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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.


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 i m p ( t ) = n = - x s ( n ) δ ( t - n T s )

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 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.

x ˜ ( t ) = ( x i m p * g ) ( t ) = - x i m p ( τ ) g ( t - τ ) d τ = - n = - x s ( n ) δ ( τ - n T s ) g ( t - τ ) d τ = n = - x s ( n ) - δ ( τ - n T s ) g ( t - τ ) d τ = n = - x s ( n ) g ( t - n T s )
Block diagram of reconstruction process for a given lowpass filter G .

Reconstruction filters

In order to guarantee that the reconstructed signal 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 Z ,

g ( n T s ) = 1 n = 0 0 n 0 = δ ( n ) .

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

x ˜ ( n T s ) = m = - x s ( m ) g ( n T s - m T s ) = m = - x s ( m ) g ( ( n - m ) T s ) = m = - x s ( m ) δ ( n - m ) = x s ( n ) ,

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 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 { B n ( t / T s - k ) | k Z } where

Questions & Answers

what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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