# Time Shifting, Reversal and Delay

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).

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.

• 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. Commented Jan 14, 2022 at 18:26
• You are rights of course.
– Max
Commented Jan 14, 2022 at 18:54