Cartesian Representation

Polar Representation

Complex Conjugate

Choosing a Representation

Useful Information

Polar Representation Ambiquity

The Imaginary Unit

Derivatives of Complex Exponentials

Converting Betweeen Representations

Complex Numbers

Exponentials as Vectors

Eulers Formula

Radial Frequency

Derivatives

The Complex Exponential

In Continuous Time, we use the \emph{Dirac Delta} function. (Technically not a function, and the height is actually infinite)

%\begin{figure}[htp]
%\centering
%\includegraphics{images/dirac/dirac.ps}
%\caption{ The Dirac Delta }
%\label{ fig:dirac }
%\end{figure}

\begin{center}
\begin{pspicture}(-3,-1)(3,1.2)
%  position
\rput(-2.571,-0.4){-3}
%  position
\rput(-1.714,-0.4){-2}
%  position
\rput(-0.857,-0.4){-1}
% line
\psline{->}(0.000,0)(0.000,1)
%  value
\rput(0.35,1){(1)}
%  position
\rput(0.000,-0.4){0}
%  position
\rput(0.857,-0.4){1}
%  position
\rput(1.714,-0.4){2}
%  position
\rput(2.571,-0.4){3}
% axes
\psline{->}(-3.428,0)(3.428,0)
\rput(3.428,-0.3){ $t$ }
	
\end{pspicture}
\end{center}

In Discrete Time, we use the \emph{Kronecker Delta} function.

%\begin{figure}[htp]
%\centering
%\includegraphics{images/kronecker.ps}
%\caption{ The Unit Impulse, a.k.a Kronecker Delta }
%\label{ fig:kronecker }
%\end{figure}

\begin{center}
\begin{pspicture}(-3,-1)(3,1.2)
% line
\psline{-*}(-2.571,0)(-2.571,0)
%  position
\rput(-2.571,-0.4){-3}
% line
\psline{-*}(-1.714,0)(-1.714,0)
%  position
\rput(-1.714,-0.4){-2}
% line
\psline{-*}(-0.857,0)(-0.857,0)
%  position
\rput(-0.857,-0.4){-1}
% line
\psline{-*}(0.000,0)(0.000,1)
%  value
\rput(0.25,1){(1)}
%  position
\rput(0.000,-0.4){0}
% line
\psline{-*}(0.857,0)(0.857,0)
%  position
\rput(0.857,-0.4){1}
% line
\psline{-*}(1.714,0)(1.714,0)
%  position
\rput(1.714,-0.4){2}
% line
\psline{-*}(2.571,0)(2.571,0)
%  position
\rput(2.571,-0.4){3}
% axes
\psline{->}(-3.428,0)(3.428,0)
\rput(3.428,-0.3){ $n$ }
\rput(2.928,1.857){ $\delta(n)$ }
	
\end{pspicture}
\end{center}

The Kronecker Delta function is used in the linear combination:
$$ \sum_{m \in M} x(m) \delta(n-m)  $$

Another signal that we use to decompose signals is the \emph{Complex Exponential}.

This signal is given by

$$ e^{i \theta t};  t \in \R$$ 
$$ e^{i \theta n};  n \in \Z$$ 


Signals to Decompose Signals

Week 2

2 Week 2