# Continuous-time signals  (Page 2/5)

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$x\left(t\right)=\frac{a\left(0\right)}{2}+\sum _{k=1}^{\infty }a\left(k\right)cos\left(\frac{2\pi }{T}kt\right)+b\left(k\right)sin\left(\frac{2\pi }{T}kt\right).$

where ${x}_{k}\left(t\right)=cos\left(2\pi kt/T\right)$ and ${y}_{k}\left(t\right)=sin\left(2\pi kt/T\right)$ are the basis functions for the expansion. The energy or power in an electrical,mechanical, etc. system is a function of the square of voltage, current, velocity, pressure, etc. For this reason, the natural setting for arepresentation of signals is the Hilbert space of ${L}^{2}\left[0,T\right]$ . This modern formulation of the problem is developed in [link] , [link] . The sinusoidal basis functions in the trigonometric expansion form a completeorthogonal set in ${L}^{2}\left[0,T\right]$ . The orthogonality is easily seen from inner products

$\left(cos\left(\frac{2\pi }{T}kt\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}cos\left(\frac{2\pi }{T}\ell t\right)\right)={\int }_{0}^{T}\left(cos\left(\frac{2\pi }{T}kt\right)\phantom{\rule{0.277778em}{0ex}}cos\left(\frac{2\pi }{T}\ell t\right)\right)\phantom{\rule{0.277778em}{0ex}}dt=\delta \left(k-\ell \right)$

and

$\left(cos\left(\frac{2\pi }{T}kt\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}sin\left(\frac{2\pi }{T}\ell t\right)\right)={\int }_{0}^{T}\left(cos\left(\frac{2\pi }{T}kt\right)\phantom{\rule{0.277778em}{0ex}}sin\left(\frac{2\pi }{T}\ell t\right)\right)\phantom{\rule{0.277778em}{0ex}}dt=0$

where $\delta \left(t\right)$ is the Kronecker delta function with $\delta \left(0\right)=1$ and $\delta \left(k\ne 0\right)=0$ . Because of this, the $k$ th coefficients in the series can be foundby taking the inner product of $x\left(t\right)$ with the $k$ th basis functions. This gives for the coefficients

$a\left(k\right)=\frac{2}{T}{\int }_{0}^{T}x\left(t\right)cos\left(\frac{2\pi }{T}kt\right)dt$

and

$b\left(k\right)=\frac{2}{T}{\int }_{0}^{T}x\left(t\right)sin\left(\frac{2\pi }{T}kt\right)dt$

where $T$ is the time interval of interest or the period of a periodic signal. Because of the orthogonality of the basis functions, afinite Fourier series formed by truncating the infinite series is an optimal least squared error approximation to $x\left(t\right)$ . If the finite series is defined by

$\stackrel{^}{x}\left(t\right)=\frac{a\left(0\right)}{2}+\sum _{k=1}^{N}a\left(k\right)cos\left(\frac{2\pi }{T}kt\right)+b\left(k\right)sin\left(\frac{2\pi }{T}kt\right),$

the squared error is

$\epsilon =\frac{1}{T}{\int }_{0}^{T}{|x\left(t\right)-\stackrel{^}{x}\left(t\right)|}^{2}dt$

which is minimized over all $a\left(k\right)$ and $b\left(k\right)$ by [link] and [link] . This is an extraordinarily important property.

It follows that if $x\left(t\right)\in {L}^{2}\left[0,T\right]$ , then the series converges to $x\left(t\right)$ in the sense that $\epsilon \to 0$ as $N\to \infty$ [link] , [link] . The question of point-wise convergence is more difficult. A sufficient condition that is adequate for mostapplication states: If $f\left(x\right)$ is bounded, is piece-wise continuous, and has no more than a finite number of maxima over an interval, the Fourierseries converges point-wise to $f\left(x\right)$ at all points of continuity and to the arithmetic mean at points of discontinuities. If $f\left(x\right)$ is continuous, the series converges uniformly at all points [link] , [link] , [link] .

A useful condition [link] , [link] states that if $x\left(t\right)$ and its derivatives through the $q$ th derivative are defined and have bounded variation, the Fourier coefficients $a\left(k\right)$ and $b\left(k\right)$ asymptotically drop off at least as fast as $\frac{1}{{k}^{q+1}}$ as $k\to \infty$ . This ties global rates of convergence of the coefficients to local smoothness conditions of the function.

The form of the Fourier series using both sines and cosines makes determination of the peak value or of the location of a particularfrequency term difficult. A different form that explicitly gives the peak value of the sinusoid of that frequency and the location or phase shift ofthat sinusoid is given by

$x\left(t\right)=\frac{d\left(0\right)}{2}+\sum _{k=1}^{\infty }d\left(k\right)cos\left(\frac{2\pi }{T}kt+\theta \left(k\right)\right)$

and, using Euler's relation and the usual electrical engineering notation of $j=\sqrt{-1}$ ,

${e}^{jx}=cos\left(x\right)+jsin\left(x\right),$

the complex exponential form is obtained as

$x\left(t\right)=\sum _{k=-\infty }^{\infty }c\left(k\right)\phantom{\rule{0.166667em}{0ex}}{e}^{j\frac{2\pi }{T}kt}$

where

$c\left(k\right)=a\left(k\right)+j\phantom{\rule{0.166667em}{0ex}}b\left(k\right).$

The coefficient equation is

$c\left(k\right)=\frac{1}{T}{\int }_{0}^{T}x\left(t\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-j\frac{2\pi }{T}kt}dt$

The coefficients in these three forms are related by

${|d|}^{2}={|c|}^{2}={a}^{2}+{b}^{2}$

and

$\theta =arg\left\{c\right\}={tan}^{-1}\left(\frac{b}{a}\right)$

It is easier to evaluate a signal in terms of $c\left(k\right)$ or $d\left(k\right)$ and $\theta \left(k\right)$ than in terms of $a\left(k\right)$ and $b\left(k\right)$ . The first two are polar representation of a complex value and the last is rectangular. Theexponential form is easier to work with mathematically.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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what is the actual application of fullerenes nowadays?
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What is lattice structure?
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or in general
Ebrahim
in general
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Graphene has a hexagonal structure
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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what is biological synthesis of nanoparticles
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