Real and Complex Signals

Real signals have amplitudes that are real numbers at every instant, while complex signals carry both magnitude and phase information simultaneously in a single mathematical object. Phasors in AC circuit analysis and baseband IQ signals in software-defined radios are everyday examples of complex signals.

Core Concept

A real signal x(t) produces a single real number at each time instant. Most physical measurements — voltage, current, pressure — are real signals. Their Fourier transforms have Hermitian symmetry: X(-f) = X*(f), meaning the negative-frequency content carries no new information.

A complex signal x(t) = x_R(t) + j x_I(t) carries two real-valued waveforms simultaneously. The real part x_R(t) and imaginary part x_I(t) are independent. Complex exponentials e^(j omega t) = cos(omega t) + j sin(omega t) are the building blocks of Fourier analysis precisely because they are complex signals with constant magnitude and linearly changing phase.

The conjugate of a complex signal is x*(t) = x_R(t) - j x_I(t). The magnitude is |x(t)| = sqrt(x_R^2 + x_I^2) and the instantaneous phase is angle(x(t)) = arctan(x_I / x_R). In digital communications, the in-phase (I) and quadrature (Q) channels form the real and imaginary parts of the complex baseband signal.

Key Formula

Euler's formula connects real and complex representations: e^(j omega t) = cos(omega t) + j sin(omega t). This gives: cos(omega t) = [e^(j omega t) + e^(-j omega t)] / 2 and sin(omega t) = [e^(j omega t) - e^(-j omega t)] / (2j). Conjugate symmetry of Fourier transform of a real signal: X(-f) = X*(f).

Given: x(t) = e^(j 2 pi 5 t). Find real part, imaginary part, magnitude, phase.

Formula: e^(j theta) = cos(theta) + j sin(theta)

Step 1: Identify theta
  theta = 2 pi * 5 * t = 10 pi t

Step 2: Real part
  x_R(t) = cos(10 pi t)

Step 3: Imaginary part
  x_I(t) = sin(10 pi t)

Step 4: Magnitude
  |x(t)| = sqrt(cos^2 + sin^2) = sqrt(1) = 1
  (constant magnitude = 1 for all t)

Step 5: Instantaneous phase
  angle x(t) = arctan(sin/cos) = 10 pi t  (increases linearly)

Final Answer:
  Real part: cos(10 pi t)
  Imaginary part: sin(10 pi t)
  Magnitude: 1
  Phase: 10 pi t radians

Exam Tip: When asked whether a signal has conjugate symmetry in its Fourier transform, the answer is yes only if the original time-domain signal is real-valued. A complex signal like e^(j omega0 t) has a single-sided spectrum — one delta function at +f0 only, not at -f0. This distinction appears in GATE problems on spectrum plotting and Fourier transform properties. Also note: |x(t)|^2 = x(t) x*(t), which is the correct way to compute energy for complex signals.

Properties Summary

• Real signal: x(t) is real for all t; Fourier transform satisfies X(-f) = X*(f).
• Complex signal: x(t) = x_R(t) + j x_I(t); no conjugate symmetry in general.
• Euler: e^(j omega t) = cos(omega t) + j sin(omega t); magnitude always 1.
• Energy of complex signal: E = integral |x(t)|^2 dt = integral x(t)x*(t) dt.
• Conjugate: x*(t) obtained by replacing j with -j everywhere.
• Real part: x_R = [x + x*]/2; imaginary part: x_I = [x - x*]/(2j).

Quick Revision

• Real signal: single real number at each t.
• Complex signal: ordered pair (real part, imaginary part) at each t.
• e^(j omega t) has magnitude 1 and phase omega t.
• Fourier transform of a real signal is conjugate symmetric: X(-f) = X*(f).
• Energy computation requires |x(t)|^2 = x(t) x*(t).
• IQ modulation in radio uses complex baseband: I + jQ.
• Exam trap: plotting the spectrum of e^(j omega0 t) as two lines at +/-f0. It has only one line at +f0. Only real sinusoids have two-sided spectra.