Questions: System Classification and Properties

Question 1

A system is defined by y(t) = x(t) · sin(2πt). Is this system time-invariant?

• A. Yes — the output depends only on the current input, not on future values
• B. No — shifting the input by T gives x(t−T)·sin(2πt), which is not equal to the shifted output y(t−T) = x(t−T)·sin(2π(t−T))
• C. Yes — multiplication by a fixed sinusoid is a linear operation, and linear systems are always time-invariant
• D. No — all systems involving multiplication fail time-invariance automatically

Answer: (B)

Time-invariance test: if x(t) → y(t), does x(t−T) → y(t−T)? Feeding x(t−T) into the system produces x(t−T)·sin(2πt), but the expected output would be y(t−T) = x(t−T)·sin(2π(t−T)). These are not equal in general (sin(2πt) ≠ sin(2π(t−T)) for arbitrary T). The multiplier sin(2πt) creates an explicit time-dependence — the system 'knows what time it is.' Option C is a common misconception: linearity and time-invariance are completely independent properties. A system can be linear and time-varying, or nonlinear and time-invariant.

Question 2

Why does classifying a system as LTI (Linear Time-Invariant) matter so much for subsequent analysis?

• A. LTI systems are the only ones guaranteed to produce bounded outputs for bounded inputs
• B. Once you know the impulse response h(t) of an LTI system, you can compute the output for any input via convolution — h(t) completely characterizes the system
• C. LTI systems are physically easier to construct than nonlinear or time-varying systems
• D. LTI classification means the system is causal and therefore safe to implement in real time

Answer: (B)

The power of LTI is analytical: linearity means outputs superpose, and time-invariance means the response to a delayed impulse δ(t−τ) is simply a delayed impulse response h(t−τ). Since any input can be decomposed into weighted, shifted impulses, the output is their weighted, shifted sum — which is precisely the convolution y(t) = ∫x(τ)h(t−τ)dτ. Break either property and convolution fails. Option A conflates LTI with BIBO stability (stability is a separate property). Option D conflates time-invariance with causality (also independent).

Question 3

A causal system — one whose output at any time depends primarily on present and past inputs — is expected to be BIBO stable.

• T. True
• F. False

Answer: False

False. Causality and BIBO stability are independent properties. A system can be causal and unstable: for example, y(t) = e^t · u(t) * x(t) has an impulse response that grows without bound, producing unbounded output for a bounded input. Conversely, a non-causal system (depending on future inputs) can be perfectly stable. Causality constrains the direction of time dependence; stability constrains whether the output magnitude is bounded. The two conditions involve completely different aspects of the system's behavior.

Question 4

To verify that a system satisfies the superposition principle, it is sufficient to confirm that doubling the input usually doubles the output.

• T. True
• F. False

Answer: False

False. Superposition requires both homogeneity (scaling: α·x → α·y) and additivity (x₁ + x₂ → y₁ + y₂). Checking only scaling is insufficient because a system can satisfy homogeneity without being additive. For example, y(t) = x(t)·|x(t)| satisfies homogeneity for real scalars but violates additivity. You must test both conditions independently — or test the combined form y(a·x₁ + b·x₂) = a·y(x₁) + b·y(x₂) directly — to confirm linearity.

Question 5

Explain why finding that a system is NOT time-invariant means you cannot use convolution with the impulse response to predict its output.

Model answer: The derivation of convolution relies on time-invariance in a critical step. Any input x(t) can be written as a sum (or integral) of weighted, time-shifted impulses. By linearity, the output is the same weighted sum of the system's responses to those impulses. Time-invariance guarantees that the response to δ(t−τ) is simply h(t−τ) — a time-shifted copy of h(t). Without time-invariance, the response to δ(t−τ) could be some entirely different function g(t, τ) that changes depending on when τ occurs. The convolution formula y(t) = ∫x(τ)h(t−τ)dτ breaks down: you would need a two-argument kernel h(t, τ) rather than a single impulse response h(t−τ), dramatically complicating analysis.

This is why the LTI class is so foundational. The moment time-invariance fails, the entire toolkit of convolution, transfer functions, and frequency response no longer applies directly. Time-varying systems — like a communications channel whose properties change with time, or a control system with changing parameters — require more complex analytical frameworks (time-varying state-space models, Volterra series, etc.). LTI is the condition under which the simplest, most powerful analysis tools are valid.