This module will take the ideas of sampling CT signals further by examining how such operations can be performed in the frequency domain and by using a computer.
Introduction
We just covered ideal (and non-ideal) (time)
sampling of CT signals .
This enabled DT signal processing solutions for CTapplications (
):
Much of the theoretical analysis of such systems relied on
frequency domain representations. How do we carry out thesefrequency domain analysis on the computer? Recall the
following relationships:
$$x(n)\stackrel{\mathrm{DTFT}}{}X()$$$$x(t)\stackrel{\mathrm{CTFT}}{}X()$$ where
$$ and
$$ are continuous frequency
variables.
Sampling dtft
Consider the DTFT of a discrete-time (DT) signal
$x(n)$ . Assume
$x(n)$ is of finite duration
$N$ (
i.e. , an
$N$ -point signal).
$X()=\sum_{n=0}^{N-1} x(n)e^{-in}$
where
$X()$ is the continuous function that is indexed by thereal-valued parameter
$-\pi \le \le \pi $ . The other function,
$x(n)$ , is a discrete function that is indexed by
integers.
We want to work with
$X()$ on a computer. Why not just
sample$X()$ ?
In
we sampled at
$=\frac{2\pi}{N}k$ where
$k=\{0, 1, , N-1\}$ and
$X(k)$ for
$k=\{0, , N-1\}$ is called the
Discrete Fourier Transform (DFT) of
$x(n)$ .
The DTFT of the image in
is written as follows:
$X()=\sum_{n=0}^{N-1} x(n)e^{-in}$
where
$$ is any
$2\pi $ -interval, for example
$-\pi \le \le \pi $ .
where again we sampled at
$=\frac{2\pi}{N}k$ where
$k=\{0, 1, , M-1\}$ . For example, we take
$$M=10$$ . In the
following section we will discuss in
more detail how we should choose
$M$ , the number of samples in
the
$2\pi $ interval.
(This is precisely how we would plot
$X()$ in Matlab.)
Given
$N$ (length of
$x(n)$ ), choose
$(M, N)$ to obtain a dense sampling of the DTFT (
):
Case 2
Choose
$M$ as small as
possible (to minimize the amount of computation).
In general, we require
$M\ge N$ in order to represent all information in
$$\forall n, n=\{0, , N-1\}\colon x(n)$$ Let's concentrate on
$M=N$ :
$$x(n)\stackrel{\mathrm{DFT}}{}X(k)$$ for
$n=\{0, , N-1\}$ and
$k=\{0, , N-1\}$$$\mathrm{numbers}\mathrm{Nnumbers}$$
Discrete fourier transform (dft)
Define
$X(k)\equiv X(\frac{2\pi k}{N})$
where
$N=\mathrm{length}(x(n))$ and
$k=\{0, , N-1\}$ . In this case,
$M=N$ .
Represent
$x(n)$ in terms of a sum of
$N$complex sinusoids of amplitudes
$X(k)$ and frequencies
$$\forall k, k\in \{0, , N-1\}\colon {}_{k}=\frac{2\pi k}{N}$$
Fourier Series with fundamental frequency
$\frac{2\pi}{N}$
Remark 1
IDFT treats
$x(n)$ as though it were
$N$ -periodic.
Think of sampling the continuous function
$X()$ , as depicted in
.
$S()$ will represent the sampling function applied to
$X()$ and is illustrated in
as well. This will result in our
discrete-time sequence,
$X(k)$ .
Remember the multiplication in the frequency domain is equal
to convolution in the time domain!
Inverse dtft of s()
$\sum $∞∞2kN
Given the above equation, we can take the DTFT and get thefollowing equation:
$N\sum $∞∞nmNSn
Why does
equal
$S(n)$ ?
$S(n)$ is
$N$ -periodic,
so it has the following
Fourier Series :
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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?