# 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.

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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!