The ideas of using the DFT to filter a signal and recover a signal from a noisy transmission are addressed based on the ideas of the DFT and convolution.
Introduction
Assume that
is specified.
How can we implement
in a computer?
Discretize (sample)
and
. In order to do this, we should take the DFTs
of
and
to get
and
. Then we will compute
Does
?
Recall that the DFT treats
-point sequences as if they are
periodically extended (
):
Compute idft of y[k]
And the IDFT periodically extends
:
This computes as shown in
:
is called
circular convolution and is denoted by
.
The above symbol for the circular convolution is for an
-periodic extension.
Dft pair
Note that in general:
Regular vs. circular convolution
To begin with, we are given the following two length-3
signals:
We can zero-pad these signals so that we have the following
discrete sequences:
where
and
.
Regular Convolution:
Using the above convolution formula (refer to the
link if you need a review of
convolution ), we can
calculate the resulting value for
to
. Recall that because we have two length-3
signals, our convolved signal will be length-5.
Regular convolution result
Result is finite duration, not periodic!
Circular Convolution:
And now with circular convolution our
changes and becomes a periodically extended
signal:
Circular convolution result
Result is 3-periodic.
illustrates the
relationship between circular convolution and regularconvolution using the previous two figures:
Circular convolution from regular
The left plot (the circular convolution results) has a
"wrap-around" effect due to periodic extension.
"Zero-pad"
and
to avoid the overlap (wrap-around) effect. We
will zero-pad the two signals to a length-5 signal (5being the duration of the regular convolution result):
Now take the DFTs of the zero-padded signals:
Now we can plot this result (
):
The sequence from 0 to 4 (the underlined part of the
sequence) is the regular convolution result. From thisillustration we can see that it is 5-periodic!
We can compute the regular convolution result of a
convolution of an
-point
signal
with an
-point
signal
by padding each signal with zeros to obtain two
length sequences and computing the circular
convolution (or equivalently computing the IDFT of
, the product of the DFTs of the zero-padded
signals) (
).
Note that the lower two images are simply the top images
that have been zero-padded.
Dsp system
The system has a length
impulse response,
Sample finite duration continuous-time input
to get
where
.
Zero-pad
and
to length
.
Compute DFTs
and
Compute IDFTs of
where
.
Reconstruct
Questions & Answers
where we get a research paper on Nano chemistry....?
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
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?