# 1.5 Energy and power

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Energy and power for analog and discrete time signals

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 verydifferent physical entities. We can say that energy and power are a measure of the signal's "size".

## Analog signals

Since we often think of a signal as a function of varying amplitude through time, it seems to reason that a goodmeasurement 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 absolutevalue, then finding the area under that curve. It turns out that what we call the energy of a signal is thearea under the squared signal, see

${E}_{a}=\int_{()} \,d t$ 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 magnitudeoperation.

## 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

${E}_{d}=\sum_{n=()}$ x n 2

Given the sequence $y(l)=b^{l}u(l)$ , 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, ${E}_{d}=\sum_{l=0}$ y l 2 1 1 b 2 . This expression is only valid for $\left|b\right|< 1$ . 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 wehave infinite energy for any such signal. Does this mean that a fifty hertz sine wave feeding into your headphones is asstrong 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.

## Analog signals

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

${P}_{a}=\frac{1}{{T}_{0}}\int_{-\left(\frac{{T}_{0}}{2}\right)}^{\frac{{T}_{0}}{2}} \left|x(t)\right|^{2}\,d t$

## Discrete signals

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

${P}_{d}=\frac{1}{N}\sum_{n={N}_{1}}^{{N}_{1}+N-1} \left|x(n)\right|^{2}$

Given the signals ${x}_{1}(t)=\sin (2\pi t)$ and ${x}_{2}(n)=\sin (\pi \frac{1}{10}n)$ , shown in , calculate the power for one period.

For the analog sine we have ${P}_{a}=\frac{1}{1}\int_{0}^{1} \sin (2\pi t)^{2}\,d t=\frac{1}{2}$ .

For the discrete sine we get ${P}_{d}=\frac{1}{20}\sum_{n=1}^{20} \sin (\frac{1}{10}\pi n)^{2}=0.500$ . Download power_sine.m for plots and calculation.

## Matlab files

• Introduction
• Discrete time signals
• Analog signals
• Discrete vs Analog signals
• Frequency definitions and periodicity
• Exercises

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