# 0.15 Appendix a

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Just because some of us can read and write and do a little math, that doesn't mean we deserve to conquer the Universe.

—Kurt Vonnegut, Hocus Pocus , 1990

This appendix gathers together all of the math facts used in the text. They are divided into six categories:

• Trigonometric identities
• Fourier transforms and properties
• Energy and power
• Z-transforms and properties
• Integral and derivative formulas
• Matrix algebra

So, with no motivation or interpretation, just labels, here they are:

## Trigonometric identities

• Euler's relation
${e}^{±jx}=\mathrm{cos}\left(x\right)±j\mathrm{sin}\left(x\right)$
• Exponential definition of a cosine
$\mathrm{cos}\left(x\right)=\frac{1}{2}\left({e}^{jx},+,{e}^{-jx}\right)$
• Exponential definition of a sine
$\mathrm{sin}\left(x\right)=\frac{1}{2j}\left({e}^{jx},-,{e}^{-jx}\right)$
• Cosine squared
${\mathrm{cos}}^{2}\left(x\right)=\frac{1}{2}\left(1,+,\mathrm{cos},\left(,2,x,\right)\right)$
• Sine squared
${\mathrm{sin}}^{2}\left(x\right)=\frac{1}{2}\left(1,-,\mathrm{cos},\left(,2,x,\right)\right)$
• Sine and Cosine as phase shifts of each other
$\begin{array}{ccc}\hfill \mathrm{sin}\left(x\right)& =& \mathrm{cos}\left(\frac{\pi }{2},-,x\right)=\mathrm{cos}\left(x,-,\frac{\pi }{2}\right)\hfill \\ \hfill \mathrm{cos}\left(x\right)& =& \mathrm{sin}\left(\frac{\pi }{2},-,x\right)=-\mathrm{sin}\left(x,-,\frac{\pi }{2}\right)\hfill \end{array}$
• Sine–cosine product
$\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)=\frac{1}{2}\left[\mathrm{sin},\left(,x,-,y,\right),+,\mathrm{sin},\left(,x,+,y,\right)\right]$
• Cosine–cosine product
$\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)=\frac{1}{2}\left[\mathrm{cos},\left(,x,-,y,\right),+,\mathrm{cos},\left(,x,+,y,\right)\right]$
• Sine–sine product
$\mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)=\frac{1}{2}\left[\mathrm{cos},\left(,x,-,y,\right),-,\mathrm{cos},\left(,x,+,y,\right)\right]$
• Odd symmetry of the sine
$\mathrm{sin}\left(-x\right)=-\mathrm{sin}\left(x\right)$
• Even symmetry of the cosine
$\mathrm{cos}\left(-x\right)=\mathrm{cos}\left(x\right)$
• Cosine angle sum
$\mathrm{cos}\left(x±y\right)=\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)\mp \mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)$
• Sine angle sum
$\mathrm{sin}\left(x±y\right)=\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)±\mathrm{cos}\left(x\right)\mathrm{sin}\left(y\right)$

