Fourier Transform Cheat Sheet | Formulas & Properties

🎯 What is Fourier Transform?

The Fourier Transform decomposes a function into its frequency components. It transforms a time-domain signal into its frequency-domain representation, revealing which frequencies are present and their amplitudes.

Forward Fourier Transform (Time → Frequency)

F(ω) = ∫_-∞^∞ f(t) e^-iωt dt

From time domain f(t) to frequency domain F(ω)

Inverse Fourier Transform (Frequency → Time)

f(t) = (1/2π) ∫_-∞^∞ F(ω) e^iωt dω

From frequency domain F(ω) back to time domain f(t)

📝 Notation Guide

Time Domain

f(t) = Original signal/function

t = Time variable (seconds)

x(t) = Alternative time notation

Frequency Domain

F(ω) = Fourier Transform of f(t)

ω = Angular frequency (rad/sec)

X(ω) = Alternative frequency notation

Relationship

f(t) ⇌ F(ω)

Time domain ⇔ Frequency domain

Forward: ℱ{f(t)} = F(ω)

Inverse: ℱ⁻¹{F(ω)} = f(t)

🔑 Essential Transform Pairs

Time Domain f(t)	Frequency Domain F(ω)	Notes
δ(t)	1	Dirac delta (impulse) - contains all frequencies equally
1	2πδ(ω)	Constant DC signal
e^iω₀t	2πδ(ω - ω₀)	Complex exponential (single frequency)
cos(ω₀t)	π[δ(ω - ω₀) + δ(ω + ω₀)]	Cosine wave - two impulses at ±ω₀
sin(ω₀t)	iπ[δ(ω + ω₀) - δ(ω - ω₀)]	Sine wave
u(t)e^-at	1/(a + iω)	Exponential decay (a > 0), u(t) is unit step
rect(t/τ)	τ sinc(ωτ/2)	Rectangular pulse of width τ
sinc(at)	(π/a) rect(ω/(2a))	Sinc function → rectangular in frequency
e^-a\|t\|	2a/(a² + ω²)	Two-sided exponential
e^-πt²	e^-πω²	Gaussian → Gaussian (self-reciprocal)

📐 Fundamental Properties

Property	Time Domain	Frequency Domain	Description
Linearity	af(t) + bg(t)	aF(ω) + bG(ω)	Transform of sum = sum of transforms
Time Shifting	f(t - t₀)	e^-iωt₀ F(ω)	Delay adds linear phase shift
Frequency Shifting	e^iω₀t f(t)	F(ω - ω₀)	Multiplication by complex exponential
Time Scaling	f(at)	(1/\|a\|) F(ω/a)	Compress time ↔ Expand frequency
Differentiation	f'(t)	iω F(ω)	Derivative ↔ Multiply by iω
Integration	∫_-∞^t f(τ) dτ	F(ω)/(iω) + πF(0)δ(ω)	Integral ↔ Divide by iω
Convolution	f(t) * g(t)	F(ω) G(ω)	Convolution ↔ Multiplication
Multiplication	f(t) g(t)	(1/2π) F(ω) * G(ω)	Multiplication ↔ Convolution
Parseval's Theorem	∫ \|f(t)\|² dt	(1/2π) ∫ \|F(ω)\|² dω	Energy conservation
Duality	F(t)	2π f(-ω)	Swap time and frequency domains

🧮 Worked Examples

Example 1: Fourier Transform of Rectangular Pulse

Given: f(t) = rect(t/τ) = { 1 for |t| < τ/2, 0 otherwise }

F(ω) = ∫_-τ/2^τ/2 e^-iωt dt

= [e^-iωt/(-iω)]_-τ/2^τ/2

= (e^-iωτ/2 - e^iωτ/2)/(-iω)

= (2 sin(ωτ/2))/ω

= τ sinc(ωτ/2π) where sinc(x) = sin(πx)/(πx)

Result: ℱ{rect(t/τ)} = τ sinc(ωτ/2)

Example 2: Time Shifting Property

Given: ℱ{f(t)} = F(ω). Find ℱ{f(t - t₀)}

ℱ{f(t - t₀)} = ∫ f(t - t₀) e^-iωt dt

Let u = t - t₀, then t = u + t₀, dt = du

= ∫ f(u) e^{-iω(u + t₀)} du

= e^-iωt₀ ∫ f(u) e^-iωu du

= e^-iωt₀ F(ω)

Result: Time delay introduces linear phase shift e^-iωt₀

🎭 Special Cases & Variations

Fourier Series (Periodic Signals)

f(t) = Σ_n=-∞^∞ cₙ e^inω₀t

cₙ = (1/T) ∫_T f(t) e^-inω₀t dt

For periodic signals with period T

Discrete Fourier Transform (DFT)

X[k] = Σ_n=0^N-1 x[n] e^-i2πkn/N

x[n] = (1/N) Σ_k=0^N-1 X[k] e^i2πkn/N

For discrete-time signals

Fast Fourier Transform (FFT)

Efficient algorithm for DFT

Complexity: O(N log N) vs DFT O(N²)

Used in digital signal processing

💡 Key Insights & Applications

Time-frequency tradeoff: Narrow in time → Wide in frequency, and vice versa
Convolution theorem: Filtering in time domain = Multiplication in frequency domain
Parseval's theorem: Total energy in time domain = Total energy in frequency domain
Applications: Signal processing, image compression (JPEG), audio analysis, communications, quantum mechanics
Common mistake: Forgetting the 2π factor in inverse transform or Parseval's theorem

⚖️ Symmetry Properties

Time Domain	Frequency Domain
f(t) real	F(-ω) = F*(ω) (Conjugate symmetric)
f(t) real and even	F(ω) real and even
f(t) real and odd	F(ω) imaginary and odd
f(t) imaginary	F(-ω) = -F*(ω)