# 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] :

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
ya I also want to know the raman spectra
Bhagvanji
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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