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

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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