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The amplitude response

If the real and imaginary parts of H f are given by

H f
the magnitude and phase are defined as H f 2 2 p so that
H f H f p
With this definition, H f is never negative and p is usually discontinuous, but it can be very helpful to write H f as
H f A
where A can be positive and negative, and continuous. A is called the amplitude response . illustrates the difference between H f and A .

A linear-phase phase filter is one for which the continuous phase is linear. H f A with M B We assume in the following that the impulse response h n is real-valued.

Why linear-phase?

If a discrete-time cosine signal x 1 n 1 n 1 is processed through a discrete-time filter with frequency response H f A then the output signal is given by y 1 n A 1 1 n 1 1 or y 1 n A 1 1 n 1 1 1 The LTI system has the effect of scaling the cosine signal and delaying it by 1 1 .

When does the system delay cosine signals with different frequencies by the same amount?

  • constant
  • K
  • The phase is linear.
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The function is called the phase delay . A linear phase filter therefore has constant phase delay.

Why linear-phase: example

Consider a discrete-time filter described by the difference equation

y n -0.1821 x n 0.7865 x n 1 0.6804 x n 2 x n 3 0.6804 y n 1 0.7865 y n 2 0.1821 y n 3
When 1 0.31 , then the delay is 1 1 2.45 . The delay is illustrated in :

Notice that the delay is fractional --- the discrete-time samples are not exactly reproduced in the output.The fractional delay can be interpreted in this case as a delay of the underlying continuous-time cosine signal.

Why linear-phase: example (2)

Consider the same system given on the previous slide, but let us change the frequency of the cosine signal.

When 2 0.47 , then the delay is 2 2 0.14 .

For this example, the delay depends on the frequency, because this system does not have linear phase.

Why linear-phase: more

From the previous slides, we see that a filter will delay different frequency components of a signal by the same amountif the filter has linear phase (constant phase delay).

In addition, when a narrow band signal (as in AM modulation) goes through a filter, the envelop will be delayed by the group delay or envelop delay of the filter. The amount by which the envelop is delayed isindependent of the carrier frequency only if the filter has linear phase.

Also, in applications like image processing, filters with non-linear phase can introduce artifacts that are visuallyannoying.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
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Maciej
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s. Reply
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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CYNTHIA
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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.
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Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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I'm interested in nanotube
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preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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