Time Scaling, Shifting, and Reversal: Fourier Tranfiorm

[Hip2dagame]

Homework Statement

Let's say I've got the typical triangular waveform with function x(t) = t, goes from 0 to 1 on the x and y axes. How would I manipulate x(t) and the given X(ω) to, say,

1)Stretch the function on the x-axis from 0 to 2, but keep the slope as 1?
2)Flip the function upside down, then shift it up, to get a square?

Homework Equations

We're given X(ω) = [e^(-jω)+jωe^(-jω)-1]/ ω^2 (standard FT of triangular wave function)

Scaling: x(at) = 1/a * X(ω/a)

Shifting: x(t-t0) = X(ω)e^(-jωt0)

The Attempt at a Solution

For 1, wouldn't it just be x(t/2), and if i wanted to shift it up, let's say Z units, x((t/2 + Z)? But what would the X(ω) look like?

For 2, if i just wanted a square I could just do 2X(ω), right?
Thanks.

 

[KataKoniK]

To stretch the function on the x-axis from 0 to 2, while keeping the slope as 1, you can use the scaling equation you provided. In this case, you would use x(2t) = 1/2 * X(ω/2). This will stretch the function on the x-axis from 0 to 2, while keeping the slope as 1. To shift the function up, you can use the shifting equation you provided. In this case, you would use x(2t + Z) = X(ω/2)e^(-jωZ). This will shift the function up by Z units.

For flipping the function upside down and then shifting it up to get a square, you are correct in using 2X(ω). This will flip the function upside down, but it will not shift it up. To shift it up, you can use the shifting equation again. In this case, you would use 2X(ω)e^(-jωZ). This will shift the function up by Z units. Keep in mind that the value of Z will determine the height of the square.

I hope this helps. Let me know if you have any other questions.