Continuous Time Signals

Continuous-time signals are functions of a real-valued time variable t that take values at every instant on a continuous time axis. They are the natural mathematical model for physical quantities such as voltage, pressure, and temperature measured by sensors in control and communication systems.

Core Concept

A continuous-time signal x(t) is defined for every real value of t in some interval. The time variable t is continuous, meaning there are no gaps. The signal amplitude may be any real or complex number. This is in contrast to discrete-time signals, where x[n] is only defined at integer values of n.

Continuous-time signals are the natural representation of physical quantities. Voltage across a capacitor, sound pressure at a microphone, and position of a mechanical arm are all continuous-time signals. Mathematical models for these signals include sinusoids, exponentials, unit step functions, and impulse functions.

Signals are classified by their properties. An energy signal has finite total energy. A power signal has finite and nonzero average power. A periodic signal repeats with a fixed period T. A causal signal is zero for t less than zero. These classifications determine which analysis tools apply.

Key Formula

Energy and power of a continuous-time signal x(t):

Energy: E = integral from -infinity to +infinity of |x(t)|^2 dt

Power: P = lim_{T->inf} (1/2T) * integral from -T to T of |x(t)|^2 dt

A signal is an energy signal if E is finite (P = 0). It is a power signal if P is finite and nonzero (E = infinity). The unit step u(t) = 1 for t >= 0 and 0 otherwise. The impulse delta(t) satisfies integral of delta(t) dt = 1 and x(t)*delta(t-a) = x(a)*delta(t-a).

Problem: Classify x(t) = e^(-2t) * u(t) as energy or power signal.

Step 1: Compute energy E
  E = integral from 0 to inf of |e^(-2t)|^2 dt
    = integral from 0 to inf of e^(-4t) dt

Step 2: Evaluate the integral
  E = [-1/4 * e^(-4t)] from 0 to inf
    = 0 - (-1/4)
    = 1/4

Step 3: Energy is finite (E = 0.25 J)
  Therefore x(t) is an ENERGY signal.

Step 4: Power of an energy signal
  P = E / infinite duration = 0

Final Answer: x(t) = e^(-2t)*u(t) is an energy signal with E = 1/4.

Exam Tip: Sinusoids and unit step signals have infinite energy but finite average power, so they are power signals. Signals that decay to zero (like e^(-at)*u(t) with a > 0) are energy signals. A signal cannot be both. Signals that grow without bound (like e^(at)*u(t) with a > 0) are neither energy nor power signals. This classification appears frequently in GATE.

Properties Summary

• Defined for all real t: x(t) is a function R -> R or R -> C, continuous in the time variable.
• Energy: E = integral |x(t)|^2 dt; finite for energy signals.
• Power: P = lim (1/2T) * integral_{-T}^{T} |x(t)|^2 dt; finite for power signals.
• Periodicity: x(t) = x(t+T) for all t and some smallest positive T (the fundamental period).
• Causality: x(t) = 0 for all t < 0. Causal signals are right-sided.
• Even/Odd decomposition: any signal splits as x(t) = x_e(t) + x_o(t), where x_e(t) = [x(t)+x(-t)]/2 and x_o(t) = [x(t)-x(-t)]/2.
• Impulse sifting: integral of x(t)*delta(t-t0) dt = x(t0). This is used constantly in convolution.

Quick Revision

• Continuous-time means defined for every real t, not just at integers.
• Energy signal: finite energy, zero average power. Example: e^(-at)*u(t), a > 0.
• Power signal: finite nonzero average power, infinite energy. Example: sinusoid, unit step.
• Neither: growing exponential e^(at)*u(t) with a > 0 is neither.
• Causality: x(t) = 0 for t < 0. Relevant for Laplace transform ROC and system stability.
• Impulse delta(t): zero everywhere except t=0, unit area. Not a conventional function.
• Exam trap: concluding that a bounded signal must be a power signal. The signal x(t) = 1/(1+t^2) is bounded but is actually an energy signal.