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Zero locations of linear-phase filters

The zeros of the transfer function H z of a linear-phase filter lie in specific configurations.

We can write the symmetry condition h n h N 1 n in the Z domain. Taking the Z -transform of both sides gives

H z z N 1 H 1 z
Recall that we are assuming that h n is real-valued. If z 0 is a zero of H z , H z 0 0 then H z 0 0 (Because the roots of a polynomial with real coefficients exist in complex-conjugate pairs.)

Using the symmetry condition, , it follows that H z 0 z N 1 H 1 z 0 0 and H z 0 z N 1 H 1 z 0 0 or H 1 z 0 H 1 z 0 0

If z 0 is a zero of a (real-valued) linear-phase filter, then so are z 0 , 1 z 0 , and 1 z 0 .

Zeros locations

It follows that

  • generic zeros of a linear-phase filter exist in sets of 4.
  • zeros on the unit circle ( z 0 0 ) exist in sets of 2. ( z 0 1 )
  • zeros on the real line ( z 0 a ) exist in sets of 2. ( z 0 1 )
  • zeros at 1 and -1 do not imply the existence of zeros at other specific points.

Examples of zero sets

Zero locations: automatic zeros

The frequency response H f of a Type II FIR filter always has a zero at : h n

    h 0 h 1 h 2 h 2 h 1 h 0
H z h 0 h 1 z -1 h 2 z -2 h 2 z -3 h 1 z -4 h 0 z -5 H -1 h 0 h 1 h 2 h 2 h 1 h 0 0 H f H H -1 0
H f 0 always for Type II filters.
Similarly, we can derive the following rules for Type III and Type IV FIR filters.
H f 0 H f 0 always for Type III filters.
H f 0 0 always for Type IV filters.
The automatic zeros can also be derived using the characteristics of the amplitude response A seen earlier.

Type automatic zeros
IV 0

Zero locations: examples

The Matlab command zplane can be used to plot the zero locations of FIR filters.

Note that the zero locations satisfy the properties noted previously.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Introduction about quantum dots in nanotechnology
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Damian Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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