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Qwestrum Engineering360 · Electrical & Electronics · Signals & Systems

Continuous Time Signals

Continuous-time signals are classified by properties — periodic/aperiodic, even/odd, energy/power — and built from the elementary step, impulse and ramp; energy signals have finite ∫|x|² dt while power signals have finite average power.
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.

  • Even: x(−t) = x(t); odd: x(−t) = −x(t)
  • Periodic:x(t+T)=x(t);fundamentalperiodT0Periodic: x(t+T) = x(t); fundamental period T_{0}
  • Exponential and complex exponential e^(st)

Topic details

Introduction

A signal is a periodic if it repeats with period T₀; the smallest such T₀ is the fundamental period, and ω₀ = 2π/T₀. Sinusoids and complex exponentials are the building blocks of Fourier analysis.

Scope in B.Tech and GATE syllabus

An energy signal has finite total energy ∫|x|² dt (and zero average power) — typical of transient pulses; a power signal has finite non-zero average power (and infinite energy) — typical of periodic signals. A signal cannot be both.

Key relations & formulas

x(t)=Acos(ωt+ϕ)x(t) = A cos(\omega t + \phi)
(sinusoidal)

Formulas (Indian textbook notation)

  • EnergyE=∫∣x(t)∣2dt;PowerP=lim(1T)∫∣x(t)∣2dtEnergy E = \int |x(t)|^{2} dt; Power P = lim (\frac{1}{T})\int |x(t)|^{2} dt

Formulas (Indian textbook notation)

  • Unitstepu(t);impulseδ(t);rampr(t)Unit step u(t); impulse \delta(t); ramp r(t)

Notation and sign conventions

Relation 1 —
xx
x(t)=Acos(ωt+ϕ)x(t) = A cos(\omega t + \phi)
(sinusoidal)
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 —
EnergyE=∫∣xEnergy E = \int |x

Formulas (Indian textbook notation)

  • EnergyE=∫∣x(t)∣2dt;PowerP=lim(1T)∫∣x(t)∣2dtEnergy E = \int |x(t)|^{2} dt; Power P = lim (\frac{1}{T})\int |x(t)|^{2} dt
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 —
UnitstepuUnit step u

Formulas (Indian textbook notation)

  • Unitstepu(t);impulseδ(t);rampr(t)Unit step u(t); impulse \delta(t); ramp r(t)
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

The unit impulse δ(t) has the sifting property ∫ x(t)δ(t−t₀) dt = x(t₀), picking out the value at t₀; the unit step u(t) is its integral, and the ramp is the integral of the step.

Governing relations in practice

Any signal decomposes into even and odd parts: x_e(t) = ½[x(t)+x(−t)] and x_o(t) = ½[x(t)−x(−t)]; this symmetry simplifies Fourier coefficients.

Design and analysis considerations

For a periodic signal the power is computed over one period; for a finite-energy pulse the energy is the integral over all time. Deciding the class first tells you which measure to compute.

Assumptions and validity limits

State assumptions explicitly before using any relation for continuous time signals — 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 continuous time signals.
4. Use equation 1:
xx
.
5. Use equation 2:
EnergyE=∫∣xEnergy E = \int |x
.
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

Continuous Time Signals 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 continuous time signals with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use continuous time signals?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Computing energy for a periodic (power) signal (it is infinite)
• Taking the fundamental period as any period rather than the smallest
• Forgetting the impulse sifting property when evaluating integrals
• Mixing average power (1/T factor) with energy (no 1/T)

Quick revision checklist

Before attempting continuous time signals problems, confirm you can:
1. Even: x(−t) = x(t); odd: x(−t) = −x(t)
2.
Periodic:x(t+T)=x(t);fundamentalperiodT0Periodic: x(t+T) = x(t); fundamental period T_{0}

3. Exponential and complex exponential e^(st)
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.

Power of a sinusoid

Problem

Find the average power of the signal x(t) = 5 cos(100πt).

Solution

For a sinusoid A cos(ωt), average power = A²/2.
A = 5, so P = 5²/2 = 25/2.
P = 12.5 W (per ohm).
Since the power is finite and non-zero, this is a power signal (its energy is infinite).

Conceptual check — Continuous Time Signals

Problem

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

Exams & GATE

Oppenheim — classify signal energy vs power type.

📖 Standard books (India)

  • Signals & Systems — Oppenheim & Willsky

    Read: Syllabus unit

    Laplace, Fourier, and LTI systems

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