<< Chapter < Page Chapter >> Page >

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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'An interactive approach to signals and systems laboratory' conversation and receive update notifications?