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w ( t ) = A 2 j e j 2 π f 0 t - e - j 2 π f 0 t .

Applying the linearity property [link] and the result of Exercise  [link] gives

F { w ( t ) } = A 2 j F { e j 2 π f 0 t } - F { e - j 2 π f 0 t } = j A 2 - δ ( f - f 0 ) + δ ( f + f 0 ) .

Thus, the spectrum of a sine wave is a pair of δ functions with opposite signs, located symmetrically about zero frequency. The corresponding magnitude spectrum,shown in [link] , is at the heart of one importantinterpretation of the Fourier transform: it shows the frequency content of any signal by displayingwhich frequencies are present (and which frequencies are absent) from the waveform. For example, [link] (a) shows the magnitude spectrum W ( f ) of a real-valued signal w ( t ) . This can be interpreted as saying that w ( t ) contains (or is made up of) “all the frequencies” up to B Hz, and that it contains no sinusoids with higher frequency. Similarly,the modulated signal s ( t ) in [link] (b) contains all positive frequencies between f c - B and f c + B , and no others.

Note that the Fourier transform in [link] is purely imaginary, as it must be because w ( t ) is odd (see  [link] ). The phase spectrum is a flat line at - 90 because of the factor  j .

The magnitude spectrum of a sinusoid with frequency f_0 and amplitude A contains two δ function spikes, one at f=f_0 and the other at f=-f_0.
The magnitude spectrum of a sinusoid with frequency f 0 and amplitude A contains two δ function spikes, one at f = f 0 and the other at f = - f 0 .

What is the magnitude spectrum of sin ( 2 π f 0 t + θ ) ? Hint: Use the frequency shift property [link] . Show that the spectrum of cos ( 2 π f 0 t ) is 1 2 ( δ ( f - f 0 ) + δ ( f + f 0 ) ) . Compare this analytical result to the numerical resultsfrom Exercise  [link] .

The magnitude spectrum of a message signal w(t) is shown in (a). When w(t) is modulated by a cosine at frequency f_c, the spectrum of the resulting signal s(t)=w(t)cos(2πf_ct+Φ) is shown in (b).
The magnitude spectrum of a message signal w ( t ) is shown in (a). When w ( t ) is modulated by a cosine at frequency f c , the spectrum of the resulting signal s ( t ) = w ( t ) cos ( 2 π f c t + Φ ) is shown in (b).

Let w i ( t ) = a i sin ( 2 π f i t ) for i = 1 , 2 , 3 . Without doing any calculations, write down the spectrum of v ( t ) = w 1 ( t ) + w 2 ( t ) + w 3 ( t ) . Hint: Use linearity. Graph the magnitude spectrum of v ( t ) in the same manner as in [link] . Verify your results with a simulation mimicking that in  [link] .

Let W ( f ) = sin ( 2 π f t 0 ) . What is the corresponding time function?

Convolution in time: it's what linear systems do

Suppose that a system has impulse response h ( t ) , and that the input consists of a sum of three impulses occurring at times t 0 , t 1 , and t 2 , with amplitudes a 0 , a 1 , and a 2 (for example, the signal w ( t ) of [link] ). By linearity of the Fourier transform, property [link] , the output is a superpositionof the outputs due to each of the input pulses. The output due to the first impulse is a 0 h ( t - t 0 ) , which is the impulse response scaled by the size of the input and shifted to beginwhen the first input pulse arrives. Similarly, the outputs to the second and thirdinput impulses are a 1 h ( t - t 1 ) and a 2 h ( t - t 2 ) , respectively, and the complete output is the sum a 0 h ( t - t 0 ) + a 1 h ( t - t 1 ) + a 2 h ( t - t 2 ) .

Now suppose that the input is a continuous function x ( t ) . At any time instant λ , the input can be thought of as consisting of an impulse scaled by the amplitude x ( λ ) , and the corresponding output will be x ( λ ) h ( t - λ ) , which is the impulse response scaled by thesize of the input and shifted to begin at time λ . The complete output is then given by integrating over all λ

y ( t ) = - x ( λ ) h ( t - λ ) d λ x ( t ) * h ( t ) .

Questions & Answers

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
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.
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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