# Continuous-time signals  (Page 3/5)

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Although the function to be expanded is defined only over a specific finite region, the series converges to a function that is defined over thereal line and is periodic. It is equal to the original function over the region of definition and is a periodic extension outside of the region.Indeed, one could artificially extend the given function at the outset and then the expansion would converge everywhere.

## A geometric view

It can be very helpful to develop a geometric view of the Fourier series where $x\left(t\right)$ is considered to be a vector and the basis functions are the coordinate or basis vectors. The coefficients become the projections of $x\left(t\right)$ on the coordinates. The ideas of a measure of distance, size, and orthogonality are important and the definition of error is easy topicture. This is done in [link] , [link] , [link] using Hilbert space methods.

## Properties of the fourier series

The properties of the Fourier series are important in applying it to signal analysis and to interpreting it. The main properties are given hereusing the notation that the Fourier series of a real valued function $x\left(t\right)$ over $\left\{0\le t\le T\right\}$ is given by $\mathcal{F}\left\{x\left(t\right)\right\}=c\left(k\right)$ and $\stackrel{˜}{x}\left(t\right)$ denotes the periodic extensions of $x\left(t\right)$ .

1. Linear: $\mathcal{F}\left\{x+y\right\}=\mathcal{F}\left\{x\right\}+\mathcal{F}\left\{y\right\}$
Idea of superposition. Also scalability: $\mathcal{F}\left\{ax\right\}=a\mathcal{F}\left\{x\right\}$
2. Extensions of $x\left(t\right)$ : $\stackrel{˜}{x}\left(t\right)=\stackrel{˜}{x}\left(t+T\right)$
$\stackrel{˜}{x}\left(t\right)$ is periodic.
3. Even and Odd Parts: $x\left(t\right)=u\left(t\right)+jv\left(t\right)$ and $C\left(k\right)=A\left(k\right)+jB\left(k\right)=|C\left(k\right)|\phantom{\rule{0.166667em}{0ex}}{e}^{j\theta \left(k\right)}$
 $u$ $v$ $A$ $B$ $|C|$ $\theta$ even 0 even 0 even 0 odd 0 0 odd even 0 0 even 0 even even $\pi /2$ 0 odd odd 0 even $\pi /2$
4. Convolution: If continuous cyclic convolution is defined by
$y\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}h\left(t\right)\circ x\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\int }_{0}^{T}\stackrel{˜}{h}\left(t-\tau \right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{x}\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}d\tau$

then $\mathcal{F}\left\{h\left(t\right)\circ x\left(t\right)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{F}\left\{h\left(t\right)\right\}\phantom{\rule{0.166667em}{0ex}}\mathcal{F}\left\{x\left(t\right)\right\}$
5. Multiplication: If discrete convolution is defined by
$e\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}d\left(n\right)*c\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum }_{m=-\infty }^{\infty }d\left(m\right)\phantom{\rule{0.166667em}{0ex}}c\left(n-m\right)$

then $\mathcal{F}\left\{h\left(t\right)\phantom{\rule{0.166667em}{0ex}}x\left(t\right)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{F}\left\{h\left(t\right)\right\}*\mathcal{F}\left\{x\left(t\right)\right\}$
This property is the inverse of property 4 and vice versa.
6. Parseval: $\frac{1}{T}{\int }_{0}^{T}{|x\left(t\right)|}^{2}dt\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum }_{k=-\infty }^{\infty }{|C\left(k\right)|}^{2}$
This property says the energy calculated in the time domain is the same as that calculated in the frequency (or Fourier) domain.
7. Shift: $\mathcal{F}\left\{\stackrel{˜}{x}\left(t-{t}_{0}\right)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}C\left(k\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-j2\pi {t}_{0}k/T}$
A shift in the time domain results in a linear phase shift in the frequency domain.
8. Modulate: $\mathcal{F}\left\{x\left(t\right)\phantom{\rule{0.166667em}{0ex}}{e}^{j2\pi Kt/T}\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}C\left(k-K\right)$
Modulation in the time domain results in a shift in the frequency domain. This property is the inverse of property 7.
9. Orthogonality of basis functions:
${\int }_{0}^{T}{e}^{-j2\pi mt/T}\phantom{\rule{0.166667em}{0ex}}{e}^{j2\pi nt/T}\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}T\phantom{\rule{0.277778em}{0ex}}\delta \left(n-m\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{cc}T\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}n=m\hfill \\ 0\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}n\ne m.\hfill \end{array}\right)$
Orthogonality allows the calculation of coefficients using inner products in [link] and [link] . It also allows Parseval's Theorem in property 6 . A relaxed version of orthogonality is called “tight frames" and is importantin over-specified systems, especially in wavelets.

## Examples

• An example of the Fourier series is the expansion of a square wave signal with period $2\pi$ . The expansion is
$x\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\frac{4}{\pi }\left[sin\left(t\right)+\frac{1}{3}sin\left(3t\right)+\frac{1}{5}sin\left(5t\right)\cdots \right].$
Because $x\left(t\right)$ is odd, there are no cosine terms (all $a\left(k\right)=0$ ) and, because of its symmetries, there are no even harmonics (even $k$ terms are zero). The function is well defined and bounded; its derivative is not,therefore, the coefficients drop off as $\frac{1}{k}$ .
• A second example is a triangle wave of period $2\pi$ . This is a continuous function where the square wave was not. The expansion of thetriangle wave is
$x\left(t\right)=\frac{4}{\pi }\left[sin\left(t\right)-\frac{1}{{3}^{2}}sin\left(3t\right)+\frac{1}{{5}^{2}}sin\left(5t\right)+\cdots \right].$
Here the coefficients drop off as $\frac{1}{{k}^{2}}$ since the function and its first derivative exist and are bounded.

how do they get the third part x = (32)5/4
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+_
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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
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
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Yasmin
what is the function of carbon nanotubes?
Cesar
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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
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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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