Fourier Transform Definition

The Fourier transform converts a time-domain signal into its frequency-domain representation, revealing which sinusoidal components make up the signal. Audio engineers use it to identify and filter unwanted frequencies in recordings.

Core Concept

A signal made up of multiple sinusoids can be described either by how it varies over time, or by listing which frequencies it contains and how strong each one is. The Fourier transform provides the second description. It decomposes x(t) into an infinite sum of complex exponentials, each at a different frequency.

The transform works because complex exponentials e^(j2*pi*f*t) form an orthogonal basis. Multiplying x(t) by e^(-j2*pi*f*t) and integrating over all time isolates the contribution at frequency f. The result X(f) is generally complex, carrying both amplitude and phase information at every frequency.

The Fourier transform exists for signals that are absolutely integrable, meaning the integral of |x(t)| over all time is finite. Signals like the unit step or sinusoids require special treatment using impulse functions in the frequency domain.

Key Formula

The forward Fourier transform is defined as the integral of x(t) multiplied by the complex exponential e^(-j2*pi*f*t) over all time. X(f) = integral from -inf to +inf of x(t) * e^(-j2*pi*f*t) dt. Here x(t) is the time-domain signal, f is frequency in Hz, j is the imaginary unit, and X(f) is the complex spectrum. An equivalent form uses angular frequency omega = 2*pi*f. Common pairs: rect(t/tau) <-> tau * sinc(f*tau), and e^(-at)u(t) <-> 1/(a + j2*pi*f) for a > 0.

Given: x(t) = e^(-3t) u(t)
Formula: X(f) = integral from 0 to inf of e^(-3t) * e^(-j2*pi*f*t) dt

Step 1: Combine exponents
  X(f) = integral from 0 to inf of e^(-(3 + j2*pi*f)t) dt

Step 2: Evaluate integral
  X(f) = [-1/(3 + j2*pi*f)] * e^(-(3+j2*pi*f)t) from 0 to inf

Step 3: Apply limits
  At t = inf: e^(-(3+j2*pi*f)*inf) = 0  (since real part 3 > 0)
  At t = 0:   e^0 = 1

Step 4: Result
  X(f) = 1 / (3 + j2*pi*f)

Final Answer: X(f) = 1 / (3 + j2*pi*f)
Magnitude: |X(f)| = 1 / sqrt(9 + 4*pi^2*f^2)

Exam Tip: The Fourier transform exists only if x(t) is absolutely integrable. For signals like u(t) or cos(2*pi*f0*t) that are not absolutely integrable, the transform contains Dirac delta terms. The angular-frequency form X(omega) = integral of x(t)*e^(-j*omega*t) dt differs from X(f) by the factor 2*pi. Always check which convention the question uses, since X(omega) = X(f)|_{f=omega/2*pi}.

Properties Summary

• Linearity: a*x1(t) + b*x2(t) <-> a*X1(f) + b*X2(f), so the transform of a sum is the sum of transforms.
• Time shifting: x(t - t0) <-> X(f)*e^(-j2*pi*f*t0), a delay in time is a phase shift in frequency.
• Frequency shifting: x(t)*e^(j2*pi*f0*t) <-> X(f - f0), multiplication by a complex exponential shifts the spectrum.
• Time scaling: x(at) <-> (1/|a|)*X(f/a), compressing in time expands in frequency and reduces amplitude.
• Convolution theorem: x(t)*h(t) <-> X(f)*H(f), convolution in time becomes multiplication in frequency.
• Parseval's theorem: integral |x(t)|^2 dt = integral |X(f)|^2 df, total energy is the same in both domains.
• Duality: if X(f) is the FT of x(t), then x(f) as a time signal has FT equal to X(-f).

Quick Revision

• X(f) is generally complex; |X(f)| is the magnitude spectrum and angle(X(f)) is the phase spectrum.
• A real and even x(t) gives a real and even X(f).
• A real and odd x(t) gives a purely imaginary and odd X(f).
• Sufficient condition for existence: x(t) is absolutely integrable, meaning integral of |x(t)| dt < infinity.
• The two-sided exponential e^(-a|t|) has FT 2a/(a^2 + (2*pi*f)^2).
• Differentiation in time: dx/dt <-> j2*pi*f * X(f).
• Exam trap: confusing X(f) with X(omega). The factor 2*pi appears differently depending on the convention, so substituting one for the other without adjustment gives wrong answers.