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Two-minute miniquiz problem

Problem 6-1

Given

H ( s ) = s ( s + 1 ) ( s + 10 ) size 12{H \( s \) = { {s} over { \( s+1 \) \( s+"10" \) } } } {}

Determine the magnitude of the frequency response in the form of a Bode diagram.

Solution

H ( ) = 1 10 ( + 1 ) ( / 10 + 1 ) size 12{H \( jω \) = { {1} over {"10"} } left [ { {jω} over { \( jω+1 \) \( jω/"10"+1 \) } } right ]} {}

V. SIGNAL PROCESSING WITH FILTERS

1/ Separation of narrowband signals

The input consists of a sum of two sinusoids, more generally the sum of two narrow-band signals such as the signals transmitted by two radio stations. The objective is to devise filters

to separate these two signals. All waveforms have normalized amplitudes.

2/ Extraction of narrow-band signal from wide-band noise

The input consists of a sinusoid, more generally a narrow-band signal, plus some wide-band noise. The objective is to extract the signal from the noise. All waveforms have normalized amplitudes.

3/ Reduction of narrow-band noise from wide-band signal

The input consists of a wide-band signal, in this case an ecg signal recorded from the surface of the chest, plus some narrowband noise, such as pickup from the power lines. The objective is to remove the narrow-band noise. All waveforms have normalized amplitudes.

VI. LOWPASS AND HIGHPASS FILTERS

1/ First-order lowpass and highpass filters

where H(s) = Y (s)/X(s) and RC = 1.

2/ Use of first-order lpf and hpf for signal separation

How much unwanted signal occurs in each output channel? That depends on the frequency separation between the signals. If the two signals are about a decade apart then the attenuation of the unwanted signal will be at most 20 dB with first-order lowpass and highpass filters.

3/ Higher-order lowpass filters, Butterworth filters

If two signals have a small frequency separation or if attenuation of the unwanted signal needs to be very large, higher-order filters are required. The pole-zero diagrams of the class of lowpass Butterworth filters of order 1-9 are shown below.

The system function for the nth-order Butterworth filter, H n ( s ) size 12{H rSub { size 8{n} } \( s \) } {} , is obtained from the property

H n ( s ) H n ( s ) = 1 1 + s c 2n size 12{H rSub { size 8{n} } \( s \) H rSub { size 8{n} } \( - s \) = { {1} over {1+ left [ { {s} over {jω rSub { size 8{c} } } } right ] rSup { size 8{2n} } } } } {}

where ω c size 12{ω rSub { size 8{c} } } {} is the cut-off frequency.

Each additional order of Butterworth filter adds an additional attenuation of −20 dB/decade. The frequency response for the nth-order Butterworth filter, H n ( ) size 12{H rSub { size 8{n} } \( jω \) } {} , is obtained from the property

( H n ( ) ) 2 = 1 1 + ω ω c 2n size 12{ \( \lline H rSub { size 8{n} } \( jω \) \lline \) rSup { size 8{2} } = { {1} over {1+ left [ { {ω} over {ω rSub { size 8{c} } } } right ] rSup { size 8{2n} } } } } {}

The frequency responses are shown for n in the range 1-9 and for ω c = 1 size 12{ω rSub { size 8{c} } "= "1} {} .

VII. RESONANCE AND BANDPASS FILTERS

1/ Resonant systems arise in many physical contexts

M is mass, B is a friction constant, K is a spring constant, f(t) is an external force, and v(t) is the velocity of the mass.

A second-order system function relates the velocity to the force

H ( s ) = V ( s ) F ( s ) = 1 M s s 2 + B M s + K M size 12{H \( s \) = { {V \( s \) } over {F \( s \) } } = { { { {1} over {M} } s} over {s rSup { size 8{2} } + { {B} over {M} } s+ { {K} over {M} } } } } {}

Such a mechanical system yields a damped oscillation in response to a force provided the damping is not too large. For example, a tuning fork may be modeled by such a mechanical system.

2/ RLC filter

Electric networks also show a similar system function. The impedance of the RLC circuit is also of second-order

Z ( s ) = V ( s ) I ( s ) = 1 sC + 1 R + 1 sL = sL s 2 LC + s L R + 1 size 12{Z \( s \) = { {V \( s \) } over {I \( s \) } } = { {1} over { ital "sC"+ { {1} over {R} } + { {1} over { ital "sL"} } } } = { { ital "sL"} over {s rSup { size 8{2} } ital "LC"+s { {L} over {R} } +1} } } {}

3/ Resonance parameterized

We can parameterize these and any second-order systems efficiently. We illustrate with the electrical network.

Questions & Answers

what is the stm
Brian Reply
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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?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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Damian Reply
absolutely yes
Daniel
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Maciej
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Abigail
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Anassong
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Lily
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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Devang Reply
are you nano engineer ?
s.
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
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Tarell
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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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