# 5.4 Linear-phase fir filters: amplitude formulas

 Page 1 / 1
(Blank Abstract)

## Summary: amplitude formulas

Type $()$ $A()$
I $-(M)$ $h(M)+2\sum_{n=0}^{M-1} h(n)\cos ((M-n))$
II $-(M)$ $2\sum_{n=0}^{\frac{N}{2}-1} h(n)\cos ((M-n))$
III $-(M)+\frac{\pi }{2}$ $2\sum_{n=0}^{M-1} h(n)\sin ((M-n))$
IV $-(M)+\frac{\pi }{2}$ $2\sum_{n=0}^{\frac{N}{2}-1} h(n)\sin ((M-n))$

where $M=\frac{N-1}{2}$

## Amplitude response characteristics

To analyze or design linear-phase FIR filters, we need to know the characteristics of the amplitude response $A()$ .

Type Properties
I $A()$ is even about $=0$ $A()=A(-)$
$A()$ is even about $=\pi$ $A(\pi +)=A(\pi -)$
$A()$ is periodic with $2\pi$ $A(+2\pi )=A()$
II $A()$ is even about $=0$ $A()=A(-)$
$A()$ is odd about $=\pi$ $A(\pi +)=-A(\pi -)$
$A()$ is periodic with $4\pi$ $A(+4\pi )=A()$
III $A()$ is odd about $=0$ $A()=-A(-)$
$A()$ is odd about $=\pi$ $A(\pi +)=-A(\pi -)$
$A()$ is periodic with $2\pi$ $A(+2\pi )=A()$
IV $A()$ is odd about $=0$ $A()=-A(-)$
$A()$ is even about $=\pi$ $A(\pi +)=A(\pi -)$
$A()$ is periodic with $4\pi$ $A(+4\pi )=A()$

## Evaluating the amplitude response

The frequency response ${H}^{f}()$ of an FIR filter can be evaluated at $L$ equally spaced frequencies between 0 and $\pi$ using the DFT. Consider a causal FIR filter with an impulse response $h(n)$ of length- $N$ , with $N\le L$ . Samples of the frequency response of the filter can be written as $H(\frac{2\pi }{L}k)=\sum_{n=0}^{N-1} h(n)e^{-i\frac{2\pi }{L}nk}$ Define the $L$ -point signal $\{g(n)\colon 0\le n\le L-1\}$ as $g(n)=\begin{cases}h(n) & \text{if 0\le n\le N-1}\\ 0 & \text{if N\le n\le L-1}\end{cases}$ Then $H(\frac{2\pi }{L}k)=G(k)={\mathrm{DFT}}_{L}(g(n))$ where $G(k)$ is the $L$ -point DFT of $g(n)$ .

## Types i and ii

Suppose the FIR filter $h(n)$ is either a Type I or a Type II FIR filter. Then we have from above ${H}^{f}()=A()e^{-iM}$ or $A()={H}^{f}()e^{iM}$ Samples of the real-valued amplitude $A()$ can be obtained from samples of the function ${H}^{f}()$ as: $A(\frac{2\pi }{L}k)=H(\frac{2\pi }{L}k)e^{iM\frac{2\pi }{L}k}=G(k){W}_{L}^{Mk}$ Therefore, the samples of the real-valued amplitude function can be obtained by zero-padding $h(n)$ , taking the DFT, and multiplying by the complex exponential. This can be written as:

$A(\frac{2\pi }{L}k)={\mathrm{DFT}}_{L}()$
h n 0 L - N
W L M k

## Types iii and iv

For Type III and Type IV FIR filters, we have ${H}^{f}()=ie^{-iM}A()$ or $A()=-i{H}^{f}()e^{iM}$ Therefore, samples of the real-valued amplitude $A()$ can be obtained from samples of the function ${H}^{f}()$ as: $A(\frac{2\pi }{L}k)=-iH(\frac{2\pi }{L}k)e^{iM\frac{2\pi }{L}k}=-iG(k){W}_{L}^{Mk}$ Therefore, the samples of the real-valued amplitude function can be obtained by zero-padding $h(n)$ , taking the DFT, and multiplying by the complex exponential.

$A(\frac{2\pi }{L}k)=-i{\mathrm{DFT}}_{L}()$
h n 0 L - N
W L M k

## Evaluating the amp resp (type i)

In this example, the filter is a Type I FIR filter of length 7. An accurate plot of $A()$ can be obtained with zero padding.

The following Matlab code fragment for the plot of $A()$ for a Type I FIR filter.

h = [3 4 5 6 5 4 3]/30;N = 7; M = (N-1)/2;L = 512; H = fft([h zeros(1,L-N)]); k = 0:L-1;W = exp(j*2*pi/L); A = H .* W.^(M*k);A = real(A); figure(1)w = [0:L-1]*2*pi/(L-1);subplot(2,1,1) plot(w/pi,abs(H))ylabel('|H(\omega)| = |A(\omega)|') xlabel('\omega/\pi')subplot(2,1,2) plot(w/pi,A)ylabel('A(\omega)') xlabel('\omega/\pi')print -deps type1

The command A = real(A) removes the imaginary part which is equal to zero to within computerprecision. Without this command, Matlab takes A to be a complex vector and the following plot command will not be right.

Observe the symmetry of $A()$ due to $h(n)$ being real-valued. Because of this symmetry, $A()$ is usually plotted for $0\le \le \pi$ only.

## Evaluating the amp resp (type ii)

The following Matlab code fragment produces a plot of $A()$ for a Type II FIR filter.

h = [3 5 6 7 7 6 5 3]/42;N = 8; M = (N-1)/2;L = 512; H = fft([h zeros(1,L-N)]); k = 0:L-1;W = exp(j*2*pi/L); A = H .* W.^(M*k);A = real(A); figure(1)w = [0:L-1]*2*pi/(L-1);subplot(2,1,1) plot(w/pi,abs(H))ylabel('|H(\omega)| = |A(\omega)|') xlabel('\omega/\pi')subplot(2,1,2) plot(w/pi,A)ylabel('A(\omega)') xlabel('\omega/\pi')print -deps type2

The imaginary part of the amplitude is zero. Notice that $A(\pi )=0$ . In fact this will always be the case for a Type II FIR filter.

An exercise for the student: Describe how to obtain samples of $A()$ for Type III and Type IV FIR filters. Modify the Matlab code above for these types. Do you notice that $A()=0$ always for special values of  ?

## Modules for further study

• Zero Locations of Linear-Phase Filters
• Design of Linear-Phase FIR Filters by Interpolation
• Linear-Phase FIR Filter Design by Least Squares

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
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
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By JavaChamp Team By Jonathan Long By OpenStax By Jams Kalo By OpenStax By David Geltner By OpenStax By OpenStax By P. Wynn Norman By OpenStax