You will be able to
- Map a DFT bin to physical frequency.
- Distinguish a transform definition from a fast implementation.
A reversible change of coordinates
The DFT maps N samples to N complex coefficients. With the forward transform unnormalised, the inverse divides by N. For real input, negative-frequency coefficients are conjugate mirrors. A one-sided amplitude plot doubles interior positive-frequency magnitudes but not DC or the even-length Nyquist bin.
FFT is an algorithm
A fast Fourier transform calculates the DFT by exploiting structure; it is not a different signal model. For radix-2 lengths, the staged butterfly graph gives O(N log N) arithmetic rather than the direct O(N²) sum. Frequency spacing is fₛ/N; higher-index bins beyond half the rate represent negative frequencies under the conventional ordering.
Worked example
N=64 at fₛ=64 Hz gives 1 Hz bin spacing. A unit-amplitude bin-centred sine has |X[k]|=32 at its positive bin, so the interior one-sided amplitude is 2×32/64=1.
A common trap
The magnitude spectrum alone does not generally let you reconstruct the original waveform; the phase coordinates matter.
Predict, then experiment
Which bin should contain a 7 Hz tone when fₛ=N=64? 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
Set Tone A to 7 Hz, amplitude 1, phase 0°; set Tone B amplitude to 0. Inspect the 7 Hz bar, then change phase while keeping frequency fixed.
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.
Reference answers (after your attempt)
- a fast algorithm for the DFT — Correct: same mathematical coefficients, different computation.
- 8Hz — 800/100=8 Hz.
Apply it: Audit transform scaling
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.
- Specify forward and inverse normalisation.
- Calculate the one-sided amplitude of an interior bin with |X|=16 and N=64.
- State the DC/Nyquist exception.
Reference solution & review criteria
With an unnormalised forward DFT, amplitude=2×16/64=0.5 at an interior positive bin. DC and Nyquist use |X|/N; other software normalisations must be accounted for.
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.
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.