# 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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Tarell
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