Energy and Power

Number of goals scored by player 2 in each match

Plotting a discrete-time signal or writing it as a sequence of numbers seems good enough, but then how do we make a comparison between two signals? For example, in this Figure, we need to know how this player stands against other players participating in the tournament. It therefore seems that we should devise some measure of “strength” or “size” of a signal.

One simple method can be just adding all the values on the amplitude axis, which is our target dependent variable. So our player 1 has a total number of 10 goals in the tournament. Now we can easily compare his performance with others, provided that the signal always stays positive.

Imagine another footballer who is so “energetic” that he just cannot resist possessing the ball during the game and sometimes even scores own goals, as shown in this Figure. From his team’s point of view, his net total by adding all goals is just 7. However, from an energy perspective in the field, he is much different than player 1.

Therefore, simple addition does not work for signals that assume both positive and negative values, such as a voltage varying with time. Addition of both positive and negative values cancels and diminishes the calculated signal strength, although logically it should not.

This suggests that strength of a signal can be measured by taking the absolute value of the signal and then adding all the values. Or square of the absolute value, or the fourth power of the absolute value, and so on. Due to its mathematical tractability (an exciting must-have for communication theorists), square of the absolute value is the preferred choice.

Hence, the energy of a discrete-time signal is defined as

    \begin{align*}   E_s &= \cdots + |s[-2]|^2 + |s[-1]|^2 + |s[0]|^2 + |s[1]|^2 + |s[2]|^2 + \cdots \nonumber \\     &= \sum \limits _n |s[n]|^2 \end{align*}

where the term \sum \limits _n denotes summation over all values of n.

Example


As an example, signal energy in of player 1 in this Figure is

    \begin{align*}   E_{P1} &= 2^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 \\     &= 14 \end{align*}

and that of player 2 in this Figure is

    \begin{align*}   E_{P2} &= 2^2 + -1^2 + 1^2 + 1^2 + -1^2 + 1^2 + ... \\          &\quad -1^2 + 2^2 + 1^2 + 1^2 + 2^2 + -1^2 \\     &= 21 \end{align*}

Using such a definition, everyone’s actual contribution is truly reflected.

 

Similarly, the energy in quadratic signal of this Figure is infinite as the signal values extend from -\infty to +\infty.

Finally, power is defined as energy per unit time.

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