Energy Signals

Energy signals have finite total energy, making them the natural model for pulse-shaped waveforms used in radar and digital communications. Knowing a pulse is an energy signal tells you it carries a definite, measurable amount of work.

Core Concept

The energy of a signal is the integral of its squared magnitude over all time: E = integral from -infinity to +infinity of |x(t)|^2 dt. Think of it as the total work the signal could do if applied across a 1-ohm resistor. If this integral converges to a finite positive number, the signal is an energy signal.

For an energy signal, average power is zero. Power is energy divided by the observation interval T as T goes to infinity. If energy is finite but T grows without bound, the ratio goes to zero. This is the key distinguishing fact — not both finite energy and finite power simultaneously.

Discrete-time energy is defined as E = sum over all n of |x[n]|^2. The classification rule is identical: finite sum means energy signal. Any finite-duration sequence is automatically an energy signal because it has finitely many nonzero terms.

Key Formula

Continuous-time energy: E = integral(-inf to +inf) |x(t)|^2 dt. Discrete-time energy: E = sum(n=-inf to +inf) |x[n]|^2. Condition for energy signal: 0 < E < infinity, which forces average power P = 0.

Given: x(t) = e^(-3t) u(t). Find signal energy.

Formula:
  E = integral(0 to inf) |e^(-3t)|^2 dt

Step 1: Square the magnitude
  |e^(-3t)|^2 = e^(-6t)

Step 2: Integrate
  E = integral(0 to inf) e^(-6t) dt
    = [-1/6 * e^(-6t)] from 0 to inf

Step 3: Apply limits
  At t = inf: e^(-inf) = 0
  At t = 0:   e^(0)    = 1
  E = 0 - (-1/6) = 1/6

Final Answer:
  E = 1/6 joule (finite, positive)
  => x(t) is an energy signal
  Average power P = 0

Exam Tip: A common VTU question asks you to classify a signal as energy, power, or neither. Check energy first by integrating |x(t)|^2. If the result is finite and nonzero, stop — it is an energy signal and power is automatically zero. Periodic signals like sin(t) are never energy signals because their energy integral diverges. Do not confuse the 1-ohm normalisation convention with actual circuit power.

Properties Summary

• Definition: E = integral |x(t)|^2 dt; signal is energy type if 0 < E < inf.
• Average power of an energy signal is always zero: P = lim(T->inf) E/(2T) = 0.
• All finite-duration signals with bounded amplitude are energy signals.
• Decaying exponentials like e^(-at)u(t) for a > 0 are energy signals.
• Periodic signals are never energy signals; they are power signals.
• Discrete-time: E = sum |x[n]|^2; any finite-length sequence qualifies.

Quick Revision

• Energy signal: 0 < E < infinity; P = 0.
• Compute E by integrating |x(t)|^2 over all time.
• Decaying exponential e^(-at)u(t), a > 0 gives E = 1/(2a).
• Any finite-duration bounded signal is an energy signal.
• Periodic signals have infinite energy — never energy signals.
• Unit impulse delta(t) is treated as an energy signal with E = 1 by convention.
• Exam trap: concluding a signal is an energy signal just because its amplitude decays. Always verify the integral converges; e^(-at) with a <= 0 gives infinite energy.