How do I determine the linear/non-linearity of this problem?

In summary: y(t_n)## where each `t_i` is a given point in time, and (2) that the transformation is a linear function.
  • #1
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Homework Statement
Determine whether the system is (a) linear, (b) time-invariant:
Relevant Equations
y[n] = T{x[n]} = x(t)
y(t) = cos(3t)x(t)
A system is linear if it satisfies the properties of superposition and homogeneity.

Superposition: adding the inputs of two systems results in the addition of the two outputs.
Ex) x1(t) + x2(t) = y1(t) + y2(t)

Homogeneity: multiplying the input by some scalar value is equal to the output multiplied by that same value.
Ex) ax1(t) = ay2(t)

After doing some internet searching, I also found this equation for superposition: f(a+b) = f(a) + f(b)

However, I don't know how to apply these rules to determine whether my equations are linear or not.
My attempt for determining SP:
y1[n] + y2[n] = T{x1[n] + x2[n]} = n(x1[n] + x2[n])

If my attempt at SP is correct, what is the next step? If it is wrong, what is the right way to write it, and then what is the next step?
 
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  • #2
What is the operator ##T\left\{\cdot \right\}## ? ##n## is natural ? ##y[n]## is discrete and ##y(t)## is continuous ? Can you define better this system before to start to prove the two conditions ?
Ssnow
 
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  • #3
Ssnow said:
What is the operator ##T\left\{\cdot \right\}## ? ##n## is natural ? ##y[n]## is discrete and ##y(t)## is continuous ? Can you define better this system before to start to prove the two conditions ?
Ssnow
Thank you for your reply Ssnow,

The second equation is a separate problem; apologies for being unclear. We can ignore that.

So for now we are just dealing with the discrete problem. Unfortunately, I am unsure what the T{} operator is for. I would guess that it is an arbitrary value representing a function of a function; i.e. x[n] is a function of T. Again guessing, I think n is some arbitrary value; it could be a, or b, or whatever else.
 
  • #4
Lapse said:
The second equation is a separate problem; apologies for being unclear. We can ignore that.
So this is the one you are concerned with?
y[n] = T{x[n]} = x(t)

I understand the first part, y[n] = T{x[n]}, but not why this would be equal to x(t).

Here T is some transformation that is applied to x[n] to produce y[n].
Lapse said:
So for now we are just dealing with the discrete problem. Unfortunately, I am unsure what the T{} operator is for. I would guess that it is an arbitrary value representing a function of a function; i.e. x[n] is a function of T.
No, you have this backwards. y[n] is a function of x[n] via the transformation T. T is a transformation that operates on a sequence or function x[n]. It is not a value.
Lapse said:
Again guessing, I think n is some arbitrary value; it could be a, or b, or whatever else.
Finally, if you are not given a formula for T, there is no way you can tell whether it is linear or time-invariant. Are you sure that the second equation, y(t) = cos(3t)x(t), isn't part of this problem?

For a transformation T, Tis linear if T(a + b) = T(a) + T(b) and if T(ka) = kT(a), where a and b are in the domain of T, and k is a scalar constant.
 
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  • #5
Mark44 said:
So this is the one you are concerned with?
y[n] = T{x[n]} = x(t)

I understand the first part, y[n] = T{x[n]}, but not why this would be equal to x(t).

Here T is some transformation that is applied to x[n] to produce y[n].
No, you have this backwards. y[n] is a function of x[n] via the transformation T. T is a transformation that operates on a sequence or function x[n]. It is not a value.

Finally, if you are not given a formula for T, there is no way you can tell whether it is linear or time-invariant. Are you sure that the second equation, y(t) = cos(3t)x(t), isn't part of this problem?

For a transformation T, Tis linear if T(a + b) = T(a) + T(b) and if T(ka) = kT(a), where a and b are in the domain of T, and k is a scalar constant.
Thank you for your reply Mark,

I am positive that the second equation is for a different problem because it is delineated as (2); instead of being a part of problem 1. However, maybe we should switch over to the 2nd problem if we can't discover the answer to the first one.

So, do I start by creating y1 & y2 and add them together?

y1(t) + y2(t) = cos(3t)[x1(t) + x2(t)]

What do I do next?
 
  • #6
Lapse said:
Thank you for your reply Mark,

I am positive that the second equation is for a different problem because it is delineated as (2); instead of being a part of problem 1. However, maybe we should switch over to the 2nd problem if we can't discover the answer to the first one.

So, do I start by creating y1 & y2 and add them together?

y1(t) + y2(t) = cos(3t)[x1(t) + x2(t)]

What do I do next?
You're still going at this backwards.
From post #1: y(t) = cos(3t)x(t)
For linearity, you need to show (1) that ##y(t_1 + t_2) = y(t_1) + y(t_2)## and (2) that ##y(kt_1) = ky(t_1)##.
So for (1), ##y(t_1 + t_2) = \cos(3(t_1 + t_2))x(t_1 + t_2)##. Does that work out to ##y(t_1) + y(t_2)##?

It would be helpful if you showed us the exact wording of the problem. If it's a problem in a textbook, upload a photo of the problem. As presented in this thread, there seems to be a lot of missing information.
 

Related to How do I determine the linear/non-linearity of this problem?

1. How do I determine if a problem is linear or non-linear?

The linearity of a problem can be determined by examining the relationship between the variables in the problem. If the relationship follows a straight line, it is considered linear. If the relationship follows a curve, it is considered non-linear.

2. What is the difference between a linear and non-linear problem?

A linear problem has a constant rate of change between the variables, while a non-linear problem has a changing rate of change. In other words, the variables in a linear problem are directly proportional, while in a non-linear problem, they are not.

3. How do I graphically determine the linearity of a problem?

To graphically determine the linearity of a problem, plot the given data points on a graph and observe the pattern. If the points form a straight line, the problem is linear. If the points form a curve, the problem is non-linear.

4. Can a problem be both linear and non-linear?

No, a problem can only be either linear or non-linear. If a problem contains both linear and non-linear relationships, it is considered to be a multi-linear problem.

5. What are some real-life examples of linear and non-linear problems?

Linear problems can be found in scenarios such as calculating distance traveled at a constant speed, while non-linear problems can be found in scenarios such as compound interest calculations or population growth. Other examples of linear and non-linear problems can be found in economics, physics, and engineering.

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