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I was going over some review problems and came across an interesting one.

Using the techniques of (linearity, time shifting, and time scaling) what are some approaches I could use to turn the fourier transform from the left graph to the right?

enter image description here

X(ω) = 4* (sin w)^2 / w^2

My understanding so far:

  • Using the time shift property for fourier transforms I know the we can use
    x(t -1) to move the graph to start at t=1
  • We can use x(t/3) to extend or "stretch" the base to the to t=7

What is eluding me is how to form that "middle" or clipped top of the z(t) graph. Any insight?

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  • $\begingroup$ You to want to use linearity, time-shifting, and time-scaling to get $z(t)$ from $x(t)$ right? What's the link with the Fourier transform here? $\endgroup$ – Gilles Mar 5 '18 at 14:41
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HINT:

Note that $x(t)+x(t-1)$ has the required shape. Now it's only a matter of shifting, time scaling, and amplitude scaling to arrive at the exact expression for $z(t)$.

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It would appear that:

$$ z(t) = 3 \cdot x( (t - 4) / 3 ) - x ( t - 4 ) $$

You should be able to work it from there.

Hope this helps,

Ced

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  • $\begingroup$ Could you walk through the rationale for each augmentation? $\endgroup$ – crazyCoder Mar 4 '18 at 23:43
  • $\begingroup$ @crazyCoder, It's pretty simple. I observed that the figure on the right was the triangle on the left blown up, with one triangle removed from the top. From there is was just a matter of figuring out the values to slide and scale. So, just look at my equation as a big triangle minus a little triangle. $\endgroup$ – Cedron Dawg Mar 5 '18 at 0:06

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