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Qwestrum Engineering360 · Electrical & Electronics · Signals & Systems

LTI System Response

An LTI system is completely described by its impulse response h(t); the output is the convolution of input and impulse response, which becomes simple multiplication in the Laplace or Fourier domain.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Impulse response h(t) fully characterises LTI system
  • Frequency response |H(jω)| and ∠H(jω) from Bode plot
  • Groupdelayτg=−d∠HdωGroup delay \tau_{g} = -d∠\frac{H}{d\omega}

Topic details

Introduction

Because an LTI system is linear and time-invariant, its response to any input is the convolution y(t) = x(t)*h(t) with the impulse response h(t). Convolution in time corresponds to multiplication of transforms, so working in the s- or ω-domain is usually easier.

Scope in B.Tech and GATE syllabus

The transfer function H(s) captures the same information; evaluating it on the imaginary axis (s = jω) gives the frequency response — magnitude and phase versus frequency.

Key relations & formulas

y(t)=x(t)∗h(t)y(t) = x(t) * h(t)
(convolution)

Formulas (Indian textbook notation)

  • Y(s)=H(s)X(s);Y(jω)=H(jω)X(jω)Y(s) = H(s) X(s); Y(j\omega) = H(j\omega) X(j\omega)

Formulas (Indian textbook notation)

  • H(s)polesinLHP→BIBOstablecausalsystemH(s) poles in LHP → BIBO stable causal system

Notation and sign conventions

Relation 1 —
yy
y(t)=x(t)∗h(t)y(t) = x(t) * h(t)
(convolution)
Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
YY

Formulas (Indian textbook notation)

  • Y(s)=H(s)X(s);Y(jω)=H(jω)X(jω)Y(s) = H(s) X(s); Y(j\omega) = H(j\omega) X(j\omega)
Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
HH

Formulas (Indian textbook notation)

  • H(s)polesinLHP→BIBOstablecausalsystemH(s) poles in LHP → BIBO stable causal system
Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Stability (bounded-input bounded-output) requires the impulse response to be absolutely integrable, equivalently all poles of H(s) in the left half-plane for a causal system. A pole on the imaginary axis gives marginal stability.

Governing relations in practice

The magnitude |H(jω)| shows how each frequency is amplified or attenuated (filtering), and the phase ∠H(jω) shows the delay; a linear phase corresponds to a constant group delay and no waveform distortion.

Design and analysis considerations

Convolution can be done graphically (flip, shift, multiply, integrate) for simple signals, or algebraically via transforms for complex ones — choose the easier route under exam time.

Assumptions and validity limits

State assumptions explicitly before using any relation for lti system response — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Signals & Systems viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Signals & Systems papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to lti system response.
4. Use equation 1:
yy
.
5. Use equation 2:
YY
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

LTI System Response appears in communications and control. In Indian electrical curricula this topic is tested because it connects theory to continuous and discrete signals.
GATE and semester exams often combine lti system response with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use lti system response?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Forgetting to flip one signal before shifting in graphical convolution
• Declaring a system stable with a pole on the imaginary axis
• Confusing the frequency response (magnitude/phase) with the impulse response
• Ignoring limits of integration when convolving finite-duration signals

Quick revision checklist

Before attempting lti system response problems, confirm you can:
1. Impulse response h(t) fully characterises LTI system
2. Frequency response |H(jω)| and ∠H(jω) from Bode plot
3.
Groupdelayτg=−d∠HdωGroup delay \tau_{g} = -d∠\frac{H}{d\omega}
Revise the solved examples in Signals & Systems — Oppenheim & Willsky and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Stability from pole locations

Problem

A causal LTI system has H(s) = (s + 1)/[(s + 2)(s − 3)]. Determine whether it is BIBO stable.

Solution

Poles are the denominator roots: s = −2 and s = +3.
For a causal system BIBO stability requires all poles in the left half-plane (negative real part).
The pole at s = +3 is in the right half-plane.
Therefore the system is NOT BIBO stable.

Conceptual check — LTI System Response

Problem

In a Signals & Systems semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of lti system response." What should a complete answer include?

Exams & GATE

Oppenheim — output via convolution or Laplace method.

📖 Standard books (India)

  • Signals & Systems — Oppenheim & Willsky

    Read: Syllabus unit

    Laplace, Fourier, and LTI systems

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