Time Invariance Test

A system is time-invariant if its behavior does not change with a shift in the input's starting time. This property determines whether a filter designed at t=0 performs identically when applied to a signal arriving later, which matters in every real-time DSP and control application.

Core Concept

A system takes an input signal and produces an output. Time invariance asks: if you wait T seconds before feeding the input, does the output simply shift by T seconds with no other change? If yes, the system is time-invariant.

The formal test compares two outputs. First, delay the input by T and then apply the system to get y1(t). Second, apply the system first and then delay the output by T to get y2(t). If y1(t) equals y2(t) for every possible T and every possible input, the system passes the test.

Systems that multiply the input by a time-varying coefficient like t or cos(t) always fail this test. An amplifier whose gain changes with time is not time-invariant. A fixed RC lowpass filter is time-invariant because its component values do not change with time.

Key Formula

Let H denote the system operator. The system is time-invariant if and only if:

If x(t) -> y(t), then x(t - T) -> y(t - T) for every real constant T.

Equivalently, H{x(t - T)} = y(t - T) for all T.

x(t): input signal. T: arbitrary time shift in seconds. y(t): output corresponding to x(t). H{}: system operator applied to its argument.

Problem: Test whether y(t) = t * x(t) is time-invariant.

Step 1 — Delay input first:
  x_delayed(t) = x(t - T)
  Apply system: y1(t) = t * x(t - T)

Step 2 — Apply system first, then delay:
  Apply system to x(t): y(t) = t * x(t)
  Delay output: y2(t) = y(t - T) = (t - T) * x(t - T)

Step 3 — Compare:
  y1(t) = t * x(t - T)
  y2(t) = (t - T) * x(t - T)
  y1(t) != y2(t) because t != (t - T) for T != 0

Final Answer: y(t) = t * x(t) is NOT time-invariant.
The time-multiplying coefficient breaks the symmetry.

Exam Tip: In VTU and Anna University papers, time-varying coefficients are the most common way examiners construct failing systems. Check every coefficient: if it contains t or n explicitly, the system is time-varying. A coefficient like 3 or a constant k always passes. For discrete systems y[n] = x[n - n0] is time-invariant, but y[n] = x[-n] is not, because negating the index changes with the reference point differently for each input.

Properties Summary

• Definition: H{x(t-T)} = y(t-T) must hold for all T and all x(t).
• Failing condition: any explicit t or n multiplying the input makes the system time-varying.
• Delay operator: y(t) = x(t - t0) for fixed t0 is time-invariant; y(t) = x(t - t) (delay equals current time) is not.
• Accumulator: y[n] = sum_{k=-inf}^{n} x[k] is time-invariant and LTI.
• Sampling or down-sampling y[n] = x[Mn] is time-varying because the output spacing changes with M.
• Cascade of two time-invariant systems is also time-invariant.

Quick Revision

• Time invariance: shifting the input shifts the output by the same amount, nothing else changes.
• Test method: compare H{x(t-T)} with y(t-T) for arbitrary T.
• Any coefficient containing t or n in front of x(t) or x[n] breaks time invariance.
• y(t) = x(2t) is time-varying: it compresses time and fails the delay-commutativity test.
• y[n] = x[n] * cos(w0*n) is time-varying because the cosine multiplier shifts phase differently for each T.
• LTI requires both linearity AND time invariance; failing either disqualifies the system.
• Exam trap: students often test only one specific input like u(t) and declare the system time-invariant; the test must hold for every possible input.