You will be able to
- Calculate a folded frequency for a real sinusoid.
- State the band-limit and filtering assumptions behind reconstruction.
Samples can hide cycles
A sampled complex exponential repeats when you add an integer multiple of fₛ to its frequency. Real sinusoids also have positive/negative-frequency symmetry. The apparent magnitude of a frequency folds into 0 to fₛ/2; the sign and phase must still be considered to match a sine waveform exactly.
A condition, not magic
For an ideal real baseband signal band-limited below B, a rate strictly greater than 2B permits ideal reconstruction under the sampling theorem. Real sensors require an anti-alias filter and a transition band. At exact Nyquist, a sine with zero phase can produce all zero samples; equality is not a safe universal rule.
Worked example
A 7 Hz sine sampled at 10 Hz is indistinguishable from a −3 Hz sine at those timestamps, so its apparent frequency magnitude is 3 Hz. A phase-adjusted +3 Hz sine also matches.
A common trap
An aliased component cannot generally be removed after sampling without also removing legitimate signal at the same apparent frequency.
Predict, then experiment
Will a 7 Hz source sampled at 10 Hz appear above or below 5 Hz? Before touching the controls, say or write your prediction.
Interactive experiment
Frequency folding
When does a sampled sinusoid become indistinguishable from a slower one?
Blue is the source, dashed amber is the apparent sinusoid, and circles are samples shared by both. The display covers two seconds.
Your experiment
Use 7 Hz and 10 samples/s, then source 13 Hz. Inspect how source and apparent curves agree at sample markers. Try source 5 Hz and sample rate 10 Hz to expose the phase-sensitive boundary.
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)
- Before sampling — Correct: prevent out-of-band components from folding.
- 3Hz — 17 Hz folds to −3 Hz, with magnitude 3 Hz.
Apply it: Budget a transition band
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.
- Preserve a signal up to 8 kHz at 24 ksample/s.
- Find Nyquist and the available filter transition interval.
- Explain why the analogue filter cannot be an infinitely sharp wall.
Reference solution & review criteria
Nyquist is 12 kHz; 8–12 kHz is a 4 kHz transition budget. A real finite-order filter trades attenuation, transition width and phase/delay; unwanted energy must be sufficiently attenuated before sampling.
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.