For a given signal X(t), why is it that if I were to do a time scaling and shifting such that Y(t) = -2x [(t-2)/2], the order at which operation is done first matters ?



For example if I were to do the time scaling first before the time shifting, then X(t) will be expanded time scale wise by 2 after which there will be time shift of two units to the right. But if I were to reverse the order by doing the time shifting first, then this time round, X(t) will be shifted to the right by one unit to the right before the time scale is expanded by 2. Why is this so mathematically ?

 

This happens with non-linear operations. They are not commutative in the same way that matrices are not commutative.

 

The give signal can be written as : y(t) = -2 * x(0.5t -1)
So there are 3 things to be done, Amplitude scaling, time scaling & time shifting.

If time scaling is done first, then the sequence of events are as follows :
1. Time scaled version of x(t) = a(t) = x(0.5t)
2. Time shifted version of a(t) = b(t) = a(t-1) = x( 0.5(t-1) ) = x(0.5t -0.5)
3. Finally doing amplitude scaling will result in y(t) = -2*x(0.5t - 0.5)
which is clearly not the signal which we require

If time shifting is done first, then the sequence of events are as follows :
1. Time shifted version of x(t) = a(t) = x(t - 1)
2. Time scaled version of a(t) = b(t) = a(0.5t)
{which means replace t by 0.5t in the fn a(t) }
Therfore b(t) = a(0.5t) = x(0.5t -1)
3. Finally doing amplitude scaling will result in y(t) = -2*x(0.5t - 1)
which is the signal which we require

Moral : Do time shifting first before u do time scaling

 

Is x(-2t+5) advanced with respect to x(-2t) or delayed

 

You can figure this one out just by coming up with a simple example! It would be advanced, because you are referencing a future part of the signal for the current time.