Even/odd parts; time scaling and folding

Even and odd parts of $x(t)$

Let any signal be written as the sum of an even part and an odd part:

$$x(t) = x_e(t) + x_o(t) \qquad (1)$$

where $x_e(-t) = x_e(t)$ and $x_o(-t) = -x_o(t)$. Replace $t$ by $-t$:

$$x(-t) = x_e(-t) + x_o(-t) = x_e(t) - x_o(t) \qquad (2)$$

Adding (1) and (2), and subtracting (2) from (1):

$$\begin{aligned} x_e(t) &= \frac{1}{2}\left[x(t) + x(-t)\right] \\ x_o(t) &= \frac{1}{2}\left[x(t) - x(-t)\right] \end{aligned}$$

These parts always exist for any $x(t)$, so every signal can be decomposed into even and odd components. The same holds for DT signals: $x_e[n] = \frac{1}{2}(x[n]+x[-n])$, $x_o[n] = \frac{1}{2}(x[n]-x[-n])$.

Time scaling

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

• $a > 1$: compression – the signal happens faster. If $x(t)$ is a pulse on $0 \le t \le 2$, $x(2t)$ is on $0 \le t \le 1$.
• $0 < a < 1$: expansion – $x(t/2)$ is on $0 \le t \le 4$.

For DT signals, $x[2n]$ keeps only every second sample (decimation), and $x[n/2]$ inserts samples (interpolation).

Time folding (reflection)

$y(t) = x(-t)$: the signal is reflected about the vertical axis $t = 0$. A value at $t = t_1$ moves to $t = -t_1$. Example: a ramp $t$ on $0 \le t \le 1$ becomes a ramp on $-1 \le t \le 0$. Folding is a key step in convolution.

 x(t)          x(2t)        x(-t)
      /|           /|        |\
    /  |          / |        |  \
  /    |         /  |        |    \
-+-----+--> t  -+---+--> t  -+-----+--> t
 0     2        0   1       -2     0