## Fourier transforms and properties

• Definition of Fourier transform
$W\left(f\right)={\int }_{-\infty }^{\infty }w\left(t\right){e}^{-j2\pi ft}dt$
• Definition of Inverse Fourier transform
$w\left(t\right)={\int }_{-\infty }^{\infty }W\left(f\right){e}^{j2\pi ft}df$
• Fourier transform of a sine
$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{A\mathrm{sin}\left(2\pi {f}_{0}t+\Phi \right)\right\}\hfill \\ & & =j\frac{A}{2}\left[-,{e}^{j\Phi },\delta ,\left(f-{f}_{0}\right),+,{e}^{-j\Phi },\delta ,\left(f+{f}_{0}\right)\right]\hfill \end{array}$
• Fourier transform of a cosine
$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{A\mathrm{cos}\left(2\pi {f}_{0}t+\Phi \right)\right\}\hfill \\ & & =\frac{A}{2}\left[{e}^{j\Phi },\delta ,\left(f-{f}_{0}\right),+,{e}^{-j\Phi },\delta ,\left(f+{f}_{0}\right)\right]\hfill \end{array}$
• Fourier transform of impulse
$\mathcal{F}\left\{\delta \left(t\right)\right\}=1$
• Fourier transform of rectangular pulse
• With
$\Pi \left(t\right)=\left\{\begin{array}{cc}1\hfill & -T/2\le t\le T/2\\ 0\hfill & \text{otherwise}\end{array}\right),$
$\mathcal{F}\left\{\Pi \left(t\right)\right\}=T\frac{\mathrm{sin}\left(\pi fT\right)}{\pi fT}\equiv T\text{sinc}\left(fT\right).$
• Fourier transform of sinc function
$\mathcal{F}\left\{\text{sinc}\left(2Wt\right)\right\}=\frac{1}{2W}\Pi \left(\frac{f}{2W}\right)$
• Fourier transform of raised cosine
• With
$w\left(t\right)=2{f}_{0}\left(\frac{\mathrm{sin}\left(2\pi {f}_{0}t\right)}{2\pi {f}_{0}t}\right)\left[\frac{\mathrm{cos}\left(2\pi {f}_{\Delta }t\right)}{1-{\left(4{f}_{\Delta }t\right)}^{2}}\right],$
$\mathcal{F}\left\{w\left(t\right)\right\}=\left\{\begin{array}{cc}1\hfill & |f|<{f}_{1}\hfill \\ \frac{1}{2}\left(1,+,\mathrm{cos},\left[\frac{\pi \left(|f|-{f}_{1}\right)}{2{f}_{\Delta }}\right]\right)\hfill & {f}_{1}<|f|B\hfill \end{array}\right),$
with the rolloff factor defined as $\beta ={f}_{\Delta }/{f}_{0}$ .
• Fourier transform of square-root raised cosine (SRRC)
• With $w\left(t\right)$ given by
$\left\{\begin{array}{cc}\frac{1}{\sqrt{T}}\frac{\mathrm{sin}\left(\pi \left(1-\beta \right)t/T\right)+\left(4\beta t/T\right)\mathrm{cos}\left(\pi \left(1+\beta \right)t/T\right)}{\left(\pi t/T\right)\left(1-{\left(4\beta t/T\right)}^{2}\right)}\hfill & t\ne 0,±\frac{T}{4\beta }\hfill \\ \frac{1}{\sqrt{T}}\left(1-\beta +\left(4\beta /\pi \right)\right)\hfill & t=0\hfill \\ \frac{\beta }{\sqrt{2T}}\left[\left(1,+,\frac{2}{\pi }\right),\mathrm{sin},\left(\frac{\pi }{4\beta }\right),+,\left(1,-,\frac{2}{\pi }\right),\mathrm{cos},\left(\frac{\pi }{4\beta }\right)\right]\hfill & t=±\frac{T}{4\beta }\hfill \end{array}\right),$
$\mathcal{F}\left\{w\left(t\right)\right\}=\left\{\begin{array}{cc}1\hfill & |f|<{f}_{1}\hfill \\ {\left[\frac{1}{2},\left(1,+,\mathrm{cos},\left[\frac{\pi \left(|f|-{f}_{1}\right)}{2{f}_{\Delta }}\right]\right)\right]}^{1/2}\hfill & {f}_{1}<|f|B\hfill \end{array}\right).$
• Fourier transform of periodic impulse sampled signal
• With
$\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right),$
and
$\begin{array}{ccc}\hfill {w}_{s}\left(t\right)& =& w\left(t\right)\sum _{k=-\infty }^{\infty }\delta \left(t-k{T}_{s}\right),\hfill \\ \hfill \mathcal{F}\left\{{w}_{s}\left(t\right)\right\}& =& \frac{1}{{T}_{s}}\sum _{n=-\infty }^{\infty }W\left(f-\left(n/{T}_{s}\right)\right).\hfill \end{array}$
• Fourier transform of a step
• With
$\begin{array}{ccc}\hfill w\left(t\right)& =& \left\{\begin{array}{cc}A\hfill & t>0\hfill \\ 0\hfill & t<0\hfill \end{array}\right),\hfill \\ \hfill \mathcal{F}\left\{w\left(t\right)\right\}& =& A\left[\frac{\delta \left(f\right)}{2},+,\frac{1}{j2\pi f}\right].\hfill \end{array}$
• Fourier transform of ideal $\pi /2$ phase shifter (Hilbert transformer) filterimpulse response
• With
$\begin{array}{ccc}\hfill w\left(t\right)& =& \left\{\begin{array}{cc}\frac{1}{\pi t}& t>0\\ 0& t<0\end{array}\right\},\hfill \\ \hfill \mathcal{F}\left\{w\left(t\right)\right\}& =& \left\{\begin{array}{cc}-j& f>0\hfill \\ j& f<0\hfill \end{array}\right).\hfill \end{array}$
• Linearity property
• With $\mathcal{F}\left\{{w}_{i}\left(t\right)\right\}={W}_{i}\left(f\right)$ ,
$\mathcal{F}\left\{a{w}_{1}\left(t\right)+b{w}_{2}\left(t\right)\right\}=a{W}_{1}\left(f\right)+b{W}_{2}\left(f\right).$
• Duality property With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\mathcal{F}\left\{W\left(t\right)\right\}=w\left(-f\right).$
• Cosine modulation frequency shift property
• With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{w\left(t\right)\mathrm{cos}\left(2\pi {f}_{c}t+\theta \right)\right\}\hfill \\ & & =\frac{1}{2}\left[{e}^{j\theta },W,\left(f-{f}_{c}\right),+,{e}^{-j\theta },W,\left(f+{f}_{c}\right)\right].\hfill \end{array}$
• Exponential modulation frequency shift property
• With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\mathcal{F}\left\{w\left(t\right){e}^{j2\pi {f}_{0}t}\right\}=W\left(f-{f}_{0}\right).$
• Complex conjugation (symmetry) property If $w\left(t\right)$ is real valued,
${W}^{*}\left(f\right)=W\left(-f\right),$
where the superscript $*$ denotes complex conjugation (i.e.,  ${\left(a+jb\right)}^{*}=a-jb\right)$ . In particular, $|W\left(f\right)|$ is even and $\angle W\left(f\right)$ is odd.
• Symmetry property for real signals Suppose $w\left(t\right)$ is real.
$\begin{array}{ccc}& \text{If}& w\left(t\right)=w\left(-t\right),\phantom{\rule{4.pt}{0ex}}\text{then}\phantom{\rule{4.pt}{0ex}}W\left(f\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{real.}\phantom{\rule{4.pt}{0ex}}\hfill \\ & \text{If}& w\left(t\right)=-w\left(-t\right),\hfill \\ & & W\left(f\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{purely}\phantom{\rule{4.pt}{0ex}}\text{imaginary.}\phantom{\rule{4.pt}{0ex}}\hfill \end{array}$
• Time shift property
• With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\mathcal{F}\left\{w\left(t-{t}_{0}\right)\right\}=W\left(f\right){e}^{-j2\pi f{t}_{0}}.$
• Frequency scale property
• With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\mathcal{F}\left\{w\left(at\right)\right\}=\frac{1}{a}W\left(\frac{f}{a}\right).$
• Differentiation property
• With $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ ,
$\frac{dw\left(t\right)}{dt}=j2\pi fW\left(f\right).$
• Convolution $↔$ multiplication property
• With $\mathcal{F}\left\{{w}_{i}\left(t\right)\right\}={W}_{i}\left(f\right)$ ,
$\mathcal{F}\left\{{w}_{1}\left(t\right)*{w}_{2}\left(t\right)\right\}={W}_{1}\left(f\right){W}_{2}\left(f\right)$
and
$\mathcal{F}\left\{{w}_{1}\left(t\right){w}_{2}\left(t\right)\right\}={W}_{1}\left(f\right)*{W}_{2}\left(f\right),$
where the convolution operator “ $*$ ” is defined via
$x\left(\alpha \right)*y\left(\alpha \right)\equiv {\int }_{-\infty }^{\infty }x\left(\lambda \right)y\left(\alpha -\lambda \right)d\lambda .$
• Parseval's theorem
• With $\mathcal{F}\left\{{w}_{i}\left(t\right)\right\}={W}_{i}\left(f\right)$ ,
${\int }_{-\infty }^{\infty }{w}_{1}\left(t\right){w}_{2}^{*}\left(t\right)dt={\int }_{-\infty }^{\infty }{W}_{1}\left(f\right){W}_{2}^{*}\left(f\right)df.$
• Final value theorem
• With ${lim}_{t\to -\infty }w\left(t\right)=0$ and $w\left(t\right)$ bounded,
$\underset{t\to \infty }{lim}w\left(t\right)=\underset{f\to 0}{lim}j2\pi fW\left(f\right),$
where $\mathcal{F}\left\{w\left(t\right)\right\}=W\left(f\right)$ .

