You will be able to
- Identify harmonics generated by a cubic nonlinearity.
- Explain oversampling/filtering order and limitations.
New frequencies inside the pipeline
A fixed linear time-invariant filter changes amplitudes/phases of existing frequency components. A nonlinearity can generate new ones. For x=sin θ and g(x)=x+a x³, the identity sin³θ=(3sin θ−sin 3θ)/4 creates a fundamental and a third harmonic. That third harmonic may exceed the sample-domain Nyquist frequency.
Protect the nonlinear stage
A common strategy interpolates to a higher rate, applies the nonlinearity, filters unwanted high-frequency output and then decimates. Filters, oversampling factor, latency and nonlinear bandwidth determine residual aliases. Antiderivative methods integrate an assumed interpolated input segment and require stable handling near equal samples. The lab uses an analytic ideal-filter comparison, not a practical finite-filter implementation.
Worked example
A 12 Hz source at 64 samples/s produces a 36 Hz third harmonic, above Nyquist 32 Hz. Direct sampling folds it to a magnitude frequency of 28 Hz. At a=1 its third-harmonic coefficient is −0.25.
A common trap
Filtering after aliases have folded into desired frequencies cannot generally identify their original provenance. Oversampling must protect the stage that generates them.
Predict, then experiment
When source crosses 32/3 Hz, what happens to its third harmonic? Before touching the controls, say or write your prediction.
Interactive experiment
A third harmonic crosses Nyquist
Where does the harmonic created by a cubic nonlinearity appear?
Rate is 64 Hz. Blue circles mark direct nonlinear samples; the solid line joins them. Dashed amber is the ideally low-pass-filtered continuous cubic result evaluated at the same sample times. This analytic comparison has no finite-filter approximation.
Your experiment
At cubic coefficient 1, compare source frequencies 10,11,12 Hz. Inspect where direct samples disagree with the analytically filtered reference. Then set coefficient to 0.
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)
- generate new harmonics — Correct: products of oscillations create harmonics.
- 36Hz — 3×12=36 Hz.
Apply it: Specify an oversampled nonlinearity
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.
- Draw interpolation → nonlinearity → low-pass → decimation.
- State rate, factor, filter stopband and latency.
- Define comparison against a high-rate or analytic reference.
Reference solution & review criteria
A defensible design discloses filter coefficients, factor and delay compensation. Test residual in-band error against an appropriately aligned reference; do not call the ideal analytical lab a measured production filter.
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.