How does multiplying a signal $g(t)$ by the rotating complex unit vector e^{-2πift} facilitate frequency detection in the winding picture?

Conditions

• $g(t)$ is a real-valued time-domain signal.
• $f$ represents cycles per unit time.
• The averaging is performed over a finite or infinite time window.

Reasoning, step by step

1. Scale the rotating unit vector $e^{-2\pi i ft}$ by the instantaneous value $g(t)$.
2. Observe that positive values follow the rotation direction, while negative values reverse it.
3. Sample these complex points uniformly in time.
4. Compute the average (centroid) of these sampled points.
5. If $g(t)$ contains a component at frequency $f$, the vectors align consistently, yielding a large magnitude for the average.
6. If $g(t)$ lacks frequency $f$, the vectors rotate rapidly relative to the sampling, causing cancellation and a near-zero average.

Example

The script states: 'Multiply $g(t)$ by the rotating complex unit vector $e^{-2\pi i ft}$... Matching oscillations can accumulate while many other contributions cancel.'

Common misconceptions

• Believing the winding process physically bends the sound wave into a circle; it is a mathematical representation.
• Assuming the result always lies on the unit circle; $g(t)$ changes the length and sign reverses direction.
• Thinking that any non-zero average implies an exact frequency match; finite windows yield nonzero responses away from exact matches.

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