Pulse trains via wavetable stretching

In the wavetable formulation, the pulse train can be made by a stretched wavetable:

$${M_a}(\phi) = W (a \phi),$$

where $-\pi \le \phi \le \pi$ is the phase. The function $W$ should be zero at and beyond the points $-\pi$ and $\pi$, and rise to a maximum at 0. A possible choice for the function $W$ is

$$W(\phi) = {1\over2} \left ( \cos(\phi) + 1 \right )$$

which is graphed in part (a) of Figure 6.4. This is known as the Hanning window function; it will come up again in chapter 9.

Figure: Pulse width modulation using the Hanning window function: a. the function $W(\phi)=(1+\cos(\phi))/2$; b. the function repeated at a duty cycle of 100% (modulation index $a=1$); c. the function at a 50% duty cycle ($a=2$).

Realizing this as a repeating waveform, we get a succession of (appropriately sampled) copies of the function $W$, whose duty cycle is $1/a$ (parts b and c of the figure). If you don't wish the copies to overlap we require $a$ to be at least 1. If you want overlap the best strategy is to duplicate the block diagram (Figure 6.3) out of phase, as described in Section 2.4 and realized in Section 2.6.5.