# 0.2 Digital filtering

 Page 1 / 1

## Purpose of upsampling in digital filters in communication

In digital communications filters are important in the process of upsampling. Upsampling is required for transmission because we want the signal's frequency representation to be narrow and confined to frequencies around the carrier frequency. By upsampling the signal, the frequency response of the signal to be transmitted gets compressed and becomes band limited to a significantly smaller range of frequencies, which is necessary for transmission.

As can be seen from the figure above, (a) shows the frequency domain representation of the signal to be transmitted, (b) shows the upsampled version of the frequency response, and upon low pass filtering the upsampled signal we get only two spikes. When the signal is modulated to the carrier frequency, both spikes appear at the corresponding carrier frequency in (c). This is important because we don't want information to be spread across the frequency spectrum, rather we want to transmit the signal at a specific carrier frequency.

The upsampling process is accomplished by taking a signal, inserting L zeros between each sample and then low pass filtering the result. Below is a description of possible low pass filters that can be used to achieve this result.

## Raised cosine filter

The raised cosine filter is a type of low pass filter that accomplishes the interpolation necessary after inserting the L zeros between each sample. It's frequency response is given by:

In the filter α is a parameter which is between 0 and 1 and is called the rolloff factor. The larger α is, the wider the bandwidth of the filter. As α approaches zero the filter will become a brick wall and will look like a box in the frequency domain. The Impulse response of the filter is shown in the figure below:

As can be seen above, the time domain representation is given by a sinc and so in reality the raised cosine filter would extend to plus and minus infinity. However, above the impulse response was truncated and selected to have a length of 41 for the purposes of our digital communication scheme. The raised cosine filter is the best filter for digital communication, specifically 16 QAM, because it removes interference that may occur from one symbol to the next. This means that the waveform can be recovered perfectly at the receiver and this is why the raised cosine filter is typically used in digital communication.

In order to complete the upsampling process it is necessary to convolve the impulse response of the raised cosine filter and the vector that contains the signal with zeros inserted between the samples. The output of this convolution will be the upsampled signal. Below is the output of the real part of a [1 0 1 0] sequence convolved with the raised cosine filter.

As can be seen above, the absolute value of the maximum value is -3. This is because the first two bits, [1 0] correspond to the real part of the sequence and they get mapped to a value of I=-3. Below is the imaginary part of the signal, and as we can see the absolute value of the maximum value is 1. This is because the last to bits, [1 0]correspond to the imaginary part of the sequence and they get mapped to a value of Q=1.

The raised cosine filter is used because it limits the bandwidth of the signal and decays quickly in the time domain. The advantages of this is that it allows for data transmission in specific frequency ranges with an insignificant amount of information spread out across large frequencies.

## Butterworth filter

Butterworth filters are another type of low pass filter which can be used to complete the upsampling process. The frequency response of the Butterworth filter is given by:

$H\left(\mathrm{j\omega }\right)=\frac{1}{\sqrt{1+\left(\frac{\omega }{{\omega }_{c}}{\right)}^{2N}}}$

Where N is a parameter which is called the order of the filter and ω c is the cutoff frequency. The Butterworth filter acts like a low pass filter because it has a flat frequency response that is usually unity gain in the passband and gradually rolls off to zero in the stopband. In between the passband and the stop band we have the cutoff frequency which will occur at the point where the gain is equal to 0.707 (1/sqrt(2)). Butterworth filters have a relatively slow roll off, especially when compared to the raised cosine filter. Below is the impulse response of the Butterworth filter:

Given a sequence of [1 0 1 0] the output of the low pass filtering of the real part of the signal will be:

The output of low pass filtering the imaginary part of the signal with a Butterworth filter will be:

As can be seen the output of the Butterworth filter is similar to that of the raised cosine filter, where both map the real part of the sequence to -3 and the imaginary part of the sequence to 1. The main difference is that the raised cosine filter's output has ripples extending on both sides, while the Butterworth filter does not. This means that the Butterworth filter's output does not need to be truncated since it does not extend to infinity like the output of the raised cosine filter does. The output of these filters would then be modulated appropriately as described, summed together and then transmitted through the channel as described in the QAM module.

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
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 is the latest information on a no technology how can I find it
William
currently
William
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
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

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Digital filters in 16-qam communication. OpenStax CNX. Dec 11, 2011 Download for free at http://cnx.org/content/col11384/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital filters in 16-qam communication' conversation and receive update notifications?

 By Brooke Delaney By John Gabrieli By Madison Christian By David Corey By OpenStax By OpenStax By OpenStax By Hannah Sheth By Brooke Delaney By Saylor Foundation