How do I know if this discrete-time system is stable?

In summary, to prove if a system is stable or unstable, the input must be bounded in order for the output to be bounded. In this case, the input term is a unit step function multiplied by a decaying exponential, making it a bounded input and therefore a stable system.
  • #1
interxavier
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Homework Statement


Given the following system:

[tex]s[n] = (\frac{1}{4})^{n}u[n+10]u[n] [/tex]

Prove if this is a stable or unstable system.


Homework Equations





The Attempt at a Solution



I know that in order for a system to be stable, the input must be bounded in order for the output to be bounded. For example,

[tex] x[n] \leq B_{x} [/tex]
[tex]y[n] \leq B_{y} [/tex]
[tex]So, y[n] = \sum_{k = 0}^{\infty}x[n]h[n][/tex]
[tex]y[n] = \sum_{k = 0}^{\infty}B_{x}h[n] = B_{y}[/tex]
 
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  • #2
So, since the input term is a unit step function multiplied by a decaying exponential, this should be a bounded input, and thus a stable system.
 

FAQ: How do I know if this discrete-time system is stable?

What is a discrete-time system?

A discrete-time system is a mathematical model that represents a physical or abstract system where the input and output are discrete quantities, such as numbers or signals, and are processed in discrete time steps. This means that the system's input and output values are only observed and calculated at specific time intervals.

How is stability defined for a discrete-time system?

A discrete-time system is considered stable if its output remains bounded for any bounded input. In other words, if the input to the system does not cause the output to grow infinitely, the system is stable. This is often represented mathematically as a bounded input-bounded output (BIBO) stability condition.

What are the different types of stability for a discrete-time system?

There are three types of stability for a discrete-time system:
1. BIBO stability: The system's output remains bounded for any bounded input.
2. Absolute stability: The system's output approaches zero as time goes to infinity for a bounded input.
3. Asymptotic stability: The system's output approaches a finite value as time goes to infinity for a bounded input.

How do I determine if a discrete-time system is stable?

One way to determine the stability of a discrete-time system is to analyze its impulse response or transfer function. If the impulse response or transfer function is bounded, then the system is stable. Another approach is to check the eigenvalues of the system's state transition matrix. If all eigenvalues have a magnitude less than one, the system is stable.

What are the implications of a stable or unstable discrete-time system?

The stability of a discrete-time system has important implications for its performance and behavior. A stable system will produce a predictable output for a given input, while an unstable system may produce unpredictable or oscillating output. Additionally, a stable system is easier to analyze and control, making it more desirable for practical applications.

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