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For a signal, $s(t)$ undergoing multiple transformations of time scaling, reversal and delay, how should I approach the problem of finding the resultant output signal?

$$s\left(\pm \frac{t-t_0}{T}\right)$$

My approach of the problem was to

  1. First, shift the signal by $t_0$ (moving it right if $t_0>0$ and moving it left if $t_0<0$)
  2. Second, scale the signal by $T$ (expand the signal if $T>1$ and compress the signal if $T<1$)
  3. Third, flip/no-flip the signal around the value $t_0$ (Flip the signal if -ve signal is time is -ve and No Change if time is +ve).
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As Robert pointed out in the the comments, this is not true. The operations are not commutative. Thus, the order of reversal operations can only be guessed.

Initial Answer below - NOT CORRECT

All the operations are commutative, the order in which you employ them is irrelevant.

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  • $\begingroup$ That is not exactly true. Offset by $\tau$ followed by reversal or scaling by $\alpha$ is not the same as scaling first followed by offset. $\endgroup$ Commented Jan 14, 2022 at 18:26
  • $\begingroup$ You are rights of course. $\endgroup$
    – Max
    Commented Jan 14, 2022 at 18:54
  • $\begingroup$ Your answer now says "True" followed by "Not True". How about editing your answer to put the correct answer first? It would help the casual reader who reads only the first paragraph of an answer to get the correct information instead of moving on with the wrong answer as the sole take away.... $\endgroup$ Commented Feb 13, 2022 at 21:01
  • $\begingroup$ I've edited the answer according to your suggestion. $\endgroup$
    – Max
    Commented Feb 14, 2022 at 7:41

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