Linearity, time invariance, causality

In summary, for each of the given systems, it was determined that they are all causal. To determine linearity, the input x[n] is scaled by a constant A and if the output y[n] is also scaled by the same constant, the system is linear. For time invariance, the output should be shifted by the same time T if the input is shifted by the same time T. Based on these tests, it was concluded that the first system is linear but not time-invariant, while the second and third systems are not linear or time-invariant. The fourth system is not specifically mentioned, but it is noted that the linear scaling test must work for all values of A and the time sifting must work for all
  • #1
Quincy
228
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Homework Statement


For each of the following systems, determine whether or not the system is linear, time-invariant, and causal.

a) y[n] = x[n]cos(0.2*PI*n)
b) y[n] = x[n] - x[n-1]
c) y[n] = |x[n]|
d) y[n] = Ax[n] + B, where A & B are constants.

Homework Equations





The Attempt at a Solution



I know that they're all causal, because they all depend on present or past values of n, I don't know how to determine if they're linear and time-invariant, the book is terrible at explaining it.
 
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  • #2
L/TI are really important concepts for signals and systems - and they're pretty easy to get too.

To test for linearity: does scaling the input x[n] by a constant A scale the output by the same constant. in equation form:

so for the first one:

Code:
y[n] = x[n]cos(0.2*PI*n)

Scaling the input by A:

= Ax[n]cos(0.2*PI*n)
= Ay[n]

so this system is linear.

To satisfy time invariance, the output of the system should be shifted by a time T if the input is shifted by the same time T.

Code:
y[n] = x[n]cos(0.2*PI*n)

Sift the input signal by a time T

= x[n + T]cos(0.2*PI*n)
!= y[n+T]
because of the cosine's dependence on n this system is not time invariant.

Hope this helps, applying these rules to the systems are fairly simple and will tell you if a system is Linear/TI. Keep in mind that the linear scaling test must work for all (real and complex) values of A and the time sifting must work for all T in order for the system to L/TI.
 

Related to Linearity, time invariance, causality

1. What is linearity and why is it important in scientific research?

Linearity refers to the property of a system where the output of the system is directly proportional to the input. In other words, if the input is doubled, the output will also be doubled. This is important in scientific research because it allows for easier analysis and prediction of the system's behavior.

2. How can we determine if a system is time-invariant?

A system is considered time-invariant if its output remains the same regardless of when the input is applied. This can be determined by testing the system at different points in time and comparing the outputs. If the outputs are the same, the system is time-invariant.

3. What is causality and how does it relate to linearity and time-invariance?

Causality refers to the relationship between cause and effect in a system. In a linear and time-invariant system, the output is directly caused by the input, making it easier to understand and predict the behavior of the system.

4. What are some examples of systems that exhibit linearity, time-invariance, and causality?

Some examples of linear, time-invariant, and causal systems include simple pendulums, electric circuits, and linear motion systems. In these systems, the output is directly proportional to the input, remains the same regardless of when the input is applied, and the output is caused by the input.

5. How can we use the concepts of linearity, time-invariance, and causality in real-world applications?

These concepts are essential in various fields such as engineering, physics, and economics. They allow for the analysis and prediction of systems, which is crucial in designing and improving technologies and processes. For example, understanding the linearity and time-invariance of a circuit can help engineers design more efficient electrical systems.

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