\documentclass[10pt,a4paper]{article}

% Packages
\usepackage{fancyhdr}           % For header and footer
\usepackage{multicol}           % Allows multicols in tables
\usepackage{tabularx}           % Intelligent column widths
\usepackage{tabulary}           % Used in header and footer
\usepackage{hhline}             % Border under tables
\usepackage{graphicx}           % For images
\usepackage{xcolor}             % For hex colours
%\usepackage[utf8x]{inputenc}    % For unicode character support
\usepackage[T1]{fontenc}        % Without this we get weird character replacements
\usepackage{colortbl}           % For coloured tables
\usepackage{setspace}           % For line height
\usepackage{lastpage}           % Needed for total page number
\usepackage{seqsplit}           % Splits long words.
%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{bdorules}
\pdfinfo{
  /Title (signals.pdf)
  /Creator (Cheatography)
  /Author (bdorules)
  /Subject (signals Cheat Sheet)
}

% Lengths and widths
\addtolength{\textwidth}{6cm}
\addtolength{\textheight}{-1cm}
\addtolength{\hoffset}{-3cm}
\addtolength{\voffset}{-2cm}
\setlength{\tabcolsep}{0.2cm} % Space between columns
\setlength{\headsep}{-12pt} % Reduce space between header and content
\setlength{\headheight}{85pt} % If less, LaTeX automatically increases it
\renewcommand{\footrulewidth}{0pt} % Remove footer line
\renewcommand{\headrulewidth}{0pt} % Remove header line
\renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit
% This two commands together give roughly
% the right line height in the tables
\renewcommand{\arraystretch}{1.3}
\onehalfspacing

% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
\newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols
\newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns
\newcommand{\tn}{\tabularnewline} % Required as custom column type in use

% Font and Colours
\definecolor{HeadBackground}{HTML}{333333}
\definecolor{FootBackground}{HTML}{666666}
\definecolor{TextColor}{HTML}{333333}
\definecolor{DarkBackground}{HTML}{A3A3A3}
\definecolor{LightBackground}{HTML}{F3F3F3}
\renewcommand{\familydefault}{\sfdefault}
\color{TextColor}

% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{signals Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{bdorules} via \textcolor{DarkBackground}{\uline{cheatography.com/120488/cs/21920/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}bdorules \\
  \uline{cheatography.com/bdorules} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 3rd March, 2020.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.09848 cm} x{1.73926 cm} x{1.73926 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{properties of the DTFT}}  \tn
% Row 0
\SetRowColor{LightBackground}
Property & Aperidic signal & FT \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
 & x{[}n{]},y{[}n{]} & X(e\textasciicircum{}jw\textasciicircum{}),Y(e\textasciicircum{}jw\textasciicircum{}) peroid of 2PI \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{linearity} & ax{[}{\emph{n}}{]}+by{[}{\emph{n}}{]} & aX(e\textasciicircum{}jw\textasciicircum{})+bY(e\textasciicircum{}jw\textasciicircum{}) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
time sifthing & x{[}n-n0{]} & e\textasciicircum{}-jwn0\textasciicircum{}X(e\textasciicircum{}jw\textasciicircum{}) \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\seqsplit{frequncey} shift & e\textasciicircum{}jw0n\textasciicircum{}x{[}n{]} & X(e\textasciicircum{}j(w-wo)\textasciicircum{}) \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
\seqsplit{conjugation} & x*{[}n{]} & X*(e\textasciicircum{}-jw\textasciicircum{}) \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
time reversal & x{[}-n{]} & X(e\textasciicircum{}jw\textasciicircum{}) \tn 
% Row Count 14 (+ 2)
% Row 7
\SetRowColor{white}
time \seqsplit{expansion} & x(k){[}n{]}=\{x{[}n/k{]}, if n=multiple of k \{\{nl\}\}o, if n=/ multiple of k & X(e\textasciicircum{}jkw\textasciicircum{}) \tn 
% Row Count 19 (+ 5)
% Row 8
\SetRowColor{LightBackground}
\seqsplit{convolution} & x{[}n{]}*y{[}n{]} & X(e\textasciicircum{}jw\textasciicircum{})Y(e\textasciicircum{}jw\textasciicircum{}) \tn 
% Row Count 21 (+ 2)
% Row 9
\SetRowColor{white}
\seqsplit{Multiplication} & x{[}n{]}y{[}n{]} & 2pi\textbackslash{}1integral\textless{}2pi\textgreater{}X(e\textasciicircum{}jØ\textasciicircum{})Y(e\textasciicircum{}j(w-Ø)\textasciicircum{})dØ \tn 
% Row Count 24 (+ 3)
% Row 10
\SetRowColor{LightBackground}
\seqsplit{differencing} in time & x{[}n{]}-x{[}n-1{]} & (1-e\textasciicircum{}-jw\textasciicircum{})X(e\textasciicircum{}jw\textasciicircum{}) \tn 
% Row Count 27 (+ 3)
% Row 11
\SetRowColor{white}
\seqsplit{accumulation} & (k=-∞∑n) x{[}k{]} & (1-e\textasciicircum{}-jw\textasciicircum{})\textbackslash{}1X(e\textasciicircum{}jw\textasciicircum{})+piX(e\textasciicircum{}j0\textasciicircum{})(k=-∞∑∞) δ(w-2pik) \tn 
% Row Count 31 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{x{1.09848 cm} x{1.73926 cm} x{1.73926 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{properties of the DTFT (cont)}}  \tn
% Row 12
\SetRowColor{LightBackground}
\seqsplit{differenition} in frequncy & nx{[}n{]} & J(dX(e\textasciicircum{}jw\textasciicircum{}))/dw \tn 
% Row Count 3 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.41887 cm} x{2.19696 cm} x{0.96117 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{DTFT pairs}}  \tn
% Row 0
\SetRowColor{LightBackground}
singnal & fourier tran & \seqsplit{coefficients} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
k=\textless{}N\textgreater{}∑ake\textasciicircum{}jk(2n/N)n\textasciicircum{} & (2pi)k=-∞∑∞(akδ(w-N\textbackslash{}2pik))) & ak \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
e\textasciicircum{}jwon\textasciicircum{} & \seqsplit{(2pi)L=-∞∑∞(δ(w-wo-2piL)} & w0=2pim/N\{\{nl\}\} \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}