Fourier Transform Properties

The Fourier transform exposes the frequency content hidden inside any signal. Audio engineers use these properties to design equalizers and noise filters without recomputing transforms from scratch.

Core Concept

A Fourier transform property is a rule that predicts how modifying a signal in time changes its spectrum. Instead of computing the transform integral again, you apply a simple algebraic operation to the existing transform.

Linearity says that scaling and adding signals in time produces the same scaled sum in frequency. This means you can break a complicated signal into simpler pieces, transform each piece, then combine.

Duality reveals that the transform and inverse transform have the same mathematical structure. If you know X(f), you can write down the transform of x(-t) immediately, without integration.

Key Formula

The Fourier transform pair uses angular frequency ω = 2πf. For a signal x(t), the transform is X(f) = ∫ x(t) e^(-j2πft) dt. The inverse recovers x(t) = ∫ X(f) e^(j2πft) df. Here t is time in seconds, f is frequency in hertz, and j = √(-1). Key property pairs: time shift x(t-t₀) ↔ e^(-j2πft₀)X(f); frequency shift x(t)e^(j2πf₀t) ↔ X(f-f₀); differentiation dx/dt ↔ j2πf·X(f).

Problem: Find the Fourier transform of x(t) = e^(-at)u(t) and apply the time-shift property to y(t) = x(t-2).

Given:
  x(t) = e^(-at)u(t), a > 0

Step 1 — Direct transform of x(t):
  X(f) = ∫₀^∞ e^(-at) e^(-j2πft) dt
       = ∫₀^∞ e^(-(a+j2πf)t) dt
       = 1 / (a + j2πf)

Step 2 — Apply time-shift property to y(t) = x(t-2):
  Time-shift property: x(t-t₀) ↔ e^(-j2πft₀) X(f)
  Here t₀ = 2

Step 3 — Write Y(f):
  Y(f) = e^(-j4πf) · X(f)
       = e^(-j4πf) / (a + j2πf)

Final Answer:
  X(f) = 1 / (a + j2πf)
  Y(f) = e^(-j4πf) / (a + j2πf)
  Magnitude |Y(f)| = |X(f)| — time shift only rotates phase, not magnitude.

Exam Tip: Time shift changes phase but never changes |X(f)|. Questions often ask you to find |Y(f)| after a shift — the answer equals |X(f)|. Duality is tested as: if x(t) ↔ X(f), then X(t) ↔ x(-f). Students often write x(f) instead of x(-f) and lose marks. For differentiation, every factor of d/dt in time multiplies by j2πf in frequency, not jω — check which convention your paper uses.

Properties Summary

• Linearity: ax(t)+by(t) ↔ aX(f)+bY(f) — superposition holds in frequency domain.
• Time shift: x(t-t₀) ↔ e^(-j2πft₀)X(f) — magnitude unchanged, phase rotated linearly.
• Frequency shift (modulation): x(t)e^(j2πf₀t) ↔ X(f-f₀) — spectrum shifts by f₀.
• Time scaling: x(at) ↔ (1/|a|)X(f/a) — compression in time means expansion in frequency.
• Differentiation: d^n x/dt^n ↔ (j2πf)^n X(f) — each derivative multiplies spectrum by j2πf.
• Convolution: x(t)*h(t) ↔ X(f)·H(f) — convolution in time equals multiplication in frequency.
• Duality: if x(t) ↔ X(f), then X(t) ↔ x(-f) — swap time and frequency roles.

Quick Revision

• Time and frequency variables are reciprocal — wider signal, narrower spectrum.
• Time shift = linear phase slope in frequency; frequency shift = modulation in time.
• Differentiation property converts differential equations to algebraic equations.
• Duality: X(t) ↔ x(-f); if x(t) is even, then X(t) ↔ x(f).
• Parseval theorem is a special case of the energy-preserving nature of the FT.
• Convolution theorem: the transform of a convolution is a pointwise product.
• Multiplication theorem: the transform of a product is a convolution of spectra.
• Exam trap: writing x(t-t₀) ↔ e^(+j2πft₀)X(f) with the wrong sign in the exponent — the exponent must be negative for a delay.