# 0.15 Appendix a

 Page 1 / 1

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,
${A}^{-1}A=A{A}^{-1}=I.$

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!