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Front Panel of CTFT and Its Properties: Combination of Input Signals Tab

Varying pulse width

Keep the default values of Time shift (=0) and Time scaling (=1) and vary the Pulse width of the rectangular pulse. First, set the value of the Pulse width to its minimum value (=0.01) and then increase it. Observe that increasing the Pulse width in the time domain decrements the width in the frequency domain (see [link] ). When the Pulse width is set to its maximum value (=1) in the frequency domain, only one value can be seen at the center frequency indicating the signal is of DC type (refer to Properties of CTFT section of Chapter 5).

Magnitude Spectrum for Different Pulse Widths: (a) 0.01, (b) 0.2, (c) 0.5, (d) 1

Time shift

Next, for a fixed pulse width, vary the time shift. Observe that the phase spectrum changes but the magnitude spectrum remains the same. If the signal x ( t ) size 12{x \( t \) } {} is shifted by a constant t 0 size 12{t rSub { size 8{0} } } {} , its FT magnitude does not change, but the term ωt 0 size 12{ - ωt rSub { size 8{0} } } {} gets added to its phase angle. This verifies the time-shifting property of FT as stated in Properties of CTFT section of Chapter 5 (see [link] ).

Magnitude and Phase Spectrum for Different Time Shifts: (a) 0, (b) 0.2, (c) 0.5, (d) 0.7

Time scaling

Observe that increasing the control Time scaling makes the spectrum wider. This indicates that compressing the signal in the time domain leads to expansion in the frequency domain. This verifies the time-scaling property of FT as stated in Properties of CTFT section of Chapter 5 (see [link] ).

Magnitude Spectrum for Different Time Scalings: (a) 1, (b) 2, (c) 3, (d) 4

Linearity

Here, combine two signals to examine the linearity property of FT. Select Linear Combination for the Time domain and Frequency domain combination method. This selection combines two time signals, x 1 ( t ) size 12{x rSub { size 8{1} } \( t \) } {} and x 2 ( t ) size 12{x rSub { size 8{2} } \( t \) } {} , linearly with the scaling factors, a 1 size 12{a rSub { size 8{1} } } {} and a 2 size 12{a rSub { size 8{2} } } {} , producing a new signal, a 1 x 1 ( t ) + a 2 x 2 ( t ) size 12{a rSub { size 8{1} } x rSub { size 8{1} } \( t \) +a rSub { size 8{2} } x rSub { size 8{2} } \( t \) } {} . [link] displays the FT of this linear combination. The linear combination in the frequency domain produces a new signal, a 1 X 1 ( ω ) + a 2 X 2 ( ω ) size 12{a rSub { size 8{1} } X rSub { size 8{1} } \( ω \) +a rSub { size 8{2} } X rSub { size 8{2} } \( ω \) } {} . [link] also displays the inverse FT of this combination. Observe that both combinations produce the same result in the time and frequency domains, as indicated by the linearity property stated in Properties of CTFT section of Chapter 5.

Verifying the Linearity Property of CTFT

Time convolution

In this part, convolve two signals in the time domain to examine the time-convolution property of FT. Select Convolution for Time domain and Multiplication for Frequency domain. This selection produces and displays a new signal, x 1 ( t ) x 2 ( t ) size 12{x rSub { size 8{1} } \( t \) * x rSub { size 8{2} } \( t \) } {} , by convolving the two time signals x 1 ( t ) size 12{x rSub { size 8{1} } \( t \) } {} and x 2 ( t ) size 12{x rSub { size 8{2} } \( t \) } {} . Multiplication in the frequency domain produces a new signal, X 1 ( ω ) X 2 ( ω ) size 12{X rSub { size 8{1} } \( ω \) X rSub { size 8{2} } \( ω \) } {} . The inverse FT of this multiplied signal is also displayed on the right. Note that both combinations produce the same outcome in the time and frequency domains. This verifies the time-convolution property stated in the Properties of CTFT section of Chapter 5 (see [link] ).

Verifying the Time-Convolution Property of CTFT

Frequency convolution

Convolve two signals in the frequency domain to examine the frequency-convolution property of FT. Select Convolution for Frequency domain and Multiplication for Time domain. This selection convolves the two time signals X 1 ( ω ) size 12{X rSub { size 8{1} } \( ω \) } {} and X 2 ( ω ) size 12{X rSub { size 8{2} } \( ω \) } {} to produce a new signal, X 1 ( ω ) X 2 ( ω ) size 12{X rSub { size 8{1} } \( ω \) * X rSub { size 8{2} } \( ω \) } {} . The inverse FT of the convolved signal is displayed. Multiplication in Time domain produces a new signal, x 1 ( t ) x 2 ( t ) size 12{x rSub { size 8{1} } \( t \) x rSub { size 8{2} } \( t \) } {} . The FT of this multiplied signal is also displayed. Note that both combinations produce the same outcome in the time and frequency domains. This verifies the frequency-convolution property stated in the Properties of CTFT section of Chapter 5 (see [link] ).

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Differences Between Laspeyres and Paasche Indices
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
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Azam
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I'm interested in Nanotube
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Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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