# 9.2 Common discrete time fourier transforms

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This module includes a listing of commonly encountered discrete time fourier transforms.

## Common dtfts

 Time Domain x[n] Frequency Domain X(w) Notes $\delta \left[n\right]$ 1 $\delta \left[n-M\right]$ ${e}^{-jwM}$ integer M ${\sum }_{m=-\infty }^{\infty }\delta \left[n-Mm\right]$ ${\sum }_{m=-\infty }^{\infty }{e}^{-jwMm}=\frac{1}{M}{\sum }_{k=-\infty }^{\infty }\delta \left(\frac{w}{2\pi }-\frac{k}{M}\right)$ integer M ${e}^{-jan}$ $2\pi \delta \left(w+a\right)$ real number a $u\left[n\right]$ $\frac{1}{1-{e}^{-jw}}+{\sum }_{k=-\infty }^{\infty }\pi \delta \left(w+2\pi k\right)$ ${a}^{n}u\left(n\right)$ $\frac{1}{1-a{e}^{-jw}}$ if $|a|<1$ $cos\left(an\right)$ $\pi \left[\delta \left(w-a\right)+\delta \left(w+a\right)\right]$ real number a $W·sin{c}^{2}\left(Wn\right)$ $tri\left(\frac{w}{2\pi W}\right)$ real number W, $0 $W·sinc\left[W\left(n+a\right)\right]$ $rect\left(\frac{w}{2\pi W}\right)·{e}^{jaw}$ real numbers W,a $0 $rect\left[\frac{\left(n-M/2\right)}{M}\right]$ $\frac{sin\left[w\left(M+1\right)/2\right]}{sin\left(w/2\right)}{e}^{-jwM/2}$ integer M $\frac{W}{\left(n+a\right)}\left\{cos\left[\pi W\left(n+a\right)\right]-sinc\left[W\left(n+a\right)\right]\right\}$ $jw·rect\left(\frac{w}{\pi W}\right){e}^{j}aw$ real numbers W,a $0 $\frac{1}{\pi {n}^{2}}\left[{\left(-1\right)}^{n}-1\right]$ $|w|$ $\begin{array}{c}\left\{\begin{array}{cc}0\hfill & n=0\hfill \\ \frac{{\left(-1\right)}^{n}}{n}\hfill & \text{elsewhere}\hfill \end{array}\right)\hfill \end{array}$ $jw$ differentiator filter $\begin{array}{c}\left\{\begin{array}{cc}0\hfill & n\text{odd}\hfill \\ \frac{2}{\pi n}\hfill & n\text{even}\hfill \end{array}\right)\hfill \end{array}$ $\begin{array}{c}\left\{\begin{array}{cc}j\hfill & w<0\hfill \\ 0\hfill & w=0\hfill \\ -j\hfill & w>0\hfill \end{array}\right)\hfill \end{array}$ Hilbert Transform

## Notes

rect(t) is the rectangle function for arbitrary real-valued $t$ .

$\text{rect(t)}=\begin{array}{c}\left\{\begin{array}{cc}0\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|>1/2\hfill \\ 1/2\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|=1/2\hfill \\ 1\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|<1/2\hfill \end{array}\right)\hfill \end{array}$

tri(t) is the triangle function for arbitrary real-valued $t$ .

$\text{tri(t)}=\begin{array}{c}\left\{\begin{array}{cc}1+t\hfill & \text{if}-1\le t\le 0\hfill \\ 1-t\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0

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?
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What is meant by 'nano scale'?
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scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
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?
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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
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absolutely yes
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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
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How can I make nanorobot?
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Do somebody tell me a best nano engineering book for beginners?
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NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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