Discrete-Summation Formulae are based on the closed-form expression for the geometric series:
Dodge and Jerse (1985) provide the following formula for the synthesis of a bandlimited impulse train:
(1)
Figure 9 provides a time- and frequency-domain plot demonstrating the use of this formula.
Figure 9:
The time and frequency magnitude response for an impulse train waveform (440 Hz fundamental and 40 partials) created using the DSF of Dodge & Jerse.
Typically, there are numerical problems when the denominators of these formulae get close to values of zero.
Dodge and Jerse (1985) provide the following formula for use when the denominator of Eq. (1) gets close to a value of zero:
Other waveform shapes can be generated from the bandlimited impulse train, as will be described below.