Speech Processing CheatSheet
Reading Time: ~10 min • Last Modified: July 29, 2025
Keywords: #SignalProcessing, #SpeechProcessing, #FFT, #STFT.
| Expression | Definition | Meaning or Interpretation | |
| \(x[n]\) | — | Discrete signal in waveform | V |
| \( E(x[n]) \) | \( E(x[n]) = \sum_{n=0}^{N-1} |x[n]|^2 \) | Energy of the signal in time domain | s |
| \( P(x[n]) \) | \( P(x[n]) = \frac{E(x[n])}{N} = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 \) | Power of the signal in time domain | s |
| \(X[f]\) | \( X[f] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j 2\pi f n / N} \) | Fast Fourier transform (FFT) | V |
| \(|X[f]|\) | \( |X[f]| = \sqrt{\text{Re}(X[f])^2 + \text{Im}(X[f])^2} \) | Magnitude spectrum | V |
| \(|X[f]|^2\) | \( |X[f]|^2 = \text{Re}(X[f])^2 + \text{Im}(X[f])^2 \) | Power spectrum | V |
| \( |X[f]|^{\gamma} \) | \( |X[f]|^{\gamma} = (\text{Re}(X[f])^2 + \text{Im}(X[f])^2)^{\frac{\gamma}{2}} \) | Power-law spectrum | V |
| \( E(X[f]) \) | \( E(X[f]) = \sum_{n=0}^{N-1} |X[f]|^2 \) | Energy of the signal in frequency domain | s |
| \( P(X[f]) \) | \( P(X[f]) = \frac{E(X[f])}{N} = \frac{1}{N} \sum_{n=0}^{N-1} |X[f]|^2 \) | Power of the signal in frequency domain | s |
| \( ||X[f])||_2 \) | \( ||X[f])||_2 = \sqrt{\sum_{n=0}^{N-1} |X[f]|^2} = \sqrt{E(X[f])} \) | L2-norm (or Euclidean norm) of the FFT or the square root of the energy | s |
| \( ||X[f])||_2^2 \) | \( ||X[f])||_2^2 = \sum_{n=0}^{N-1} |X[f]|^2 = E(X[f]) \) | Squared L2-norm (or Euclidean norm) of the FFT or the energy | s |
| \(X[t,f]\) | \( X[t, f] = \sum_{n=0}^{N-1} x[n + tH] \cdot w[n] \cdot e^{-j \frac{2\pi fn}{N}} \) | Short-time Fourier transform (STFT) | M |
| \(|X[t,f]|\) | \( |X[t,f]| = \sqrt{\text{Re}(X[t,f])^2 + \text{Im}(X[t,f])^2} \) | Magnitude spectrogram | M |
| \(|X[t,f]|^2\) | \( |X[t,f]|^2 = \text{Re}(X[t,f])^2 + \text{Im}(X[t,f])^2 \) | Power spectrogram (commonly called "spectrogram" in DSP) | M |
| \(|X[t,f]|^{\gamma}\) | \( |X[t,f]|^{\gamma} = (\text{Re}(X[t,f])^2 + \text{Im}(X[t,f])^2)^{\frac{\gamma}{2}} \) | Power-law spectrogram | M |
s = scale, V = vector, M = Matrix
Mathematical Functions
| \(|\cdot|\) | Absolute value, modulus or magnitude (\( \forall z \in \mathbb{C}; \; |z| = \sqrt{\text{Re}(z)^2 + \text{Im}(z)^2})\) |
| \(|| \cdot ||_2\) | Euclidean norm or \(\mathcal{L}_{2}\)-norm ( \(A \in \mathbb{R}^{m \times n} \text{or} \mathbb{C}^{m \times n};\;\ ||A||_F = \sqrt{\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2} \)) |
| \(|| \cdot ||_F\) | Frobenius norm |