Cheatsheet — Fourier Quick References

Core identities

Euler's formula

$$e^{j\theta} = \cos\theta + j\sin\theta,\qquad$$ $$e^{-j\theta} = \cos\theta - j\sin\theta$$

Polar / phasor

$$r e^{j\theta} \quad\text{magnitude } r,\ \text{phase } \theta$$

Continuous Fourier Transform (engineering convention)

Forward

$$X(f)=\int_{-\infty}^{\infty} x(t)\,e^{-j2\pi f t}\,dt$$

Inverse

$$x(t)=\int_{-\infty}^{\infty} X(f)\,e^{+j2\pi f t}\,df$$

Discrete Fourier Transform (N-point, engineering / FFT convention)

Forward

$$X[k]=\sum_{n=0}^{N-1} x[n]\,e^{-j\frac{2\pi}{N}kn},\qquad k=0,\dots,N-1$$

Inverse

$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1} X[k]\,e^{+j\frac{2\pi}{N}kn},\qquad n=0,\dots,N-1$$

Frequency of bin $k$ (sample rate $f_s$):

$$f_k = k\frac{f_s}{N},\quad\Delta f=\frac{f_s}{N}$$

Bins with $k>N/2$ correspond to negative frequencies at $f_k-f_s$.

Useful properties (quick)

Conjugate symmetry (real input)

$$x[n]\in\mathbb{R}\ \Rightarrow\ X[N-k]=\overline{X[k]}$$

Parseval / energy conservation (DFT)

$$\sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2$$

(depends on chosen normalization — check convention)

Linearity

$$FT\{a·x + b·y\} = a·FT\{x\} + b·FT\{y\}$$

Time shift:

$$x[n-n_0] \quad ↔ \quad X[k] e^{-j\frac{2\pi}{N}k n_0}$$

Frequency shift (modulation):

$$e^{j\frac{2\pi}{N} m n}x[n] \quad \text{shifts bins}$$

Practical tips & gotchas

• DFT implicitly assumes periodicity of the finite sequence → circular convolution.
  • Use zero-padding to reduce wrap-around / increase frequency resolution.
  • Use windowing (Hann, Hamming, Blackman) to reduce spectral leakage for non-periodic segments.
• Zero-padding: increases bin count (interpolates spectrum) but does not add new information.
• fftshift / ifftshift: center the zero-frequency bin for visualization.
• Normalization conventions vary:
  • Engineering/FFT: forward has no $\frac{1}{N}$, inverse has $\frac{1}{N}$ (this cheatsheet's default).
  • Other conventions: symmetric $\frac{1}{\sqrt{N}}$ on both transforms, or $\frac{1}{N}$ on forward. Always check library docs.
• Nyquist / sampling theorem:
  • To avoid aliasing: $f_s \ge 2 f_{\text{max}}$ (sample at least twice the highest signal frequency).
  • If undersampled, high-frequency content folds into lower frequencies (aliasing).

Quick NumPy / SciPy commands (examples)

• 1-D FFT / inverse (NumPy / SciPy follow engineering convention)
  • X = np.fft.fft(x)
  • x_rec = np.fft.ifft(X) (if x real, x_rec.real equals original within numerical error)
• Zero-pad to length M:
  • x_padded = np.pad(x, (0, M-len(x))) then np.fft.fft(x_padded)
• Shift for plotting:
  • Xc = np.fft.fftshift(np.fft.fft(x))
  • Frequency axis: freqs = np.fft.fftshift(np.fft.fftfreq(N, d=1/f_s))
• Real-input optimized transforms:
  • X = np.fft.rfft(x) and inverse x = np.fft.irfft(X, n=N)
• 2-D FFT (images):
  • F = np.fft.fft2(image)
  • Fshift = np.fft.fftshift(F)
  • image_rec = np.fft.ifft2(F)

Quick visual checklist

• Before taking FFT: detrend / remove DC if needed.
• Use window when analyzing short, non-periodic segments.
• Use zero-padding for smoother spectral plots (visual refinement).
• For filters: remember multiplication in frequency = convolution in time (and vice versa). For linear convolution via DFT, pad signals to length ≥ N1+N2−1.

Minimal cheat summary (one-liners)

• FT kernel: $e^{-j2\pi f t}$ (analysis), inverse uses opposite sign.
• DFT maps time samples $n$ ↔ bins $k$ with phase factor $e^{-j2\pi kn/N}$.
• Nyquist: $f_s \ge 2 f_{\max}$.
• np.fft.fft, np.fft.ifft, np.fft.fftshift are your go-to tools.