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X ( e j ω ) = X ( z ) z = e j ω = n = - x ( n ) e - j ω n

From the definition of the Z-transform, a change of variable m = n - K shows that a delay of K samples in the time domain is equivalent to multiplication by z - K in the Z-transform domain.

x ( n - K ) Z n = - x ( n - K ) z - n = m = - x ( m ) z - ( m + K ) = z - K m = - x ( m ) z - m = z - K X ( z )

We may use this fact to re-write [link] in the Z-transform domain, by taking Z-transforms of both sides ofthe equation:

Y ( z ) = i = 0 N - 1 b i z - i X ( z ) - k = 1 M a k z - k Y ( z ) Y ( z ) 1 + k = 1 M a k z - k = X ( z ) i = 0 N - 1 b i z - i H ( z ) = Y ( z ) X ( z ) = i = 0 N - 1 b i z - i 1 + k = 1 M a k z - k

From this formula, we see that any filter which can be represented by a linear difference equation with constant coefficientshas a rational transfer function (i.e. a transfer function which is a ratio of polynomials).From this result, we may compute the frequency response of the filter by evaluating H ( z ) on the unit circle:

H ( e j ω ) = i = 0 N - 1 b i e - j ω i 1 + k = 1 M a k e - j ω k .

There are many different methods for implementing a general recursive difference equation of the form [link] . Depending on the application, some methods may be more robustto quantization error, require fewer multiplies or adds, or require less memory. [link] shows a system diagram known as the direct form implementation; it works for any discrete-time filterdescribed by the difference equation in [link] . Note that the boxes containing the symbol z - 1 represent unit delays, while a parameter written next to a signal path represents multiplication by that parameter.

Direct form implementation for a discrete-time filter described by a linear recursive difference equation.

Design of a simple fir filter

Download the files, nspeech1.mat and DTFT.m for the following section.

Location of two zeros for simple a FIR filter.

To illustrate the use of zeros in filter design, you will design a simple second order FIR filterwith the two zeros on the unit circle as shown in [link] . In order for the filter's impulse response to be real-valued,the two zeros must be complex conjugates of one another:

z 1 = e j θ
z 2 = e - j θ

where θ is the angle of z 1 relative to the positive real axis. We will see later that θ [ 0 , π ] may be interpreted as the location of the zeros in the frequency response.

The transfer function for this filter is given by

H f ( z ) = ( 1 - z 1 z - 1 ) ( 1 - z 2 z - 1 ) = ( 1 - e j θ z - 1 ) ( 1 - e - j θ z - 1 ) = 1 - 2 cos θ z - 1 + z - 2 .

Use this transfer function to determine the difference equation for this filter.Then draw the corresponding system diagram and compute the filter's impulse response h ( n ) .

This filter is an FIR filter because it has impulse response h ( n ) of finite duration. Any filter with only zeros and no poles other than thoseat 0 and ± is an FIR filter. Zeros in the transfer function represent frequencies that arenot passed through the filter. This can be useful for removing unwanted frequencies in a signal.The fact that H f ( z ) has zeros at e ± j θ implies that H f ( e ± j θ ) = 0 . This means that the filter will not pass pure sine waves at a frequencyof ω = θ .

Use Matlab to compute and plot the magnitude of the filter's frequency response | H f ( e j ω ) | as a function of ω on the interval - π < ω < π , for the following three values of θ :

  • θ = π / 6
  • θ = π / 3
  • θ = π / 2

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Source:  OpenStax, Purdue digital signal processing labs (ece 438). OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10593/1.4
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