Signal Properties
=================



Even Functions
    A function $x$ is called **even** in case


   .. list-table:: 
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	
	- DT
	  
      * - $\forall t: x(-t) = x(t)$
	
	- $\forall n: x[-n] = x[n]$


Odd Functions
    A function $x(t)$ is called **odd** in case

    .. list-table:: 
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	
	- DT
	  
      * - $\forall t: x(-t) = -x(t)$

	- $\forall n: x[-n] = -x[n]$


In some cases it is useful to write an arbitrary signal $x(t)$ as the
sum of an even and an odd signal. The following formula's achieve that
goal.

.. math::
   x_e(t) &= \frac{x(t)+x(-t)}{2}\\
   x_o(t) &= \frac{x(t)-x(-t)}{2}\\
   x(t) &= x_e(t) + x_o(t)


       
Periodic Function
    A function $x(t)$ is called **periodic with period** $T$ in case

    .. list-table:: 
      :header-rows: 1
      :class: tablefullwidth

      * - CT
	
	- DT
	  
      * - $\forall t: x(t+T) = x(t)$

	- $\forall n: x[n+T] = x[n]$
       
    Note that for a DT signal the period has to be an integer number.

Observe that in case $x(t)$ is a periodic CT signal, its sampled
discrete version $x[n]=x(n\Delta t)$ need not be periodic.