How to express y(t) as function of x(t) from a given two signal


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is there a simple way to answer ?
I couldn't do it by finding the equation of the two signal and then proceed like any problem in math

Best Answer

Looks like a quiz about transformation. We don't do homework problems here, but I'll try to explain how this problem works. All of the choices have

$$ y(t) = -x(-2t + toffset) + 1.5 $$

Where Toffset is 0, +2, -2, -1, or something else. So first look at the transformations that are common to all of the choices, and understand what each of them does. Try to visualize what the graph should look like.

Adding a constant offset should shift everything up:

$$ y(t) = x(t) + 1.5 $$

Multiplying the x(t) by -1 is a verical flip:

$$ y(t) = -x(t) $$

Multiplying the x argument by 2 makes time go faster:

$$ y(t) = x(2t) $$

Multiplying the x() argument by -2 makes time go faster, but backwards:

$$ y(t) = x(-2t) $$

And now just adding an offset to the x() argument shifts when time 0 starts.

$$ y(t) = x(t+toffset) $$

Note that the graph of x(t) has a convenient, distinct feature at time 0, so it helps if you can find where that feature is on the y(t) graph that you're trying to match.

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