Why sinusoids? Signals as vectors

Before any transform: why decompose signals into sinusoids at all? The answer is linear algebra wearing headphones — signals are vectors, sinusoids are a basis, and “finding the spectrum” is projection. Once you see this, Fourier analysis stops being a formula and becomes geometry.

Build the intuition

A signal is a very long vector

A 1-second audio clip at 44,100 samples is a list of 44,100 numbers — a vector in 44,100-dimensional space. Adding signals adds vectors; scaling volume scales them. Everything from linear algebra applies: bases, projections, dot products. Dimension count changed; rules didn't.

Inner products measure resemblance

The dot product told you how aligned two arrows are. For signals: multiply two signals point by point and total — large when they rise and fall together, zero when unrelated. “How much 440 Hz is in this clip?” becomes “what's the inner product of the clip with a 440 Hz sinusoid?” Correlation as geometry.

\langle x, y \rangle = \sum_n x[n]\, y[n]

Sinusoids are perpendicular — that's the miracle

Sinusoids of different (harmonic) frequencies have inner product zero: they're orthogonal, like perpendicular axes. So they form a basis where each coordinate can be measured independently — project onto each frequency and no measurement disturbs another. Add that sinusoids pass through LTI systems unbent (last course), and they're not just a basis, they're the right basis.

\langle \sin(m\omega t), \sin(n\omega t) \rangle = 0 \quad (m \neq n)

Best approximation = projection = least squares

Can't keep all coordinates? Projection onto fewer basis directions gives the best possible approximation in the squared-error sense — the least-squares idea. Keeping a signal's largest frequency components (how MP3 and JPEG compress) is literally orthogonal projection. Compression is geometry.

See it move

InteractiveVectors: arrows that add

a →2

a ↑1

b →-1

b ↑2

a = (2, 1) + b = (-1, 2) = (1, 3) — tip to tail. Dot product 0: the two are nearly perpendicular — they share almost nothing.

The geometry in miniature: the dot-product readout is your “how much of one signal lives in another” meter. Perpendicular = zero = independent coordinates.

A worked example

Measure a frequency by inner product

Signal: x = [0, 1, 0, −1] (one cycle of a slow wave, 4 samples). Probe: s = [0, 1, 0, −1], a matching sinusoid.
Inner product: 0 + 1 + 0 + 1 = 2 — strong match.
Probe with the faster wave c = [1, −1, 1, −1]: inner product 0 + (−1) + 0 + 1 = 0 — none of that frequency present.
You just computed two bins of a Fourier transform by hand. The DFT (lesson 4) is exactly this, systematized.

Out in the world

MP3: keep the big coordinates

An MP3 encoder transforms audio into a frequency basis, keeps the components your ear notices, and discards the rest — a projection chosen by psychoacoustics. A 90% smaller file that sounds identical: orthogonal bases doing commercial labor since 1993.

Common confusion, cleared

“Fourier analysis is a clever trick someone invented.”

It's a change of basis — the same operation as rotating coordinates in geometry. The cleverness is in choosing sinusoids; the machinery is linear algebra you already know.

“Orthogonality is abstract bookkeeping.”

It's what makes the analysis practical: orthogonal components can be measured independently and discarded independently. Without it, removing one frequency would corrupt the others.

Recap

Signals are vectors; inner products measure resemblance.
Sinusoids are orthogonal — independent, separately measurable coordinates.
Spectra are coordinates; compression is projection; best-fit is least squares.

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