<< Chapter < Page Chapter >> Page >
A more complete model of how discrete time processing of continuous time signals is implemented in practice. Notice the addition of anti-aliasing and anti-imaging filters to promote input and output bandlimitedness. The ADC is shown to perform sampling with quantization. The digital filter is further specified to be causal. The DAC is shown to perform imperfect reconstruction, a zero order hold in this case.

Anti-aliasing filter

In reality, we cannot typically guarantee that the input signal will have a specific bandlimit, and sufficiently high sampling rates cannot necessarily be produced. Since it is imperative that the higher frequency components not be allowed to masquerade as lower frequency components through aliasing, anti-aliasing filters with cutoff frequency less than or equal to ω s / 2 must be used before the signal is fed into the ADC. The block diagram in [link] reflects this addition.

As described in the previous section, an ideal lowpass filter removing all energy at frequencies above ω s / 2 would be optimal. Of course, this is not achievable, so approximations of the ideal lowpass filter with low gain above ω s / 2 must be accepted. This means that some aliasing is inevitable, but it can be reduced to a mostly insignificant level.

Signal quantization

In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the ADC performs sampling exactly. However, while an ADC does convert a continuous time signal to a discrete time signal, it also must convert analog values to digital values for use in a digital logic device, a phenomenon called quantization. The ADC subsystem of the block diagram in [link] reflects this addition.

The data obtained by the ADC must be stored in finitely many bits inside a digital logic device. Thus, there are only finitely many values that a digital sample can take, specifically 2 N where N is the number of bits, while there are uncountably many values an analog sample can take. Hence something must be lost in the quantization process. The result is that quantization limits both the range and precision of the output of the ADC. Both are finite, and improving one at constant number of bits requires sacrificing quality in the other.

Filter implementability

In real world circumstances, if the input signal is a function of time, the future values of the signal cannot be used to calculate the output. Thus, the digital filter H 2 and the overall system H 1 must be causal. The filter annotation in [link] reflects this addition. If the desired system is not causal but has impulse response equal to zero before some time t 0 , a delay can be introduced to make it causal. However, if this delay is excessive or the impulse response has infinite length, a windowing scheme becomes necessary in order to practically solve the problem. Multiplying by a window to decrease the length of the impulse response can reduce the necessary delay and decrease computational requirements.

Take, for instance the case of the ideal lowpass filter. It is acausal and infinite in length in both directions. Thus, we must satisfy ourselves with an approximation. One might suggest that these approximations could be achieved by truncating the sinc impulse response of the lowpass filter at one of its zeros, effectively windowing it with a rectangular pulse. However, doing so would produce poor results in the frequency domain as the resulting convolution would significantly spread the signal energy. Other windowing functions, of which there are many, spread the signal less in the frequency domain and are thus much more useful for producing these approximations.

Anti-imaging filter

In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the DAC performs perfect reconstruction. However, when considering practical matters, it is important to remember that the sinc function, which is used for Whittaker-Shannon interpolation, is infinite in length and acausal. Hence, it would be impossible for an DAC to implement perfect reconstruction.

Instead, the DAC implements a causal zero order hold or other simple reconstruction scheme with respect to the sampling rate ω s used by the ADC. However, doing so will result in a function that is not bandlimited to ( - ω s / 2 , ω s / 2 ) . Therefore, an additional lowpass filter, called an anti-imaging filter, must be applied to the output. The process illustrated in [link] reflects these additions. The anti-imaging filter attempts to bandlimit the signal to ( - ω s / 2 , ω s / 2 ) , so an ideal lowpass filter would be optimal. However, as has already been stated, this is not possible. Therefore, approximations of the ideal lowpass filter with low gain above ω s / 2 must be accepted. The anti-imaging filter typically has the same characteristics as the anti-aliasing filter.

Discrete time processing of continuous time signals summary

As has been show, the sampling and reconstruction can be used to implement continuous time systems using discrete time systems, which is very powerful due to the versatility, flexibility, and speed of digital computers. However, there are a large number of practical considerations that must be taken into account when attempting to accomplish this, including quantization noise and anti-aliasing in the analog to digital converter, filter implementability in the discrete time filter, and reconstruction windowing and associated issues in the digital to analog converter. Many modern technologies address these issues and make use of this process.

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask