\[% to cope with definition mismatch between MathJax and LaTeX
\newcommand{\coloneq}{\mathrel{:=}}
\newcommand{\eqcolon}{\mathrel{=:}}
% general purpose
\newcommand{\parens}[1]{\left(#1\right)}
\newcommand{\bracks}[1]{\left[#1\right]}
\newcommand{\braces}[1]{\left\{#1\right\}}
\newcommand{\Verts}[1]{\left\|#1\right\|}
\newcommand{\naturalNumbers}{\mathbb{N}}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\realNumbers}{\mathbb{R}}
\newcommand{\complexNumbers}{\mathbb{C}}
\newcommand{\EuclideanSpace}{\mathbb{E}}
% number theory
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\GaloisField}[1]{\mathrm{GF}\parens{#1}}
% real analysis
\newcommand{\NapierE}{\mathrm{e}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\norm}[1]{\Verts{#1}}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\sinc}{\operatorname{sinc}}
% derivative
\newcommand{\deriv}[3]{\frac{\operatorname{d}^{#3}#1}{\operatorname{d}{#2}^{#3}}}
\newcommand{\derivLong}[3]{\frac{\operatorname{d}^{#3}}{\operatorname{d}{#2}^{#3}}{#1}}
\newcommand{\partDeriv}[3]{\frac{\operatorname{\partial}^{#3}#1}{\operatorname{\partial}{#2}^{#3}}}
\newcommand{\partDerivLong}[3]{\frac{\operatorname{\partial}^{#3}}{\operatorname{\partial}{#2}^{#3}}{#1}}
% integral
\newcommand{\integrate}[5]{\int_{#1}^{#2}{#3}{\mathrm{d}^{#4}}#5}
% complex analysis
\newcommand{\conj}[1]{\overline{#1}}
\renewcommand{\Re}[1]{{\operatorname{Re}{\parens{#1}}}}
\renewcommand{\Im}[1]{{\operatorname{Im}{\parens{#1}}}}
\newcommand{\Arg}{\operatorname{Arg}}
% discrete fourier transform
\newcommand{\DFT}[1]{\mathrm{DFT}\parens{#1}}
\newcommand{\IDFT}[1]{\mathrm{IDFT}\parens{#1}}
% Z-transform
\newcommand{\ZTrans}[1]{\operatorname{\mathcal{Z}}\parens{#1}}
\newcommand{\IZTrans}[1]{\operatorname{\mathcal{Z}}^{-1}\parens{#1}}
\newcommand{\SSZTrans}[1]{\underset{\text{s.s.}}{\mathcal{Z}}\parens{#1}} % single-sided Z-transform
% linear algebra
\newcommand{\bm}[1]{{\boldsymbol{#1}}}
\newcommand{\diag}[1]{\operatorname{diag}\parens{#1}}
\newcommand{\transpose}[1]{{#1}^\top}
\newcommand{\HerConj}[1]{{#1}^*}
\newcommand{\tr}[1]{\mathrm{tr}\parens{#1}}
% unit of measurement
\newcommand{\second}{\text{s}}
\newcommand{\hertz}{\text{Hz}}
\newcommand{\decibel}{\text{dB}}\]
Last active
July 19, 2026 10:10
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