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Two different cases:

  1. We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$.

  2. We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$.

I'm looking for possible restrictions (e.g. on the input / impulse response) such that the $y(t)$ and $z(t)$ are equal, if possible.

So far I have this:

$$ y(t) = x(t) * h(2t) \implies Y(j\omega) = X(j\omega) \frac{1}{2} H(j\omega/2) $$

$$ z(t) = x(2t) * h(t) \implies Z(j\omega) = \frac{1}{2} X(j\omega/2)H(j\omega) $$

If we take, for example, $X(j\omega)$ from the first equation and plug into the other one, we get

$$ Z(j\omega) = \frac{Y(j\omega/2)H(j\omega)}{H(j\omega/4)} $$

I'm stuck at this point to comment on the relation between $y(t)$ and $z(t)$ and assert some restrictions to make them equal possibly.

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  • $\begingroup$ $w$ is not the same as $\omega$. $\endgroup$ Mar 28 at 14:43
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    $\begingroup$ @robertbristow-johnson "nit" is nit the same as "not" :-). Dilip Nitpicker $\endgroup$ Mar 28 at 14:48
  • $\begingroup$ :-)..... ........ $\endgroup$ Mar 28 at 17:23
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Aside from

  1. Input that is only a constant value or
  2. a System that does nothing to the signal, i.e. $H = \text{const.}$,

these things are not the same.

Sketch of a proof:

  • Scaling input shifts energy to frequency components that hadn't energy before,
  • Scaling the system means changes the Transfer Function, but keeps the system LTI, which means it can't add energy to any components that had none before.
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  • $\begingroup$ I don't understand: if the input is constant, then outputs are not the same but time-scaled versions of each other, right? Also, if a system is doing nothing, it means its impulse response must be unit impulse, right (and not any other impulse)? In this case, still outputs are not the same but time scaled versions of each other? $\endgroup$
    – fpp
    Mar 28 at 18:28
  • $\begingroup$ As for the sketch of a proof: how come scaling with some number greater than 1 results in giving energy to the frequencies that didn't have any energy before; doesn't frequency response get shrunk, and not expanded? $\endgroup$
    – fpp
    Mar 28 at 18:29
  • $\begingroup$ I'm confused by your comments: let me address them individually: $\endgroup$ Mar 28 at 18:32
  • $\begingroup$ if the input is constant, then outputs are not the same but time-scaled versions of each other, right? wrong, if the input is a constant, time-scaling doesn't change it. $\endgroup$ Mar 28 at 18:32
  • $\begingroup$ if a system is doing nothing, it means its impulse response must be unit impulse, right (and not any other impulse)? right $\endgroup$ Mar 28 at 18:33

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