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Energy and Power

Module by: Anders Gjendemsjø, Melissa Selik, Richard Baraniuk. E-mail the authors

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Summary: Energy and power for analog and discrete time signals

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From physics we've learned that energy is work and power is work per time unit. Energy was measured in Joule (J) and work in Watts(W). In signal processing energy and power are defined more loosely without any necessary physical units, because the signals may represent very different physical entities. We can say that energy and power are a measure of the signal's "size".

Signal Energy

Analog signals

Since we often think of a signal as a function of varying amplitude through time, it seems to reason that a good measurement of the strength of a signal would be the area under the curve. However, this area may have a negative part. This negative part does not have less strength than a positive signal of the same size. This suggests either squaring the signal or taking its absolute value, then finding the area under that curve. It turns out that what we call the energy of a signal is the area under the squared signal, see Figure 1

Energy - Analog signal:

Ea=∫-∞∞|xt|2dt Ea t x t 2
Note that we have used squared magnitude(absolute value) if the signal should be complex valued. If the signal is real, we can leave out the magnitude operation.
Figure 1: Sketch of energy calculation
(a) Signal x(t)
Figure 1(a) (signal.png)
(b) The energy of x(t) is the shaded region
Figure 1(b) (signal_energy.png)

Discrete signals

For time discrete signals the "area under the squared signal" makes no sense, so we will have to use another energy definiton. We define energy as the sum of the squared magnitude of the samples. Mathematically

Energy - Discrete time signal:

Ed=∑n=-∞∞|xn|2 Ed n x n 2

Example 1

Given the sequence yl=blul y l b l u l , where u(l) is the unit step function. Find the energy of the sequence.

We recognize y(l) as a geometric series. Thus we can use the formula for the sum of a geometric series and we obtain the energy, Ed=∑l=0∞yl2=11−b2 Ed l 0 y l 2 1 1 b 2 . This expression is only valid for |b|<1 b 1 . If we have a larger |b|, the series will diverge. The signal y(l) then has infinite energy. So let's have a look at power...

Signal Power

Our definition of energy seems reasonable, and it is. However, what if the signal does not decay fast enough? In this case we have infinite energy for any such signal. Does this mean that a fifty hertz sine wave feeding into your headphones is as strong as the fifty hertz sine wave coming out of your outlet? Obviously not. This is what leads us to the idea of signal power, which in such cases is a more adequate description.

Figure 2: Signal with inifinite energy
Figure 2 (infinite_energy.png)

Analog signals

For analog signals we define power as energy per time interval.

Power - analog signal:

Pa=1T0∫-T02T02|xt|2dt Pa 1 T0 t T0 2 T0 2 x t 2

Discrete signals

For time discrete signals we define power as energy per sample.

Power - Discrete time:

Pd=1N∑n=N1N1+N−1|xn|2 Pd 1 N n N1 N1 N 1 x n 2

Example 2

Given the signals x1t=sin2πt x1 t 2 t and x2n=sinπ110n x2 n 1 10 n , shown in Figure 3, calculate the power for one period.

For the analog sine we have Pa=11∫01sin22πtdt=12 Pa 1 1 t 0 1 2 t 2 1 2 .

For the discrete sine we get Pd=120∑n=120sin2110πn=0.500 Pd 1 20 n 1 20 1 10 n 2 0.500 . Download power_sine.m for plots and calculation.

Figure 3: Analog and discrete time sine.
(a) Analog sine
Figure 3(a) (sine_analog.png)
(b) Discrete time sine
Figure 3(b) (sine_discrete.png)

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