The energy of a continuous-time signal $x(t)$
$$E_x=\int_{-\infty}^{\infty} |x(t)|^2 dt$$
The energy of a discrete-time signal $x[n]$ is
$$E_x=\sum_{n=-\infty}^{\infty} |x[n]|^2$$
The Dirac signal (continuous-time impulse signal) is defined by
$$\delta(t) = \left\{
\begin{eqnarray}
\infty &\quad& \textrm{for} \quad t = 0 \\\
0 &\quad& \textrm{for} \quad t \neq 0
\end{eqnarray}
\right.$$
where
$$\int_{-\infty}^{\infty} \delta(t) dt = 1$$
The Kronecker signal (discrete-time impulse signal) is defined by
$$\delta[n] = \left\{
\begin{eqnarray}
1 &\quad& \textrm{for} \quad n = 0 \\\
0 &\quad& \textrm{for} \quad n \neq 0
\end{eqnarray}
\right.$$
The properties of impulse signals:
Energy: $E_x =$ not well defined
Power: $P_x = 0$
Even / Odd?: Even (Understanding why is beyond the purview of this course. You will not be tested on this.)
Periodic?: No
Causal?: Yes
The continuous-time step function $u(t)$ is defined by
$$u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau
= \left\{
\begin{eqnarray}
1 &\quad& \textrm{for} \quad t \geq 0 \\\
0 &\quad& \textrm{for} \quad t < 0
\end{eqnarray}
\right.$$
The discrete-time step function $u[n]$ is defined by
$$u[n] = \sum_{k = -\infty}^{n} \delta[k]
= \left\{
\begin{eqnarray}
1 &\quad& \textrm{for} \quad n \geq 0 \\\
0 &\quad& \textrm{for} \quad n < 0
\end{eqnarray}
\right.$$
The properties of the Heaviside step functions:
Energy: $E_x=\infty$
Power: $P_x=1/2$
Even / Odd?: Neither
Periodic?: No
Causal?: Yes
Property
Explanation
Energy
$E_x=\infty$
Power
$P_x=1/2$