Waveforms

Introduction

If we have a fundamental sine wave (1^st harmonic) of the following form (for time-domain waves, x may be replaced by ft, where f is the fundamental frequency and t is the time point):

and add n^th harmonics of the form:

(where A₁ = 1 and B₁ = 0 for the fundamental) then the waveform resulting from summing the first N harmonics (including the fundamental) can be expressed as:

We define w_∞ to be the limiting case when an infinite number of harmonics are added:

Examples

Figure 1: Harmonic sum w_N(x) compared to limiting case w_∞(x).

Figure 2: Individual harmonics h_n(x) which form w_N(x) in figure 1.

Notes

• The modulo operator used here binds tighter than plus and minus, and is defined as: (x and y don't need to be integers).
• X(x) is a helper function used to achieve periodicity in the definitions of w_∞(x). X(x) is identical to x in the range -0.5 ≤ x < 0.5 and repeats itself periodically:
• J₁(x) is the first-order Bessel function of the first kind.