## Energy and power

• Energy of a continuous time signal $s\left(t\right)$ is
$E\left(s\right)={\int }_{-\infty }^{\infty }{s}^{2}\left(t\right)dt$
if the integral is finite.
• Power of a continuous time signal $s\left(t\right)$ is
$P\left(s\right)=\underset{T\to \infty }{lim}\frac{1}{T}{\int }_{-T/2}^{T/2}{s}^{2}\left(t\right)dt$
if the limit exists.
• Energy of a discrete time signal $s\left[k\right]$ is
$E\left(s\right)=\sum _{-\infty }^{\infty }{s}^{2}\left[k\right]$
if the sum is finite.
• Power of a discrete time signal $s\left[k\right]$ is
$P\left(s\right)=\underset{N\to \infty }{lim}\frac{1}{2N}\sum _{k=-N}^{N}{s}^{2}\left[k\right]$
if the limit exists.
• Power Spectral Density
• With input and output transforms $X\left(f\right)$ and $Y\left(f\right)$ of a linear filter with impulse response transform $H\left(f\right)$ (such that $Y\left(f\right)=H\left(f\right)X\left(f\right)$ ),
${\mathcal{P}}_{y}\left(f\right)={\mathcal{P}}_{x}\left(f\right)\phantom{\rule{4pt}{0ex}}{|H\left(f\right)|}^{2},$
where the power spectral density (PSD) is defined as
${\mathcal{P}}_{x}\left(f\right)=\underset{T\to \infty }{lim}\frac{|{X}_{T}{\left(f\right)|}^{2}}{T}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\mathrm{Watts}/\mathrm{Hz}\right),$
where $\mathcal{F}\left\{{x}_{T}\left(t\right)\right\}={X}_{T}\left(f\right)$ and
${x}_{T}\left(t\right)=x\left(t\right)\phantom{\rule{4pt}{0ex}}\Pi \left(\frac{t}{T}\right),$
where $\Pi \left(·\right)$ is the rectangular pulse [link] .

