Sub-Space Signals: Fourier Wave Synthesis

Sub-space transmitters operating at the edge of the galaxy communicate by sending electromagnetic pulses across lightyears. To pierce through intense cosmic dust interference, these transmitters must send flat, crisp square wave pulses. However, physical antennas can only generate smooth, undulating sine waves.

To bridge this barrier, engineers rely on one of the most stunning mathematical insights ever conceived: Fourier’s Theorem.

Fourier proved that any periodic wave, no matter how complex or sharp-edged (like a square, triangle, or sawtooth wave), can be built by summing up an infinite series of basic, smooth sine waves of increasing frequencies and decreasing amplitudes.

The Trigonometric Sum

Suppose we want to synthesize a square wave of amplitude A and frequency f. A square wave flips instantly between +A and -A, producing a sharp vertical wall.

Fourier proved that we can construct this shape by starting with a fundamental sine wave at frequency f, and then adding only odd harmonics (3f, 5f, 7f, …) with scaling amplitudes:

Square(t) = (4/π) · [ sin(ωt) + (1/3)·sin(3ωt) + (1/5)·sin(5ωt) + (1/7)·sin(7ωt) + … ]

  Wave Component Frequencies:
  1st Term:  ~ sin(x)         (Fundamental)
  3rd Term:  ~ (1/3)*sin(3x)  (Adds steepness)
  5th Term:  ~ (1/5)*sin(5x)  (Flattens the peak)

As we sum more terms (harmonics), the vertical edges of our approximation steepen and the peaks flatten, closer approaching a perfect square wave.

Sawtooth and Triangle Series

By changing the amplitude scales and harmonic terms, we can synthesize other periodic shapes:

1. Sawtooth Wave (Linear Ramp)

Contains both odd and even harmonics, summing all frequencies with alternating signs: Sawtooth(t) = (2/π) · [ sin(ωt) - (1/2)·sin(2ωt) + (1/3)·sin(3ωt) - (1/4)·sin(4ωt) + … ]

2. Triangle Wave (Symmetric Slopes)

Contains only odd harmonics, but the amplitudes decrease much faster (1/n²), leading to rapid convergence with very few terms: Triangle(t) = (8/π²) · [ sin(ωt) - (1/9)·sin(3ωt) + (1/25)·sin(5ωt) - (1/49)·sin(7ωt) + … ]