Skip to main content

Fourier Transform Cheat Sheet

Everything you need on one page. Bookmark this or print it for exams and lab work.

① Key Formulas

Continuous Fourier Transform

Forward

F(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-j\omega t}\,dt

Inverse

f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)\,e^{j\omega t}\,d\omega

Discrete Fourier Transform

Forward (DFT)

X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j2\pi kn/N}

Inverse (IDFT)

x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]\,e^{j2\pi kn/N}

Discrete-Time Fourier Transform

Forward

X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]\,e^{-j\omega n}

Inverse

x[n] = \frac{1}{2\pi}\int_{-\pi}^{\pi} X(e^{j\omega})\,e^{j\omega n}\,d\omega

FFT Complexity

Direct DFT

O(N^2) \text{ multiplications}

FFT (Cooley-Tukey)

O(N \log_2 N) \text{ multiplications}

For N = 1024: 1,048,576 → 10,240 — a 100× speedup.

② Essential Pairs

The 8 pairs you'll use most often. See the full table →

Signal f(t) F(\omega)
Impulse \delta(t) 1
Constant 1 2\pi\delta(\omega)
Rect \text{rect}(t/\tau) \tau\,\text{sinc}(\omega\tau/2\pi)
Gaussian e^{-\alpha t^2} \sqrt{\pi/\alpha}\,e^{-\omega^2/4\alpha}
Exponential e^{-\alpha t}u(t) 1/(\alpha + j\omega)
Cosine \cos(\omega_0 t) \pi[\delta(\omega{-}\omega_0)+\delta(\omega{+}\omega_0)]
Comb \sum\delta(t - nT) \frac{2\pi}{T}\sum\delta(\omega - 2\pi k/T)
Signum \text{sgn}(t) 2/(j\omega)

③ Key Properties

The 6 properties you'll reach for most. See all 15 properties →

Linearity

af + bg \leftrightarrow aF + bG

Time Shifting

f(t-t_0) \leftrightarrow e^{-j\omega t_0}F(\omega)

Convolution

f * g \leftrightarrow F \cdot G

Time Scaling

f(at) \leftrightarrow \frac{1}{|a|}F(\omega/a)

Differentiation

f'(t) \leftrightarrow j\omega\,F(\omega)

Parseval's Theorem

\int|f|^2 dt = \frac{1}{2\pi}\int|F|^2 d\omega

④ Common Relationships

Frequency Resolution

\Delta f = \frac{f_s}{N} = \frac{1}{T_{\text{record}}}
  • f_s = sample rate, N = number of points
  • More samples → finer resolution
  • Higher sample rate alone does NOT improve resolution

Zero-Padding Effect

N_{\text{padded}} = 2^{\lceil\log_2 N\rceil} \text{ (next power of 2)}
  • Interpolates the spectrum (smoother appearance)
  • Does not add new information
  • Does not improve true frequency resolution
  • Useful for FFT efficiency (radix-2)

Windowing Trade-offs

Window Main Lobe Side Lobes Best For
RectangularNarrowest−13 dBResolution
HannMedium−31 dBGeneral
HammingMedium−42 dBSidelobe control
BlackmanWide−58 dBDynamic range
Flat TopWidest−93 dBAmplitude accuracy

Nyquist-Shannon Sampling

f_s \ge 2\,f_{\max}
  • f_{\max} = highest frequency present in the signal
  • Violating this causes aliasing — high frequencies fold into low frequencies
  • Anti-aliasing filter must precede the ADC
  • Nyquist frequency: f_N = f_s / 2

⑤ Practical Tips

Choosing FFT Size

  • Use power-of-2 sizes (N = 2^k) for fastest computation
  • Want finer resolution? Collect more data (longer recording), not just more zero-padding
  • Common sizes: 256, 512, 1024, 2048, 4096, 8192
  • For real-time: balance latency (N/f_s seconds) vs. resolution (f_s/N Hz)

Handling Real Signals

  • FFT of a real signal is conjugate symmetric: only bins 0 to N/2 are unique
  • Use rfft for efficiency (half the output)
  • Bin k corresponds to frequency f_k = k \cdot f_s / N
  • DC component at bin 0, Nyquist at bin N/2

Interpreting Magnitude & Phase

  • Magnitude |X[k]|: strength of each frequency component
  • Phase \angle X[k]: timing offset of each component
  • Power spectrum: |X[k]|^2 / N or in dB: 10\log_{10}|X[k]|^2
  • Phase is often noisy — only meaningful where magnitude is significant
  • Use unwrap on phase to remove 2\pi discontinuities

Spectral Leakage

  • Occurs when signal frequency doesn't land on an FFT bin center
  • Energy "leaks" into adjacent bins
  • Fix: apply a window function before the FFT
  • Trade-off: windowing improves leakage but widens spectral peaks

Overlap-Add / Overlap-Save

  • For real-time filtering of long (streaming) signals
  • Break input into overlapping blocks, FFT each, multiply by filter, IFFT
  • Overlap-add: zero-pad and sum adjacent output blocks
  • Overlap-save: discard edge samples, keep valid region

Quick Conversions

  • \omega = 2\pi f (angular ↔ ordinary freq)
  • T = 1/f (period ↔ frequency)
  • dB magnitude: 20\log_{10}|H(f)|
  • dB power: 10\log_{10}|H(f)|^2
  • 3 dB down = half power = 1/\sqrt{2} amplitude

Need More Detail?

This cheat sheet covers the essentials. For the complete tables, see the full references.

Full Fourier Pairs Table All 15 Properties FFT Calculator Visualizer