Systems Classification - Signals and Systems

What is systems Classification?

Systems classification is done in to below categories:
  • Liner and Non-liner Systems
  • Time Variant and Time Invariant Systems
  • Liner Time variant and Liner Time invariant systems
  • Static and Dynamic Systems
  • Causal and Non-causal Systems
  • Invertible and Non-Invertible Systems
  • Stable and Unstable Systems

Liner and Non-liner Systems

A system is described as linear as it satisfies superposition and homogenate principles. Let’s take below systems with inputs as x1(t), x2(t), and outputs as y1(t), y2(t) respectively. Go as per the superposition and homogenate principles,

liner

You can find that response of overall system is equal to response of individual system.

Example:

liner

Which is not equal toa1y1(t) + a2y2(t). Hence the system is said to be non linear.

Time Variant and Time Invariant Systems

Here the system is called as time variant if its input and output characteristics change along with time, other than the system is called as time invariant.
The condition for time invariant system is:
y (n , t) = y(n-t)
The condition for time variant system is:
y (n , t) ≠≠ y(n-t)
Where y (n , t) = T[x(n-t)] = input change
y (n-t) = output change

Example:

time

Liner Time variant (LTV) and Liner Time Invariant (LTI) Systems

Incase if a system is both liner and time variant, then it is called liner time variant (LTV) system.
Incase if a system is both liner and time Invariant then the system is considered as liner time invariant (LTI) system.

Static and Dynamic Systems

Static system is known as memory-less whereas dynamic system is considered as memory system.
Example 1: y(t) = 2 x(t)
For present value t=0, the system output is y(0) = 2x(0). Here, the output is mainly depends on present input. Then the system is memory less or static.
Example 2: y(t) = 2 x(t) + 3 x(t-3)
For present value t=0, the system output is y(0) = 2x(0) + 3x(-3).
Here x(-3) is past value for the present input for which the system requires memory to get this output. Hence, the system is a dynamic system.

Causal and Non-Causal Systems

A system is said to be causal if its output depends upon present and past inputs, and does not depend upon future input.
For non causal system, the output depends upon future inputs also.
Example 1 :y(n) = 2 x(t) + 3 x(t-3)
For present value t=1, the system output is y(1) = 2x(1) + 3x(-2).
Here, the system output only depends upon present and past inputs. Hence, the system is causal.
Example 2: y(n) = 2 x(t) + 3 x(t-3) + 6x(t + 3)
For present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4) Here, the system output depends upon future input. Hence the system is non-causal system.

Invertible and Non-Invertible systems

A system is said to invertible if the input of the system appears at the output.
invertible_system

stable

Hence, the system is invertible.
invertthen the system is said to be non-invertible.

Stable and Unstable Systems

The system is also called as stable when the output is bounded for bounded input. In case the bounded input, if the output is unbounded in the system then it is said to be unstable.
Note: For a bounded signal, amplitude is finite.
Example 1:y (t) = x2(t)
Let the input is u(t) (unit step bounded input) then the output y(t) = u2(t) = u(t) = bounded output.
Hence, the system is stable.
Example 2: y (t) = ∫x(t)dt
Let the input is u (t) (unit step bounded input) then the output cc ramp signal (unbounded because amplitude of ramp is not finite it goes to infinite when t →→ infinite).
Hence, the system is unstable.

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