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I am a little confused about the properties (is it Linear, Causal, Time-Invariant, Stable?) of this T system.

$$T[x(t)]=\sum_{k=t_0}^{t}{x(k)}$$

Some are obvious (it is linear), but can’t come up with the rest of them… any tips on this would be appreciated.

Marconius
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Angel Rmz
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1 Answers1

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The system can't be time-invariant, because the interval over which the input is accumulated depends on the time $t$.

The formula implies that the system's output is zero if $t<t_0$. If this is the case then the system is causal, because no future values of the input signal are necessary for computing the output signal.

As for stability, the system is definitely BIBO stable because a bounded input will always produce a bounded output.

Matt L.
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  • Sounds logical. Thank you for the great tip. – Angel Rmz Oct 27 '15 at 21:46
  • How about if I change it for a transcendental, say "e", for an equation like this... http://3.bp.blogspot.com/-zYfPS_sAHOI/Vi_yQxyObvI/AAAAAAAAAvM/oTS4FmLjDq0/s1600/Information2%2Bjpg.jpg – Angel Rmz Oct 27 '15 at 21:56
  • @AngelRmz: That's an entirely different system; you should try to understand the basic system properties and how to test them, otherwise you will need to ask a new question for each new system you come up with. – Matt L. Oct 27 '15 at 22:43
  • I will. Thanks!! – Angel Rmz Nov 05 '15 at 01:47