## Z-transforms and properties

• Definition of the Z-transform
$X\left(z\right)=\mathcal{Z}\left\{x\left[k\right]\right\}=\sum _{k=-\infty }^{\infty }x\left[k\right]{z}^{-k}$
• Time-shift property
• With $\mathcal{Z}\left\{x\left[k\right]\right\}=X\left(z\right)$ ,
$\mathcal{Z}\left\{x\left[k-\Delta \right]\right\}={z}^{-\Delta }X\left(z\right).$
• Linearity property
• With $\mathcal{Z}\left\{{x}_{i}\left[k\right]\right\}={X}_{i}\left(z\right)$ ,
$\mathcal{Z}\left\{a{x}_{1}\left[k\right]+b{x}_{2}\left[k\right]\right\}=a{X}_{1}\left(z\right)+b{X}_{2}\left(z\right).$
• Final Value Theorem for $z$ -transforms If $X\left(z\right)$ converges for $|z|>1$ and all poles of $\left(z-1\right)X\left(z\right)$ are inside the unit circle, then
$\underset{k\to \infty }{lim}x\left[k\right]=\underset{z\to 1}{lim}\left(z-1\right)X\left(z\right).$

## Integral and derivative formulas

• Sifting property of impulse
${\int }_{-\infty }^{\infty }w\left(t\right)\delta \left(t-{t}_{0}\right)dt=w\left({t}_{0}\right)$
• Schwarz's inequality
${\left|{\int }_{-\infty }^{\infty },a,\left(x\right),b,\left(x\right),d,x\right|}^{2}\le \left\{{\int }_{-\infty }^{\infty },{|a\left(x\right)|}^{2},d,x\right\}\left\{{\int }_{-\infty }^{\infty },{|b\left(x\right)|}^{2},d,x\right\}$
and equality occurs only when $a\left(x\right)=k{b}^{*}\left(x\right)$ , where superscript $*$ indicates complex conjugation (i.e.,  ${\left(a+jb\right)}^{*}=a-jb$ ).
• Leibniz's rule
$\begin{array}{cc}\hfill \frac{d\left[{\int }_{a\left(x\right)}^{b\left(x\right)},f,\left(\lambda ,x\right),d,\lambda \right]}{dx}& =f\left(b\left(x\right),x\right)\frac{db\left(x\right)}{dx}-f\left(a\left(x\right),x\right)\frac{da\left(x\right)}{dx}\hfill \\ & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+{\int }_{a\left(x\right)}^{b\left(x\right)}\frac{\partial f\left(\lambda ,x\right)}{\partial x}d\lambda \hfill \end{array}$
• Chain rule of differentiation
$\frac{dw}{dx}=\frac{dw}{dy}\frac{dy}{dx}$
• Derivative of a product
$\frac{d}{dx}\left(wy\right)=w\frac{dy}{dx}+y\frac{dw}{dx}$
• Derivative of signal raised to a power
$\frac{d}{dx}\left({y}^{n}\right)=n{y}^{n-1}\frac{dy}{dx}$
• Derivative of cosine
$\frac{d}{dx}\left(\mathrm{cos},\left(,y,\right)\right)=-\left(\mathrm{sin}\left(y\right)\right)\frac{dy}{dx}$
• Derivative of sine
$\frac{d}{dx}\left(\mathrm{sin},\left(,y,\right)\right)=\left(\mathrm{cos}\left(y\right)\right)\frac{dy}{dx}$

## Matrix algebra

• Transpose transposed
${\left({A}^{T}\right)}^{T}=A$
• Transpose of a product
${\left(AB\right)}^{T}={B}^{T}{A}^{T}$
• Transpose and inverse commutativity If ${A}^{-1}$ exists,
${\left({A}^{T}\right)}^{-1}={\left({A}^{-1}\right)}^{T}.$
• Inverse identity If ${A}^{-1}$ exists,
${A}^{-1}A=A{A}^{-1}=I.$

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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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what king of growth are you checking .?
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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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Tarell
what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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