Inverse Fourier Transform

The inverse Fourier transform reconstructs a time-domain signal from its frequency-domain representation, allowing you to recover x(t) from its spectrum X(f). Communications receivers use this to decode modulated signals back into audio or data.

Core Concept

The inverse Fourier transform is the reverse operation of the Fourier transform. Given the spectrum X(f), it reconstructs the original signal x(t). It works by adding together sinusoids at every frequency, each weighted by the complex amplitude X(f) at that frequency.

Think of X(f) as a recipe: it says how much of each frequency ingredient to add. The IFT mixes those ingredients to recreate the original signal. The integration over all frequencies is a continuous version of the superposition principle used in Fourier series.

The forward and inverse transforms form a transform pair. Applying the FT followed by the IFT returns the original signal, provided x(t) is well-behaved. At points of discontinuity, the IFT converges to the average of the left and right limits.

Key Formula

The inverse Fourier transform is x(t) = integral from -inf to +inf of X(f) * e^(j2*pi*f*t) df. Here X(f) is the complex spectrum, f is frequency in Hz, t is time in seconds, and the exponential e^(j2*pi*f*t) is the basis sinusoid at frequency f. In angular frequency notation: x(t) = (1/2*pi) * integral of X(omega) * e^(j*omega*t) d*omega. Notice the 1/2*pi factor that appears in the omega convention but not the f convention.

Given: X(f) = 1/(2 + j2*pi*f)
Find:  x(t) using the known inverse FT pair

Step 1: Recognise the standard form
  X(f) = 1/(a + j2*pi*f)  with a = 2

Step 2: Recall the inverse pair
  1/(a + j2*pi*f)  <-->  e^(-at) u(t)

Step 3: Apply the pair directly
  x(t) = e^(-2t) u(t)

Verification: FT of e^(-2t) u(t)
  = integral from 0 to inf of e^(-2t) e^(-j2*pi*f*t) dt
  = 1/(2 + j2*pi*f)  [confirmed]

Final Answer: x(t) = e^(-2t) u(t)

Exam Tip: The IFT in angular frequency form carries a (1/2*pi) factor: x(t) = (1/2*pi) integral X(omega) e^(j*omega*t) d*omega. In the f-Hz form there is no such factor. Questions often mix the two conventions, so identify the convention first. Also remember that the IFT of a real and symmetric spectrum is always a real signal.

Properties Summary

• Inverse linearity: IFT{a*X1(f) + b*X2(f)} = a*x1(t) + b*x2(t).
• Frequency shift in IFT: IFT{X(f - f0)} = x(t) * e^(j2*pi*f0*t), spectrum shift multiplies x(t) by a complex exponential.
• Time delay from IFT: IFT{X(f) * e^(-j2*pi*f*t0)} = x(t - t0).
• Conjugate symmetry: if x(t) is real, then X(-f) = X*(f), so the IFT of a conjugate-symmetric spectrum is always real.
• Duality: IFT{x(f)} = X(-t), the IFT of x used as a frequency function equals X evaluated at -t.
• Parseval via IFT: integral |x(t)|^2 dt = integral |X(f)|^2 df, energy is preserved through both directions.

Quick Revision

• IFT in f-convention: x(t) = integral X(f) e^(j2*pi*f*t) df (no scaling factor).
• IFT in omega-convention: x(t) = (1/2*pi) integral X(omega) e^(j*omega*t) d*omega (1/2*pi factor present).
• At a jump discontinuity, the IFT gives the average: [x(t+) + x(t-)]/2.
• Applying FT then IFT returns x(t) exactly at all points of continuity.
• Real and even X(f) gives a real and even x(t) after IFT.
• The IFT of a delta function delta(f - f0) is the complex sinusoid e^(j2*pi*f0*t).
• Exam trap: forgetting the (1/2*pi) factor when working in the omega domain leads to a scaled answer that is wrong in magnitude but correct in shape.