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## Pulse trains via waveshaping

When we use waveshaping the shape of the formant is determined by a modulation term

For small values of the index , the modulation term varies only slightly from the constant value , so most of the energy is concentrated at DC. As increases, the energy spreads out among progressively higher harmonics of the fundamental . Depending on the function , this spread may be orderly or disorderly. An orderly spread may be desirable and then again may not, depending on whether our goal is a predictable spectrum or a wide range of different (and perhaps hard-to-predict) spectra.

The waveshaping function , analyzed on Page , gives well-behaved, simple and predictable results. After normalizing suitably, we got the spectra shown in Figure 5.13. A slight rewriting of the waveshaping modulator for this choice of (and taking the renormalization into account) gives:

where so that is proportional to the bandwidth. This can be rewritten as

with

Except for a missing normalization factor, this is a Gaussian distribution, sometimes called a "bell curve". The amplitudes of the harmonics are given by Bessel "I" type functions.

Another fine choice is the (again unnormalized) Cauchy distribution:

which gives rise to a spectrum of exponentially falling harmonics:

where and are functions of the index (explicit formulas are given in [Puc95a]).

In both this and the Gaussian case above, the bandwidth (counted in peaks, i.e., units of ) is roughly proportional to the index , and the amplitude of the DC term (the apex of the spectrum) is roughly proportional to . For either waveshaping function ( or ), if is larger than about 2, the waveshape of is approximately a (forward or backward) scan of the transfer function, so the resulting waveform looks like pulses whose widths decrease as the specified bandwidth increases.

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