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
Is there any normative that regulates the use of silver nanoparticles?
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?