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
$y(n)=(x(n), h(n))=\sum $∞∞xkhnk
$Y()=X()H()$
Assume that
$H()$ is specified.
How can we implement
$X()H()$ in a computer?
Discretize (sample)
$X()$ and
$H()$ . In order to do this, we should take the DFTs
of
$x(n)$ and
$h(n)$ to get
$X(k)$ and
$X(k)$ . Then we will compute
$$\stackrel{~}{y}(n)=\mathrm{IDFT}(X(k)H(k))$$ Does
$\stackrel{~}{y}(n)=y(n)$ ?
is called
circular convolution and is denoted by
.
Dft pair
Note that in general:
Regular vs. circular convolution
To begin with, we are given the following two length-3
signals:
$$x(n)=\{1, 2, 3\}$$$$h(n)=\{1, 0, 2\}$$ We can zero-pad these signals so that we have the following
discrete sequences:
$$x(n)=\{, 0, 1, 2, 3, 0, \}$$$$h(n)=\{, 0, 1, 0, 2, 0, \}$$ where
$x(0)=1$ and
$h(0)=1$ .
Regular Convolution:
$y(n)=\sum_{m=0}^{2} x(m)h(n-m)$
Using the above convolution formula (refer to the
link if you need a review of
convolution ), we can
calculate the resulting value for
$y(0)$ to
$y(4)$ . Recall that because we have two length-3
signals, our convolved signal will be length-5.
"Zero-pad"
$x(n)$ and
$h(n)$ 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):
$$x(n)=\{1, 2, 3, 0, 0\}$$$$h(n)=\{1, 0, 2, 0, 0\}$$
We can compute the regular convolution result of a
convolution of an
$M$ -point
signal
$x(n)$ with an
$N$ -point
signal
$h(n)$ by padding each signal with zeros to obtain two
$M+N-1$ length sequences and computing the circular
convolution (or equivalently computing the IDFT of
$H(k)X(k)$ , the product of the DFTs of the zero-padded
signals) (
).
Dsp system
Sample finite duration continuous-time input
$x(t)$ to get
$x(n)$ where
$n=\{0, , M-1\}$ .
Zero-pad
$x(n)$ and
$h(n)$ to length
$M+N-1$ .
Compute DFTs
$X(k)$ and
$H(k)$
Compute IDFTs of
$X(k)H(k)$$$y(n)=\stackrel{~}{y}(n)$$ where
$n=\{0, , M+N-1\}$ .
Reconstruct
$y(t)$
Questions & Answers
what is variations in raman spectra for nanomaterials
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
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?