You will be able to
- Test linearity through superposition.
- Distinguish linearity, time invariance and causality.
Two independent properties
A system T is linear if it preserves scaled sums: T(ax+by)=aT(x)+bT(y). It is time-invariant if shifting the input shifts the output by the same amount. These properties are distinct. A time-varying gain can be linear but not time-invariant; squaring is time-invariant but nonlinear.
The LTI shortcut
For a discrete linear time-invariant system with appropriate zero-state assumptions, the impulse response h describes the output through convolution. Causality means no dependence on future input, and is another separate property. Nonzero initial state can add a response that is not captured by convolving the new input alone.
Worked example
T(x)[n]=2x[n] is linear and time-invariant. T(x)[n]=n x[n] is linear but time-varying. T(x)[n]=x[n]² is time-invariant but nonlinear.
A common trap
“Linear” does not mean a plotted output must be a straight line. It describes how an operator combines inputs.
Predict, then experiment
Should doubling every input to a fixed convolution double every output? Before touching the controls, say or write your prediction.
Interactive experiment
One output, multiply by multiply
Which input values contribute to one sample of a three-tap smoothing filter?
Blue is the input and dashed amber is the full output. The moving window and filled blue points show the samples under the kernel; the large amber point is the selected output. The fixed kernel is [0.25, 0.5, 0.25].
Your experiment
Inspect the fixed kernel and contributions at several output indices. Write what would happen if all input samples doubled. Contrast this with the cubic nonlinearity in the linked essay.
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)
- nonlinear and time-invariant — Correct: shifting still commutes with pointwise squaring.
- 18 — 6+12=18, which equals T(6).
Apply it: Classify an operator
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.
- Consider y[n]=x[n]+x[n−1] with zero extension.
- Test superposition and a one-sample shift.
- Decide whether it is causal.
Reference solution & review criteria
It is linear, time-invariant and causal: it uses present/past samples and fixed coefficients. Finite-array boundary policies should be distinguished from the infinite-sequence operator.
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.