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

• Comp Sci
• Lapse
In summary: y(t_n)## where each t_i is a given point in time, and (2) that the transformation is a linear function.
Lapse
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?

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

Lapse and BvU
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

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.

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.

Last edited:
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.

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?

Lapse said:

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.

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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