Time Invariance Result
Sample Comparison Rows
| t | T{x(t - t0)} | y(t - t0) | Error | Status |
|---|
Advanced Time Invariance Test
Enter a signal, select a system rule, choose a time shift, and compare both timing paths.
Example Data Table
| Signal | System Rule | Shift | Expected Result | Reason |
|---|---|---|---|---|
| Sine wave | Gain, bias, and pure delay | 1 | Time invariant | The delay moves with the input and output. |
| Gaussian pulse | Clock multiplier | 1 | Time varying | The absolute time value changes the output. |
| Polynomial | Moving average window | 0.5 | Time invariant | The averaging window shifts with the signal. |
| Step signal | Clock gate | 2 | Time varying | The gate stays tied to zero time. |
Formula Used
A system is time invariant when shifting the input before the system gives the same output as shifting the original output after the system.
T{x(t - t0)} = y(t - t0)
The calculator samples many time points and checks:
Error(ti) = |T{x(ti - t0)} - y(ti - t0)|
Max Error = max(Error(ti))
Mean Error = sum(Error(ti)) / n
RMS Error = sqrt(sum(Error(ti)^2) / n)
If maximum error and RMS error are within the selected tolerance, the tested system passes for the chosen signal and settings.
How to Use This Calculator
- Select the input signal type.
- Enter amplitude, frequency, offset, or other signal values.
- Choose the system rule you want to test.
- Enter the time shift value t0.
- Set the time range, sample count, and tolerance.
- Click Calculate to compare both timing paths.
- Review the decision, errors, and sample table.
- Use CSV or PDF buttons to save the report.
Understanding Time Invariance
Time invariance means a system reacts the same way today, tomorrow, or after any chosen delay. Only the input shift should move the output. The shape, gain, and rule should stay unchanged. This idea matters in signals, controls, audio, finance models, and many simulation tools.
Why the Test Matters
A time invariant system is easier to predict. You can study one response and reuse it after a delay. Filters, amplifiers, and pure delay blocks often pass this test. Systems that multiply by the current clock usually fail. Their output changes because the clock value changes.
How the Calculator Works
This calculator builds a signal from your selected waveform. It then applies the chosen system rule in two paths. The first path shifts the input before the system. The second path applies the system first, then shifts the output. Both paths are compared at many sample points. The tool reports maximum error, average error, root mean square error, and a clear decision.
Choosing Good Inputs
Use a time range that covers the important part of the signal. Increase sample count for curves with fast changes. Use a smaller tolerance when values are exact or smooth. Use a larger tolerance when numerical derivatives or moving averages are used. Try different signals before trusting the conclusion.
Reading the Result
A pass means the two paths matched within your tolerance. It does not prove every possible case. It proves the chosen system behaved as time invariant for your selected data. A fail means the timing rule likely depends on the absolute clock. Review the sample table to see where the mismatch grows.
Practical Uses
Engineers use this check before building filters. Students use it to verify system theory. Analysts use it to compare delayed scenarios. Developers use it when testing models that process streams. A reliable time invariant rule keeps behavior stable when data starts later.
Common Mistakes
Do not judge from one point only. A single value can hide a timing problem. Avoid ranges that are too narrow. They may miss important changes. Always test positive and negative shifts. This reveals hidden clock dependence in many practical systems very quickly.
FAQs
What is time invariance?
Time invariance means a system gives the same shaped response after a delay. If the input shifts by t0, the output should shift by the same t0 without changing the rule.
What does this calculator compare?
It compares the output from a shifted input with the shifted version of the original output. If both match within tolerance, the tested system is treated as time invariant.
Can a nonlinear system be time invariant?
Yes. Nonlinearity and time invariance are different properties. A square law system can be nonlinear and still time invariant when it does not depend on absolute time.
Why can clock multiplier systems fail?
A clock multiplier uses the current time value directly. When the signal shifts, the clock term does not shift in the same way. This creates different output values.
What tolerance should I use?
Use a small tolerance for smooth exact functions. Use a slightly larger value for derivative and moving average tests because numerical sampling can create tiny rounding differences.
Does one pass prove the system forever?
No. A pass confirms the selected signal, range, shift, and settings. For stronger confidence, test several signals, shifts, ranges, and parameter values.
Why should I increase sample count?
More samples catch more changes across the time range. This helps detect mismatches in fast signals, narrow pulses, sharp steps, and changing system rules.
Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report with the decision, errors, settings, and sample rows.