1.2.2. 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 \(y\):

Vertical Translation

CT	DT
\(y(t)=x(t)+h\)	\(y[n]=x[n]+h\)

Vertical Scaling

Let \(x\) be a signal, then vertical scaling with factor \(a\) gives the signal \(y\):`

Vertical Scaling

CT	DT
\(y(t)=a\,x(t)\)	\(y[n]=a\,x[n]\)

Horizontal Translation

Translating (shifting) signal \(x\) to the right gives the signal \(y\):

Horizontal Translation

CT	DT
\(y(t)=x(t-u)\)	\(y[n]=x[n-m]\)

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.

Horizontal Scaling

CT	DT
\(y(t)=x(\frac{t}{b})\)	no unique definition