (negative frequencies may be thought of as time reversal, as regularly seen in stroboscopic observations and sampled-signal aliasing effects). The animation on the left shows the two source (+/- bipolar) waveforms. The animation to its right shows the (unipolar) logical AND of these and the resulting through-zero modulation effect. The third animation to the right shows the Exclusive-OR through-zero modulation effect, which is equivalent to the multiplication of the two source (+/- bipolar) waveforms, (i.e., classical amplitude modulation). The animation at the far right shows the numerical result of adding the two source waveforms together with the Exclusive-OR through-zero modulation effect and an offsetting constant, which scales by a factor of 1/4 to reconstruct the logical AND through-zero modulation effect, verifying the transformations described above. Thus the logical AND through-zero modulation effect is perceived as equivalent to the equally weighted sum of the two source waveforms and the Exclusive-OR (classical amplitude modulation) through-zero modulation effect.

    Note these two types of through-zero pulse-width modulation behave in the following manner:

    • Both processes match the behavior of the triangle-wave centered-modulation representation
    • The "XOR" process involves two pulses per period centered at two different positions
    • The "AND" process involves only one pulse per period, jumping between the two above positions

    These phenonmena, together with spectral partitioning invoked by sub-octave square-wave frequency division [3],[9], give the sub-octave square-wave cross-product signal processing technology of U.S. Patent 6,849,795 [2], original publication [3], and commercial product [4] their characteristic sounds and variational mixing properties. Additionally, the simple operations involved may be readily realized in inexpensive logic circuitry or dramatically modest line-count software.

      REFERENCES

      [1] L. Ludwig and B. Hutchins, "A New Look at Pulse-Width Modulation -- Part 3," Electronotes, Vol. 12, No.118, October 1980, pp. 3-18.

      [2] U.S. Patent 6,849,795 "Controllable Frequency-Reducing Cross-Product Chain," February 11, 2005.

      [3] L. Ludwig, "A Square-Wave Frequency-Division Sub-Octave Cross-Product Module," Electronotes, Vol. 11, No.98, February 1980, pp. 9-15.

      [4] Synthesis Technology, "MOTM-120 Sub-Octave Multiplexer Module," product description, http://www.synthtech.com/motm120.html.

      [5] Beyer, William H, CRC Standard Mathematical Tables, CRC press, Inc., Boca Raton, 1987 ISBN: 0-8493-0628-0

      [6] Eric W. Weisstein. "Prosthaphaeresis Formulas." From MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

      [7] B. Hutchins and L. Ludwig, "A New Look at Pulse-Width Modulation -- Part 2," Electronotes, Vol. 12, No.110, February 1980, pp. 3-18.

      [8] L. Ludwig and B. Hutchins, "Multi-Phase Scanning with Sawtooth Waveforms," Electronotes Application Note, No. 73, February 4, 1978.

      [9] NRI Whitepaper, "Controllable Frequency-Reducing Cross-Product Chain," available at XXXXXXXXXXXXXXX, October 2005.