Discrete time signal

A time-discrete signal , sometimes just referred to as a discrete signal or discontinuous signal , is a special form of a signal that is only defined at certain, usually equidistant points in time. It is obtained from a continuous-time signal by sampling , a signal value being taken from the continuous-time signal curve at certain points in time. On the one hand, each signal value can be value-continuous and its resolution can be as precise as desired. On the other hand, a time-discrete signal can be converted into a digital signal by an additional quantization of the individual signal values, that means a reduction of the value stock to a certain, finite number of levels .

Discrete-time signals play an important role in signal theory and information theory for system description and as a preliminary stage for digital signal processing .

General

The continuous signal curve in gray; the vertical red lines represent the time-discrete signal formed from them.

A continuous-value discrete-time signal can mathematically as a sequence x [ n ] of real numbers with are described. The index n represents the time variable standardized to the sampling rate - sampling usually takes place at constant time intervals T _s . The reciprocal value is referred to as the sampling rate or the sampling frequency f _s . The values of the time-discrete signal between two sampling times and are not zero, but not defined . ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle p}$ ${\ displaystyle p + 1}$

In this case, the Nyquist-Shannon sampling theorem describes the effect that the sequence x [n] then contains the complete information of the continuous signal curve if its highest frequency components f _{a are} less than half the sampling frequency f _s :

{\ displaystyle f_ {a} <{\ frac {f_ {s}} {2}}}

A continuous signal can, as an example and shown in the right figure, by the function

{\ displaystyle 2 ^ {- x}}

to be discribed. The time-discrete signal derived from this is marked with red vertical lines and can be expressed as:

{\ displaystyle x [n] = {\ begin {cases} 0, & {\ text {for}} (x <0) \\ 2 ^ {- n}, & {\ text {with}} n = {\ frac {1} {2}} p {\ text {; for}} p = 0,1,2, \ ldots, N \ end {cases}}}

literature

Karl-Dirk Kammeyer, Kristian Kroschel: Digital signal processing . 6th edition. Teubner, Stuttgart / Leipzig / Wiesbaden 2006, ISBN 978-3-8351-0072-5 .

Individual evidence

↑ Rolf Unbehauen : Systems Theory 1 . Oldenbourg Verlag, Munich Vienna 2002, ISBN 978-3-486-25999-5 .

[1] Rolf Unbehauen : Systems Theory 1 . Oldenbourg Verlag, Munich Vienna 2002, ISBN 978-3-486-25999-5 .