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: $x[n]$: …
Read MoreMeasure 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 = …
Read MoreTransform 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 …
Read MoreAnalyze 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 $s$ $u(t)$ $\frac{1}{s}$ $\text{Re}(s) > 0$ $e^{-at}u(t)$ $\frac{1}{s+a}$ $\text{Re}(s) > -a$ …
Read MoreConvert 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) $$ x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot \text{sinc}\left(\frac{t - …
Read MoreFundamental concepts for understanding and processing signals. Signal Types Continuous-Time vs Discrete-Time Continuous-Time Signal: $x(t)$, defined for all $t \in \mathbb{R}$ Discrete-Time Signal: $x[n]$, defined only at integer values $n \in \mathbb{Z}$ Analog vs Digital Analog: Continuous in both time and amplitude …
Read MoreFundamental concepts and mathematical tools for signal processing, including Fourier analysis, convolutions, and correlations.
Read MoreReduce spectral leakage in FFT analysis. Common Windows Rectangular (No Window)
$$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) + …
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