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=== System Properties === | === System Properties === | ||
| − | == Memoryless Systems == | + | == <span style="color:red"> Memoryless </span> 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. | + | - 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. <br /> |
| − | Systems which '''ARE''' memoryless are: | + | Systems which '''ARE''' memoryless are: <br /> |
| − | + | ||
| − | + | ||
| − | + | y(t) = x(t)<br /> | |
| − | y(t) = x(t | + | |
| − | + | ||
| − | = | + | y[n] = 2x[n] - 1<br /> |
| − | + | Systems which '''ARE NOT''' memoryless are: <br /> | |
| − | = | + | y(t) = x(t-1)<br /> |
| − | = | + | y[n] = 2x[2n]<br /> |
| − | == Time-Invariant Systems == | + | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| + | == <span style="color:blue"> Invertible </span> Systems == | ||
| + | - These are systems in which one distinct input signal produces one distinct output signal. <br /> | ||
| + | |||
| + | Systems which '''ARE''' invertible are: <br /> | ||
| + | |||
| + | y(t) = 2x(t) + 3 <br /> | ||
| + | |||
| + | y[n] = x[n] <br /> | ||
| + | |||
| + | Systems which '''ARE NOT''' invertible are: <br /> | ||
| + | |||
| + | y(t) = |x(t)| <br /> | ||
| + | |||
| + | y[n] = (x[n])<sup>2</sup> <br /> | ||
| + | |||
| + | These are not invertible because for each system, multiple inputs result in one output. <br /> | ||
| + | |||
| + | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ||
| + | == <span style="color:green"> Causal </span> Systems == | ||
| + | - These are systems in which the output signal depends only on the input signal in past or present times. <br /> | ||
| + | |||
| + | Systems which '''ARE''' causal are: <br /> | ||
| + | |||
| + | y(t) = 2x(t) + 3 <br /> | ||
| + | |||
| + | y[n] = x[n-1] <br /> | ||
| + | |||
| + | Systems which '''ARE NOT''' causal are: <br /> | ||
| + | |||
| + | y(t) = x(t+1) <br /> | ||
| + | |||
| + | y[n] = x[2|n|] <br /> | ||
| + | |||
| + | These are not causal because for each system, the output depends on the state of the input signal at a future time. <br /> | ||
| + | |||
| + | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ||
| + | == <span style="color:orange"> (BIBO) Stable </span> Systems == | ||
| + | - These are systems in which bounded inputs yield bounded outputs. <br /> | ||
| + | |||
| + | Systems which '''ARE''' stable are: <br /> | ||
| + | |||
| + | y(t) = sin(x(t)) <br /> | ||
| + | |||
| + | Systems which '''ARE NOT''' stable are: <br /> | ||
| + | |||
| + | y[n] = n*(x[n|) <br /> | ||
| + | |||
| + | This system is not stable because as n tends to infinity, the output has no bound. <br /> | ||
| + | |||
| + | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ||
| + | == <span style="color:purple"> Linear </span> Systems == | ||
| + | - These are systems in which inputs x<sub>1</sub> and x<sub>2</sub>, when passed through a system result in corresponding outputs y<sub>1</sub> and y<sub>2</sub>, and the system's response to ax<sub>1</sub> + bx<sub>2</sub> are outputs ay<sub>1</sub> + by<sub>2</sub> for any constants a and b.<br /> | ||
| + | |||
| + | Systems which '''ARE''' linear are: <br /> | ||
| + | |||
| + | y[n] = 2x[n] <br /> | ||
| + | |||
| + | Systems which '''ARE NOT''' linear are: <br /> | ||
| + | |||
| + | y[n] = (x[n])<sup>2</sup> <br /> | ||
| + | |||
| + | This system is not linear because if ax<sub>1</sub>[n] + bx<sub>2</sub>[n] were passed through the system, the output would be (ay<sub>1</sub>[n] + by<sub>2</sub>[n])<sup>2</sup> rather than ay<sub>1</sub>[n]<sup>2</sup> + by<sub>1</sub>[n]<sup>2</sup><br /> | ||
| + | |||
| + | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ||
| + | == <span style="color:yellow"> Time-Invariant </span> Systems == | ||
| + | - These are systems in which the shifted input signal results in the corresponding output of the system shifted.<br /> | ||
| + | |||
| + | Systems which '''ARE''' time-invariant are: <br /> | ||
| + | |||
| + | y(t) = 2x(t) <br /> | ||
| + | y[n] = x[n] - 4 <br /> | ||
| + | |||
| + | Systems which '''ARE NOT''' time-invariant are: <br /> | ||
| + | |||
| + | y(t) = x(2t) <br /> | ||
| + | y[n] = x[n<sup>2</sup>] <br /> | ||
| + | |||
| + | These systems are not time-invariant because depending on whether the input is shifted first or passed through the system first, different output signals are produced. <br /> | ||
Latest revision as of 20:12, 17 November 2018
Contents
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 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
- These are systems in which the shifted input signal results in the corresponding output of the system shifted.
Systems which ARE time-invariant are:
y(t) = 2x(t)
y[n] = x[n] - 4
Systems which ARE NOT time-invariant are:
y(t) = x(2t)
y[n] = x[n2]
These systems are not time-invariant because depending on whether the input is shifted first or passed through the system first, different output signals are produced.
