Energy/power signals; shifting and scaling

Energy and power signals

For a signal $x(t)$, the total energy and average power are

$$E = \int_{-\infty}^{\infty} |x(t)|^2\,dt, \qquad P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2\,dt$$

(for discrete time, replace the integrals with sums over $n$ and $1/2T$ with $1/(2N+1)$).

• Energy signal: $0 < E < \infty$, so $P = 0$. These are usually time-limited or decaying signals. Example: $x(t) = e^{-2t}u(t)$ has $E = 1/4$ J, $P = 0$; a single rectangular pulse.
• Power signal: $0 < P < \infty$, so $E = \infty$. These are usually periodic or lasting forever. Example: $x(t) = A\cos(\omega_0 t)$ has $P = A^2/2$, $E = \infty$; the unit step $u(t)$ has $P = 1/2$.
• Some signals are neither, e.g. the ramp $r(t) = t\,u(t)$ (both $E$ and $P$ are infinite).

Time shifting

Time shifting moves a signal along the time axis without changing its shape: $y(t) = x(t - t_0)$.

• $t_0 > 0$: delay, signal moves right.
• $t_0 < 0$: advance, signal moves left.
• Discrete time: $y[n] = x[n - n_0]$ with integer $n_0$.

Example: if $x(t)$ is a pulse on $0 \le t \le 2$, then $x(t-3)$ is the same pulse on $3 \le t \le 5$. Delays appear in echoes and transmission lines.

Time scaling

Time scaling compresses or expands a signal in time: $y(t) = x(at)$, $a > 0$.

• $a > 1$: compression, the signal is squeezed (plays faster). $x(2t)$ on $0 \le t \le 1$ if $x(t)$ was on $0 \le t \le 2$.
• $0 < a < 1$: expansion, the signal is stretched (plays slower). $x(t/2)$ lies on $0 \le t \le 4$.
• $a < 0$ also reflects the signal (time reversal).
• Discrete time: $x[2n]$ keeps only even samples (decimation, information can be lost); $x[n/2]$ needs interpolation.

x(t):    ____          x(2t):  __        x(t/2): ________
        |    |                |  |              |        |
   -----+----+---       ------+--+---     ------+--------+--
        0    2                0  1              0        4

When shifting and scaling are combined, $y(t) = x(at - b)$: first shift by $b$, then scale by $a$ (or scale first and shift by $b/a$).