Aperiodic Signals

An aperiodic signal does not repeat with any finite period, and its total energy may be finite or infinite depending on its shape. Aperiodic signals are analyzed with the Fourier transform rather than the Fourier series, which gives a continuous spectrum instead of discrete harmonic lines.

Core Concept

An aperiodic signal has no period T. It may exist only for a finite duration (a rectangular pulse, a single impulse), or it may extend infinitely but without repetition (a decaying exponential). Aperiodic signals that have finite energy are the most important class in signal processing.

The Fourier transform generalises the Fourier series to aperiodic signals by letting the period go to infinity. As T increases, the harmonic spacing 1/T shrinks until the discrete spectral lines merge into a continuous spectrum. The Fourier transform X(f) is a continuous function of frequency describing the signal's spectral density.

Aperiodic signals can be energy signals or power signals. An aperiodic signal with finite energy (such as a rectangular pulse) is an energy signal. An aperiodic signal that has finite average power but infinite energy (such as a single-sided white noise process) is a power signal. A growing exponential is neither.

Key Formula

Fourier transform pair for an aperiodic energy signal:

X(f) = integral from -inf to +inf of x(t) * exp(-j*2*pi*f*t) dt

x(t) = integral from -inf to +inf of X(f) * exp(j*2*pi*f*t) df

Parseval's theorem for aperiodic signals: integral |x(t)|^2 dt = integral |X(f)|^2 df

Key pairs: rectangular pulse of width tau transforms to A*tau*sinc(f*tau). Decaying exponential e^(-a*t)*u(t) transforms to 1/(a + j*2*pi*f).

Problem: Find the Fourier transform of x(t) = e^(-3t) * u(t).

Step 1: Apply Fourier transform definition
  X(f) = integral_0^inf e^{-3t} * e^{-j2*pi*f*t} dt
       = integral_0^inf e^{-(3 + j2*pi*f)*t} dt

Step 2: Evaluate the integral
  Let s = 3 + j2*pi*f (Re{s} > 0, so integral converges)
  X(f) = [-1/s * e^{-st}] from 0 to inf
       = 0 - (-1/s)
       = 1/s

Step 3: Substitute back
  X(f) = 1 / (3 + j*2*pi*f)

Step 4: Magnitude spectrum
  |X(f)| = 1 / sqrt(9 + (2*pi*f)^2)
  This is a Lorentzian (bell-shaped) function.

Step 5: Energy via Parseval
  E = integral |x(t)|^2 dt = integral_0^inf e^{-6t} dt = 1/6

Final Answer: X(f) = 1/(3 + j2*pi*f), energy E = 1/6.

Exam Tip: Parseval's theorem is often the fastest way to compute the energy of a signal whose Fourier transform is known. If X(f) is simpler to integrate in frequency than x(t) is to integrate in time, use E = integral |X(f)|^2 df. Also note: an aperiodic signal has a continuous spectrum, while a periodic signal has a line (discrete) spectrum. GATE questions sometimes ask you to identify the type of spectrum from the time-domain description.

Properties Summary

• Aperiodic signals have a continuous Fourier spectrum X(f), not discrete harmonic lines.
• Finite-energy aperiodic signals satisfy integral |x(t)|^2 dt < infinity.
• Time-bandwidth product: a shorter pulse in time has a broader spectrum. tau * BW = constant.
• Fourier transform pair: x(t) <-> X(f); synthesis integral uses e^{+j2*pi*f*t}.
• Parseval: integral |x(t)|^2 dt = integral |X(f)|^2 df; energy is the same in both domains.
• Rectangular pulse of duration tau: X(f) = A*tau*sinc(f*tau). First null at f = 1/tau.
• Exponential e^{-at}u(t): X(f) = 1/(a+j2*pi*f). Magnitude is Lorentzian with 3dB bandwidth a/(2*pi) Hz.

Quick Revision

• Aperiodic = no repeating pattern. Fourier transform gives a continuous spectrum.
• Energy signal: finite integral of |x(t)|^2. Examples: rectangular pulse, e^{-at}u(t).
• Power signal: finite average power, infinite energy. Example: unit step.
• Fourier transform exists for signals satisfying the Dirichlet conditions (absolutely integrable, finite extrema, finite discontinuities).
• Shorter pulse => wider spectrum. This is the time-bandwidth trade-off.
• Parseval connects time-domain energy and frequency-domain energy.
• Periodic signals viewed as limits: as T -> inf, Fourier series becomes Fourier transform.
• Exam trap: using the Fourier series coefficient formula c_n = (1/T)*integral for an aperiodic signal. Aperiodic signals need the Fourier transform integral, not the series formula.