Calculator Inputs
Formula Used
The time invariant test compares two routes for the same shift k.
Route one: shift the input first, then apply the system: T{x[n-k]}.
Route two: apply the system first, then shift the output: y[n-k].
The system passes the entered sample test when |T{x[n-k]} - y[n-k]| is less than or equal to the selected tolerance for every checked index.
How to Use This Calculator
- Enter discrete signal samples in the signal box.
- Select a system rule from the dropdown list.
- Set the time shift k and other optional values.
- Choose the number of samples to compare.
- Press Calculate and review the verdict table.
- Use the CSV or PDF buttons to save the result.
Example Data Table
| Example | Input x[n] | System rule | Shift k | Expected result |
|---|---|---|---|---|
| Linear gain | 1, 2, 4, 8 | y[n] = 2x[n] + 3 | 1 | Usually passes |
| Index scaled | 1, 2, 4, 8 | y[n] = n x[n] | 1 | Usually fails |
| Input delay | 3, 5, 7, 9 | y[n] = x[n-1] | 2 | Usually passes |
Time Invariance in Signals
A time invariant system keeps the same behavior after a signal is shifted. The shape of the response does not depend on the calendar position of the input. It depends only on the input values and the rule of the system. This idea is important in signal processing, control work, communication models, and sampled data checks.
What the Test Means
The calculator compares two paths. First, it shifts the input signal and then applies the selected system. Second, it applies the system first and then shifts the original output. A system is time invariant when both paths give the same result for every checked sample. Small floating point differences can be accepted through the tolerance field.
Why This Calculator Helps
Manual testing can be slow when many samples are involved. It is also easy to shift the wrong direction. This tool lists both comparison paths in one table. It also shows the difference for each index. You can quickly see where the rule passes or fails. This is useful when learning system theory or checking a model before further analysis.
Good Input Practice
Use enough samples to cover the expected delay, advance, or averaging window. Add zeros at the start or end when boundary behavior matters. Pick a shift value that creates a clear comparison. Positive shifts move the signal to the right in the common discrete time form. Negative shifts move it to the left.
Reading the Result
The final verdict depends on every compared row. If each difference is within tolerance, the sample test passes. A pass does not prove the rule for all possible signals. It proves the entered signal and settings matched the time invariant condition. A fail means at least one sample changed after the timing shift. Systems that multiply by n usually fail. Systems that use only delayed input values often pass.
Practical Notes
Use the custom expression option for memoryless systems such as 2*x+3, x*x, or sin(x). Use built in modes for delay, average, modulation, and n based rules. Keep expressions simple. Review the formula section before trusting a result. The table and downloads help you document the test. It also supports repeatable reports well.
FAQs
What is a time invariant system?
A time invariant system gives the same shaped output when the input is shifted in time. The response shifts by the same amount. The rule itself does not change with the index n.
Does a passed test prove the system is always time invariant?
No. It proves the selected rule passed for the entered samples, shift, and tolerance. A formal proof may still be needed for every possible input signal.
What does shift k mean?
The shift k represents x[n-k]. A positive k moves the signal to the right. A negative k moves the signal to the left in discrete time notation.
Why do n based systems often fail?
Rules that directly use n can change when the signal is shifted. The same input value may be multiplied by a different index, so the two comparison paths may not match.
What tolerance should I use?
Use a small tolerance such as 0.000001 for decimal calculations. Increase it only when rounding, trigonometric values, or long averages create tiny numeric differences.
Can I test continuous time systems?
This tool uses discrete samples. You can approximate a continuous signal with sampled values, but the result is still a sampled test, not a full continuous proof.
What does the boundary value do?
The boundary value fills samples outside the provided signal range. Zero is common. Change it when your model assumes another outside value.
Which custom functions are allowed?
The custom field supports x, n, pi, arithmetic operators, and functions such as sin, cos, tan, sqrt, abs, log, exp, pow, min, max, floor, ceil, and round.