This module describes FFT, convolution, filtering, LTI systems,
digital filters and circular convolution.
Important application of the fft
How many complex multiplies and adds are required to
convolve two
$N$ -pt
sequences?
$$y(n)=\sum_{m=0}^{N-1} x(m)h(n-m)$$
There are
$2N-1$ non-zero output points and each will be computed
using
$N$ complex mults and
$N-1$ complex adds. Therefore,
$$\text{Total Cost}=(2N-1)(N+N-1)\approx O(N^{2})$$
Zero-pad these two sequences to length
$2N-1$ , take DFTs using the FFT algorithm
$$x(n)\to X(k)$$$$h(n)\to H(k)$$ The cost is
$$O((2N-1)\lg (2N-1))=O(N\lg N)$$
Multiply DFTs
$$X(k)H(k)$$ The cost is
$$O(2N-1)=O(N)$$
Inverse DFT using FFT
$$X(k)H(k)\to y(n)$$ The cost is
$$O((2N-1)\lg (2N-1))=O(N\lg N)$$
So the total cost for direct convolution of two
$N$ -point sequences is
$O(N^{2})$ . Total cost for convolution using FFT algorithm is
$O(N\lg N)$ . That is a
huge savings (
).
Summary of dft
$x(n)$ is an
$N$ -point signal
(
).
$$X(k)=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi}{N}kn)}=\sum_{n=0}^{N-1} x(n){W}_{N}^{(kn)}$$ where
${W}_{N}=e^{-(i\frac{2\pi}{N})}$ is a "twiddle factor" and the first part is the basic DFT.
What is the dft
$$X(k)=X(F=\frac{k}{N})=\sum_{n=0}^{N-1} x(n)e^{-(i\times 2\pi Fn)}$$ where
$X(F=\frac{k}{N})$ is the DTFT of
$x(n)$ and
$\sum_{n=0}^{N-1} x(n)e^{-(i\times 2\pi Fn)}$ is the DTFT of
$x(n)$ at digital frequency
$F$ . This is a sample of the
DTFT. We can do frequency domain analysis on a computer!
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?
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