# 4.1 Lab 3: prelab (part 1)

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You will derive the transfer function of a second-order, Direct Form II, infinite impulse response (IIR) filter. Then you will create a fourth-order IIR filter, plot its frequency response, and decompose the fourth-order filter into two second-order sections, choosing an appropriate gain for each stage to prevent overflow.

The transfer function for the second-order section shown in IIR Filtering: Introduction is

$H(z)=G\frac{1+{b}_{1}z^{-1}+{b}_{2}z^{-2}}{1+{a}_{1}z^{-1}+{a}_{2}z^{-2}}$

## Exercise

First, derive the above transfer function. Begin by writing the difference equations for $w(n)$ in terms of the input and past values ( $w(n-1)$ and $w(n-2)$ ). Then write the difference equation for $y(n)$ also in terms of the past samples of $w(n)$ . After finding the two difference equations, compute the corresponding Z-transforms and use the relation $H(z)=\frac{Y(z)}{X(z)}=\frac{Y(z)W(z)}{W(z)X(z)}$ to verify the IIR transfer function in .

Next, design the coefficients for a fourth-order filter implemented as the cascade of two bi-quad sections. Write aMATLAB script to compute the coefficients. Begin by designing the fourth-order filter and checking the responseusing the MATLAB commands

[B,A] = ellip(4,.25,10,.25) freqz(B,A)

MATLAB's freqz command displays the frequency responses of IIR filters and FIR filters. For more information about this, type help freqz . Be sure to look at MATLAB's definition of the transfer function.
If you use the freqz command as shown above, without passing its returned data to anotherfunction, both the magnitude (in decibels) and the phase of the response will be shown.

Next you must find the roots of the numerator, zeros , and roots of the denominator, poles , so that you can group them to create two second-order sections. The MATLAB commands roots and poly will be useful for this task. Save the scripts you use to decompose yourfilter into second-order sections; they will probably be useful later.

Once you have obtained the coefficients for each of your two second-order sections, you are ready to choose a gain factor, $G$ , for each section. As part of your MATLAB script, use freqz to compute the response $\frac{W(z)}{X(z)}$ with $G=1$ for each of the sets of second-order coefficients. Recall that on the DSP we cannot represent numbers greaterthan or equal to 1.0. If the maximum value of $\left|\frac{W(z)}{X(z)}\right|$ is or exceeds 1.0, an input with magnitude less than one could produce $w(n)$ terms with magnitude greater than or equal to one; this is overflow . You must therefore select a gain values for each second-order section such that theresponse from the input to the states, $\frac{W(z)}{X(z)}$ , is always less than one in magnitude. In other words, set the value of $G$ to ensure that $\left|\frac{W(z)}{X(z)}\right|< 1$ .

## Preparing for processor implementation

As the processor exercises become more complex, it will become increasingly important to observe good programmingpractices. Of these, perhaps the most important is careful planning of your program flow, memory and register use, andtesting procedure. Write out pseudo-code for the processor implementation of a bi-quad. Make sure you consider the wayyou will store coefficients and states in memory. Then, to prepare for testing, compute the values of $w(n)$ and $y(n)$ for both second-order sections at $n=\{0, 1, 2\}$ using the filter coefficients you calculated in MATLAB. Assume $x(n)=(n)$ and all states are initialized to zero. You may also want to create a frequency sweep test-vector like theone in DSP Development Environment: Introductory Exercise for TI TMS320C54x and use the filter command to find the outputs for that input. Later,you can recreate these input signals on the DSP and compare the output values it calculates with those you find now. Ifyour program is working, the values will be almost identical, differing only slightly because of quantizationeffects, which are considered in IIR Filtering: Filter-Coefficient Quantization Exercise inMATLAB .

#### 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
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