1.2.2. Signal Properties

Even Functions

A function \(x\) is called even in case

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

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.

\[\begin{split}x_e(t) &= \frac{x(t)+x(-t)}{2}\\ x_o(t) &= \frac{x(t)-x(-t)}{2}\\ x(t) &= x_e(t) + x_o(t)\end{split}\]

Periodic Function

A function \(x(t)\) is called periodic with period \(T\) in case

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.