Difference between revisions of "SystemPropertiesFall2018" - Rhea

Revision as of 19:44, 17 November 2018

System Properties

Memoryless Systems

- These are systems whose output signals only depend upon input signals at the specific time. These systems do not depend upon the past input signal, nor the future.

Systems which ARE memoryless are:

y(t) = x(t)

y[n] = 2x[n] - 1

Systems which ARE NOT memoryless are:

y(t) = x(t-1)

y[n] = 2x[2n]

Invertible Systems

- These are systems in which one distinct input signal produces one distinct output signal

Systems which ARE invertible are:

y(t) = 2x(t) + 3

y[n] = x[n]

Systems which ARE NOT invertible are:

y(t) = |x(t)|

y[n] = (x[n])²

These are not invertible because for each system, multiple inputs result in one output.

Causal Systems

- These are systems in which the output signal depends only on the input signal in past or present times.

Systems which ARE causal are:

y(t) = 2x(t) + 3

y[n] = x[n-1]

Systems which ARE NOT causal are:

y(t) = x(t+1)

y[n] = x[2|n|]

These are not causal because for each system, the output depends on the state of the input signal at a future time.

(BIBO) Stable Systems

- These are systems in which bounded inputs yield bounded outputs.

Systems which ARE stable are:

y(t) = sin(x(t))

Systems which ARE NOT stable are:

y[n] = n*(x[n|)
^{This system is not stable because as n tends to infinity, the output has no bound.}

== Linear Systems == - These are systems in which inputs x₁ and x₂, when passed through a system result in corresponding outputs y₁ and y₂, and the system's response to ax₁ + bx₂ are the outputs ay₁ + by₂ for any constants a and b.

Systems which ARE linear are:

y[n] = 2x[n]

Systems which ARE NOT linear are:

y[n] = (x[n])²

This system is not linear because if ax₁[n] + bx₂[n] were passed through the system, the output would be (ay₁[n] + by₂[n])² rather than ay₁[n]² + by₁[n]²

Time-Invariant Systems