Euler's helix

As time runs, the point e^iθ with θ = 2πft + φ goes round the unit circle in the complex plane, f times a second. Stretch that motion out along a time axis and the point traces a helix. Look at it end-on and you see the circle; look from the side or from above and you see a sine and a cosine. Drag the view to turn it.

The real part of the helix is A·e^σt·cos θ and the imaginary part is A·e^σt·sin θ. A quarter turn separates them: the sine is the cosine 90° later. That pair is exactly what a radio's I and Q channels carry.

A helix that turns the other way, e^−iθ, has a negative frequency. Add the two and the imaginary parts cancel: cos θ = (e^iθ + e^−iθ) / 2. A real cosine is two counter-rotating helices, which is why the spectrum of a real signal is always mirrored about 0 Hz, and why it takes I and Q together to tell +f from −f.

Sample the helix fs times a second and you only see the point once per sample. Between samples it turns 2πf/fs radians, but any turn more than half a revolution looks like a smaller turn the other way. So a complex signal can show only −fs/2 to +fs/2, and everything else folds into that range. A real signal can't tell the two directions apart, so it only has 0 to fs/2. Turn the frequency past half the sample rate in the side panel and watch it fold.

The FFT of N samples, from −fs/2 to +fs/2. A complex helix shows one line, at its (folded) frequency. A real cosine shows two half-height lines at ±f, the two helices from section 3. Growth or decay (σ ≠ 0) widens the line, because the helix isn't a single steady frequency any more.

Amplitude-calibrated: a steady helix of amplitude A on a bin centre reads 20 log₁₀ A dB. Each half of a real cosine reads 6 dB lower.