Time-Invariant Systems

Introduction

A time-invariant system behaves the same regardless of when you apply the input. If you feed a signal into the system today and get a certain output, feeding the exact same signal tomorrow will produce the exact same output (just delayed by one day). The system itself does not change over time — its characteristics remain constant. This property, combined with linearity, defines the Linear Time-Invariant (LTI) class of systems that forms the core of classical signal processing theory.

Time-invariance is a modeling assumption that works well for many engineering systems: a well-designed amplifier, a digital filter with fixed coefficients, a stable communication channel. However, many real-world systems are time-varying: a fading wireless channel, a biological system whose parameters change with health or activity, or an adaptive filter that updates its coefficients in real-time.

Formal Definition

A system is time-invariant if a time shift in the input produces an identical time shift in the output:

$$\text{If } T\{x(t)\} = y(t), \quad \text{then } T\{x(t - t_0)\} = y(t - t_0) \text{ for all } t_0$$

In discrete-time: $$\text{If } T\{x[n]\} = y[n], \quad \text{then } T\{x[n - n_0]\} = y[n - n_0] \text{ for all } n_0$$

The system commutes with the delay operator — it does not matter whether you delay first and then process, or process first and then delay. The result is identical.

Testing Procedure

Step 1: Find the output $y(t) = T\{x(t)\}$ for a general input $x(t)$.

Step 2: Compute the delayed output $y(t - t_0)$ by replacing $t$ with $t - t_0$ in the expression for $y(t)$.

Step 3: Compute the response to a delayed input: find $T\{x(t - t_0)\}$ by replacing $x(t)$ with $x(t - t_0)$ in the system equation.

Step 4: Compare the results from Steps 2 and 3. If they are equal for all $t_0$, the system is time-invariant. Otherwise, it is time-varying.

Example 1: $y(t) = 3x(t)$ — Time-Invariant

Step 2: $y(t-t_0) = 3x(t-t_0)$ Step 3: Input $x(t-t_0)$ gives output $3x(t-t_0)$ Equal ✓ — Time-Invariant

Example 2: $y(t) = tx(t)$ — Time-Varying

Step 2: $y(t-t_0) = (t-t_0)x(t-t_0)$ Step 3: Input $x(t-t_0)$ gives output $t \cdot x(t-t_0)$ $(t-t_0)x(t-t_0) \neq t \cdot x(t-t_0)$ ✗ — Time-Varying

The coefficient $t$ changes with time, so the system behaves differently at different moments.

Example 3: $y[n] = x[n] - x[n-1]$ — Time-Invariant

Step 2: $y[n-n_0] = x[n-n_0] - x[n-n_0-1]$ Step 3: Input $x[n-n_0]$ gives $x[n-n_0] - x[(n-n_0)-1] = x[n-n_0] - x[n-n_0-1]$ Equal ✓ — Time-Invariant

Example 4: $y[n] = x[-n]$ — Time-Varying

Step 2: $y[n-n_0] = x[-(n-n_0)] = x[-n+n_0]$ Step 3: Input $x[n-n_0]$ gives output $x[-(n)-n_0] = x[-n-n_0]$ $x[-n+n_0] \neq x[-n-n_0]$ ✗ — Time-Varying

Time reversal is a time-varying operation because it does not commute with delay.

Physical Interpretation

Time-invariance means the system\'s internal parameters do not change over time:

• Resistor values remain constant
• Filter coefficients are fixed
• Channel characteristics are stationary
• Algorithm parameters do not adapt

If any system parameter is a function of time, the system is time-varying.

Examples of Time-Varying Systems

Fading wireless channel: The channel impulse response $h(t, \tau)$ changes as the transmitter, receiver, or scatterers move.

Adaptive filter: Coefficients are updated at each sample based on an error signal: $w[n+1] = w[n] + \mu \cdot e[n] \cdot x[n]$.

Modulator: $y(t) = x(t)\cos(\omega_c t)$ — the carrier acts as a time-varying gain.

Biological systems: Heart rate, muscle response, and neural signals vary with time, activity, and health.

Switched systems: A system whose mode changes based on a clock or external command (e.g., time-division multiplexing).

Significance for LTI Theory

When a system is both linear AND time-invariant, the complete theory of LTI systems applies:

• The system is completely characterized by its impulse response $h(t)$
• Output is computed via convolution: $y(t) = x(t) * h(t)$
• Sinusoids are eigenfunctions: input $e^{j\omega t}$ produces output $H(j\omega)e^{j\omega t}$
• Transfer function $H(s)$ or $H(z)$ provides complete frequency-domain characterization
• Stability is determined by pole locations

Without time-invariance, the impulse response becomes a function of two variables $h(t, \tau)$, convolution does not apply in its standard form, and the analysis becomes significantly more complex.

Key Takeaways

• Time-invariant: delaying the input delays the output by the same amount — system behavior does not change over time
• Test: compare $y(t-t_0)$ with the response to $x(t-t_0)$; if equal for all $t_0$, time-invariant
• Systems with explicitly time-dependent coefficients (like $tx(t)$) are time-varying
• Time reversal $x[-n]$ and time scaling $x[2n]$ are time-varying operations
• LTI = Linear + Time-Invariant — the most analytically powerful class of systems
• Time-varying systems require more general analysis tools (time-varying impulse response, state-space)