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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 ( t ) = A cos ( ωt + φ ) size 12{x \( t \) =A"cos" \( ωt+φ \) } {} . Sampling this signal at t = nT s size 12{t= ital "nT" rSub { size 8{s} } } {} , with the sampling time interval of T s size 12{T rSub { size 8{s} } } {} , generates the discrete-time signal

x [ n ] = A cos ( ω nT s + φ ) = A cos ( θn + φ ) , n = 0,1,2, . . . , size 12{x \[ n \] =A"cos" \( ω ital "nT" rSub { size 8{s} } +φ \) =A"cos" \( θn+φ \) , matrix {{} # n=0,1,2, "." "." "." ,{} } } {}

where θ = ωT s = 2πf f s size 12{θ=ωT rSub { size 8{s} } = { {2πf} over {f rSub { size 8{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 ωT s size 12{ωT rSub { size 8{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 size 12{f} {} and the sampling frequency f s size 12{f rSub { size 8{s} } } {} define the digital frequency θ size 12{θ} {} of the corresponding digital signal.

Sampling of Two Different Analog Signals Leading to the Same Digital Signal

Sampling of the Same Analog Signal Leading to Two Different Digital Signals

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 pairs for analog and digital signals
Fourier transform pair for analog signals { X ( ) = x ( t ) e jωt dt x ( t ) = 1 X ( ) e jωt size 12{ left lbrace matrix { X \( jω \) = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {x \( t \) e rSup { size 8{ - jωt} } ital "dt"} {} ##x \( t \) = { {1} over {2π} } Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {X \( jω \) e rSup { size 8{jωt} } dω} } right none } {}
Fourier transform pair for discrete signals { X ( e ) = n = x [ n ] e jn θ , θ = ωT s x [ n ] = 1 π π X ( e ) e jn θ size 12{ left lbrace matrix { X \( e rSup { size 8{jθ} } \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \[ n \]e rSup { size 8{ - ital "jn"θ} } } matrix { , {} # θ=ωT rSub { size 8{s} } {}} {} ## x \[ n \]= { {1} over {2π} } Int rSub { size 8{ - π} } rSup { size 8{π} } {X \( e rSup { size 8{jθ} } \) e rSup { size 8{ ital "jn"θ} } dθ} } right none } {}

(a) Fourier Transform of a Continuous-Time Signal, (b) Its Discrete-Time Version

As illustrated in [link] , when an analog signal with a maximum bandwidth of W size 12{W} {} (or a maximum frequency of f max size 12{f rSub { size 8{"max"} } } {} ) is sampled at a rate of T s = 1 f s size 12{T rSub { size 8{s} } = { {1} over {f rSub { size 8{s} } } } } {} , its corresponding frequency response is repeated every size 12{2π} {} radians, or f s size 12{f rSub { size 8{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 0, f s / 2 size 12{ left [0,f rSub { size 8{s} } /2 right ]} {} .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, An interactive approach to signals and systems laboratory. OpenStax CNX. Sep 06, 2012 Download for free at http://cnx.org/content/col10667/1.14
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