Fundamental operation for LTI systems and filtering. Definition Continuous-Time $$ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau $$ Discrete-Time $$ y[n] = (x * h)[n] = \sum_{k=-\infty}^{\infty} x[k] h[n - k] $$ Physical Interpretation Convolution represents the output of an LTI system: $x[n]$: …

Measure similarity between signals for pattern matching and analysis. Cross-Correlation $$ (x \star y)[n] = \sum_{k=-\infty}^{\infty} x[k] y[n + k] $$ Auto-Correlation $$ R_{xx}[n] = (x \star x)[n] = \sum_{k=-\infty}^{\infty} x[k] x[n + k] $$ Implementation 1import numpy as np 2 3# Cross-correlation 4corr = …

Transform signals between time and frequency domains. Overview The Fourier Transform decomposes a signal into its constituent frequencies, revealing the frequency content of time-domain signals. Continuous Fourier Transform (CFT) Forward Transform $$ X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt $$ Inverse …

Analyze LTI systems in the s-domain. Definition $$ X(s) = \mathcal{L}{x(t)} = \int_0^{\infty} x(t) e^{-st} dt $$ Where $s = \sigma + j\omega$ is a complex frequency. Common Transforms $x(t)$ $X(s)$ ROC $\delta(t)$ $1$ All $s$ $u(t)$ $\frac{1}{s}$ $\text{Re}(s) > 0$ $e^{-at}u(t)$ $\frac{1}{s+a}$ $\text{Re}(s) > …

Convert between continuous and discrete-time signals. Nyquist-Shannon Sampling Theorem A bandlimited signal with maximum frequency $f_{max}$ can be perfectly reconstructed if: $$ f_s \geq 2f_{max} $$ Reconstruction Ideal (Sinc Interpolation) $$ x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot \text{sinc}\left(\frac{t - …

Fundamental concepts for understanding and processing signals. Signal Types Continuous-Time vs Discrete-Time Continuous-Time Signal: $x(t)$, defined for all $t \in \mathbb{R}$ Discrete-Time Signal: $x[n]$, defined only at integer values $n \in \mathbb{Z}$ Analog vs Digital Analog: Continuous in both time and amplitude …

Fundamental concepts and mathematical tools for signal processing, including Fourier analysis, convolutions, and correlations.

Reduce spectral leakage in FFT analysis. Common Windows Rectangular (No Window) $$w[n] = 1$$ Hanning $$w[n] = 0.5 \left(1 - \cos\left(\frac{2\pi n}{N-1}\right)\right)$$ Hamming $$w[n] = 0.54 - 0.46\cos\left(\frac{2\pi n}{N-1}\right)$$ Blackman $$w[n] = 0.42 - 0.5\cos\left(\frac{2\pi n}{N-1}\right) + …