Energy and Power of Signals

### Motivating question

Consider a simple circuit where a DC voltage source with $v$ volts is connected to a 1 Ohm resistor. How much power is dissipated by the resistor? It is clear that a power is $v^2/R = v^2$ watts or joules/sec. How much energy is dissipated in the resistor over a 1 minute time duration? Since energy is the integral of power, the energy dissipated is $v^2 \times 60 = 60 v^2$ joules. Now consider the same circuit but with a voltage source whose voltage varies with time, i.e., the voltage at time $t$ is $x(t)$. Let us now consider the question of how much energy is dissipated from the resistor over the entire time interval from $-\infty,\infty$. At any given time $t$, the power is given by $x^2(t)$ and the overall energy is given by $\int_{-\infty}^{\infty} x^2(t) \ dt$. Notice that we said that the power dissipated at time $t$ is $x^2(t)$, but can we define one value for the power of the entire signal $\underline{x}(t)$?

### Definition of Energy and Power

The energy and power of a CT signal $\underline{x}(t)$ and DT signal $\underline{x}[n]$ are defined as

(1)
\begin{align} \hbox{Energy of a CT signal} : & E_x=\int_{-\infty}^{\infty} |x(t)|^2 \ dt \\ \hbox{Energy of a DT signal} : & E_x=\sum_{n=-\infty}^{\infty} |x[n]|^2 \\ \hbox{Power of a CT signal} : & P_x= \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \ dt \\ \hbox{Power of a DT signal} : & P_x=\lim_{N \rightarrow \infty} \frac{1}{2N+1} \sum_{n=-\infty}^{\infty} |x[n]|^2 \\ \end{align}

These defintions apply to both real and complex signals $x(t)$ and $x[n]$.

#### Power of periodic signals

Consider a is a periodic CT signal $\underline{x}(t)$ with time period $T_0$ such as the example shown in the figure below.

For such a signal, the energy of the signal given by $\int_{-\infty}^{\infty} |x(t)|^2 dt$ is infinite. Since the power is the average energy
per period, the power is given by

(2)
\begin{align} \hbox{Power}: P_{x}=\frac{1}{T_0} \int_{t_0}^{t_0+T_0} |x(t)|^2 dt. \end{align}

where $t_0$ is any arbitrary time instant starting from which we measure the time period.
If $x(t)$ is a periodic DT signal with time period $N_0$,

(3)
\begin{align} \hbox{Power}: P_x=\frac{1}{N_0} \sum_{n_0}^{n_0+N_0-1} |x[n]|^2 \end{align}

### Energy as the strength of a signal

Even though we used the circuit example as a motivation to define the energy of a signal, the definition of energy is not confined only to signals which can be interpreted as a voltage waveform. Rather, the energy of a signal can be used as a measure of strength of a signal. Often, we encounter situations where we would like to measure the strength of a signal or compare the strengths of two signals and the energy of the signal provides a quantitative measure of the strength of the signal. The above definition of energy to measure the strength of a signal is indeed only one of many possible choices and there are other ways to define the energy or strength of a signal. For example, one can look at the maximum value taken by the signal as one measure of strength, one can look at the sum of the absolute values of a DT signal as another choice. All these measures are meaningful and depending on the decision that we would like to make, we must choose the measure. The energy of a signal defined as in (1) is commonly used and in this course, this will be our default definition of energy. The definition of energy is closely related to what is called in mathematics as the norm of a vector

### Energy type and power type signals

1. $x(t)$ is an energy type signal if $0<E_x<\infty$
2. $x(t)$ is a power type signal if $0<P_x<\infty$

Clearly, for any periodic signal $E_x$ is not bounded and hence, periodic signals cannot be energy type signals. If the energy within one period of the signal is bounded, then the power will be bounded and hence, such signals will be power type signals.

### Example Problems

• Example 1: Consider the signal given below. Is this power or energy type signal?
• Example 2: Let $x(t) = A \cos\left(\omega_0t+ \theta\right)$? Is this a power signal or energy signal?
• What is the power of the signal $x(t) = e^{j\omega_0t}$? where $(T_0=\frac{2\pi}{\omega_0})$
• What is the energy of the signal $x[n]$ given by
(8)
\begin{align} x[n]= \begin{cases} \big( \frac{1}{3}\big)^{n} & \text{if}\ n\geq0 \\ 0 & \text{otherwise} \end{cases} \end{align}