Signals to learning / 4. Change coordinates

Lesson 13 of 40 · Intermediate · About 25 minutes

Projection onto waves

How can a dot product detect a frequency?

You will be able to

  • Explain a Fourier coefficient as a projection.
  • Recognise why both sine and cosine coordinates are needed.

Ask the signal a precise question

A dot product against a reference wave measures alignment with that wave over a record. Orthogonal reference waves separate coordinates: their dot product is zero. For the N-point Fourier basis, integer bin frequencies complete whole cycles in the record and are orthogonal under the discrete inner product.

Do not lose phase

A sine can be poorly aligned with a sine reference yet strongly aligned with its cosine partner. Using a complex exponential bundles both projections. Magnitude summarises the pair; angle describes relative phase. Outside the exact bin grid, energy generally appears across several finite-record coefficients.

X[k] = Σₙ₌₀ᴺ⁻¹ x[n](cos(2πkn/N) − i sin(2πkn/N))

Worked example

For x=[1,0,−1,0], the k=1 cosine reference is [1,0,−1,0]. Their dot product is 2; the matching sine projection is zero.

A common trap

A transform does not discover a unique physical decomposition without assumptions. Its finite basis describes the supplied record exactly, not everything outside it.

Predict, then experiment

If phase changes by 90° at a bin-centred frequency, should its spectrum magnitude disappear? Before touching the controls, say or write your prediction.

Interactive experiment

Harmonic composer

How do amplitude, frequency and phase appear in a discrete spectrum?

The waveform is the sample-wise sum; its vertical scale is automatic and labelled. The spectrum view shows 0–16 Hz of the available 0–32 Hz band. Bars use the left amplitude scale; phase points use the right −π to +π radian scale. Phase is omitted below amplitude 0.03.

Your experiment

Use one 4 Hz tone and no Tone B. Compare phase 0° and 90°. Observe unchanged amplitude with changing phase; then move the tone to 4.25 Hz.

Change one parameter at a time. Compare what you observed with your prediction and record an explanation in your notes.

Check your understanding

Try from memory first. Hints and feedback are part of learning, not a penalty. These checks are for self-study, not a graded qualification.

1. Why include both sine and cosine at a frequency?
Give me a hint

Imagine a cosine compared with a zero-phase sine.

2. Compute [1,0,−1,0]·[1,0,−1,0].

Accepted absolute tolerance: ±0.001 .

Give me a hint

Square each nonzero coordinate.

Reference answers (after your attempt)
  1. To capture arbitrary phase — Correct: they span both phase coordinates.
  2. 2 — 1+0+1+0=2.

Apply it: Calculate a projection

Use paper, your notes, or a local notebook/editor. Code examples are Python 3 starting points to run yourself; this page does not execute code or automatically assess the practical task.

  1. Use x=[0,1,0,−1].
  2. Project against sine [0,1,0,−1] and cosine [1,0,−1,0].
  3. Describe what the results say about phase.
Reference solution & review criteria

Sine dot product=2; cosine dot product=0. Under the negative-exponential DFT convention, k=1 has imaginary coefficient −2, not +2.

Your notebook

Notes and progress stay in this browser. They are not sent to the server; ordinary web-server request logs still operate. Export a backup from the course page.

Review list

Completion requires both correct checks and your practical-task acknowledgement.

Go deeper

Sources support further study; the worked examples and questions here are original. Browser experiments are small, disclosed models—not evidence of production-model quality or hardware performance.