# 1.5 Energy and power

 Page 1 / 1
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

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!