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I've found this diagram in a slide, but i have no backround to properly understand it. Could you explain to me, in simple terms, how this process behaves and what's the purpose of the blocks $z^{-1}$ ?

enter image description here

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That's a simple process, where $y(k)=u(k-3)$, as the $z^{-1}$ blocks just perform a delay by one sample. $x_1(k)$, $x_2(k)$ and $x_3(k)$ are just arbitrary names. In other words: output will be exactly like input, only delayed by three samples.

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