Signal Transformations
======================

$\newcommand{\setR}{\mathbb R}$
$\newcommand{\setC}{\mathbb C}$

We discuss the basis signal transformations. You have learned these in
highschool. We start with CT signals.


Vertical Translation
   Let $x$ be a signal, then translating it upwards over vertical
   distance $h>0$ gives the signal $y$:

   .. list-table:: **Vertical Translation**
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	- DT
      * - :math:`y(t)=x(t)+h`
	- :math:`y[n]=x[n]+h`


Vertical Scaling
   Let $x$ be a signal, then vertical scaling with factor $a$ gives
   the signal $y$:`

   .. list-table:: **Vertical Scaling**
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	- DT
      * - :math:`y(t)=a\,x(t)`
	- :math:`y[n]=a\,x[n]`


Horizontal Translation
   Translating (shifting) signal $x$ to the right gives the signal
   $y$:

   .. list-table:: **Horizontal Translation**
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	- DT
      * - :math:`y(t)=x(t-u)`
	- :math:`y[n]=x[n-m]`

   In CT the translation is over $u\in\setR$ whereas in DT the
   translation is over $m\in\setZ$.


Horizontal Scaling
   Horizontal scaling of signal CT signal with factor $b$ is easily
   defined. For a DT signal a generic definition is not feasible.

   .. list-table:: **Horizontal Scaling**
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	- DT
      * - :math:`y(t)=x(\frac{t}{b})`
	- no unique definition