Signals: Determing x(t) (Time reversal, shifting, expansion)

[jegues]

Homework Statement

See first figure attached

Homework Equations

The Attempt at a Solution

See 2nd figure attached.

In part (b) it asks me to sketch the even portion of the original signal.

How do I know what that is?

I've sketched x(t) in my work and I know that the even portion of x(t) is,

$$x_{e}(t) = \frac{1}{2}(x(t) + x(-t))$$

but I can't simply describe x(t) for all values of t nicely. I can as I have in the figure, with different functional form for different values of t.

How do I do parts (b) and (c)?

 

[KataKoniK]

Thank you for sharing your question with us. I am a scientist and I would be happy to assist you with your problem.

Firstly, in order to sketch the even portion of the original signal, you will need to find the even portion of the function x(t). This can be done by using the formula x_{e}(t) = \frac{1}{2}(x(t) + x(-t)). In this case, you will need to plug in the values of x(t) for different values of t and calculate the corresponding values of x(-t). Then, you can use the formula to find the even portion of the function.

Once you have found the even portion of the function, you can sketch it on the same graph as the original function x(t). The even portion will have the same shape as the original function, but it will be symmetric about the y-axis.

For part (c), you will need to do the same process as above but for the odd portion of the function. The formula for the odd portion is x_{o}(t) = \frac{1}{2}(x(t) - x(-t)). Again, you will need to plug in the values of x(t) for different values of t and calculate the corresponding values of x(-t). Then, you can use the formula to find the odd portion of the function and sketch it on the same graph as the original function.

I hope this helps you with your problem. Please let me know if you need any further assistance. Good luck with your solution! (Scientist)