# Continuous-time signals

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Signals occur in a wide range of physical phenomenon. They might be human speech, blood pressure variations with time, seismic waves,radar and sonar signals, pictures or images, stress and strain signals in a building structure, stock market prices, a city'spopulation, or temperature across a plate. These signals are often modeled or represented by a real or complex valued mathematicalfunction of one or more variables. For example, speech is modeled by a function representing air pressure varying with time. Thefunction is acting as a mathematical analogy to the speech signal and, therefore, is called an analog signal. For these signals, the independent variable is time and it changescontinuously so that the term continuous-time signal is also used. In our discussion, we talk of the mathematical function asthe signal even though it is really a model or representation of the physical signal.

The description of signals in terms of their sinusoidal frequency content has proven to be one of the most powerful tools ofcontinuous and discrete-time signal description, analysis, and processing. For that reason, we will start the discussion ofsignals with a development of Fourier transform methods. We will first review the continuous-time methods of the Fourier series (FS),the Fourier transform or integral (FT), and the Laplace transform (LT). Next the discrete-time methods will be developed in moredetail with the discrete Fourier transform (DFT) applied to finite length signals followed by the discrete-time Fourier transform(DTFT) for infinitely long signals and ending with the Z-transform which allows the powerful tools of complex variable theory to beapplied.

More recently, a new tool has been developed for the analysis of signals. Wavelets and wavelet transforms [link] , [link] , [link] , [link] , [link] are another more flexible expansion system that also can describe continuousand discrete-time, finite or infinite duration signals. We will very briefly introduce the ideas behind wavelet-based signal analysis.

## The fourier series

The problem of expanding a finite length signal in a trigonometric series was posed and studied in the late 1700's by renowned mathematicians suchas Bernoulli, d'Alembert, Euler, Lagrange, and Gauss. Indeed, what we now call the Fourier series and the formulas for the coefficients were used byEuler in 1780. However, it was the presentation in 1807 and the paper in 1822 by Fourier stating that an arbitrary function could be represented bya series of sines and cosines that brought the problem to everyone's attention and started serious theoretical investigations and practicalapplications that continue to this day [link] , [link] , [link] , [link] , [link] , [link] . The theoretical work has been at the center of analysis and the practical applications havebeen of major significance in virtually every field of quantitative science and technology. For these reasons and others, the Fourier seriesis worth our serious attention in a study of signal processing.

## Definition of the fourier series

We assume that the signal $x\left(t\right)$ to be analyzed is well described by a real or complex valued function of a real variable $t$ defined over a finite interval $\left\{0\le t\le T\right\}$ . The trigonometric series expansion of $x\left(t\right)$ is given by

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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