10.2 Signal reconstruction

 Page 1 / 2
This module describes the reconstruction, also known as interpolation, of a continuous time signal from a discrete time signal, including a discussion of cardinal spline filters.

Introduction

The sampling process produces a discrete time signal from a continuous time signal by examining the value of the continuous time signal at equally spaced points in time. Reconstruction, also known as interpolation, attempts to perform an opposite process that produces a continuous time signal coinciding with the points of the discrete time signal. Because the sampling process for general sets of signals is not invertible, there are numerous possible reconstructions from a given discrete time signal, each of which would sample to that signal at the appropriate sampling rate. This module will introduce some of these reconstruction schemes.

Reconstruction process

The process of reconstruction, also commonly known as interpolation, produces a continuous time signal that would sample to a given discrete time signal at a specific sampling rate. Reconstruction can be mathematically understood by first generating a continuous time impulse train

${x}_{imp}\left(t\right)=\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)\delta \left(t-n{T}_{s}\right)$

from the sampled signal ${x}_{s}$ with sampling period ${T}_{s}$ and then applying a lowpass filter $G$ that satisfies certain conditions to produce an output signal $\stackrel{˜}{x}$ . If $G$ has impulse response $g$ , then the result of the reconstruction process, illustrated in [link] , is given by the following computation, the final equation of which is used to perform reconstruction in practice.

$\begin{array}{cc}\hfill \stackrel{˜}{x}\left(t\right)& =\left({x}_{imp}*g\right)\left(t\right)\hfill \\ & ={\int }_{-\infty }^{\infty }{x}_{imp}\left(\tau \right)g\left(t-\tau \right)d\tau \hfill \\ & ={\int }_{-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)\delta \left(\tau -n{T}_{s}\right)g\left(t-\tau \right)d\tau \hfill \\ & =\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right){\int }_{-\infty }^{\infty }\delta \left(\tau -n{T}_{s}\right)g\left(t-\tau \right)d\tau \hfill \\ & =\sum _{n=-\infty }^{\infty }{x}_{s}\left(n\right)g\left(t-n{T}_{s}\right)\hfill \end{array}$

Reconstruction filters

In order to guarantee that the reconstructed signal $\stackrel{˜}{x}$ samples to the discrete time signal ${x}_{s}$ from which it was reconstructed using the sampling period ${T}_{s}$ , the lowpass filter $G$ must satisfy certain conditions. These can be expressed well in the time domain in terms of a condition on the impulse response $g$ of the lowpass filter $G$ . The sufficient condition to be a reconstruction filters that we will require is that, for all $n\in \mathbb{Z}$ ,

$g\left(n{T}_{s}\right)=\left\{\begin{array}{cc}1& n=0\\ 0& n\ne 0\end{array}\right)=\delta \left(n\right).$

This means that $g$ sampled at a rate ${T}_{s}$ produces a discrete time unit impulse signal. Therefore, it follows that sampling $\stackrel{˜}{x}$ with sampling period ${T}_{s}$ results in

$\begin{array}{cc}\hfill \stackrel{˜}{x}\left(n{T}_{s}\right)& =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)g\left(n{T}_{s}-m{T}_{s}\right)\hfill \\ & =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)g\left(\left(n-m\right){T}_{s}\right)\hfill \\ & =\sum _{m=-\infty }^{\infty }{x}_{s}\left(m\right)\delta \left(n-m\right)\hfill \\ & ={x}_{s}\left(n\right),\hfill \end{array}$

which is the desired result for reconstruction filters.

Cardinal basis splines

Since there are many continuous time signals that sample to a given discrete time signal, additional constraints are required in order to identify a particular one of these. For instance, we might require our reconstruction to yield a spline of a certain degree, which is a signal described in piecewise parts by polynomials not exceeding that degree. Additionally, we might want to guarantee that the function and a certain number of its derivatives are continuous.

This may be accomplished by restricting the result to the span of sets of certain splines, called basis splines or B-splines. Specifically, if a $n\mathrm{th}$ degree spline with continuous derivatives up to at least order $n-1$ is required, then the desired function for a given ${T}_{s}$ belongs to the span of $\left\{{B}_{n}\left(t/{T}_{s}-k\right)|k\in \mathbb{Z}\right\}$ where

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
Got questions? Join the online conversation and get instant answers!