Systems
=======

In these lecture notes we restrict ourselves to the most simple
systems: **SISO systems**. Systems with one input signal (Single
Input) and one output signal.

When the system is fed with an input signal $x(t)$ it outputs the
signal $y(t)$. For now a system is just a black box for us. We will
often use the **block diagram** representation of systems.

   .. tikz:: A one input, one output (SISO) system
      :xscale: 60
	       
      \bXInput{A}
      \bXBloc[8]{B}{System}{A}
      \bXLink[{Input $x(t)$}]{A}{B}
      \bXOutput[8]{C}{B}
      \bXLink[{Output $y(t)$}]{B}{C}



More complex systems have multiple inputs and multiple outputs: **MIMO
systems**.


      
We will often find the need to use several systems and combine them
into a new systems. 

Cascaded Systems
  Two systems connected in such a way that the output of one system is
  the input to the second system are cascaded or connected in series.

      .. tikz:: Cascaded (or serial) Systems 
	 :xscale: 60
		   
	 \bXInput{A}
	 \bXBloc[4]{B}{$S_1$}{A}
	 \bXLink[$x(t)$]{A}{B}
	 \bXBloc[4]{C}{$S_2$}{B}
	 \bXLink[$y(t)$]{B}{C}
	 \bXOutput[4]{D}{C}
	 \bXLink[$z(t)$]{C}{D}

Addition / Subtraction
   One signal $x(t)$ can be splitted to function as the input of two
   systems $S_1$ and $S_2$. The output of the two systems can then, for
   instance, be added together to result in one signal again.

   .. tikz:: **Adding signals.** $y(t) = x_1(t) + x_2(t) + x_3(t)$.
      :xscale: 35

      \draw (0,0) node[adder] (mix) {};
      \draw (-2,1) node[above right] {$x_1(t)$}  -- ++(1.5,0) coordinate(xone);
      \draw[->] (xone) to[short]   (mix);
      \draw (-2,0) node[above right] {$x_2(t)$} -- ++(1.5,0) coordinate(xtwo);
      \draw[->] (xtwo) --  (mix);
      \draw (-2,-1) node[above right] {$x_3(t)$} -- ++(1.5,0) coordinate(xthr);
      \draw[->] (xthr) -- (mix);
      %\draw[->] (mix) -- ++(2,0) node[above] {$y(t)$};
      \draw[->] (mix) to[short, l=$y(t)$] ++(2,0);
      
Feedback Loops
   When an output signal of a system is fed into the input of the
   system we talk of a **feedback loop**. Most often a negative
   feedback loop is used where the output signal $y(t)$ is subtracted
   from the input signal $x(t)$ and $x(t)-y(t)$ is fed into the
   system.

      .. tikz:: Feedback Loop
	 :xscale: 45
		  
	 \bXInput{A}
	 \bXComp[4]{B}{A}
	 \bXLink[$x(t)$]{A}{B}
	 \bXBloc[2]{C}{System}{B}
	 \bXLink{B}{C}
	 \bXOutput[4]{E}{C}
	 \bXLink[$y(t)$]{C}{E}
	 \bXReturn{C-E}{B}{}

   The feedback loop will often contain a second system. Feedback
   loops are very often used in signal processing. We will see
   examples when building systems for a special purpose and also when
   discussing classical control systems.

In a later section when we restrict ourselves to linear systems we
will look again at the block diagrams to construct composite systems.