LTI Systems

Almost every practical signal processing system is analyzed under the framework of Linear Time-Invariant (LTI) systems. This framework is powerful because it allows complete system characterization using just the impulse response. Before applying this framework, a system must satisfy two fundamental properties: linearity and time-invariance. Causality and stability are additional constraints that determine real-world applicability.

Core Concept Explanation

Linearity

A system T is called linear if it satisfies two conditions simultaneously. First, additivity: the response to a sum of inputs equals the sum of individual responses, T{x₁[n] + x₂[n]} = T{x₁[n]} + T{x₂[n]}. Second, homogeneity or scaling: multiplying the input by a constant multiplies the output by the same constant, T{a·x[n]} = a·T{x[n]}. Together these form the superposition principle. If initial conditions are non-zero, the system may still be linear if only the zero-state response is considered.

Time-Invariance

A system is time-invariant if a time shift in the input causes an identical time shift in the output, without any other change in the output waveform. Formally, if T{x[n]} = y[n], then T{x[n−k]} = y[n−k] for any integer k. A practical way to test this is to pass x[n−k] through the system and see if the result equals y[n−k]. If coefficients of the system depend on n explicitly, such as y[n] = n·x[n], the system is time-variant.

Causality

A causal system produces output that depends only on present and past input values, never on future inputs. In terms of the impulse response, causality requires h[n] = 0 for all n < 0. All real-time physical systems must be causal because future inputs have not yet occurred. Non-causal systems can be implemented in offline processing where the entire signal is available, such as in image processing or offline audio filtering.

BIBO Stability

A system is BIBO stable (Bounded Input Bounded Output) if every bounded input produces a bounded output. For an LTI system, the necessary and sufficient condition for BIBO stability is that the impulse response h[n] must be absolutely summable: Σ|h[n]| from n = −∞ to +∞ must be finite. In the z-domain, this translates to the requirement that the region of convergence of H(z) includes the unit circle.

Mathematical Expression

For an LTI system, once the impulse response h[n] is known, the output for any input x[n] is computed by the convolution sum: y[n] = x[n] * h[n] = Σ x[k]·h[n−k] summed over all k. In the frequency domain, this becomes Y(e^jω) = X(e^jω)·H(e^jω), meaning convolution in time equals multiplication in frequency. This frequency-domain representation is the reason LTI systems can be designed as frequency-selective filters.

Practical Understanding

To test a system for linearity and time-invariance, a structured approach is needed. For linearity, compute the output for x₁, x₂, and for a·x₁ + b·x₂ separately and check if the last equals a·y₁ + b·y₂. For time-invariance, replace x[n] with x[n−k] in the system equation, compute the output, and compare with y[n−k]. If both match, the system is LTI. Systems with variable coefficients such as y[n] = x[n]·cos(ω₀n) or systems with absolute value nonlinearities such as y[n] = |x[n]| are not LTI.

Given:
System: y[n] = n·x[n]
Test for time-invariance.

Why this formula applies:
For TI: applying delay to input must equal delaying the output.

Formula:
Step 1: Find output for delayed input x[n-k]: y₁[n] = n·x[n-k]
Step 2: Delay original output y[n]=n·x[n] by k: y[n-k] = (n-k)·x[n-k]

Substitution:
y₁[n] = n·x[n-k]
y[n-k] = (n-k)·x[n-k]

Calculation:
y₁[n] ≠ y[n-k] because n ≠ (n-k) for k ≠ 0

Final Answer:
The system y[n] = n·x[n] is TIME-VARIANT (not time-invariant).

Exam Tip: In GATE, the most common trap is y[n] = n·x[n] which looks simple but is time-variant. Always substitute x[n−k] into the system equation and separately delay the output, then compare. Also, a system with non-zero initial conditions is typically NOT linear even if the differential equation is linear.

LTI System Testing Checklist

• Linearity: test both additivity and homogeneity using superposition.
• Time-invariance: pass delayed input through system; compare with delayed output.
• Causality: check if h[n] = 0 for all n < 0, or if output depends on future input.
• BIBO stability: verify Σ|h[n]| converges absolutely.
• An LTI system is fully characterized by its impulse response h[n].

Quick Revision

• Linearity = additivity + homogeneity (superposition principle).
• Time-invariance: T{x[n−k]} = y[n−k]. Test by substituting delayed input.
• Causality: h[n] = 0 for n < 0. Output depends only on present and past inputs.
• BIBO stability: Σ|h[n]| < ∞. ROC of H(z) includes unit circle.
• LTI output: y[n] = x[n] * h[n] (convolution sum).
• Trap: y[n] = n·x[n] is linear but time-variant. y[n] = x²[n] is time-invariant but nonlinear.
• Frequency domain: Y(e^jω) = X(e^jω)·H(e^jω).