# 0.4 Lab 4 - sampling and reconstruction

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Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu

## Introduction

It is often desired to analyze and process continuous-time signals using a computer.However, in order to process a continuous-time signal, it must first be digitized.This means that the continuous-time signal must be sampled and quantized, forming a digital signal that can be stored in a computer.Analog systems can be converted to their discrete-time counterparts, and these digital systems then process discrete-time signalsto produce discrete-time outputs. The digital output can then be converted back to an analog signal, or reconstructed , through a digital-to-analog converter. [link] illustrates an example, containing the three general components described above: a sampling system,a digital signal processor, and a reconstruction system.

When designing such a system, it is essential to understand the effects of the sampling and reconstruction processes.Sampling and reconstruction may lead to different types of distortion, including low-pass filtering, aliasing, and quantization.The system designer must insure that these distortions are below acceptable levels,or are compensated through additional processing.

## Sampling overview

Sampling is simply the process of measuring the value of a continuous-time signal at certain instants of time.Typically, these measurements are uniformly separated by the sampling period, ${T}_{s}$ . If $x\left(t\right)$ is the input signal, then the sampled signal, $y\left(n\right)$ , is as follows:

$y\left(n\right)={\left(x,\left(,t,\right)|}_{t=n{T}_{s}}\phantom{\rule{4pt}{0ex}}.$

A critical question is the following: What sampling period, ${T}_{s}$ , is required to accurately represent the signal $x\left(t\right)$ ? To answer this question, we need to look at thefrequency domain representations of $y\left(n\right)$ and $x\left(t\right)$ . Since $y\left(n\right)$ is a discrete-time signal, we represent its frequency content with the discrete-time Fourier transform (DTFT), $Y\left({e}^{j\omega }\right)$ . However, $x\left(t\right)$ is a continuous-time signal, requiring the use of the continuous-time Fourier transform (CTFT), denoted as $X\left(f\right)$ . Fortunately, $Y\left({e}^{j\omega }\right)$ can be written in terms of $X\left(f\right)$ :

$\begin{array}{ccc}\hfill Y\left({e}^{j\omega }\right)& =& \frac{1}{{T}_{s}}\sum _{k=-\infty }^{\infty }{\left(X,\left(,f,\right)|}_{f=\frac{\omega -2\pi k}{2\pi {T}_{s}}}\hfill \\ & =& \frac{1}{{T}_{s}}\sum _{k=-\infty }^{\infty }X\left(\frac{\omega -2\pi k}{2\pi {T}_{s}}\right)\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$

Consistent with the properties of the DTFT, $Y\left({e}^{j\omega }\right)$ is periodic with a period $2\pi$ . It is formed by rescaling the amplitude and frequency of $X\left(f\right)$ , and then repeating it in frequency every $2\pi$ . The critical issue of the relationship in [link] is the frequency content of $X\left(f\right)$ . If $X\left(f\right)$ has frequency components that are above $1/\left(2{T}_{s}\right)$ , the repetition in frequency will cause these components to overlap with (i.e. add to) the components below $1/\left(2{T}_{s}\right)$ . This causes an unrecoverabledistortion, known as aliasing , that will prevent a perfect reconstruction of $X\left(f\right)$ . We will illustrate this later in the lab. The $1/\left(2{T}_{s}\right)$ “cutoff frequency” is known as the Nyquist frequency .

To prevent aliasing, most sampling systems first low pass filter the incoming signalto ensure that its frequency content is below the Nyquist frequency. In this case, $Y\left({e}^{j\omega }\right)$ can be related to $X\left(f\right)$ through the $k=0$ term in [link] :

#### Questions & Answers

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
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
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
characteristics of micro business
Abigail
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Anassong
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Lily
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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