Time shifting, scaling and inversion

These are basic operations on the independent variable $t$.

Time shifting

$y(t) = x(t - t_0)$.

• $t_0 > 0$: the signal is delayed (shifted right) by $t_0$.
• $t_0 < 0$: the signal is advanced (shifted left).

Example: if $x(t)$ is a pulse from 0 to 1, then $x(t-2)$ is the same pulse from 2 to 3. A radar echo is a delayed copy of the transmitted signal.

Time scaling

$y(t) = x(at)$, $a > 0$.

• $a > 1$: signal is compressed in time (plays faster).
• $0 < a < 1$: signal is expanded (plays slower).

Example: for the pulse $x(t)$ on $0 \le t \le 2$, $x(2t)$ lies on $0 \le t \le 1$ and $x(t/2)$ on $0 \le t \le 4$. Playing an audio tape at double speed gives $x(2t)$.

Time inversion (reflection/folding)

$y(t) = x(-t)$: the signal is mirrored about $t = 0$. A value at $t = 2$ in $x(t)$ appears at $t = -2$ in $x(-t)$. Example: playing a recording backwards.

 x(t)            x(t-2)          x(2t)          x(-t)
 1 +--+          1    +--+      1 +-+       1 +--+
   |  |               |  |        | |         |  |
 --+--+---t    --+----+--+-t  ----+-+---t  ---+--+-+--t
   0  1          0    2  3        0 .5       -1  0

Combined operation $x(at - b)$: first shift by $b$ to get $x(t-b)$, then scale by $a$ (replace $t$ by $at$). Example: $x(2t-2)$ for the pulse on $[0,1]$ lies on $1 \le t \le 1.5$.