Sound, images, sensor readings — almost every signal you meet is
really a recipe. Take simple ingredients, pure oscillations at fixed frequencies,
and add them with different strengths and phases. The Fourier transform
is the tool that reads that recipe back out.
Here is a deliberately small example: two cosine waves added together,
two and three hertz. On a graph it just looks like
a busy squiggle. Nothing screams two separate tones — yet they
are both in there, waiting to be separated.
This curve is g of t, height versus time.
Our job is to discover which pure frequencies are
actually present, without guessing from the shape alone.
The trick is geometric. Pick a test frequency, and wrap the wave
around the origin. Each moment contributes a point in the plane: radius
from the signal's value, angle spinning at your chosen rate. As time
runs forward, you trace a closed-looking path — the wound curve.
For most test rates the path is symmetric enough that
everything cancels. The average position, the center of mass of
that winding, sits almost at the origin. That means this
test frequency is not a strong ingredient.
But when the wrapping rate lines up with a frequency that
really lives inside the signal, the symmetry breaks. The path bulges
to one side, and the center of mass jumps outward. You
have found a spike of energy at that rate.
Sweep the test frequency and record how far the center drifts
from zero. That trace is the magnitude spectrum: a plot of
how much each frequency contributes. Every spike marks a pure tone
that was hiding in the squiggle all along.
The same idea lives in three dimensions. Stack time vertically, and let each
slice be the rotating complex value g of t times e to the
minus two pi i f t. From the side you see the familiar
wound curve; from above you sense the drift of the center. When rotation
rate matches content, the path fails to close neatly through the bulk —
that off-center drift is exactly what the integral measures.
In one line: multiply by a spinning complex exponential, integrate, and
you extract each frequency component. This is the Fourier transform.