Through-Zero Pulse-Width Modulations

Through-Zero Pulse-Width Modulations

Demonstration 1: Centered-, Left-, and Right-Modulated Periodic Pulse Waveforms

Periodic binary-valued pulse waveforms, with adjustable widths are commonly used in communications, control and storage systems as well as in electronic music sound synthesis. Here the spectral properties of center-modulated, left-modulated, and right-modulated periodic pulse waveforms are considered. As an opening remark, it is noted that as an attribute of a periodic waveform is modulated, the resulting waveform is typically no longer periodic (at least with the same period). If the rate of modulation is slow, the spectrum can be modeled as a sequence of spectral "snapshots", each snapshot being the spectrum of a periodic waveform with the modulated attribute set to various fixed values, as suggested in the Figure below.

 

 

Such models are particularly useful in analyzing and waveforms when the modulation rate varies slowly with respect to the lowest audible frequency (16-20Hz). Related tools and expressions for this type of analysis are "short-term Fourier transform" and " time-varying Fourier series." Formal mathematicians may find these notions somewhat uncomfortable, but they are akin to other forms of "local analysis" employed in geometry, topology, and functional analysis and can be formalized for that audience. The " time-varying Fourier series" viewpoint will be adopted in the material that follows.

The Fourier series for a periodic pulse waveform whose ratio d (0 < d <1) of pulse width to overall waveform period is [1]:

This waveform can be made to be periodic in time t with frequency f by setting x = ft. If d is varied slowly, the ear will not perceive the sub-audio variation: of the first (so-called "constant" or "D.C. term" and thus the human ear perceives at a given instant a sound associated with the periodic waveform:


Additionally, the human ear is under most circumstances unable to perceive absolute phase, so the amplitude of the nth harmonic cos(2nπft) is perceived as

where |x| denotes the absolute value of x .

The animation below shows snapshots of one period of a pulse waveform whose width d is symmetrically and smoothly decreased over time and the spectrum associated with each of the snapshots.

Notice that a periodic pulse waveform of width d can be multiplied by -1 to create a periodic waveform of width (1-d):

Since the ear does not perceive any spectral amplitude difference from this sign change, one would expect the Fourier series for a periodic pulse of width d and for a period pulse of width (1-d) to be the same for all values of d between 0 and 1. Mathematically this can be shown using the angle difference formula for since

The amplitude of the nth harmonic for pulse width d is

and the amplitude of the same harmonic for pulse width (1-d) is

Apply the angle difference formula to the latter one finds

Since for integer n one has:

this becomes

so thus

which is remarkable as for d close to zero, (1-d) is close to one, causing the spectral envelope to oscillate at dramatically different rates. The animation below illustrates this phenomenon. the two spectral envelope curves cross at exactly at the integer-valued harmonic numbers..

Note the envelope curves meet when

Thus far, the only pulse width modulation situation considered has been symmetric about the center. It is also possible to modulate the pulse asymmetrically. If the extreme, one edge of the pulse is fixed and only the position of the other edge moves. These cased are compared below.

Centered pulse-width modulation can be thought of a s the result of comparing a symmetric triangle waveform with variable-level threshold.

Left pulse width modulation may be thought of as the result of replacing the symmetric triangle waveform with a rising periodic sawtooth waveform. Similarly, right pulse width modulation may be thought of as the result of replacing the symmetric triangle waveform width a falling periodic sawtooth waveform

These are illustrated in action in the animations below, each giving a high pulse value when the source waveform (triangle or sawtooth) is exceeds the reference value. The first three animations show arrangements where the pulse-width increasing as the reference threshold increases.

The next three animations show arrangements where the pulse-width increasing as the reference threshold decreases.

Other similar comparison-type arrangements are possible employing alternate comparison logic. Such modulation can also be generated by other arrangements and processes, such as timers, logic gate operations, stroboscopic operations, etc.

The relative motions can be seen in more detail below for a common variation in pulse-width. The centered case edges move at half the speed of the other two uncentered cases

This difference turns out to be related to a doppler-shift-like effect associated with the two uncentered cases. To explore this, the animation below compares the location of the pulse center as the modulation occurs. Note that for left-modulation the pulse center-line is displaced backwards to earlier in the cycle, while for right-modulation the pulse center-line is displaced forward to a later time in the cycle.

The instantaneous rate of center-line shift is exactly half of the instantaneous rate of pulse width modulation. This time shifting of the centered modulation case thus invokes a frequency shifting effect (similar to Doppler-shifting) atop the spectral behavior of the centered pulse width modulation case. This can be clearly heard when using a low-frequency triangle wave as a modulation source for each of the cases below. The centered case features spectral variation but retains the perceived pitch of the source, while the left and right cased shift pitch alternatively up and down corresponding to the slope of the low frequency modulation waveform.

The animation below shows the case where the pulse width is increasing rather than decreasing as in the animation above. In this case, the Doppler shift effects are reversed.

    REFERENCES

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