# Digital signals and their transforms  (Page 2/2)

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In this lab, we learn how to compute the continuous-time Fourier transform (CTFT), normally referred to as Fourier transform, numerically and examine its properties. Also, we explore noise cancellation and amplitude modulation as applications of Fourier transform.

In the previous labs, different mathematical transforms for processing analog or continuous-time signals were covered. Now let us explore the mathematical transforms for processing digital signals. Digital signals are sampled (discrete-time) and quantized version of analog signals. The conversion of analog-to-digital signals is implemented with an analog-to-digital (A/D) converter, and the conversion of digital-to-analog signals is implemented with a digital-to-analog (D/A) converter. In the first part of the lab, we learn how to choose an appropriate sampling frequency to achieve a proper analog-to-digital conversion. In the second part of the lab, we examine the A/D and D/A processes.

## Sampling and aliasing

Sampling is the process of generating discrete-time samples from an analog signal. First, it is helpful to mention the relationship between analog and digital frequencies. Consider an analog sinusoidal signal $x\left(t\right)=A\text{cos}\left(\mathrm{\omega t}+\phi \right)$ . Sampling this signal at $t={\text{nT}}_{s}$ , with the sampling time interval of ${T}_{s}$ , generates the discrete-time signal

$x\left[n\right]=A\text{cos}\left(\omega {\text{nT}}_{s}+\phi \right)=A\text{cos}\left(\mathrm{\theta n}+\phi \right),\begin{array}{cc}& n=0,1,2,\text{.}\text{.}\text{.},\end{array}$

where $\theta ={\mathrm{\omega T}}_{s}=\frac{2\pi f}{{f}_{s}}$ denotes digital frequency with units being radians (as compared to analog frequency ω with units being radians/second).

The difference between analog and digital frequencies is more evident by observing that the same discrete-time signal is obtained from different continuous-time signals if the product ${\mathrm{\omega T}}_{s}$ remains the same. (An example is shown in [link] .) Likewise, different discrete-time signals are obtained from the same analog or continuous-time signal when the sampling frequency is changed. (An example is shown in [link] .) In other words, both the frequency of an analog signal $f$ and the sampling frequency ${f}_{s}$ define the digital frequency $\theta$ of the corresponding digital signal.

It helps to understand the constraints associated with the above sampling process by examining signals in the frequency domain. The Fourier transform pairs for analog and digital signals are stated as

 Fourier transform pair for analog signals $\left\{\begin{array}{c}X\left(\mathrm{j\omega }\right)={\int }_{-\infty }^{\infty }x\left(t\right){e}^{-\mathrm{j\omega t}}\text{dt}\\ x\left(t\right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(\mathrm{j\omega }\right){e}^{\mathrm{j\omega t}}\mathrm{d\omega }\end{array}$ Fourier transform pair for discrete signals $\left\{\begin{array}{c}X\left({e}^{\mathrm{j\theta }}\right)=\sum _{n=-\infty }^{\infty }x\left[n\right]{e}^{-\text{jn}\theta }\begin{array}{cc},& \theta ={\mathrm{\omega T}}_{s}\end{array}\\ x\left[n\right]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X\left({e}^{\mathrm{j\theta }}\right){e}^{\text{jn}\theta }\mathrm{d\theta }\end{array}$

As illustrated in [link] , when an analog signal with a maximum bandwidth of $W$ (or a maximum frequency of ${f}_{\text{max}}$ ) is sampled at a rate of ${T}_{s}=\frac{1}{{f}_{s}}$ , its corresponding frequency response is repeated every $2\pi$ radians, or ${f}_{s}$ . In other words, the Fourier transform in the digital domain becomes a periodic version of the Fourier transform in the analog domain. That is why, for discrete signals, one is interested only in the frequency range $\left[0,{f}_{s}/2\right]$ .

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
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
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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