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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 we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
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Damian Reply
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Stoney Reply
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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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.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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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